Advances in Intelligent Systems and Computing 741
Neha Yadav · Anupam Yadav
Jagdish Chand Bansal · Kusum Deep
Joong Hoon Kim Editors
Harmony
Search and
Nature Inspired
Optimization
Algorithms
Theory and Applications, ICHSA 2018
Advances in Intelligent Systems and Computing
Volume 741
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail: kacprzyk@ibspan.waw.pl
The series “Advances in Intelligent Systems and Computing” contains publications on theory,
applications, and design methods of Intelligent Systems and Intelligent Computing. Virtually all
disciplines such as engineering, natural sciences, computer and information science, ICT, economics,
business, e-commerce, environment, healthcare, life science are covered. The list of topics spans all the
areas of modern intelligent systems and computing such as: computational intelligence, soft computing
including neural networks, fuzzy systems, evolutionary computing and the fusion of these paradigms,
social intelligence, ambient intelligence, computational neuroscience, artificial life, virtual worlds and
society, cognitive science and systems, Perception and Vision, DNA and immune based systems,
self-organizing and adaptive systems, e-Learning and teaching, human-centered and human-centric
computing, recommender systems, intelligent control, robotics and mechatronics including
human-machine teaming, knowledge-based paradigms, learning paradigms, machine ethics, intelligent
data analysis, knowledge management, intelligent agents, intelligent decision making and support,
intelligent network security, trust management, interactive entertainment, Web intelligence and multimedia.
The publications within “Advances in Intelligent Systems and Computing” are primarily proceedings
of important conferences, symposia and congresses. They cover significant recent developments in the
field, both of a foundational and applicable character. An important characteristic feature of the series is
the short publication time and world-wide distribution. This permits a rapid and broad dissemination of
research results.
Advisory Board
Chairman
Nikhil R. Pal, Indian Statistical Institute, Kolkata, India
e-mail: nikhil@isical.ac.in
Members
Rafael Bello Perez, Universidad Central “Marta Abreu” de Las Villas, Santa Clara, Cuba
e-mail: rbellop@uclv.edu.cu
Emilio S. Corchado, University of Salamanca, Salamanca, Spain
e-mail: escorchado@usal.es
Hani Hagras, University of Essex, Colchester, UK
e-mail: hani@essex.ac.uk
László T. Kóczy, Széchenyi István University, Győr, Hungary
e-mail: koczy@sze.hu
Vladik Kreinovich, University of Texas at El Paso, El Paso, USA
e-mail: vladik@utep.edu
Chin-Teng Lin, National Chiao Tung University, Hsinchu, Taiwan
e-mail: ctlin@mail.nctu.edu.tw
Jie Lu, University of Technology, Sydney, Australia
e-mail: Jie.Lu@uts.edu.au
Patricia Melin, Tijuana Institute of Technology, Tijuana, Mexico
e-mail: epmelin@hafsamx.org
Nadia Nedjah, State University of Rio de Janeiro, Rio de Janeiro, Brazil
e-mail: nadia@eng.uerj.br
Ngoc Thanh Nguyen, Wroclaw University of Technology, Wroclaw, Poland
e-mail: Ngoc-Thanh.Nguyen@pwr.edu.pl
Jun Wang, The Chinese University of Hong Kong, Shatin, Hong Kong
e-mail: jwang@mae.cuhk.edu.hk
More information about this series at http://www.springer.com/series/11156
Neha Yadav Anupam Yadav
Jagdish Chand Bansal Kusum Deep
Joong Hoon Kim
•
•
Editors
Harmony Search and Nature
Inspired Optimization
Algorithms
Theory and Applications, ICHSA 2018
123
Editors
Neha Yadav
School of Engineering and Technology
BML Munjal University
Gurgaon, Haryana
India
Anupam Yadav
Department of Sciences and Humanities
National Institute of Technology
Srinagar, Uttarakhand
India
Kusum Deep
Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee, Uttarakhand
India
Joong Hoon Kim
School of Civil, Environmental
and Architectural Engineering
Korea University
Seoul
Korea (Republic of)
Jagdish Chand Bansal
Department of Mathematics
South Asian University
New Delhi
India
ISSN 2194-5357
ISSN 2194-5365 (electronic)
Advances in Intelligent Systems and Computing
ISBN 978-981-13-0760-7
ISBN 978-981-13-0761-4 (eBook)
https://doi.org/10.1007/978-981-13-0761-4
Library of Congress Control Number: 2018943721
© Springer Nature Singapore Pte Ltd. 2019
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
It is a matter of pride that 4th International Conference on Harmony Search, Soft
Computing and Applications (ICHSA 2018) is being organized in India for the very
first time. It is noted that earlier editions of this conference were held at South
Korea and Spain. This annual event of ICHSA is a joint effort of many reputed
institutes: BML Munjal University, Gurugram; National Institute of Technology
Uttarakhand; and Korea University. The first and second series of this conference
were held at Korea University, Seoul, Republic of Korea. Professor Joong Hoon
Kim, Korea University, has successfully organized first two versions in his Parent
University. The third conference of the series was organized at Tecnalia, Bilbao,
Spain. Keeping the legacy of the conference on it was a proud moment to organize
it in India at BML Munjal University in collaboration with NIT Uttarakhand, Korea
University, and Soft Computing Research Society during 7–9 February 2018. The
focus of ICHSA 2018 is to provide a common platform for all the researchers
working in the area of harmony search and other soft computing techniques and
their applications to diverse areas of control systems, data mining, game theory,
supply chain management, signal processing, pattern recognition, big data applications, cloud computing, defence disaster modelling, renewable energy, robotics
water and waste management, structural engineering, etc. ICHSA 2018 attracted a
wide spectrum of thought-provoking articles. A total of 117 high-quality research
articles were selected for the appearance in the form of this proceedings.
We strongly hope that the papers published in this proceedings will be helpful
for improving the understating of various soft computing methods, and it will
inspire many upcoming researchers in this field as a torchbearer. The real-life
applications presented in this proceedings show the contemporary significance and
future scope of soft computing methods. The editors express their sincere gratitude
to ICHSA 2018, Chief Patron, Patron, Keynote Speakers, Chairs of the conference,
reviewers and local organizing committee; without their support, it would be
impossible to maintain the quality and standards of this conference series. We pay
our sincere thanks to the Springer and its team for their invaluable support in the
v
vi
Preface
preparation and publication of this conference proceedings. Over and above, we
express our deepest sense of gratitude to the ‘BML Munjal University’ for facilitating the hosting of the conference.
Gurgaon, India
Srinagar (Garhwal), India
New Delhi, India
Roorkee, India
Seoul, Korea (Republic of)
Neha Yadav
Anupam Yadav
Jagdish Chand Bansal
Kusum Deep
Joong Hoon Kim
Organizing Committee
Chief Patron
Mr. Akshay Munjal, President, BMU
Patrons
Prof. (Dr.) B. S. Satyanarayana, Vice Chancellor, BMU
Prof. (Dr.) M. B. Srinivas, Dean SOET, BMU
Honorary Chair
Prof. Joong Hoon Kim, Korea University, Seoul, South Korea
General Chairs
Prof. Kusum Deep, Professor, Mathematics, IIT Roorkee
Dr. Jagdish Chand Bansal, South Asian University, New Delhi
Dr. Kedar Nath Das, NIT Silchar
Conveners & Organizing Chairs
Dr. Neha Yadav, Assistant Professor, Mathematics, BMU
Dr. Anupam Yadav, Assistant Professor, NIT Uttarakhand
vii
viii
Local Organizing Committee
Dr. Ziya Uddin, BMU
Dr. Rishi Asthana, BMU
Dr. Ranjib Banerjee, BMU
Dr. Akhlaq Husain, BMU
Dr. Kalluri Vinayak, BMU
Dr. Maheshwar Dwivedi, BMU
Dr. Rakesh Prasad Badoni, BMU
Dr. Mukesh Mann, BMU
Dr. Pradeep Arya, BMU
Dr. Sumit Roy, BMU
Prof. Goldie Gabrani, BMU
Dr. Swati Jha, BMU
Dr. Deepti Sharma, BMU
Dr. Vaishali Sharma, BMU
Mr. Nilaish, BMU
Dr. Ashok Kumar Suhag, BMU
Dr. Sanmitra Barman, BMU
Dr. Nandita Choudhary
Dr. Sanjay Kashyap, BMU
Mr. Jai Prakash Bhardwaj, BMU
Ms. Neera Sood, BMU
Publicity Chairs
Mr. Nilaish, BMU
Dr. Shwetank Avikal, Graphic Era University
Organizing Committee
Advisory Committee
National Advisory Committee
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Prof. Kusum Deep, IIT Roorkee
Prof. Swagatam Das, ISI Kolkata
Prof. Laxmidhar Behera, IIT Kanpur
Prof. Ajit Kumar Verma, IIT Bombay
Prof. Mohan K. Kadalbajoo, LNMIIT, Jaipur
Dr. Manoj Kumar, MNNIT Allahabad
Dr. J. C. Bansal, South Asian University, New Delhi
Dr. Kedar Nath Das, NIT Silchar
Dr. Manoj Thakur, IIT Mandi
Dr. Krishna Pratap Singh, IIIT Allahabad
Dr. Harish RTU, Kota
Dr. Amreek Singh, DRDO Chandigarh
Prof. Sangeeta Sabharwal, NSIT Delhi
Prof. U. C. Gupta, IIT Kharagpur
Dr. Nagendra Pratap Singh, NCBS
Dr. Harish, RTU Kota
International Advisory Committee
•
•
•
•
•
•
•
Prof. J. H. Kim, Korea University, South Korea
Prof. Z. W. Geem, Gachon University, South Korea
Prof. Javier Del Ser, Tecnalia Research and Innovation, Spain
Dr. Lipo Wang, Nanyang Technological University, Singapore
Dr. Patrick Siarry, Universit de Paris 12, France
Prof. Xin-She Yang, Middlesex University, UK
Prof. Chung-Li Tseng, University of New South Wales, Australia
ix
x
•
•
•
•
•
•
•
Advisory Committee
Prof. I. Kougias, European Commission, Joint Research Centre
Prof. K. S. McFall, Kennesaw State University, USA
Dr. D. G. Yoo, Korea University, South Korea
Dr. Ali Sadollah, Iran
Dr. Donghwi Jung, Korea University, South Korea
Prof. A. K. Nagar, Liverpool Hope University, UK
Prof. Andres Iglesias, University of Cantabria, Spain
Contents
Privacy Preserving Data Mining: A Review of the State
of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shivani Sharma and Sachin Ahuja
1
An MCDM-Based Approach for Selecting the Best State
for Tourism in India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rashmi Rashmi, Rohit Singh, Mukesh Chand and Shwetank Avikal
17
Gravitational Search Algorithm: A State-of-the-Art Review . . . . . . . . . .
Indu Bala and Anupam Yadav
27
Investigating the Role of Gate Operation in Real-Time Flood
Control of Urban Drainage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fatemeh Jafari, S. Jamshid Mousavi, Jafar Yazdi and Joong Hoon Kim
39
Molecular Dynamics Simulations of a Protein in Water and
in Vacuum to Study the Solvent Effect . . . . . . . . . . . . . . . . . . . . . . . . . .
Nitin Sharma and Madhvi Shakya
49
An Exploiting Neighboring Relationship and Utilizing an Overhearing
Concept for Improvement Routing Protocol in Wireless Mesh
Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mohammad Meftah Alrayes, Neeraj Tyagi, Rajeev Tripathi
and Arun Kumar Misra
57
A Comparative Study of Machine Learning Algorithms
for Prior Prediction of UFC Fights . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hitkul, Karmanya Aggarwal, Neha Yadav and Maheshwar Dwivedy
67
Detection of a Real Sinusoid in Noise using Differential
Evolution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gayathri Narayanan and Dhanesh G. Kurup
77
xi
xii
Contents
Inherited Competitive Swarm Optimizer for Large-Scale
Optimization Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Prabhujit Mohapatra, Kedar Nath Das and Santanu Roy
Performance Comparison of Metaheuristic Optimization Algorithms
Using Water Distribution System Design Benchmarks . . . . . . . . . . . . . .
Ho Min Lee, Donghwi Jung, Ali Sadollah, Eui Hoon Lee
and Joong Hoon Kim
85
97
Comparison of Parameter-Setting-Free and Self-adaptive
Harmony Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Young Hwan Choi, Sajjad Eghdami, Thi Thuy Ngo,
Sachchida Nand Chaurasia and Joong Hoon Kim
Copycat Harmony Search: Considering Poor Music Player’s
Followship Toward Good Player . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Sang Hoon Jun, Young Hwan Choi, Donghwi Jung and Joong Hoon Kim
Fused Image Separation with Scatter Graphical Method . . . . . . . . . . . . 119
Mayank Satya Prakash Sharma, Ranjeet Singh Tomar, Nikhil Paliwal
and Prashant Shrivastava
Ascending and Descending Order of Random Projections:
Comparative Analysis of High-Dimensional Data Clustering . . . . . . . . . 133
Raghunadh Pasunuri, Vadlamudi China Venkaiah and Bhaskar Dhariyal
Speed Control of the Sensorless BLDC Motor Drive Through
Different Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Vikas Verma, Nidhi Singh Pal and Bhavnesh Kumar
Urban Drainage System Design Minimizing System Cost Constrained
to Failure Depth and Duration Under Flooding Events . . . . . . . . . . . . . 153
Soon Ho Kwon, Donghwi Jung and Joong Hoon Kim
Analysis of Energy Storage for Hybrid System Using FLC . . . . . . . . . . 159
Ayush Kumar Singh, Aakash Kumar and Nidhi Singh Pal
Impact of Emission Trading on Optimal Bidding of Price
Takers in a Competitive Energy Market . . . . . . . . . . . . . . . . . . . . . . . . 171
Somendra P. S. Mathur, Anoop Arya and Manisha dubey
Impact of NOVEL HVDC Superconducting Circuit Breaker
on HVDC Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Tarun Shrivastava, A. M. Shandilya and S. C. Gupta
Palmprint Matching based on Normalized Correlation Coefficient
and Mean Structural Similarity Index Measure . . . . . . . . . . . . . . . . . . . 193
Deval Verma, Himanshu Agarwal and A. K. Aggarwal
Contents
xiii
A Comparative Study on Feature Selection Techniques
for Multi-cluster Text Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Ananya Gupta and Shahin Ara Begum
Fuzzy Decision Tree with Fuzzy Particle Swarm Optimization
Clustering for Locating Users in an Indoor Environment Using
Wireless Signal Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Swathi Jamjala Narayanan, Boominathan Perumal, Cyril Joe Baby
and Rajen B. Bhatt
Optimization Approach for Bounds Involving Generalized
Normalized d-Casorati Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Pooja Bansal and Mohammad Hasan Shahid
Particle Swarm Optimization with Probabilistic Inertia Weight . . . . . . . 239
Ankit Agrawal and Sarsij Tripathi
An Evolutionary Algorithm Based Hyper-heuristic for the Job-Shop
Scheduling Problem with No-Wait Constraint . . . . . . . . . . . . . . . . . . . . 249
Sachchida Nand Chaurasia, Shyam Sundar, Donghwi Jung, Ho Min Lee
and Joong Hoon Kim
An Evolutionary Algorithm Based Hyper-heuristic for the Set
Packing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Sachchida Nand Chaurasia, Donghwi Jung, Ho Min Lee
and Joong Hoon Kim
Developing a Decision-Making Model Using Interval-Valued
Intuitionistic Fuzzy Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
Syed Abou Iltaf Hussain, Uttam Kumar Mandal and Sankar Prasad Mondal
A Multi-start Iterated Local Search Algorithm with Variable
Degree of Perturbation for the Covering Salesman Problem . . . . . . . . . 279
Pandiri Venkatesh, Gaurav Srivastava and Alok Singh
A New Approach to Soft Hyperideals in LA-Semihypergroups . . . . . . . 293
Sabahat Ali Khan, M. Y. Abbasi and Aakif Fairooze Talee
Adjusted Artificial Bee Colony Algorithm for the Minimum
Weight Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Adis Alihodzic, Haris Smajlovic, Eva Tuba, Romana Capor Hrosik
and Milan Tuba
Decision-Making Proposition of Fuzzy Information Measure
with Collective Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Anjali Munde
Exact Algorithm for Lð2; 1Þ Labeling of Cartesian Product Between
Complete Bipartite Graph and Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
Sumonta Ghosh, Prosanta Sarkar and Anita Pal
xiv
Contents
The Forgotten Topological Index of Graphs Based on New Operations
Related to the Join of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
Prosanta Sarkar, Nilanjan De and Anita Pal
Clustering and Auction in Sequence: A Two Fold Mechanism
for Participatory Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Jaya Mukhopadhyay, Vikash Kumar Singh, Sajal Mukhopadhyay
and Anita Pal
High-Order Compact Finite Difference Scheme for Euler–Bernoulli
Beam Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Maheshwar Pathak and Pratibha Joshi
Test Case Optimization and Prioritization Based on Multi-objective
Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
Deepti Bala Mishra, Rajashree Mishra, Arup Abhinna Acharya
and Kedar Nath Das
PSO-SVM Approach in the Prediction of Scour Depth Around
Different Shapes of Bridge Pier in Live Bed Scour Condition . . . . . . . . 383
B. M. Sreedhara, Geetha Kuntoji, Manu and S. Mandal
Replenishment Policy for Deteriorating Items Under Price
Discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Anubhav Namdeo and Uttam Kumar Khedlekar
Performance Emission Characterization of a LPG-Diesel Dual Fuel
Operation: A Gene Expression Programming Approach . . . . . . . . . . . . 405
Amitav Chakraborty, Sumit Roy and Rahul Banerjee
Comprehensive Survey of OLAP Models . . . . . . . . . . . . . . . . . . . . . . . . 415
Harkiran Kaur and Gursimran Kaur
Energy Efficiency in Load Balancing of Nodes Using Soft
Computing Approach in WBAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Rakhee and M. B. Srinivas
Single Image Defogging Based on Local Extrema and Relativity
of Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
R. Vignesh and Philomina Simon
Improved Edge-Preserving Decomposition Based on Single Image
Dehazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
S. K. Anusuman and Philomina Simon
Global and Local Neighborhood Based Particle Swarm
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
Shakti Chourasia, Harish Sharma, Manoj Singh and Jagdish Chand Bansal
Contents
xv
Rough Set Theoretic and Logical Study of Some Approximation
Pairs Due to Pomykala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
Pulak Samanta
The Benefits of Carrier Collaboration for Capacity Shortage
Under Incomplete Advance Demand Information . . . . . . . . . . . . . . . . . 471
Arindam Debroy and S. P. Sarmah
Allocation of Bins in Urban Solid Waste Logistics System . . . . . . . . . . . 485
P. Rathore and S. P. Sarmah
Image Segmentation Through Fuzzy Clustering: A Survey . . . . . . . . . . 497
Rashi Jain and Rama Shankar Sharma
Study of Various Technologies in Solar Power Generation . . . . . . . . . . 509
Siddharth Gupta, Pratibha Tiwari and Komal Singh
Reduction of Test Data Volume Using DTESFF-Based Partial
Enhanced Scan Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
Ashok Kumar Suhag
Performance Analysis and Optimization of Vapour Absorption
Refrigeration System Using Different Working Fluid Pairs . . . . . . . . . . 527
Paras Kalura, Susheem Kashyap, Vishal Sharma, Geetanjali Raghav
and Jasmeet Kalra
Vehicle Routing Problem with Time Windows Using Meta-Heuristic
Algorithms: A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
Aditya Dixit, Apoorva Mishra and Anupam Shukla
Design and Aerodynamic Enhancement of Wing for
BMW 5 Series Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
A. Agrawal, A. Juneja, A. Gupta, R. Mathur and G. Raghav
Semi-distributed Modelling of Stormwater Drains Using
Integrated Hydrodynamic EPA-SWM Model . . . . . . . . . . . . . . . . . . . . . 557
M. K. Sinha, K. Baier, R. Azzam, T. Baghel and M. K. Verma
A MCDM-Based Approach for Selection of a Sedan Car
from Indian Car Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
Rohit Singh, Rashmi and Shwetank Avikal
Design and Simulation of Photovoltaic Cell Using Simscape
MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Sucheta Singh, Shubhra Aakanksha, Manisha Rajoriya and Mohit Sahni
A Regulated Computer Cooling Method: An Eco-Friendly
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
Kumar Gourab Mallik and Sutirtha Kumar Guha
xvi
Contents
Robust Control Techniques for Master–Slave Surgical Robot
Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
Mohd Salim Qureshi, Gopi Nath Kaki, Pankaj Swarnkar
and Sushma Gupta
OLAP Approach to Visualizations and Digital ATLAS for
NRIs Directory Database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
Harkiran Kaur, Kawaljeet Singh and Tejinder Kaur
Problems Associated with Hydraulic Turbines . . . . . . . . . . . . . . . . . . . . 621
Aman Kumar, Kunal Govil, Gaurav Dwivedi and Mayank Chhabra
A Sine-Cosine Optimizer-Based Gamma Corrected Adaptive
Fractional Differential Masking for Satellite Image Enhancement . . . . . 633
Himanshu Singh, Anil Kumar and L. K. Balyan
Electrical Conductivity Sensing for Precision Agriculture:
A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 647
Sonia Gupta, Mohit Kumar and Rashmi Priyadarshini
Spam Detection Using Ensemble Learning . . . . . . . . . . . . . . . . . . . . . . . 661
Vashu Gupta, Aman Mehta, Akshay Goel, Utkarsh Dixit
and Avinash Chandra Pandey
A Coupled Approach for Solving a Class of Singular Initial Value
Problems of Lane–Emden Type Arising in Astrophysics . . . . . . . . . . . . 669
Pratibha Joshi and Maheshwar Pathak
Identification of Hindi Plain Text Using Artificial Neural
Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
Siddheshwar Mukhede, Amol Prakash and Maiya Din
A Variable Dimension Optimization Approach for Text
Summarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
Pradeepika Verma and Hari Om
Minimizing Unbalance of Flexible Manufacturing System
by Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
Kritika Gaur, Indu and Vivek Chawla
“Big” Data Management in Cloud Computing Environment . . . . . . . . . 707
Mohit Agarwal and Gur Mauj Saran Srivastava
Automatic Optimization of Test Path Using Firefly Algorithm . . . . . . . . 717
Nisha Rathee, Rajendra Singh Chillar, Sakshi Vij and Sakshi Kukreja
Image Denoising Techniques: A Brief Survey . . . . . . . . . . . . . . . . . . . . 731
Lokesh Singh and Rekhram Janghel
Contents
xvii
Applying PSO Based Technique for Analysis of Geffe Generator
Cryptosystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741
Maiya Din, Saibal K. Pal and S. K. Muttoo
An Agent-Based Simulation Modeling Approach for Dynamic
Job-Shop Manufacturing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
Om Ji Shukla, Gunjan Soni, Rajesh Kumar, A. Sujil and Surya Prakash
Risk Analysis of Water Treatment Plant Using Fuzzy-Integrated
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
Priyank Srivastava, Mohit Agrawal, G. Aditya Narayanan, Manik Tandon,
Mridul Narayan Tulsian and Dinesh Khanduja
Keyframes and Shot Boundaries: The Attributes of Scene
Segmentation and Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
N. Kumar and N. Sukavanam
Toward Human-Powered Lower Limb Exoskeletons: A Review . . . . . . 783
Ashish Singla, Saurav Dhand, Ashwin Dhawad and Gurvinder S. Virk
An Efficient Bi-Level Discrete PSO Variant for Multiple
Sequence Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797
Soniya Lalwani, Harish Sharma, M. Krishna Mohan and Kusum Deep
System Identification of an Inverted Pendulum Using Adaptive
Neural Fuzzy Inference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809
Ishan Chawla and Ashish Singla
Dynamic Modeling of Flexible Robotic Manipulators . . . . . . . . . . . . . . . 819
Ashish Singla and Amardeep Singh
Academic Performance Prediction Using Data Mining Techniques:
Identification of Influential Factors Effecting the Academic
Performance in Undergrad Professional Course . . . . . . . . . . . . . . . . . . . 835
Preet Kamal and Sachin Ahuja
An Area IF-Defuzzification Technique and Intuitionistic
Fuzzy Reliability Assessment of Nuclear Basic Events of Fault
Tree Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845
Mohit Kumar
Spotted Hyena Optimizer for Solving Complex and Non-linear
Constrained Engineering Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857
Gaurav Dhiman and Vijay Kumar
Reconfiguration of PTZ Camera Network with Minimum
Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869
xviii
Contents
Sanoj Kumar, Claudio Piciarelli and Harendra Pal Singh
Performance Evaluation of Optimization Techniques with Vector
Quantization Used for Image Compression . . . . . . . . . . . . . . . . . . . . . . 879
Rausheen Bal, Aditya Bakshi and Sunanda Gupta
Single Multiplicative Neuron Model in Reinforcement Learning . . . . . . 889
Shobhit Nigam
Analysis of Educational Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . 897
Ravinder Ahuja, Animesh Jha, Rahul Maurya and Rishabh Srivastava
A Review on Search-Based Tools and Techniques to Identify Bad
Code Smells in Object-Oriented Systems . . . . . . . . . . . . . . . . . . . . . . . . 909
Amandeep Kaur and Gaurav Dhiman
Feature Selection Using Metaheuristic Algorithms on Medical
Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923
Shivam Mahendru and Shashank Agarwal
Improved Mutation-Based Particle Swarm Optimization for Load
Balancing in Cloud Data Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939
Neha Sethi, Surjit Singh and Gurvinder Singh
Computational Intelligence Tools for Protein Modeling . . . . . . . . . . . . . 949
Rajesh Kondabala and Vijay Kumar
Performance Analysis of Space Time Trellis Codes in Rayleigh
Fading Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
Shakti Raj Chopra, Akhil Gupta and Himanshu Monga
Neural Network Based Analysis of Lightweight Block Cipher
PRESENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969
Girish Mishra, S. V. S. S. N. V. G. Krishna Murthy and S. K. Pal
User Profile Matching and Identification Using TLBO
and Clustering Approach Over Social Networks . . . . . . . . . . . . . . . . . . 979
Shruti Garg, Sandeep K. Raghuwanshi and Param Deep Singh
Hybrid Metaheuristic Based Scheduling with Job Duplication for
Cloud Data Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989
Rachhpal Singh
Total Fuzzy Agility Evaluation Using Fuzzy Methodology:
A Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999
Priyank Srivastava, Dinesh Khanduja, Vishnu P. Agrawal and Neeraj Saini
Black-Hole Gbest Differential Evolution Algorithm for Solving
Robot Path Planning Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009
Prashant Sharma, Harish Sharma, Sandeep Kumar and Kavita Sharma
Contents
xix
Fibonacci Series-Inspired Local Search in Artificial Bee Colony
Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023
Nirmala Sharma, Harish Sharma, Ajay Sharma and Jagdish Chand Bansal
Analysis of Lightweight Block Cipher FeW on the Basis
of Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041
Aayush Jain and Girish Mishra
Analysis of RC4 Crypts Using PSO Based Swarm Technique . . . . . . . . 1049
Maiya Din, Saibal K. Pal and S. K. Muttoo
Pipe Size Design Optimization of Water Distribution Networks
Using Water Cycle Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057
P. Praneeth, A. Vasan and K. Srinivasa Raju
An Improved Authentication and Data Security Approach
Over Cloud Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069
Ramraj Dangi and Satish Pawar
Second Derivative-Free Two-Step Extrapolated Newton’s
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077
V. B. Kumar Vatti, Ramadevi Sri and M. S. Kumar Mylapalli
Review of Deep Learning Techniques for Gender Classification
in Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089
Neelam Dwivedi and Dushyant Kumar Singh
A Teaching–Learning-Based Optimization Algorithm for the
Resource-Constrained Project Scheduling Problem . . . . . . . . . . . . . . . . 1101
Dheeraj Joshi, M. L. Mittal and Manish Kumar
A Tabu Search Algorithm for Simultaneous Selection
and Scheduling of Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111
Manish Kumar, M. L. Mittal, Gunjan Soni and Dheeraj Joshi
A Survey: Image Segmentation Techniques . . . . . . . . . . . . . . . . . . . . . . 1123
Gurbakash Phonsa and K. Manu
Analysis and Simulation of the Continuous Stirred Tank
Reactor System Using Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 1141
Harsh Goud and Pankaj Swarnkar
Fuzzy Logic Controlled Variable Frequency Drives . . . . . . . . . . . . . . . . 1153
Kartik Sharma, Anubhav Agrawal and Shuvabrata Bandopadhaya
Butterfly-Fat-Tree Topology-Based Fault-Tolerant Network-on-Chip
Design Using Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . 1165
P. Veda Bhanu, Pranav Venkatesh Kulkarni, U. Anil Kumar
and J. Soumya
xx
Contents
Big Data Classification Using Scale-Free Binary Particle Swarm
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177
Sonu Lal Gupta, Anurag Singh Baghel and Asif Iqbal
Face Recognition: Novel Comparison of Various Feature
Extraction Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1189
Yashoda Makhija and Rama Shankar Sharma
Performance Analysis of Hidden Terminal Problem in VANET
for Safe Transportation System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199
Ranjeet Singh Tomar, Mayank Satya Prakash Sharma, Sudhanshu Jha
and Brijesh Kumar Chaurasia
Effect of Various Distance Classifiers on the Performance of Bat
and CS-Based Face Recognition System . . . . . . . . . . . . . . . . . . . . . . . . . 1209
Preeti and Dinesh Kumar
An Improved TLBO Leveraging Group and Experience
Learning Concepts for Global Functions . . . . . . . . . . . . . . . . . . . . . . . . 1221
Jatinder Kaur, Surjeet Singh Chauhan and Pavitdeep Singh
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235
About the Editors
Dr. Neha Yadav is an assistant professor in School of Engineering & Technology,
BML Munjal University, Gurugram. She worked as research professor in School of
Civil, Environmental and Architectural Engineering at Korea University, South
Korea. She received her Ph.D. from Motilal Nehru National Institute of
Technology, Allahabad, India; M.Sc. in Mathematical Sciences from Banasthali
University, Jaipur; and B.Sc. in Mathematics from Dr. R.M.L. Avadh University,
Faizabad, in 2013, 2009 and 2007, respectively. Her research interests are real-time
flood forecasting, mathematical modelling, numerical analysis, soft computing
techniques, differential equations, boundary value problems, mathematical modelling, optimization. She has several journal papers and one book to her credit.
Dr. Anupam Yadav is an assistant professor of Mathematics at National Institute
of Technology Uttarakhand. His research areas are numerical optimization,
high-order graph matching and operations research. He received his Ph.D. from
Indian Institute of Technology Roorkee and M.Sc. in Mathematics from Banaras
Hindu University, Varanasi, India. He has one book, one chapter, few invited talks,
several journal and conference papers to his credit.
Dr. Jagdish Chand Bansal is an assistant professor in South Asian University,
New Delhi, India. Holding an excellent academic record, he is an excellent
researcher in the field of swarm intelligence at national and international levels,
having several research papers in journals of national and international repute.
Prof. Kusum Deep is working as a full-time professor in the Department of
Mathematics at Indian Institute of Technology Roorkee, Roorkee, India. Over the
last 25 years, her research is increasingly well cited, making her a central international figure in the areas of nature-inspired optimization techniques, genetic
algorithms and particle swarm optimization.
xxi
xxii
About the Editors
Prof. Joong Hoon Kim is associated with School of Civil, Environmental and
Architectural Engineering, Korea University, South Korea. His major areas of
interest include optimal design and management of water distribution systems,
application of optimization techniques to various engineering problems, and
development and application of evolutionary algorithms. He has 216 journal publications, 262 conference proceedings, several books/chapters to his credit. His
publications include A New Heuristic Optimization Algorithm: Harmony Search,
Simulation, February 2001, Vol. 76, pp 60–68, which has been cited over 2,500
times by other journals of diverse research areas.
Privacy Preserving Data Mining:
A Review of the State of the Art
Shivani Sharma and Sachin Ahuja
Abstract Safeguarding of security in information mining has risen as an outright
essential for trading secret data as far as information investigation, approval, and
distributing. Constantly raising web phishing postured serious danger on across the
board proliferation of delicate data over the web. Then again, the questionable sentiments and conflicts intervened unwillingness of different data suppliers towards the
unwavering quality insurance of information from exposure frequently comes about
absolute dismissal in information sharing or off base data sharing. This article gives
an all-encompassing outline on new point of view and precise translation of a rundown distributed literary works through their fastidious association in subcategories.
The crucial ideas of the current protection safeguarding information mining strategies, their benefits, and deficiencies are displayed. The present security protecting
information mining methods are ordered in light of contortion, affiliation administer,
shroud affiliation control, scientific categorization, bunching, cooperative characterization, outsourced information mining, disseminated, and k-anonymity, where their
remarkable points of interest and hindrances are underlined. This watchful investigation uncovers the past improvement, show examine challenges, future patterns,
the holes and weaknesses. Promote huge improvements for more powerful security
insurance and safeguarding are confirmed to be compulsory.
Keywords Association · Classification · Clustering · Data mining · Distortion
K-anonymity · Outsourcing · Privacy preserving
S. Sharma (B) · S. Ahuja
Chitkara University, Chandigarh, Punjab, India
e-mail: shivani.sharma@chitkara.edu.in
S. Ahuja
e-mail: sachin.ahuja@chitkara.edu.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_1
1
2
S. Sharma and S. Ahuja
1 Introduction
Preeminent web security against web ridiculing has turned into a need. The dangers forced due to always expanding trick assaults with cutting-edge disloyalty have
turned into another test as far as moderation. Recently, web mocking brought on
critical security and financial worries on the clients and endeavors around the world.
Variegated correspondence channels through web administrations, for example, webbased business, web managing an account, investigate, and online merchant has misused both human and programming powerlessness experienced enormous budgetary
misfortune. So there is an improved need of protection saving information digging
strategies for secured and dependable data trade over the web. The expansion of
putting away clients’ individual information prompted an enhanced information mining calculation with pointed effect on the data sharing. The security must ensure three
mining angles completely that contains affiliation tenets, order, and bunching [47].
The challenging issues of information mining are deliberated in numerous groups
[37]. The data sharing for aggregate interests in now possible due to the advancement
in distributed computing innovation.
Presently, various privacy preservation data mining methods are available. The
methods that are available are association rule, classification, clustering, condensation, and cryptographic, distributed privacy preservation, K-anonymity etc. [47].
Privacy-preserving approaches in data mining ensure the information by adjusting
them to cover or eradicating the first delicate one to be hidden. Essentially, the strategies depend on the ideas of protection disappointment, the degree to decide the first
information gave by the client from the changed one, and estimation of data misfortune and information precision [66]. The fundamental reason for all the current
strategies is to contribute a smaller among exactness and security. Different methodologies that make utilization of cryptographic procedures to safeguard the individual
data are extremely costly [6]. In some cases, the people are apathetic to share the
whole informational collection and may wish to shroud the data utilizing assortments
of assertion. The fundamental purpose behind executing such procedures is to keep
up people’s protection while removing aggregate outcomes over the whole information [1]. It is critical to secure the information conveyed to different suppliers. For
protection, customers’ data should be distinguished before imparting to the doubtful
clients who are not specifically permitted to get to the applicable information.
1.1 Privacy Preserving Data Mining (PPDM)
Raju et al. [46] graphed the usage for including or copying the tradition based homomorphic encryption close by the surviving thought of automated envelope technique
in achieving shared information mining while in the meantime keeping the private
information unblemished among the normal social occasions. The proposed strategy
presented rich effect on different applications. Ashok and Mukkamala [34] perceived
Privacy Preserving Data Mining: A Review of the State of the Art
3
a plan of soft based mapping procedures as to security saving qualities and the ability
to keep up a comparable relationship with various fields. Zong and Qi [43] outlined
particular existing techniques of information digging for the confirmation of protection depending upon information transport, mutilation, mining computations, and
information or rules stowing without end. About information flow, less counts are
starting late used for security confirmation information mining on brought together
and dispersed information. Matwin [32] broke down and analyzed the propriety
of protection saving information mining techniques. Usage of specific techniques
unveiled their ability to block the uneven use of information mining. Vatsalan et al.
[58] analyzed ‘Protection Preserving Record Linkage’ (PPRL) system, that empowered the relationship to interface their databases by safeguarding the security. Sachan
et al. [47] and Malina and Hajny [31] explored the present protection saving frameworks for cloud organizations, in which the result is portrayed on bleeding edge
cryptographic sections. The course of action demonstrated the darken get to, the
unlink limit and control of cover of passed on information. At long last, this game
plan is done, the trial results are gotten and the execution is perceived.
1.2 Data Distortion Dependent PPDM
Three new models were proposed by Kamakshi and Babu [18] that included customers, data focuses, and databases of each site. Since the data focus is totally
unconcerned therefore, the customers and the site database part seem interchangeable. Brankovic and Islam [15] presented a strategy that included diverse novel
strategies that influenced every one of the components in the database. Test conclusions demonstrated that the outlined system is extremely sufficient in preserving the
first examples in a bothered dataset.
Kamakshi [17] outlined an imaginative idea to enthusiastic analyze the fragile
parts of PPDM. Finding of these perspectives relies on upon the skirt furthest reaches
of delicacy of every trademark. It is understood that the data proprietor adjusted the
incentive under grouped fragile perspective utilizing swapping system to ensure
the data privacy. The data was adjusted in a way, such that it pointed the same
underlying properties of the data. A short time later, Zhang et al. (2012a) outlined
a recently adorn authentic likelihood based commotion era system called HPNGS.
The impersonation conclusion demonstrated that the HPNGS can lessen the quantity
of commotion necessities over its arbitrary supplement till 90%. The focus was on
the privacy security along with clamor jumble in distributed computing (Zhang et al.
2012b). As an outcome, another affiliation likelihood based commotion era procedure
(APNGS) was created. The examination established that the proposed APNGS to
some degree enhanced the privacy insurance on clamor tangle including affiliation
probabilities at a direct additional cost than ordinary perfect outlines.
4
S. Sharma and S. Ahuja
1.3 Association Rule Based PPDM
Aggarwal and Yu [1] highlighted two vital parts including the connection lead mining, for instance, support and conviction. For an association control X > Y , the
support is the rate of trades in the dataset which fuses X U Y . The nature of an
association run X > Y is the extent of the trades number by X. Furthermore, Belwal
et al. [4] reduced the introduction of support and assurance of sensitive precepts
without changing the given database. Regardless, suggested adjustment can be executed through starting late including parameters interfacing with database trades and
association rules. Display day thought contain Changed support, Altered assurance
and Concealing counter. The count associated the importance of support and sureness. In this way, it shrouded the fundamental sensitive connection manage with no
horrible. Regardless, it can cover up only the precepts for single delicate thing on
the left hand side (LHS). Li and Liu [26] proposed a connection represent digging
figuring for security protecting known as DDIL. The introduced framework relies on
upon demand constraint and information unsettling impact. The honest to goodness
information can be covered up by using DDIL count to upgrade the security beneficially. This is a gainful strategy to make different things from balanced information.
Experiential results exhibit that this framework is capable for making agreeable estimations of protection alter with suitable decision of self-assertive parameters. Naeem
et al. [35] arranged a computation which separated the limited alliance standards with
thorough elimination of the alluded to disagreeable, for instance, the period of undesirable, non-veritable association rules while yielding no “covering” disillusionment.
This strategy used fundamental numerical measures in place of common structure,
especially measuring method in light of central slant.
Vijayarani et al. [59] elucidated the system for quantifiable revelation control
gathering, the database gathering, and the cryptography gathering. Less adequacy
of information needs high cost. A refreshed mutilation procedure for security safeguarding persistent thing set mining was arranged by Srivastava et al. [51], secured
fp & nfp probability guidelines. Upgraded viability is accomplished within the sight
of an irrelevant pressure security by modifying the two new parameters. Jain et al.
[16] arranged another framework to decrease the support of the left-hand side (LHS)
and right-hand side (RHS) oversee thing to cover up or guarantee the association
rules. The familiar strategy is found with be helpful as it rolled out less improvement
to the information entries to secure a course of action of rules with less CPU usage
time than the main work. It is kept to association oversee so to speak.
1.4 Hide Association Rule Based PPDM
Weng et al. [63] presented Fast Hiding Sensitive Association Rules (FHSAR) calculation. This guaranteed the delicate affiliation rules (SAR) with less unfavorable,
where an approach is intended to avoid concealed disappointments. What’s more, two
Privacy Preserving Data Mining: A Review of the State of the Art
5
heuristic methods were acquainted with upgrading the execution of the framework
to take care of the issues. The heuristic capacity is additionally connected to choose
the past weight for every particular exchange so that the request of altered exchanges
can be chosen successfully. Dehkordi et al. [7] progressed multi-target method for
ensuring the delicate affiliation leads in enhancing the security of database. The
protection and exactness of dataset progressed in proposed strategy depending on
hereditary calculation (GA) idea. Verykios et al. (2009) displayed a correct outskirt
based procedure to accomplish an ideal outcome to stow away fragile regular thing
sets with least expansion of the underlying database. This strategy applies an augmentation to the underlying database as opposed to modifying the current database.
Kasthuri and Meyyappan [20] acquainted another calculation with breaking down
the fragile things by disguising the touchy affiliation rules. This system found the
basic thing sets and delivered the affiliation rules. Average affiliation rules idea is
found the fragile things. Covering the touchy affiliation rules utilizing picked fragile
things is discovered valuable.
Quoc et al. [44] have presented a heuristic calculation in light of the convergence
cross section of regular thing sets to secure the arrangement of secret affiliation rules
utilizing bending technique. To bring down the reactions, the heuristic for support
and certainty minimization situated crossing point grid (HCSRIL) calculation are
utilized.
1.5 Classification Based PPDM
Xiong et al. [65] presented storage room neighbor grouping strategy that relies on
upon Secure Multiparty Computation (SMC) procedures to settle the protection cons
in less laps alongside the pf determination of the protection safeguarding closest
neighbor and the classification of protection preserving. This development is uniform
in regard of productivity, execution, and protection security. In addition, it is adaptable
to the various settings to accomplish distinctive enhancement condition. Singh et al.
[52] introduced novel order strategy for smooth and powerful protection for cloud
information. The evaluation of the closest neighbors for K-NN arrangement was
based on Jaccard comparability measure and the balance test is transported into
make sense between two scrambled records. This method encouraged a guaranteed
nearby neighbor calculation at every hub in the cloud and arranged the concealed
records by means of weighted K-NN order plot. It is essential to focus on authorizing
the sturdiness of the outlined calculation with the goal that speculation to various
information mining errands can be made, where security and secrecy are craved.
Baotou [3] exhibited a successful development based on arbitrary bother network to
safeguard security characterization information mining. This technique is polished on
unmistakable information of character sort, Boolean sort, grouping sort and numeric
sorts. The exploratory unveiled the to a great degree decorated components of this
new planned calculation as far as protection security and proficiency of information
mining calculation, where the processing technique is exceedingly lessened however
6
S. Sharma and S. Ahuja
at more prominent cost. Vaidya et al. [57] presented vertical apportioned information
mining approach. This plan was able to adjust and upgrade distinctive information
mining applications as choice trees. Promote powerful arrangements are required
to find tight upper bound on the multifaceted nature. Sathiyapriya and Sadasivam
[49] looked into the characterization techniques in grouping protection safeguarding
strategies and talked about the benefits and restrictions of various strategies.
1.6 Clustering Based PPDM
Yi and Zhang [67] sketched out a few points before clarifications to ensure classification of dispersed k-implies grouping and conveyed an inflexible clarification for
fairly contributing multiparty convention which implies that grouping is utilized on
vertically divided information, albeit every information site contributed k-implies
bunching uniformly. As per essential origination, information destinations cooperate to encode k values with a normal general key in each phase of grouping. At that
point, it safely looked at k values and yielded the list of the base without showing
the transitional qualities. In some setting, this is convenient and more effective than
Vaidya–Clifton convention [57].
1.7 Associative Classification Based PPDM
Raghuram and Gyani [45] presented an acquainted grouping model contingent upon
vertically apportioned datasets. A scalar item based outsider security safeguarding
model is received to keep up the protection for information sharing procedure among
various clients. The veracity of the given technique is approved on its VCI databases
with moving outcomes. Lin and Lo [27] composed an arrangement of calculations
comprising of Equal Working Set (EWS), Small Size Working Set (SSWS), Request
on Demand (ROD), and the Progressive Size Working Set (PSWS).
Harnsamut and Natwichai [13] presented novel heuristic calculation that relies
upon Classification Correction Rate (CCR) of a particular database to secure
database. The outlined strategy was tried and the exploratory outcomes are approved.
The heuristic calculation is observed to be to a great degree compelling and effective.
Seisungsittisunti and Natwichai [50] sketched out the issues identified with information change to protect security for information mining strategy and affiliated
grouping in an incremental information situation. An incremental polynomial time
calculation is intended to adjust the information to keep up a security standard called
k-namelessness.
Privacy Preserving Data Mining: A Review of the State of the Art
7
1.8 Privacy Preserving Outsourced Data Mining
Giannotti et al. [11] illustrated the issues related to the outsourcing of affiliation
control digging assignment for a corporate security saving system. An assault model
is composed in light of the foundation information for protection saving outsourced
mining depending upon one–one exchange figures of things that contained the false
exchanges to share each figure thing.
Worku et al. [64] decorated the execution of the above outline by diminishing the
computational escalated operations, for example, bilinear mapping. The technique
pronounced the outcomes to be more secure and effective after careful examination
of security execution. However, the information square inclusion resulted in conspire
non-dynamic. Along these lines, the advancement of a total fundamental and secure
general investigation technique remains an open test for a cloud framework.
1.9 Distributed Method Based PPDM
Ying-hua et al. [68] made it clear that the DPPDM is dependent upon particular essential advancements. Existent methodologies are gathered into three groups named
secure multiparty calculation, bother and confined inquiry. Li [25] compared the work
of each group by outlining and assessing a symmetric key based security safeguarding configuration to strengthen mining tallies. An allurement study is anticipated to
the investigation of the ensured calculation by exhibiting a disagreeable reputation
framework in remote system. The planned structure displayed an allure for acting
mischievously hubs to carry on legitimately. Exploratory conclusion uncovered the
framework proficiency in finding the trespass hubs and enhanced throughput of entire
system consistently. Besides, Dev et al. [9] perceived mystery risk associated with
information mining on cloud framework and outlined an appropriated system to
evacuate such perils. Tassa [56] outlined another plan for secured mining of affiliation standards in on a level plane conveyed database. The planned plan showed
benefits over better plans related than execution and security. This plan encased two
arrangement of principles. Chan and Keng [5] proposed approaches which rely upon
Field and Row-Level scattering of value-based information. The creators planned a
conveyed structure to secure outsourcing affiliation mining rules and investigated the
achievability of its appropriation. The outlined structure for allotting exchanges to
send servers relies on upon the significance of the sorts of protection idea to a client.
Xu and Yi [66] inspected the security protecting conveyed information mining that
goes through unmistakable stages and proceeded. The creators proposed scientific
categorization to insist the consistency and assessment of the conventions effectiveness. Inan and Saygin [14] planned a strategy to assemble disparity designs for
flat conveyed information mining. Nanavati and Jinwala [36] illustrated distinctive
methodology of co-agent setup for the protection of the specific gatherings world-
8
S. Sharma and S. Ahuja
wide and halfway cycles. The interleaved technique is extended and modified to
choose general stage in recurrent affiliation governs privately.
Wang et al. [61] presented an upgraded calculation called Privacy Preserving Frequent Data Mining (PPFDM) in reference to the Frequent Data Mining (FDM) to
protect the security. Om Kumar et al. [42] utilized WEKA for inspection of examples in a particular cloud. Cloud information merchant was utilized with a secured
circulated approach to give productive arrangement that anticipated such mining
assaults on cloud. Nix et al. [41] executed two different conventions for the scalar
(speck) result of two different vectors utilized as sub-conventions in larger information mining. Keshavamurthy et al. [22] showed that there are two potential focal
points in GA approach whereas customary successive example mining calculation
has only one. It is found that in frequent design mining, the populace is framed just
once. On the other hand, in GA strategy the populace is framed for every era that
amplifies the specimen set. Be that as it may, the real disadvantage of GA approach
is associated with the replication in its successive eras. For protection safeguarding
information mining over conveyed dataset, the key objective is to allow calculation
of cumulative measurements for final database with affirmation of the security for
private information of the contributing databases.
1.10 K-Anonymity Based PPDM
Samarati [48] introduced the concept of k-anonymity. A database is k-anonymity
regarding semi-identifier traits (an arrangement of characteristics that can be utilized
with certain outside data to recognize a particular individual) if there exist at any rate
k exchanges in the database having similar esteems as per the semi identifier qualities. Wang et al. [62] concentrated the information mining as an approach utilized for
information veiling called information mining in view of security assurance. After
information concealing, the normal information mining strategies are utilized with
no adjustment engaging the two key components, quality, and versatility. Loukides
and Gkoulalas-Divanis [28] proposed a novel system to anonymize the information
by fulfilling the information distributors’ use necessities encountering low data misfortune. Friedman et al. (2008) augmented the meanings of K-anonymity to demonstrate that the information mining model does not disregard the K-anonymity of the
customers spoken to in the learning illustrations.
To ensure the respondent’s character, the use of K-anonymity additionally consolidated with information mining was proposed by Ciriani et al. [6]. They highlighted
the potential dangers to K-anonymity, which are raised by means of the usage of
mining to gather information and examinations of two principle systems to join Kanonymity in information mining. Soodejani et al. [53] utilized a rendition of the
pursuit named as standard pursue, which put a few limitations on the conditions and
compels being certain and conjunctive. This is forthcoming zone for future review
in mastering examinations on the relevance of different forms of the pursuit in the
strategy. The anonymity guideline of their technique uncovers a few similarities to
Privacy Preserving Data Mining: A Review of the State of the Art
9
the L-diversity security show. Examination of other protection models, for example,
t-closeness may give a more dealt security model to the proposed technique with outrageous value. Loukides et al. [29] presented a decision based security display that
permitted information distributors to prompt fine-grained insurance requirements
for character and sensitive information revelation. They created two anonymization calculations. Karim et al. [19] proposed a numerical strategy to mine maximal
continuous examples with protection saving capacity. This strategy demonstrated a
productive information change procedure which is novel encoded and compacted
cross-section structure and MFPM calculation which diminished both the hunt space
and seeking time. Vijayarani et al. [60] considered K-anonymity to be a fascinating
way to deal with smaller scale information identified with open or semi-open parts
from linking attacks. The possible dangers to K-anonymity approach are depicted
in detail. Especially, the issues identified with information and the methodologies
are recognized to join K-anonymity with information mining. Nergiz et al. [39]
enhanced and augmented the meanings of K-anonymity to complex relations meanings of K-anonymity expression. It is demonstrated that before created strategies
either neglected to secure protection or all in all lessened the information use, and
information insurance in a numerous relations setting. Tai et al. [55] addressed the
issue of secured outsourcing of continuous itemset mining on the multi-cloud situations. In view of the difficulties in huge information examination, they proposed to
segment the information into a few sections and outsourced each part freely to various cloud in light of pseudoscientific classification, anonymization strategy, known
as KAT.
In view of concealment, Deivanai et al. proposed another K-anonymity method
called “kactus” [8]. Kactus performs multidimensional concealment. The qualities
are smothered to a specific record in view of different properties without utilizing
the space progression trees. Another meaning of K-anonymity demonstrate for compelling security insurance of individual consecutive information is presented [33].
Nergiz and Gök [38] and Nergiz et al. [40] played out the speculations, as well as
included the instrument for information migration. In information handle, the position of specific cells is changed to some populated indistinct information cells. The
outcomes uncovered that few movements could upgrade the utility when contrasted
with the heuristic measurements and question noting precision. A hybrid speculations system to migrate the information is presented [38]. In information migration
prepare, information cells are moved to certain populated little gatherings of tuples
which stayed discernable from each other.
Zhang et al. (2013a, 2014a) researched the issues identified with adaptability
related to sub-tree anonymization for colossal information stockpiling over the cloud.
They developed a crossbreed approach alongside Specialization and Generalization
procedures where specialization was top-down and generalization was a bottom-up.
In light of the commitments thus, it merits investigating the following stride on adaptable security protection mindful examination and planning on large-scale datasets.
Afterward, Zhang et al. (2014b) presented a two-stage TDS method in light of Map
Reduce on cloud. In the primary stage, the informational collections are anonymized
and divided in parallel for the creation of the middle of the road results. In the second
10
S. Sharma and S. Ahuja
stage, these transitional outcomes were collected for further anonymization to provide reliable K-anonymous datasets. They have exhibited a proficient semi identifier
record based method to safeguard the security over incremental datasets on cloud.
In the proposed system, QI-gatherings (QI: semi identifier) are recorded utilizing
space values in the present speculation level, which permitted the get to just to a
little bit of records in any database as opposed to induction to the entire information base (Zhang et al. 2013b, c). Moreover, Ding et al. [10] presented a dispersed
anonymization convention for security saving information distributing from different
information suppliers in a cloud framework.
2 Shortcomings of PPDM Methods
Right now, few information mining systems are accessible to secure the protection. Comprehensively, the security saving procedures are grouped by information
conveyance, information bending, information mining calculations, anonymization,
information or principles stowing away, and protection insurance. Table 1 abridges
distinctive procedures connected to secure information mining protection. Concentrated research discoveries throughout the decades uncovered that the current protection preserving information mining look methodologies are still experiencing the ill
effects of real inadequacy including the circulated customers’ information to multi
semi-genuine suppliers, the overhead of registering worldwide mining, incremental
information security issues in distributed computing, honesty of mining results, utility of information, versatility, and overhead execution. Without a doubt, K-anonymity
is a powerful technique for security insurance in information mining. In any case,
a few showed that the information handled by this technique frequently neglected
to beat a few attacks and are defenseless to web phishing. Thusly, the future security safeguarding information mining based K-anonymity needs a propel information
infrastructure to bolster the mix of present information usefulness. This would satisfy
the prerequisites of various types of customers and groups.
3 Conclusion
A comprehensive outline on PPDM strategies in view of mutilation, cooperative
classification, randomization, conveyance, and k-anonymization is introduced. It is
set up that PPDM is showed up logically basic because of simple sharing of protection touchy information for investigation. The striking favorable circumstances
and evident inconveniences of current reviews are stressed. By and by, Big Data
are frequently shared crosswise over areas, for example, well-being, Business-toBusinesses, and Government-to-Government. Therefore, conservation for security
across divulgence is basically needed. A few major associations and governments
worldwide being absolutely reliant on data correspondences by means of web com-
Privacy Preserving Data Mining: A Review of the State of the Art
11
Table 1 Explanation of PPDM methods
PPDM techniques
Explanation
Data distribution
May contain vertical or a level plane apportioned information
Data distortion
Contains blocking, accumulation or consolidating, swapping and
inspecting
Data mining algorithms
Encases grouping mining, affiliation govern mining, bunching, and
Bayesian networks and so on
Data or rules hidden
Signifies to shroud principle information or standards of creative
information
Accomplish anonymization
K-anonymity
L-diverse
Keeps the minimum gathering size K, and keeps up the diversity of
delicate qualities
Taxonomy tree
Assigns the speculation to constrain the data leakage
Randomization
An unrefined and important method to shroud the individual
information in PPDM
Ensures the security, it ought to adjust information painstakingly to
achieve ideal information utility
Privacy protection
municated grave worries over protection issues. Thus, the fast improvement of new
technologies confronted lot of difficulties. Information mining has been the capacity
to concentrate and mine immense ocean of fascinating examples or knowledge from a
gigantic measure of information requires outright security. The fundamental thought
of PPDM is to join the conventional information mining methods in changing the
information to mask delicate data. The real test is to effectively change the information and recuperate its mining result from the changed one. Moreover, the inadequacy
of past reviews demonstrated constrained us to take part in a broad investigation of
the issues of conveyed and distributed information. Subsequently, the overhead for
worldwide mining processing, saving security of developing information, the trustworthiness of mining result, the utility of information, the adaptability and overhead
execution with regards to PPDM are analyzed. There is an earnest need to build up
a solid, effective, and adaptable techniques to vanish issues. The crevices and flaw
of existent literary works has been recognized and investigated issues for critical
upgrades, vigorous security insurance, and protection. is thorough and instructive
audit article is would have liked to fill in as scientific categorization for exploring
and understanding the exploration progressions towards PPDM. As none of the current PPDM calculations can beat all the others as for every one of the criteria, we
talked about the significance of specific measurements for every particular kind of
PPDM calculations, and furthermore called attention to the objective of a decent
metric. There are a few future research bearings en route of measuring the PPDM
calculations and its techniques. There is a need to build up a new technique which
can access different PPDM algorithms. It is additionally vital to plan great measurements that can better mirror the properties of a PPDM calculation, and to develop
benchmark databases for testing a wide range of PPDM calculations.
12
S. Sharma and S. Ahuja
References
1. Aggarwal, C.C., Yu, P.S.: A general survey of privacy-preserving data mining models and
algorithms. In: Privacy Preserving Data Mining (Chap. 2), pp. 11–52. Springer, New York
(2008)
2. Arunadevi, M., Anuradha, R.: Privacy preserving outsourcing for frequent item set mining. Int.
J. Innov. Res. Comp. Commun. Eng. 2(1), 3867–3873 (2014)
3. Baotou, T.: Research on privacy preserving classification data mining based on random perturbation. In: X. Zhang, H. Bi pp. 1–6 (2010)
4. Belwal, R., Varshney, J., Khan, S.: Hiding sensitive association rules efficiently by introducing
new variable hiding counter. In: IEEE International Conference on Service Operations and
Logistics, and Informatics, vol. 1, pp. 130–134, IEEE/SOLI 2008 (2013)
5. Chan, J., Keng, J.: Privacy protection in outsourced association rule mining using distributed
servers and its privacy notions, pp. 1–5 (2013)
6. Ciriani, V., Vimercati, S.D.C., Foresti, S., Samarati, P.: k-anonymous data mining: a survey.
In: Privacy-Preserving Data Mining, pp. 105–136. Springer, New York (2008)
7. Dehkordi, M.N.M., Badie, K., Zadeh, A.K.A.: A novel method for privacy preserving in association rule mining based on genetic algorithms. J. Softw. 4(6), 555–562 (2009)
8. Deivanai, P., Nayahi, J., Kavitha, V.: A hybrid data anonymization integrated with suppression
for preserving privacy in mining multi party data. In: IEEE International Conference on Recent
Trends in Information Technology (ICRTIT) (2011)
9. Dev, H., Sen, T., Basak, M., Ali, M.E.: An approach to protect the privacy of cloud data
from data mining based attacks. In: IEEE 2012 SC Companion High Performance Computing,
Networking, Storage and Analysis (SCC) (2012)
10. Ding, X., Yu, Q., Li, J., Liu, J., Jin, H.: Distributed anonymization for multiple data providers
in a cloud system. In: Database Systems for Advanced Applications. Springer, Berlin (2013)
11. Giannotti, F., Lakshmanan, L.V.S., Monreale, A., Pedreschi, D., Wang, H.: Privacy-preserving
mining of association rules from outsourced transaction databases. IEEE Syst. J. 7(3), 385–395
(2013)
12. Gkoulalas-Divanis, A., Verykios, V.S.: Exact knowledge hiding through database extension.
IEEE Trans. Knowl. Data Eng. 21(5), 699–713 (2009)
13. Harnsamut, N., Natwichai, J.: A novel heuristic algorithm for privacy preserving of associative
classification. In: PRICAI 2008: Trends in Artificial Intelligence, pp. 273–283. Springer, Berlin
(2008)
14. Inan, A., Saygin, Y.: Privacy preserving spatio-temporal clustering on horizontally partitioned
data. In: Lecture Notes in Computer Science (including Subseries Lecture Notes in Artificial
Intelligence and Lecture Notes in Bioinformatics), 6202 LNAI, pp. 187–198 (2010)
15. Islam, M.Z., Brankovic, L.: Privacy preserving data mining: a noise addition framework using
a novel clustering technique. Knowl Based Syst. 24(8), 1214–1223 (2011)
16. Jain, Y.Y.K., Yadav, V.K.V.V.K., Panday, G.G.S.: An efficient association rule hiding algorithm
for privacy preserving data mining. Int. J. Comp. Sci Eng. 3(7), 2792–2798 (2011)
17. Kamakshi, P.: Automatic detection of sensitive attribute in PPDM. In: IEEE International
Conference on Computational Intelligence & Computing Research (ICCIC) (2012)
18. Kamakshi, P., Babu, A.V.: Preserving privacy and sharing the data in distributed environment
using cryptographic technique on perturbed data 2(4) (2010)
19. Karim, R., Rashid, M., Jeong, B., Choi, H.: Transactional databases, pp. 303–319 (2012)
20. Kasthuri, S., Meyyappan, T.: Detection of sensitive items in market basket database using
association rule mining for privacy preserving. In: IEEE International Conference on Pattern
Recognition, Informatics and Mobile Engineering (PRIME) (2013)
21. Kerschbaum, F., Julien, V.: Privacy-preserving data analytics as an outsourced service. In:
Proceedings of the 2008 ACM Workshop on Secure Web Services. ACM (2008)
22. Keshavamurthy, B.N., Khan, A.M., Toshniwal, D.: Privacy preserving association rule mining
over distributed databases using genetic algorithm. Neural Comput. Appl. pp. 351–364 (2013)
Privacy Preserving Data Mining: A Review of the State of the Art
13
23. Kumbhar, M.N., Kharat, R.: Privacy preserving mining of association rules on horizontally and
vertically partitioned data: a review paper. In: 12th IEEE International Conference on Hybrid
Intelligent Systems (HIS), pp. 231–235 (2012)
24. Lai, J., Li, Y., Deng, R.H., Weng, J., Guan, C., Yan, Q.: Towards semantically secure outsourcing
of association rule mining on categorical data. Inf. Sci. (NY) 267, 267–286 (2014)
25. Li, Y.: Privacy-Preserving and Reputation System in Distributed Computing with Untrusted
Parties, no. July. Pro- Quest LLC (2013). Copyright in the Dissertation held by the Author.
Microform Edition © ProQuest LLC. All rights reserved (2013)
26. Li, W., Liu, J.: Privacy preserving association rules mining based on data disturbance and
inquiry limitation. In: 2009 Fourth International Conference on Internet Computer Science
Engineering, pp. 24–29 (2009)
27. Lin, K.W., Lo, Y.-C.: Efficient algorithms for frequent pattern mining in many-task computing
environments. Knowl Based Syst. 49, 10–21 (2013)
28. Loukides, G., Gkoulalas-divanis, A.: Expert systems with applications utility-preserving transaction data anonymization with low information loss. Expert Syst. Appl. 39(10), 9764–9777
(2012)
29. Loukides, G., Gkoulalas-Divanis, A., Shao, J.: Efficient and flexible anonymization of transaction data. Knowl. Inf. Syst. 36(1), 153–210 (2012)
30. Machanavajjhala, A., Kifer, D., Gehrke, J., Venkitasubramaniam, M.: l-diversity: privacy
beyond k-anonymity. ACM Trans. Knowl. Discov. Data 1(1), 3 (2007)
31. Malina, L., Hajny, J.: Efficient security solution for privacy-preserving cloud services. In: 36th
International Conference on Telecommunications and Signal Processing (TSP), pp. 23–27
(2013)
32. Matwin, S.: Privacy-preserving data mining techniques: survey and challenges. In: Discrimination & Privacy in the Information Society, pp. 209–221. Springer, Berlin (2013)
33. Monreale, A., Pedreschi, D., Pensa, R.G., Pinelli, F.: Anonymity preserving sequential pattern
mining. Artif. Intell. Law 22(2), 141–173 (2014)
34. Mukkamala, R., Ashok, V.G.: Fuzzy-based methods for privacy-preserving data mining. In:
IEEE Eighth International Conference on Information Technology: New Generations (ITNG)
(2011)
35. Naeem, M., Asghar, S., Fong, S.: Hiding sensitive association rules using central tendency.
In: 6th International Conference on Advanced Information Management and Service (IMS),
pp. 478–484 (2010)
36. Nanavati, N., Jinwala, D.: Privacy preservation for global cyclic associations in distributed
databases. Procedia Technol. 6, 962–969 (2012)
37. Nayak, G., Devi, S.: A survey on privacy preserving data mining: approaches and techniques.
Int. J. Eng. Sci. Tech. 3(3), 2117–2133 (2011)
38. Nergiz, M.E., Gök, M.Z.: Hybrid k-anonymity. Comput. Secur. 44, 51–63 (2014)
39. Nergiz, M.E., Christopher, C., Ahmet, E.N.: Multirelational k-anonymity. IEEE Trans. Knowl.
Data Eng. 21(8), 1104–1117 (2009)
40. Nergiz, M.E., Gök, M.Z., Özkanlı, U.: Preservation of utility through hybrid k-anonymization.
In: Trust, Privacy, and Security in Digital Business, pp. 97–111. Springer, Berlin (2013)
41. Nix, R., Kantarcioglu, M., Han, K.J.: Approximate privacy-preserving data mining on vertically
partitioned data. In: Data and Applications Security and Privacy XXVI, pp. 129–144. Springer,
Berlin (2012)
42. Om Kumar, C.U., Tejaswi, K., Bhargavi, P.: A distributed cloud—prevents attacks and preserves
user privacy. In: 15th International Conference on Advanced Computing Technologies, ICACT
(2013)
43. Qi, X., Zong, M.: An overview of privacy preserving data mining. Procedia Environ. Sci.
12(Icese 2011), 1341–1347 (2012)
44. Quoc, H., Arch-int, S., Xuan, H., Arch-int, N.: Computers in industry association rule hiding
in risk management for retail supply chain collaboration. Comput. Ind. 64(7), 776–784 (2013)
45. Raghuram, B., Gyani, J.: Privacy preserving associative classification on vertically partitioned
databases. In: IEEE International Conference on Advanced Communication Control and Computing Technologies (ICACCCT), pp. 188–192 (2012)
14
S. Sharma and S. Ahuja
46. Raju, R., Komalavalli, R., Kesavakumar, V.: Privacy maintenance collaborative data mining: a
practical approach. In: 2nd International Conference on Emerging Trends in Engineering and
Technology (ICETET), pp. 307–311 (2009)
47. Sachan, A., Roy, D., Arun, P.V.: An analysis of privacy preservation techniques in data mining.
In: Advances in Computing and Information Technology, vol. 3, pp. 119–128. Springer, Berlin
(2013)
48. Samarati, P.: Protecting respondents’ identities in microdata release. IEEE Trans. Knowl. Data
Eng. (TKDE) 13(6), 1010–1027 (2001)
49. Sathiyapriya, K., Sadasivam, G.S.: A survey on privacy preserving association rule mining.
Int. J. Data Min Knowl. Manag. Process 3(2), 119–131 (2013)
50. Seisungsittisunti, B., Natwichai, J.: Achieving k-anonymity for associative classification
in incremental-data scenarios. In: Security-Enriched Urban Computing and Smart Grid,
pp. 54–63. Springer, Berlin (2011)
51. Shrivastava, R., Awasthy, R., Solanki, B.: New improved algorithm for mining privacy—preserving frequent itemsets. Int. J. Comp. Sci. Inform. 1, 1–7 (2011)
52. Singh, M.D., Krishna, P.R., Saxena, A.: A cryptography based privacy preserving solution to
mine cloud data. In: Proceedings of Third Annual ACM Bangalore Conference. ACM (2010)
53. Soodejani, A.T., Hadavi, M.A., Jalili, R.: k-anonymity-based horizontal fragmentation to preserve privacy in data outsourcing. In: Data and Applications Security and Privacy XXVI,
pp. 263–273. Springer, Berlin (2012)
54. Sweeney, L.: Achieving k-anonymity privacy protection using generalization and suppression.
Int. J. Uncertain Fuzziness Knowl. Based Syst. 10(5), 571–588 (2002)
55. Tai, C.-H., Huang, J.-W., Chung, M.-H.: Privacy preserving frequent pattern mining on multicloud environment. In: 2013 International Symposium on Biometrics and Security Technologies (ISBAST) (2013)
56. Tassa, T.: Secure mining of association rules in horizontally distributed databases. IEEE Trans.
Knowl. Data Eng. 26(4), 970–983 (2014)
57. Vaidya, J., Clifton, C., Kantarcioglu, M., Patterson, A.S.: Privacy-preserving decision trees
over vertically partitioned data. ACM Trans. Knowl. Discov. Data 2(3), 1–27 (2008)
58. Vatsalan, D., Christen, P., Verykios, V.S.: A taxonomy of privacy-preserving record linkage
techniques. Inf. Syst. 38(6), 946–969 (2013)
59. Vijayarani, S., Tamilarasi, A., Seethalakshmi, R.: Privacy preserving data mining based on
association rule: a survey. In: IEEE International Conference on Communication and Computational Intelligence (INCOCCI) (2010a)
60. Vijayarani, S., Tamilarasi, A., Sampoorna, M.: Analysis of privacy preserving k-anonymity
methods and techniques. In: IEEE International Conference on Communication and Computational Intelligence (INCOCCI), pp. 540–545 (2010b)
61. Wang, H., Hu, C., Liu, J.: Distributed mining of association rules based on privacy- preserved method. In: International Symposium on Information Science & Engineering (ISISE),
pp. 494–497 (2010)
62. Wang, K., Yu, P.S., Chakraborty, S.: Bottom-up generalization: a data mining solution to privacy
protection. In: IEEE Fourth International Conference on Data Mining (ICDM’04) (2004)
63. Weng, C., Chen, S., Lo, H.: A novel algorithm for completely hiding sensitive association rules.
In: Eighth International Conference on Intelligent Systems Design and Applications, ISDA’08,
vol. 3, pp. 202–208 (2008)
64. Worku, S.G., Xu, C., Zhao, J., He, X.: Secure and efficient privacy-preserving public auditing
scheme for cloud storage. Comput. Electr. Eng. 40(5), 1703–1713 (2014)
65. Xiong, L., Chitti, S., Liu, L.: k nearest neighbor classification across. In: Proceedings of the
15th ACM International Conference on Information & Knowledge Management CIKM’06,
pp. 840–841 (2006)
66. Xu, Z., Yi, X.: Classification of privacy-preserving distributed data mining protocols. In: Sixth
International Conference on Digital Information Management, pp. 337–342 (2011)
67. Yi, X., Zhang, Y.: Equally contributory privacy-preserving k-means clustering over vertically
partitioned data. Inf. Syst. 38(1), 97–107 (2013)
Privacy Preserving Data Mining: A Review of the State of the Art
15
68. Ying-hua, L., Bing-ru, Y., Dan-yang, C., Nan, M.: State-of-the-art in distributed privacy preserving data mining. In: IEEE 3rd International Conference Communication Software and
Networks, pp. 545–549 (2011)
An MCDM-Based Approach for Selecting
the Best State for Tourism in India
Rashmi Rashmi, Rohit Singh, Mukesh Chand and Shwetank Avikal
Abstract In today’s era, there are many fastest growing industries in this world
and Tourism is one of them. Tourism plays a crucial role in the economic growth
and advance development of a Nation. India is one of the most well-liked tourist
destinations in Asia. Tourism is one of the major sources of foreign exchange. It helps
in the development of an international understanding of our culture and heritage.
Every year thousands of foreigners come to India as a result of which we earn a
lot of foreign exchange. Selection of best place for traveling is a decision-making
problem based on a number of criteria that reflects the preferences of the traveler.
In presented work, a Fuzzy-AHP and TOPSIS approach has been proposed to solve
above discussed problem. In this work., Fuzzy-AHP approach helps to evaluate
the weight of different criteria and TOPSIS method helps to recognize the most
favourable tourist place (states) all over the India and ranks each state accordingly.
Keywords Tourism · MCDM · Fuzzy-AHP · TOPSIS
1 Introduction
Tourism denotes to individual’s temporary motion from their abidance to a destination and tourism industry provided each facility or services affiliated with the
destination to the tourists [1]. In 2005, The Indian Tourism Development Corporation (ITDC) begins a movement that is known as “Incredible India” to enhance the
growth of tourism in India. The tourism industry provides a job to the large number of
individuals, whether skilled or not. Tourism industry is very beneficial for the growth
of hostels, travel agencies, transport including airlines. Tourism encourages national
and international understanding. A productive tourism industry can increase regional
R. Rashmi · R. Singh · S. Avikal (B)
Department of Mechanical Engineering, Graphic Era Hill University, Dehradun, India
e-mail: shwetank.avikal@gmail.com
M. Chand
Department of Supply Chain Management, Infosys, Mangalore, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_2
17
18
R. Rashmi et al.
economic growth and developing a source of valuable foreign exchange income [2,
3]. Nowadays, there are many nations and areas where, tourism can be consider as
one of the leading growth industries [4]. According to Dellaert et al. [5], decision
taken by tourists are complex miscellaneous decisions, where the alternatives for
various factors are interconnected. In presented work, to solve the above-discussed
MCDM problem, a Fuzzy-AHP and TOPSIS based approach has been proposed. In
this work, the best state for traveling in India has been selected with the help of above
MCDM-based approach. The presented work contains 30 several states of India, with
five several criteria. At the end of the work best state is selected among these states.
This paper has been prepared as follows: the second section represents the literature review; the third section represents methodology, while fourth section represents
problem definition, fifth section represent calculations and result, and the final section
represents the conclusion.
2 Literature Review
Mohamad et al. [6] have shown an evaluation of the censorious factors that affecting
the preferred destination chosen by local tourist’s in Kedha and used Fuzzy Hierarchical TOPSIS (FHTOPSIS) method, for resolve the tourists’ strong liking for
destinations with respect to these factors.
Liu et al. [7] have applied the hybrid MCDM method to analyze the dependent
relationship within different properties and criteria of tourism policies and finally, to
propose a most favorable development plan for Taiwan tourism policy.
Hsu et al. [8] have recognized those factors which affect the preferred destination
selected by tourists and determine the right choice of tourism for preferred destination.
Cracolici et al. [4] have evaluated the comparative attractiveness of challenging
tourist destinations with the help of perceptual experience of individual visitors about
holiday destination.
Chen et al. [9] have recognized those factors that affecting lake surroundings and
to determine a multi-criteria evaluation configuration for tourist.
Alptekin et al. [10] have suggested an intelligent framework for travel authority
that is based on Web which provides quick and trustable response service to the people
in a smaller amount. The suggested framework incorporates case-based reasoning
(CBR) system with a widely known critical decision making (MCDM) method, that
is Analytic Hierarchy Process to increase the accuracy and fastness in case similar
in tourist destination planning.
Stauvermann et al. [11] have suggested a model for the tourism requirements in
the circumstance of a fast growing country. The model parameters are a tourist region
describe through noncompetitive, while the primary element of production is human
capital and hostels have market power.
An MCDM-Based Approach for Selecting the Best State …
19
3 Methodology
3.1 AHP and Fuzzy AHP Approach
Satty [12, 13] proposed a method namely, Analytic Hierarchy Process (AHP) that has
been effectively used in many areas like evaluation, selection, ranking, forecasting,
etc. AHP is a organized method to determine the final importance using pairwise
comparisons among various attributes [14]. Regardless of its beneficial features and
liked by many people, it is also finding faults for its failure to effectively manage
inherent doubt and vagueness of evaluated things. To solve this type of uncertainty
AHP is integrated with fuzzy set theory proposed by [15]. Fuzzy-AHP method has
been mostly used by several researchers and turn out to be one of the best methods
to solve decision-making problems. In this study, Fuzzy-AHP proposed by Avikal
et al. [16] has been used as reference.
3.2 TOPSIS Method (Technique for Order Performance
by Similarity to Ideal Solution)
TOPSIS is a multi-criteria decision-making technique used to rank a finite set of
alternatives. By TOPSIS, the best alternatives should have the shortest distance from
the positive ideal solution (PIS) and the farthest distance from the negative ideal
solution (NIS). The PIS is formed as a composite of the best performance values
exhibited by any alternative for each criterion. The NIS is the composite of the
worst performance values [17]. The ranking of alternatives is based on the relative
similarity to the ideal solution that avoids the conditions of the alternative having
the same similarity to both PIS and NIS. In this study, TOPSIS proposed by Avikal
et al. [16] has been used as reference.
4 Problem Definition
The main objective of this work is to solve the problem of tourist for the selection of
best tourist place in India according to their needs, and help them where to travel and
where not to. During holidays mostly people make plans for trip and get confused by
thinking that which place is best for them and are not able to select a suitable tourist
place. For this study, five several criteria have been selected and discussed in Table 1.
A survey has been conducted with the help of these selected criteria among tourist
experts. On the basis of this survey, the weight of each criterion has been computed
using Fuzzy-AHP and the computed weights have been used for further calculation
and eventually TOPSIS method for ranking the states according to selected criteria.
20
R. Rashmi et al.
Table 1 Several criteria and their definition
No. Criteria
Definition
C1
Visual value
There are definite attractive things that have a power to attract the
tourist and appeal to them, some are natural, cultural or historical
C2
C3
No. of
attractions
Ease of access
Amount of tourist attractions. For example, no. of natural and cultural
attractions
Access to tourist terminus. To reach the expected terminus either by
car, taxi, train or plane
C4
Security
Security for women, night security, and crime rates in tourism places
and also the current existence of police forces to provide security
C5
Enviromental
impact
Enviromental affect like waste disposal system, noise pollution,
enviromental pollution
Table 2 Results obtained
with fuzzy AHP
Criteria
Weights
ňmax, CI, RI
C1
C2
0.1566
0.0485
ňmax 5.4176
C3
C4
C5
0.4479
0.2545
0.0923
CI 0.1044
RI 1.12
CR
CR 0.0932
5 Calculation
Singh et al. [18] have presented the rating of all the 30 states. The weight calculated by
Singh et al. using Fuzzy AHP has been used for further study and has been presented
in Table 2. Finally, each state has been ranked by means of TOPSIS method. The
steps of TOPSIS method have been presented in the following Tables 3 and 4. Step 1
has been solved in the table in 3, Step 2 has been solved in the table in 4, Step 3 has
been solved in Table 5, Step 4 has been solved in Tables 6 and 7, and final ranking
has been shown in Table 8.
6 Conclusion
This work shows an MCDM-based technique and has been proposed to determine the
most prestigious tourist state for tourism in India. Fuzzy-AHP method has been used
to calculate the weight of each criterion and TOPSIS method has been used to rank
all states. The result shows that Maharashtra is the most preferred state for tourism
and Telangana is the least preferred states for tourism. Maharashtra is most preferred
because of its prominent rating in all criteria and Telangana is least preferred because
of its least rating in all criteria.
An MCDM-Based Approach for Selecting the Best State …
Table 3 Data normalization for TOPSIS
States
C1
C2
21
C3
C4
C5
Uttarakhand
Madhya
Pradesh
Maharashtra
Kerala
Jammu
Kashmir
Delhi
Andhra
Pradesh
Arunachal
Pradesh
Assam
Bihar
Chhattisgarh
1
0.8
0.5
0.166667
0.666667
1
0.769231
0.384615
0.25
0.75
1
0.8
1
1
0.75
0
1.333333
0.666667
0.333333
0.769231
0.576923
0.384615
0.75
0.5
0.25
0.8
0.6
0.416667
0.083333
1
0.666667
0.384615
0.461538
0.5
0.8
0.8
0.25
1
0.769231
0.5
0.4
0
0
0.166667
0
0.166667
1
0.333333
0
0.615385
0.846154
0.884615
0.5
1.5
0
Goa
Haryana
1
0.2
0.333333
0
1
0.666667
0.576923
0.692308
0.5
0.7
Himachal
Pradesh
Jharkhand
Karnataka
Gujarat
1
0.5
0.666667
0.769231
0.8
0.4
0.8
1
0.416667
0.333333
0.833333
0.666667
1
1
0.576923
0.769231
0.384615
1
1
1
Manipur
0.8
0.666667
0.333333
0.769231
0.25
Meghalaya
0.8
0.583333
0
0.769231
0.5
Mizoram
Nagaland
0.6
1
0.75
0.75
0.333333
0
0.692308
0.846154
1.25
0.4
Odisha
Punjab
0.52
0.8
0.333333
0.333333
0.666667
0.666667
0.384615
0.307692
1
1
Rajasthan
0.6
0.583333
0.666667
0.384615
0.75
Sikkim
Tamil Nadu
Telangana
0.8
0.4
0.2
0.25
0.75
0
0.333333
1
0
0.461538
0.346154
0
0.5
0.5
1
Tripura
0.4
0.166667
0
0.423077
1
Uttar Pradesh
West Bengal
0.6
0.4
0.583333
0.083333
1
1
1
0.692308
0.25
0.5
22
R. Rashmi et al.
Table 4 Weight decision matrix
States
C1
C2
C3
C4
C5
Uttarakhand
Madhya
Pradesh
Maharashtra
Kerala
Jammu
Kashmir
Delhi
Andhra
Pradesh
Arunachal
Pradesh
Assam
Bihar
Chhattisgarh
0.1566
0.12528
0.02425
0.008083
0.2986
0.4479
0.195769
0.097885
0.023075
0.069225
0.1566
0.12528
0.1566
0.0485
0.036375
0
0.5972
0.2986
0.1493
0.195769
0.146827
0.097885
0.069225
0.04615
0.023075
0.12528
0.09396
0.020208
0.004042
0.4479
0.2986
0.097885
0.117462
0.04615
0.07384
0.12528
0.012125
0.4479
0.195769
0.04615
0.06264
0
0
0.008083
0
0.008083
0.4479
0.1493
0
0.156615
0.215346
0.225135
0.04615
0.13845
0
Goa
Haryana
0.1566
0.03132
0.016167
0
0.4479
0.2986
0.146827
0.176192
0.04615
0.06461
Himachal
Pradesh
Jharkhand
Karnataka
Gujarat
0.1566
0.02425
0.2986
0.195769
0.07384
0.06264
0.12528
0.1566
0.020208
0.016167
0.040417
0.2986
0.4479
0.4479
0.146827
0.195769
0.097885
0.0923
0.0923
0.0923
Manipur
0.12528
0.032333
0.1493
0.195769
0.023075
Meghalaya
0.12528
0.028292
0
0.195769
0.04615
Mizoram
Nagaland
0.09396
0.1566
0.036375
0.036375
0.1493
0
0.176192
0.215346
0.115375
0.03692
Odisha
Punjab
0.081432
0.12528
0.016167
0.016167
0.2986
0.2986
0.097885
0.078308
0.0923
0.0923
Rajasthan
0.09396
0.028292
0.2986
0.097885
0.069225
Sikkim
Tamil Nadu
Telangana
0.12528
0.06264
0.03132
0.012125
0.036375
0
0.1493
0.4479
0
0.117462
0.088096
0
0.04615
0.04615
0.0923
Tripura
0.06264
0.008083
0
0.107673
0.0923
Uttar Pradesh
West Bengal
0.09396
0.06264
0.028292
0.004042
0.4479
0.4479
0.2545
0.176192
0.023075
0.04615
MAX
MIN
0.1566
0
0.0485
0
0.5972
0
0.2545
0
0
0.13845
Table 5 Positive ideal solution (PIS) and negative ideal solution (NIS)
PIS
NIS
0.1566
0
0.0485
0
0.5972
0
0.2545
0
0
0.13845
An MCDM-Based Approach for Selecting the Best State …
23
Table 6 Separation distance of alternative from positive idle solution (K+)
States
C1
C2
C3
C4
C5
SUM
K+
Uttarakhand
Madhya
Pradesh
Maharashtra
Kerala
Jammu
Kashmir
Delhi
Andhra
Pradesh
Arunachal
Pradesh
Assam
Bihar
Chhattisgarh
0.1566
0.12528
0.02425 0.2986
0.008083 0.4479
0.195769 0.023075
0.097885 0.069225
0.093732
0.054225
0.306156
0.232864
0.1566
0.12528
0.1566
0.0485
0.5972
0.036375 0.2986
0
0.1493
0.195769 0.069225
0.146827 0.04615
0.097885 0.023075
0.008241
0.104013
0.228027
0.090782
0.322511
0.477522
0.12528
0.09396
0.020208 0.4479
0.004042 0.2986
0.097885 0.04615
0.117462 0.07384
0.05073
0.119294
0.225233
0.34539
0.12528
0.012125 0.4479
0.195769 0.04615
0.030174
0.173706
0.06264
0
0
0.008083 0.4479
0
0.1493
0.008083 0
0.156615 0.04615
0.215346 0.13845
0.225135 0
0.044464
0.248192
0.383667
0.210864
0.498188
0.619409
Goa
Haryana
0.1566
0.03132
0.016167 0.4479
0
0.2986
0.146827 0.04615
0.176192 0.06461
0.037059
0.117516
0.192508
0.342806
Himachal
Pradesh
Jharkhand
Karnataka
Gujarat
0.1566
0.02425
0.195769 0.07384
0.098652
0.314089
0.06264
0.12528
0.1566
0.020208 0.2986
0.016167 0.4479
0.040417 0.4479
0.146827 0.0923
0.195769 0.0923
0.097885 0.0923
0.118904
0.036285
0.055403
0.344824
0.190487
0.235379
0.2986
Manipur
0.12528
0.032333 0.1493
0.195769 0.023075
0.205838
0.453694
Meghalaya
0.12528
0.028292 0
0.195769 0.04615
0.363616
0.603006
Mizoram
Nagaland
0.09396
0.1566
0.036375 0.1493
0.036375 0
0.176192 0.115375
0.215346 0.03692
0.224129
0.359691
0.473422
0.599742
Odisha
Punjab
0.081432 0.016167 0.2986
0.12528 0.016167 0.2986
0.097885 0.0923
0.078308 0.0923
0.128905
0.130751
0.359034
0.361596
Rajasthan
0.09396
0.028292 0.2986
0.097885 0.069225
0.122815
0.350449
Sikkim
0.12528
Tamil Nadu 0.06264
Telangana
0.03132
0.012125 0.1493
0.036375 0.4479
0
0
0.117462 0.04615
0.088096 0.04615
0
0.0923
0.223828
0.061086
0.447985
0.473104
0.247156
0.669317
Tripura
0.06264
0.008083 0
0.107673 0.0923
0.397187
0.630228
Uttar
Pradesh
West
Bengal
0.09396
0.028292 0.4479
0.2545
0.027155
0.164788
0.06264
0.004042 0.4479
0.176192 0.04615
0.041357
0.203365
0.023075
24
R. Rashmi et al.
Table 7 Separation distance of alternative from negative idle solution (K−)
States
C1
C2
C3
C4
C5
SUM
K−
Uttarakhand
Madhya
Pradesh
Maharashtra
Kerala
Jammu
Kashmir
Delhi
Andhra
Pradesh
Arunachal
Pradesh
Assam
Bihar
Chhattisgarh
0.024524 0.000588 0.089162 0.038326 0.013311 0.165911
0.015695 6.53E−05 0.200614 0.009581 0.004792 0.230748
0.407321
0.480363
0.024524 0.002352
0.015695 0.001323
0.024524 0
0.356648 0.038326 0.004792 0.426641
0.089162 0.021558 0.008519 0.136258
0.02229 0.009581 0.013311 0.069707
0.653178
0.369131
0.264021
0.015695 0.000408 0.200614 0.009581 0.008519 0.234819
0.008828 1.63E−05 0.089162 0.013797 0.004174 0.115978
0.484581
0.340556
0.015695 0.000147
0.200614 0.038326 0.008519 0.263301
0.513129
0.003924 6.53E−05 0.200614 0.024528 0.008519 0.237651
0
0
0.02229 0.046374 0
0.068664
0
6.53E−05 0
0.050686 0.019168 0.069919
0.487495
0.262039
0.264423
Goa
Haryana
0.024524 0.000261
0.000981 0
0.200614 0.021558 0.008519 0.255477
0.089162 0.031044 0.005452 0.126639
0.505447
0.355864
Himachal
Pradesh
Jharkhand
Karnataka
Gujarat
0.024524 0.000588
0.089162 0.038326 0.004174 0.156774
0.395946
0.003924 0.000408
0.015695 0.000261
0.024524 0.001634
0.089162 0.021558 0.00213
0.200614 0.038326 0.00213
0.200614 0.009581 0.00213
0.117182
0.257026
0.238483
0.342319
0.506978
0.488347
Manipur
0.015695 0.001045
0.02229
0.038326 0.013311 0.090668
0.301111
Meghalaya
0.015695 0.0008
0
0.038326 0.008519 0.06334
0.251675
Mizoram
Nagaland
0.008828 0.001323
0.024524 0.001323
0.02229
0
0.031044 0.000532 0.064018
0.046374 0.010308 0.082529
0.253018
0.287279
Odisha
Punjab
0.006631 0.000261
0.015695 0.000261
0.089162 0.009581 0.00213
0.089162 0.006132 0.00213
0.107766
0.11338
0.328277
0.33672
Rajasthan
0.008828 0.0008
0.089162 0.009581 0.004792 0.113164
0.336399
0.02229 0.013797 0.008519 0.060449
0.200614 0.007761 0.008519 0.222142
0
0
0.00213 0.003111
0.245864
0.471319
0.055774
Sikkim
0.015695 0.000147
Tamil Nadu 0.003924 0.001323
Telangana 0.000981 0
Tripura
0.003924 6.53E−05 0
Uttar
Pradesh
West
Bengal
0.008828 0.0008
0.011593 0.00213
0.017712
0.133088
0.013311 0.288325
0.536959
0.003924 1.63E−05 0.200614 0.031044 0.008519 0.244118
0.494083
0.200614 0.06477
An MCDM-Based Approach for Selecting the Best State …
Table 8 Final rank of states in India
States
K+
25
K−
SCORE
RANK
Uttarakhand
Madhya Pradesh
0.306156
0.232864
0.407321
0.480363
0.570896
0.673507
12
10
Maharashtra
Kerala
Jammu Kashmir
Delhi
Andhra Pradesh
Arunachal
Pradesh
Assam
Bihar
Chhattisgarh
0.090782
0.322511
0.477522
0.225233
0.34539
0.173706
0.653178
0.369131
0.264021
0.484581
0.340556
0.513129
0.877974
0.533702
0.356042
0.682687
0.496477
0.747092
1
14
22
8
17
3
0.210864
0.498188
0.619409
0.487495
0.262039
0.264423
0.698058
0.344685
0.299178
7
24
27
Goa
Haryana
0.192508
0.342806
0.505447
0.355864
0.724183
0.509345
5
15
Himachal
Pradesh
Jharkhand
Karnataka
Gujarat
0.314089
0.395946
0.557644
13
0.344824
0.190487
0.235379
0.342319
0.506978
0.488347
0.498177
0.726886
0.674767
16
4
9
Manipur
0.453694
0.301111
0.398926
21
Meghalaya
0.603006
0.251675
0.294467
28
Mizoram
Nagaland
0.473422
0.599742
0.253018
0.287279
0.348299
0.323869
23
25
Odisha
Punjab
0.359034
0.361596
0.328277
0.33672
0.477625
0.482189
20
19
Rajasthan
0.350449
0.336399
0.489772
18
Sikkim
Tamil Nadu
Telangana
0.473104
0.247156
0.669317
0.245864
0.471319
0.055774
0.341968
0.655999
0.07692
26
11
30
Tripura
0.630228
0.133088
0.174355
29
Uttar Pradesh
West Bengal
0.164788
0.203365
0.536959
0.494083
0.765175
0.708415
2
6
SCORE denotes the relative closeness to the ideal solution for each competitive design alternative
RANK denotes the ranking of all the states according to relative closeness
26
R. Rashmi et al.
References
1. Matheison, A., Wall, G.: Tourism: Economic, Physical and Social Impacts. Longman, New
York (1982)
2. Chang, J.R., Chang, B.: The development of a tourism attraction model by using fuzzy theory.
Math. Prob. Eng. pp. 1–10 (2015)
3. Wu, W.-W.: Beyond travel & tourism competitiveness ranking using DEA, GST, ANN and
Borda count. Expert Syst. Appl. 38(10), 12974–12982 (2011)
4. Cracolici, M.F., Nijkamp, P.: The attractiveness and competitiveness of tourist destinations: a
study of Southern Italian regions. Tour. Manag. 30, 336–344 (2008)
5. Dellaert, B.G.C., Etterma, F., Lindh, C.: Multi-faceted tourist travel decisions: a constraintbased conceptual framework to describe tourist sequential choice of travel components. Tour.
Manag. 19(4), 313–320 (1998)
6. Mohamad, D., Jamil, R.M.: A preference analysis model for selecting tourist destinations
based on motivational factors: a case study in Kedah, Malaysia. Procedia—Social Behav. Sci.
65, 20–25 (2012)
7. Liu, C.H., Tzeng, G.H., Lee, M.H.: Improving tourism policy implementation—The use of
hybrid MCDM models. Tour. Manag. 33, 413–426 (2012)
8. Hsu, T.K., Tsai, Y.F., Wu, H.H.: The preference analysis for tourist choice of destination: a
case study of Taiwan. Tour. Manag. 30, 288–297 (2009)
9. Chen, C.L., Bau, Y.P.: Establishing a multi-criteria evaluation structure for tourist beaches in
Taiwan: a foundation for sustainable beach tourism. Ocean Coast. Manag. 121, 88–96 (2016)
10. Alptekin, G.I., Buyukozkan, G.: An integrated case-based reasoning and MCDM system for
web based tourism destination planning. Expert Syst. Appl. 38, 2125–2132 (2011)
11. Stauvermann, P.J., Kumar, R.R.: Productivity growth and income in the tourism sector: role of
tourism demand and human capital investment. Tour. Manag. 61, 426–433 (2017)
12. Saaty, T.L.: A scaling method for priorities in hierarchical structures. J. Math. Psychol. 15(3),
234–281 (1977)
13. Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)
14. Ng, C.Y.: Evidential reasoning-based fuzzy AHP approach for the evaluation of design alternatives environmental performances. Appl. Soft Comput. 46, 381–397 (2016)
15. Zadeh, L.: Fuzzy sets. Inf. Control 8, 338–353 (1965)
16. Avikal, S., Jain, R., Mishra, P.K.: A Kano model, AHP and M- TOPSIS method-based technique for disassembly line balancing problems under fuzzy environment. Appl. Soft Comput.
pp. 519–525 (2014)
17. Terol, A.B., Parra, M.A., Fernández, V.C., Ibias, J.A.: Using TOPSIS for assessing the sustainability of government bond funds. Int. J. Manag. Sci. (2014)
18. Singh, S., Mundepi, V., Hatwal, D., Raturi, V., Chand, M., Rashmi, Sharma, S., Avikal, S.:
Selection of best state for tourism in India by fuzzy approach. Adv. Comput. Comput. Sci.
pp. 549–557 (2017)
Gravitational Search Algorithm:
A State-of-the-Art Review
Indu Bala and Anupam Yadav
Abstract Gravitational search algorithm (GSA) is a recent algorithm introduced in
2009 by Rashedi et al. It is a heuristic optimization algorithm based on Newton’s
laws of motion and law of Gravitation. Till now, a lot of changes have been done in
original GSA to improve its speed of convergence and its quality of solution; also
this algorithm is still exploring in many fields. Therefore, this article is intended to
provide the current state of algorithm, modifications, advantages, disadvantages, and
its future possibilities of research.
Keywords Gravitational search algorithm (GSA) · Applications · Hybridization
Modification GSA · Evolutionally optimization · Nature inspired computational
search
1 Introduction
Gravitational search algorithm (GSA) is a heuristic technique in the field of numerical optimization. It is scholastic and swarm-based search for hard combinational
problems. GSA is based on law of gravity and law of motion [1]. It is comprised
of masses (agents) in which heavier masses consider as a prominent solution of the
problem. Due to gravitational law of motion, each mass attracts towards each other
that cause a global movement. Also lighter masses attract towards the heavier mass
which gives an optimal solution. Every heuristics algorithm follows exportation and
exploitation criteria. Similar in GSA, algorithm first explores the search region then
laps of iterations; it converges to a solution called the exploitation step. GSA is a
very fast growing algorithm which helps to find optimal or near-optimal solutions.
GSA has been used by many applications of several problems. It can apply on conI. Bala (B)
Northcap University, Gurgaon 122017, India
e-mail: iinduyadav@gmail.com
A. Yadav
National Institute of Technology Uttarakhand, Srinagar 246174, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_3
27
28
I. Bala and A. Yadav
tinuous as well as binary search space [2]. The various versions of GSA have been
developed which helped to improve the efficiency of exploration and exploitation.
Study of development of GSA is necessary to know about, how far its development,
advantage, disadvantages and how much has been used to solve an optimization
problem. This article will describe about its advantages, disadvantages, and modification that have been made till now to make a conclusive remark on the ability of
GSA. In Sect. 2, standard GSA is described, Sect. 3 covers modification in GSA till
then. Section 4 describes all hybrid form of heuristic algorithm with GSA, Sect. 5
tells it advantages and disadvantages with its criticism and in Sect. 6 we wrap our
work and discuss conclusion and future scope.
2 Standard GSA
GSA was first introduced in 2009 by Rashedi et al. [1]. The aim of this algorithm is to
solve hard combinational optimization problem with reasonable cost. GSA simulates
a set of agents that work as point masses in a N dimensional space. xi represents
position and m i represents the mass of agent i. In GSA, positions are considered
as candidate solutions and masses are correlated with the quality of the candidate
solutions; means if quality is high then mass would be large. Due to gravitational
law, a force of attraction between mass i and j at time step t and dimension d is
given as:
Fidj (t) G(t) ·
Mi (t) × M j (t) d
(x j (t) − xid (t))
Ri j (t) + ε
(1)
G(t) is gravitational constant which controls the process using variable α and
decreases with time as
G(t) G 0 × exp(−α × iter/max iter)
(2)
ε is a small constant, and Rik (t) is Euclidian distance between agents i and k.
Hence, the total force of mass i at time t is given as
Fid (t) N
randk Fikd (t)
(3)
k∈K best,ki
where randk represents random number in the interval [0, 1], Kbest is the set of best
fitness value of first K agents.
If all the agents attract each other that cause a global movement of object towards
the heavy masses, hence position of particle influenced by the velocity (veli ) and
accelerations aci as:
Gravitational Search Algorithm: A State-of-the-Art Review
29
velid (t + 1) randi × velid (t) + acid (t)
xid (t
+ 1) xid (t)
+
velid (t
(4)
+ 1)
(5)
By Newton’s Law of motion, the acceleration of object i in dth dimension is given
as
acid (t) Fid (t)
Mi (t)
(6)
By the help of velocity and position equations, we can update the position of
agents; it helps to move masses towards the heavier mass which considered as a
prominent solution. After running prescribes iterations or lapse of time, all masses
converge to heavier mass that follow optimal solution. We can update the agents as
m i (t) fiti (t) − worst(t)
,
best(t) − worst(t)
m i (t)
Mi (t) N
j1 m j (t)
(7)
where fiti (t) represents the fitness value of the objects i and best(t) and worst(t) are
given for maximization case as
best(t) max fiti (t) worst(t) i∈(1,2...N )
min
i(1,2...N )
fiti (t)
3 Modification of GSA
The modifications of GSA consist of three categories: modification and extension of
parameters, extension of searching space, and hybridization with another technique.
The modification in GSA can improve its speed and performance.
1. Binary GSA [2]
GSA helps to solve optimization problems in real continuous space, while BGSA [2]
can solve it for discrete space. In discrete binary space, every dimension can take only
0 or 1 or vice versa. In this algorithm, force, acceleration and velocity
are calculated
same as GSA continuous algorithm, its only update position xid by switching bit
between 0 and 1. The position update
carried in a manner that the current bit value
is changed with a probability S vid , which is calculated by mass velocity as
S vid (t) tanh vid (t) Once probability of mass velocity is calculated, then the objects will move as
if rand < S vid (t + 1) ; then xid (t + 1) complement xid (t) ;
else xid (t + 1) xid (t)
(8)
30
I. Bala and A. Yadav
Probability of changing position must be near zero when velocity is ‘small’ and
hence got optimum solution.
• A small absolute value of the velocity must provide a small probability of changing
the position. In other words, a zero value of the velocity represents that the mass
position is good and must not be changed.
2. Single Objective GSA [3]
A large number of GSA variables can be found to locate single solutions. They were
specially developed to find single solution in continuous unconstrained optimization
problems. Most of these algorithms can also be applied to other types of problem.
3. Multi-objective GSA (MOGSA) [4]
This algorithm helps to find multiple non-dominating solutions. It is also called as
niching algorithm. In MOGSA [4], for all iterations, a randomly selected object
is considered as leader and other objects seek following it. For better exploration
criteria, a grid structure has been created and stored in “achieve”. The grid structure
is created as follows: Divided each dimension
in the objective space into 2ni equal
k
ni
i1
divisions and hence for a k-objects 2
, where i denotes dimension index. As
long as the archive is not full, new non-dominated solutions are added to it. Hence
laps of time, it converges to a solution. Also during the time gravitational constant
must decrease which implies a finer search of optima in last iteration.
4. Piecewise-based GSA (PFGSA) [5]
PFGSA [5] improves the searching ability of GSA. It is more flexible to control the
decreasing rate of gravitational constant (G). It divides G into three stages: Coarse,
moderate, and fine search stage. In coarse stage, G decreases to a larger rate and
reduces the search space quickly. In moderate stage, decreasing rate of G becomes
slow and gradually it comes close to global optima. In fine stage, G is quite small
due to low decreasing rate of G and it searches the global optima in a meticulous
way.
5. Disruption operator with GSA or Improved GSA (IGSA) [6]
A nature-inspired operator named “Disruption” is introduced to improve the performance of standard GSA. It helps to improve the ability of exploration and exploitation
in search space. All masses (solutions) converge towards an optimum solution; a new
operator D [6] is introduced
Ri j · U (−0.5, 0.5) if Ri,best ≥ 1
D
(9)
1 + ρ · U (−0.5, 0.5) otherwise
U (−0.5, 0.5) is uniform distributed pseudo-random number in. (−0.5, 0.5). Exploration and exploitation process depends upon operator D, if Ri,best ≥ 1, D explores
search space and if Ri,best < 1, D converges to the best solution.Ri,best is the distance
between mass i and best solution so far.
Gravitational Search Algorithm: A State-of-the-Art Review
31
6. Quantum-Based GSA (QGSA) [7]
QGSA [7] is based on dynamic of quantum. In this algorithm, each object has quantum behavior, which means each object is expressed by a wave function. In standard
GSA, kbest set contains all prominent solution of the problem whereas in QGSA
each kbest member is the center of an attractive potential field which is called the
delta potential well. In which each agent chooses a kbest member by probabilistic
mechanism. It guarantees to limit the quantum boundary of the object.
7. Adaptive GSA [8]
In QGSA [7], selection process of kbest member for delta potential well was not
properly defined and hence exploration process can be uncontrolled. But adaptive
GSA helps to overcome this problem. This algorithm reduces parametric sensitivity
by the help of fuzzy controller. It uses two depreciation laws of the gravitational
constant G and also it considers a parameter in the weighted sum of all forces exerted
from the other agents to the iteration index. This algorithm controls the searching
ability of GSA and gives high convergence rate
8. Fast Discrete GSA (FDGSA) [9]
GSA was originally introduced for continuous-valued space. Many problems are
however defined for discrete value and then binary GSA came into existence. The
main difference between the fast discrete GSA and binary GSA is that position
of masses is updated by its direction and velocity. Both the direction and velocity
determine the candidates of integer values for the position update of the masses
and then selection process is completed randomly. FDGSA [10] converges faster as
compared to BGSA.
9. Synchronous versus Asynchronous GSA (A-GSA) [11]
In standard GSA, the velocity and position of whole population are updated after
evaluating the agent’s performance and then worst and best performing agents are
recognized. This updating method is classified as synchronous update. But in A-GSA
[12], agent’s velocity and position are updated parallel with agent’s performance,
without waiting for the entire population to be evaluated. Therefore, the best and
worst agents are recognized using mix information of previous and current iterations.
This updating method encourages agent’s more exploration in search space.
10. Modified GSA (MGSA) [12]
This algorithm contributes effective modification in standard GSA. This algorithm
modifies maximum velocity constraints which help to control exploration process of
standard GSA. MGSA searching criteria are based on two factors: minimum factor
of safety and minimum reliability index. It increases the convergence rate and helps
to obtain a solution with a lower number of iterations.
32
I. Bala and A. Yadav
11. Improved QGSA (IQGSA) [13]
It is a new version of QGSA. The proposed algorithm improves the efficiency of
QGSA by replacing fitness function of QGSA into new fitness function. It has given
better result than original GSA and QGSA.
12. Multi-agents based GSA [14]
In proposed algorithm, operations have implemented in parallel way, not sequentially.
This algorithm has ability to solve non-convex objective function in optimization
problem. It also reduces parametric sensitivity and performs very well.
13. Fuzzy GSA [10]
In this algorithm, a fuzzy logic controller is (FLC) introduced which improves the
convergence rate and give better result. FLC controls GSA’s parameters G and α, and
also balances the exploration and exploitation search process.
14. Groping GSA (GGSA) [15]
This algorithm is introduced for data clustering problem, this refers to the process of
grouping a set of data objects into clusters in which the data of a cluster must have
great similarity and data of different clusters have high dissimilarity. The performance
of GGSA evaluated through many benchmark datasets from the well-known UCI
machine learning repository and found good convergence rate.
15. Adaptive Centric GSA (AC-GSA) [16]
This algorithm introduces velocity update formula and weight function to improve
standard GSA efficiency. Also Kbest can be found as
Kbest finalper + 1 −
iteration
max itri
∗ 100 − finalper
where finalper is the particle which can attribute other in the last generation.
16. Locally Informed GSA (LIGSA) [17]
In LIGSA, each agent learns from its unique neighborhood formed by k local neighbor and gbest formed from the kbest group. It avoids premature convergence and
explores the search space quickly. Also gbest agent accelerates the convergence
speed.
17. Fitness Based GSA (FBGSA) [18]
In FBGSA, new velocity function is introduced, in which new velocity depends upon
the previous velocity and acceleration based on the fitness of the solutions. Also, the
high fit solution converges to promising search region where low fit solution explores
the search space.
Gravitational Search Algorithm: A State-of-the-Art Review
33
4 Hybrid Versions of GSA
The hybridization of an algorithm makes the algorithm more effective and improves
the ability of an algorithm. Due to hybridization, exploring area of algorithm can be
enhanced and can solve more problems. Hybridizations with GSA are given below:
1. Hybrid Particle swarm optimization and GSA (PSOGSA) [19]
This hybridization [12] is based on PSO and GSA function optimization problem.
PSO algorithm is based on natural phenomena of birds flocking. This algorithm
introduced global best, i.e., gbest concept of PSO in GSA which gives best current
position of agents, and most of the function provides faster convergence speed.
2. Modified PSO and GSA (MPSOGSA) [20]
Standard PSO has features of saving previous local optimum and global optimum
solutions which are referred as memory of PSO. In this hybridization, PSO puts
particle memory in GSA. The particle memory in GSA is revised its own global
and local optimum solutions in the updating process. MPSOGSA [21] gives better
performance and high accuracy of selection process.
3. Genetic Algorithm and GSA (GAGSA) [21]
GA is based on the fact “Fitness for survival” which considers three operators: Natural
selection, crossover, and mutation. In GAGSA [22], mutation and crossover operators
of GA support to find the global optimum solution in GSA and also improve GSA’s
speed displacement formula. This algorithm makes the convergence faster and it is
comparable to PSO and GSA.
4. Gravitational Particle Swarm (GPS) [22]
This algorithm is the hybridization of PSO and GSA. In GPS [23], agents update their
corresponding positions with PSO’s velocity and GSA’s acceleration. It is applied
on 23 benchmark functions and better performance was obtained.
5. Artificial Bee Colony and GSA (ABCGSA) [23]
Artificial bee colony algorithm inspired by foraging behavior of honey bee. This
algorithm divides searching process in three steps; first employed bees go to food
source in her memory and evaluate its nectar amount then onlooker bees provide a
better source of this food and scouts discovered the new food sources and replace
abandoned food source into a new one. ABCGSA [24] combines the search mechanism of the three steps of ABC with the moving method of GSA and obtained better
results.
6. K-Mean and GSA (GSA-KM) [25]
34
I. Bala and A. Yadav
GSA-KM [26] gives another approach to generate initial population and supports
K-Mean algorithm to escape from the local optima. K-Mean algorithm generates
appropriate initial population in GSA which provides solution in least possible iteration. It encourages the quality of solution and convergence speed.
7. Hybrid Neural Network and GSA (HNNGSA) [24]
GSA techniques are applied to a multilayer artificial neural network. It is used to
stimulate the adaptable parameters and an approximate solution of Wessinger’s equation is also obtained . The performance of HNNGSA [24] is compared with R-K,
Euler and improved Euler methods and obtained better results.
8. K Harmonic Mean and Improved GSA (IGSAKHM) [26]
This hybrid form was introduced to solve clustering problems in data mining. The
proposed algorithm [26] is improved version of GSA into KHM. It provided better
result than Standard KHM and PSOKHM.
9. Differential Evolution and GSA (DE-GSA) [27]
In this algorithm, two strategies are used for update the agent’s search: DE strategy
and GSA strategy, for the avoidance of local minima on boundary, it restricts the
searching speed first and if objects move outside the boundary, algorithm scatters
them in a feasible region away from the boundary, instead of stopping them on the
boundary. The performance of DE-GSA is evaluated through several benchmark
functions and gets better results.
5 Advantages and Disadvantages of GSA
GSA is a recently developed algorithm which solves many complex nonlinear optimization problems. It has the ability to solve complex problem while somewhere it
takes more time to execute some iterations. It contains some advantages and disadvantages, these are:
5.1 Advantages
• GSA could produce result with high accuracy [10].
• GSA has good local minima avoidance as compared to other heuristic techniques
like PSO and DE [19].
• GSA generates better quality solution and gives stable convergence.
Gravitational Search Algorithm: A State-of-the-Art Review
35
Publications
60
26
20
0
47
46
40
4
33
34
8
33
Publications
11
2009 2010 2011 2012 2013 2014 2015 2016 2017
Fig. 1 Year-wise publication of articles on GSA by leading international journals
5.2 Disadvantages
• GSA uses complex operator and long computational time, and it suffers from slow
searching speed in last few iteration [19].
• Selection of gravitational constant parameter G is not appropriate. Although G
controls the search accuracy but still does not guarantee a global solution at alltime [8].
• It is not flexible, if premature convergence happens, there will not be any recovery
for this algorithm. In other words, after becoming converged, the algorithm loses
their ability to explore and become inactive [6].
• GSA is memoryless algorithm, only the current position of agents plays a role in
update procedure [1].
5.3 Criticism of GSA
Apart from merits and applications of GSA, it has also faced the criticism on its
fundamental idea. Gauci et al. [28] had claimed that GSA does not take the distance
between solutions into account and therefore it cannot be considered to be based on
the law of gravity.
6 Conclusion and Future Scope
In this paper, the development of GSA has been presented, Also year-wise publication
of GSA’s papers till June, 2017 in leading journals is presented in Fig. 1. Although
GSA is a newly developed algorithm but it has been applied in many areas in such
a short time. This shows the promising future of GSA. It has been applied in many
areas like clustering, image processing, neural network training, controller design,
and filter modeling and so on. But still there are many areas like finance, military,
36
I. Bala and A. Yadav
economics are not yet penetrating. More studies can also be done in these areas.
More development could be done to the structure of GSA and a lot of possible hybrid
techniques could be explored such as hybridization of GSA with ACO, Artificial Fish
School, Artificial Immune System, etc. GSA is an open problem, and it is expected
to produce new techniques of GSA with better performance in future.
Acknowledgements This research is supported by National Institute of Technology Uttarakhand
and North-cap university (NCU) Gurgaon.
References
1. Rashedi, E., Nezamabadi, H-pour, Saryazdi, S.: GSA: a gravitational search algorithm. Inf.
Sci. 179(13), 2232–2248 (2009)
2. Rashedi, E., Nezamabadi, H-pour, Saryazdi, S.: BGSA: binary gravitational search algorithm.
Nat. Comput. 9(3) (2009)
3. Amoozegar, M., Nezamabadi, H.-pour: Software performance optimization based on constrained GSA. In: The 16th CSI International Symposium on Artificial Intelligence and Signal
Processing (AISP), pp. 134–139 (2012)
4. Hassanzadeh, H.R., Rouhani, M.: MOGSA: multi objective gravitational search algorithm.
In: 2nd International Conference of Computational Intelligence, Communication System and
Networks (2010)
5. Li, C., Li, H., Kou, P.: Piecewise function based gravitational search algorithm and its application on parameter identification of AVR system. Neurocomputing 124, 139–148 (2014)
6. Sarafrazi, S., H-pour, Nezamabadi, Saryazdi, S.: Disruption: a new operator in gravitational
search algorithm. Scientia Iranica 18(3), 539–548 (2011)
7. Soleimanpour, M., Nezamabadi, H-pour, Farsangi, M.M.: A quantum behaved gravitational
search algorithm. In: Proceeding of International Conference on Computational Intelligence
and Software Engineering, Wuhan, China (2011)
8. David, R.-C., Precup, R.-E., Petriu, E., Rdac, M.-B., Purcaru, C, Dragos, C.-A., Preitl, S.:
Adaptive gravitational search algorithm for PI-fuzzy controller tuning. In: Proceedings of the
9th International Conference on Informatics in Control, Automation and Robotics, pp. 136–141
(2012)
9. Shamsudin, H.C., Irawan, A., Ibrahim, Z., Abidin, A.F.Z., Wahyudi, S., Rahim, M.A.A., Khalil,
K.: A fast discrete gravitational search algorithm. In: 2012 Fourth International Conference on
Computational Intelligence, Modelling and Simulation (CIMSiM), pp. 24–28 (2012)
10. Precup, R.M., David, R.C., Petriu, E.M., Preitl, S., Paul, A.S.: Gravitational search algorithmbased tuning of fuzzy control systems with a reduced parametric sensitivity. Adv. Intell. Soft
Comput. 96, 141–150 (2011)
11. Azlina, N., Ibrahim, Z., Nawawi, S.W.: Synchronous versus asynchronous gravitational search
algorithm. In: First International Conference on Artificial Intelligence, Modelling & Simulation
(2013)
12. Khajehzadeh, M., Taha, M.R., El-Shafie, A., Eslami, M.: A modified gravitational search
algorithm for slope stability analysis. Eng. Appl. Artif. Intell. 25(8), 1589–1597 (2012)
13. Soleimanpour moghadam M., Nezamabadi, H- pour: An improved quantum behaved gravitational search algorithm. In: Proceeding of 20th Iranian Conference on Electrical Engineering,
(ICEE2012), pp. 711–715 (2012)
14. Nanji, H.R., Mina, S., Rashedi, E.: A high-speed, performance-optimization algorithm based
on a gravitational approach. J. Comput. Sci. Eng. 14(5), 56–62 (2012)
15. Dowlatshahi, Bagher, M., Nezamabadi, H-pour: GGSA: a grouping gravitational search algorithm for data clustering. Eng. Appl. Artif. Intell. 36, 114–121 (2014)
Gravitational Search Algorithm: A State-of-the-Art Review
37
16. Wu, Z., Hu, D., Tec, R.: An adaptive centric gravitational search algorithm for complex multimodel problems. Tec. Ing. Univ. 39, 123–134 (2016)
17. Sun, G., Zhang, A., Wang, Z., Yao, Y., Ma, J.: Locally informed gravitational search algorithm.
Knowl. Based Syst. 104, 134–144 (2016)
18. Gupta, A., Sharma, N., Sharma, H.: Fitness based gravitational search algorithm. Comput.
Commun. Autom. IEEE (2017)
19. Mirjalili, S., Hashim, S.Z., Sardroudi, H.M.: Training feedforward neural networks using hybrid
particle swarm optimization and gravitational search algorithm. Appl. Math. Comput. 218(22),
11125–11137 (2012)
20. Jiang, S., Ji, Z., Shen, Y.: A novel hybrid particle swarm optimization and gravitational search
algorithm for solving economic emission load dispatch problems with various practical constraints. Int. J. Electr. Power Energy Syst. 55, 628–644 (2014)
21. Sun, G., Zhang, A.: A hybrid genetic algorithm and gravitational using multilevel thresholding.
Pattern Recognit. Image Anal. 7887, 707–714 (2013)
22. Tsai, H.C., Tyan, Y.-Y., Wu, Y.-W., Lin, Y.-H.: Gravitational particle swarm. Appl. Math.
Comput. 219(17), 9106–9117 (2013)
23. Guo, Z.: A hybrid optimization algorithm based on artificial bee colony and gravitational search
algorithm. Int. J. Digital Content Technol. Appl. 6(17), 620–626 (2012)
24. Ghalambaz, M., Noghrehabadi, A.R., Behrang, M.A., Assareh, E., Ghanbarzadeh, A., Hedayat,
N.: A hybrid neural network and gravitational search algorithm (HNNGSA) method to solve
well known Wessinger’s equation. World Acad. Sci. Eng. Technol. pp. 803–807 (2011)
25. Hatamlou, A., Abdullah, S., H-pour, Nezamabadi: A combined approach for clustering based
on K-means and gravitational search algorithms. Swarm Evol. Comput. 6, 47–55 (2012)
26. Yin, M., Hu, Y., Yang, F., Li, X., Gu, W.: A novel hybrid K-harmonic means and gravitational
search algorithm approach for clustering. Expert Syst. Appl. 38(8), 9319–9324 (2011)
27. Xiangtao, L., Yin, M., Ma, Z.: Hybrid differential evolution and gravitation search algorithm
for unconstrained optimization. Int. J. Phys. Sci. 6(25), 5961–5981 (2011)
28. Gauci, M., Dodd, T.J, Groß, R.: Why ‘GSA: A Gravitational Search Algorithm’ is Not Genuinely Based on the Law of Gravity. Springer Science & Business Media, Berlin (2012)
Investigating the Role of Gate Operation
in Real-Time Flood Control of Urban
Drainage Systems
Fatemeh Jafari, S. Jamshid Mousavi, Jafar Yazdi and Joong Hoon Kim
Abstract Flooding is a potential risk to human beings life and assets, and the environment in urban areas. To mitigate such a phenomenon and related damages, structural and nonstructural options can be considered. This study investigates the effect
of gate operation on flood mitigation during intense rainfall events. A prototype
network, consisting of a detention reservoir located in a portion of Tehran, the capital city of Iran, is considered. Different operational scenarios are examined using
an optimal real-time operation model. An SWMM model of the system, simulating
rainfall–runoff and hydraulic routing processes, is built and is linked to the harmony
search optimization algorithm, evaluating the system operation performance for different scenarios. Results demonstrate that there is stillroom to increase the potential
flood regulation capacity of the studied system by equipping it with more controllable
apparatus.
Keywords Urban drainage system · Flood control · Real-time optimization
Detention reservoir
1 Introduction
Climate change and exponential growth of impervious surfaces in urban regions due
to excessive development of man-made structures, such as buildings, squares, and
F. Jafari · S. J. Mousavi (B)
Department of Civil and Environmental Engineering, Amirkabir University of Technology,
Tehran, Iran
e-mail: jmosavi@aut.ac.ir
F. Jafari
e-mail: Jafari.f@aut.ac.ir
J. Yazdi
College of Engineering, Shahid Beheshti University, Tehran, Iran
J. H. Kim
School of Civil and Architectural Engineering, Korea University, Seoul, Korea
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_4
39
40
F. Jafari et al.
roads, have led to a remarkable rise in the rate and volume of surface runoff and
flooding. During severe rainfall events, the urban drainage system (UDS) becomes
overloaded, causing flood occurrence [1]. To prevent flooding in urban areas, offline
storage installations are applied which temporarily store stormwater volume. This
solution is often expensive due to the high costs of construction and maintenance [1].
On the contrary, nonstructural approaches which are utilized to manage flood with the
existing facilities are avoiding large investments [2–4]. Real-time control (RTC) is
among the nonstructural approaches that broadly used to manage UDSs. This method
let the network to be real-time monitored and regulated so as to compatibly work in
various situations and different rainfall events [5]. In RTC procedure, controllable
elements, such as gates and pumps, are regulated using operation policies that come
from an optimization strategy so as to obtain the desired UDS working behavior [1].
According to the literature, many research studies focus on RTC approach to manage UDSs. In the study of Pleau et al. [6], a global optimal control (GOC) system,
consisting of a nonlinear hydrologic–hydraulic model and a nonlinear programming
optimization algorithm, was applied to the Quebec westerly sewer network. The
objectives of optimization problem were the minimization of set points variations
in real time and minimization of the frequency and volume of sewer overflows discharged into the basin’s rivers. To adjust flows and inline storages in combined sewer
systems, Darsono and Labadie [7] proposed a neural-optimal algorithm-based RTC
approach to regulate flows and inline storages in combined sewer systems. Beeneken
et al. [4] also applied a global RTC approach to the combined sewer system of Dresden
city, Germany with the hydrodynamic pollution load calculations module to improve
the efficiency of the sewer system. The performances of two real-time models for
pump station operation, namely a historical adaptive-network-based fuzzy inference
system (ANFIS–His) and an optimized ANFIS (ANFIS-Opt), were compared to find
optimal operational policies for flood mitigation in urban areas [8]. Yazdi and Kim
[9] suggested a harmony search algorithm-based predictive control model to obtain
optimal operational policies. They considered the coordinated operation of drainage
facilities in a river diversion and an urban drainage system. Using a gossip-based
algorithm linked to the SWMM hydrodynamic simulation model, Garofalo et al. [1]
developed a distributed RTC model.
In this study, an online RTC model is applied to a portion of urban drainage
system consisting of a detention reservoir with controllable and uncontrollable gates
and openings. The way how the online real-time optimal operation model is applied
to the studied system and practical suggestion to improve system’s performance is
discussed in the following sections.
2 Methodology
According to Fig. 1, suppose a detention reservoir with an outflow gate located at B
meters above the surface. If the maximum depth of the reservoir is discretized into
n levels, the operation model’s aim is to reduce flood inundation at downstream of
Investigating the Role of Gate Operation in Real-Time …
41
Fig. 1 Discrete water level in the detention reservoir
G1
G2
G3
…
Gj
…
Gm
Fig. 2 Decision variable vector
the system. In this case, the outflow gate plays a significant role in flood control.
Therefore, the optimization problem of system’s operation performance is solved by
considering decision variables as a policy on how to regulate
gate openings. In other
words, the decision variable vector includes variables G j each of which represents
the percentage of gate opening corresponding to water levels within the interval
[d j , d j+1 ) (Fig. 2).
The number of decision variables is dependent on height B since the gate starts
working as the water level exceeds the bottom edge elevation of the gate. Obviously,
adding more gates to the system will increase the number of decision variables.
Extracting operation policy (percentage of gate openings) for evacuating water
out of the system can be obtained via an optimal real-time operation model.
2.1 Optimal Real-Time Operation (RTOP) Model
In the RTOP model, the operation policies of regulators are updated periodically,
so that the time horizon D is divided into a number of decision time intervals Ti ,
and a particular control rule Ri is derived for each decision time. As a result, a
finite sequence of operating policies R1 , R2 , . . . , Ri , . . . , R H is determined over time
horizon D, where each Ri alludes to a vector of optimal policies for gate operation to
be applied during the interval Ti . The model formulation of RTOP model is presented
below.
RTOP model formulation
MIN :
TH
FT
(1)
T Ti
Subject to:
FT f R, h t , G j , . . . Ti
(2)
42
F. Jafari et al.
0 ≤ h t ≤ HMAX
h t f h t−1 , Q in,t , G j Ti
0
if h t ≤ B
[G j ]Ti 11
Z 1 Z z × Pz otherwise
11
Zz 1
(3)
(4)
(5)
(6)
z1
Note that the formulation represents a multi-period optimization model as it considers the state of the system from the current decision time Ti to the end of horizon
time TH in the evaluation of objective function (Eq. 1). However, the found optimal
operation rule is applied only for decision time interval Ti .
In the above formulation, FTi is the total volume of the flood in the period Ti which
is a function of a number of variables such as rainfall amount and characteristics
R, the water level at the detention reservoir h t , gate operational policy (decision
variables), and other parameters that will be determined using the rainfall–runoff and
flow routing simulation model. Equations (2)–(4) represent the SWMM simulation
module of the model that must be performed for each objective function. G j is the
gate opening percentage corresponding to water levels within an interval [d j , d j+1 )
which is accounted via Eq. (5) in which Pz is an integer variable that takes a value
among [0, 10, 20, . . . , 100%], and Z z is a binary variable. h t is the reservoir’s water
level at a time t, which is a function of the inflow discharge to the detention reservoir
at the time t, Q in,t , the water level at a previous time step, h t−1 , and G j .
The popular metaheuristic harmony search (HS) algorithm was used to solve
the aforementioned optimization problem. Suitable values of optimization algorithm
parameters were determined after some trial runs of the HS algorithm for several
flood scenarios as summarized in Table 1. Optimization–simulation models were
solved using an Intel Core i7 3.4 GHz system with 8 GB of random access memory
(RAM).
Table 1 Parameters used in
HS algorithm
Parameter
Value
HM size
HMCR
PAR
FW
100
0.98
0.1
0.02 × variable ranges
Investigating the Role of Gate Operation in Real-Time …
43
3 Study Area
The studied system is located in the south part of the main drainage system of
Tehran, the capital of Iran. The network covers an area of 156 km2 and includes
42 sub-catchments and 132 conduits. The considered drainage network consists of
116 km underground tunnels approximately 15.6 km of which does not have enough
capacity to safely transfer stormwater runoff of a 50-year design rainfall. As shown
in Fig. 3, because of lack of hydraulic capacity, a detention reservoir, characteristics
of which are presented in Table 2, has been built to temporarily store excess storm
runoff.
The concrete outlet intake structure of the detention reservoir equipped with three
controllable steel sluice gates with 1.6 × 1.6 m2 size. Additionally, eight rectangular
openings with 0.6 × 0.9 m2 size at the upper part and a three-diameter octagonal
opening on the roof of the structure act as spillway while the water level rises.
Therefore, physical characteristics of the system do not provide the ability to control
these openings and they automatically work as the water level exceeds their bottom
elevation.
Fig. 3 Schematic representation of the studied network and outlet intake structure of the detention
reservoir
Table 2 Detention reservoir
characteristics
Maximum depth (m)
7.5
Area (m2 )
EL 0: 85,000
EL 1: 160,000
EL 7.5: 160,000
44
F. Jafari et al.
Fig. 4 Precipitation hydrograph for investigated events
Table 3 Historical storm events studied
Event
Duration (min)
29/12/1976
26/01/1980
07/12/1984
28/03/2002
04/05/2004
15/07/2012
610
235
230
190
405
270
Accumulation of precipitation
(mm)
21.22
15.4
25.71
9.14
8.75
28.7
Six severe historical rainfall events were utilized to examine model performance.
The hyetographs of the events and their characteristics are presented in Fig. 4 and
Table 3, respectively.
4 Results and Discussion
The simulation model of the system is developed using SWMM as shown in Fig. 5
using the aforementioned system’s data, features and characteristics, collected by MG
Consulting Engineers (MGCE) [10]. To reduce the executing runtime, the network
is separated into two sub-models. In this way, for each decision time, the upstream
sub-model is run just for one time and the downstream sub-model is called for
Investigating the Role of Gate Operation in Real-Time …
(a)
45
(b)
Separated node
Fig. 5 Simulation sub-models a upstream sub-model b downstream sub-model
each function evaluation. The model is separated into two sub-models based on
the assumption that the inflow to the separated node is independent of the gate
performance and the gravity flow is formed in the upstream model. Figure 6 confirms
the validity of this assumption by comparing the results obtained for integrated
(Fig. 3) and separated (Fig. 5a) networks for different events.
According to Fig. 3, the reservoir contains three sluice gates and eight openings,
operations of which play a significant role in flood reduction. The allowable maximum water depth in the detention reservoir was considered 7.5 m which was divided
into 15 discrete values with 0.5 m increments. The decision variables are considered
as the percentages of openings corresponding to each discrete water level. According
to the previous explanation about the number of decision variables in Sect. 2, the
total number of 38 variables is defined with 15, 12, 8, and 3 variables for sluice gates
1, 2, 3, and all openings together (4), respectively.
To investigate the importance of each gate operation, three operational scenarios
are defined as follows:
Scenario 1:
In this scenario, all the gates and openings are considered to be fully open all the
time without any controlling rule. This is the procedure that currently is in practice.
46
F. Jafari et al.
Fig. 6 Flood hydrographs at separated node validating the gravity flow assumption in the model
of upstream network
Table 4 Comparison of three
scenarios in terms of flood
volume
Event
29/12/1976
26/01/1980
07/12/1984
28/03/2002
04/05/2004
15/07/2012
Flooding (1000 m3 )
Scenario 1
Scenario 2
Scenario 3
1059
659.65
1080
339.23
304.25
1120
204.24
0
224.14
0
0
267.84
34.55
0
45
0
0
86.93
Scenario 2:
In this scenario, the operation of sluice gates 1, 2, and 3 are controlled using RTOP
model, but eight openings at an elevation of 4.5 m are considered to be fully open
all the time without any control rule.
Scenario 3:
In this scenario, all sluice gates and openings are regulated using RTOP model. In
other words, all the gates and openings in the system are assumed to be controllable.
Table 4 displays the flood volumes resulting from three scenarios for six different
rainfall events. It can be inferred that the real-time optimal control approach performs
quite well by reducing the negative consequences of flooding and flood volume
significantly.
Additionally, comparison of the outcomes related to the scenarios 2 and 3 presented in Table 5 demonstrates that controlling the system partially, compared with
the case of full regulation of the system components, results in an increase in flood
inundation up to 17%. This shows the importance of gate operation in flood management. The ability to regulate all controllable elements in the system leads to an
Investigating the Role of Gate Operation in Real-Time …
Table 5 Percentage of flood
reduction resulting from
scenarios 2 and 3
Event
29/12/1976
26/01/1980
07/12/1984
28/03/2002
04/05/2004
15/07/2012
47
Percentage of flood reduction (%)
Scenario 2
Scenario 3
81
100
79
100
100
76
97
100
96
100
100
92
Variation of
reduction
16
0
17
0
0
16
Fig. 7 Comparison of different scenarios in terms of reservoir depth
efficient use of the system’s regulation capacity. According to Fig. 7, applying scenario 3 leads to the optimal utilization of the reservoir capacity, where excess water
is temporarily stored and is used later for other purposes such as irrigation of urban
green landscape.
48
F. Jafari et al.
5 Conclusion
Flood is one of the natural disasters that cause damage to human life and assets and
the environment. The flood control system including drainage network and detention
reservoir is highly dependent on the operation of controllable elements such as gates
and pump stations. In this study, the importance of operation of outflow gates of a
detention reservoir located in a portion of the urban drainage system of the capital city of Iran was investigated. Three different operational scenarios, representing
zero to full utilization of regulating the capacity of the system’s components, were
defined and investigated using an optimal real-time operation model. In the real-time
operation model, the SWMM simulation model was linked to harmony search optimization algorithm to find a real-time optimal policy for gate operation. Comparing
results of different scenarios showed that we can significantly reduce flood inundation using the real-time control approach presented. Moreover, the operation of each
individual gate will significantly impact the operation of the whole system as partial
control of the system, compared with a fully controlled case, led to 17% increase in
flood inundation. Therefore, to utilize the system’s regulation capacity during floods,
the studied system may be equipped with more controllable apparatus while taking
economic considerations into account.
References
1. Garofalo, G., Giordano, A., Piro, P., Spezzano, G., Vinci, A.: A distributed real-time approach
for mitigating CSO and flooding in urban drainage systems. J. Netw. Comput. Appl. 78, 30–42
(2017)
2. Schütze, M., Campisano, A., Colas, H., Schilling, W., Vanrolleghem, P.A.: Real time control
of urban wastewater systems—where do we stand today? J. Hydrol. 299(3), 335–348 (2004)
3. Bach, P.M., Rauch, W., Mikkelsen, P.S., McCarthy, D.T., Deletic, A.: A critical review of
integrated urban water modelling–Urban drainage and beyond. Environ. Model Softw. 54,
88–107 (2014)
4. Beeneken, T., Erbe, V., Messmer, A., Reder, C., Rohlfing, R., Scheer, M.: Real time control
(RTC) of urban drainage systems–a discussion of the additional efforts compared to conventionally operated systems. Urban Water J. 10(5), 293–299 (2013)
5. Dirckx, G., Schütze, M., Kroll, S., Thoeye, C., De Gueldre, G., Van De Steene, B.: RTC versus
static solutions to mitigate CSO’s impact. In: 12th International Conference on Urban Drainage,
2011b. Porto Alegre, Brazil (2011, September)
6. Pleau, M., Colas, H., Lavallée, P., Pelletier, G., Bonin, R.: Global optimal real-time control of
the Quebec urban drainage system. Environ. Model Softw. 20(4), 401–413 (2005)
7. Darsono, S., Labadie, J.W.: Neural-optimal control algorithm for real-time regulation of in-line
storage in combined sewer systems. Environ. Model Softw. 22(9), 1349–1361 (2007)
8. Hsu, N.S., Huang, C.L., Wei, C.C.: Intelligent real-time operation of a pumping station for an
urban drainage system. J. Hydrol. 489, 85–97 (2013)
9. Yazdi, J., Kim, J.H.: Intelligent pump operation and river diversion systems for urban storm
management. J. Hydrol. Eng. 20(11), 04015031 (2015)
10. MGCE: Tehran Stormwater Management Master Plan, Vol. 4: Existing Main Drainage Network, Part 2: Hydraulic Modeling and Capacity Assessment, December 2011, MG Consultant
Engineers, Technical and development deputy of Tehran municipality, Tehran, Iran (2011a)
Molecular Dynamics Simulations
of a Protein in Water and in Vacuum
to Study the Solvent Effect
Nitin Sharma and Madhvi Shakya
Abstract Molecular dynamics simulation shows the motions of distinct molecules
in models of liquids, solids, and gases. The motion of a molecule defines how its
positions, velocities, and orientations change with time. In this study, an attempt has
been made to study the solvent effect on the dynamics of Major Prion protein. By
keeping the focus mainly on united motions of the molecule, molecular dynamics
simulations of Major Prion protein in vacuum and water are performed up to 100 ps.
The results obtained from these two simulations are compared to study the solvent
effect on the dynamics of Major Prion protein. Energy minimization and molecular
dynamics simulation have been done through GROMACS using OPLS-AA force
field.
Keywords Molecular dynamics · OPLS-AA force field · RMSD · RMSF
MSD
1 Introduction
Protein is a molecule made up of amino acids that are needed for the body to function
properly. The tertiary structure of protein is the native and functional state of protein
[1]. To see the functions and study of protein structure, Molecular dynamics (MD)
simulations are extensively used. The outcomes of a given simulation depend on a
number of factors, such as the quality of the molecular force field, the behavior of
solvent, the time period of the simulation, and the sampling ability of the simulation
procedure. There has been massive investment in the basic technology in each of these
areas, and the range of application of molecular dynamics simulations has extended
N. Sharma (B)
Department of Sciences and Humanities, NIT Uttrakhand, 246174 Srinagar,
Garhwal, India
e-mail: nitinsharma@nituk.ac.in
M. Shakya
Department of Mathematics, MANIT Bhopal, Bhopal 462051, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_5
49
50
N. Sharma and M. Shakya
significantly since the technique was first applied [2]. The initial interpretation of
proteins as comparatively firm structures has been swapped by a dynamic model
in which the internal motions and succeeding conformational variations play an
indispensable role in their function.
A solvent plays an imperative part in the study of structure and dynamics of
a complex molecule like protein. Numerous approaches have been recommended
for the computational simulation of a protein, which can include the effect of a
solvent straight or incidentally [3]. The molecular dynamics (MD) or the Newtonian
dynamics, that openly contains water molecules and other environmental elements
such as ions, is theoretically forthright and has been applied by numerous writers in
an effort to replicate the solvent environment [4–7]. Examines in these mechanisms
mainly focused on oscillations in the atomic positions to compare on the behavior of
the inter and intramolecular hydrogen bonding, or on the conformational dynamics
close to energetic location and with the X-ray crystallography [8]. The overview of
the process allows us to presume in standard all dynamics and structural aspects of
a complex/protein molecule in solution. At minimum as vital are the united motions
in proteins, as those modes with low frequencies give vital contributions to the
scale of the oscillations of atoms. By a few lowest frequency modes more than half
of the magnitude of the root mean square oscillations of atoms can be expressed
[1, 6]. Consequently, it is fascinating to understand how the solvent effects this
type of low frequency modes. In this work, an attempt has been made to see and
study the solvent effects on the dynamics and structure of a complex Major Prion
protein molecule mainly by aiming on the united modes. By projecting the molecular
dynamics trajectory onto a set of orthogonal principal axes [8], it can be achieved.
Projection method has been efficaciously implemented to the study of the united
motions of Major Prion protein in vacuum [9]. In the motionless and in the dynamic
properties of the protein, the effects of the solvent are embodied when the degrees
of freedom of the solvent are projected on those of the protein, In the potential of
the mean force for the protein, the static effect is simulated which defines, among
others, the transmission of the hydrophobic and electrostatic potential interactions.
By Pettitt and Karplus [10], this adjustment of the potential surface because of the
solvent has been verified for alanine dipeptide in water by a treatment based on
the prolonged RISM theory. Whereas the potential surface has two deep minima in
vacuum in the dihedral angle space, there are numerous minima in solvent which are
parted by considerably lesser potential barricades. This modification in the potential
surface must affect the fluctuation and conformation of the protein ominously.
2 Methodology
In the present study, an attempt has been made to study the solvent effect in the
motion of Major Prion protein. To see the solvent effects on the dynamics of Major
Prion protein, we simulated the above mention protein in water and vacuum. The
initial idea for all parameter interpretations, the experimental structure of Major Prion
Molecular Dynamics Simulations of a Protein in Water …
51
protein obtained by Saviano, G., and Tancredi, T. and accessible from the Protein
Data Base under the code 2IV4. All computational and simulation were done by
using the GROMACS simulation software and OPLS-AA force field, by using SPC
water model for which we used the flexible system as mentioned by the GROMACS
constraint files.
For the first molecular dynamics simulation, one Major Prion protein molecule
was equilibrated composed with 1650 water molecules in a cubic box with periodic
boundary conditions in an NpT ensemble at a temperature of 300 K and a reference
pressure of 1 bar and a simulation of 100 ps was performed for analysis. The Newton equation of motion is integrated using Leap-Frog algorithm using time step of
0.002 ps.
For the second molecular dynamics simulation, Major Prion protein was equilibrated in vacuum at unbroken temperature 300 K. The other simulation constraints
are alike to those of the first simulation.
3 Results and Discussion
First, we simulated the Major Prion protein up to 100 ps in solvent (SPC water
model) with 1650 solvent particles after that we simulated it in vacuum for the
same duration of time, i.e., up to 100 ps with the same parameter that we used for
solvent simulation. After that we determined the important parameters like RMSF
(root mean square fluctuation), projection on principal axes for both the simulation,
RMSD (Root means square deviation) and then we compared the results to see
the change and effect of solvent on Major Prion protein while simulation, and for
validation, we calculated and plotted MSD (mean square displacement) for both
the simulation. For the molecular dynamics simulation in water for Cα atoms and
side chain, it is established that the root mean square fluctuations (RMSF) are much
smaller than those in the molecular dynamics in vacuum. This evidence shows from
already established facts that potential surface for the protein has reformed in vacuum
due to the existence of water solvent [11].
3.1 Atoms Fluctuations
In this part, the root mean square fluctuations of atoms are discussed. It is observed
that RMSF is much smaller in water than that in vacuum. The RMSF in the molecular
dynamics simulation in water and molecular dynamics simulation in vacuum are
shown for Cα and side chain in Figs. 1 and 2 respectively, where the black curve is
showing the RMSF in solvent and the red curve showing the RMSF in vacuum.
(1) First, we have calculated and plotted RMSF for Cα.
(2) Now we have calculated and plotted RMSF for side chain.
52
N. Sharma and M. Shakya
Fig. 1 RMSF for Cα
Fig. 2 RMSF for side chain
3.2 Molecular Dynamics Trajectories Projection
on the Principal Axes
Now we have plotted the projection of the molecular dynamics trajectory on to
the principal axes up to 100 ps. Molecular dynamics trajectories projections on
to the first three principal axes in vacuum and in water are shown in Figs. 3 and
4, correspondingly. The projections in vacuum are smooth curves while there is
significant noise in water and the periodicities do not seem.
(1) Molecular dynamics trajectories projection on to the three principal axes in
vacuum
(2) Molecular dynamics trajectories projection on to the three principal axes in
solvent.
Molecular Dynamics Simulations of a Protein in Water …
53
Fig. 3 Projection on to the three principal axes in vacuum
3.3 Root Mean Square Deviation (RMSD)
The easiest approach to verify the accuracy of simulation is to define the point
to which the motion causes a collapse of the X-ray structure. In vacuum, it takes
approximately 20 ps afore the root mean square deviation from initial structure
touches a steady point, while in solvent, the structure becomes stable more rapidly
around 5 ps and remains closer to the X-ray structure (Fig. 5).
4 Validations
Mean square displacement
Now we calculated and plotted mean square displacement against time in solvent
and vacuum shown in Fig. 6a, b respectively.
It is clear from Fig. 6a that mean square displacement for solvent propagates linearly
with time and for vacuum from Fig. 6b, it is not growing with time.
54
Fig. 4 Projection on to the three principal axes in solvent
Fig. 5 RMSD in vacuum
(Red) and RMSD in solvent
(Black)
N. Sharma and M. Shakya
Molecular Dynamics Simulations of a Protein in Water …
55
Fig. 6 a Mean square displacement against time in solvent b Mean square displacement against
time in vacuum
5 Conclusions
In the present work, an attempt has been made to study the solvent effect on the
dynamics of Major Prion protein for this we simulated the protein in water and
vacuum. It is observed from Figs. 1 and 2 that RMSF for Cα atoms and side chains
is smaller than that in vacuum which shows that the potential surface of the protein
is transformed due to the existence of water solvent from that in vacuum [8]. It is
observed from Figs. 3 and 4 that the projection in vacuum has smooth curves than
that of solvent and include heavy noise in solvent due to the presence of water. Also,
we showed that (Fig. 5) molecular dynamics simulation of protein motion is more
accurate when solvent is incorporated, in that the structure remains nearer to the Xray structure. Finally, we calculated and plotted MSD (Mean Square Displacement)
against time up to 100 ps. It is known from already established facts that MSD for
solvent simulation should increase linearly with time which is shown in Fig. 6a that
the mean square displacement for solvent grows linearly with time [12] and it does
not show the same behavior (Fig. 6b) for vacuum simulation which is validating the
results.
References
1. Dill, K., et al.: The protein folding problem. Annu. Rev. Biophys. 37, 289–316 (2008)
2. Fan, H.: Comparative Study of Generalized Born Models: Protein Dynamics. PNAS (2005)
3. McCammon, J.A., Harvey, S.C.: Dynamics of Proteins and Nucleic Acids, ch. 4. Cambridge
University Press, Cambridge (1987)
4. Brooks III, C.L., Karplus, M.: Solvent effects on protein motion and protein effects on solvent
motion. Dynamics of the active site region of lysozyme. J. Mol. Biol. 208, 159 (1989)
5. Go, N.: A theorem on amplitudes of thermal atomic fluctuations in large molecules molecules
assuming specific conformations calculated by normal mode analysis. Biophys. Chem. 35, 105
(1990)
6. Go, N., Noguti, T., Nishikawa, T.: Latent dynamics of a protein molecule observed in dihedral
angle space. Proc. Natl. Acad. Sci. U.S.A. 80, 3696 (1983)
56
N. Sharma and M. Shakya
7. Jorgensen, W.L., TiradoRives, J.: Chem. Scripta A 29, 191 (1989)
8. Kitao, A., Hirata, F., Go, N.: The effects of solvent on the conformation and the collective
motions of protein: normal mode analysis and molecular dynamics simulations of mellittin in
water and in vacuum. Chem. Phy. 158(1991), 447–472
9. Horiuchi, T., Go, N.: Projection of Monte Carlo and molecular dynamics trajectories onto the
normal mode axes: human lysozyme. Proteins 10, 106–116 (1991)
10. Pettitt, B.M., Karplus, M.: Chem. Phys. Lett. 121, 194 (1985)
11. Levitt, M., Sharon, R.: Accurate simulation of protein dynamics in solution. Proc. Natl. Acad.
Sci. U.S.A. 85, 7557–7561 (1988)
12. Leach, AR.: Molecular Modeling Principal and Application, 2nd ed. Prentice Hall (2001)
An Exploiting Neighboring Relationship
and Utilizing an Overhearing Concept
for Improvement Routing Protocol
in Wireless Mesh Network
Mohammad Meftah Alrayes, Neeraj Tyagi, Rajeev Tripathi
and Arun Kumar Misra
Abstract Reduction in control packets and minimization of setting-up time of the
route are two challenging issues in wireless mesh networks. Solutions to these two
issues are expected to save channel bandwidth and decrease the time delay, which
in turn will improve the quality of services. In this paper, a mechanism, based on
exploitation of local connectivity and overhearing concept, has been proposed for
route discovery and route repair in the well known AODV (i.e., Ad Hoc On-demand
Distance Vector) routing protocol. In this proposed work, any neighboring mesh node
of the destination mesh node can provide a route to the source mesh node, even if the
neighboring node does not have route entry about that destination in routing table.
The promiscuous mode (overhearing concept) has been applied to reduce the number
of duplicate control packets sent by neighbors of same destination nodes. Simulation
results demonstrate how the proposed work outperforms the AODV under routing
overhead, end to end delay, throughput, and packet delivery ratio in wireless mesh
networks.
Keywords AODV · Wireless mesh networks · Promiscuously mode
Neighboring table
M. M. Alrayes (B)
Applied Research and Development Organization, Tripoli, Libya
e-mail: moha872@yahoo.co.uk
N. Tyagi · A. K. Misra
Department of Computer Science and Engineering, Motilal Nehru
National Institute of Technology, Allahabad 211004, India
e-mail: neeraj@mnnit.ac.in
A. K. Misra
e-mail: akm@mnnit.ac.in
R. Tripathi
Department Electronics and Communication and Engineering, Motilal Nehru National Institute of
Technology, Allahabad 211004, India
e-mail: rt@mnnit.ac.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_6
57
58
M. M. Alrayes et al.
Internet
Gateway Mesh Routers
Backbone Mesh
Routers
Border mesh
Routers
Mesh
Clients
Wireless Links
Wired Links
Fig. 1 Architecture of wireless mesh network
1 Introduction
Wireless mesh networks (WMNs) are one of the promising candidates for next generation wireless networks. These provide cost-effective connectivity solutions, whereas
other existing technologies fail to do. Wireless mesh networks have adopted valuable characteristics of ad hoc network and traditional wired and wireless networks.
It helps to increase the capacity and coverage area and provides high connectivity to
end users in a pervasive manner [1]. Wireless mesh networks as shown in Fig. 1 are
composed of mesh routers and mesh clients, such that the mesh clients are mobile in
nature and mesh routers are static. Routing in WMNs is a challenging problem and
a good routing solution should be fast adaptable for any change in topology as well
as changes in wireless link conditions. It should have characteristics of decentralizing, self-organizing, self-healing, scalability, and robustness. The improvement of
network layer mechanisms in WMNs is very important and it is challenging issue,
and it will offer a better quality of service to different types of traffic.
In wireless mesh network, most of the traffic in wireless mesh network travels
from gateway to mesh clients and vice versa. The path length between gateway and
client mesh is very long and size of network is also large. The control packets take
long time and consume a lot of bandwidth for transmitting in case of route discovery,
route maintenance and route repair, and the increase in packet overhead in wireless
mesh network has significant impact as compared to that in ad hoc networks, because
the wireless mesh network supplies backhaul connectivity to different technologies.
Thus, the control routing overhead should reduce [2]. Nemours routing protocols have
been developed till date, it is basically based on AODV and OLSR (i.e., Optimized
An Exploiting Neighboring Relationship and Utilizing …
59
link state Routing protocol [3]. Some of the prior research works focus on design
approach for repairing a route failure [6], the QoS (i.e., quality of services) [4]
and security routing [5]. In this paper, the proposed method to improve existing
on-demand routing in wireless mesh networks has been presented with following
contributions:• The proposed mechanism utilizes local connectivity and overhearing concept of
AODV protocol for route discovery and local route repair.
• The proposed M-AODV routing protocol based on AODV has been implemented
in NS-2 [7] (i.e., network simulator).
The rest of the paper is organized as follows: Sect. 2 presents proposed work. The
detailed analysis of results and discussions is given in Sect. 3 followed by conclusion
and references.
2 Proposed Work
In the present work, a mechanism for route discovery and route repairing has been
proposed by exploiting overhearing packets, where this concept has been applied
to reduce a number of duplicated control packets during constructing the alternative route [6] and for adopting a local route recovery [8]. An overhearing table has
been constructed that helps in reducing the number of duplicate route reply packets (RREP) which have been sent by neighbors of same destination nodes. Further,
the number of route request packets (RREQ) that are rebroadcasted by neighbors of
same destination will also reduce. Overhearing table has been used in recording, the
source and destination addresses of data packet and route reply packet (RREP). It
uses the following fields in every route entry:
1. Source IP address field consists of the source IP address of RREP packet or data
packet.
2. Destination IP address field consists of the destination IP address of RREP packet
or data packet.
3. Sequence number field consists of destination sequence number of RREP packet.
In the case of data packets, sequence number is nil.
4. Next hop consisting of mesh node sends RREP message, either to an intermediate
mesh node or to destination mesh node itself.
An example of overhearing table is given in Fig. 2, A mesh node purges a route
entry for keeping only a fresh information in overhearing table in the following
cases:• If overhearing mesh node has not heard the data packets or any packets for same
source/destination for active route timeout.
• Overhearing route error packet (i.e., RERR) has been sent to same source and
destination.
60
M. M. Alrayes et al.
s
A
B
C
D
F
RREP PACKET
E
OVERHEAR ING
Source IP address
Destination IP address
Sequence number
D’s IP address
S’s IP address
Seq no of RREP
Next hope IP address
F’s IP address
Fig. 2 Mesh node E overhears RREP packet from mesh node F and then create route entry in
overhear table
A neighboring table has also been constructed, which is used in tracking the
neighboring mesh nodes, even if it still has neighboring relationship or not using the
information from the fields of hello message of AODV routing protocol, as follows:
1. Source ID: Source address of hello message.
2. Sequence number: Latest destination sequence number of hello sender.
3. Life time: ALLOWED_HELLO_LOSS * HELLO_INTERVAL.
The life time value is updated when a mesh node receives a next hello message
from same sender node. When a mesh node receives a first hello message from its
neighbor, it checks neighboring table, and if it does not have route entry, it creates
route entry into neighboring table, with source address of hello sender, last destination sequence number and life time, otherwise it will update life time, which
makes a route entry valid. If a mesh node fails to receive any hello message in
ALLOWED_HELLO_LOSS* HELLO_INTERVAL milliseconds or get indication
that a link with its neighbor has been broken, then route entry for this neighbor will be
deleted from neighboring table. Construction of neighboring table is given in Fig. 3.
When an intermediate mesh node receives a new RREQ packet with a new sequence
number from the same source or different sources to the same destination or different
destinations, it will check the overhearing table, whether a source IP address of RREQ
packet is destination IP address field in overhearing table and destination IP address
of RREQ packet is source address field in overhearing table. If an intermediate node
has route entry for this route in overhearing table, then, it will not rebroadcast RREQ
packet, and drop it. This process helps in reducing overhead packets in the network
and saves bandwidth consumption. Whenever a route entry is found in overhearing
table it means that a route is already established. As defined in Fig. 4, a mesh node F
receives a fresh RREQ packet late form source mesh node S for destination D after
the route has been established. Data traffic has started exchanges. Mesh node F will
not send route reply packet, and it will only append that in overhearing table.
If an intermediate mesh node has no route entry in overhearing table, then through
a lookup into neighboring table, it will check that whether destination node is neighboring or not. In case the destination is not neighbor of intermediate mesh node, then
the intermediate mesh node will rebroadcast RREQ packet and if an intermediate
An Exploiting Neighboring Relationship and Utilizing …
61
B
C
Hello message
F
A
E
Source IP address
Destination IP address
F’s IP address
Life time
Last Sequence number of mesh node
Allowed hello loss*Hello interval
Fig. 3 Mesh node E receive hello packet from mesh node F, then create route entry in neighboring
table
S
A
B
C
D
E
J
Data Packet
RREQ Packet
F
Fig. 4 Mesh node F has received a fresh RREQ of mesh node S as source & destination mesh node
D, after data packets has started flowing from source to destination
Fig. 5 Mesh node C has
send unicast RREP packet on
behave mesh node
A
B
C
D
RREQ PACKET
RREP PACKET
mesh node is neighbor of destination, it sends RREP packet to the mesh node that
has broadcasted the RREQ packet on behalf of destination node. An intermediate
mesh node generates a new destination sequence number based on previous destination sequence number of destination node, the sequence number of destination
mesh node can be obtained by destination sequence number that is available in last
received hello message from destination itself, and available in a neighboring route
table. This is done to prevent routing loops and ensuring that fresh route has been
generated. After this an intermediate mesh node generates RREP and sends it back
to its neighbor from which it received the RREQ. In the present case, we do not
need to send a gratuitous RREP to the destination node, because a destination will
overhear RREP packet. For example, mesh node C can send route reply packet on
behalf of the mesh node D, when it receives RREQ packet from mesh node B, instead
of rebroadcasting the RREQ packet, the mesh node C sends route reply packet to
mesh node A via mesh node B. The above scenario is shown in Fig. 5.
To prevent more than one RREP packet being sent by all neighbors of the same
destination which have received same RREQ packet, first mesh node that receives an
RREQ packet should only send RREP packet. This will also avoid unnecessary traffic.
Once other neighboring nodes that have neighboring relationship with destination
and have received same RREQ get overhear route reply, they will make entry in
overhearing table and drop RREQ packet. Further, they also not send route reply
62
Fig. 6 Mesh node E and
mesh node F has overhearing
route replay packet from
mesh node C
Fig. 7 Mesh node E will
send route reply after
receiving RREQ packet.
Mesh node A will receive
duplicated route reply
M. M. Alrayes et al.
RREP PACKET.
E
A
B
C
D
F
A
I
B
K
C
J
D
E
RREP Packet
F
packet, as shown in Fig. 6. Mesh node A intends for getting a path to mesh node D.
Mesh nodes E and F are neighbors of both mesh nodes C and D, once mesh nodes E
and F overhear RREP packet that send from mesh node C, mesh node E and F will
not send RREP and will drop RREQ packet in case it is received form predecessor
nodes. This way helps to eliminate a number of control packets and it can reduce a
time delay during route creation, and the amount of decrease is one hop away from
a destination.
This proposed method suggests that an intermediate mesh node, which is generating a RREP packet for destination neighbor, is not necessary to be stored in a route
table for forward route, because destination sequence numbers that keeps a route is
fresh and generated by this mesh node itself and destination IP address is that of
its neighbor. This in turn slightly reduces the size of route table in comparison with
AODV routing protocol and can clearly appear when more than one route is created.
Lookup time at the route table in case of data packets being sent to the destination is
also reduced. A typical situation can arise when a mesh node is unable to hear RREP
from a neighbor of destination, which is also not neighbor of it. But, meanwhile, this
mesh node becomes neighbor of destination. In this situation, mesh node will send
RREP packet and a source mesh node of route (i.e., source mesh node of RREQ
packet) will receive multiple route reply packets and chooses the best one based on
least hop count and newest destination sequence number, as shown in Fig. 7. Mesh
node E is a neighbor of mesh node F but not neighbor of mesh node D, and is unable
to hear route reply packet (i.e., RREP). This packet has been sent by mesh node D for
setting up the route between mesh node A and F. Mesh node E has received RREQ
packet for mesh node F as destination and mesh node D as source from mesh node
J. It thinks that no mesh node has send route reply packet. Mesh node E will send
unicast route reply packet. Mesh node A will receive two route reply packets from
both mesh nodes D and E.
Mesh node A will choose better route based on hop count and fresh destination
sequence number. Local route repair is also modified using the same idea of replies
from destination neighboring. This destination neighboring mesh node has arrived
and makes neighboring relationship after the route was established; whereas this
neighboring mesh node belongs to other route. This mesh node can send route reply
An Exploiting Neighboring Relationship and Utilizing …
Fig. 8 Proposed modified
local rout repair
A
B
63
C
D
E
RREQ Packet
Flow of path
RREP Packet
N
packet to sender of RREQ packet once it receives RREQ. Figure 8 illustrates a
modified local route repairing in our proposed method. When the route is broken
between mesh node D and mesh node E, a mesh node N which is a neighbor of
destination mesh node E receives RREQ Packet. It establishes alternative route by
send route reply packet to mesh node D.
3 Simulation Results and Discussion
Our proposed work has been simulated by using NS-2 version 2.33 for evaluating the
performance, and we consider packet delivery fraction, an end to end delay, average
route overhead, and average throughput [9].
3.1 Simulation Results and Analysis by Varying Number
of Mesh Clients
In this scenario, the network density has been varied by varying the number of mesh
clients from 5 to 65. The simulation results of this scenario are shown in Figs. 9, 10,
11 and 12. It can be observed from Fig. 9 that the proposed method has less delay in
comparison to AODV standard over wireless mesh network. Time latency for new
route or repairing of route break will be saved at least one hop in the cases of lower
as well as higher density of nodes. The proposed scheme also exhibits reduction in
an averaging end to end delay over varying number of mesh clients by 14.665%
when compared with AODV routing protocol. It can be seen from Fig. 10 that our
proposed method has better delivery fraction than AODV standard. With the help of
overhearing and neighboring tables, reduction in flooding of RREQ packets helps to
increase the chance for other neighbors to exchange data packets.
AODV standard has more routing packet overhead than our proposed method in
all our experiments as can be seen from Fig. 11. Our proposed method has reduced
averaging overhead by 8.095%.
From Fig. 12, it can be observed that our proposed method successfully achieves
a better throughput than AODV standard. The improvement in throughput is by
3.680%.
64
M. M. Alrayes et al.
Fig. 9 Number of mesh
clients versus end to end
delay
Fig. 10 Number of mesh
clients versus packet delivery
fraction
Fig. 11 Number of mesh
clients versus routing packet
overhead
Fig. 12 Throughput versus
number of mesh clients
4 Conclusion
The proposed method has suggested the use of advantages of local connectivity
(neighboring relationship) and promiscuously mode (overhearing concept). It has
aided to enhance routing protocol in route discovery phase and route repair phase.
An Exploiting Neighboring Relationship and Utilizing …
65
The simulation results under different number of mobile mesh clients show us that
significant improvement in key performance metrics in terms of delay, throughput,
packet delivery fraction, and route packet overhead that have been achieved as compared to that of AODV standard.
References
1. Akyildiz, I., Wang, X., Wang, W.: Wireless mesh networks: a survey. Comput. Netw. 47(4),
445–487(2005). Elsevier
2. Campista, M.E.M., Costa, L.H.M.K., Duarte, O.C.: A routing protocol suitable for backhaul
access in wireless mesh networks. Comput. Netw. 56(2), 703–718 (2012)
3. Alotaibi, E., Mukherjee, B.: A survey on routing algorithms for wireless Ad-Hoc and mesh
networks. Comput. Netw. 56(2), 940–965 (2012). Elsevier
4. Paris, S., Nita-Rotaru, C., Martignon, F., Capone, A.: Cross-layer metrics for reliable routing in
wireless mesh networks. IEEE/ACM Trans. Networking 21, 1003–101 (2013)
5. Khan, S., Loo, J.: Cross layer secure and resource-aware on-demand routing protocol for hybrid
wireless mesh networks. Wireless Pers. Commun. 62(1), 201–214(2012). Springer
6. Jeon, J., Lee, K., Kim, C.: Fast route recovery scheme for mobile ad hoc networks. In: IEEE
International Conference on Information Networking (ICOIN), pp. 419–423 (2011)
7. The Network Simulator NS, https://www.isi.edu/nsnam/ns
8. Youn, J.-S., Lee, J.-H., Sung, D.-H., Kang, C.-H.: Quick local repair scheme using adaptive
promiscuous mode in mobile ad hoc networks. J. Netw. 1, 1–11(2006)
9. Alrayes, MM., Tripathi, R., Tyagi, N., Misra, A.K.: Exploiting neighboring relationship for
enhancement of AODV in hybrid wireless mesh network. In: 17th IEEE International Conference
on Networks (ICON), pp. 71–76 (2011)
A Comparative Study of Machine
Learning Algorithms for Prior Prediction
of UFC Fights
Hitkul, Karmanya Aggarwal, Neha Yadav and Maheshwar Dwivedy
Abstract Mixed Martial Arts is a rapidly growing combat sport that has a highly
multi-dimensional nature. Due to a large number of possible strategies available to
each fighter, and multitude of skills and techniques involved, the potential for upset
in any fight is very high. That is the chance of a highly skilled, veteran athlete being
defeated by an athlete with significantly less experience is possible. This problem
is further exacerbated by the lack of a well-defined, time series database of fighter
profiles prior to every fight. In this paper, we attempt to develop an efficient model
based on the machine learning algorithms for the prior prediction of UFC fights. The
efficacy of various machine learning models based on Perceptron, Random Forests,
Decision Trees classifier, Stochastic Gradient Descent (SGD) classifier, Support Vector Machine (SVM), and K-Nearest Neighbor (KNN) classifiers is tested on a time
series set of a fighter’s data before each fight.
Keywords Machine learning algorithms · Mixed martial arts · Classifiers
1 Introduction
Mixed Martial Arts (MMA) is currently one of the fastest growing sports in the
world. The UFC or Ultimate Fighting Championship is currently the largest fight
promotion in the mixed martial arts world. Between 2013 and 2017, the promotion
Hitkul · K. Aggarwal · N. Yadav (B) · M. Dwivedy
School of Engineering and Technology, BML Munjal University,
Gurugram 122413, Haryana, India
e-mail: neha.yadav@bmu.edu.in
Hitkul
e-mail: hitkul.bmu.14cse@bmu.edu.in
K. Aggarwal
e-mail: karmanya.aggarwal.14cse@bmu.edu.in
M. Dwivedy
e-mail: maheshwar.dwivedy@bmu.edu.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_7
67
68
Hitkul et al.
had presented over 1400 fights and counting, with an event being held bi-monthly
and having multiple fights per event.
We attempted to evaluate the accuracy of multiple machine learning algorithms
in order to determine which method is best suited to predict fight results given both
competitors’ records prior to the fight. Though several works have been published that
seek to forecast performance of an MMA fighter prior to the fight [1], we attempted to
create a dataset that reflects each fighter’s statistical record prior to each fight and build
a predictive model. Thus, we should ideally be able to predict a fighter’s performance.
Intuitively, an experienced fighter would most certainly have an advantage over a
novice provided the age difference is not large enough to affect athletic performance.
We evaluated many different machine learning models, and charted their performance
over the dataset. It was found that the Random Forests and SVM gave the best results
in terms of prediction accuracy.
For a brief background of a UFC event, the UFC is a fighting promotion. MMA
employs various techniques from an ensemble of different martial arts such as Jiu
Jitsu, Boxing, Taekwondo, and Wrestling. This allows for a wide variety of strikes
and tactics to be employed by the fighters depending on their expertise in each
art. A typical UFC event has multiple fights on a particular day—these events take
place roughly once every 2 weeks. Each fight typically lasts three rounds of 5 min
each. However, major fights will last five rounds. The two fighters are denoted red
and blue side, with the better known fighter being allocated the red side. There are
multiple ways to win a fight, via Knockout/Technical Knockout wherein the fighter
overwhelms his opponent with strikes until he is unable to continue, via submission;
wherein a fighter cedes victory, or finally by decision, when the fight reaches the end
of the allotted time for the fight and the fighters are judged by a panel of three judges
on factors such as damage inflicted, aggression, and ring control. Decision victories
are the most common, however these are the hardest to judge, as the judging process
tends to be rather opaque [2, 3].
Today, statistical modeling and its applications in the UFC are in its infancy [4–6].
No thoroughly rigorous statistical models have been published till date to predict the
UFC fights previously. In this paper, we attempt to correct this imbalance—while
there remains insufficient data available to build fighter specific models (with the
UFC publishing granular fight data only since 2013 and each fighter fighting less
than 10 times every year). We have attempted to build a model to predict which fighter
is more likely to emerge victorious. In order to create the dataset, we retrieved each
fighter’s current statistics and subtracted their per fight statistics in order to create a
sort of time-dependent dataset—reflecting what each fighter’s statistics were prior to
each fight, in terms of strikes, takedowns, styles, etc. In an ideal world, this model can
be used to create matchups where both fighters are equally likely to win, as having
this sort of equity in winning chance will most likely correlate with more exciting
fights, as well as equalizing betting odds for fighters prior to each fight.
The organization of the paper is as follows: brief description of the models used
is given in Sect. 2. Section 3 describes about the data exploration and feature manipulation. Statistical models along with the results are given in Sect. 4. Further Sect. 5
continues with results and discussion and finally Sect. 6 concludes the study.
A Comparative Study of Machine Learning Algorithms for Prior …
69
2 Models Used
2.1 Random Forests
Random Forests is an ensemble classification technique consisting of a collection of
tree-structured classifiers where random vectors are distributed independently and
each tree casts a unit vote for the most popular class for a particular input [7].
2.2 Support Vector Machine (SVM)
SVMs are set of related supervised learning methods used for classification and
regression. The input vector is mapped to a higher dimensional space where a maximal separating hyperplane is constructed [8].
2.3 K-Nearest Neighbors (KNN)
KNN is a classification technique that assigns points in our input set to the dominant
class amongst its nearest neighbors, as determined by some distance metric [9].
2.4 Decision Tree
Decision trees are sequential models, which logically combine a sequence of simple
tests. Each test compares a numeric attribute against a threshold value or a nominal
attribute against a set of possible values [10].
2.5 Naive Bayes
A Naive Bayes classifier is a simple probabilistic classifier based on applying Bayes
theorem (from Bayesian statistics) with strong independence assumptions. An advantage of the naive Bayes classifier is that it only requires a small amount of training
data to estimate the parameters necessary for classification [11].
70
Hitkul et al.
2.6 Perceptron
A Perceptron is composed of several layers of neurons: an input layer, possibly one
or several hidden layers and an output layer. Each neuron’s input is connected with
the output of the previous layer’s neurons whereas the neurons of the output layer
determine the class of the input feature vector [9].
2.7 Stochastic Gradient Descent (SGD)
SGD, also known as incremental gradient descent, is a stochastic approximation of
the gradient descent optimization method for minimizing an objective function that
is written as a sum of differentiable functions [12].
3 Data Exploration and Feature Manipulation
Granular fight data is available for UFC fighters by FightMetric LLC. Highly granular
data is only available post 2013, thus an assumption has been made that all fighters
from that period and beyond start at 0. By collecting and summing statistics per fight,
we were able to assemble a tabulation of each fighter’s statistics prior to each fight.
From this set, we can see that we have a total of 895 columns and one dependent
variable. The columns themselves have 13 integer types (Streaks, Previous Wins,
etc.), 9 object types (Names, Winner, Winby, etc.) and 873 Float types. The features
for data set are represented by Figs. 1, 2 and 3. Some quick observations from the
raw dataset1.
2.
3.
4.
Red side seems to win slightly more than blue (867/1477 58.7%).
There are more fighters fighting debut fights.
Most fights are won by decision, and 2015 had the most fights.
The features seek to accommodate different fighter’s styles (including both
attempted strikes/takedowns versus significant or landed strikes/takedowns in
an effort to quantify strike/takedown volume as a meaningful statistic.
We then filled all the Null values in our dataset with 0 values and assigned numeric
codes to all categorical values. As one can see from Fig. 1 that the highest correlations
are with Round 4 and Round 5 features, since most fights do not have Round 4 and
Round 5. To deal with this sparsity, we summed the respective features of each round.
Finally, we then attempt to half the number of features again, by taking the ratio of
features from red and blue side fighters.
A Comparative Study of Machine Learning Algorithms for Prior …
71
Fig. 1 A heatmap of the highest 10 correlations with our target variable
4 Modeling
Performance of multiple machine learning models on this dataset is then evaluated
and explored by a variety of statistical methods described in Sect. 2. Table 1 describes
the performance of our chosen models on the raw dataset. Table 2 describes the
performance of the same models after we summed respective round features and
Table 3 describes the performance of the models post taking the ratio of red and blue
side fighters’ respective features (Figs. 4, 5 and 6).
5 Results and Discussion
From Fig. 7, it is evident that random Forests and SVM showed the most consistent
results against the dataset. Models like Naive Bayes and simple decision trees showed
72
Hitkul et al.
Fig. 2 A heatmap of linear correlations between our target variable, post feature reduction by
summing rounds
Table 1 Prediction accuracy
of our machine learning
models on the data set before
any feature manipulation
Model
Prediction accuracy
KNN
Decision tree
SGD classifier
Random forests
SVM
Bayes
0.554054054054
0.533783783784
0.530405405405
0.581081081081
0.628378378378
0.35472972973
Perceptron
0.537162162162
very poor results does not show good result. The dataset itself has much room for
improvement, and the assumption that all fighters start from 0 in 2013 coupled with
the rise in debut fights for new fighters means that our dataset is very sparse. However,
from simply examining the dataset, one can easily see that factors such as fighter age
are very relevant to the eventual winner of the fight. Moreover, the Red Side Fighter
A Comparative Study of Machine Learning Algorithms for Prior …
73
Fig. 3 Correlation matrix heatmap post feature reduction by taking the ratio of features amongst
red and blue fighters
Table 2 Prediction accuracy
for each of our models upon
the dataset with summed
features
Model
Prediction accuracy
KNN
Decision tree
SGD classifier
Random forests
SVM
Bayes
0.557432432432
0.516891891892
0.550675675676
0.584459459459
0.577702702703
0.202702702703
Perceptron
0.557432432432
tends to win more frequently. Depending on the model and feature, we exhibit about a
3–6% increase in prediction accuracy from zeroR policy. Our best predictive model is
SVM by far—using hyperparameter optimization we were able to get very consistent
results with a predictive accuracy of 61% and a best observed accuracy of 62.8%.
74
Table 3 Prediction accuracy
of each model on the data
post ratio of features
Hitkul et al.
Model
Prediction accuracy
KNN
Decision tree
SGD classifier
Random forests
SVM
Bayes
0.543918918919
0.503378378378
0.543918918919
0.597972972973
0.611486486486
0.212837837838
Perceptron
0.560810810811
Fig. 4 Confusion matrices for each model on the dataset
Fig. 5 Confusion matrix for each predictor post feature reduction by summing
A Comparative Study of Machine Learning Algorithms for Prior …
75
Fig. 6 Confusion matrix for each predictor after all the feature manipulations
Fig. 7 A bar graph of prediction accuracy of each model over all three sets of data instances, the
baseline, the summed rounds and the ratio of features
Moreover, the robustness of SVM can be validated by the drop in prediction accuracy
as the features were reduced.
76
Hitkul et al.
6 Conclusion
In conclusion, SVM proved to be the most resilient of machine learning models for
this type of dataset or problem domain, while we did perform some small amount of
hyperparameter optimization and feature engineering, it is worth noting that SVM
with the RBF kernel performed very well on the dataset straight out of the box.
Thus, for sports where a lot of statistical data is not available, it might be a very
valuable classifier. In the future, one can also employ some sort of feature selection
mechanism to reduce the overfitting in the dataset.
References
1. Johnson, J.D.: Predicting outcomes of mixed martial arts fights with novel fight variables.
Master Thesis, University of Georgia, Athens, Georgia (2012)
2. Gift, P.: Performance evaluation and favoritism: evidence from mixed martial arts. J. Sports
Econ. (2014). https://doi.org/10.1177/1527002517702422
3. Collier, T., Johnson, A., Ruggiero, J.: Aggression in Mixed Martial Arts: An Analysis of the
Likelihood of Winning a Decision. Violence and Aggression in Sporting Contests: Economics,
History and Policy, pp. 97–109 (2012)
4. Betting on UFC Fights—A Statistical Data Analysis, https://partyondata.com/2011/09/21/bet
ting-on-ufc-fights-a-statistical-data-analysis, last accessed 12 June 2017
5. Goel, E., Abhilasha, E.: Random forest: a review. Int. J. Adv. Res. Comput. Sci. Software Eng.
7(1), 251–257 (2017)
6. Bhavsar, H., Panchal, M.H.: A review on support vector machine for data classification. Int. J.
Adv. Res. Comput. Eng. Technol. (IJARCET) 1(10), 185–189 (2012)
7. Lotte, F., Congedo, M., Lécuyer, A., Lamarche, F., Arnaldi, B.: A review of classification
algorithms for EEG-based brain–computer interfaces. J. Neural Eng. 4(2), R1 (2007)
8. Kotsiantis, S.B.: Decision trees: a recent review. Artif. Intell. Rev. 39(4), 261–283 (2013)
9. Kaur, G., Oberai, N.: A review article on Naïve Bayes classifier with various smoothing techniques. Int. J. Comput. Sci. Mobile Comput. 3(10), 864–868 (2014)
10. Lessmann, S., Sung, M., Johnson, J.E.: Alternative methods of predicting competitive events:
an application in horserace betting markets. Int. J. Forecast. 26(3), 518–536 (2010)
11. Lock, D., Nettleton, D.: Using random forests to estimate win probability before each play of
an NFL game. J. Quant. Anal. Sports 10(2), 197–205 (2014)
12. Bottou, L.: Large scale machine learning with stochastic gradient descent. In: Proceedings of
COMPSTAT’2010. Physica-Verlag HD, pp. 177–186 (2010)
Detection of a Real Sinusoid in Noise
using Differential Evolution Algorithm
Gayathri Narayanan and Dhanesh G. Kurup
Abstract Detection of sinusoidal signals embedded in noise is a pertinent problem
in applications such as radar and sonar, communication systems and defense, to name
a few. This paper, describes the detection of a real sinusoid in additive white Gaussian
noise (AWGN) using the Differential Evolution Algorithm (DE). The performance
of DE is evaluated for different sampling rates and also for different signal-to-noise
ratios (SNR). The proposed DE which combines two DE strategies enhances the
detection performance compared to the original DE algorithm. We show that the
detection performance of the proposed algorithm is superior to previously reported
methods, especially at low SNR.
Keywords Differential Evolution (DE) · Fast Fourier Transform (FFT)
Cramer-Rao Lower Bound (CRLB) · Detection
1 Introduction
Detection of sinusoidal signals in noise has numerous applications such as sonar,
radar, communication systems, spectroscopy, image analysis, and instrumentation
systems. Although the DFT-based method is a simple and fundamental method in
this regard, the frequency resolution one can detect using Discrete Fourier Transform
(DFT) is limited by the sampling frequency. One of the popular approach to overcome
this problem, is by using three samples around the maximum absolute value as
described in [1]. Based on the method detailed in [2], where in the author estimates
the frequency with an arbitrary number of DFT coefficients, Candan [3] proposed an
G. Narayanan (B)
Department of Electronics and Communication Engineering, Amrita Vishwa Vidyapeetham,
Amritapuri, India
e-mail: gayathrin@am.amrita.edu; gaya321@gmail.com
D. G. Kurup
Department of Electronics and Communication Engineering, Amrita Vishwa Vidyapeetham,
Bangalore, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_8
77
78
G. Narayanan and D. G. Kurup
approach for frequency with reduced complexity. In [4], an estimator with improved
bias performance using Fourier Coefficient representations is presented. Candan’s
estimators in [4, 5] approach the theoretical Cramer-Rao Lower Bound (CRLB). In
[6], a fine estimation frequency estimator for a real sinusoid is presented based on
estimators developed for complex sinusoids and a filtering method. These methods
of estimation are more conceptual and do not necessarily provide the optimum value
of the frequency estimated.
Optimization algorithms which are inspired from natural evolution have been successfully applied to many engineering disciplines. Some of these methods include
Genetic Algorithm and Particle Swarm Optimization algorithm (PSO) [7, 8]. The
primary advantage of these evolutionary algorithms is that it has the ability to find
the global minimum or maximum of an objective function with single or multiple
parameters. Of all the algorithms under the category of Genetic Algorithms, Differential evolution (DE) is one among the most powerful and simple stochastic real
parameter optimization method which is available today [9].
In this paper, we apply a variant of Differential Evolution algorithm (DE) which
incorporates multiple strategies for evolution of population for the problem of sinusoid detection in noise. This version of DE, which we refer to as the Modified
Differential Evolution Algorithm (MDE) hereafter, is described in [10] and applied
for optimizing antenna arrays. We compare the results obtained using MDE with
other frequency estimation methods as well as Cramer-Rao Lower Bound (CRLB).
2 Proposed Method
Figure 1 illustrates the steps involved in applying the Modified Differential Evolution
Algorithm (MDE) as described in [10] for detecting the sinusoid signal embedded
in noise. As can be seen in Fig. 1, the first step is to initialize a parent population
p̄i , where i ∈ [1, N p ], in the parameter space. In the case of sinusoid detection, the
parameter space spans the frequencies around frequency f max corresponding to the
maximum FFT bin as,
Fs
Fs
: f max +
(1)
p̄i = f max −
N
N
In the Modified Differential Evolution Algorithm [10], as can be seen from the
following equations, each strategy can be expressed as the linear combination of the
differences of vectors, which are a subset of the parent population, along with the
parent entries p̄i and p̄b . It is to be noted that the size of the population in the subset
will be significantly smaller than the size of the parent population. This would imply
that the number of parents who are partaking in the evolution process could be more
than two, unlike the GA, which typically makes use of only two entries from the parent
Detection of a Real Sinusoid in Noise using Differential …
79
Fig. 1 Modified differential evolution algorithm
population for the point and the uniform crossovers. For the MDE as described in
[10], applied to the sinusoid detection problem, the vector transformations of parent
population are as follows:
t¯1 = p̄b + F ( p̄i − p̄r )
(2)
t¯2 = p̄r + F( p̄i − p̄s )
(3)
In the above equations, F is a constant which controls the differential variations
( p̄i − p̄r ) and ( p̄i − p̄s ). The members p̄r and p̄s constitute the subset of parent
population. It should satisfy the condition that the indexes r and s (r = s) are different
and that they are also different from the running index i. As shown in Fig. 1, once
the children corresponding to the next generation are obtained as described above,
these children constitute the parent population for computing the set for the following
generation [10].
80
G. Narayanan and D. G. Kurup
The results that are obtained using MDE at different SNR levels have been compared with other existing frequency estimation methods as well as with the CramerRao Lower Bound (CRLB). An approximation of CRLB is given by the following
expression [4]:
2
=
σCRLB
6
2π 2 N (N 2 − 1)SNR
(4)
where N denotes the number of samples and SNR is the signal-to-noise ratio.
3 Results
In the simulations, real sinusoidal signals are generated randomly according to,
f (i) = [0.1Fs : 0.4Fs ], where Fs is the sampling frequency. Noise, according to
normal distribution as per the AWGN assumption, and for different SNR, is added to
the signal and the noisy data is applied to the algorithm. In order to assess the performance of the MDE algorithm, FFT-based estimation, and other frequency estimation
methods have been implemented [4]. The FFT-based estimation method locates the
frequency corresponding to the peak absolute value of FFT. The Mean-Squared Error
(MSE) is calculated for each method as follows:
MSE =
Ne −1
1 (i) 2
| f (i) − f est
|
Ne n=0
(5)
(i)
are the actual and
where Ne is the number of Monte Carlo experiments and f (i) , f est
the estimated frequency for ith experiment. Simulations are performed for different
resolutions corresponding to N = [64, 128, 256, 512].
Figure 2 shows the mean-squared error (MSE) for frequency resolution corresponding to N = 64. In order to compare MDE with other standard estimation techniques, the results using FFT and Candan’s method [4] are added along with CRLB.
From, Fig. 2, we can conclude that the performance of MDE is better than FFT as
well as Candan’s method, especially for low SNR values.
Figures 3, 4 and 5 show the mean-squared error (MSE) for frequency resolution corresponding to N = [128, 256, 512] respectively. Similar to earlier results, to
compare MDE with other standard estimation techniques, the results using FFT and
Candan’s method [4] have been included along with the CRLB. From the results, we
can conclude that the performance of MDE is better than FFT as well as Candan’s
method, especially for low SNR values.
Detection of a Real Sinusoid in Noise using Differential …
81
0.1
0.01
0.001
MSE
0.0001
1e-05
1e-06
1e-07
1e-08
1e-09
-20
CRLB
FFT
Candan
MDE
-15
-10
-5
0
5
10
15
20
SNR(dB)
Fig. 2 Comparison of MSE (mean-squared error) for MDE, Candan [4] and FFT along with CRLB
for frequency resolution corresponding to N = 64
0.1
0.01
0.001
MSE
0.0001
1e-05
1e-06
1e-07
1e-08
1e-09
1e-10
-20
CRLB
FFT
Candan
MDE
-15
-10
-5
0
5
10
15
20
SNR(dB)
Fig. 3 Comparison of MSE (mean-squared error) for MDE, Candan [4] and FFT along with CRLB
for frequency resolution corresponding to N = 128
82
G. Narayanan and D. G. Kurup
0.1
0.01
0.001
0.0001
MSE
1e-05
1e-06
1e-07
1e-08
1e-09
1e-10
1e-11
-20
CRLB
FFT
Candan
MDE
-15
-10
-5
0
SNR(dB)
5
10
15
20
Fig. 4 Comparison of MSE (mean-squared error) for MDE, Candan [4] and FFT along with CRLB
for frequency resolution corresponding to N = 256
0.1
0.01
0.001
0.0001
MSE
1e-05
1e-06
1e-07
1e-08
1e-09
1e-10
1e-11
-20
CRLB
FFT
Candan
MDE
-15
-10
-5
0
SNR(dB)
5
10
15
20
Fig. 5 Comparison of MSE (mean-squared error) for MDE, Candan [4] and FFT along with CRLB
for frequency resolution corresponding to N = 512
Detection of a Real Sinusoid in Noise using Differential …
83
4 Conclusion
Through this work, we show that the performance of Modified Differential Evolution
Algorithm outperforms other detection strategies, especially at low SNR values. It
is also seen that at high SNR, the Mean-Squared Error (MSE) closely approaches
Cramer-Rao Lower Bound (CRLB). The proposed method has the potential to be
applied to real-world sinusoid detection applications.
References
1. Quinn, B.G.: Recent advances in rapid frequency estimation. Digital Signal Proc. 19, 942–948
(2009)
2. Jacobsen, E., Kootsookos, P.: Fast accurate frequency estimators [DSP Tips & Tricks]. IEEE
Signal Proc. Mag. 24, 123–125 (2007)
3. Candan, C.: A method for fine resolution frequency estimation from three DFT sample. IEEE
Signal Proc. Lett. 18, 351–354 (2011)
4. Candan, C.: Analysis and further improvement of fine resolution frequency estimation method
from three DFT samples. IEEE Signal Proc. Lett. 20(9), 913–916 (2013)
5. Orguner, U., Candan, C.: A fine resolution frequency estimator using an arbitrary number of
DFT coefficients. Signal Proc. 105, 17–21 (2014)
6. Djukanovic, S.: An accurate method for frequency estimation of a real sinusoid. IEEE Signal
Proc. Lett. 23 (2016)
7. Man, K.F., Tang, K.S., Kwong, S.: Genetic algorithms: concepts and applications. IEEE Trans.
Ind. Electron. 43 (1996)
8. Das, S., Konar, A., Chakraborty, U.K.: Improving particle swarm optimization with differentially perturbed velocity. In: Proceedings on Genetic and Evolutionary Computation Conference, pp. 177–184 (2005)
9. Price, K., Storn, R., Lampinen, J.: Differential Evolution A Practical Approach to Global
Optimization. Springer, Berlin, Germany (2005)
10. Dhanesh, D.G., Himdi, M., Rydberg, A.: Synthesis of uniform amplitude unequally spaced
antenna arrays using the differential evolution algorithm. IEEE Trans. Antennas Propogation
51, 2210–2217 (2003)
Inherited Competitive Swarm Optimizer
for Large-Scale Optimization Problems
Prabhujit Mohapatra , Kedar Nath Das
and Santanu Roy
Abstract In this paper, a new Inherited Competitive Swarm Optimizer (ICSO) is
proposed for solving large-scale global optimization (LSGO) problems. The algorithm is basically motivated by both the human learning principles and the mechanism
of competitive swarm optimizer (CSO). In human learning principle, characters pass
on from parents to the offspring due to the ‘process of inheritance’. This concept of
inheritance is integrated with CSO for faster convergence where the particles in the
swarm undergo through a tri-competitive mechanism based on their fitness differences. The particles are thus divided into three groups namely winner, superior loser,
and inferior loser group. In each instances, the particles in the loser group are guided
by the winner particles in a cascade manner. The performance of ICSO has been
tested over CEC2008 benchmark problems. The statistical analysis of the empirical results confirms the superiority of ICSO over many state-of-the-art algorithms
including the basic CSO.
Keywords Competitive swarm optimizer · Evolutionary algorithms
Large-scale global optimization · Particle swarm optimization
Swarm intelligence
1 Introduction
Particle swarm optimization (PSO), proposed by Eberhart and Kennedy [1] is a
stochastic and population-based self-adaptive global optimization technique inspired
from social and competitive behavior of bird flocking and fish schooling. The PSO
simulates the swarm behavior to steer the particles in locating the global optimal
solution. Particles tune their path in search space dynamically using the personal
best (pbest) position and the global best (gbest) position of the whole swarm. Due
to its simplicity and ease implementation, PSO has gained wide popularity over the
P. Mohapatra (B) · K. N. Das · S. Roy
National Institute of Technology, Silchar 788001, Assam, India
e-mail: prabhujit.mohapatra@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_9
85
86
P. Mohapatra et al.
past few decades. However, while solving multimodal functions, PSO gets trapped
into local optima, resulting with a premature convergence [2, 3]. Over the time,
researchers attempted to face the challenge of reforming PSO to get rid of it. As
a result, numerous PSO variants are developed in the literature [4–8]. Since these
variants use the modified mechanisms with new operators, they mostly become computationally expensive. Moreover, the existence of ‘gbest’ operator in PSO helps in
faster convergence, but mostly leads to premature convergence. Hence, Liang [9]
suggested a new PSO variant, deprived of the gbest terms and the update approach
relies only on pbest position. Later, some other alternate techniques are proposed in
which the concept of neither gbest nor pbest is employed. In 2013, an effort was made
with a multi-swarm structure built on a feedback mechanism [10], where particles are
rationalized by a pairwise competition among particles of two unlike swarms. Similar approaches have been proposed by other researchers [11–13] too. This idea of
competitive mechanism predominantly marks two significances. Firstly, as a convergence approach, the weak solutions get an opportunity to learn from the stronger
ones of the other swarm. Secondly, as a mutation scheme the stronger particles
self-inspired by the earlier experiences to yield improved results. These tactics collectively impacts on retaining proper balance between exploration and exploitation.
Using this concept, another algorithm namely competitive swarm optimizer (CSO)
[14] is suggested in the recent literature. In CSO, after each pairwise competition
between particles, the loser particle learns from the winner one, instead of from pbest
or gbest. The working principle of CSO algorithm is very simple, yet influential to
solve LSGO problems. Although the CSO mechanism has established many success
milestones in the recent evolutionary world, the improved quality in the solution and
greater rate of convergence [15, 16] are yet to be addressed.
In this paper, a new CSO algorithm inspired by human learning principles has been
proposed. The proposed algorithm employs the process of inheritance that allows
the particles to improve the search capabilities by utilizing the experience of more
efficient particles. The basic idea is to allow the average and below average solutions
to converge towards good solutions in a cascade manner. As a result, an improved
rate of convergence is expected through a better rate of exploration in the search
space.
The paper is structured as follows. The related works to large-scale optimization
problem are being reviewed in Sect. 2. The motivation behind the proposition and
the proposed algorithm are outlined in Sect. 3. In Sect. 4, the comparative studies of
the experimental results are carried out. Lastly, the conclusion of the paper is drawn
in Sect. 5.
2 Large-Scale Optimization Problems
The real-world problems arise around is mostly complex structured due to the presence of a large number of decision variables and hence takes huge time for getting solved. Such problems are usually called as Large-Scale Global Optimization
Inherited Competitive Swarm Optimizer for Large-Scale …
87
(LSGO) problems. In fact, proposing an efficient algorithm for solving LSGO is a
greater challenge among researchers. Hence over the time, quite a large number of
metaheuristic algorithms are proposed in the literature to solve LSGO. Based on
the decomposition of the problem dimension, such algorithms could be categorized
into two kinds. The first kind is ‘Decomposition based algorithms’ which are also
known as Cooperative Coevolution (CC) [17–19] algorithms. In such kind, the highdimensional problems are decomposed into subproblems of low dimension. Each
subproblem is being solved initially by some traditional optimization algorithm for a
fixed number of generations in a round-robin approach. Then the solution from each
sub-problem is combined to form an n-dimensional solution. However, Yang et al.
[20] integrated a DE- based CC method called DECC-G [21] which inspires with the
notion of random grouping of decision variables to solve LSGO problems of 500 and
1000 dimensions. Later, it has been modified to a multilevel CC algorithm (MLCC)
[22] that uses a decomposer pool. It works with dynamic group size of variables
that rely on the past performance of the decomposer. Gradually, similar algorithms
namely CCPSO2 [23] and CC-CMA-ES [24] are being proposed to solve LSGO
problems.
The second kind is the ‘Non-Decomposition based algorithms’. Instead of the
divide-and-conquer approach, it uses different effective approaches in order to
improve the performance. These methods are mainly categorized as local search
based [25, 26], evolutionary computation based [27, 28] and swarm intelligence
based methods [29]. This present work proposes a modified CSO [14, 30] namely
Inherited CSO (ICSO) based on human learning principle. Both CSO and ICSO here
belong to the swarm intelligence approach. The motivation behind proposing such
an algorithm is as follows.
3 Motivation and Proposition
3.1 Motivation
Human beings have good social cognizance and are most intelligent creature in the
society. Probably for this reason, the algorithms inspired by human thoughts are
superior to those inspired by other creatures [31, 32]. In a family; the beliefs, the
ideas, the customs and the cultures usually inherited from one generation to the other.
The most experienced person acts as a guide and others attempt to learn from him
directly or indirectly. A son learns from his father and father from the grandfather.
Sometimes, son used to learn from the grandfather too. This process is known as
‘method of inheritance’. This concept of inheritance is presented in Fig. 1, which
became the major motivation of the proposed algorithm.
88
P. Mohapatra et al.
Fig. 1 Graphical illustration
of the concept of inheritance
3.2 Proposition
In CSO, only half of the swarm gets the opportunity to improve their solution,
which results high diversity and slow rate of convergence. In order to balance the
exploration and exploitation a new tri-competitive scenario along with the method
of inheritance is introduced here. The tri-competitive scenario allows 2/3rd of the
swarms to participate in the upgradation process whereas it passes the rest 1/3rd
directly to the next generation to retain the necessity of swarm diversity [33]. Further,
the learning abilities of the collaborators are again more strengthened through the
method of inheritance. In this process, the offspring continuously learn from their
parents. This healthy learning process effectively passes the good qualities of the
elders to the younger ones. As a result, the self and social cognizance of human
thoughts leads towards better solution over the search space.
Selection Strategy
In a swarm of size m, three randomly selected particles are undergone through
a tri-competition in terms of their fitness values, resulting with one winner and two
losers. The superior loser is symbolized as l1 and the inferior as l2 . Eventually, through
this selection process, there will be a K ( m/3) number of distinct competitions
possible. Therefore, three distinct groups namely winner group, superior loser group
and inferior loser group will be formed, each of size K . Let X w,k (t), X l1 ,k (t), X l2 ,k (t)
and Vw,k (t), Vl1 ,k (t), Vl2 ,k (t) represents the position and velocity of the winner and
two losers respectively in the k-th round of competition (k 1, 2, . . . , K ) at iteration
t. The selection strategy of particles under tri-competition along with their inherited
learning strategy is presented in Fig. 2.
Inherited Competitive Swarm Optimizer for Large-Scale …
89
Fig. 2 Swarm’s tri-competition mechanism in ICSO and the upgradation of winners and losers
Inherited Learning Strategy
The particles in each distinct group formed by selection strategy learn through
different inherited learning strategies as discussed below, which are mainly motivated
by the concept of inheritance.
Winner group: Since it includes the top performing particles (viz. the winner of
each tri-competition), they act as a guide (the most experienced person like grandfather in a family) for the loser particles. These particles, being the best individuals
in the swarm, need least attention for improvement. Therefore, the particles in the
winner group are directly allowed to transfer to the next generation without any
alteration.
Superior loser group: The particles in this group are the average individuals. They
are assigned to perform two tasks. Firstly they improve themselves by learning from
the winner and secondly they guide the inferior loser to improve their performance
(like father simultaneously learns from grandfather and teaches to the son). The
velocity and position of superior loser (l1 ) are updated by (1) and (2) respectively as
follows.
vl1,k (t + 1) R1 (k, t)vl1,k (t) + R2 (k, t) xw,k (t) − xl1,k (t)
+ ϕ1 R3 (k, t) xk (t) − xl1,k (t)
(1)
xl1,k (t + 1) xl1,k (t) + vl1,k (t + 1)
(2)
Here X k (t) is the mean position of the whole swarm. The factor ϕ1 governs the effect
of the mean position in maintaining the diversity that helps in escaping from getting
90
P. Mohapatra et al.
trapped into the local optima. Moreover, R1 (k, t), R2 (k, t) and R3 (k, t) represent
three randomly generated vectors at the k-th round competition in generation t.
Inferior loser group: The particles in this group are the least efficient and hence
require special guidance for performance improvement. These particles do not have
any other social responsibilities except improving themselves. Thus, it utilizes the
experience of superior loser as well as the winner (like son learns from the father
and the grandfather as well). Here superior loser acts as a primary mentor, which is
reflected in the middle term of (3). Since the inexperienced individuals also need to
be guided by the experienced individuals, the inferior losers are additionally allowed
to learn from the mean of the winners as given in the last term of (3). The velocity
and position of inferior loser (l2 ) are updated by (3) and (4) respectively as follows.
vl2,k (t + 1) R4 (k, t)vl2,k (t) + R5 (k, t) xl1,k (t) − xl2,k (t)
+ ϕ2 R6 (k, t) xw (t) − xl2,k (t)
(3)
xl2,k (t + 1) xl2,k (t) + vl2,k (t + 1)
(4)
Here ϕ2 is the factor that helps in governing the effect of X w (t). R4 (k, t), R5 (k, t),
and R6 (k, t) are three randomly generated vectors at the k-th round competition in
generation t.
The above strategies are incorporated to construct the proposed ICSO algorithm.
The entire working mechanism of ICSO algorithm is presented through a flow diagram in Fig. 3.
4 Experimental Results and Discussions
4.1 Experimental Setup
In order to evaluate the performance of the proposed algorithm ICSO, a set of 7
benchmark functions of CEC2008 are considered. The reason behind considering
such sets is to test the efficiency of ICSO in solving problems of different taste.
The ICSO algorithm is implemented in Matlab R2013a on a PC with a Dual Core
i7 2.00 GHz CPU having 4 GB RAM, Microsoft Windows 7, and 32-bit operating
system. The experiments were conducted 25 times and in each run the maximum
function evaluations (FEs) for CEC2008 are fixed using (5).
Maximum_FEs 3000 ∗ Dimension of the problem
(5)
The benchmark functions are all scalable i.e. the dimension can be user-defined.
In this study, the dimension of all the benchmark functions is fixed at 1000. The
parameters ϕ1 , ϕ2 and m in ICSO are considered here as reported in [14].
Inherited Competitive Swarm Optimizer for Large-Scale …
91
Fig. 3 Flowchart of ICSO algorithm
4.2 Performance Comparison
In this section, ICSO is deployed to solve CEC2008 LSGO problems under the
parameter setting recommended in the last section. The optimum solutions achieved
by ICSO are compared with CSO [14] and some other state-of-the-art algorithms like
CCPSO2 [23], multilevel cooperative co-evolution (MLCC) [22], separable covariance matrix adaption strategy (sep-CMA-ES) [34], efficient population utilization
strategy for particle swarm optimizer (EPUS-PSO) [29] and DMS-PSO [25].
Statistical Tests
Mean, Standard Deviation and t-test: To analyze and investigate the results, three
types of statistical measures are considered. The experimental outcomes in terms of
92
P. Mohapatra et al.
mean, standard deviation (Std) and t-values of errors are reported in the Table 1. The
overall best mean and least Std are emphasized with boldface letters. To confirm the
existence of significant differences between ICSO and other algorithms, t-test with
a significance level α 0.05 has been carried out. ICSO is significantly better over
another algorithm if the equivalent t-value is boldfaced. In case of a tie, the values
are tinted with bold italic. Further, the last column of each of these tables under the
heading w/t/l denotes the win, tie and loss totals of ICSO over that specific algorithm
in the sense of t-values. The algorithms with high win values are again emphasized
with bold letters. From the last column it is observed that the win total of ICSO is
maximum.
Average Ranking test according to Friedman test: Due to Friedman Test, the
average ranking of ‘n’ different algorithm in solving ‘m’ different functions can be
calculated through following steps.
a. First, each of ‘n’ algorithm is used to solve all m functions to form an m− tuple
vectors of solutions for a particular algorithm.
b. Against each function, the relative ranking (from 1 to n) is made.
c. The average ranks for each function over all algorithms will be calculated through
the mean value of relative rankings.
The average ranking comparison of ICSO with all rest algorithms is reflected in
Table 1. It is observed from Table 1 that ICSO attains the best ranking and supersedes
others including CSO.
Best Count test: The ‘Best Count’ of an algorithm is the number of functions for
which the algorithm provides the best results as compared to the rest algorithms. For
each algorithm the Best Count is reported just right to the average ranking in Table 1.
The highest count of ICSO indicates that it outperforms over others everywhere.
Convergence Analysis: Convergence comparison of ICSO is made with its immediate competitor CSO by allowing both of them to run from the same seed in order
to ensure a fair comparison. The seven benchmark functions of CEC2008 are taken
into consideration and the convergence graphs are pictured in Fig. 4, in which each
subfigure is responsible for one function. From this figure it can be concluded that
sooner or later, ICSO converges closer towards optimal solution as compared with
CSO. In few cases where ICSO initially could not beat CSO, gradually could do it
later.
5 Conclusion
In this paper, a new inherited competitive swarm optimizer (namely ICSO) is proposed. The synergy of ‘method of inheritance’ in human learning principle along with
CSO beautifies the strength of the proposed algorithm. Unlike CSO, ICSO updates
2/3rd of the population strings using an inherited technique in a cascade manner. It is
especially designed to handle large-scale global optimization problems. The experimental results and statistical analysis concludes that ICSO delivers the supreme
results and outclasses many state-of-the-art algorithms including CSO in terms of
1.52E−15
−2.57E+01
5.53E+02
2.86E+01
−9.67E+01
0.00E+00
0.00E+00
2.13E+02
Std.
t-Values
EPUS-PSO Mean
Std.
t-Values
DMS-PSO Mean
Std.
t-Values
Sep-CMAES
MLCC
CCPSO2
CSO
9.50E−25
2.23E−26
–
1.66e−22
1.18e−23
−6.99E+01
5.18E−13
9.61E−14
−2.70E+01
8.45E−13
5.00E−14
−8.44E+01
7.81E−15
Mean
Std.
t-Values
Mean
Std.
t-Values
Mean
Std.
t-Values
Mean
Std.
t-Values
Mean
ICSO
f1
9.02E+00
−1.92E+02
4.66E+01
4.00E−01
−2.05E+02
9.15 E+01
7.13E−01
−4.01E+02
1.76E+01
5.84E−01
–
3.76e+01
1.18e+00
−7.63E+01
7.82E+01
4.25E+01
−7.13E+00
1.087E+02
4.754E+00
−9.51E+01
3.65E+02
f2
4.54E+01
7.38E+00
8.37E+05
1.52E+05
−2.75E+01
8.98E+09
4.38E+08
−1.02E+02
9.77E+02
9.23E−02
–
9.81E+02
6.49E−01
−2.98E+01
1.33E+03
2.63E+02
−6.71E+00
1.79E+03
1.58E+02
−2.60E+01
9.10E+02
f3
2.48E+02
−9.86E+01
7.58E+03
1.51E+02
−2.37E+02
3.83E+03
1.70E+02
−1.00E+02
4.14E+02
1.03E+01
–
5.21e+02
2.95e+01
−1.72E+01
1.99E−01
4.06E−01
2.01E+02
1.37E−10
3.37E−10
2.01E+02
5.31E+03
f4
1.97E−03
−1.00E+00
5.89E+00
3.91E−01
−7.53E+01
0.00E+00
0.00E+00
7.85E+84
2.22E−16
0.00e+00
–
2.22e−16
0.00e+00
0.00E+00
1.18E−03
3.27E−03
−1.80E+00
4.18E−13
2.78E−14
−7.51E+01
3.94E−04
f5
Table 1 Comparison of ICSO versus others in solving CEC 2008 benchmark problems
3.19E−01
−3.37E+02
1.89E+01
2.49E+00
−3.80E+01
7.75E+00
8.92E−02
−4.35E+02
7.81E−14
3.41E−15
–
8.306e−13
1.673e−14
−2.20E+02
1.02E−12
1.68E−13
−2.80E+01
1.06E−12
7.68E−14
−6.39E+01
2.15E+01
f6
5/0/2
5/0/2
5/2/0
–
w/t/l
9.36E+01
5/1/1
−6.67E+01
−6.62E+03
3.18E+01
7/0/0
−5.28E+02
−7.50E+03
1.63E+01
5/0/2
−5.04E+02
−1.40E+04
6.23E+01
–
−1.38e+04
3.37e+02
−1.60E+00
−1.43E+04
8.27E+01
1.45E+01
−1.47E+04
1.51E+01
5.48E+01
−1.25E+04
f7
1
0
2
6
4.28
2
0
0
2
Best
count
4.85
3.71
3.57
2.85
2
Average
ranking
Inherited Competitive Swarm Optimizer for Large-Scale …
93
94
P. Mohapatra et al.
Fig. 4 The convergence profiles during 5 × 106 Fitness Evaluations (FEs) of CSO and ICSO on
1000D CEC2008 benchmark functions
both the solution excellence and the rate of convergence. The Inherited mechanism
indeed shapes the proposed algorithm to become more robust and effective.
References
1. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948. IEEE (1995)
2. Yang, Y., Pedersen, J.O.: A comparative study on feature selection in text categorization. In:
Proceedings of International Conference on Machine Learning, pp. 412–420. Morgan Kaufmann Publishers (1997)
3. Chen, W.N., Zhang, J., Lin, Y., Chen, E.: Particle swarm optimization with an aging leader and
challengers. IEEE Trans. Evol. Comput. 17(2), 241–258 (2013)
4. Ratnaweera, A., Halgamuge, S.K., Watson, H.C.: Self-organizing hierarchical particle swarm
optimizer with time-varying acceleration coefficients. IEEE Trans. Evol. Comput. 8(3),
240–255 (2004)
5. Zhan Z.-H., Zhang, J., Li, Y., Chung, H.-H.: Adaptive particle swarm optimization. IEEE Trans.
Systems Man Cybern. B Cybern. 39(6), 1362–1381 (2009)
6. Hu, M., Wu, T., Weir, J.D.: An adaptive particle swarm optimization with multiple adaptive
methods. IEEE Trans. Evol. Comput. 17(5), 705–720 (2013)
7. Juang C.-F.: A hybrid of genetic algorithm and particle swarm optimization for recurrent
network design. IEEE Trans. Systems Man Cybern. B Cybern. 34(2), 997–1006 (2004)
8. Mendes, R., Kennedy, J., Neves, J.: The fully informed particle swarm: simpler, maybe better.
IEEE Trans. Evol. Comput. 8(3), 204–210 (2004)
9. Liang, J.J., Qin, A., Suganthan, P.N., Baskar, S.: Comprehensive learning particle swarm optimizer for global optimization of multimodal functions. IEEE Trans. Evol. Comput. 10(3),
281–295 (2006)
10. Cheng, R., Sun, C., Jin, Y.: A multi-swarm evolutionary framework based on a feedback
mechanism. In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 718–724.
IEEE (2013)
11. Goh, C., Tan, K., Liu, D., Chiam, S.: A competitive and cooperative co-evolutionary approach
to multi-objective particle swarm optimization algorithm design. Eur. J. Oper. Res. 202(1),
42–54 (2010)
Inherited Competitive Swarm Optimizer for Large-Scale …
95
12. Hartmann, S.: A competitive genetic algorithm for resource-constrained project scheduling.
Naval Res. Logistics (NRL) 45(7), 733–750 (1998)
13. Whitehead, B., Choate, T.: Cooperative-competitive genetic evolution of radial basis function
centers and widths for time series prediction. IEEE Trans. Neural Netw. 7(4), 869–880 (1996)
14. Ran, C., Yaochu, J.: A competitive swarm optimizer for large scale optimization. IEEE Trans.
Cybern. 45(2), 191–204 (2015)
15. Clerc, M., Kennedy, J.: The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002)
16. Trelea, I.C.: The particle swarm optimization algorithm: convergence analysis and parameter
selection. Inf. Process. Lett. 85(6), 317–325 (2003)
17. Tseng, L.-Y., Chen, C.: Multiple trajectory search for large scale global optimization. In:
Proceedings of IEEE Congress on Evolutionary Computation, pp. 3052–3059. IEEE (2008)
18. LaTorre, A., Muelas, S., Pena, J.M.: Large scale global optimization: experimental results with
MOS-based hybrid algorithms. In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 1–8. IEEE (2013)
19. Potter, M.A., Jong, K.A.D.: A cooperative coevolutionary approach to function optimization.
In: Proceedings of the International Conference on Evolutionary Computation, pp. 249–257
(1994)
20. Yang, Z., Tang, K., Yao, X.: Differential evolution for high-dimensional function optimization.
In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 3523–3530. IEEE (2007)
21. Yang, Z., Tang, K., Yao, X.: Large scale evolutionary optimization using cooperative coevolution. Inf. Sci. 178(15), 2985–2999 (2008)
22. Yang, Z., Tang, K., Yao, X.: Multilevel cooperative coevolution for large scale optimization.
In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 1663–1670. IEEE (2008)
23. Li, X., Yao, Y.: Cooperatively coevolving particle swarms for large scale optimization. IEEE
Trans. Evol. Comput. 16(2), 1–15 (2011)
24. Liu, J., Tang, K.: Scaling up covariance matrix adaptation evolution strategy using cooperative
coevolution. In: Proceedings of International Conference on Intelligent Data Engineering and
Automated Learning, pp. 350–357. Springer (2013)
25. Liang, J., Suganthan, P.: Dynamic multi-swarm particle swarm optimizer. In: Proceedings of
IEEE Swarm Intelligence Symposium, pp. 124–129. IEEE (2005)
26. LaTorre, A., Muelas, S., Peña, J.-M.: A MOS-based dynamic memetic differential evolution
algorithm for continuous optimization: a scalability test. Soft. Comput. 15(11), 2187–2199
(2011)
27. Yang, Z., Tang, K., Yao, X.: Scalability of generalized adaptive differential evolution for largescale continuous optimization. Soft. Comput. 15, 2141–2155 (2011)
28. Brest, J., Maucec, M.S.: Self-adaptive differential evolution algorithm using population size
reduction and three strategies. Soft. Comput. 15(11), 2157–2174 (2011)
29. Hsieh, S.-T., Sun, T.-Y., Liu, C.-C., Tsai, S.-J.: Solving large scale global optimization using
improved particle swarm optimizer. In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 1777–1784. IEEE (2008)
30. Mohapatra, P., Das, K.N., Roy, S.: A modified competitive swarm optimizer for large scale
optimization problems. Appl. Soft Comput. 59, 340–362 (2017)
31. Tanweer, M.R., Suresh, S., Sundararajan, N.: Human meta-cognition inspired collaborative
search algorithm for optimization. In: Proceedings of the IEEE International Conference on
Multisensor Fusion and Information Integration for Intelligent Systems, pp. 1–6. IEEE (2014)
32. Shi, Y.: Brain storm optimization algorithm. An optimization algorithm based on brainstorming
process. Int. J. Swarm Intell. Res. (IJSIR) 2(4), 35–62 (2011)
33. Olorunda, O., Engelbrecht, A.P.: Measuring exploration/exploitation in particle swarms using
swarm diversity. In: Proceedings of IEEE Congress on Evolutionary Computation, pp. 1128–34.
IEEE (2008)
34. Ros, R., Hansen, N.: A simple modification in cma-es achieving linear time and space complexity. In: Parallel Problem Solving from Nature–PPSN X, pp. 296–305 (2008)
Performance Comparison
of Metaheuristic Optimization
Algorithms Using Water Distribution
System Design Benchmarks
Ho Min Lee, Donghwi Jung , Ali Sadollah , Eui Hoon Lee
and Joong Hoon Kim
Abstract Various metaheuristic optimization algorithms are being developed and
applied to find optimal solutions of real-world problems. Engineering benchmark
problems have been often used for the performance comparison among metaheuristic algorithms, and water distribution system (WDS) design problem is one of the
widely used benchmarks. However, only few traditional WDS design problems have
been considered in the research community. Thus, it is very challenging to identify
an algorithm’s better performance over other algorithms with such limited set of
traditional benchmark problems of unknown characteristics. This study proposes an
approach to generate WDS design benchmarks by changing five problem characteristic factors which are used to compare the performance of metaheuristic algorithms.
Obtained optimization results show that WDS design benchmark problems generated
with specific characteristic under control help identify the strength and weakness of
reported algorithms. Finally, guidelines on the selection of a proper algorithm for
WDS design problems are derived.
Keywords Metaheuristic optimization algorithms · Performance measurement
Water distribution systems
H. M. Lee · D. Jung · E. H. Lee
Research Center for Disaster Prevention Science and Technology, Korea University,
Seoul, South Korea
A. Sadollah
Department of Mechanical Engineering, Sharif University of Technology, Tehran,
Iran
J. H. Kim (B)
School of Civil Environmental and Architectural Engineering, Korea University,
Seoul, South Korea
e-mail: jaykim@korea.ac.kr
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_10
97
98
H. M. Lee et al.
1 Introduction
Optimization can be defined as the process to find the solution having the best fitness satisfying a set of constraints. Various metaheuristic optimization algorithms
are being developed and applied to real-world engineering problems such as truss
structure design, dam operation, parameter estimation, and traffic engineering. Mathematical benchmark problems have been used for performance comparison among
metaheuristic algorithms, however, engineering optimization problems have their
own characteristics. Good performance on the mathematical benchmark problems
does not guarantee good performance in real-world engineering problems. Therefore,
to evaluate the performance of the metaheuristic algorithm for the real problem, it
should be verified by applying the engineering problem with specific characteristics.
The water distribution system (WDS) design problem is one of the widely used
engineering benchmark problems. Several metaheuristic optimization algorithms
have been applied to optimal design of WDSs with various characteristics. Simpson
et al. [1] applied genetic algorithms (GAs), and Maier et al. [2] applied ant colony
optimization (ACO) for optimal design of WDS. The particle swarm optimization
(PSO) and the harmony search (HS) were applied by Montalvo et al. [3] and Geem
[4] respectively. More recently, the water cycle algorithm (WCA) and the mine blast
algorithm (MBA) were applied by Sadollah et al. [5, 6] to find an optimal design of
WDS.
However, only few traditional WDS design problems (e.g., New York tunnels,
Hanoi, and Balerma network) have been considered in the research community [7–9]
Therefore, it is very challenging to identify a metaheuristic algorithm’s better performance over others with such limited set of traditional WDS design problems of
unknown characteristics. Thus, in this study, engineering design problems are generated by modifications of existing WDS design benchmark and applied to performance
measurement of metaheuristic algorithms.
2 WDS Design Benchmark Generation
WDS is one of the most critical infrastructures for human activity. The main purpose
of WDSs is to supply the required quantity of water from source to users while
ensuring appropriate water quality and pressure [10]. The object of optimal design of
WDS is finding the most cost-effective design among various alternative designs with
satisfying hydraulic requirements. The objective function for the least-cost design
of WDSs with nodal pressure constraint is calculated from diameter and length of
pipes, as shown in Eq. (1):
Min.Cost N
i1
Cc (Di ) × L i +
M
j1
Pj
(1)
Performance Comparison of Metaheuristic Optimization Algorithms …
99
where, C c (Di ) is the construction cost according to pipe diameter per unit length;
L i is the pipe length; Di is the pipe diameter; Pj is the penalty function for ensuring
pressure constraints are satisfied; N is the number of pipes; M is the number of nodes.
If a design solution does not meet the pressure nodal pressure requirements, the
penalty function is added into the objective function, as shown in Eq. (2):
P j α(h min − h j ) + β
if h j < h min
(2)
where hj is the nodal pressure at node j; hmin is the minimum pressure requirement
at node j; α and β are constants in penalty function.
In this study, the GoYang network design problem first introduced by Kim et al.
[11] is used as reference benchmark problem to generate WDS design benchmarks.
The GoYang network in South Korea is one of the well-known benchmark WDSs.
It consists of 21 demand nodes, one zero demand node, 30 pipes, one constant
pump of 4.52 kW in the downstream of a single reservoir adding a constant head
gain of 71 m and nine loops as shown in Fig. 1. Total eight commercial pipes with
internal diameters from 80 to 350 mm have to be selected for the GoYang network
in the original design problem. Therefore, the number of candidate designs of whole
network is 830 .
The WDS design benchmarks in this study are generated by modifying five individual characteristics based on the GoYang network design problem. The number of
pipes (n) and the number of candidate pipe diameter options (m) are used as problem size modification factors. The pressure constraint (p), the roughness coefficient
(c) and the nodal demand multiplier (d) are also considered as problem complexity modification factors. The default problem characteristic factors are set as bold
numbers given in Table 1. In addition, four values are considered for each problem
characteristic factor and 20 benchmark problems are generated in this study.
Fig. 1 Layout of the GoYang network
100
H. M. Lee et al.
3 Performance Measurement Results
In this study, we compared four algorithms: random search (RS), genetic algorithms
(GAs) [12], simulated annealing (SA) [13], harmony search (HS) [14], and water
cycle algorithm [15]. Each metaheuristic algorithm is tested with 20 independent
runs for each of 20 cases shown in Table 1 and the maximum number of function
evaluations is set to 20,000 considered as stopping criterion. The ratio of an optimal
solution cost obtained from an algorithm to the known worst solution cost is defined
as the improvement ratio, because the global optimal solution of WDS design problems is generally unknown, and the global optimal solution changes as the problem
characteristics changes.
The RS founds feasible solution in 99% of the total cases, and the other algorithms
found feasible solutions in all individual runs of each case. Figures 2 and 3 show the
average and standard deviation of the average improvement ratios. Note that, average
and standard deviation are calculated from feasible solutions.
First, when average values of average improvement ratio are compared, it is found
that the RS has the lowest performance to search optimal design in 20 design benchmarks. The RS shows the smallest standard deviation among applied metaheuristic
algorithms in terms of variation of roughness coefficient and nodal demand multiplier. However, it is found that the RS searches the solution with low fitness, and the
reliability of its performance is low.
Even though the SA finds feasible design solutions in all cases, the SA shows
second worst results in terms of average of average improvement ratio. The GAs
shows average performance among applied metaheuristic algorithms in the average
and standard deviation. The GAs obtained better optimal solutions compared with
the SA, however, it shows lower performance and reliability with variation of number
of pipes and nodal demand multiplier to compare with the HS and the WCA.
The HS and WCA show similar performance and reliability in the modified
GoYang network design problems. The HS and WCA have the lowest performance
and reliability with variation of nodal demand multiplier to compare with variation
of the other factors.
Meanwhile, the metaheuristic algorithms have its own strength and weaknesses.
Furthermore, as the complexity and the difficulty of design benchmarks are increased,
the performance and reliability of applied algorithms are weakened consistently.
Thus, it is important to select proper design algorithm for a given engineering prob-
Table 1 Applied factors for
benchmark generation
Factors
Used values
n
30, 60, 90, 120
m
8, 10, 12, 14
p
15, 17, 19, 21
c
100, 90, 80, 70
d
1.00, 1.25, 1.50, 1.75
Performance Comparison of Metaheuristic Optimization Algorithms …
(a) RS
(b) GAs
(c) SA
(d) HS
101
(e) WCA
Fig. 2 Performance of metaheuristic algorithms (average of average improvement ratio)
lem, and to improve existing algorithms by enhancement of optimization process
with considering problem characteristics.
102
H. M. Lee et al.
(a) RS
(b) GAs
(c) SA
(d) HS
(e) WCA
Fig. 3 Performance of metaheuristic algorithms (standard deviation of average improvement ratio)
4 Conclusions
Engineering benchmark problems can be used for performance comparison among
metaheuristic algorithms and the water distribution system (WDS) design problem is
one of the widely used benchmarks. However, the traditional WDS design problems
have limitation in set of problem characteristics.
Performance Comparison of Metaheuristic Optimization Algorithms …
103
Therefore, engineering design problems are generated by modifications of existing
WDS design benchmarks and applied to performance measurement of metaheuristic
algorithms in this study. Each applied algorithm shows its own strength and weakness,
and the performances of algorithms are weakened as the size and the complexity of
problems are increased. It implies that finding optimal solutions for engineering
problems using a metaheuristic algorithm requires an efficient approach considering
characteristics of the problem.
In addition, the cost minimization is selected as an objective function, and the
nodal pressure requirement is used as a hydraulic constraint. However, there exist
several objectives (e.g., system reliability and greenhouse gas emission) and constraints (e.g., water flow velocity limitation and water quality requirement) in the
WDS design. Therefore, various combinations of objectives and constraints will be
considered, and also other problem modification factors can be used to benchmark
problem generation in future studies.
Acknowledgements This work was supported by a grant from The National Research Foundation
(NRF) of Korea, funded by the Korean government (MSIP) (No. 2016R1A2A1A05005306).
References
1. Simpson, A.R., Dandy, G.C., Murphy, L.J.: Genetic algorithms compared to other techniques
for pipe optimization. J. Water Resour. Plan. Manag. 120(4), 423–443 (1994)
2. Maier, H.R., Simpson, A.R., Zecchin, A.C., Foong, W.K., Phang, K.Y., Seah, H.Y., Tan, C.L.:
Ant colony optimization for design of water distribution systems. J. Water Resour. Plan. Manag.
129(3), 200–209 (2003)
3. Montalvo, I., Izquierdo, J., Pérez, R., Tung, M.M.: Particle swarm optimization applied to the
design of water supply systems. Comput. Math Appl. 56(3), 769–776 (2008)
4. Geem, Z.W.: Optimal cost design of water distribution networks using harmony search. Eng.
Optim. 38(03), 259–277 (2006)
5. Sadollah, A., Yoo, D.G., Yazdi, J., Kim, J.H., Choi, Y.: Application of water cycle algorithm
for optimal cost design of water distribution systems. In: International Conference on Hydroinformatics (2014)
6. Sadollah, A., Yoo, D.G., Kim, J.H.: Improved mine blast algorithm for optimal cost design of
water distribution systems. Eng. Optim. 47(12), 1602–1618 (2015)
7. Schaake, J.C., Lai, F.H.: Linear programming and dynamic programming application to water
distribution network design. MIT Hydrodynamics Laboratory (1969)
8. Fujiwara, O., Khang, D.B.: A two-phase decomposition method for optimal design of looped
water distribution networks. Water Resour. Res. 26(4), 539–549 (1990)
9. Reca, J., Martínez, J.: Genetic algorithms for the design of looped irrigation water distribution
networks. Water Resour. Res. 42(5) (2006)
10. Lee, H.M., Yoo, D.G., Sadollah, A., Kim, J.H.: Optimal cost design of water distribution
networks using a decomposition approach. Eng. Optim. 48(12), 2141–2156 (2016)
11. Kim, J.H., Kim, T.G., Kim, J.H., Yoon, Y.N.: A study on the pipe network system design using
non-linear programming. J. Korean Water Resour. Assoc. 27(4), 59–67 (1994)
12. Goldberg, D.E., Holland, J.H.: Genetic algorithms and machine learning. Mach. Learn. 3(2),
95–99 (1988)
13. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. science
220(4598), 671–680 (1983)
104
H. M. Lee et al.
14. Geem, Z.W., Kim, J.H., Loganathan, G.V.: A new heuristic optimization algorithm: harmony
search. Simulation 76(2), 60–68 (2001)
15. Eskandar, H., Sadollah, A., Bahreininejad, A., Hamdi, M.: Water cycle algorithm–a novel
metaheuristic optimization method for solving constrained engineering optimization problems.
Comput. Struct. 110, 151–166 (2012)
Comparison of Parameter-Setting-Free
and Self-adaptive Harmony Search
Young Hwan Choi , Sajjad Eghdami , Thi Thuy Ngo ,
Sachchida Nand Chaurasia and Joong Hoon Kim
Abstract This study compares the performance of all parameter-setting-free and
self-adaptive harmony search algorithms proposed in the previous studies, which
do not ask for the user to set the algorithm parameter values. Those algorithms
are parameter-setting-free harmony search, Almost-parameter-free harmony search,
novel self-adaptive harmony search, self-adaptive global-based harmony search algorithm, parameter adaptive harmony search, and adaptive harmony search, each of
which has a distinctively different mechanism to adaptively control the parameters
over iterations. Conventional mathematical benchmark problems of various dimensions and characteristics and water distribution network design problems are used
for the comparison. The best, worst, and average values of final solutions are used
as performance indices. Computational results show that the performance of each
algorithm has a different performance indicator depending on the characteristics of
optimization problems such as search space size. Conclusions derived in this study
are expected to be beneficial to future research works on the development of a new
optimization algorithm with adaptive parameter control. It can be considered to
improve the algorithm performance based on the problem’s characteristic in a much
simpler way.
Keywords Harmony search · Parameter-setting-free · Self-adaptive
Y. H. Choi
Department of Civil, Environmental and Architectural Engineering, Korea University,
Seoul 136-713, South Korea
S. Eghdami · T. T. Ngo · S. N. Chaurasia
Research Center for the Disaster and Science Technology, Korea University, Seoul 136-713,
South Korea
J. H. Kim (B)
School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 136-713,
Korea
e-mail: jaykim@korea.ac.kr
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_11
105
106
Y. H. Choi et al.
1 Introduction
Optimization problems involve in various fields such as mathematical and engineering problems. Many optimization algorithms have been developed and applied to
solve these optimization problems. However, the performances of these optimization algorithms such as exploitation and exploration ability have much variance
depend on the algorithm own parameters setting which are required adaptable algorithm parameter values. These algorithms depend on the parameter and their values
directly affect the performance of the algorithm in hand. Finding the best set of parameter values is itself a challenging task. To overcome the drawback, many studies with
relative parameters control method such as parameter-setting-free and self-adaptive
approach have proposed. Likewise, Harmony Search (HS) performs these improvements in order to enhance the performance of the algorithm. HS is proposed by [1]
and [2]. It is a method to find the solution using musical improvisations. HS’s improvisation method is inspired by the musical improvisation technique. Previous studies
state that HS takes fewer mathematical operations compared to other optimization
algorithms and can be easily adapted for solving various kind of optimization [3, 4].
However, the HS algorithm is that three parameters are constant values and it is hard
to decide the values rely on the different problems. To improve the performance of
HS algorithm, a variety of parameter-setting-free and self-adaptive HS have been
proposed as below.
The proposed parameter-setting-free and self-adaptive HS algorithm improve
operation and dynamic of three parameters [i.e., harmony memory considering rate
(HMCR), pitch adjusting rate (PAR), and bandwidth (Bw)] and applied in various
fields (i.e., mathematics, multidisciplinary engineering problems).
The parameter-setting-free (PSF) method automatically updates the parameters
values after each iteration by using structure formation [5]. The study introduced a
new operating type memory (OTM) and this memory updates considering number
of operating type such as harmony memory considering or pitch adjusting.
Shivaie et al. [6] developed a self-adaptive global best harmony search algorithm
(SGHSA) inspired by global harmony search algorithm [4]. SGHSA employed a new
improvisation scheme about an adaptive bandwidth, depends on the rate of generation
compared to the total number of iterations. In the early generation (less than the half
of iteration), the bandwidth is calculated by dynamic bandwidth formulation and
above the half of iteration, the bandwidth used lower boundary bandwidth value.
Luo [7] developed the novel self-adaptive harmony search (NSHS) algorithm that
considered HMCR, PAR, and bandwidth for suitable parameters setting. HMCR sets
a constant value according to the number of decision variables. The PAR procedure
is replaced considering the variance of fitness and new solutions are generated by
boundary condition of decision variable. NSHS applies a dynamic bandwidth which
the bandwidth value decreases gradually by increasing number of iterations and
increases as the range of boundary condition expands.
Comparison of Parameter-Setting-Free and Self-adaptive …
107
The previous PSF approach considered only HMCR and PAR. However, almostparameter-setting-free Harmony Search (APS-HS) proposed by [8] includes the Bw.
It is controlled by min/max decision variable value.
In this study, we compare the PSF and self-adaptive harmony search method
developed for improving the quality of the solution and it is applied on mathematical
benchmark problems. In addition, various performance indexes are used to compare the quantitative performance of each algorithm. It is expected to benefit future
research work which formulates the approaches. Especially the performance of the
newly proposed algorithms can be rigorously tested in a much simpler way.
2 Parameter-Setting-Free and Self-adaptive Harmony
Search
2.1 Harmony Search
HS can be explained as the improvisation process by a musician. The technique to
search for an optimum harmony in music is equivalent to the optimum solution. When
many different musicians play their instruments, all the various sounds generate one
single harmony. The musicians may change gradually to a suitable harmony, and
finally find an aesthetically pleasing harmony. In other words, HS is an approach
which finds the optimum harmony in the music. In the HS, four parameters are used
to search for the optimum solution [i.e., harmony memory (HM), harmony memory
considering rate (HMCR), pitch adjusting rate (PAR), and bandwidth (Bw)]. These
parameters set a constant value. A search space for instrument is limited to some
memory space and is described as harmony memory (HM), where harmony memory
size (HMS) represents the maximum number of harmonies to be saved in the memory
space. The main operators of HS are random selection (RS), memory consideration
(MC), and pitch adjustment (PA), to find better solutions among the HM.
2.2 Parameter-Setting-Free Harmony Search
The parameter-setting-free harmony search (PSF-HS) was developed to reduce the suitable parameter setting [5]. PSF-HS modifies the improvisation step of HS by updating the HMCR and PAR
on the every iteration for each decision variable. This study introduced operation type memory (OTM) to update the parameters. It was a memory that is
used to generate a new solution among HS operators (i.e., RS, MC, and PA) and
the parameters (i.e., HMCR and PAR) are updated using the number of selected
operators. As the number of iterations increases, the HMCR generally increases, but
108
Y. H. Choi et al.
the PAR decreases. And, this trend can excess HMCR to 1 and PAR to 0. To prevent
this problem, noise value is used to control the HMCR and PAR between 0 and 1.
2.3 Almost-Parameter-Free Harmony Search
Almost-parameter-free harmony search (APS-HS) is the modified version of original
PSF-HS [5] that is additionally considered dynamic Bw including automatic HMCR
and PAR setting. It also applied OTM to calculate the adopted HMCR and PAR by
using the same formulation. In the APS-HS, Bw is dynamically updated according
to the maximum and minimum values in the HM.
2.4 Novel Self-adaptive Harmony Search
Novel self-adaptive harmony search (NSHS) is the modified process of determining
HMCR, PAR, and Bw from constant values [7]. HMCR is set according to the
dimensions of the problem and it is analogous for example complex problem has
large HMCR. In the original HS, setting the Bw is important to convergence of
optimal solution. Therefore, NSHS used dynamic Bw to do fine-tune and the tuning
range is wider in the beginning and narrower at the end of simulation. PA is replaced
with considering the variance of fitness and a new solution is generated by boundary
condition of decision variable.
2.5 Self-adaptive Global-Based Harmony Search Algorithm
Self-adaptive global-based harmony search (SGHSA) to find a better solution and
more effective parameter tuning [6]. SGHSA changed pitch adjustment rule to avoid
falling into a local optimum solution. The value of the Bw parameter is dynamically
reduced by subsequent generations.
2.6 Parameter Adaptive Harmony Search
Kumar et al. [9] proposed a dynamic change in the values of HMCR and PAR,
consequently modifying the improve version of harmony search called parameter
adaptive harmony search (PAHS). PAHS keeps the value of HMCR small so as to
make the algorithm explores each solution. The best obtained solutions are stored in
HM as the algorithm proceeds with the increase in number of generations. During the
final generations, the value of HMCR increases to make the search restricted to HM
Comparison of Parameter-Setting-Free and Self-adaptive …
109
that the solutions could be obtained from within HM only. Similarly, PAR has high
value during earlier generations that it makes the algorithm to modify the solutions
either stored in HM or from the feasible range.
3 Application Results
To evaluate parameter-setting-free and self-adaptive harmony search, the mathematical benchmark problems are a used measure the performance. The individual
simulation is repeated 50 times, and each simulation performs 50,000 function evaluations (NFEs) for each problem. To eliminate the influence of initial condition, same
initial solution is used for all of the initial solutions by a random generation.
In application of single-objective optimization problems, 30 (=5 benchmark functions × 6 kinds of decision variables: 2, 5, 10, 30, 50, 100) case of simulations are
employed to compare the performance of these approaches in Table 1. The performance measures for quantifying the performance of the compared algorithms in this
section are best, mean, and worst using 50 times individual run and to fair comparison the initial solution of five algorithms used same value generated by random.
Based on the Appendix 1–5 for relative low dimensional cases (DV 2, 5, 10),
the SGHSA outperforms comparing with the other algorithm. Among the 20 cases
[i.e., three kinds of decisions variable (2, 5, and 10) × 4 benchmark problems (i.e.,
Rosenbrock, Rastrigin, Griewank, and Ackley)], SGHSA achieved first rank in 9
cases. In case of large benchmark problems (DV 30, 50, 100), the NSHS is shown
the best performance.
Table 1 Test problem for single-objective optimization
Name
Dimensions
Search domain
Global optimum
∞]n
0
Rosenbrock function
(Valley-shaped)
[−30, 30]n
0
Rastrigin function
(Many local optimum)
[−5.12, 5.12]n
0
Griewank function
(Many local optimum)
[−600, 600]n
0
Ackley function
(Distinction global
optimum)
[−32.768, 32.768]n
0
Sphere function
(Bowl-shaped)
2, 5, 10, 30, 50, 100
[−∞,
110
Y. H. Choi et al.
4 Discussion
This study presents a comparison of parameter-setting-free with self-adaptive harmony search to show the effect of their own optimization operators and improving
the performance of finding the best solution. It applied on famous mathematical
benchmark problems and evaluates fairly using statistical analysis, using same initial solutions.
As a result, among these parameters control optimization algorithms, in most of
the cases NSHS shows the best performance, especially, in the higher dimension
benchmark problems. NSHS is an improved harmony search method that modifies
the harmony memory considering and dynamic bandwidth. This approach has the
ability of avoid being stuck into local optimal by using fstd (standard deviation of
fitness) of the decision variable(s). NSHS has several boundary conditions (min/max
decision variable range and bandwidth range) and uses a method to take into the
gap of decision variables. Therefore, this algorithm can find a good solution for
continuous problems, but the performance of detecting for discrete problems is not
vilified. So, a discrete problem is worth simulating to evaluate its detecting ability.
The algorithm can be extended for discrete optimization problems.
This study shows special feature between the finding operator and problem characteristics. It would be helpful for proposing the new algorithms for testing much
simpler way. In the future study, by considering the characteristic of these parameter
control optimization algorithm, the new self-adaptive algorithm can be developed and
applied on various benchmark problems (e.g., continuity and discrete mathematical
problem, real-world engineering problem).
Acknowledgements This work was supported by a grant from The National Research Foundation
(NRF) of Korea, funded by the Korean government (MSIP) (No. 2016R1A2A1A05005306).
Appendix
See Tables 2, 3, 4, 5 and 6.
Table 2 The Sphere function optimization results (Average)
Algorithm Dimension
2
5
10
30
50
100
Simple HS
7.53.E−09
2.30.E−07
1.34.E−06
3.43.E−03
1.86.E−02
5.42.E−02
PSF-HS
APF-HS
SGHSA
NSHS
PAHS
8.20.E−09
4.69.E−09
8.44.E−09
2.17.E−12
2.61.E−05
3.76.E−05
2.41.E−05
1.24.E−08
6.15.E−10
1.82.E−03
1.18.E−03
7.73.E−04
7.58.E−07
4.90.E−09
9.50.E−03
2.09.E−02
2.23.E−02
6.34.E−03
6.76.E−08
5.66.E−02
4.72.E−02
5.60.E−02
2.72.E−02
2.03.E−07
1.17.E−01
1.26.E−01
1.45.E−01
8.80.E−02
9.43.E−07
2.60.E−01
Comparison of Parameter-Setting-Free and Self-adaptive …
111
Table 3 The Rosenbrock function optimization results
Algorithm Dimension
2
5
10
30
50
100
Simple HS
5.10.E−08
1.70.E−05
2.09.E−05
2.19.E−02
5.69.E−02
1.59.E−01
PSF-HS
APF-HS
SGHSA
NSHS
PAHS
3.21.E−08
3.78.E−08
0.00.E+00
0.00.E+00
4.77.E−03
3.12.E−05
1.83.E−05
1.36.E−05
1.61.E−05
6.37.E−03
2.21.E−03
1.70.E−03
7.94.E−05
1.95.E−05
8.65.E−03
4.35.E−02
3.53.E−02
2.20.E−02
2.13.E−05
2.05.E−02
1.06.E−01
1.01.E−01
6.52.E−02
1.82.E−05
3.29.E−02
2.80.E−01
2.93.E−01
2.36.E−01
2.60.E−05
5.91.E−02
Table 4 The Rastrigin function optimization results
Algorithm Dimension
2
5
10
30
50
100
Simple HS
2.06.E−04
4.35.E−04
7.19.E−04
1.44.E−03
1.68.E−03
3.09.E−03
PSF-HS
APF-HS
SGHSA
NSHS
PAHS
3.05.E−04
2.33.E−04
2.10.E−05
1.26.E−04
4.77.E−03
5.99.E−04
5.21.E−04
2.07.E−06
2.30.E−04
6.37.E−03
7.26.E−04
1.01.E−03
1.73.E−05
4.37.E−04
8.65.E−03
1.02.E−03
1.08.E−03
1.10.E−03
8.64.E−04
2.05.E−02
2.25.E−03
2.09.E−03
1.48.E−03
1.07.E−03
3.29.E−02
4.58.E−03
2.53.E−03
2.06.E−03
1.46.E−03
5.91.E−02
Table 5 The Griewank function optimization results
Algorithm Dimension
2
5
10
30
50
100
Simple HS
1.97.E−10
7.73.E−06
2.62.E−04
1.47.E−03
2.08.E−03
3.09.E−03
PSF-HS
APF-HS
SGHSA
NSHS
PAHS
2.39.E−09
3.46.E−09
1.17.E−13
2.23.E−13
7.76.E−06
5.71.E−06
5.71.E−06
2.73.E−11
6.86.E−11
4.43.E−04
1.07.E−04
9.27.E−05
2.14.E−04
5.58.E−10
1.20.E−03
9.58.E−04
1.03.E−03
1.58.E−03
3.05.E−09
2.76.E−03
1.48.E−03
1.65.E−03
2.07.E−03
5.34.E−09
3.54.E−03
2.28.E−03
2.42.E−03
3.15.E−03
1.20.E−08
4.95.E−03
Table 6 The Ackley function optimization results
Algorithm Dimension
2
5
10
30
50
100
Simple HS
5.22.E−05
8.15.E−03
6.53.E−02
1.32.E−01
1.51.E−01
1.76.E−01
PSF-HS
APF-HS
SGHSA
NSHS
PAHS
1.57.E−04
1.57.E−04
1.01.E−06
1.25.E−06
1.55.E−02
1.01.E−02
9.83.E−03
7.40.E−06
7.59.E−06
7.88.E−02
3.81.E−02
3.59.E−02
6.37.E−02
2.14.E−05
1.19.E−01
1.03.E−01
1.10.E−01
1.33.E−01
7.18.E−05
1.71.E−01
1.25.E−01
1.32.E−01
1.47.E−01
1.28.E−04
1.83.E−01
1.45.E−01
1.52.E−01
1.77.E−01
2.90.E−04
2.00.E−01
112
Y. H. Choi et al.
References
1. Geem, Z.W., Kim, J.H., Loganathan, G.V.: A new heuristic optimization algorithm: harmony
search. Simulation 76(2), 60–68 (2001)
2. Kim, J.H., Geem, Z.W., Kim, E.S.: Parameter estimation of the nonlinear Muskingum model
using harmony search. JAWRA J. Am. Water Resour. Assoc. 37(5), 1131–1138 (2001)
3. Mahdavi, M., Fesanghary, M., Damangir, E.: An improved harmony search algorithm for solving
optimization problems. Appl. Math. Comput. 188(2), 1567–1579 (2007)
4. Omran, M.G., Mahdavi, M.: Global-best harmony search. Appl. Math. Comput. 198(2), 643–656
(2008)
5. Geem, Z.W.: Parameter estimation of the nonlinear Muskingum model using parameter-settingfree harmony search. J. Hydrol. Eng. 16(8), 684–688 (2010)
6. Shivaie, M., Ameli, M.T., Sepasian, M.S., Weinsier, P.D., Vahidinasab, V.: A multistage framework for reliability-based distribution expansion planning considering distributed generations
by a self-adaptive global-based harmony search algorithm. Reliab. Eng. Syst. Saf. 139, 68–81
(2015)
7. Luo, K.: A novel self-adaptive harmony search algorithm. J. Appl. Math. (2013)
8. Jiang, S., Zhang, Y., Wang, P., Zheng, M.: An almost-parameter-free harmony search algorithm
for groundwater pollution source identification. Water Sci. Technol. 68(11) (2013)
9. Kumar, V., Chhabra, J.K., Kumar, D.: Parameter adaptive harmony search algorithm for unimodal
and multimodal optimization problems. J. Comput. Sci. 5(2), 144–155 (2014)
Copycat Harmony Search: Considering
Poor Music Player’s Followship Toward
Good Player
Sang Hoon Jun , Young Hwan Choi , Donghwi Jung
and Joong Hoon Kim
Abstract Harmony Search (HS), one of the most popular metaheuristic
optimization algorithms, is inspired by musical improvisation process. HS operators mimic music player’s different behaviors to make the best harmony. For example, harmony memory considering realizes the player’s utilization of a combination
of sounds among the good harmony found in the past whereas pitch adjustment is
derived from fine pitch tuning. However, at the authors’ best knowledge, there is no
harmony search which takes into account the fact that poor music player improves
as he/she follows from the good performer. This study proposes a new improved
version of HS called Copycat Harmony Search (CcHS) which employs a novel pitch
adjustment approach for dynamic bandwidth change and poor solution’s followship
toward a good solution. The performance of CcHS is compared to that of the original
HS and HS variants with modified pitch adjustment in a set of well-known mathematical benchmark problems. Results obtained show that CcHS outperforms other
algorithms in most problems finding the known global optimum.
Keywords Copycat harmony search · Improved pitch adjustment
Poor solution’s followship
S. H. Jun · Y. H. Choi
Department of Civil, Environmental and Architectural Engineering, Korea University,
Seoul 136-713, South Korea
D. Jung
Research Center for Disaster Prevention Science and Technology, Korea University,
Seoul, South Korea
J. H. Kim (B)
School of Civil, Environmental and Architectural Engineering, Korea University,
Seoul 136-713, South Korea
e-mail: jaykim@korea.ac.kr
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_12
113
114
S. H. Jun et al.
1 Introduction
In mathematics and computer science, optimization refers to the process of finding
the best element from some sets of available alternatives. For optimization, mathematical methods (e.g., linear programming, non-linear programming, dynamic programming) were traditionally used to solve problems. However, because of their
drawbacks such as requiring large number of evaluations for applying to the complex mathematical problems and real-life optimization problems, metaheuristic algorithms which mimic natural and behavioral phenomena are now widely used. Harmony Search (HS) [1] is one of the most famous metaheuristic algorithms for its
simplicity and efficiency. It is inspired by the musical improvisation process that
musicians search for the best harmony adjusting the pitches of instrument. Since HS
is developed, there were many improved versions of the algorithm to make its performance better such as Improved Harmony Search (IHS) [2], Global-best Harmony
Search (GHS) [3], and Self-Adaptive Harmony Search (SaHS) [4]. These algorithms
modified the pitch adjustment of the original HS to eliminate the drawback that
occurs by fixing the value of parameters.
In this study, a new variant of HS called Copycat Harmony Search (CcHS) is
proposed to enhance the skill for searching global optimum. Also, to reduce the
dependency on the selection of different parameters, CcHS searches the solution
automatically based on its Harmony Memory (HM). The details of the algorithm are
explained in Sect. 2. The performance of CcHS is examined and compared to other
HS algorithms using eight mathematical benchmark problems of 30 dimensions.
2 Copycat Harmony Search
In this study, CcHS makes poor solutions mimic good solutions in HM. A new
harmony is improvised as optimization proceeds, and the new one is changed with
the worst harmony in HM if it is better than the worst. However, if the harmonies
in HM are nearly identical, it is more likely that the best and worst harmonies in
HM are not changed over iterations. To overcome the limitation, a novel adjustment
is employed when the best and worst harmonies are not changed for a predefined
number of iteration.
CcHS has two different ways of generating new harmony when pitch adjusting. If
the best harmony in HM does not change during a predefined number of iterations,
a new harmony is formed considering the range of good harmonies. The number of
good harmonies to consider for optimization process is decided by the user. In this
research, the number of good harmonies (NGH ) is set to 3, which means that a new
harmony is the created within the range of top 3 solutions in HM.
Other strategy is applied when the worst harmony is fixed for specific iteration.
The worst harmony in HM will not be updated if the newly searched harmony is not
better than the worst. In CcHS, these new improvised harmonies are considered as
Copycat Harmony Search: Considering Poor Music Player’s …
115
the bad solutions because the good harmony is not generated anymore. To improvise
a better solution, the harmony is formed considering the best harmony in HM. For the
pitch adjustment process, the new harmony tries to move forward to the best value in
HM. This concept is from swarm intelligence to make the bad solutions mimic the
good one. Unlike GHS [3], the new solution is generated between the best in HM
and the random value in HM.
For the proposed method of adjustment, new parameters, update counting of
best (UCB ), update counting of worst (UCW ), and fixed number of iteration (FI )
are introduced. The number of iterations during which the best and worst harmony
remains unchanged is stored in the counter parameters Ucb and Ucw, respectively.
When UCB and UCW exceeds its FI , new harmony is adjusted as Eqs. (1) and (2).
xi,new min H M Ni G H + (max H M Ni G H − min H M Ni G H ) ∗ rand()
xi,new xi + xbest,i − xi ∗ rand()
(1)
(2)
When the best harmony
is fixed
until F
I of the best,
a new harmony is generated
as Eq. (1) while min H M Ni G H and max H M Ni G H are the smallest and the largest
values of NGH in HM, respectively. NGH is for deciding how many good harmonies to
be considered including the best harmony in HM. For the adjustment while the worst
harmony is fixed until FI of the worst, Eq. (2) is applied. To update the worst harmony
(i.e., the poorest music player), each decision variable mimics the best solution (i.e.,
the good player).
Besides bad solution’s followship to the good solution, to eliminate the inconvenience of setting fixed values of the parameters (e.g., PAR, BW), PAR is linearly increased from 0.0 to 1.0 during the iteration and BW is dynamically changed
according to HM at each iteration. Wang and Huang [4] suggested that PAR should
be decreased linearly as iteration proceeds, but based on the results of preliminary
tests, increasing PAR during optimization outperformed in most cases. Also, to avoid
setting a constant value of BW, a novel pitch adjustment is suggested. In CcHS, BW
dynamically changes considering the values of variables in HM at each iteration as
follows:
bwi,t max H M i − min(H M i )
(3)
The BW of ith variable at tth iteration is determined as Eq. (3) while max H M i
and min(H M i ) mean the largest and the smallest values of ith variable in HM. By
proposed pitch adjustment, the inconvenience of setting specific value for BW is
solved. Also, by adjusting the size of ith variable in HM for its BW at each iteration,
it is possible to apply pitch adjustment considering memories found before. Decision
variables in HM would have different ranges at each iteration. The variable which
shows big difference between its maximum and minimum value would mean that it
has not converged yet. It needs more exploration, searching globally by large BW,
which is calculated considering its own range. Meanwhile, when the maximum and
minimum values are nearly same, it represents that the decision variable converged
116
S. H. Jun et al.
to specific value. So exploitation should be performed, and the small BW of the
variable will help. Therefore, BW of variables changing dynamically regarding their
own status in HM seems reasonable for both global and local search.
3 Application and Results
In this study, the proposed algorithm is applied in seven 30-dimensional and one
2-dimensional mathematical benchmark problems (Table 1).
The performance of CcHS is compared to that of the original HS, IHS, GHS, and
SaHS with respect to the final solution’s quality. Thirty independent optimizations
are conducted to calculate the mean, best, and worst solution value, starting with randomly generated HM. The parameters sets suggested in previous studies are adopted
for the comparison (Table 2). The consistent value of FI for UCB , UCW and NGH is
used for all problems (FI of best 40, FI of worst 20, NGH 3). The total number
of function evaluations allowed is set to 50,000 for all algorithms.
Table 3 shows the obtained results from eight mathematical benchmark problems.
In most problems, CcHS outperforms other variants of HS finding the known global
optimum. However, SaHS achieved better results for mean and worst value than
Table 1 The details of 8 mathematical benchmark problems (D 30)
Name
Function
Range
D 2
Sphere
f1(x) i1 x
−100 < xi < 100
D
D
Schwefel function 2.22
f2(x) i1 |x| + i1 x
−10 < x i < 10
( f min : 0)
Rosenbrock’s valley
f3(x) D−1
2 2 + (x − 1)2
i
i1 100 x i+1 − x i
Step function
f4(x) Schwefel function 2.26
Rastrigin function
f5(x) √
D 418.98289 ∗ D + i1
−xi sin |xi |
D 2
f6(x) i1
xi − 10 cos(2π xi ) + 10
Ackley function
f7(x) −20 ∗ exp −0.2
exp
Six-Hump Camel-Back
function
1
D
D
i1 [x i
D
−30 < x i < 30
( f min : 0)
−100 < x i < 100
( f min : 0)
+ 0.5]2
1
D
D
i1
xi2 −
cos(2π xi ) + 20 + e
−512 < x i < 512
( f min : 0)
−5.12 < x i < 5.12
( f min : 0)
−32 < x i < 32
( f min : 0)
i1
f8(x) 4x12 − 2.1x14 + 13 x16 + x1 x2 − 4x22 − 4x24
−3 < x 1 < 3,
−2 < x2 < 2,
( f min : −1.03162845)
Copycat Harmony Search: Considering Poor Music Player’s …
Table 2 Parameter data in HS variants
Parameter
HS
IHS
117
GHS
SaHS
CcHS
HMS
HMCR
PAR
PARmin
PARmax
BW
BWmin
5
0.9
0.3
–
–
0.01
–
5
0.9
–
0.01
0.99
–
0.0001
5
0.9
–
0.01
0.99
–
–
50
0.99
–
0.0
1.0
–
–
10
0.99
–
0.0
1.0
–
–
BWmax
–
(xU −x L )
20
–
–
–
CcHS in the Ackley problem. Harmony Memory Size (HMS) in SaHS is 50 as
suggested in previous study. Large size of HM could consider various combinations
of the decision variables but requires more time for comparing the generated solution
with the solutions in HM. The running time of optimization is important factor to
consider the performance of algorithms. Although SaHS showed better results, it
has deficiency with the evaluating time. The effect of HMS in CcHS should be
investigated later for the best performance.
4 Conclusions
Since its introduction, HS have gained its popularity and have been applied to many
complex problems. To enhance the performance and to solve disadvantages of the
original HS, a lot of improved version of HS have been invented until today. In this
study, a new version of HS called Copycat Harmony Search was proposed with a
novel pitch adjustment strategy. When the solution is not generated for predefined
number of iteration, the bad solution mimics the good solution, with dynamic bandwidth considering the values in Harmony Memory. The performance of proposed
algorithms was compared to that of other improved versions of HS in a set of benchmark problems. The results showed that CcHS outperformed other algorithms. By
the followship of the bad solution toward the good solution, the algorithms showed
enhancement. In future research, verification of CcHS’s performance on real-life
optimization problems should be implemented.
Acknowledgements This research was supported by a grant [13AWMP-B066744-01] from
Advanced Water Management Research Program funded by the Ministry of Land, Infrastructure,
and Transport of the Korean government.
118
S. H. Jun et al.
Table 3 Optimization results
Parameter
HS
Sphere
IHS
GHS
SaHS
CcHS
Mean
5.67E+00
1.14E−06
2.52E−03
2.64E−11
1.34E−12
Best
Worst
Mean
2.72E+00
7.33E+00
8.19E−02
6.96E−07
1.44E−06
4.23E−03
6.24E−05
6.20E−03
2.29E−02
2.18E−14
1.41E−10
5.12E−05
1.78E−14
4.44E−12
5.63E−09
Best
Worst
Rosenbrock’s Mean
valley
5.76E−02
1.03E−01
1.88E+02
3.57E−03
4.67E−03
1.07E+02
2.35E−03
4.32E−02
1.94E+01
5.93E−07
1.52E−04
2.66E+01
6.21E−10
1.70E−08
7.94E+00
Best
Worst
Step function Mean
8.95E+01
2.49E+02
8.29E−02
2.29E+01
1.70E+02
1.09E−06
1.16E−01
3.03E+01
1.08E−04
2.50E+01
2.75E+01
1.03E−12
3.91E−02
1.79E+01
2.04E−13
Best
Worst
Mean
4.60E−03
1.67E−01
2.16E+01
6.04E−07
1.32E−06
2.52E−02
3.42E−07
5.47E−04
1.55E−02
2.10E−14
5.69E−12
1.08E+00
6.66E−15
8.03E−13
8.18E−05
Best
Worst
Mean
1.40E+01
2.82E+01
3.57E−01
3.38E−03
1.78E−01
2.34E+00
3.38E−03
4.39E−02
1.93E−03
4.43E−03
3.35E+00
2.61E+00
8.18E−05
8.18E−05
2.81E−09
Best
Worst
Mean
4.37E−02
1.05E+00
7.23E−01
5.55E−01
5.03E+00
7.80E−04
5.47E−05
7.41E−03
1.20E−02
1.29E+00
3.83E+00
9.89E–06
1.07E−14
2.59E−08
1.93E−05
Best
Worst
Mean
1.87E−01
6.13E−04
2.47E−03
1.94E−07
7.54E−08
1.08E+00
8.62E−04
2.25E−02
2.77E–05
1.06E−04
−1.03162845 −1.03162843 −1.03162522 −1.03162846 −1.03162845
Best
Worst
−1.03162845 −1.03162855 −1.03162845 −1.03162846 −1.03162845
−1.03162845 −1.03162843 −1.03162203 −1.03162846 −1.03162845
Schwefel
function 2.22
Schwefel
function 2.26
Rastrigin
function
Ackley
function
Six-Hump
Camel-Back
function
References
1. Geem, Z.W., Kim, J.H., Loganathan, G.V.: A new heuristic optimization algorithm: harmony
search. Simulation 76, 60–68 (2001)
2. Mahdavi, M., Fesanghary, M., Damangir, E.: An improved harmony search algorithm for solving
optimization problems. Appl. Math. Comput. 188, 1567–1579 (2007)
3. Omran, M.G.H., Mahdavi, M.: Global-best harmony search. Appl. Math. Comput. 198, 643–656
(2008)
4. Wang, C.M., Huang, Y.F.: Self-adaptive harmony search algorithm for optimization. Expert Syst.
Appl. 37, 2826–2837 (2010)
Fused Image Separation with Scatter
Graphical Method
Mayank Satya Prakash Sharma, Ranjeet Singh Tomar, Nikhil Paliwal
and Prashant Shrivastava
Abstract Image fusion and its separation is a frequently arising issue in Image processing field. In this paper, we have described image fusion and its Separation using
Scatter graphical method and Joint Probability Density Function. Fused image separation using Scatter Graphical Method depend on Joint Probability density function
of fused image. This technique gives batter result of other technique based on Signal
Interference ratio and peak signal-to-noise ratio.
Keywords Real image · Scatter · BSS · PSNR · SIR real mixture
1 Introduction
Separation of merged and overlapped images is a frequently arising issue in image
processing field such as separation of fused and overlapped images achieved from
many applications. In which we get a mixture which contains of two or more than
two images and for identification we essential to separate them. In this paper, it
is supposed that original images are mutually statistically independent and identifiable at the time of mixing and merging, and the difficulty is solved by applying
Scatter graphical method, To apply Scatter Graphical in frequency domain, Equivarient Adaptive Separation Via Independence (EASI) algorithm was extended to
separate complex valued signals when photographing objects placed behind a glass
window or windscreen, since most varieties of glass have semi reflecting properties
[1]. The need to separate the contributions of the original and the virtual images to
the combined, superimposed, images is important in applications where reflections
may create ambiguity in scene analysis. In which we get a mixture which contain of
M. S. P. Sharma (B) · N. Paliwal · P. Shrivastava
Rustan Ji Institute of Technology, Tekanpur, India
e-mail: mayanksintal@gmail.com
R. S. Tomar
ITM University Gwalior, Gwalior, India
e-mail: er.ranjeetsingh@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_13
119
120
M. S. P. Sharma et al.
two or more than two merged images and for identification we necessary to separate
them. Algebraically, image mixture can be given below
X KS
K k11 k12
,
k21 k22
(1)
(2)
where, X [x1 , x2 ]T are mixed images, S [s1 , s2 ]T are the real images and K is
a combined matrix. Blind source separation (BSS) problem is depend on fused real
image separation, because neither source 2d real signal, nor mixed coefficients are
known. The observed images are define is weighted linear combination of source
2d signals and mixing weights are also not given [2]. If we can calculate mixing
weighted matrix than the original unmixed images can also be define as
S k −1 X.
(3)
There are many other applications of image separation namely, image denoising
[3, 4], medical signal processing like FMRI, ECG, EEG [5–7] feature extraction in
Content-Based Image Retrieval (CBIR) [8–10], face recognition [4, 9], compression redundancy reduction [11], watermarking [12, 13], remote sensing in cloud
prediction and detection [14], where VHRR (very high-resolution radiometer) is a
technique of cloud detection in remote sensing, scientific data mining [15], finger
print separation (in crime branch) [16]. There are less technique and systems which
permit separates the particular speaker from different merged image information and
data which is contained at some unwanted noisy environments. The similar application is studies in digital hearing aid system, TV meeting area, image recognition
system, etc. In particular, Independent Component Analysis (ICA) Technique and
microphone array based approaches are target. Microphone array approach permit
enhances an object and target image from the merged images and discard noises and
the phase difference among different image sources which relates to the distance
between the microphone and the position of the different image sources. There are
many algorithms for digital merged image separation namely scatter graphical technique, Singular decomposition based independent component analysis technique,
and principal component analysis (PCA), etc. There are many approaches for digital
mixed image separation namely (1) scatter graphical technique (2) SVD-based ICA
technique (3) Convolutive mixture separation. These techniques are based on BSS
(Blind source separation).
Fused Image Separation with Scatter Graphical Method
121
2 Scatter-Geometrical Based Method
Scatter graphical method is an efficient technique for separation. In this paper we
will use scatter graphical technique for image separation. The two-dimensional blind
separation problem consist of the input 2d signals (i.e., mixtures) to be the linear combination of two different source signals. Scatter graphical approach is applicable for
non-sparse signal. The fused mixtures are accordingly described by Eqs. (4) and (5)
X 1 (x, y) k11 s1 (x, y) + k12 s2 (x, y)
(4)
X 2 (x, y) k12 s2 (x, y) + k21 s2 (x, y),
(5)
where si and X i are the sources and fused mixtures signals, respectively. The signal
si are supposed to be nonnegative and normalized, i.e., 0 ≤ S ≤ 1. The gain of the
signals and dynamic range are integrated into the mixing matrix. Dependencies are
presented. The Problem of Blind Source Separation (BSS) when the hidden images
are Nonnegative (N-BSS). In this case, the scatter plot of the mixed data is contained
within the simplified parallelogram generated by the columns of the mixing matrix.
Shrinking Algorithm for not mixing Nonnegative Sources, aims at calculate the
mixing matrix and the sources by parallelogram
X a max(w1 )
(6)
ya max(w2 ),
(7)
where w1 and w2 are one dimensional image vector.
Further calculation depends on the assumption that Q1 < Q2 , where Q1 and Q2 are
defined by
K 21
K 22
K 22
Q2 K 12
Q1 (8)
(9)
Another mathematical concept of scatter approach
A [(k11 + k12 )k, [(k21 + k22 )]k
(10)
B [(k12 − k11 )k, [−(k21 − k22 )]k
(11)
C [−(k11 + k12 )k, [−(k21 + k22 )]k
(12)
D [−(k12 + k11 )k, [(k21 − k22 )]k,
(13)
where ABCD is a parallelogram edges
122
M. S. P. Sharma et al.
[(K 11 + K 12 )]K xa ,
[(K 21 + K 22 )]k yb
(14)
[(K 12 − K 11 )]K xb ,
[−(k21 − k22 )]K yb
(15)
[−(K 11 + K 12 )]K xc ,
[−(k21 + A22 )]K yc
(16)
[(K 12 − K 11 )]K xd ,
[(K 21 − K 22 )]K yd
(17)
We will estimate the mixing coefficient with some algebraic equation. These
equation are given below
xa + xd
2
xa + xb
k12 k 2
ya − yd
k22 k 2
ya − yb
k21 k 2
k11 k xa − xb
2
xa − xd
k12 k 2
ya + yb
k22 k 2
ya + yd
k21 k .
2
k11 k (18)
(19)
(20)
(21)
3 Work Done
We will take four different images. We will fuse these images with help Scatter
Graphical method make six combinations of these images according to c2n where n number of images. We will separate these images with help of Scatter method, then
calculate the Peak to signal ratio and Signal interference ratio (SIR) of difference
between the original image and separated image. In this paper, a scatter graphical
method of blind source separation is introduced on images. Result of experiment
shows the scatter approach can separate images. And show proposed approach can
separate every image.
Fused Image Separation with Scatter Graphical Method
123
4 Image Separation with Scatter-Geometrical Method
We will take four different gray images size 512 * 512 bmp images. So our aim is to
estimate the mixing matrix from original image. Let us take two images IM(1) and
IM(2) in Fig. 1 (Fig. 2).
Two histogram equalized real images are linearly mixed in Eqs. (22 and 23)
then the predicted and observed real images will no longer have uniform probability
distributions function
x1 k11 I M1 + K 12 I M2
(22)
x2 k21 IM1 + K 22 IM2.
(23)
In vector matrix form the above equation can be written
1M2 K IM,
(24)
where, mixing coefficient is given by
K K 11 K 12
K 21 K 22
(25)
Fig. 1 Original image
X2
Fig. 2 Fused image IM1 and IM2
124
M. S. P. Sharma et al.
Fig. 3 Probability density function (PDF) of independent component x 1 and x 2
IM1 and IM2 are independent to each other. Then we will take histeqlization
(uniform distribution) of given image
1
, if IMi ∈ [−kk]
2k
fIM(IMi ) .
(26)
0
elsewhere
Graphical, both the source IM1 and IM2 and fuse image x1 , x2 are independent
with each other and having the uniform distribution within range [−kk] is shown
below (Fig. 3).
Uniform distribution of independent component x1 and x2 having uniform distribution within the range of −k to k and magnitude of uniform distribution is 2k1
X1
1
X1
1
∗
(27)
F(x1 ) f x1
f x2
k11
k11
k12
k12
X2
1
X2
1
∗
,
F(x2 ) f x2
f x2
k11
k21
k12
k22
where ‘*’ operator the convolution let us assume that
Scaling of the fused data
X1
1
f g1 (g1 )
f x1
k11
k11
x2
1
f g1 (g2 )
f x1
k12
k12
f x1 (x1 ) f g1 (g1 ) ∗ f g2 (g2 )
(28)
(29)
(30)
(31)
Mathematically, we get the expression for the probability density function of the
mixture x1 and likewise for mixture x2 graphical probability density function (pdf)
of mixture x1 and mixture x2 [21] (Fig. 4).
Fused Image Separation with Scatter Graphical Method
125
Fig. 4 Probability distribution function of fused image x1 and X2
Different Fused image
Fig. 5 Fused image of 2M3
Then the resultant distribution of the observed images for k12 > k11 and k22 > k21
is given function w1 and w2. Where, w1 and w2 is given below (Figs. 5 and 6).
⎤
⎡
1
k
+
k
k
+
w
+
k
≤
−(k
−
k
−(k
≤
w
(k
)
)k
)k
11
12
1
11
12
1
12
1
2
⎥
⎢ 4k11 k12 k
⎥
⎢
1
⎢
−(k12 + k11 )k ≤ w1 ≤ (k12 − k11 ) ⎥
2k12 k
⎥
F(w1 ) ⎢
⎥
⎢ 1
⎥
⎢
k
+
k
k
+
w
+
k
≤
+
k
−(k
≤
w
(k11
12
1)
11
12 )k
1
12 )k ⎦
⎣ 4k11 k12 k 2 (k11
otherwise
⎡
⎢
⎢
⎢
f(w2 ) ⎢
⎢
⎢
⎣
1
k
4k21 k22 k 2 (k21
0
+ k22 k + w2 ) −(k21 + k22 )k ≤ w2 ≤ −(k22 − k21 )k
1
2k22 k
1
k
4k11 k12 k 2 (k11
+ k + y1 )
otherwise
⎤
⎥
⎥
−(k22 − k21 )k ≤ w2 ≤ (k22 − k21 ) ⎥
⎥
⎥
(k22 + k21 )k ≤ w2 ≤ (k21 + k22 )k ⎥
⎦
0
126
M. S. P. Sharma et al.
Different Fused image
Fig. 6 Fused Image 3M4
5 Scatter Plot of Mixed Image
Show uncorrelated mixture of those independent components, when the mixture are
uncorrelated that the distribution is not same. The independent components are mixed
using orthogonal mixing matrix, which corresponds rotation of plane. The edge of
the square, we are estimate the rotation that gives the original component nonlinear
correlation that gives the original component Using two independent components
with uniform distribution (Figs. 7, 8, 9, 10 and 11).
Fig. 7 Scatter plot of mixture X 1 and X 2 (1M2) (K 11 0.467 K 12 0.23 K 21 0.33 K 22 0.667) Horizontal axis is labeled as X 1 and vertical axis X 2
Fused Image Separation with Scatter Graphical Method
127
Fig. 8 Scatter plot of mixture X 1 and X 2 (2M33M4) (K 11 0.467K 12 0.23K 21 0.33K 22 0.667) Horizontal axis is labeled as X 1 and vertical axis X 2
Fig. 9 Separated
image 1M2
psnr=8.1890
SIR=1.7532E+003
psnr=15.2778
SIR=1.732E+003
6 Results and Discussion
Real images separation of fused images, this technique has been evaluated on six
fused real-image pairs and performance is analyzed in terms of signal to interference
ration (SIR) and peak signal to noise ratio (PSNR). These merged images for k11 0:467; k12 0:29; k21 0:33; and k22 0:67 are generated using randomly chosen
four real images in the bitmap form (Tables 1 and 2).
128
M. S. P. Sharma et al.
Fig. 10 Separated
image 2M3
psnr=8.1890
Fig. 11 Separated
image 3M4
psnr=8.7393 SIR = 2.247E+003
Table 1 Estimated matrix
coefficient for 4 combination
of image
SIR=1.7532E+003
psnr=15.2778
SIR=1.732E+003
psnr=16.0909 SIR=2.2254E+005
Mixture
k11
k12
k21
k22
1M2
1M3
1M4
2M3
2M4
0.52
0.52
0.52
0.52
0.52
0.23
0.23
0.23
0.23
0.23
3.30E−01
0.33
3.30E−01
3.31E−01
3.31E−01
6.62E−01
6.62E−01
6.62E−01
6.62E−01
6.62E−01
K K 11 K 12
K 21 K 22
Fused Image Separation with Scatter Graphical Method
Table 2 Result with scatter
method
129
Mixture
Scatter method
PSNR1
PSNR2
SIR1
SIR2
1M2
1M3
1M4
2M3
2M4
9.6594
8.6195
8.9315
8.189
8.3965
1.97E + 01
17.9967
2.00E + 01
1.86E + 01
1.92E + 01
2.35E + 01
2.31E + 01
2.35E + 01
2.29E + 01
2.32E + 01
Actual matrix 17.0704
14.9952
17.2053
15.2778
16.4529
0.465 0.23
0.33 0.667
7 Conclusion
The given technique for image separation depends on scatter graphical plot successfully and separates the histogram equalized for fused real images and in this paper,
we have to separate image with scatter graphical method. Main problem of how can
we estimate the mixing matrix? Since the image separation aims at estimating both
130
M. S. P. Sharma et al.
the original image separation and the mixing matrix using only the observation, our
aim to estimate mixing matrix gives estimate of source 2d signal. With some information about the source and on the basis of information we are trying to calculate
mixing coefficient with the help of scatter graphical method. Some limitations of
finding the mixing matrix are—(1) Image sources are independent to each other (2)
fused images are noise-free. In this paper, we assume that we have the idea about the
distribution of sources, different type of graphical structures and by analysis of these
structures; we can estimate the mixing coefficient easily. We can take two images
having weighting coefficient, i.e., (k11 k12 k21 k22 ). All the different cases for the all
two observed fused image.
Mixture
Structure
Estimating coefficient
X1
X2
Straight line
k11 k12 k21 k22
X1
X2
Rhombus
k11 k22 , k12 k21
We have carefully chosen—several different fused image combination of four
different samples of proportionate mixtures of mixed image and then has calculated
the PSNR and signal interference ratio of difference between the original image
and separated image by scatter graphical method. In this paper, scatter graphical
algorithms give better result, compared to any other technique based on PSNR and
SIR.
References
1. Singh, D.K., Tripathi, S., Kalra, P.K.: Separation of image mixture using complex ICA. In:
Proceedings of ASID ’06, pp. 8–12. New Delhi
2. Tonazzini, A., Bedini, L., Salerno, E.: A Markov model for blind image separation by a meanfield EM algorithm. IEEE Trans. Image Process. 15(2) (2006)
3. Kumari, M., Wajid, M.: Source separation of independent components. LRC, JUIT, 2013, SPR
621 KUM, SPM1327
4. Carasso, D., Vizel, E., Zeevi, Y.Y.: Blind source separation using mixtures scatter plot properties. In: 2009 16th International Conference on Digital Signal Processing. IEEE (2009)
5. Sternberg, S.R.: Biomedical image processing. Computer 16(1), 22–34 (1983)
6. Parra, L., Sajda, P.: Blind source separation via generalized eigenvalue decomposition. J. Mach.
Learn. Res. 4, 1261–1269 (2003) Submitted 10/02; Published 12/03
7. Zarzoso, V., Comon, P.: Robust independent component analysis for blind source separation
and extraction with application in electrocardiography. In: 30th Annual International IEEE
EMBS Conference Vancouver. British Columbia, Canada, August 20–24, 2008
8. Choras, R.S.: Image feature extraction techniques and their applications for CBIR and biometrics systems. Int. J. Biol. Biomed. Eng. 1(1), 6–16 (2007)
9. Ziquan, H.: Algebraic feature extraction of image forrecognition. Pattern Recogn. 24(3),
211–219 (1991)
10. Zou, W., Li, Y., Lo, K.C., Chi, Z.: Improvement of image classification with wavelet and independent component analysis (ICA) based on a structured neural network. In: 2006 International
Fused Image Separation with Scatter Graphical Method
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
131
Joint Conference on Neural Networks Sheraton Vancouver Wall Centre Hotel, Vancouver, BC,
Canada July 16–21, 2006
Chye, K.C., Mukherjee, J., Mitra, S.K.: New efficient methods of image compression in digital
cameras with color filter array. Consum. Electron. IEEE Trans. 49(4), 1448–1456 (2003)
Jadhav, S.D., Bhalchandra, A.S.: Blind source separation based robust digital image watermarking using wavelet domain embedding. In: 2010 IEEE Conference on Cybernetics and
Intelligent Systems (CIS). IEEE (2010)
kundur, D., Hatzinakos, D.: A robust digital image watermarking scheme using the waveletbased fusion. In: Image Processing, International Conference on vol. 1. IEEE Computer Society
(1997)
Huadong, D., Yongqi, W., Yaming, C.: Studies on cloud detection of atmospheric remote
sensing image using ICA algorithm. In: 2nd International Congress on Image and Signal
Processing, 2009 CISP’09. IEEE (2009)
Ye, N. (ed.): The Handbook of Data Mining, vol. 24. Lawrence Erlbaum Associates, Publishers
(2003)
Zhao, Q., Jain, A.K.: Model based separation of overlapping latent fingerprints. Inf. Forensics
Secur. IEEE Trans. 7(3), 904–918 (2012)
Tonazzini, A., Bedini, L., Salerno, E.: A Markov model for blind image separation by a meanfield EM algorithm. Image Process. IEEE Trans. 15(2), 473–482 (2006)
Chen, F., et al.: Separating overlapped fingerprints. Inf. Forensics Secur. IEEE Trans. 6(2),
346–359 (2011)
Hyvärinen, A., Karhunen, J., Oja, E.: Independent Component Analysis, vol. 46. Wiley (2004)
Hyvarinen, A.: Blind source separation by nonstationarity of variance: a cumulant-based
approach. Neural Networks IEEE Trans. 12(6), 1471–1474 (2001)
Ascending and Descending Order of
Random Projections: Comparative
Analysis of High-Dimensional Data
Clustering
Raghunadh Pasunuri, Vadlamudi China Venkaiah and Bhaskar Dhariyal
Abstract Random Projection has been used in many applications for dimensionality reduction. In this paper, a variant to the iterative random projection K-means
algorithm to cluster high-dimensional data has been proposed and validated experimentally. Iterative random projection K-means (IRP K-means) method [1] is a
fusion of dimensionality reduction (random projection) and clustering (K-means).
This method starts with a chosen low-dimension and gradually increases the dimensionality in each K-means iteration. K-means is applied in each iteration on the
projected data. The proposed variant, in contrast to the IRP K-means, starts with the
high dimension and gradually reduces the dimensionality. Performance of the proposed algorithm is tested on five high-dimensional data sets. Of these, two are image
and three are gene expression data sets. Comparative Analysis is carried out for the
cases of K-means clustering using RP-Kmeans and IRP-Kmeans. The analysis is
based on K-means objective function, that is the mean squared error (MSE). It indicates that our variant of IRP K-means method is giving good clustering performance
compared to the previous two (RP and IRP) methods. Specifically, for the AT & T
Faces data set, our method achieved the best average result (9.2759 × 109 ), where as
IRP-Kmeans average MSE is 1.9134 × 1010 . For the Yale Image data set, our method
is giving MSE 1.6363 × 108 , where as the MSE of IRP-Kmeans is 3.45 × 108 . For
the GCM and Lung data sets we have got a performance improvement, which is a
multiple of 10 on the average MSE. For the Luekemia data set, the average MSE is
3.6702 × 1012 and 7.467 × 1012 for the proposed and IRP-Kmeans methods respectively. In summary, our proposed algorithm is performing better than the other two
methods on the given five data sets.
Keywords Clustering · High-dimensional data · K-means · Random Projection
R. Pasunuri (B) · V. China Venkaiah · B. Dhariyal
School of Computer and Information Sciences, University of Hyderabad, Hyderabad, India
e-mail: raghupasunuri@gmail.com
V. China Venkaiah
e-mail: venkaiah@hotmail.com
B. Dhariyal
e-mail: bdhariyal94@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_14
133
134
R. Pasunuri et al.
1 Introduction
The K-means [2] is a clustering algorithm in which the data with n points given in
R d and san integer K is specified. The algorithm finds K cluster centers such that
the mean squared error is minimized. It starts by initializing the centers by randomly
selected K points. The initial centers are updated regularly after each iteration by
taking the mean of each cluster. For every iteration the means are recalculated and all
the points are reassigned to its closest center, which is the mean of the cluster points.
The total squared error is reduced in each of the iteration. The algorithm converges
when it reaches the minimum squared error. The disadvatage of K-means is that it
can be caught in local minimum.
Random Projection (RP) [3] is a is a very famous and powerful technique for dimensionlaity reduction, which uses matrix multiplication to project the data into
lower dimensional space. It uses a random matrix to project the original highdimensional data into a low-dimensional subspace, by which the distance between
the data points is approximately preserved. Fradkin and Madigan [4] have done a
comparative analysis on the combination of PCA and RP with SVM, decision trees
and nearest neighbor methods. Bingham and Mannila [5] is another work in the literature, in which different dimensionality reduction methods have been compared for
image nad text data. The distortion rate and computational complexity is reported
as performance. Fern and Brodley [6] have used RP and ensemble methods to improve the clustering performance of high-dimensional data. Deegalla and Bostrom
[7] applied PCA and RP for Nearest Neighbor Classifier to report the advantage of
performance increse when dimensions grow fastly. An iterative version of RP Kmeans algorithm is given in Cardoso and Wichert [2], which got some improvement
over the RP K-means.
A variant to IRP K-means method that performs clustering of the high-dimensional
data using random projections in the iterative dimensions of IRP K-means algorithm
[2] is proposed in this work. We call this method Variant of IRP K-means (VIRP
K-means). The performance of VIRP Kmeans is compared with the related methods
namely, IRP K-means, RP K-means. From the empirical results, we can say that the
performance (mean squared error) of VIRP Kmeans is improved when compared to
RP Kmeans and IRP Kmeans methods. Results of the conducted experiments reveal
that gradual decrease in the reduced dimensionality and then clustering on that lowdimensions gives us better solution than clustering on the original high-dimensional
data.
The remaining contents of this paper is organized as: In Sect. 2, we describe Kmeans clustering. In Sect. 3, Random Projection (RP), RP K-means and IRP K-means
algorithms are presented. Section 4 presents the proposed VIRP K-means. Section 5
reports our experimental results. Section 6 ends with conclusion and some future
directions.
Ascending and Descending Order of Random …
135
2 K-means Algorithm
K-means performs cluster analysis on low and high dimensional data. It is basically
an iterative algorithm which takes input as N observations and divide them across K
non-overlapping clusters. The clusters are identified by initializing K random points
as centroids and iterating them over N observations. The centroids for K clusters
are calculated by minimising the error function used to discriminate a point from its
cluster, in this case euclidean distance. The lesser the error, more is the “goodness”
of that cluster.
Let X = {xi , i = 1, . . . , N } be the set of N observations, and these observations
are going to be grouped into K clusters, C = {ck , k = 1, . . . , K }, where K N .
The main goal of K-means is to reduce the squared Euclidean distance between the
center of a cluster and the observations in the cluster. The mean of a cluster ck is
denoted by μk and is defined as [8, 9]:
μk =
1 xi
Nk x ∈c
i
(1)
k
where Nk is the number of observations in cluster ck .
Minimisation of error function can be done using gradient descent approach. It
scales well with large datasets and is considered to be a good heuristic for optimising
the distance. The following are the steps involved in K-means algorithm:
1.
2.
3.
4.
5.
Randomly initialize K cluster centroids.
Calculate euclidean distance between each observation and each cluster center.
Find the closest center for each point and assign it to that cluster.
Find the new center or mean of each cluster using Eq. (1).
Repeat steps 2 and 3 until no change in the mean.
3 Random Projection
Random Projection (RP) is a dimensionality reduction method, that projects the data
into lower dimensional space by using a random matrix multiplication. It approximately preserves the distance between the points [4, 5].
Random Projection method projects the original d−dimensioanl data to a Ddimensional subspace (D d) using a random d × D orthogonal matrix P. The
orthogonal matrix Pd X D is having unit length columns. Symbolically, it can be written as:
P
= X N ×d Pd×D
X NR ×D
(2)
The theme of Random Projection is based on the Johnson-Lindenstrauss (JL) lemma.
Johnson-Lindenstrauss lemma [3] states that if N points in vector space of dimension
136
R. Pasunuri et al.
d are projected onto a randomly selected subspace of dimension D, then the Euclidean
distance between the points are approximately preserved. We can find more details
about JL lemma in [10]. The statement of the JL lemma given in [10] is as follows:
Theorem 1 [JL Lemma] For any 0 < < 1 and any integer N , let D be a positive
integer such that
2
3
D≥4
−
2
3
−1
ln N .
Then for any set V of N points in R d , we can find a map f : R d → R D such that
for all u, v ∈ V ,
(1 − ) u − v2 ≤ f (u) − f (v)2 ≤ (1 + ) u − v2 .
Here, the map f can be constructed in randomized polynomial time.
Proof of this lemma is given in [10, 11]. Many researchers proposed different methods
for generating the random matrix [11, 12]. The computational cost of projecting
the data is reduced by using integers and by using sparseness in random matrix P
generation. Tha matrix P is actually not orthogonal, but it incurs a large amount
of computational cost to make it orthogonal. However, there are almost orthogonal
directions are present in the high-dimensional space [13]. Hence, we can say that
these vectors that are having random directions are considered as orthogonal.
In the Literature, there are many algorithms are present to generate random projections which satisfy JL Lemma. Of these, Achlioptas [11] algorithm is very famous
and used widely. In [11], the elements of a random vector P are defined as:
pi j =
or
+1 with Pr = 21 ;
−1 with Pr = 21 .
⎧ √
1
⎪
⎨+ 3 with Pr = 6 ;
pi j =
0 with Pr = 23 ;
⎪
⎩ √
− 3 with Pr = 16 .
(3)
(4)
The computational complexity of random projection is O(d D N ) where d represents
the original high-dimension of the input, D represents reduced dimensionality of the
projected subspace and N is the size of the input data that is the number of samples it
contains. It becomes O(cD N ), when the input X has c non-zero entries per column
and is sparse [14].
Ascending and Descending Order of Random …
3.1
137
RP K-means
Several researchers combined the K-means clustering algorithm with random projection [12, 15, 16]. The basic idea here is project the original high-dimensional data
into a low-dimensional space and then perform clustering on this low-dimensional
subspace. This reduces the K-means iteration cost effectively. The solution we get in
low-dimensional space is same as the one in the high-dimension. The RP K-means,
first initializes cluster membership G randomly. Select K points randomly as cluster
centers. Then generates a random matrix P to project the input data. Project the input
data X N ×d to D dimensions where D < d, using the projection matrix Pd×D . The
initial cluster centers C R P defined by the mean of each cluster in X R P with the help
P
and G. We apply K-means clustering upto convergence
of projected data data X NR ×D
or we will stop based on some stopping condition. The details of this method is
described in Algorithm 1.
Algorithm 1 RP K-means[1]
Input: Dimension D, Data Set X N ×d , No. of clusters K
Output: cluster membership G.
begin
1: Set G as K by taking random points from X .
2: Set a random matrix Pd×D
P = X
3: Set X NR ×D
N ×d Pd×D
R
P
4: Set Ck×D by finding the mean of each cluster in X R P according to G.
5: Find G with K-means on X R P with C R P as initialization.
6: return G
3.2
Iterative Version of RP K-means
It is an iterative algorithm [1], the dimension of the space is increased in each iteration so that the local minimums are avoided in the original space. Each solution
constructed in one iteration can be used in the following iterations thereby saving the
computations. This is same as cooling in simulated annealing clustering [17]. The
wrong cluster assignments are reduced as dimensionality increased. A wrong cluster
is defined by the Euclidean distance from center to the point in the original space.
The algorithm is same as RP K-means, but here the projection and clustering is applied in many iterations. The projection dimension is increased in each iteration. The
clusters in the previous iterative dimension are the base for initializing the clusters
in the present dimension.
The algorithm randomly selects K points from the input data set X and initializes
as cluster membership G. The algorithm starts in dimension D1 , Initial centroids are
the randomly selected K points. The input data X is projected into a D1 (D1 < d)
138
R. Pasunuri et al.
dimension space by random projection P1 , obtaining X R P1 . K-means clustering is
performed in X R P1 to get the new cluster membership G, and this G will become the
basis for next dimension (D2 ) for initilizaing K-means. We recalculate the centroids
now in dimension D2 (D1 ≤ D2 < d) by using the cluster membership G obtained
from K-means in dimension D1 and X R P2 to obtain the new initial centroids C R P2 ,
in a new D2 dimensional space. Now in D2 , we perform K-means clustering again
using C R P2 as initialization. This process is repeated until the last Dl (D1 ≤ D2 ≤
· · · ≤ Dl < d) is reached, returning the cluster membership from Dl . This algorithm
is based on a heuristic relation D1 ≤ D2 ≤ · · · ≤ Dl < d which is analogous to
simulated annealing cooling. The procedure is presented in Algorithm 2.
Algorithm 2 Iterative RP K-means
Input: list of dimensions Da = 1, 2, 3, ..., l, Data Set X N ×d , No. of clusters K
Output: G which is cluster membership.
begin
1: Select K random points from X and assign as G.
2: for Da = 1 to l do
3: Define Pa (d × Da ) (random matrix)
4: Set X R Pa (N × Da ) = X Pa
5: Set C R Pa (K × Da ) by finding the mean of each cluster in X R Pa according to G.
6: Apply K-means on X R Pa with C R Pa as initialization to get G.
7: end for
8: return G
4 Proposed Variant of IRP-Kmeans
The proposed variation is based on Iterative dimension reduction using random projections K-means(Algorithm 2) but instead of gradually increasing the dimension,
we decrease the dimension from the high-dimension to low-dimension in the random
projection part of the algorithm. Similar to IRP-Kmeans, we try to capture the solution
constructed in one iteration and use it in subsequent iteration. In this way, it transfers
the characteristics of previous generation to following generation. In our experiment,
we ran our method for the reduced dimensions from the list (d, d/2, d/4, d/8).
5 Experimental Study
The performance analysis is done on five high-dimensional data sets, two image (AT
& T, Yale), three micro array (also called gene expression data) data sets, which are:
GCM, Leukemia and Lung.
Ascending and Descending Order of Random …
139
Algorithm 3 Proposed Variant
Input: list of dimensions D = (d/2, d/4, d/8),
Data Set X N ×d ,
No. of clusters K
Output: G which is cluster membership.
begin
1: Select K random points from X and assign as G.
2: Set Da = d/2
3: Set a random matrix Pa (d × Da )
4: Set X R Pa (N × Da ) = X Pa
5: Set C R Pa (K × Da ) by finding the mean of each cluster in X R Pa according to G.
6: If Da < d/8
7: Da = Da /2
8: and Goto STEP 3
9: Apply K-means on X R Pa with C R Pa as initialization to get G.
10: return G
Table 1 Specifications of data sets
Data set
No. of samples
AT&T faces (ORL)
Yale
GCM
Leukemia
Lung
400
165
280
72
181
No. of features
No. of classes
10304
1024
16063
7129
12533
40
15
2
2
2
The mean squared error (MSE) which is the objective function of K-means clustering is taken as a measure to report the performance of the proposed method.
5.1 Data Sets
In this study, we considered five high-dimensional data sets to evaluate the performance of the proposed variation of IRP-K-means algorithm. A detailed specifications
of the data sets are present in Table 1. AT & T Database of Faces (formerly ORL
Database) consists a total of 400 images of 40 different persons. Global Cancer
Map (GCM) data set consists of 190 tumor samples and 90 normal tissue samples.
Leukemia data set contains 72 samples of two types: 25 acute lymphoblastic leukemia
(ALL) and 47 acute myeloid leukemia (AML). Lung cancer is a gene expression data
which contains 181 samples, which are classified into malignant pleural mesothelioma (MPM) and adenocarcinoma (ADCA). Yale data set contains 165 face images
of 15 persons and 11 images per person, with a dimensionality of 1024.
140
R. Pasunuri et al.
Table 2 MSE for several datasets
Data set
D
AT&T faces (ORL)
Yale
GCM
Leukemia
Lung
221
166
234
212
226
IRP-Kmeans
(Classical normal
matrix)
IRP-Kmeans
(Achlioptas random
matrix)
7.8850 × 108
1.2311 × 108
4.5467 × 1011
4.1263 × 1011
10.88 × 1010
8.1216 × 108
1.459 × 108
4.9832 × 1011
4.1620 × 1011
4.43 × 1010
Sample average over 20 runs
Table 3 MSE for several datasets
S.No.
Data sets
1
2
3
4
5
AT&T faces
(ORL)
Yale
GCM
Leukemia
Lung
RP
IRP
Proposed (VIRP)
8.53 × 109
19.134 × 109
9.2759 × 109
1.61 × 108
1.20 × 1013
4.17 × 1012
1.19 × 1012
3.45 × 108
1.551 × 1013
7.467 × 1012
13.3 × 1012
1.6363 × 108
0.74438 × 1013
3.6702 × 1012
1.309 × 1012
Sample average 20 runs
5.2 Results and Discussion
The system configuration used to perform the experiments is: 4 GB RAM, Intel i5third generation processor. By implementing the Theorem 1, we have calculated the
bound for the data sets that are considered for experimentation. The value is fixed at
0.99 in all the experiments. The MSE for several data sets with the implementation of
Angelo et al. [2] and by using Achlioptas Random matrix (our own implementation),
we got almost similar results, except for the Lung data set with a difference of 101
times in the MSE for AT & T Faces, Lung and GCM data sets. These results are
presented in Table 2.
The average MSE over 20 runs for the proposed variant along with three other
methods is shown in Table 3. From this, it is evident that the proposed variant outperforms the IRP-K-means method on the given five high-dimensional data sets. When
compared with RP-K-means, the performance of the proposed one is almost same
for all the data sets considered except GCM. The performance of VIRP is doubled
for GCM data set when compared with RP-Kmeans Algorithm. The performance of
VIRP is 6 times improved when GCM data set is considered. The performance of the
proposed VIRP method is double as that of IRP method on the first four data sets,
and it is 10 times improved for the Lung data set.
Ascending and Descending Order of Random …
141
6 Conclusion and Future Directions
In this paper, we have proposed a variant for IRP K-means algorithm by gradually
decreasing dimension in the iteration there by preserving the inter-point distances
efficiently. This can be confirmed by the empirical results produced above. Our
method is compared with the Single Random Projection (RP), IRP K-means (IRP)
methods. Compared to these two methods, our proposed method is giving best results
for the given high-dimensional data sets. The future course of work may involve
using any dimensionality reduction technique generate the random matrix and verify
if the method saves the inter point distances. And also to comparative analysis of the
proposed method with some standard clustering algorithms.
Acknowledgements The first author would like to thank Dr.Angelo Cardoso for providing the
IRP-Kmeans code.
References
1. Cardoso, A., Wichert, A.: Iterative random projections for high-dimensional data clustering.
Pattern Recogn. Lett. 33, 1749–1755 (2012)
2. Lloyd, S.: Least squares quantization in pcm. Inf. Theory IEEE Trans. 28, 129–137 (1982)
3. Johnson, W., Lindenstrauss, J.: Extensions of lipschitz mappings into a hilbert space. Contemp.
Math. 26, 189–206 (1984)
4. Fradkin D., Madigan D.: Experiments with random projections for machine learning. In: KDD
’03: Proceedings of the ninth ACM SIGKDD International Conference on Knowledge Discovery and Data mining (2003)
5. Bingham E., Mannila H.: Random projection in dimensionality reduction: applications to image
and text data. In: KDD ’01: Proceedings of the seventh ACM SIGKDD International Conference
on Knowledge Discovery and Data mining (2001)
6. Fern, X.Z., Brodley, C.E.: Random projection for high dimensional data clustering: a cluster
ensemble approach. In: Proceedings of the Twentieth International Conference of Machine
Learning (2003)
7. Deegalla S., Bostrom H.: Reducing high-dimensional data by principal component analysis vs.
random projection for nearest neighbor classification. In: Proceedings of the 5th International
Conference on Machine Learning and Applications (ICMLA), Fl, pp. 245–250 (2006)
8. Jain, A.K.: Data clustering: 50 years beyond K-means. Pattern Recogn. 31, 651–666 (2010)
9. Alshamiri, A.K., Singh, A., Surampudi, B.R.: A novel ELM K-means algorithm for clustering. In: Proceedings of 5th International Conference on Swarm, Evolutionary and Memetic
Computing (SEMCO), pp. 212–222. Bhubaneswar, India (2014)
10. Dasgupta, S., Gupta, A.: An elementary proof of a theorem of Johnson and Lindenstrauss.
Random Struct. Algorithms 22, 60–65 (2003)
11. Achlioptas D.: Database-friendly random projections: Johnson-lindenstrauss with binary coins.
J. Comput. Syst. Sci. 66, pp. 671–687. Special Issue on PODS 2001
12. Li P., Hastie T.J., Church K.W.: Vary sparse random projections. In: Proceedings of the 12th
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 287–
296. ACM, New York, NY, USA (2006)
13. Hecht-Nielsen R.: Context vectors: general purpose approximate meaning representations selforganized from raw data. Computational Intelligence: Imitating Life, pp. 43–56 (1994)
142
R. Pasunuri et al.
14. Papadimitriou, C.H., Raghavan, P., Tamaki, H., Vempala, S.: Latent semantic indexing: a probabilistic analysis. In: Proceedings of 17th ACM Symposium on the Principles of Database
Systems, pp. 159–168 (1998)
15. Boustsidis, C., Zouzias, A., Drineas, P.: Random projections for k-means clustering. Adv.
Neural Inf. Proc. Syst. 23, 298–306 (2010)
16. Dasgupta S.: Experiments with random projection. In: Uncertainity in Artificial Intelligence:
Proceedings of the Sixteenth Conference (UAI-2000), pp. 143–151 (2000)
17. Selim, S.Z., Alsultan, K.: A simulated annealing algorithm for the clsutering problem. Pattern
Recogn. 24, 1003–1008 (1991)
18. Megan A.: Dimensionality reductions that preserve volumes and distance to affine spaces, and
their algorithmic applications. Randomization and Approximation Techniques in Computer
Science. Springer. volume 2483 of Lecture Notes in Computer Science, pp. 239–253 (2002)
Speed Control of the Sensorless BLDC
Motor Drive Through Different
Controllers
Vikas Verma, Nidhi Singh Pal and Bhavnesh Kumar
Abstract Nowadays Brushless DC motors (BLDC) are gaining popularity and are
also replacing the motor with brushes in numerous applications due to their high efficiency, low maintenance, and effective operation. This paper presents the sensorless
speed control of the BLDC drive with the technique of zero-crossing detection of
indirect back EMF. Several controllers are employed and compared for acquiring the
effective control over the speed. The particular paper demonstrates the performances
of sensorless BLDC drive has been evaluated with different controller schemes such
as a conventional controller (PI), anti-windup PI, Fuzzy based and the Hybrid (FuzzyPI) controller at different load and speed. Their results have been compared in which
fuzzy based controller offers a better response in maximum cases. This reduces
the cost and complexity without compromising the performance. Fuzzy Logic Controller is used to enhance its robustness and reliability. The effectiveness of the work
is demonstrated through simulation done in MATLAB Version (2013) environment
and simulation results of the sensorless drive have been analyzed.
Keywords BLDC motor · Back EMF sensing · Sensorless drive
PI · Anti-windup-PI · Fuzzy logic · Hybrid (Fuzzy-PI)
1 Introduction
In industrial as well as consumer applications Brushless DC (BLDC) motors are
mostly used because it is efficient as well as reliable and also requires less maintenance and salient in operation [1, 2]. In recent times, there is high demand for
V. Verma (B) · N. S. Pal
Gautam Buddha University, Greater Noida, India
e-mail: vverma.vikas.verma@gmail.com
N. S. Pal
e-mail: nidhi@gbu.ac.in
B. Kumar
Netaji Shubhash Institute of Technology, Delhi, India
e-mail: kumar_bhavnesh@yahoo.co.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_15
143
144
V. Verma et al.
this type of permanent magnet drives. BLDC motor is a permanent magnet synchronous motor. In case of BLDC motor, the electromagnet does not move as it
is replaced by the permanent magnet which rotates and the armature is at rest [2].
The electronic commutators are employed for successful commutation of current to
the armature but information of the rotor position is required for this. The position
information is mainly obtained with position sensors, which do not perform well at
high-temperature applications [3]. Due to the sensor failure at high temperatures the
system becomes unstable. Another drawback is related to the traditional controllers
which are generally used by industries in large numbers. These controllers are simple
in structure and easy to implement but their performances are affected by the load
disturbances, nonlinearity and in conditions like variations in parameters. The problems like rollover which is related to saturation effect also compel to develop new
schemes to attain the better control on speed. For this the sensorless BLDC drive the
traditional controllers are being accomplished with the intelligent controllers such
as fuzzy controllers to optimize the system performances [4, 5]. Also, to reduce the
saturation effect, anti-windup scheme instead of the traditional scheme is used. In
recent times, different control algorithms are being incorporated with conventional
controllers to achieve better control. Nowadays, there is wide use of hybrid controllers [6]. In this, a combination of both traditional PI as well as Fuzzy controllers
acts accordingly. One reduces the error and disturbances due to load variation and
another one minimizes error due to large changes in input.
This paper develops a sensorless speed control of BLDC motor drive based on
indirect back EMF zero-crossing detection with intelligent techniques such as a fuzzy
logic controller. This also overcomes all the drawbacks related to the sensor drive
along with the use of conventional controllers. The sensorless drive is reliable and has
good tracking capability over a wide range of speed. On the other hand, it is efficient
and effective in cost aspects also which makes this proposed system economical.
2 Sensorless Operation of BLDC Motor Drive
The permanent magnet BLDC motors are controlled electronically, for this it requires
the information of rotor position to achieve proper commutation of current in the stator winding. For information of rotor position, hall sensors are used. But the use of
hall sensors is not desirable because it increases the cost, complexity in structure and
also the sensor fails at high-temperature applications. For achieving the sensorless
operation there are various methods of sensorless control such as third harmonic voltage integration method, Flux estimation method, observer-based technique, detection
of freewheeling diode conduction, back EMF sensing techniques [7–9]. Among all
methods, the most efficient is the back EMF sensing method for the proper commutation sequence in motors and it also estimates and gives the information of the rotor
position to rotate with synchronized phases [10, 11]. In sensorless drive scheme, only
electrical dimensions are being used. At the condition of standstill, the back EMF
value is zero as it is proportional to rotor’s speed. This operation shows limitation
Speed Control of the Sensorless BLDC Motor Drive …
145
Fig. 1 Simulink model of proposed sensorless speed control scheme of BLDCM drive
when the speed is low or zero. To tackle this a strategy is adopted which is based on
starting the drive in an open loop [12]. BLDC motor has magnetic saturation characteristics so with the change in inductance variation the current value also changes
this helps in determining the initial rotor’s position. The starting of this drive is done
with open-loop strategy and after that transferred to the sensorless mode [13, 14].
3 Proposed Sensorless Speed Control Scheme of BLDCM
Drive
The designed sensorless model along with different controllers is shown in Fig. 1.
3.1 Speed Control of Sensorless BLDC Motor Drive
The generated signals from the above sensing scheme are implied into different
controllers and all the output working response of this sensorless drive with the
conventional, anti-windup scheme, intelligent techniques such as fuzzy logic controllers and the hybrid (combination of conventional and fuzzy) controllers are being
analyzed and compared.
PI Controller
The structure of the traditional PI is shown in Fig. 2. It is a most common controller,
which is prominently used in industries. The values of this controller are obtained
through Ziegler Nicholas’s tuning method as shown in Table 1.
146
V. Verma et al.
Fig. 2 Traditional PI controller structure
Table 1 PI controller gain values
Controller
Kp
PI
0.033
Ki
10.61
Fig. 3 Anti-windup-PI controller structure
Anti-windup-PI controller
The performance is degraded because of the effect, which occurs from the rollover
action of the traditional PI controller. In the case of a conventional PI controller, this
problem arises only because of the saturation effect. This happens because of the large
input value of error or due to the constant input value to the integrator. So to remove
this drawback, the input value resulting from the difference of the unsaturated and
saturated output is given to the integrator. This improves the output performance. For
this, there is a modification in the conventional controller and named as anti-windup
as shown in Fig. 3.
Fuzzy Logic-Based Controller
Fuzzy logic controllers involve the control logic which is based on the approach of
the control with a linguistic variable. Figure 4 shows the steps involved in it. Fuzzy
logic involves fuzzification, inference system, and defuzzification. The Fuzzy logic
controller (FLC) is designed by using the Fuzzy Toolbox in MATLAB. In our work
the logic preferred is of Mamdani type. Change in error of speed (ce) and error of
speed (e) are the two inputs for this particular controller. The functions taken are in
triangular membership. There are totally 49 (7 * 7) rules being developed in the rule
block.
Hybrid (Fuzzy-PI) Controller The specified Fuzzy-PI controller is the type of a
hybrid controller that utilizes both PI and Fuzzy Logic Controllers, which provide
Speed Control of the Sensorless BLDC Motor Drive …
147
Fig. 4 Basic fuzzy logic controller structure
Fig. 5 Hybrid (Fuzzy-PI) controller structure
the best response in nonlinearity. Both the controllers also give the good response
during speed tracking for steady state. The hybrid structure is shown in Fig. 5. By
combining both the controllers the error and overshoot are minimized as well as it
also gives the fast output response of the system. For the hybrid (Fuzzy-PI) algorithm,
the structures are designed in such a way so that switching can occur smoothly. The
designing is done in such a manner so that the utilization of both the controllers
can be acquired by smooth switching between the lower speed and the higher speed
gains.
4 Simulation Results and Discussion
The different control strategies are applied to sensorless BLDC motor drive, which
is verified through the simulation with all the standard specifications. This sensorless
148
V. Verma et al.
Fig. 6 Speed curve at 3000 rpm (fixed speed) under no load
Fig. 7 Speed curve with 2 Nm at fixed speed 3000 rpm
drive is simulated by using different types of controllers designed for this specific
work. The output curves obtained from this sensorless BLDC drive has been compared and evaluated under the fixed speed as well as the variable speed with different
loading conditions. All these conditions are described in the figures below.
Figure 6 shows the output speed curves of four different controllers which are
hybrid (Fuzzy-PI) controller, Fuzzy controller, anti-windup-PI and the traditional
PI controller is observed at 3000 rpm (fixed speed) under the condition of no load.
The output response curves ensure that the particular Fuzzy-PI controllers is fast in
comparison with the other controllers and settling time is 0.016 s, second, the antiwindup-PI also shows fast response and less peak overshoot than a conventional PI
controller.
Figure 7 shows the output speed curves of all the four controllers used for the
condition, when the motor is rotating at 3000 rpm under the 2 Nm loading condition.
As Fuzzy-PI controller takes 0.012 s for settling, which is very less than another
controller.
Figure 8 shows the performance curves of the different controllers with the speed
change of 3000–1500 rpm in 0.5 s for the condition of no load. Particularly, Fuzzy-PI
controller gives a very fast response in comparison with others and the time taken
for settling is 0.010 s.
Speed Control of the Sensorless BLDC Motor Drive …
149
Fig. 8 Speed curve at change in speed 3000–1500 rpm in 0.5 s without any load
Fig. 9 Speed curve at changing speed 3000–1500 rpm with 2 Nm load
Figure 9 shows the performance curves of the different controllers under a fixed
load of (2 Nm) with the changing speed of 3000 rpm–1500 in 0.5 s. As under loading
condition also Fuzzy-PI controller rises fast and has minimum peak overshoot and
settling time of this controller is also less than the conventional controllers.
Figure 10 shows the output performance curves of the different controllers at the
variable changes in speed that is from 3000 rpm to 1000 rpm in 0.5 s then from
1000 rpm to 3000 in 1 s under the condition of no load. The output response shows
the reliability and tracking capability of the system. In which the Fuzzy-PI controllers
is much faster than the other controllers.
Figure 11 shows the performance curves of the different controllers at the fixed
speed of 3000 rpm under the condition of load variation from 1 to 4 Nm in 0.5 s.
As with changing load the time of settling is better in case of Fuzzy-PI. In this case,
even the anti-windup-PI also takes nearly same time as of PI controller.
For the evaluation of the output performance of all the four controllers, which is
employed in the sensorless speed control of BLDC drive is also being compared in
respect of (t r ) rise time, (t s ) settling time, (%M p ) peak overshoot is shown in Table 2.
0.056
0.053
0.052
0.051
0.54
0.036
0.039
0.036
0.033
0.034
3000 rpm with 2 Nm
load
3000–2000 rpm at no
load
3000–2000 rpm with
2 Nm load
Variable speed with no
load
3000 rpm with load
change of 1–4 Nm in
0.5 s
%mp
1.6
1.4
1.8
2.5
2.9
0.021
0.027
0.025
0.028
0.026
0.029
0.51
0.041
0.045
0.040
0.042
0.049
ts
tr
0.059
2.7
tr
0.037
ts
Anti-windup-PI
Controllers
PI
3000 rpm at no load
Parameters
Table 2 Performance comparison of different controllers
%mp
1.9
1.6
1.7
1.4
1.2
1.5
0.012
0.015
0.014
0.016
0.014
0.019
tr
Fuzzy
ts
0.021
0.023
0.020
0.021
0.022
0.024
0.01
0.04
0.05
0.03
0.11
0.02
%mp
0.006
0.004
0.003
0.005
0.007
0.009
tr
Hybrid (fuzzy-PI)
ts
0.011
0.012
0.011
0.010
0.012
0.016
%mp
0.000
0.001
0.003
0.002
0.001
0.003
150
V. Verma et al.
Speed Control of the Sensorless BLDC Motor Drive …
151
Fig. 10 Speed curve at variable speed changing: 3000–2000 rpm in 0.5 s and 3000 rpm at 1 s with
no load
Fig. 11 Speed curve at fixed speed of 3000 rpm with change in load from 1 to 4 Nm at 0.5 s
5 Conclusion
In this paper, the sensorless speed control of three-phase BLDC motor with different
types of Intelligent and conventional controllers based on the sensorless technique
of back EMF sensing have been simulated using the MATLAB Version (2013) and
their performance is observed. The simulation results show and depict output performances for conventional PI, anti-windup-PI, Fuzzy logic and hybrid (Fuzzy-PI)
controllers. Their performance is compared in the respect of time taken to rise, the
time taken to settle down and percentage of peak overshoot at a fixed speed as well
as at variable speed with different loading conditions. The results obtained from the
simulation shows that Fuzzy-PI shows the best performance among all controllers.
Both Fuzzy as well as Fuzzy-PI shows better results than conventional controllers.
Even anti-windup-PI controller also shows fast response and minimum peak overshoot than a conventional PI controller. The results of this designed model demonstrate that the system is cost-effective, reliable, and robust which makes it suitable
for robotics, fuel pumps, and the industrial automation-related applications.
152
V. Verma et al.
References
1. Miller J.E.: Brushless permanent magnet dc motor drives. Power Eng. J. 2(1) (1998)
2. Bose, B.K.: Modern Power Electronics and AC Drives. Pearson Education Asia (2002)
3. Kim, T., Lee, H.-W. Ehsani, M.: Position sensorless brushless DC motor drives: review and
future trends. IEEE, IET Electr. Power Appl. 1(4), 557–564 (2007)
4. Sriram, J., Sureshkumar, K.: Speed control Of BLDC motor using fuzzy logic controller based
on sensorless technique. In: IEEE International Conference on Green Computing Communication and Electrical Engineering (2014)
5. Anjali, A.R.: Control of three phase bldc motor using fuzzy logic controller. Int. J. Eng. Res.
Technol. 2 (2013) ISSN: 2278-0181
6. Abidin, M.F.Z., Ishak, D., Hasan, A.H.A.: A comparative study of PI, fuzzy and hybrid PI fuzzy
controller for speed control of BLDC motor drives. In: Proceedings of the IEEE International
Conference in Computer Application and Industrial Electronics Application, pp. 189–195.
Malaysia (2011)
7. Shao, J., Nolan, D., Hopkins, T.: A novel direct back EMF detection for sensorless brushless DC
motor drives. In: IEEE Conference on Applied Power Electronics Conference and Exposition
Seventeenth Annual, vol. 1 (2002)
8. Mathew, T., Sam, C.A.: Closed loop control of BLDC motor using fuzzy logic and single current sensor. In: IEEE International Conference on Advanced Computing And Communication
System, pp. 19–21 (2013)
9. Damordhan, P., Sandeep, R., Vasudevan, K.: Simple position sensorless starting method for
brushless DC motor. IET Electr. Power Appl. 2(1), 49–55 (2008)
10. Singh, S., Singh, S.: A control scheme for position sensorless PMBLDC motor from standstill
to rated speed. In: IEEE International Conference on Control, Instrumentation, Energy and
Communication (2014)
11. Somantham, R., Prasad, P.V., Rajkumar, A.D.: Modelling and simulation of sensorless control
of PMBLDC motor using zero crossing back emf detection. In: IEEE Intenational Symposiyum
on Power Electronics, Drives, Automotive Motion (2006)
12. Lad, C.K., Chudamani, R.: Sensorless brushless DC motor drive based on commutation instants
derived from the line voltages and line voltage differences. In: IEEE Annual Indian Conference
(2013)
13. Damodharan, P., Vasudevan, K.: Line voltage based indirect back-emf zero crossing detection
of bldc motor for sensorless operation. Int. J. Power Energy Syst. 28 (2008)
14. Damordhan, P., Vasudevan, K.: Sensorless brushless DC motor drive based on the zero crossing
detection of back EMF from the line voltage difference. IEEE Trans. Energy Conv. 25(3),
661–668 (2010)
Urban Drainage System Design
Minimizing System Cost Constrained
to Failure Depth and Duration Under
Flooding Events
Soon Ho Kwon , Donghwi Jung
and Joong Hoon Kim
Abstract Recently, property damages and loss of life caused by natural disasters
are increasing in urban area because of local torrential rainfall, which is mostly originated from recent global climate change. Acceleration of population concentration
and increase of impervious area from urbanization worsen the situation. Therefore,
it is highly important to consider system resilience which is the system’s ability
to prepare, react, and recover from a failure (e.g., flooding). This study proposes a
resilience-constrained optimal design model of urban drainage network, which minimizes total system cost while satisfying predefined failure depth and duration (i.e.,
resilience measures). Optimal layout and pipe sizes are identified by the proposed
model comprised of Harmony Search Algorithm (HSA) for optimization and Storm
Water Management Model (SWMM) for dynamic hydrology-hydraulic simulation.
The proposed model is applied to the design of Gasan urban drainage system in
Seoul, Korea, and the resilience-based design obtained is compared to the least-cost
design obtained with no constraint on the resilience measures.
Keywords Urban drainage system (UDS) · Resilience · Harmony search
S. H. Kwon
Department of Civil, Environmental and Architectural Engineering, Korea University,
Seoul, South Korea
D. Jung
Research Center for Disaster Prevention Science and Technology, Korea University,
Seoul, South Korea
J. H. Kim (B)
School of Civil, Environmental and Architectural Engineering, Korea University,
Seoul 136-713, South Korea
e-mail: jaykim@korea.ac.kr
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_16
153
154
S. H. Kwon et al.
1 Introduction
Urban Drainage System (UDS) is an urban water infrastructure to carry wastewater
and rainwater to the outlet of an urban basin from which they are either treated or
discharged to a river. UDS consists of various components such as drainage pipe,
detention reservoir, and pump station. Drainage pipe delivers rainwater entering into
manholes to the downstream pipes whereas detention reservoir stores the delivered
rainwater for reducing the peak discharge in downstream. Pump station expels the
stored rainwater from low to high elevation against the gravity finally to riverside land.
Therefore, determining the size and capacity of these components is an important
task for securing cost-effective functionality of a UDS.
Previous studies on the optimal design of UDS drain pipes are classified into two
groups: one that determines pipe sizes only with fixed pipe layout and the other that
optimizes both sizes and layout. [1, 2] have developed a model that minimizes total
system cost by considering both slopes of pipes and sizes in the sewer system. Other
studies have developed separate algorithms for layout generation and pipe sizing for
UDS [3, 4]. However, few efforts have been devoted to maximizing system resilience,
especially with respect to failure depth and recovery time.
In this study, we introduce a resilience-constrained UDS optimal design model
that determines both layout and pipe sizes to minimize total system cost satisfying
a predefined level of resilience. The maximum nodal flooding volume is used as
a failure depth indicator and considered in a constraint. The proposed resilienceconstrained optimal design model is demonstrated in the Gasan urban drainage network in Seoul, Korea. The optimal results obtained under different levels of failure
depths were compared with respect to total system cost and resulting total flooding
volume.
2 Resilience-Constrained Optimal Design Model
The proposed UDS design model minimizes total system cost satisfying a constraint
on the level of failure depth as follows in (1):
Minimize F N
i1
Ci (Di ) × L i +
N
Pj
(1)
j1
where C i (Di ) is the unit cost of the pipe I which is a function of Di ($); L i is the
length of the pipe (m); Di is the diameter of conduit (m); Pj is the penalty cost ($);
N is the total number of conduits in UDS. The penalty cost was calculated based on
the total flooding volume of UDS.
In addition, this study is calculated the objective function by considering the
constraints as follows in (3)–(5):
Urban Drainage System Design Minimizing System Cost Constrained …
155
di Di + 0.5(m)
(3)
failure depth < 80% × MAXMAXF
(4)
failure depth < 90% × MAXMAXF
(5)
where d i is the burial depth at each node (0.5 m is added considering the freezing
depth).
In this study, the level of failure depth is defined as the maximum value of each
time interval’s maximum nodal flooding volumes (MAXMAXF). The base level
of failure depth is obtained from the MAXMAXF for the least-cost design. The
proposed model with two different levels of MAXMAXF, i.e., 80 and 90% of the
base MAXMAXF is applied independently to the study network.
3 Study Area
The Gasan sewer network in Seoul, Korea is as shown in Fig. 1. The study network
consists of 32 pipes, 32 nodes, and sub-catchments. A pumping station is located at the
outlet of the sewer network for expelling the collected rainwater to the mainstream.
There are five pumps in the pumping station, the first to the third pumps have the
identical capacity of 100 m3 /min. The fourth and the fifth pumps have the capacity
of 170 m3 /min. The first and second pumps turn on when the water depth in the
front detention reservoir reaches at 0.6 and 0.8 m, respectively, and the third and fifth
pumps turn on at a water depth of 1 m.
4 Application Results
Table 1 indicates the unit cost of pipes. In this study, the HSA is used to the design
of minimizing system cost and to reduce flooding volume in each node for UDS.
Applied parameters on this model are HMCR 0.8, PAR 0.05, and number of
iterations 100,000. The least design cost based on proposed resilience-constrained
optimal design model is calculated by considering constraint (see Table 2). Table 2
obtained under a different level of failure depths were compared with respect to least
design cost and resulting total flooding volume. The results show that as the level of
failure depth decrease, the total flooding volume is decreased. In addition, the pipe
sizes are set larger because the total flooding volume decrease, the total system cost
increased.
156
S. H. Kwon et al.
Fig. 1 The schematic of the sewer network
5 Conclusions
This study proposes a method to apply the disaster response and management to
prepare the damage and mitigate the property losses. The design of minimizing
system cost in urban drainage system by integrating harmony search algorithm and
stormwater management model was presented. The level of failure depth based on
resilience-constrained optimal design model was calculated by considering least
design cost and total flooding volume. The results of both return periods show that as
the design cost increase, the total flooding volume decrease. Further research could
be compared different flood damages with their corresponded design system costs
by considering the importance of the buildings regarding their domestic of industrial
application.
Urban Drainage System Design Minimizing System Cost Constrained …
157
Table 1 The unit cost of pipes
Pipe size (m)
Unit cost ($/m)
0.25
0.30
0.35
0.40
0.45
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
239.20
246.68
270.04
281.11
304.86
339.10
400.25
488.14
552.89
634.52
738.17
834.09
943.18
Table 2 The result of least design cost and total flooding volume
Level of failure depth (90%)
Level of failure depth (80%)
Total system cost Total flooding
($)
volume (m3 )
50-year
6,158,821
frequency design
rainfall
100-yr frequency 6,161,564
design rainfall
Total system cost Total flooding
($)
volume (m3 )
21.181
6,177,816
15.251
22.840
6,187,743
17.047
Acknowledgements This research was supported by a grant (13AWMP-B066744-01) from the
Advanced Water Management Research Program funded by the Ministry of Land, Infrastructure,
and Transport of the Korean government.
References
1. Mays, L.W., Yen, B.C.: Optimal cost design of branched sewer systems. Water Resour. Res.
11(1), 37–47 (1975)
2. Mays, L.W., Wenzel, H.G.: A serial DDDP approach for optimal design of multi-level branching
storm sewer systems. Water Resour. Res. 12(5), 913–917 (1976)
158
S. H. Kwon et al.
3. Lui, G., Matthew, R.G.S.: New approach for optimization of urban drainage systems. J. Environ.
Eng. 116(5), 927–944 (1990)
4. Tekeli, S., Belkaya, H.: Computerized layout generation for sanitary sewers. J. Water Resour.
Planning Manage. 112(4), 500–515 (1986)
Analysis of Energy Storage for Hybrid
System Using FLC
Ayush Kumar Singh, Aakash Kumar and Nidhi Singh Pal
Abstract In this paper hybrid renewable energy resources (HRES) composed of PV,
wind, and batteries as storage units use a fuzzy logic technique to control the energy
between load demand and generation. The control technique using a fuzzy logic
controller is simulated on MATLAB, which balances the suitable power management
between intermittent energy generation by renewable sources and loads.
Keywords PV · WECS · Hybrid energy system · Fuzzy · Battery power
management
1 Introduction
Renewable energy resources (RES) such as solar, wind energy, etc., are a hopeful
option for future power generation as they are freely available and environmental
friendly. Hybrid solar PV-Wind system is an efficient resource to supply power to the
grid or an isolated load [1]. A wind turbine converts kinetic energy into mechanical
energy and further generates AC power by the generator. Solar PV modules that
convert sun energy into DC power. Use of conventional resources is not for multiple
challenges. Renewable energy is the only solution to such energy challenges [2,
3]. The major drawback of this energy is that they are nature dependent so due
to intermittence, uncertainty, and low availability of nature which makes system
A. K. Singh (B) · N. S. Pal
Department of Electrical Engineering, Gautam Buddha University,
Greater Noida, India
e-mail: ayush.singh.sm@gmail.com
N. S. Pal
e-mail: nidhi@gbu.ac.in
A. Kumar
Energy Conservation Services Jeevanam Water Technologies Maharashtra,
Pune, India
e-mail: aakashdodwal@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_17
159
160
A. K. Singh et al.
Fig. 1 Block diagram of hybrid PV-wind with battery storage system [4]
unreliable. Therefore, a hybrid topology is used to overcome intermittency and other
issues of RES and make the system more reliable [4, 5].
The basic block diagram of hybrid PV-Wind with battery storage system is shown
in Fig. 1. In this paper, the study of two renewable energy sources: PV generator
and a PMSG-based wind turbine as the primary sources and a battery storage system
as the secondary source is implemented to overcome the fluctuations from PV and
wind turbine. The intelligent control-based fuzzy is implemented to control the flow
of power between generation and load side.
2 Hybrid PV-Wind System with Battery Storage System
The hybrid system consists of renewable resources such as wind and solar PV system
and battery storage system to fullfill the desired demand. These types of systems are
not always connected to the grid, but it makes also the solution for the standalone
system. By combining the two renewable sources, solar and wind get better reliability
and is economical to run.
The PV system consists of the solar cells. Whenever a photovoltaic cell is exposed
to the sun, since it is a semiconductor material, it absorbs solar energy and converts
it into electrical energy. The basic PV cell equivalent circuit contains Rs as the
equivalent series resistance and Rp as the equivalent parallel resistance of the PV
array. PV generator is used as a renewable energy system and connected to the inverter
through DC/DC boost converter. The relationship between output voltage V PV and
load current I L can be expressed as [6].
In this paper, the wind turbine is the permanent magnet synchronous generator
(PMSG) type. The kinetic energy of wind is converted into mechanical energy with
Analysis of Energy Storage for Hybrid System Using FLC
161
the help of wind turbine (WT). Study of torque and power characteristics produced
by WT at various wind speeds and other parameter variations due to variation in
wind speed can be done as given in [2, 7]. PMSG has been used as a generator in
this paper. The power Pwind extracted from the wind is [8].
There is a need for storage device with renewable energy with the help of which the
fluctuations in renewable energy can be compensated. If excess energy is produced by
renewable sources then the battery will store the excess energy. Whenever renewable
energy is not enough to satisfy the load then the battery will provide the energy to
meet the load demand. The battery is mostly used for long-term storage. It has fixed
maximum capacity and voltage and current ratings are provided by the manufacturer.
The most important parameter is state of charge (SOC). SOC refers to the percentage
charge present in the battery. Battery calculation is taken from [9].
SOC 100% means battery is completely charged. SOC 0% means battery is
completely discharged.
To avoid operations like undercharging and overcharging, they should not be
completely discharged or overcharged. For this reason, it is necessary to determine
the maximum depth of discharge. Generally, the depth mainly varies from 30 to 80%
[10]. A good intermediate value is 50% which means that only half of the capacity
of the battery will be used.
3 Control Scheme
The operating conditions of PV on which output of PV depends are irradiance and
temperature. At a different value of irradiance and temperature, the output of PV
will be different. For the productive operation, PV system must work at maximum
power point (MPP) of P–V curve. Different types of MPPT techniques have been
introduced [11]. The incremental conductance (IC) algorithm has been carried in this
paper. The PV MPPT senses current I PV and voltage V PV of PV and according to
that change in duty cycle, so that PV extracts the maximum power throughout the
day.
Due to fluctuating wind speed, the variation in frequency and amplitude of a
variable-speed PMSG makes it unfit for the proposed operation. Here, AC output of
WT is converted into DC voltage with the help of three-phase diode bridge rectifier.
To extract the maximum power of wind turbine at any wind speed, the duty cycle of
the switch of DC/DC boost converter should be controlled. To achieve the maximum
energy from the WT below rated wind speed, a variable-speed control technique
is introduced [7, 12]. At different wind speeds (V wind ), rotor speed (ωm ) of WT is
different and the corresponding mechanical power obtained is also different. The
mechanical power of WT is depended upon the rotor speed. To achieve maximum
power from wind turbine which is of variable speed nature, the rotor of the wind
turbine is operated at optimal speed using MPPT so that the maximum power from
the turbine can be obtained at below rated wind speed.
162
A. K. Singh et al.
A three-phase six switches, pulse width modulation (PWM) VSI has been implemented for the proposed HES model. A converter must act as a unidirectional power
controller between DC and load. In this control system, desired power transfer by
hybrid PV-wind system to load as they are generated [6, 13]. There is two type of
loops in control technique which is applied to inverter control. The first control loop
is internal which control the load current and second control loop is external which
control the DC link voltage. The main function of the internal control loop is to
maintain the power quality and external control loop is to maintain the power flow
in the system. MPPT is associated with WT to get optimal power at different speeds.
4 Power Management Strategy
An overall control strategy is needed in multisource energy system for power management strategy [14]. Pitch angle controller controls the WECS and maximum
power extracted by MPPT and with the help of the maximum power point tracker,
the output of PV is controlled. A battery is used to compensate the fluctuation and
full fill the power at load side. A fuzzy controller is used with the battery to control
the power. A bi-directional converter is also used to charge and discharge the battery.
With the help of fuzzy controller, power produced by hybrid PV-Wind system and
battery system is capable of transferring the desired power to load.
4.1 Battery Power Management System
The intelligent control system is necessary for the nonlinear system. The main purpose to introduce the intelligent control system is to keep away from the insufficient
operating time and to protect the battery storage system. The intelligent control system supply desired power to load and also help to compensate the fluctuating generation by hybrid sources. The algorithm applied to this intelligent control system
provides a better management for battery storage system. The fuzzy logic controller
has two inputs and one output [15]. According to the value of SOC, fuzzy logic controller decides the battery charging and discharging operation. The net output power
produced by hybrid PV-Wind and battery are calculated as
Pnet PPV + Pwind + Pbattery
(1)
PPV —PV Power; Pwind —WECS Power; Pbattery —Battery Power.
The control strategy is that at any time if power generated by PV and wind is
excess, then is used to charge the battery. Now the equation is
PPV + Pwind Pbattery + Pload
Pnet > 0
(2)
Analysis of Energy Storage for Hybrid System Using FLC
Table 1 Fuzzy rule table
e
NB
NS
Z
PS
PB
Table 2 PV module
parameters
163
PL
NB
NS
Z
PS
PB
NB
NM
NM
NM
PB
NB
NM
NM
NM
PB
NM
NS
ZE
PS
PM
NB
PM
PM
PM
PB
NB
PM
PM
PM
PB
Maximum power (Pmax )
9.5 KW
Voltage at MPP
29 V
Current at MPP
Open-circuit voltage
7.35 A
36.3 V
Short circuit current
7.84 V
When power generated by sources is less than load required by load side then the
battery power is used to compensate the deficit power and fullfill the load side.
PPV + Pwind + Pbattery Pload
Pnet < 0
(3)
The fuzzy logic controller decides the charging and discharging operation of the
battery, which depends on the SOC.
A 5 * 5 rule base used in fuzzy controller is given in Table 1. Two inputs are load
power (PL ) and error (e) between power generated by PV-Wind system and load
power. Output is state of charge (SOC).
5 Simulation Results
The simulated responses of the implemented hybrid energy system with battery
power management using MATLAB/Simulink are studied.
5.1 PV System
The surface temperature of PV is considered to be 25 °C and irradiation varies.
IC-based MPPT tracks and controls the constant voltage throughout the day and
varies according to irradiance. Thus maximum power is extracted at each irradiation.
In this system, a 9.5 KW of PV is simulated on MATLAB. The PV module parameters
are given in Table 2. Block diagram of PV system is given in Fig. 2. Combining PV
modules in various ways of series and parallel connection gives the 500 V.
164
A. K. Singh et al.
Fig. 2 Irradiance, output
power and voltage of PV
Fig. 3 Wind speed and rotor
speed of WT
Variation of irradiance of PV is shown in Fig. 2. From time t 0 to t 2 s
irradiance is 1000 w/m2 and time t 2 to t 6 s irradiance is 850 w/m2 . The output
voltage of PV was 500 V and it was boosted up to voltage of 640 V using boost
converter. The output power of PV is 9500 W from time t 0 to t 2 s at time t 2 s power is 8100 W. After 2 s, power reduces because radiation was decreased to
the level of 850 w/m2 .
5.2 Wind System
The block diagram of WT system is shown in Fig. 3. The speed of rotor changes with
the wind speed (V wind ). As speed of wind increases, rotor speed (ωm ) also increases
and corresponding output power of WT increases and vice versa. With increases in
the output power of WT, rectified current and voltage also increases. WT parameters
used are given in Table 3.
Speed of wind is variable in nature. From time t 0 to t 0.5, wind speed is
5 m/s and from time t 0.5 s to t 3 s wind speed is 12 m/s and from time t 3 s
to t 6 s wind speed is 9 m/s. There are variations in rotor speed as the wind speed
is changed. As wind speed is very low for time t 0–0.5 s so corresponding rotor
speed is also very less. At time t 1–3 s, if speed of wind is maximum then rotor
Analysis of Energy Storage for Hybrid System Using FLC
Table 3 PMSG based wt
parameters
165
Maximum power (Pmax)
8.5 KW
Rated wind speed
12 m/s
Rectified voltage at rated wind 500 V
speed
Rectified current at rated wind 11.8 A
speed
Fig. 4 WT rectified current
and voltage
Fig. 5 Wind power
speed also reaches the maximum speed. At t 3 s, when speed of wind decreases
the rotor speed also decreases as shown in Fig. 3.
The rectified output current and voltage also varied according to wind speed as
shown in Fig. 4. Initially, the voltage is less due to less wind speed. At time t 1–3 s,
speed of wind is maximum then voltage also reaches the maximum. At t 3 s, when
speed of wind decreases the voltage also decreases.
The output power of wind turbine also varies according to wind speed as shown
in Fig. 5. Initially, the power was zero but at t 1–3 s the wind speed reaches its
maximum speed then power also reaches the maximum rated power. Further, at t 3 s, when wind speed is decreased the power also decreases.
166
Table 4 Battery parameters
A. K. Singh et al.
Battery
Ni–MH
Voltage
300 V
Current
State of charge (%)
6.5 A
60
Fig. 6 SOC (%)
5.3 Battery System
The simulation of the battery consists of the battery with the bi-directional DC–DC
converter. The battery employed in this system is Ni–MH acid battery and parameters
are given in Table 4.
Initially, when irradiance is good but wind speed is very low than PV-battery
supply, the load and battery will start discharging as shown in Fig. 6. After t 1 s,
when enough power is generated from PV-wind then the battery will start charging.
At t 2 s, the power of PV reduces due to decrease in irradiance and hence a little
decrease in state of charge because of battery getting less current. Further at t 3 s,
wind speed decreases so output power of WT also decreases, hence battery gets very
low current from PV-wind system. Due to which rate of charge is almost constant.
At t 4–5 s, an extra load is added to the system.
PV-wind system is unable to satisfy the extra added load so at that time battery
compensates the deficit power and start discharging which is reflected by a sharp
decrease during t 4–5 s.
The variation in battery charging voltage and charging current under different
load conditions will be different. When the load is connected to the battery, battery
start discharging and current will be positive otherwise negative.
The power of battery that varied according to the power required to system is
shown in Fig. 7. Initially, load is satisfied by PV and battery. Further battery is
getting charge, hence charging condition is shown at time t 1–4 s. At time t 4–5 s an extra load is added to system and PV-Wind system is not satisfy the load.
Hence battery fed power to satisfy the load is shown in this duration.
Analysis of Energy Storage for Hybrid System Using FLC
167
Fig. 7 Battery power
Fig. 8 Battery power
management
5.4 Battery Power Management
The simulation output waveform of implemented system is based on the data provided
i.e. during the time interval of 0 < t < 0.5 s the wind speed is 5 m/sec and is increased
to 12 m/sec at t 0.5 s and again decrease to 9 m/s at t 3 s. The irradiation is
initially 1000 w/m2 for 0 < t < 2 s and at t 2 s reduces to 850 w/m2 . The demand on
the load side is of 10 KW throughout time in the system but an extra load of 4 KW is
added to the system during time t 4–5. Initially, PV generator and battery fed the
power to full fill the load side demand. At t 1 s the power produced by PV-Wind is
sufficient to full fill the load demand and the remaining power is used to charge the
battery. At t 2 s, the output of PV generator is decreased due to less irradiance and
WT produce maximum power hence PV-Wind system is capable to satisfy the load
and remaining power is used to charge the battery. At t 3 s the wind power also
decreases due to less wind speed and here PV-Wind system is again capable to full
fill the load demand so the battery is in charging condition. But at t 4–5 s, a 4 KW
load is increased in the system so at that time generated power is insufficient to full
fill the load requirement so battery fed power to system to full fill the load power.
Battery feeds power to system when required by the system. Battery power management is shown in Fig. 8.
168
A. K. Singh et al.
6 Conclusion
In RES, the output power of solar and the wind are fluctuating in nature because
these energy sources are nature dependent. So, hybrid topology is used to overcome the intermittence and complement each other. In this paper, discuss control
and operation of a balanced power between sources and load is discussed. The system contains hybrid PV-Wind and battery connected to load. The hybrid PV-Wind
system and battery are connected to common DC bus in which PV-Wind is connected
through DC/DC boost converter and battery are connected through bi-directional. In
MATLAB/Simulink, 9.5 KW PV and 8.5 KW wind hybrid system have been implemented. Power generated by hybrid PV-Wind system and battery system are capable
of transferring the desired power to load. This paper implements the fuzzy control to
obtain the battery power management system applications. Such type of intelligent
control system increases the accuracy of this nonlinear system and it also obtains the
optimization and distributed energy generation by its control algorithm.
References
1. Villalva, M.G., Gazoli, J.R., Filho, E.R.: Comprehensive approach to modelling and simulation
of photovoltaic arrays. IEEE Trans. Power Electron. 24(5), 1198–1208 (2009)
2. Bae, Sungwoo, Kwasinski, Alexis: Dynamic modeling and operation strategy for a microgrid
with wind and photovoltaic resources. IEEE Trans. Smart Grid 3(4), 1867–1876 (2012)
3. Liu, X., Wang, P., Loh, P.C.: A hybrid AC/DC microgrid and its coordination control. IEEE
Trans. Smart Grid 2(2), 278–286 (2011)
4. Wang, C., Nehrir, M.H.: Power management of a stand-alone wind/photovoltaic/fuel cell energy
system. IEEE Trans. Energy Convers. 23(3), 957–967 (2008)
5. Ahmed, N.A., Miyatake, M., Al-Othman, A.K.: Power fluctuations suppression of stand-alone
hybrid generation combining solar photovoltaic/wind turbine and fuel cell systems. Energy
Convers. Manage. 49(10), 2711–2719 (2008)
6. Li, Xiangjun, Hui, Dong, Lai, Xiaokang: Battery energy storage station (BESS)-based smoothing control of photovoltaic (PV) and wind power generation fluctuations. IEEE Trans. Sustain.
Energy 4(2), 464–473 (2013)
7. Haque, MdEnamul, Negnevitsky, Michael, Muttaqi, Kashem M.: A novel control strategy for
a variable-speed wind turbine with a permanent-magnet synchronous generator. IEEE Trans.
Ind. Appl. 46(1), 331–339 (2010)
8. Kasera, J., Chaplot, A., Maherchandani, J.K.: Modeling and simulation of wind-PV hybrid
power system using Matlab/Simulink. In: 2012 IEEE Students’ Conference on Electrical, Electronics and Computer Science (SCEECS). IEEE (2012)
9. Ayush, A.K., Gautam, S., Shrivastva, V.: Analysis of energy storage system for wind power generation with application of bidirectional converter. In: 2016 Second International Conference
on Computational Intelligence &Communication Technology (CICT). IEEE (2016)
10. Tani, A., Camara, M.B., Dakyo, B.: Energy management in the decentralized generation systems based on renewable energy—ultra capacitors and battery to compensate the wind/load
power fluctuations. IEEE Trans. Indus. Appl. 51(2), 1817–1827 (2015)
11. Hussein, K.H., et al.: Maximum photovoltaic power tracking: an algorithm for rapidly changing
atmospheric conditions. IEE Proc.-Gener. Transm. Distrib. 142(1), 59–64 (1995)
12. Yang, Y. et al.: Nonlinear dynamic power tracking of low-power wind energy conversion
system. IEEE Trans. Power Electron. 30(9), 5223–5236 (2015)
Analysis of Energy Storage for Hybrid System Using FLC
169
13. Ciobotaru, M., Teodorescu, R., Blaabjerg, F.: A new single-phase PLL structure based on
second order generalized integrator. In: Power Electronics Specialists Conference (2006)
14. Wang, Trudie, O’Neill, Daniel, Kamath, Haresh: Dynamic control and optimization of distributed energy resources in a microgrid. IEEE Trans Smart Grid 6(6), 2884–2894 (2015)
15. Ruban, M.A.A.M., Rajasekaran, G.M., Rajeswari, M.N.: Implementation of Energy Management System to PV-Wind Hybrid Power Generation System for DC Microgrid Application
(2015)
Impact of Emission Trading on Optimal
Bidding of Price Takers in a Competitive
Energy Market
Somendra P. S. Mathur, Anoop Arya and Manisha dubey
Abstract All over the world, electricity sector emerged as the main source of GHG
emission. Emission trading scheme and Renewable support schemes are the main
schemes to diminish Greenhouse Gas emissions, which is adopted by various countries and some developed countries or regions are going to be implementing. In the
first part, this paper depicts the summary of several obligatory greenhouse gases
trading schemes adopted by the various countries worldwide and their future trends
in carbon trading. The second part evaluated the optimal bidding of thermal power
plant in a competitive energy market with the strategy that considering the impact of
CO2 emission in an emission trading market. In this paper, a stochastic optimization
model is presented with the theory that the pdf of rival’s bidding are known. For this
purpose, in a sealed auction with considering the impact of CO2 emission trading a
nature-inspired new genetic algorithm approach has been employed in a day-ahead
market to solve the optimization problem with symmetrical and unsymmetrical information of rivals. The feasibility of the proposed method is checked by an IEEE-30
bus system with six generators.
Keywords Competitive energy market · Emission trading schemes · Genetic
algorithm
S. P. S. Mathur (B) · A. Arya · M. dubey
Electrical Engineering Department, Maulana Azad National
Institute of Technology Bhopal, Bhopal, India
e-mail: somendra.mathur0007@gmail.com
A. Arya
e-mail: anooparya.nitb@gmail.com
M. dubey
e-mail: manishadubey6@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_18
171
172
S. P. S. Mathur et al.
1 Introduction
Electric power industry has undergone a restructuring process worldwide and competition is increased greatly from monopoly to competitive market power. The main
aim of the power industry is to establish a competitive electricity market by reformation. The reformation started around mid-1980s in the various countries of the
world. The pioneer of reformation is the Chile, where it is started in 1987. Electric
power industry reforms started in India when Electricity act 2003 and various policy
i.e. National electricity policy and Tariff policy were adopted by the government.
Two power exchanges i.e. Power Exchange India Ltd (PXIL) and Indian Energy
Exchange Ltd. (IEX) are operational in India since 2008. The endeavor of reformation is to change the economics of energy market from monopoly to competitive
market power, increased fuel availability and develops new technologies [1, 2]. In
a competitive electric power industry, all the price takers have market power and
can make the healthy profit via its strategic bidding behavior, and much research
has been undertaken. Theoretically, to maximize the profit price takers should bid at
very close to their marginal cost in the competitive energy market and when price
takers did this, then this behavior is called strategic bidding. According to the different market mechanisms and bidding protocols, various modeling techniques have
been adopted by many researchers. These modeling techniques can be classified
as Optimization models, Game theory models, agent-based simulation models, and
hybrid models. References [3, 4] describes the various price takers’ strategic bidding
modeling methods in a competitive energy market on the state of the art.
In the current energy market, various causes affect the bidding strategies of price
takers in the day-ahead market [5]. This paper considers the impact of emission trading schemes on the optimal bidding of price takers in a competitive energy market.
Currently, GHG emissions are the main environmental issues worldwide. Market
liberalization and economic development played an important role in raising the levels of CO2 emissions and other greenhouse gases in the atmosphere [6]. Worldwide,
energy market recognized as a vital cause of GHG emission. 1/3rd of CO2 emissions are accounted for Generation Company in Europe. In the Netherlands, more
than 50% of generation of emission is from the energy market, while in India this
is more than 45%. The inception of emission trading schemes into the generation
companies contributes to the cutback of emission and impact energy market process.
According to the size, scope, and design, various ETSs are operating worldwide.
Most of them are linked with the Kyoto Protocol commitments (UNFCC 1998) [7,
8]. Some schemes are mandatory, others are voluntary. However, they all are sharing
a common premise: emission reductions i.e. cutting the overall cost of skirmishing climate change. Carbon trading consist, trading of six major greenhouse gases
i.e. carbon dioxide (CO2 ), methane (CH4 ), nitrousoxide (N2 O), hydrofluorocarbons
(HFCs), perfluorocarbons (PFCs), and sulfur hexafluoride (SF6 ).
This paper is organized as follows, in Sect. 2, a mathematical model for the price
takers in a day-ahead market with the impact of emission trading in the energy market is developed, and represented as a stochastic optimization problem. Section 3
Impact of Emission Trading on Optimal Bidding of Price Takers …
173
described a computational procedure of newly developed genetic algorithm technique. Section 4 illustrates the execution of the proposed method with numerical
simulation. Finally, Sect. 5 concludes the paper with possible directions for future
research.
2 Mathematical Formulation
2.1 Market Structure
According to their characteristics in various countries, the formation of energy market primarily consists the spot market, medium and long-term trade market which
are suitable for the practical purpose. For example, PJM market in North America
operates a day-ahead, real-time, daily capacity, monthly and multi-monthly capacity,
regulation, and the (FTRs) auction energy market. The Nordic PX Pool, established
in 1993 and it operates the day-ahead energy market. Japan Electric Power Exchange
(JPEX), it starts operating in 2003 and it operates the power pool with the day-ahead
energy market [4].
Assume a power exchange operates in a day-ahead market and ISO checks the
system security and stabilization for the better operating condition. Power exchange
consists of M generating companies and N utility companies, who participate in the
demand side bidding submitting a nondecreasing demand function for the trading
time slot t ε T (1, 2… 24). If trading time slot assumes as 30 min, then T 48. M
generating companies include thermal power station submit nondecreasing supply
curve in a day-ahead market. The assumptions are made for the modeling are as
follows: (1) all price takers have no market power; (2) power outputs of price takers
can be accurately controlled, (3) load prediction error is assumed to be negligible.
2.2 Cost Model
Under ETS, price takers (agent) needs to purchase CO2 emission allowances from
an emission allowance trading market with price pCO2 , then production cost function
and a marginal cost function of the agent (i) can be represented by the following
equation:
2
Ci (qt,i ) (bi + pco2 ηi )qt,i + 0.5ci qt,i
(1)
Mi (qt,i ) (bi + pco2 ηi ) + ci qt,i
(2)
where,
C i (qt,i ) Production cost of Genco i including CO2 emission,
174
M i (qt,i )
bi , c i
qt,i
ηi
S. P. S. Mathur et al.
Marginal cost of Genco i;
Production cost constant coefficient;
Output of Genco i at hour t and
CO2 emission factor
If agent i selects jth strategy, then the corresponding coefficient is
j
D j Dmin +
(Dmax − Dmin )
K −1
(3)
Here, Dmin and Dmax are the lower and upper limit of the coefficients D.
Thus, the bidding price of agent i can be represented as
Bi (qt,i ) αi + pco2 ηi + D j βi qt,i
(4)
2.3 Bidding Model
Assume a power exchange operates in a day-ahead market consist of ‘m’ independent price takers and ‘n’ load customers participate in a competitive electricity power
market, in which price takers submit their sealed bid with a pay-as-bid MCP to the
energy market. Also, assume that each price takers and customers submit a nondecreasing supply/demand curve to power exchange. For determining the generation
outputs, minimize the total purchase cost and maximize the expected profit can be
obtained by solving the following equations:
αi + pco2 ηi + D j βi qt,i R i 1, 2 . . . m
n
qi Q(R)
(5)
(6)
i1
qi,min ≤ qi ≤ qi,max
(7)
Here, α i , β i are the bidding coefficients of agent i, R is the MCP and Q(R) is the
aggregate pool load curve which can be represented by the linear equation:
Q(R) Q 0 −k R
(8)
where Q0 is a nonnegative constant which is represent the price elasticity of the
system, for k 0 the system is largely inelastic. When the inequality constraints are
neglected, then the solution of Eq. (5) and (6) are
R
Q0 +
n
K+
(αi + pco2 ηi )
i1
D j βi
n
1
i1 D j βi
(9)
Impact of Emission Trading on Optimal Bidding of Price Takers …
Pt,i R − (αi + p co2 ηi )
D j βi
175
i 1, 2 . . . n
(10)
The solution of Eq. (10) changes according to their generation output limits (7),
i.e., if Pi exceeds the Pi,max , then Pi is set to Pi,max and if Pi is lower than the Pi,min , then
Pi is zero and the related price takers detached from the problem as a noncompetitive
contestant.
2.4 Profit Model
For ith price takers at unit time, profit maximization function can be represented as
max πi (αi , βi ) R × Pt,i − Ci (Pt,i )
(11)
From Eq. (11), our objective is to evaluate α j and β j for maximization the profit
subject to some inequality constraints expressed by Eq. (5)–(7). Price takers do not
have access to complete information of their opponent, so it is required for price
takers to estimate other participant’s unknown information. For the ith supplier’s,
bidding coefficients αi and βi can be represented by the following probability density
function (pdf).
pdf(αi , βi ) 1
×
(β)
2π σi(α) σi
1 − ρi2
⎧
⎡
⎨
α − μi(α)
1
⎣ i
exp −
⎩ 2(1 − ρi2 )
σi(α)
−
2
(β)
2ςi αi − μi(α)
βi − μi
(β)
σi(α) σi
(β)
+
βi − μi
(β)
σi
2
⎤⎫
⎬
⎦
⎭
(12)
3 Optimal Bidding by GA
3.1 Overview of GA
Genetic algorithm is a stochastic nondeterministic method, to evaluate the most
excellent solution in the complicated problem through the optimization. It is based
on the theory of survival of the fittest to get a best possible solution. GA start with
a string of solution called population (Chromosome). A string of new population
176
S. P. S. Mathur et al.
(offspring) will be generated from the solution of each population according to the
hope that the new populations have higher fitness value to reproduce [9].
The procedure of genetic algorithm can be divided into three modules, i.e., Production, evaluation and reproduction module. In the production module, the initial
population will be created using the initialization operator with randomly generated
individuals. In the evaluation module, fitness operator checks the character of each
chromosome based on maximum or minimum level to satisfy the objective. Under
the reproduction module, three operators will be used, i.e., selection, recombination,
and mutation operator.
3.2 GA Procedure
The proposed methodology for optimal bidding using newly developed GA consists
of the following steps:
Step 1 (Initialization) Read cost coefficients of price takers and their limits,
Aggregate load and price elasticity, convergence tolerance,
k, Dmin , Dmax, and emission factor.
Step 2 Set iteration count 1. Set chromosome count 1.
Step 3 (Representation) Identified the chromosomes as a parent and create a random
population of β j of Eq. (12) using Monte Carlo simulation.
Step 4 Evaluate the Market clearing price and fitness of each population using Eqs. (9) and (11) respectively.
Step 5 “Healthiest” chromosomes are sorted in decreasing order
of their fitness value.
Step 6 Calculate error function from (12). Check if error <convergence tolerance, go to 10.
Step 7 (Reproduction) Check if fit(1) fit(last). If yes go to 12.
Step 8 Copy Pe chromosomes of old population to new population
starting from the best ones from the top.
Step 9 (Crossover) In this process, chromosomes are selected from the mating
pool of parents and these are mixed with different chromosomes.
Step 10 (Mutation) A very low mutation rate is selected.
Step 11 Increase iteration count and if it is less then maximum
iteration then go for next iteration, else print “problem not
converged in maximum number of iterations”.
Step 12 (Termination) Problem converged. Print the values of bj at which suppliers
get the maximum benefit.
Impact of Emission Trading on Optimal Bidding of Price Takers …
177
Table 1 Cost coefficients, generation output limits, and emission factor of price takers
Generator
a
bi
ci
Pt,jmin
Pt,jmax
ηj
no.
1
2
3
4
5
6
0
0
0
0
0
0
6.0
5.25
3.0
9.75
9.0
9.0
0.01125
0.0525
0.1375
0.02532
0.075
0.075
40
30
20
20
20
20
160
130
90
120
100
100
Table 2 GA parameter and aggregated pool load characteristic parameter
Population
No. of
Crossover rate Mutation rate Q0
size
iteration
20
10
0.7
0.03
300
0.918
0.937
1.025
0.958
1.125
0.426
k
5
4 Numerical Results
In order to evaluate the optimal bidding with carbon emissions trading, an IEEE30 bus system is considered. This bus system consists of six price takers which are
participating in a day-ahead energy market and assume each possesses one generation
unit. Also assume that five are of coal-fired ones, and sixth is a CCGT one. The
cost coefficients, CO2 emission factors and generation output limits of IEEE-30 bus
system are listed in Table 1.
For the execution of proposed methodology, the constant parameters associated
with newly developed GA and the aggregated pool load characteristic parameter
which is described in (8) are presented in Table 2.
Assume that the CO2 emission allowances are 5 million tons and the upper limit
is 20$/ton. Set K 5, Dmin 1 and Dmax 2. Here the bidding problem is a bi-level
optimization problem, in the first level price takers possess the random samplings
of α j and β j according to their pdf’s, and in the second level profit maximization is
achieved by using optimization technique.
In this work, Monte Carlo simulation is employed for the random sampling and
a new genetic algorithm is used for the optimization of bidding problem. Genetic
algorithm is a stochastic nondeterministic method, to evaluate the most excellent
solution in the complicated problem through the optimization. It is based on the
theory of survival of the fittest to get a best possible solution. In general, the process
of genetic algorithm can be divided into three modules, i.e., production, evaluation,
and reproduction module [10].
For the random sampling assumes that the price takers fix α j bj and a Monte
Carlo simulation method is used to evaluate β j . β j, should not be less than cj and it is
searched between cj and M × cj and M 10 for all simulation. Probability density
parameter with symmetrical and with unsymmetrical information of rival’s based on
178
S. P. S. Mathur et al.
Table 3 Estimation of the rival parameters
(α)
With
symmetrical
information
With unsymmetrical
information
(β)
(β)
(α)
μi
μi
σi
σi
ji
1.2 × bi
1.2 × ci
0.0375 × bi
0.0375 × ci
−0.1
1.1 × bi
1.1 × ci
0.0375 × bi
0.0375 × ci
−0.1
Table 4 Monte Carlo and GA with symmetrical information of rivals
Monte Carlo
Proposed GA
Genco
βj
Pj (MW)
Profit ($)
βj
Pj (MW)
Profit ($)
1
2
3
4
5
6
MCP
0.0292
0.1242
0.2923
0.0743
0.1705
0.1705
16.35
160.00
89.40
45.70
88.80
43.10
43.10
1368.0
572.7
322.9
386.4
177.5
77.5
0.0647
0.1052
0.2753
0.0554
0.1508
0.1508
16.85
160.00
105.91
49.23
120.00
50.11
50.11
1372
592.6
326.4
429
181
181
their historical bidding data, which is described in Eq. (12) are shown in Table 3.
Detailed explanations of the specification of these parameters are given in ref. [11].
4.1 Without Considering the Carbon Emission Trading
The bidding parameter, MCP, generation output and expected profit of power producers by using Monte Carlo and the proposed GA method with symmetrical information and with unsymmetrical information of rivals are shown in Tables 4 and 5
respectively.
4.2 Considering the Carbon Emission Trading
Assume that the CO2 emission allowances are 5 million tons and the upper limit is
20$/ton. Set K 5, Dmin 1, and Dmax 2. Here the bidding problem is a bi-level
optimization problem, in the first level price takers possess the random samplings
of α j and β j according to their pdfs, and in the second level profit maximization is
achieved by using optimization technique. The bidding parameter, MCP, generation
output and expected profit of price takers by using proposed GA method in compari-
Impact of Emission Trading on Optimal Bidding of Price Takers …
179
Table 5 Monte Carlo and GA with unsymmetrical information of rivals
Monte Carlo
Proposed GA
Genco
βj
Pj (MW)
Profit ($)
βj
Pj (MW)
Profit ($)
1
2
3
4
5
6
MCP
0.0292
0.1536
0.2923
0.0743
0.1705
0.1705
16.72
160.00
74.21
47.45
93.50
45.8
45.8
1420
585.6
342.2
420.5
192
192
0.0292
0.1536
0.2923
0.0743
0.1705
0.1705
17.85
160
36.2
55.4
120.0
59.5
59.5
1592.5
184.2
396.4
596.5
256.4
256.4
Table 6 Monte Carlo and GA with symmetrical information of rivals
Monte Carlo
Proposed GA
Genco
βj
Pj (MW)
Profit ($)
βj
Pj (MW)
Profit ($)
1
2
3
4
5
6
MCP
0.0292
0.1242
0.2923
0.0743
0.1705
0.1705
28.28
160
120.5
90
115.2
81.2
83.5
2355.6
1185.2
615.8
680.5
315.3
419.25
0.0647
0.1052
0.2753
0.0554
0.1508
0.1508
29.65
160
122.5
90
118.2
82.5
87.3
2368.2
1195.6
623.6
695.3
325.6
445.3
Table 7 Monte Carlo method with unsymmetrical information of rival’s
Monte Carlo
Proposed GA
Genco
βj
Pj (MW)
Profit ($)
βj
Pj (MW)
Profit ($)
1
2
3
4
5
6
MCP
0.0292
0.1536
0.2923
0.0743
0.1705
0.1705
31.5
160
125.6
90
120
90.6
95.6
2370.2
1252.6
630.5
750.6
385.4
458.3
0.0292
0.1536
0.2923
0.0743
0.1705
0.1705
32.8
160
130
90
120
95.2
96.5
2375.6
1325.2
670.2
805.2
415.2
462.8
son with Monte Carlo method with symmetrical information and with unsymmetrical
information of rivals are shown in Tables 6 and 7 respectively.
After analyzing CCGT unit from the above results it is clearly shown that, with
considering carbon emission trading, bidding coefficients obtained from the proposed GA approach gives lesser values than the Monte Carlo method. From which
increased dispatched power, MCP, and actual profit are obtained. Hence optimal
bidding strategy obtained by proposed GA gives higher profit than the Monte Carlo
approach. It is observed that through strategic bidding profit obtained by Genco VI
180
S. P. S. Mathur et al.
is 256.4 $, when the CO2 emission trading not considered. However, when the CO2
emission trading is considered, its profit increased to 462.8 $, which is 1.81 times
that of the former. It is primarily because of considering the impact of CO2 emission, which has significantly enhanced the competitiveness of the price takers in the
electricity market. The time taken for the convergence of 100 generation by using
GA method is 2.10 s.
5 Conclusion
In this paper, a newly developed GA has been used to explain optimal strategic
bidding problem for price takers participating in a day-ahead electricity market by
considering the impacts of the emissions trading. For this purpose, a stochastic optimization model is presented with the assumption that the probability distribution
function of rivals bidding is known and considering the impact of CO2 emission
in an emission trading market. Here the bidding problem is a bi-level optimization
problem, in the first level price takers possess the random samplings according to
their pdf’s, and in the second level profit maximization is achieved by the proposed
GA approach. The main aim of the price takers participating in the optimal bidding
is to maximize the profit with symmetrical and with unsymmetrical information of
rivals.
References
1. Liu, Youfei, Wu, Felix F.: Generator bidding in oligopolistic electricity markets using optimal
control: fundamentals and application. IEEE Trans. Power Syst. 21(3), 1050–1061 (2006)
2. Kirschen D.S., Strbac, G.: Fundamentals of Power System Economics. Wiley (2004)
3. Mathur, S., Arya, A., Dubey, M.: Optimal Bidding Strategy for Price Takers and Customers in
a Competitive Electricity Market. Cogent engineering, Taylor Francis online http://dx.doi.org/
10.1080/23311916.2017.1358545
4. Li, G., Sh, J., Qu, X.: Modeling methods for GenCo bidding strategy optimization in the
liberalized electricity spot market: a state-of-the-art review. Energy 36, 4686–4700 (2011)
5. Prabavathi, M., Gnanadass, R.: Energy bidding strategies for restructured electricity market.
Electr. Power Energy Syst. 64, 956–966 (2015)
6. Verma, Y.P., Kumar, A.: Potential impacts of emission concerned policies on power system
operation with renewable energy sources. Electr. Power Energy Syst. 44, 520–529 (2013)
7. Chappin E.J.L., Dijkema G.P.J.: On the impact of CO2 emission-trading on the power generation
emissions. Technol. Forecast. Soc. Change 76, 358–370 (2009)
8. Perdan, S., Azapagic, A.: Carbon trading: current schemes and future developments. Energy
Policy 39, 6040–6054 (2011)
9. Dahal, K.P., Galloway, S.J., Burt, G.M., McDonald, J.R.: Generation scheduling using genetic
algorithm based hybrid techniques. In: Power Engineering. LESCOPE ‘01. 2001 Large Engineering Systems Conference, pp. 74–78 (2001)
10. Azadeh, A., Ghaderi, S.F., Pourvalikhan Nokhandan, B., Sheikhalishahi, M.: A new genetic
algorithm approach for optimizing bidding strategy viewpoint of profit maximization of a
generation company. Expert System Appl. 39(1), 1565–1574 (2012)
11. Wen, F.S., David, A.K.: Optimal bidding strategies and modeling of imperfect information’s
among competitive generators. IEEE Trans. Power Syst. 1, 15–21 (2001)
Impact of NOVEL HVDC
Superconducting Circuit Breaker
on HVDC Faults
Tarun Shrivastava, A. M. Shandilya and S. C. Gupta
Abstract The key obstacle of very large discharge current of the DC-link capacitor, Voltage Source Converter-based HVDC (VSC-HVDC) is it is vulnerable to the
DC short-circuit faults. Superconducting fault current limiter (SFCL) is an effective
option to minimize commutation failure and increase the steady-state stability to
mitigate the fault and also limit or reduce transient electrical surges that may occur
in transmission and distribution networks of high-voltage direct current (HVDC)
system. The SFCL can limit the fault current on the ac side of the converter and
thus quickly restore the HVDC system to normal status. A NOVEL HVDC superconducting circuit breaker compared with resistive and no SFCL which can more
effectively limit the amplitude of short-circuit current of the VSC-HVDC system
has been presented in this literature. Finally, the fundamental design requirements
including HVDC superconducting breaking NOVEL SFCL with current interruption
for changing the intensity of different dc fault conditions are proposed. Simulation
results have been presented for different fault condition in DC breaking combination
of NOVEL, resistive and no SFCL with simulink toolbox of MATLAB 2014a are
discussed.
Keywords HVDC circuit breaker · Superconducting fault current limiter
(SFCL) · Voltage source converter (VSC) · Fault
1 Introduction
The advanced modern power electronic system increased for further high-power
application of HVDC system based on the Voltage Source Converter (HVDC-VSC)
provides a reliable and cost-effective solution for bridging long distance for bulk
power transmission. The technical advantages of VSC-HVDC system have shown
T. Shrivastava (B) · A. M. Shandilya · S. C. Gupta
Department of Electrical Engineering, Maulana Azad National
Institute of Technology, Bhopal 462003, MP, India
e-mail: tarunmitsmanit08@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_19
181
182
T. Shrivastava et al.
them to be a best solution than thyristor-based HVDC system [1], carryout with
the independent control active and reactive power, achieve to interconnection of
AC transmission system, employ large cheaper filters, uses with light cable and
long transmission line for multiterminal application [2–5]. In this context, the VSCHVDC system relation to the advance power electronics transmission technology are
motivating researchers to study the large this system. However in a HVDC system,
the dc fault current raises rapidly with large magnitude within several milliseconds
to the ac network. Thus, the studies of VDC-HVDC protection system are necessary
and essential for this technology. Breaking this large short-circuit current limited
by dc circuit breaker can quickly and reliably isolate the faulty network. Thus, the
protection of HVDC system must be fast enough to clear the fault before it exceeds the
current limit, in order to mitigate the faults effects in VSC-HVDC systems. With the
lack of latest technology to break and isolate the huge dc fault current in multiterminal
HVDC system, it is still not possible in large HVDC system, despite many advantages
and many practical industry applications. Dc circuit breakers selectively isolated a
faulty line fast and reliably dc short-circuit current [6].
The behavior of superconducting fault current limiters to limit the direct current
directly or supply a DC current affects the superconducting material with nonlinear
response to magnetization of saturable iron core, temperature, current and magnetic
field. Increasing any of these parameters can cause a transition between the superconducting and the normal conducting regimes [7, 8]. With the major concern for the
transmission system operators as increased fault current levels represents negative
effect on the reliability and security of entire power system [9]. For the safe operation
of power system, various strategies for mitigating the fault current have been opted
in power industry as construction of new substations, splitting of existing substation
busses, upgradation of multiple circuit breakers and installation of high impedance
transformer [10–12]. The current limiting apparatus, such as series reactors, solid
state fault current limiters are widely used to reduce the fault levels in existing power
grid systems. Superconducting fault current limiters (SFCL’s) are considered as the
most desirable alternatives to the conventional protective method due to the remarkable features of the superconducting materials, also limit the faults even prior of
attaining the first peak of short-circuit current and is capable of automatically restore
to its superconducting state.
Some of the literature have focused on superconducting fault current limiter
(SFCL) has been a greatest interest for researchers recommended for fast and quick
response with less power dissipation during normal operation. Reference [13] proposed short-circuit- and ground fault analyses of VSC-based dc systems. Reference
[14] consisting a NOVEL hybrid super conducting dc circuit breaker (DCCB) model
proposed the SFCL located in reference (main) line, to limit the fault current until
the signal to the super conducting dc circuit breaker. So DC circuit breakers are
valuable for current interruption and less power consumption of an arc for all types
of circuit breaker application. Reference [10] proposed the inductive and resistive
superconducting fault current limiter to improve the transient behaviors of VSCHVDC system with wind power plants. The fault current interruption can be easily
achieved by natural zero current crossing in AC circuit breaker, while in HVDC
Impact of NOVEL HVDC Superconducting Circuit Breaker on HVDC …
183
R SC
IL
I SC
CB
I Shunt
R shunt
LShunt
Fig. 1 Resistive SFCL with shunt element
circuit breaker implemented by artificial current zero crossing for enable the fault
current interruption. To resolve these zero interruption, various HVDC circuit breakers and there topology are summarized with different fault condition in Ref. [16]. In
this paper, we have explained the effect of NOVEL HVDC superconducting circuit
breaker with different dc faults, which is useful to improve the reliability and transient stability of the system study based on equal area criteria, to reduce the fault
current capacity.
2 Resistive SFCL
Resistive SFCL brings an innovative interest in electric grids which is a need to
respond to several prototypes demand in power quality and supply secularization with
medium- and high-voltage systems. During the normal operation in HVDC system,
the current flow through the superconducting element dissipates the low energy. But
when a fault occurs, the current of superconductor increases to critical current level
than it quench with increasing resistance exponentially. The current level at which
quench occurs is determined by operating temperature and types of superconductors.
With increasing the resistance, the voltage across the superconductor also increases
the current to transfer in parallel resistance or inductive shunt that is needed to adjust
the limiting current and limit to overvoltage across the superconductor during a
quench. The resistive SFCL is much smaller and lighter other than SFCL. So it acts
like a switch with milliseconds response. The principle operation of resistive SFCL
is shown by one-line diagram in Fig. 1.
The quenching phenomenon in resistive SFCL results in the heat which is away
from the superconducting element by the cryogenic cooling system. Because of that,
cryogenic system can restore the operating temperature; this period is known as
recovery time and is a critical parameter to design the various SFCL breakers for
utility systems. Some of the resistive SFCL having fast switching components in
series with superconductor elements, which is quickly, isolates the superconductor
for allowing the element to begin recovery cycle in limited actions. The fast acting
184
T. Shrivastava et al.
Fig. 2 Simplified
configuration of NOVEL
SFCL
i2
R1
is
Rmoa
R2
i3
L1
i1
Switch K1
switches reduces the fault current with high temperature of the superconductive
material.
3 SFCL Circuit Configuration and Operational Principle
The superconducting fault current limiter (SFCL) is a very large application of superconductivity that has been a great interest in research area in last few decades for
industrial and medium- and high-voltage system application [17–20]. The main focus
of NOVEL SFCL in super-conducting dc circuit breaker is to suppress the dc shortcircuit current with minimum possible fault level and reduce current interruption
on circuit breaker. For the analysis process of SFCLs, the simplified configuration
diagram of the NOVEL SFCL is shown in Fig. 2 where
R1 Current limiting resistance
R2 Protective resistance with parallel to DC circuit breaker
Rmoa Metal oxide arrester which is suppress the switching overvoltage, which is
parallel with current limiting inductance
L 1 Current limiting inductance which is made of superconducting coils.
Figures 3 and 4 are shows the VSC-HVDC diagram integrated with SFCL with
pole-to-pole- and pole-to-ground fault. When the VSC-HVDC transmission is in
normal position, the switch K 1 is in close position and the dc current is flowing
through the inductor l1 , for the steady state condition/superconducting state. There
is no need of superconducting fault current limiter with the additional losses during
normal operations. Here L 1 can work as smoothing reactor to mitigate the harmonic
current and voltage of DC line. The equivalent circuit is shown in Fig. 1, the DC line
current (is ) can be calculated as
is
Total source voltage at rectifier side
(Total impedance at rectifier side + equivalent impedance of DC transmission line)
Impact of NOVEL HVDC Superconducting Circuit Breaker on HVDC …
185
Overhead DC line
SFCL
3 Phase
AC
System
SFCL
l1
Rectifier
Station
VSC-1
Inverter
Station
VSC-2
l2
SLCL
3 Phase
AC
System
SFCL
Fig. 3 Schematic diagram of VSC-HVDC integrated with SFCL with pole to pole fault
Overhead DC line
SFCL
3 Phase
AC
System
Rectifier
Station
VSC-1
SFCL
l1
l2
SLCL
Inverter
Station
VSC-2
3 Phase
AC
System
SFCL
Fig. 4 Schematic diagram of VSC-HVDC integrated with SFCL with pole to ground fault
When the fault occurs current i1 will increase rapidly. Here switch (K 1 ) controller
will be opened with high resistance value R1 inserted in the circuit. Most of the fault
current is passes through the resistance R1 which limit the dc short-circuit current
beyond the critical value. The fault current is, is expressed as
is i2 + i3 ,
where
i2
i3
current flow in resistance R1
current flow through controlled DC circuit breaker.
When the switch controller (K 1 ) detects the occurrence of fault, it will open and
corresponding parallel resistance R2 will be inserted in the circuit with large value of
resistance. So R2 is used to limit the superconducting current beyond the critical value.
For the working of SFCL, the current limiting inductance is detection signal, where
inductance voltage exceeds a predetermined threshold value, an external controller
sends a disconnect signal to the DC switch K 1 . After the disconnected switch K 1 ,
resistance R2 will be connected in the circuit to limit the superconducting fault current
which is below the critical value. So, in this case, reliability of SFCL will improve
for the system requirement. When the current value of DC line (is ) is lower than
the threshold value, it shows the SFCL requires the current limiting requirements
for located the fault and remove it. For the study analysis the different fault current
topologies are given in Table 1.
186
T. Shrivastava et al.
Table 1 Basic characteristics of different FCL technologies
Technology
Losses
Triggering
Recovery
Size/weight
Distortion
Relatively
small
During first
cycle of
current
limitation
During first
cycle of
current
limitation
Resistive
SFCL
AC losses
Passive
Hybrid
resistive
SFCL
AC losses
Passive
Super
conductor
must be
re-cooled
Faster than
resistive
SFCL due to
reduced
energy
dissipation
Saturablecore
SFCL
DC Power
needed to
stature the
iron core
Passive
Immediate
Large and
heavy due to
iron core and
conventional
windings
Some due to
nonlinear
magnetic
characteristics
Shielded core
SFCL
AC losses
Passive
Faster
recovery than
resistive
SFCL
Large and
heavy due to
iron core and
windings
During first
cycle of
current
limitation
Solid state
SFCL
Similar to
resistive
SFCL
Active
Immediate
Smaller to
resistive
SFCL
Fuses
Negligible
passive
Never, must
be replaced
Smallest
Switching of
power
electronics
introduces
harmonics
None
Smaller than
resistive
SFCL but
depends on
the additional
components
4 DC Current Interruption Methods
The interruption of DC fault current on rectifier side is possible when fault time
is longer at overheating equipment. The short-circuit faults can be interrupted by
either electromechanical breakers or power electronics switches. Generally power
electronics switches like diode rectifier, forced commutated rectifier as IGBT, IGCT
with free-wheeling diode have less or no fault breaking capability but thyristorcontrolled rectifier consisting fault breaking capability with first ac side current zero
interruption. There are various methods available in DC current interruption as shown
in Fig. 5.
5 Simulation Analysis and Discussion
The VSC-HVDC system consisting two terminal symmetric Monopolar-type system
with two level converters. The parameters of VSC-HVDC system are shown in
Impact of NOVEL HVDC Superconducting Circuit Breaker on HVDC …
187
DC Current
Interruption
Switch Assisted
Interruption
Source potential
Reduction
Direct current
suppression
Backup batteries
Nonlinear
Devices
Without Current
Commutation
Superconducting
current Limiter
Current
commutation
Conventional
Mechanical Breaker
Arc quenching in Air,
Oil, Vacuum and SF6
PTC Resistor
Solid State
breaker
Current
oscillation
Pyro technique
Current
Commutation
Self Oscillation
(Passive)
Uncharged
LC +Arc
Fuses
Forced Oscillation
(Active)
Conventional DC
and HVDC
breaker
Charged
LC+Arc
Hybrid breaker
Fig. 5 Classification of DC current interruption methods [15]
Table 2 Simulation
parameter of the VSC-HVDC
system
DC voltage
230 kV
Rated capacity
2000 MVA
DC capacitor
70 µf
Smoothing reactor
8 mh
Resistance of the DC line
Inductance of the DC line
Length of the DC line
0.14737 /km
46.90 mh/km
75 km
Table 2. The resistance and inductance may have a limited range when SFCL is
using in HVDC system with short-circuit capacity. The performance of VSC-HVDC
system with ±230 kV, maximum short-circuit current reaches 9 kA in pole to pole
and 16.4 kA in pole to ground with any protective device.
The value of limiting inductance is nearly 160 MH and limiting resistance is ~10 considered for the design of DC circuit breaker. The main simulation parameters are
indicated in Table 2 with the MATLAB simulation model as shown in Figs. 2 and
3. We have shown all MATLAB result diagrams current, voltage, and power with
respect to time of pole-to-pole- and pole-to-ground faults. Also simulate the three
phases AC voltage and current flowing through the rectifier ac of different limiting
condition.
In this section, it presents the simulation result of different fault condition in the
VSC-HVDC system. In order to validate the presented work with different fault
conditions three different SFCL condition. These devices have fault current at pole
to pole and pole to ground can be explained in Table 3, maximum current can be
interrupted by different SFCL breakers conditions. The pole-to-ground fault was
simulated between the DC line and limiter without any faults. The fault was initiated
188
T. Shrivastava et al.
Table 3 Simulation result of fault condition
Fault condition
DC fault current
(Amp)
DC voltage (KV)
Power consumption
(MW)
Pole to pole
Pole to ground
No SFCL
9200
9037
Resistive SFCL
Novel SFCL
No SFCL
6860
6095
101
6850
6040
101
Resistive SFCL
Novel SFCL
No SFCL
136
132
577
114
112
510
Resistive SFCL
Novel SFCL
479
492
507
479
(a)
(b)
10000
6000
4000
2000
No SFCL
Resistive SFCL
Novel SFCL
15000
Current (A)
Current (A)
8000
10000
5000
0
0
-2000
20000
No SFCL
Resistive SFCL
Novel SFCL
0
0.2
0.4
0.6
Time (sec)
0.8
1
-5000
0
0.2
0.4
0.6
0.8
1
Time (sec)
Fig. 6 DC fault current in (Positive line/phase) a Pole to pole b Pole to ground
0.5 s shown in Fig. 6b with three cases. The red curve represented the fault current
when the system operated without SFCL with 9.03 KA. The others resistives SFCL
and NOVAL SFCL are fault current 6.85 and 6.04 KA for same type of fault. It is
concluded that the value of resistance R1 reduces for the value of DC line current.
Hence it is advised to select the appropriate value of R1 to consider the current limiting
effects or interrupt the current. The current limiting breaker shows the effectiveness of
application of NOVAL SFCL mitigating the faults in VSC-HVDC system. It is noted
that the fault current exceeds the critical value, the interruption capacity of breaker
for a short period of time, after that the circuit breaker disrupts the fault current.
As we know the continuous fault current level was significantly reduced, so the
protection of VSC-HVDC system has more facilitates with protection coordination.
Figure 6 shows that the peak value of the DC voltage different condition without
SFCL, resistive SFCL, NOVAL SFCL are for pole to pole fault 101, 136, 132 kV
and pole-to-ground fault 101, 114, 112 kV respectively (Figs. 7, 8, 9 and 10).
Impact of NOVEL HVDC Superconducting Circuit Breaker on HVDC …
(a) 1.5
5
x 10
No SFCL
Resistive SFCL
Novel SFCL
0.5
0
x 10
No SFCL
Resistive SFCL
Novel SFCL
1
0.5
0
-0.5
-0.5
-1
5
(b) 1.5
Voltage (V)
Voltage (V)
1
189
0
0.2
0.4
0.6
0.8
-1
1
0
0.2
0.4
Time (sec)
0.6
0.8
1
Time (sec)
Fig. 7 DC voltage in (Positive line/phase) a Pole to ground b Pole to pole
(a) 6
8
x 10
No SFCL
Resistive SFCL
Novel SFCL
2
0
-2
-4
No SFCL
Resistive SFCL
Novel SFCL
0.5
0
-0.5
-1
-6
-8
0
x 10
1
Power (Watt)
Power (Watt)
4
9
(b) 1.5
0.2
0.4
0.6
0.8
1
-1.5
0
Time (sec)
0.2
0.4
0.6
0.8
1
Time (sec)
Fig. 8 DC Power of a Pole to pole b Pole to ground
In view of Table 3 shows the resulted parameter for no SFCL, resistive SFCL and
impact of NOVAL SFCL have compared with voltage, dc fault current, and power
consumption with different fault conditions. It is shown that the NOVAL SFCL has
lowest significant value of dc fault current and power consumption as others, so it is
required for less maintenance, low losses, and low cost in HVDC power system.
6 Conclusion
In this paper, different fault analysis of VSC-HVDC system with resistive, NOVEL
and no SFCL have been performed. The impacts of NOVEL SFCL circuit breaker
having fast fault response and the results demonstrate compensation of the DC voltage, overcome the DC fault current, and reduce the power disturbance. The proposed
NOVAL SFCL method is quite effective and importance of circuit breaker capabili-
190
T. Shrivastava et al.
0
0
AC Current
AC Current
5
0
-5
0
5
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
10
Rectifier side (using Resistive SFCL) AC Current
Pole to ground fault HVDC
0
-10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time
Rectifier side (using novel SFCL) AC Current
Pole to Pole fault HVDC
-5
0
-20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time
Rectifier side (using Resistive SFCL) AC Current
Pole to Pole fault HVDC
to ground fault HVDC
20
AC Current
-5
(b) Rectifier side (without SFCL) AC Current Pole
AC Current
to Pole fault HVDC
5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
AC Current
AC Current
(a) Rectifier side (without SFCL) AC Current Pole
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time
Rectifier side (using novel SFCL) AC Current
Pole to ground fault HVDC
5
0
-5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Fig. 9 Three-phase AC current a Pole to pole b Pole to ground
2
Rectifier side (without SFCL) AC
Voltage Pole to Pole fault HVDC
0
-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(b)
AC Voltage
AC Voltage
(a)
Rectifier side (without SFCL) AC Voltage
Pole to ground fault HVDC
5
0
-5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2
Rectifier side (using Resistive SFCL)
AC Voltage Pole to Pole fault HVDC
0
-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
5
Rectifier side (using Resistive SFCL) AC
Voltage Pole to ground fault HVDC
0
-5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
2
0
-2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
Time
AC Voltage
AC Voltage
Time
Rectifier side (using novel SFCL) AC
Voltage Pole to Pole fault HVDC
1
Time
AC Voltage
AC Voltage
Time
5
Rectifier side (using novel SFCL) AC
Voltage Pole to ground fault HVDC
0
-5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Time
Fig. 10 Three-phase AC voltage a Pole to pole b Pole to ground
Time
1
Impact of NOVEL HVDC Superconducting Circuit Breaker on HVDC …
191
ties for analyzing the SFCL in HVDC performance. The safety measures for integration with various kinds of distributed generation and loads are available in system.
The analysis of pole-to-pole and pole-to-ground fault provides critical time limits
for protective power HVDC system. In future, NOVEL SFCL topology based on
current limiting features are used to increase the transient stability and power quality without upgrading the power grinds and lead to scientific research facilities for
industrial application with VSC-HVDC system. In addition, the economic feasibility
of the superconducting materials applied in VSC-HVDC system will be evaluated
with the utility personal, planners who want to progress impact the NOVAL SFCL
technologies to move towards industrial and commercial viability.
References
1. Jovcic, D., van Hertem, D., Linden, K., Taisne, J.P., Grieshaber, W.: Feasibility of DC transmission networks. In: 2011 2nd IEEE PES International Conference and Exhibition on Innovative
Smart Grid Technologies (ISGT Europe), pp. 1–8. Manchester (2011)
2. Yang, J., O’Reilly, J., Fletcher, J.E.: An overview of DC cable modeling for fault analysis of
VSC-HVDC transmission systems. In: 2010 20th Australasian Universities Power Engineering
Conference (AUPEC), pp. 1–5. Christchurch (2010)
3. Wang, G.D., Wai, R.J., Liao, Y.: Design of back stepping power control for grid-side converter
of voltage source converter-based high voltage dc wind power generation system. IET Renew.
Power Gener. 7(2), 118–133 (2013)
4. Dinh, M.-C., Kim, S.-K., Kim, J.-G., Park, M., Yu, I.-K.: Loss characteristic analysis of an
HTS DC model cable connected to a model VSC-HVDC system. IEEE Trans. Appl. Supercond.
23(3), 5900305 (2013)
5. Nam, T., Shim, J.W., Hur, K.: Design and operation of double SMES coils for variable power
system through VSC-HVDC connections. IEEE Trans. Appl. Supercond. 23(3), 5701004
(2013)
6. Meyer, C., Kowal, M., De Doncker, R.W.: Circuit breaker concepts for future high-power
DC-applications. IEEE IAS Annu. Meet. Conf. 2, 860–866 (2005)
7. Ngamroo, I., Vachirasricirikul, S.: Coordinated control of optimized SFCL and SMES for
improvement of power system transient stability. IEEE Trans. Appl. Supercond. 22(3) (2012)
8. Li, Y., Zhang, Z., Rehtanz, C., Luo, L., Rüberg, S., Liu, F.: Study on steady-and transient-state
characteristics of a new HVDC transmission system based on an inductive filtering method.
IEEE Trans. Power Electron. 26(7), 1976–1986 (2011)
9. He, Y.: Sensor fault diagnosis of superconducting fault current limiter with saturated iron core
based on SVM. IEEE Trans. Appl. Supercond. 24(5) (2014)
10. Chen, L., Chen, H., Shu, Z., Zhang, G., Xia, T., Ren, L.: Comparison of inductive and resistive
SFCL to robustness improvement of a VSC-HVDC system with wind plants against DC fault.
IEEE Trans. Appl. supercond. 26(7) (2016)
11. Mokhberdoran, A., Carvalho, A., Silva, N.: Design and implementation of fast current releasing
DC circuit breaker. Electr. Power Syst. Res. 151, 218–232 (2017)
12. Martinez-Velasco, J.A., Magnusson, J.: Parametric analysis of the hybrid HVDC circuit breaker.
Electr. Power Energy Syst. 84, 284–295 (2017)
13. Yang, J., Fletcher, J.E., O’Reilly, J.: Short-circuit and ground fault analyses and location in
VSC-Based DC network cables. IEEE Trans. Power Electron. 59(10) (2012)
14. Khan, U.A., Lee, J.-G., Amir, F., Lee, B.-W.: A novel model of HVDC hybrid-type superconducting circuit breaker and its performance analysis for limiting and breaking DC Fault
currents. IEEE Trans. Appl. Supercond. 25(6) (2015)
192
T. Shrivastava et al.
15. Kulkarni, S., Santoso, S.: Interrupting short-circuit direct current using an AC circuit breaker
in series with a reactor. Adv. Power Electron. 2012(805958), 14 (2012). https://doi.org/10.11
55/2012/805958
16. Nguyen, V.-V., Son, H.I., Nguyen, T.-T.: A novel topology of hybrid HVDC circuit breaker for
VSC-HVDC application. Energies 10, 1675 (2017). https://doi.org/10.3390/en10101675
17. Noe, M., Steurer, M.: High-temperature superconductor fault current limiters: concepts, applications, and development status. Supercond. Sci. Technol. 20(3), R15 (2007)
18. Xin, Y. et al.: “Development of 220 kV/300 MVA superconductive fault current limiter. Supercond. Sci. Technol. 25(10), Art. ID. 105 011 (2007)
19. Eckroad, S.: Superconducting power equipment—Technology watch 2012. Electric Power
Research Institute (EPRI), Palo Alto, CA, USA (2012)
20. Kim, H., et al.: Development and grid operation of superconducting fault current limiters in
KEPCO. IEEE Trans. Appl. Supercond. 24(5), 1–4 (2014)
Palmprint Matching based on
Normalized Correlation Coefficient
and Mean Structural Similarity Index
Measure
Deval Verma, Himanshu Agarwal and A. K. Aggarwal
Abstract Biometrics is the widely used technology for identification of a person.
Palmprint is one of the most frequently used biometric identifiers. Decision for
matching of two images is a common step in all the biometric system. Similarity
comparison between two images of a biometric identifier decides that two images
are matched or not. Normalized correlation coefficient (NCC) and mean structural
similarity index measure (MSSIM) are two well-known functions that measure similarity between two images. This paper presents a comparative false-positive rate
(FPR) and the false-negative rate (FNR) analysis of these two functions for palmprint images. Experiment is done on a palmprint database containing 5502 images.
Decision for matching is taken for different threshold values. Ground truth is used to
evaluate the false-positive rate (FPR) and the false-negative rate (FNR). It is verified
that MSSIM-based similarity measure is better than NCC-based similarity measure.
Keywords Biometrics · NCC · MSSIM · FPR · FNR · ROC
Similarity measures · Threshold
1 Introduction
Nowadays, personal identification by measuring physiological and behavioral characteristics of individuals are required for various purposes in security. Biometric
personal identification [1] is a way by which a large group of people can automati-
D. Verma · H. Agarwal (B) · A. K. Aggarwal
Department of Mathematics, Jaypee Institute of Information Technology,
A-10, Sector 62, Noida 201309, Uttar Pradesh, India
e-mail: himanshu.agarwal@jiit.ac.in
D. Verma
e-mail: deval09msc@gmail.com
A. K. Aggarwal
e-mail: amrish.aggarwal@jiit.ac.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_20
193
194
D. Verma et al.
cally recognize their identity. The biometric identifiers can be iris pattern or retina,
palmprint, fingerprint, and face images [2–5].
A similarity function is used for measuring similarity between two images [6].
The normalized cross-correlation coefficient (NCC) [7] is used to compute the degree
of similarity between two images. The importance of NCC over the cross-correlation
is that it is less delicate to straight changes in the amplitude of brightening in two
compared images [8]. It is broadly utilized for pattern or object recognition [9],
detection in confounded images [10].
The another widely used similarity measure is mean structural similarity measure (MSSIM) [1, 11–16]. It is compatible with human visual system for similarity
measurement between two images.
In this paper, we have evaluated the performance of palmprint-based biometric
system. The similarity is measured by using the NCC and MSSIM [17]. The similarity is used to compute the error of biometric system. Error is computed in terms
of FPR and FNR. For aggregate performance of the biometric system, receiver operating characteristic (ROC) curve is plotted between FPR and FNR. An algorithm is
proposed for systematic plot of ROC curve.
The rest of the paper is organized as follows. Section 2 explains the detailed theory
of NCC and MSSIM. Basic equations for matching of two images and algorithm to
plot ROC curve are discussed in Sect. 3. The experiments and results are analyzed in
Sect. 4 followed by conclusions in Sect. 5.
2 Preliminaries
2.1 Normalized Correlation Coefficient (NCC)
An original image x1 of size P × Q and reference image y1 of size M × N are taken
into consideration and normalized cross-correlation is computed between them. The
dividend of the fraction η(x1 , y1 ) corresponds to the cross-correlation between the
reference image and the original image C(x1 , y1 ) and its calculation wind up by
M N
the interference in the evaluation of η(x1 , y1 ) where ||x1 ||2 =
{x1 (l, m)}2 and
||y1 ||2 =
M N
l=1 m=1
{y1 (l, m)}2 .
l=1 m=1
M N
η(x1 , y1 ) = {x1 (l, m).y1 (l, m)}
∈ [0, 1]
M M N
N
{x1 (l, m)}2 .
{y1 (l, m)}2
l=1 m=1
l=1 m=1
l=1 m=1
(1)
Palmprint Matching based on Normalized Correlation Coefficient …
195
2.2 Assessment of Image Quality Using MSSIM
The index of SSIM approach [6, 16, 18] was calculated by taking a size of local
8 × 8 or 11 × 11 square window [17]. The SSIM index is calculated on moving
these windows on the pixel by pixel from upper left corner to the down right corner
of the image. At each step, the local statistics and SSIM index are calculated within
the local window. Gaussian weighting function w = wl , l = 1, 2, . . . N and its unit
N
sum is defined as ( wl = 1) is adopted.
l=1
μx1 =
N
wl .xl
(2)
n=1
N
σx1 = {wl (xl − μx1 )2 }
(3)
l=1
σx1 y1 =
N
{wl (xl − μx1 )(yl − μ y1 )}
(4)
l=1
After combining all these image quality maps into a single quality for the complete
image, the image quality measurement system is computed. The agreeable way is to
use a weighted summation. If x1 (n) and y1 (n) are the two images being compared,
and SS I M[x1 (n), y1 (n)] defined as
SS I M(x1 , y1 ) =
(2μx1 μ y1 + C1 ) (2σx1 y1 + C2 )
(μ2x1 + μ2y1 + C1 ) (σx21 + σ y21 + C2 )
(5)
be the local SSIM index evaluated at location n, then the mean SSIM (MSSIM) index
between xn and yn is defined as
1
SS I Mn
M n=1
M
M SS I M =
(6)
where M is the total number of windows.
3 Palmprint Matching Methodology
The matching function for two images by using NCC similarity measure is defined
as follows:
196
D. Verma et al.
C N CC,τ (x1 , y1 ) =
1
0
if N CC(x1 , y1 ) ≥ τ
if N CC(x1 , y1 ) < τ
(7)
where x1 , x2 are two images and τ is a threshold value. Similarly, the matching
function for two images by using MSSIM similarity measure is defined as follows:
C M SS I M,τ (x1 , y1 ) =
1
0
if M SS I M(x1 , y1 ) ≥ τ
if M SS I M(x1 , y1 ) < τ
(8)
Let xi be a set of original images, yi be a set of reference images and τ be a
threshold value. Let true-positive values be denoted by (T P), false-negative values
be denoted by (F N ), true-negative values be denoted by (T N ), and false-positive
values is denoted by (F P). Let GT be the ground truth between the images xi and
yi , which is 1 if both palmprints are of same hand of same person, and 0 otherwise.
The algorithm to plot ROC curve is discussed as follows.
Algorithm : An algorithm to compute F P R, F N R and R OC.
Input: xi , yi , C = (C N CC , C M SS I M )
for i = 1 : 1 : N
for |τ | = 0 : 0.05 : 1
where xi = (x1 , x2 , x3 ....x N )
where yi = (y1 , y2 , y3 ....y N )
Select x1 ∈ xi , y1 ∈ yi
Calculate ground truth (GT) between x1 and y1 .
Calculate NCC (x1 , y1 )
Calculate MSSIM (x1 , y1 )
if M SS I M ≥ |τ | ;
then C = 1 else C = 0;
if N CC ≥ |τ | ;
then C = 1 else C = 0;
if ((GT==1) and (C N CC,τ /C M SS I M,τ (x1 , y1 ) == 1))
then TP =1 else TP = 0
if ((GT==1) and (C N CC,τ /C M SS I M,τ (x1 , y1 ) == 0))
then FN =1 else FN = 0
if ((GT==0) and (C N CC,τ /C M SS I M,τ (x1 , y1 ) == 1))
then FP =1 else FP = 0
if ((GT==0) and (C N CC,τ /C M SS I M,τ (x1 , y1 ) == 0)
then TN =1 else TN = 0
end
FPR and FNR for both measures are calculated by using following equations. It
is depicted from figure 3 and 4.
23. for |τ | = 0 : 0.05 : 1
(F P)
(9)
FPR =
(T P) + (F P)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Palmprint Matching based on Normalized Correlation Coefficient …
197
FNR = (F N )
(F N ) + (T N )
(10)
24. end
25. [F P R, F N R] = ROC (GT,X,Y,τ )
26. Plot ROC between F P R and F N R.
Output F P R, F N R and R OC
4 Experiments and Results
This section presents details of experiment, results and analysis. MSSIM and NCC
are used to measure the similarity between two images. The similarity values for
both the measures are ranged from [0, 1]. The similarity values are 1 for the same
images. The dataset for the experiment contains images from the database of Casia
palmprint. This dataset contains 5502 images of 312 different subjects. The palmprint
images are formatted as “xxxx_m/f_l/r_xx.jpg". These are 8-bit gray scale images.
The unique identifier of people ranging from 0000 to 0312 is denoted by “xxxx”,
gender male/female is denoted by “m/f” and left/right palm is represented by “l/r”.
The index image with the same type of palm ranges between 1 and 15 is denoted by
“xx”. The threshold τ is varied as 0:0.05:1.
FPR and FNR are computed by using Eqs. (9) and (10). FPR and FNR for the
first target palmprint number 0001 are provided in Table (1), Figs. 1 and 2. The ROC
between FPR and FNR for palmprint number 0001 is plotted in Fig. 3. For NCC-based
system, FPR is minimum in the threshold range [0.6, 0.85] and FNR is minimum
at threshold value 1. For MSSIM-based system, FPR is minimum in the threshold
range [0.3, 0.6] and FNR is minimum in the threshold range [0.8, 1.0]. FPR increases
and FNR decreases with threshold.
FPR and FNR for the second target palmprint number 0099 are provided in
Table (2) and Figs. 4 and 5. The ROC between FPR and FNR for palmprint number 0099 is plotted in Fig. 6.
The ideal point on ROC curve (FPR, FNR)=(0,0) does not exist. The MSSIMbased ROC curves are below NCC-based ROC curve, which indicates that MSSIMbased similarity measure outperforms NCC-based similarity measure. ROC curves
are slightly different for palmprint number 0001 and palmprint number 0099. This
depicts that performance of biometric system depends on target images.
198
D. Verma et al.
Table 1 FPR and FNR for palmprint number 0001
Threshold
FPR_{NCC}
FNR_{NCC}
0
0.05
0.10
0.15
0.20
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
–
–
–
–
–
–
–
–
–
–
–
–
0
0
0
0
0
0
0.0006
0.0013
0.0013
Fig. 1 Plot of FPR and FNR
against threshold for
NCC-based similarity
measure, palmprint number
0001
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9984
0.9983
0.9979
0.9965
0.9925
0.9722
0
FPR_{MSSIM}
FNR_{MSSIM}
–
–
–
–
–
–
0
0
0
0
0
0
0
0.0004
0.0009
0.0013
0.0013
0.0013
0.0013
0.0013
0.0013
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9984
0.9982
0.9979
0.9972
0.9955
0.9907
0.9758
0.9444
0
0
0
0
0
Palmprint Matching based on Normalized Correlation Coefficient …
Table 2 FPR and FNR for palmprint number 0099
Threshold
FPR_{NCC}
FNR_{NCC}
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
–
–
–
–
–
–
–
–
–
–
–
–
0
0
0
0
0
0
0.0004
0.0005
0.0013
Fig. 2 Plot of FPR and FNR
against threshold for
MSSIM-based similarity
measure, palmprint number
0001
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9984
0.9979
0.9933
0.6666
0
199
FPR_{MSSIM}
FNR_{MSSIM}
–
–
–
–
–
–
0
0
0
0
0
0
0.0002
0.0004
0.0004
0.0005
0.0011
0.0013
0.0013
0.0013
0.0013
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9985
0.9984
0.9982
0.9977
0.9966
0.9872
0.9016
0.1666
0
0
0
0
0
200
Fig. 3 ROC curve for
MSSIM- and NCC-based
similarity measures,
palmprint number 0001
Fig. 4 Plot of FPR and FNR
against threshold for
NCC-based similarity
measure, palmprint number
0099
Fig. 5 Plot of FPR and FNR
against threshold for
MSSIM-based similarity
measure, palmprint number
0099
D. Verma et al.
Palmprint Matching based on Normalized Correlation Coefficient …
201
Fig. 6 ROC curve for
MSSIM- and NCC-based
similarity measures,
palmprint number 0099
5 Conclusions
An algorithm is presented to compute FPR and FNR of similarity measure-based
biometric system. This algorithm is implemented on a dataset of palmprints. The
FPR and FNR of NCC and MSSIM based similarity measure biometric systems are
compared. The best threshold range with respect to FPR for NCC-based system is
[0.6, 0.85] and for MSSIM-based system is [0.3, 0.6]. The best threshold value with
respect to FNR for NCC based system is 1 and best range for MSSIM based system is
[0.8, 1.0]. The MSSIM based similarity measure is better than NCC based similarity
measure. The ideal point on ROC curve does not exist in any case. As a future scope,
this comparative analysis can be extended for noisy palmprint images.
References
1. Di Stefano, L.: Fast template matching using bounded partial correlation. Mach. Vision Appl.
13, 213–221 (2003)
2. Zhang, D., Kong, A., You, J., Wong, M.: Online palmprint identification. IEEE Trans. Pattern
Anal. Mach. Intell. 25(9), 1041–1050 (2003)
3. Dewangan, D.P., Pandey, A.: A survey on security in palmprint recognitio: a biometric trait.
Int. J. Adv. Res. Comput. Eng. Technol. (IJARCET) 1, 347 (2012)
4. Gupta, P., Gupta, P.: An accurate finger vein based verification system. Digit. Signal Proc. 38,
43–52 Elsevier (2015)
5. Yang, B., Xiang, X., Xu, D., Wang, X., Yang, X.: 3D palmprint recognition using shape index
representation and fragile bits. Multimedia Tools Appl. 76, 15357–15375 (2017)
6. Brooks, A.C., Zhao, X., Pappas, T.N.: Structural similarity quality metrics in a coding context:
exploring the space of realistic distortions. IEEE Trans. Image Proc. 17, 1261–1273 (2008)
202
D. Verma et al.
7. Tsai, D.M., Chien, T.L.: Fast normalized cross correlation for defect detection. Pattern Recogn.
Lett. 24, 2625–2631 (2003)
8. Tsai, D.M., Chien, T.L., Chen, J.F.: The evaluation of normalized cross correlations for defect
detection. Pattern Recogn. Lett. 24, 2525–2535 (2003)
9. Gonzalez, Woods, : Digital Image Processing using MATLAB. Prentice Hall, New Delhi, India
(2006)
10. Di Stefano, L., Mattoccia, S.: A sufficient condition based on the Cauchy-Schwarz inequality
for efficient Template Matching. In: International Conference on Image Processing (ICIP 2003),
vol. 1, pp. I-269 (2003)
11. Pappas, T.N., Safranek, R.J.: Perceptual criteria for image quality evaluation, pp. 669–684.
Academic Press, Handbook of Image and Video Processing (2000)
12. Sun, C.M.: Fast optical flow using 3D shortest path techniques. Image Vision Comput. 20,
981–991 (2002)
13. Wang, Z., Bovik, A.C.: A Universal Image Quality Index. IEEE Signal Processing Letters 9,
81–84 (2002)
14. Di Stefano, L., Mattoccia, S., Mola, M.: An efficient algorithm for exhaustive template matching
based on normalized cross correlation. In: IAPR International Conference on Image Analysis
and Processing (ICIP 2003), pp. 322–327. Mantova, Italy, Sept 17–19 2003
15. Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error
visibility to structural similarity. IEEE Trans. Image Proc. 13, 600–612 (2004)
16. Seshadrinathan, K., Pappas, T.N., Safranek, R.J., Chen, J., Wang, Z., Sheikh, H.R., Bovik,
A.C.: Image Quality Assessment the Essential Guide to Image Processing. Academic Press,
Electronics and Electrical (2009)
17. Brooks, A.C., Pappas, T.N.: Structural similarity quality metrics in a coding context: exploring
the space of realistic distortions. Human Vision and Electronic Imaging, p. 60570U (2006)
18. Sun, C.M.: Fast stereo matching using rectangular subregioning and 3D maximum-surface
techniques. Int. J. Comput. Vision 47, 99–117 (2002)
A Comparative Study on Feature
Selection Techniques for Multi-cluster
Text Data
Ananya Gupta and Shahin Ara Begum
Abstract Text clustering involves data that are of very high dimension. Feature
selection techniques find subsets of relevant features from the original feature space
that help in efficient and effective clustering. Selection of relevant features merely
on ranking scores without considering correlation interferes with the clustering performance. An efficient feature selection technique should be capable of preserving
the multi-cluster structure of the data. The purpose of the present work is to demonstrate that feature selection techniques which take into consideration the correlation
among features in multi-cluster scenario show better clustering results than those
techniques that simply rank features independent of each other. This paper compares
two feature selection techniques in this regard viz. the traditional Tf -Idf and the
Multi-Cluster Feature Selection (MCFS) technique. The experimental results over
the TDT2 and Reuters-21,578 datasets show the superior clustering results of MCFS
over traditional Tf -Idf .
Keywords Feature selection · Multi-Cluster feature selection
Tf-Idf · Clustering · Text data
1 Introduction
The recent trends of accumulation of large volumes of text documents in electronic
form has led to methods that has made the data mining and machine learning tasks of
clustering almost intractable [1]. Feature selection techniques are employed to overcome the problem of high dimensionality [2–7]. Feature selection selects a subset of
relevant features from the feature space that are discriminative features for clustering.
The features are chosen on the basis of certain relevance evaluation criterion which
A. Gupta (B) · S. A. Begum
Department of Computer Science, Assam University, Silchar 788011, India
e-mail: gupta.ananya77@gmail.com
S. A. Begum
e-mail: shahin.begum.ara@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_21
203
204
A. Gupta and S. A. Begum
leads to better performance in terms of computational cost and accuracy. The text
clustering uses the bag of words [8] representation where every document represents
a vector of real numbers.
Traditional feature selection techniques select features that have top ranks on the
basis of scores assigned to them independently of each other. The score estimates
their discriminative capability in subsequent clustering task. However, this method
of assigning score is not likely to be appropriate in multi-cluster problems where
different features have different power of discrimination in different clusters. In such
cases, the feature selection should select the subset of those features that can retain
the multi-cluster structure of the data. Real-world datasets such as the text dataset
have a multi-cluster data structure with correlation among its features.
This paper makes a comparative study between two feature selection techniques,
the Tf-Idf and MCFS to highlight their performance in multi-cluster scenario. Tf-Idf
is commonly employed term weighting scheme for quantifying the importance of the
term on the basis of scores assigned to them independent of the correlation among
them. On the contrary, MCFS takes into account the correlation between the features
to select the feature subset retaining the multi-cluster data structure based on spectral
analysis and L1-regularized model for subset selection. Although both the methods
have been proposed independently earlier, yet there is no comparative study on these
methods on multi-cluster text domain. Feature selection is carried out on benchmark
text datasets using the two methods and subsequently the feature subset is clustered
using the k-means algorithm to measure its effectivity.
The remainder of the paper is organized into the following sections: Sect. 2 gives
a brief overview of the concepts used in the paper, Sect. 3 is the experimental result
and Sect. 4 concludes the paper.
2 Brief Review of Concepts
2.1 Text Representation with Tf-Idf
Tf-Idf was proposed with a heuristic perception that query terms occurring across
many documents are not good discriminators and should not be considered as discriminative terms. They should be assigned less weight than those occurring across
few documents [9]. Term frequency (Tf ) represents the frequency of the term occurring in the document whereas Inverse Document Frequency (Idf ) gives the inverse
measure of the number of documents to which the term is assigned [10]. To express
the significance of textual data, it is expressed as the product of Tf and Idf . Tf-Idf is
given by
wi j t f i j X log
N
,
d fi
(1)
A Comparative Study on Feature Selection Techniques …
205
where, wi j is the weight of the term i in document j, N is the number of documents
in the collection, t f i j is the term frequency of term i in document j and d f i is
the document frequency of the term i in the collection. Tf-Idf is an estimate of the
relevance of the term to the document [11]. A term occurring in many documents will
have low Tf-Idf values than those appearing relatively fewer across the documents.
2.2 Multi-cluster Feature Selection (MCFS)
Multi-cluster feature selection (MCFS) [12] was proposed on the basis of spectral
analysis of the data [13, 14] and L1-regularized model for subset selection [15, 16].
This technique selects features that can preserve the multi-cluster structure of the data
[17] and covers all the structures in it. Spectral clustering involves two steps, viz.
opening the geometric structure together with flat embedding of data by manifold
algorithms [13, 18, 19] followed by traditional clustering on the data points [14]. The
underlying geometric structure of the data is modeled using the manifold learning
algorithms in terms of a nearest neighbor graph. The weight matrix W, uses the dot
product weighting method
Wi j xiT x j ,
(2)
where, x is the data point of the graph with N vertices.
Let D be a diagonal matrix whose
elements are column (or row as W is symmetric)
sums of weight matrix W , Dii j Wi j . The graph Laplacian is given by L D − W [20]. The unfolding of the data points manifold is obtained by solving the
generalized eigen-problem
L y λDy.
(3)
Let Y [y1 , . . . y K ], yk ’s are the eigen vectors of the generalized eigen-problem
in Eq. (3) with respect to the smallest eigen value. The rows of Y represent the flat
embedding of the data points. K is intrinsic dimensionality of the data and each yk
reflects its distribution along the dimension.
For a given yk , the subset of relevant features can be obtained by minimizing the
fitting error
2
(4)
min yk − X T ak + β|ak |,
ak
where, ak is an M dimensional vector containing
combination coefficient of the
different features in approximating yk and |ak | M
j1 |ak, j | is the L1-norm of ak.
When β is large enough, some of these coefficients will be zero due to L1-norm
penalty. Relevant features would be selected corresponding to non zero entries in ak
for every yk . Equation (4) is a regression problem and its equivalent formulation is
206
A. Gupta and S. A. Begum
2
min yk − X T ak ak
(5)
such that |ak | ≤ γ .
Instead of specifying γ , the optimization problem in Eq. (5) can be solved using
Least Angle Regression (LARs) [15] algorithm which sets control on the sparseness
of ak by specifying on its cardinality which is suitable for feature selection. For a data
M
∈ R M can be calculated.
containing K clusters, K sparse coefficient vectors {ak }k1
To select d features from among M candidate features, the features are selected using
MCFS value defined as
MCFS( j) maxk ak, j ,
(6)
where, ak, j is the jth element of ak . All the features are sorted in descending order
and the top d features are selected.
3 Experimental Setup
Experiments are performed on highly correlated text data using multi-cluster feature
selection (MCFS) and the traditional Tf-Idf feature selection methods. Corresponding feature subset is clustered using the k-means clustering and results are evaluated
using the evaluation metrics.
3.1 Datasets
The TDT2 corpus comprises of 11,201 on-topic documents classified into 96 categories. In these experiments, documents appearing in two or more categories were
removed and only the largest 30 categories were retained with 9394 documents.
Reuters 21,578 corpus contains 21,578 documents in 135 categories. Those documents with multiple category labels are discarded. It is left with 8293 documents
in 65 categories. After preprocessing, this corpus contains 18,933 distinct terms.
3.2 Evaluation Metric
The results of clustering are evaluated on standard measures of accuracy, precision,
recall, F-measure and Normalized Mutual Information (NMI). They are defined as
Accuracy TP + TN
TP + TN + FP + FN
A Comparative Study on Feature Selection Techniques …
Precision Recall F − Measure 2
207
TP
TP + FP
TP
TP + FN
Precision.Recall
Precision + Recall
TP,FP,TN and FN are the number of true positives, false positives, true negatives
and false negatives respectively. Normalized Mutual Information (NMI) is given by
MI C, C ,
MI max(H (C), H (C ))
where, C and C denote the set of clusters
from the ground truth and labels
obtained
after clustering respectively. H (C),H C are entropies of C and C respectively.
Their mutual information MI C, C is given by
P Ci , C j
,
MI C, C P(Ci , C j ) log2
P(Ci ).P C j
Ci ∈C,C j ∈C where, P(Ci ) and P C j are probabilities that features are selected arbitrarily from
the corpus belongs to Ci and C j respectively and P Ci , C j are joint probabilities
that the selected feature belongs to both the clusters at the same time. MI C, C ranges from 0 to 1. If MI 0 the two sets are independent and if MI 1, then the
clusters are identical.
Experimental Results
K-means clustering is performed on the feature subset with different values of K
(K = 10, 20, 30) to randomize the experiments. For a given cluster K, 30 tests are
carried out randomly and their average performance is recorded in terms of evaluation
metric. The k-means algorithm is applied 20 times with random starting points and
best result is recorded in terms of objective function of k-means. The number of
nearest neighbors is set to 5.
The Tf-Idf scores of each term are calculated for every document in the corpus.
The scores are in the range of [0,1] as the document vectors are unit normalized which
is their cosine similarity. All terms for every document are sorted on the basis of their
Tf-Idf scores. The feature subset for the entire document is obtained by combining
top scores of each document in the corpus. Features with Tf-Idf scores greater than
0.6 are candidate features of the feature subset. K-Means is applied on feature subset
for different values of k. Table 1 and 2 show the average clustering performance and
their corresponding plots in Figs. 1 and 2 respectively.
208
A. Gupta and S. A. Begum
Table 1 Clustering performance of Tf -Idf on TDT2 dataset
K
Accuracy
Precision
Recall
2
3
4
5
6
7
8
9
10
0.8215
0.7946
0.7826
0.7512
0.7472
0.7322
0.7213
0.7161
0.7071
0.6667
0.6472
0.6318
0.6479
0.6212
0.6166
0.6043
0.6025
0.6011
0.8507
0.8213
0.8092
0.7976
0.7846
0.7579
0.7488
0.7269
0.7032
F-measure
NMI
0.7479
0.7239
0.7095
0.7152
0.6976
0.6672
0.6668
0.6473
0.6495
0.7514
0.6087
0.6016
0.6010
0.6000
0.5921
0.5839
0.5766
0.5576
Table 2 Clustering performance of Tf -Idf on Reuters-21,578 dataset
K
Accuracy
Precision
Recall
F-measure
2
3
4
5
6
7
8
9
10
0.8456
0.8243
0.8156
0.8014
0.7834
0.7724
0.7678
0.7507
0.7434
0.7079
0.6943
0.6823
0.7347
0.7628
0.7137
0.6926
0.7078
0.6989
0.8789
0.8978
0.8498
0.8466
0.8324
0.8159
0.7876
0.7763
0.7579
0.7840
0.7829
0.7568
0.7865
0.7960
0.7525
0.7369
0.7404
0.7270
NMI
0.7761
0.7423
0.7387
0.7072
0.6658
0.6324
0.6229
0.6183
0.6178
The performance is tested for top score values greater than 0.6, 0.7, 0.8 and 0.9,
respectively. The results are tabulated in Table 3 for TDT2, and Reuters-21,578
respectively. Figures 3 and 4 gives their corresponding plots.
MCFS is applied to select d features and subsequently they are clustered for
different values of k. Different values of d are taken for each k (k 2, 3, 4, 5, 6, 7,
8, 9, 10) and clustering is performed. The average values of their performance are
recorded in Tables 4 and 5 on TDT2 and Reuters 21,578, respectively. Figures 5 and
6 show the corresponding plots of clustering performance versus number of clusters.
MCFS reduces the dimensionality of data significantly. It produces best results
with small number of features, typically around 250 for TDT2 dataset and 150 for
Reuters 21,578. Tables 6 and 7 record the details of clustering with different number
of features for K 10, 20, 30. Figures 7 and 8 give their corresponding plots.
A Comparative Study on Feature Selection Techniques …
209
Table 3 Variation in clustering performance of Tf -Idf with top scores on datasets
Top score
K 10
K 20
K 30
value
F-measure NMI
F-measure NMI
F-measure
NMI
TDT2
0.6
0.7168
0.7
0.7567
0.8
0.7789
0.9
0.8023
Reuters-21,578
0.6934
0.7047
0.7177
0.7253
0.7043
0.7377
0.7627
0.7845
0.6872
0.7076
0.7084
0.7166
0.6833
0.7334
0.7565
0.7733
0.6600
0.6866
0.7070
0.7080
0.6
0.7
0.8
0.9
0.6975
0.6989
0.7165
0.7255
0.7053
0.7179
0.7267
0.7523
0.6671
0.6886
0.6963
0.7037
0.7225
0.7477
0.7463
0.7589
0.6784
0.6877
0.6994
0.7005
0.7307
0.7526
0.7735
0.7833
Table 4 Clustering performance on TDT2 dataset using MCFS
K
Accuracy
Precision
Recall
F-measure
2
3
4
5
6
7
8
9
10
0.9416
0.9277
0.8916
0.8781
0.8655
0.8413
0.8257
0.8213
0.8010
0.9612
1.0000
0.9423
0.9012
0.8973
0.8892
0.8624
0.8613
0.8433
0.8500
0.8000
0.8226
0.8125
0.7989
0.7761
0.7789
0.7691
0.7689
0.8994
0.9354
0.8835
0.8523
0.8370
0.8213
0.8152
0.7873
0.8040
Table 5 Clustering performance on Reuters-21,578 dataset using MCFS
K
Accuracy
Precision
Recall
F-measure
2
3
4
5
6
7
8
9
10
0.9754
0.9589
0.9416
0.9355
0.9179
0.9009
0.8867
0.8589
0.8476
0.9578
0.9423
0.9377
0.9234
0.9067
0.9326
0.9313
0.8708
0.8655
0.9623
0.9505
0.9234
0.9457
0.9247
0.9456
0.9452
0.9063
0.7984
0.9598
0.9463
0.9304
0.9343
0.9155
0.9389
0.9380
0.9410
0.8305
NMI
0.8270
0.8010
0.7964
0.7686
0.7424
0.7375
0.7220
0.7169
0.7079
NMI
0.8572
0.8460
0.8433
0.8261
0.8243
0.8172
0.8094
0.7939
0.7743
210
A. Gupta and S. A. Begum
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
0.3
Precision
Recall
0.2
F-Score
Accuracy
0.1
0
NMI
2
3
4
5
6
7
8
9
10
Number of Clusters
Fig. 1 Performance curve of Tf -Idf on TDT2
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
0.3
Precision
0.2
Recall
0.1
Accuracy
0
F-Score
NMI
2
3
4
5
6
7
Number of Clusters
Fig. 2 Performance curve of Tf -Idf on Reuters-21,578
8
9
10
A Comparative Study on Feature Selection Techniques …
211
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
F-Measure 10Cluster
NMI 10Cluster
F-Measure 20Cluster
NMI 20Cluster
F-Measure 30Cluster
NMI 30Cluster
0.3
0.2
0.1
0
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Top Score Values
Fig. 3 Variation of clustering with top score on TDT2 with Tf -Idf
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
F-Measure 10Cluster
NMI 10Cluster
F-Measure 20Cluster
NMI 20Cluster
F-Measure 30Cluster
NMI 30Cluster
0.3
0.2
0.1
0
0.6
0.65
0.7
0.75
0.8
Top Score Values
0.85
0.9
Fig. 4 Variation of clustering with top score on Reuters-21,578 with Tf -Idf
It is observed that clustering performance on features selected with MCFS outperforms Tf-Idf on both the datasets. For highest ranked features of Tf-Idf , in range
212
A. Gupta and S. A. Begum
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
0.3
Precission
Recall
0.2
F-Score
Accuracy
0.1
0
NMI
2
3
4
5
6
7
8
9
10
Number of Clusters
Fig. 5 Performance curve TDT2 using MCFS
Table 6 Variation in clustering performance with number of features on TDT2 dataset using MCFS
Number of K 10
K 20
K 30
features
F-measure NMI
F-measure NMI
F-measure NMI
100
150
200
250
300
350
400
450
500
0.6840
0.7053
0.8245
0.844
0.8344
0.8362
0.8323
0.8366
0.8356
0.6679
0.7087
0.7182
0.7265
0.7294
0.7240
0.7222
0.7227
0.7238
0.6667
0.6978
0.8045
0.8156
0.8124
0.8134
0.812
0.8133
0.8115
0.6456
0.6844
0.6987
0.7145
0.7130
0.7122
0.7128
0.7112
0.7111
0.6737
0.6984
0.8069
0.8379
0.8355
0.8334
0.8365
0.8362
0.8322
0.6529
0.6770
0.7003
0.7188
0.7184
0.7163
0.7176
0.7159
0.7167
of [0.9–1], F-measure and NMI is not as high as that of MCFS for K = 10, 20, 30 in
both cases, although dimensionality of feature set decreases substantially.
A Comparative Study on Feature Selection Techniques …
213
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
0.3
Precission
0.2
Recall
0.1
Accuracy
0
F-Score
NMI
2
3
4
5
6
7
8
9
10
Number of Clusters
Fig. 6 Performance curve Reuters- 21,578 using MCFS
Table 7 Variation in clustering performance with number of features on Reuters-21,578 dataset
using MCFS
Number of K 10
K 20
K 30
features
F-measure NMI
F-measure NMI
F-measure NMI
100
150
200
250
300
350
400
450
500
0.7264
0.8266
0.8241
0.8176
0.822
0.8092
0.8173
0.8207
0.8228
0.7005
0.7587
0.7532
0.7565
0.7479
0.7515
0.7486
0.7463
0.7554
0.7334
0.7978
0.789
0.7966
0.7836
0.7886
0.7872
0.797
0.7937
0.6986
0.7278
0.7257
0.7189
0.7265
0.7233
0.7183
0.7266
0.7254
0.6976
0.8177
0.8164
0.815
0.8096
0.8098
0.8173
0.8168
0.8157
0.6981
0.7366
0.7354
0.7356
0.7289
0.7293
0.7345
0.7357
0.735
4 Conclusion
This paper highlights the significance of multi-cluster feature selection on real world
text datasets. It is observed that MCFS outperforms Tf-Idf in terms of clustering
performance. It can be concluded that top score features selected by Tf-Idf based
214
A. Gupta and S. A. Begum
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
F-Measure 10Cluster
NMI 10Cluster
F-Measure 20Cluster
NMI 20Cluster
F-Measure 30Cluster
NMI 30Cluster
0.3
0.2
0.1
0
100
150
200
250
300
350
Number of Features
400
450
500
Fig. 7 Variation of clustering with number of features on TDT2 with MCFS
1
0.9
0.8
Performance
0.7
0.6
0.5
0.4
F-Measure 10Cluster
NMI 10Cluster
F-Measure 20Cluster
NMI 20Cluster
F-Measure 30Cluster
NMI 30Cluster
0.3
0.2
0.1
0
100
150
200
250
300
350
Number of Features
400
450
500
Fig. 8 Variation of clustering with number of features Reuters-21,578 with MCFS
on ranks are not always the best discriminative features. Feature subset produced
by MCFS consists of features wherein correlation among the features are conserved
which leads to better performance. Therefore, in multi-cluster datasets, multi-cluster
feature selection techniques are preferable to the traditional ranking feature selection
techniques.
A Comparative Study on Feature Selection Techniques …
215
References
1. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley-Interscience, Hoboken, NJ (2000)
2. Boutemedjet, S., Bouguila, N., Ziou, N.: A hybrid feature extraction selection approach for
high-dimensional non-gaussian data clustering. IEEE Trans. Pattern Anal. Mach. Intell. 31(8),
1429–1443 (2009)
3. Boutsidis, C., Mahoney, M. W., Drineas, P.: Unsupervised feature selection for principal components analysis. In: Proceeding of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD’08), pp. 61–69 (2008)
4. He, X., Cai, D., Niyogi, P.: Laplacian score for feature selection. In: Advances in Neural
Information Processing Systems, 18 (2005)
5. Liu, H., Yu, L.: Toward integrating feature selection algorithms for classification and clustering.
IEEE Trans. Knowl. Data Eng. 17(4), 491–502 (2005)
6. Wolf, L., Shashua, A.: Feature selection for unsupervised and supervised inference: The emergence of sparsity in a weight-based approach. J. Mach. Learn. Res. 6, 1855–1887 (2005)
7. Zhao, Z., Liu, H.: Spectral feature selection for supervised and unsupervised learning. In:
Proceedings of the 24th Annual International Conference on Machine Learning (ICML’07),
pp. 1151–1157 (2007)
8. Salton, G., McGill, M.J.: An Introduction to Modern Information Retrieval. McGraw-Hill,
1983
9. Sparck Jones, K.: IDF term weighting and IR research lessons. J. Documentation 60(6),
521–523 (2004)
10. Lee, D.L., Chuang, H., Seamons, K.: Document ranking and vector space models. IEEE Softw.
14(2), 67–75 (1997)
11. Roberston, S.: Understanding inverse document frequency: on theoretical argument for IDF. J.
Documentation 60(5), 503–520 (2004)
12. Cai, D., Zhang, C., He, X.: Unsupervised feature selection for multi-cluster data. In: Proceeding
of the 16th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD’10),
(2010)
13. Dy, J.G., Brodley, C.E.: Feature selection for unsupervised learning. J. Mach. Learn. Res. 5,
845–889 (2004)
14. Ng, A. Y., Jordan, M., Weiss, Y.: On spectral clustering: Analysis and an algorithm. In: Advances
in Neural Information Processing Systems, vol. 14, pp. 849–856. MIT Press, Cambridge, MA
(2001)
15. Efron, B., Hastie, T., Johnstone, I., Tibshirani, R.: Least angle regression. Ann. Stat. 32(2),
407–499 (2004)
16. Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining,
Inference, and Prediction. Springer, New York (2001)
17. Li, H., Xiang, S., Zhong, Z., Ding, K., Pan, C.: Multicluster spatial–spectral unsupervised
feature selection for hyperspectral image classification. IEEE Geo Sci. Remote Sens. Lett.
12(8) (2015)
18. Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science
290(5500), 2323–2326 (2000)
19. Tenenbaum, J., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)
20. Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, 92, AMS
(1997)
Fuzzy Decision Tree with Fuzzy Particle
Swarm Optimization Clustering
for Locating Users in an Indoor
Environment Using Wireless Signal
Strength
Swathi Jamjala Narayanan, Boominathan Perumal, Cyril Joe Baby
and Rajen B. Bhatt
Abstract Wireless networks play a vital role towards elevating great interest among
researchers in developing a smart indoor environment using the handheld devices of
the users. Tracing the user’s location in an indoor environment can enable several
services for automating several activities like automating switch on/off the room
lights, air conditioning, etc., which makes the environment smart. In this paper, we
propose to apply fuzzy decision tree which utilizes the fuzzy memberships generated
from fuzzy particle swarm optimization clustering technique for the user localization
application. Here, we consider the user localization problem as a pattern classification
problem, where based on the signal strengths received from mobile devices, the
location of the user is predicted as in conference room, kitchen area, sports hall,
and work area in an indoor environment. The dataset of wireless signal strength is
taken from the physical facility at our research facility. From the results obtained, we
observe that the proposed algorithm has given highly encouraging results towards
user localization.
S. J. Narayanan (B) · B. Perumal
School of Computer Science and Engineering, VIT University, Vellore 632014, India
e-mail: swathi.jns@gmail.com
B. Perumal
e-mail: boomi051281@gmail.com
C. J. Baby
School of Electronics and Communication Engineering, VIT University, Vellore 632014, India
e-mail: cyrilbabyjoe@gmail.com
R. B. Bhatt
Robert Bosch Research and Technology Center, Pittsburgh, USA
e-mail: rajen.bhatt@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_22
217
218
S. J. Narayanan et al.
Keywords Fuzzy decision tree · Wi-Fi
Fuzzy particle swarm optimization clustering · Classification · User localization
1 Introduction
Predicting the location of the mobile users is one of the most important information
for the mobile service providers in indoor and outdoor environment. For example, the
major companies like AT & T, Google, Microsoft offers several special services like
a specialized web search, navigation and nearby-friend-finding, etc. In the recent
trend of smart environments, there is lot of IOT applications coming into use to
provide service to the users with smart handheld devices. Few applications are indoor
navigation aid system for the blind, controlling smart appliances, and handling smart
indoor environments based on the user location. In this regard, few researchers have
come with strategies to predict user location based on the Wi-Fi connectivity of smart
phones.
In recent literature, in the year 2013, Lu et al. [1] produced an IOT application
to locate iPhone users and to provide healthcare services. The algorithm used for
their experiments are NN, KNN, etc. Galvan-Tejeda et al. [2] tested algorithms like
random forest, nearest centroid, KNN and ANN classifiers for locating users in
indoor environment and has shown nearest centroid is the best in predicting user
location. Further, Gaxiola-Pacheco and Licea [3] has come up with Type-2 fuzzy
inference system to determine the user zone location. Methods like SVM and radial
basis neural network is used by YuFeng et al. [4] for indoor user localization. In 2015,
Zou et al. [5] proposed online sequential extreme learning machine algorithm and
has shown better performance in accuracy. Using Signal Strength Indicator, Finkel
et al. [6] identified the user’s location in the Ian Potter Museum art using machine
learning techniques. The best results were given by random forest algorithm, for both
indoor and outdoor user’s localization, Cho [7] came up with methods like KNN and
multiple decision trees. Here they have used smart phone logs to predict the user
localization.
In order to improve the accuracy and stability of the fuzzy decision tree (FDT)
towards Pedestrian Dead Reckoning (PDR), Chiang et al. [8] introduced FDT supported with map information. Three step processes followed in this method estimates indoor navigation solutions at real time. The proposed method had shown
good improvements in reducing the computational complexity over traditional finger
printing methods used. In this paper, we address locating users in an indoor environment using wireless signal strengths as a pattern classification problem addressed
using Fuzzy decision tree with particle swarm optimization clustering mechanism.
Fuzzy Decision Tree with Fuzzy Particle Swarm Optimization …
219
The rest of the manuscript is organized as follows. Section 2 describes user localization as a classification problem. In Sect. 3, the proposed methodology is presented.
In Sect. 4, computational experimental results are given followed by conclusion and
references.
2 User Localization as a Classification Problem
The main idea behind user localization using wireless signal strength is to predict
the location of users in an indoor environment using the Wi-Fi signal strengths
obtained or received by the smart handheld devices in their indoor environments.
For this research experiment, we have used Android phone and have observed the
wireless signal strengths received over the phone for seven wireless routers. The
signal strengths have been measured at different room in door such as conference
room, kitchen, indoor sports room, and work area. At each location, many signal
strength readings were collected by polling Wi-Fi signal strengths at every one second
interval. Then the same process was carried out in another location to collect signal
strengths.
A total of 2000 signal strengths were observed using seven wireless routers. The
reason for choosing seven routers is due to the physical layout of our research facility,
where the data is collected. In the indoor environment, the mobile devices receive
see seven wireless signals. The sample data of the observations made are given in
Table 1. In this table, WSS1 is the signal received from wireless router1, WSS2 is
the signal received from wireless router2, and so on. The class labels represent the
user location in indoor office environment. The locations are labeled as conference
room, kitchen, indoor sports room, and work area.
Table 1 Sample data
WSS1 WSS2 WSS3
WSS4
WSS5
WSS6
WSS7
Class label
−64
−68
−17
−16
−52
−56
−57
−66
−70
−48
−61
−61
−61
−58
−56
−66
−65
−37
−14
−53
−71
−71
−68
−73
−62
−82
−85
−75
−71
−78
−81
−85
−77
−80
−81
Conference room
Conference room
Kitchen area
Kitchen area
Sports hall
−49
−55
−51
−49
−63
−81
−73
Sports hall
−65
−64
−57
−59
−45
−46
−69
−65
−48
−48
−91
−91
−94
−91
Work area
Work area notations and their
descriptions
220
S. J. Narayanan et al.
Table 2 Notations and their descriptions
Notation
Description
X
Input attribute set
xi
ith training pattern
xj
jth attribute
x ij
ith pattern of x j attribute
c
F jk
Number of clusters
Fuzzy set of the attribute x j representing kth cluster
μ F jk (x ij )
Membership degree of the ith value of attribute x j on the fuzzy set F jk
cj
Cluster center of the jth attribute
m
X(t)
Fuzziness coefficient
Position matrix
V(t)
Velocity matrix
w
Weight factor
c1, c2
Constants
pbest
Particle best
gbest
Global best
K
Jm
Constant
Objective function
FCM
Fuzzy c means
FPSO
Fuzzy particle swarm optimization
FDT
Fuzzy decision tree
The problem of locating users using the wireless signal strength received in their
mobiles is converted into a supervised pattern classification problem where user
location is the decision variable, and the signal strengths received are the input
variables (Table 2).
3 Proposed Methodology
The architecture for the proposed user localization prediction using Fuzzy decision
tree with fuzzy particle swarm optimization clustering is given in Fig. 1. The obtained
signal strengths are passed on to FPSO for clustering and later to FDT for induction.
The details of clustering and the FDT induction process is given below.
Fuzzy Decision Tree with Fuzzy Particle Swarm Optimization …
Data Collection
Pre Processing
and
221
Apply tenfold cross
Validation to obtain Training Data and Test Data
Apply Fuzzy Particle Swarm
Optimization Clustering
Construct Fuzzy Decision Tree
using Fuzzy ID3 with FPSO
Fuzzy Classification Rules
Wireless signal
StrengthsActual
Class label
Apply Product- ProductSum Reasoning
Estimated Class label
Classification Accuracy
Fig. 1 Process flow for user localisation using FDT
3.1 Fuzzy Particle Swarm Optimization (FPSO)
Fuzzy particle swarm optimization proposed by Pang et al. [9] is a hybrid evolutionary
optimization algorithm in which the position and the velocity of the particles are
denoted as fuzzy relations between variables. The position of the particle represented
as X, denotes the relation between set of n data points and c cluster centers.
⎤
⎡
μ11 . . . μ1c
⎥
⎢
. .
. ⎥
(1)
X ⎢
⎣ .. . . .. ⎦
μn1 · · · μnc
222
S. J. Narayanan et al.
μi j is the membership of the data object i in jth cluster. This is exactly same as
the membership values given by FCM clustering algorithm. For a given data point
xi , its membership to cluster j is calculated as follows:
μi j 1
C
k1
xi −c j xi −c j 2m−1
,
(2)
where, m is the fuzziness coefficient and the centre vector c j is calculated as follows:
N
m
i1 μi j .x i
z j N
(3)
m
i1 μi j
μimj is the value of the degree of membership that is calculated in the previous
iteration. Note that at the start of the algorithm, the degree of membership
for data
point i to cluster j is initialized with a random value θi j , 0 ≤ 1 such that C j δi j 1.
The velocity of the particle is also represented as a matrix of n × c size having the
values range between [−1,1]. The updating of position matrix and velocity matrix is
performed using Eqs. (2 and 3).
V (t + 1) w ∗ V (t) + (c1r 1) ∗ (pbest(t) − X (t)) + (c2r 2) ∗ (gbest(t) − X (t))
(4)
X (t + 1) X (t) ⊕ V (t + 1)
(5)
The fitness function used for evaluating the solutions obtained is given in Eq. (6)
f (x) K
,
Jm
(6)
where K is a constant and Jm is the objective function of the FCM clustering algorithm. As the Jm value is smaller, the fitness will be high and the clustering effect is
better. After updating, the membership values are normalized to bring them back to
a specific range.
Modified FPSO Algorithm
The pseudocode of the modified FPSO algorithm for generating fuzzy partition space
for each attribute individually is given below.
Input: Set of feature Descriptors
Output: Optimized Fuzzy Partition Space
Fuzzy Decision Tree with Fuzzy Particle Swarm Optimization …
223
Algorithm:
Initialize particle size P=10, c1=2, c2=2, w=0.9, max_Iterations=1000
For each feature descriptor
Create swarm (X, pbest, gbest and V are nxc matrices) of particles
For each particle P
Initialize X, V, pbest
End For
End For
While max_Iterations is not met Do
For each feature descriptor
Initialize gbest for the swarm
For each particle P
Calculate the cluster centres as in FCM given in Eq. (2-3)
Calculate the fitness value using Eq. (6).
Find pbest for the particle.
End For
Find gbest for the swarm.
For each particle P
Update the velocity matrix using Eq. (4).
Update the position matrix using Eq. (5).
End For
End For
End While
The termination condition in proposed method is the maximum number of iterations or no improvement in gbest in a number of iterations. The obtained fuzzy
membership values are then passed onto FDT induction process.
3.2 Fuzzy ID3 Induction Process
The procedure used for generating FDT using Fuzzy ID3 is outlined as follows [10]:
Prerequisites: optimal Fuzzy partition space, leaf selection threshold βth , best node
selection criterion.
Procedure:
While there exist candidate nodes
DO
Select one of them using average fuzzy classification entropy search strategy,
Create its child-nodes,
Child-nodes meeting the leaf threshold have to be levelled as leaf-nodes,
otherwise the remaining child-nodes are regarded as new candidate nodes
and the procedure is repeated until the stopping criterion is met.
END
224
S. J. Narayanan et al.
4 Computational Experimental Results
The experiment carried out is based on fuzzy decision tree. The FDT takes fuzzy
partitions, leaf selection threshold, and attribute selection criterion as input. The
fuzzy partitions for our experiment are obtained using both fuzzy c-means clustering
and our proposed fuzzy particle swarm optimization clustering. For each of the input
variables, the fuzzy clusters are obtained and passed to FDT for its induction process.
The leaf selection threshold value is set to 0.75 for FDT and the following parameter
values namely particles P 10, c1 2, c2 2 and w 0.9 are passed on to FPSO
algorithm. During the process of FDT generation, the dataset is divided into tenfolds where ninefolds act as training and the tenth fold act as testing. This process
is repeated for 10 times so that each part falls in both training set as well as testing
set. Percentage classification accuracy has been calculated by nnc × 100 %; where n
is the total number of test patterns and n c is the number of test patterns classified
correctly [11] (Fig. 2; Table 3).
Accuracy %- Best
Accuracy %- Average
Accuracy %- Worst
Accuracy %
100
98
96
94
92
90
88
86
84
82
FCM Clustering FPSO Clustering
Fuzzy Decision
Tree- FCM
Fuzzy Decision
Tree-FPSO
Fig. 2 Accuracy graph for the algorithms considered
Table 3 User localization classification accuracy in terms of best, average and worst of 10-fold
execution
Algorithm
Accuracy %
Best
Average
Worst
FCM clustering
94.0
92.0
88.0
FPSO clustering
95.5
93.5
89.0
Fuzzy decision
tree-FCM
Fuzzy decision
tree-FPSO
97.0
92.5
90.0
99.5
97.0
94.0
Fuzzy Decision Tree with Fuzzy Particle Swarm Optimization …
225
5 Conclusion and Future Work
In this research work, we propose to use Fuzzy particle swarm optimization clustering
method to develop fuzzy partitions which play a major role in the induction process
of FDT and its accuracy. For the user localization dataset, the results show that in
terms of all best, average, and worst accuracies of 10-fold validation results FDT
developed with FPSO clustered membership values have shown better performance
than other methods considered. In future, we would like to compare the results of
Fuzzy decision trees where optimization will be done after induction and before
induction of Fuzzy decision trees.
References
1. Lu, X., Liu, W., Guan, Y.: iPhone independent real time localization system research and its
healthcare application. Adv. Internet Things 3(4), 53–65 (2013)
2. Galván-Tejada, C.E., García-Vázquez, J.P., García-Ceja, E., Carrasco-Jiménez, J.C., Brena,
R.F.: Evaluation of four classifiers as cost function for indoor location systems. Procedia Comput. Sci. 32, 453–460 (2014)
3. Gaxiola-Pacheco, C., Licea, G.: Localization of indoor areas using WiFi signals, Type-2 fuzzy
inference systems and data mining. In: Proceedings of the World Congress on Engineering, 1,
(2014)
4. Yu, F., Jiang, M, Liang, J., Qin, X., Hu, M., Peng, T., Hu, X.: An Improved indoor localization
of WiFi based on support vector machin. Int. J. Future Gener. Commun. Networking 7(5),
191–206 (2014)
5. Zou, H., Lu, X., Jiang, H., Xie, L.: A fast and precise indoor localization algorithm based on
an online sequential extreme learning machine. Sensors 15(1), 1804–1824 (2015)
6. Finkel, A., Harwood, A., Gaunt, H., Antig, J.: Optimizing indoor location recognition through
wireless fingerprinting at the Ian Potter Museum of Art. In: International Conference on Indoor
Positioning and Indoor Navigation (IPIN), pp. 210–219. IEEE (2014)
7. Cho, S.B.: Exploiting machine learning techniques for location recognition and prediction with
smartphone logs. Neurocomputing 176, 98–106 (2016)
8. Chiang, K.W., Liao, J.K., Tsai, G.J., Chang, H.W. The performance analysis of the map-aided
fuzzy decision tree based on the pedestrian dead reckoning algorithm in an indoor environment.
Sensors 16(1), 34 (2015)
9. Pang, W., Wang, K.P., Zhou, C.G., Dong, L.J.: Fuzzy discrete particle swarm optimization for
solving traveling salesman problem. In: The Fourth International Conference on Computer and
Information Technology, pp. 796–800 (2004)
10. Yuan, Y., Shaw, M.J.: Induction of fuzzy decision trees. Fuzzy Sets Syst. 69(2), 125–139 (1995)
11. Narayanan, S.J., Bhatt, R.B., Paramasivam, I., Khalid, M., Tripathy, B.K.: Induction of fuzzy
decision trees and its refinement using gradient projected-neuro-fuzzy decision tree. Int. J. Adv.
Intell. Paradigms 6(4), 346–369 (2014)
Optimization Approach for Bounds
Involving Generalized Normalized
δ-Casorati Curvatures
Pooja Bansal and Mohammad Hasan Shahid
Abstract By using T. Oprea’s optimization method on a real hypersurfaces of
complex quadric Qm with QSMC, we prove extremal inequalities concerning normalized scalar curvature and generalized normalized δ-Casorati curvatures. Moreover, we show the equilibrium cases at all points which signalize the invariantly
quasi-umbilical real hypersurfaces. Finally, applications of this technique as a constrained programming problem.
Keywords Optimization methods · Programming problems · Real hypersurface
Complex quadric · Scalar curvature · Generalized normalized δ-Casorati curvature
1 Introduction
H. A. Hayden in 1932 gave the concept of metric connection with torsion in a Riemannian manifold [9]. In [8], Golab stated and examined quarter-symmetric connection
on a differentiable manifold with affine connection, which generalizes the notion
of semi-symmetric connection. Various properties of quarter-symmetric metric connection have been examined by several geometers [11, 13, 16].
A linear connection ∇ˆ on Riemannian manifold (Mn , g) is called a quartersymmetric connection [8] if its torsion tensor T̂ meets
T̂ (U, V ) = ∇ˆ U V − ∇ˆ V U − [U, V ],
satisfies T̂ (U, V ) = η(U )φV − η(V )φU
(1)
(2)
P. Bansal (B) · M. H. Shahid
Department of Mathematics, Jamia Millia Islamia, New Delhi 110025, India
e-mail: poojabansal811@gmail.com
M. H. Shahid
e-mail: hasan_jmi@yahoo.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_23
227
228
P. Bansal and M. H. Shahid
where U, V ∈ Tp M, η ∈ (T (0,1) M) and φ ∈ (T (1,1) M). Ancillary, a quartersymmetric linear connection ∇ˆ holds (∇ˆ U g)(U, V ) = 0 ∀ U, V ∈ Tp M, then ∇ˆ
is called quarter-symmetric metric connection which is formalized as
∇ˆ U V = ∇U V − η(U )φV,
for U, V ∈ Tp M.
(3)
Moreover, in 1993, the study of Chen invariants proposed by B. Y. Chen [4] and
he obtained some optimal inequalities consisting of intrinsic and some extrinsic
invariants for any Riemannian submanifolds [5]. Additionally, Casorati proposed
the Casorati curvature which enlarge the notion of the principal direction [3]. Some
optimal inequalities containing Casorati curvatures were examined for submanifolds
of real, complex, and quaternionic space forms [6, 7, 10].
A lot of work has been done by many geometers on real hypersurfaces of Qm .
Y. J. Suh obtained some results on real hypersurfaces in Qm with some geometric
conditions like parallel Ricci tensor [14] and Reeb parallel shape operator [15].
From now on for the sake of simplicity, throughout a paper we denote quartersymmetric metric connection and Levi-Civita connection by QSMC and LCC,
respectively.
2 Geometry of Complex Quadric Qm
The complex hypersurface of CP m+1 is said to be complex quadric Qm given by the
2
= 0, where z1 , . . . , zm+1 are homogeneous coordinates on
relation z12 + · · · + zm+1
m+1
CP . The Kähler structure on CP m+1 induces canonically (J , g) on Qm . Q1 is
isometric to S 2 and Q2 is isometric to S 2 × S 2 . Thus, throughout this paper we take
m ≥ 3.
Separated from the induced complex structure J , we have one more geometric structure on Qm , viz., a parallel rank two-vector bundle U which contains a
complex conjugation A on Tp Qm . We symbolize Ap the shape operator of Qm in
CP m+1 satisfying Ap w = w ∀ w ∈ Tp Qm , p ∈ Qm , that is, Ap is just complex
conjugation restricted to Tp Qm which is an involution. The tangent space Tp Qm
can be expressed as Tp Qm = V(Ap ) ⊕ J V(Ap ), where V(Ap ) = (+1)-eigenspace,
J V(Ap ) = (−1)-eigenspace of Ap .
The classification of singular tangent vectors is given as follows [14]:
1. If ∃ A ∈ U: X is an eigenvector corresponding to an eigenvalue (+1), then X is
singular tangent vector which is known as U-principal.
√
2. If ∃ A ∈ U and orthonormal vectors U, V ∈ V(A) : X /||X || = (U + J V )/ 2,
then X is known as U-isotropic.
For X ∈ Tp Qm , ∃ A ∈ U and orthonormal vectors U, V ∈ V(A) satisfying X =
cos(t)U + sin(t)J V , 0 ≤ t ≤ π/4. Both these above-defined classifications correspond to t = 0 and t = π/4.
Optimization Approach for Bounds Involving Generalized …
229
3 Some General Fundamental Formulas
Here, we remind some notions for a real hypersurface M in Qm .
Consider a real hypersurface M of Qm with a connection ∇ induced from the LCC
∇ in Qm . For U ∈ Tp M, J U = φU + η(U )N where N ∈ Tp⊥ M and φU is the
tangential part of J U . Here, M associates an induced almost contact metric structure
(φ, ξ, η, g) satisfying [2]
ξ = −J N , η(ξ ) = 1, η ◦ φ = 0, φ 2 U = −U + η(U )ξ, φξ = 0
g(φU, φV ) = g(U, V ) − η(U )η(V )
Moreover, the Gauss and Weingarten formulas for M are given as, respectively,
∇ U V = ∇U V + h(U, V ) and ∇ U N = −SU
for U, V ∈ Tp M and N ∈ Tp⊥ M. The second fundamental form h and the shape
operator S of M are grouped by g(h(U, V ), N ) = g(SN U, V ) = g(SU, V ).
Moreover, the structure (φ, ξ, η, g) satisfies
∇U ξ = φSU
Now, we take into account A ∈ Up with N = cos(t)Z1 + sin(t)J Z2 where Z1 , Z2
are orthonormal vectors in V(A) and 0 ≤ t ≤ π4 which is a function on M. Since
we know that ξ = −J N , we have
N = cos(t)Z1 + sin(t)J Z2 , AN = cos(t)Z1 − sin(t)J Z2 ,
ξ = sin(t)Z2 − cos(t)J Z1 , Aξ = sin(t)Z2 + cos(t)J Z1
which follows that g(ξ, AN ) = 0. From Codazzi equation [14]
g((∇U S)V, W ) − g((∇V S)U, W ) = g(U, AN )g(AV, W ) − g(V, AN )g(AU, W )
+g(U, Aξ )g(J AV, W ) − g(V, Aξ )g(J AU, W ) + η(U )g(φV, W ) − η(V )g(φU, W )
−2η(W )g(φU, V ).
(4)
Now, the Gauss equation for U, V, W ∈ Tp M yields
R(U, V )W = g(V, W )U − g(U, W )V + g(φV, W )φU − g(φU, W )φV
−2g(φU, V )φW + g(AV, W )AU − g(AU, W )AV + g(J AV, W )J AU
−g(J AU, W )J AV + g(S V, W )SU − g(SU, W )S V
Relation (5) can be reworked as
(5)
230
P. Bansal and M. H. Shahid
g(R(U, V )W, W ) = g(V, W )g(U, W ) − g(U, W )g(V, W ) + g(φV, W )g(φU, W )
−g(φU, W )g(φV, W ) − 2g(φU, V )g(φW, W ) + g(AV, W )g(AU, W )
−g(AU, W )g(AV, W ) + g(J AV, W )g(J AU, W ) − g(J AU, W )g(J AV, W )
+g(S V, W )g(SU, W ) − g(SU, W )g(S V, W )
(6)
∀ U, V, W, W ∈ Tp M. Then, we can see
g(R(U, V )W + R(V, W )U + R(W, U )V, W ) = 0,
(7)
i.e., the first Bianchi identity holds for M with respect to LCC.
4 Curvature Tensor of M in Qm Endowed with QSMC
Here, we first obtain the curvature tensor of a real hypersurface M in Qm with
respect to QSMC and then we find the intrinsic scalar curvature with respect to
QSMC. Consider complex quadric Qm endowed with QSMC ∇ˆ and the LCC ∇.
Now, let M be a real hypersurface of Qm with induced QSMC ∇ˆ and the induced
LCC ∇. Put R̂ as the curvature tensor of ∇ˆ and R as the curvature tensor of ∇ on M.
The Gauss formulae for ∇ˆ and ∇, respectively, given by
∇ˆ U V = ∇ˆ U V + ĥ(U, V ) and ∇ U V = ∇U V + h(U, V )
where ĥ is a (0,2)-tensor on M and from these two relations, one can easily get
ĥ(U, V ) = h(U, V ). Moreover, the Codazzi equation of QSMC is disposed by
g((∇ˆ U S)V − (∇ˆ V S)U, W ) = g((∇U S)V − (∇V S)U, W )
−η(U )g((φS − Sφ)V, W ) + η(V )g((φS − Sφ)U, W )
(8)
Theorem 1 Let M be a real hypersurface of Qm with QSMC such that M has
isometric Reeb flow. Then, Codazzi equation with respect to QSMC coincides with
the Codazzi equation with respect to LCC.
Proof Let us assume that M has isometric Reeb flow, i.e., φS = Sφ. Then, (8)
immediately follows the result.
Now, we know the curvature tensor can be calculated by
R̂(U, V )W = ∇ˆ U ∇ˆ V W − ∇ˆ V ∇ˆ U W − ∇ˆ [U,V ] W
Optimization Approach for Bounds Involving Generalized …
231
So, using the relation (3), the expression for the curvature tensor of M admitting
QSMC has the expression
R̂(U, V )W = R(U, V )W + η(U )[η(W )SV − g(SV, W )ξ ] − g(φSU, V )φW
−η(V )[η(W )SU − g(SU, W )ξ ] + g(φSV, U )φW
(9)
Then, from (9), one can easily obtain
g(R̂(V, U )W, W ) = −g(R̂(U, V )W, W ), g(R̂(U, V )W , W ) = −g(R̂(U, V )W, W )
Moreover,
g(R̂(W, W )U, V ) = g(R̂(U, V )W, W ) + g(φSU, V )g(φW, W )
− g(φSV, U )g(φW, W ) − g(φSW, W )g(φU, V ) + g(φSW , W )g(φU, V )
(10)
and
g(R̂(U, V )W + R̂(V, W )U + R̂(W, U )V, W ) =
g(R(U, V )W + R(V, W )U + R(W, U )V, W ) − g((φS + Sφ)U, V )g(φW, W )
−[g(φS V, W ) − g(φSW, V )]g(φU, W ) − [g(φSW, U ) − g(φSU, W )]g(φV, W )
By the virtue of (7), above relation reduces to
g(R̂(U, V )W + R̂(V, W )U + R̂(W, U )V, W ) =
−{[g(φSU, V ) − g(φS V, U )]g(φW, W ) + [g(φS V, W ) − g(φSW, V )]g(φU, W )
+[g(φSW, U ) − g(φSU, W )]g(φV, W )}
(11)
Thus, we have the following theorems.
Theorem 2 Curvature tensor of a real hypersurface M in Qm with respect to
QSMC satisfies the following:
(a) Curvature tensor of M with QSMC is given by (9)
(b) R̂(V, U )W = −R̂(U, V )W
(c) g(R̂(U, V )W , W ) + g(R̂(U, V )W, W ) = 0
∀ U, V, W, W ∈ Tp M.
Theorem 3 Let M be a real hypersurface of Qm with QSMC together with the
geometric condition φS + Sφ = 0. Then
(a) g(R̂(W, W )U, V ) = g(R̂(U, V )W, W ) ∀ U, V, W, W ∈ Tp M
(b) M holds the first Bianchi identity with respect to QSMC.
Proof By using the assumption, (a) and (b) follows from (10) and (11), respectively.
232
P. Bansal and M. H. Shahid
5 Inequalities for Generalized Normalized δ-Casorati
Curvature with QSMC
Here, by using the T. Oprea’s technique, we obtain some inequalities for scalar
curvature, normalized scalar curvature, and the extrinsic generalized normalized δCasorati curvature for a real hypersurfaces M of Qm with respect to induced QSMC.
of Tp M and a local orthonormal
Consider a local orthonormal tangent frame {ei }2m−1
1
normal frame {e2m = N } of Tp⊥ M. The scalar curvature τ̂ of M can be formalized
by
K(ei ∧ ej ),
τ̂ =
1≤i<j≤2m−1
where K(π ) denotes the sectional curvature of plane section π ⊂ Tp M and is
spanned by tangent vectors {ei , ej } and K(ei ∧ ej ) = g(R(ei , ej )ej , ei ) for 1 ≤ i <
j ≤ 2m − 1.
The normalized scalar curvature ρ̂ of M is
ρ̂ =
2τ̂
(2m − 1)(2m − 2)
The expression for the mean curvature vector field Ĥ of M is
Ĥ =
2m−1
1
h(ei , ei )
2m − 1 i=1
Conveniently, let hαij = g(h(ei , ej ), eα ) = g(h(ei , ej ), N ) for i, j ∈ {1, . . . , 2m − 1}
and α = 2m. Then, we have the squared mean curvature ||H ||2 and the squared norm
||h||2 of h, respectively, as follows:
||Ĥ ||2 =
2m−1
2
2m−1
1
α
2
h
and
||h||
=
(hαij )2
ij
2
(2m − 1) i,j=1
i,j=1
where α = 2m, hαij = g(h(ei , ej ), N ) and here, Ĥ of ∇ˆ and H of ∇ are invariant.
It is well known that the squared norm of h over dimension 2m − 1 is called the
Casorati curvature of M in Qm and is denoted by C . Thus, we have
C=
n
1
||h||2
=
(hα )2
2m − 1
2m − 1 i,j=1 ij
1
which can be rewritten as C = 2m−1
tr(S 2 ).
m
The real hypersurface M of Q is said to be invariantly quasi-umbilical if ∃ a local
orthonormal normal frame {e2m } of M in Qm such that the shape operators Se2m have
Optimization Approach for Bounds Involving Generalized …
233
an eigenvalue of multiplicity 2m − 2 for α = 2m and the distinguished eigendirection
of Se2m is the same for α = 2m [1].
Now, let L be a k-dimensional subspace of Tp M, k ≥ 2, with an orthonormal basis
{ei }k1 . Then, the scalar curvature τ̂ (L) is formalized by
τ̂ (L) =
K(ei ∧ ej )
1≤i<j≤k
and the Casorati curvature C(L) is given by C(L) = k1 ki,j=1 (hαij )2 . We set D =
{C(L)|L : hyperplane of Tp M}. Then, the normalized δ-Casorati curvatures δc (2m −
2) and δ̂c (2m − 2) of M in Qm are given by [10]
2m
1
inf D
δc (2m − 2) (p) = C(p) +
2
2(2m − 1)
[2(2m − 1) − 1]
supD
δ̂c (2m − 2) (p) = 2C(p) −
2(2m − 1)
(12)
(13)
Now, the generalized normalized δ-Casorati curvatures δc (r; 2m − 2) and δ̂c (r; 2m −
2
−(2m−1)−r]
2) of M in Qm for B(r, 2m − 2) = (2m−2)(2m−1+r)[(2m−1)
such that r <
r(2m−1)
(2m − 1)(2m − 2) and r > (2m − 1)(2m − 2) are, respectively, given as [10]
δc (r; 2m − 2) (p) = rCp + B(r, 2m − 2)inf D
δ̂c (r; 2m − 2) (p) = rCp + B(r, 2m − 2)supD.
(14)
(15)
From these two relations, one can note that δc (r; 2m − 2) and δ̂c (r; 2m − 2) are the
generalized versions of δc (2m − 2) and δ̂c (2m − 2), respectively, by substituting r
as
to (2m−1)(2m−2)
2
(2m − 1)(2m − 2)
; 2m − 2) (p) = (2m − 1)(2m − 2)[δc (2m − 2)](p) (16)
δc (
2
(2m − 1)(2m − 2)
δ̂c (
; 2m − 2) (p) = (2m − 1)(2m − 2)[δ̂c (2m − 2)](p) (17)
2
Now, relation (9) can be rewritten as
g(R̂(U, V )W, W ) = g(R(U, V )W, W ) + η(U )[η(W )g(S V, W ) − g(S V, W )η(W )]
−g(φSU, V )g(φW, W ) − η(V )[η(W )g(SU, W )
−g(SU, W )η(W )] + g(φS V, U )g(φW, W )
Now, on contracting U and W in above-defined relation, we derive
234
P. Bansal and M. H. Shahid
ˆ
Ric(V,
W ) = (2m − 1)g(V, W ) − 3η(V )η(W ) − g(AN , N )g(AV, W ) + g(S V, W )
+g(AW, N )g(AN , V ) + g(Aξ, W )g(Aξ, V ) + tr(S)g(S V, W )
−g(S 2 V, W ) − η(V )η(W )tr(S) + g((Sφ + φS)V, φW )
(18)
ˆ
where Ric(V,
W ) and Ric(V, W ) are the Ricci tensors of the connection ∇ˆ and ∇,
respectively.
Theorem 4 Let M be a real hypersurface of Qm with QSMC. Then, the generalized
normalized δ-Casorati curvature δc (r; 2m − 2) and δ̂c (r; 2m − 2) holds
(i)ρ̂ ≤
2m
g(AN , N )2
g((φS + Sφ)ei , φei )
δc (r; 2m − 2)
+
+
+
(2m − 1)(2m − 2) 2m − 1 (2m − 1)(2m − 2)
(2m − 1)(2m − 2)
(ii)ρ̂ ≤
δ̂c (r; 2m − 2)
2m
g(AN , N )2
g((φS + Sφ)ei , φei )
+
+
+
.
(2m − 1)(2m − 2) 2m − 1 (2m − 1)(2m − 2)
(2m − 1)(2m − 2)
Moreover, both relations (i) and (ii) hold the equalities iff M is an invariantly quasiumbilical real hypersurface with flat normal connection in Qm : for some orthonormal
of Tp M and {e2m = N } of Tp⊥ M, the matrix of the shape operator
frame {ei }2m−1
1
SN is
M
0
0
, where M is scalar matrix with entries a.
(19)
2τ̂ = (2m − 1)2 + g(AN , N )2 − 1 + (2m − 1)2 ||Ĥ ||2 − (2m − 1)C
+g((φS + Sφ)ei , φei )
(20)
SN =
(2m−1)(2m−2)
a
r
Proof From (9), we deduce that
We now consider the quadratic polynomial P with n = 2m − 1
P = rC + B(r, n − 1)C(L) − 2τ̂ (p) + n2 + g(AN , N )2 + g((φS + Sφ)ei , φei ) − 1,
where L is a hyperplane of Tp M. Let {ei }n−1
to be the orthonormal basis of L and
1
putting eα = N = en+1 for α = n + 1, it gives
P=
n
n−1
r α 2 B(r, n − 1) α 2
(hij ) +
(h ) − 2τ̂ (p) + n2 + g(AN , N )2 − 1
n i,j=1
n − 1 i,j=1 ij
+g((φS + Sφ)ei , φei )
From (20) and (21), we obtain
(21)
Optimization Approach for Bounds Involving Generalized …
P=
=
≥
235
n
n−1
r α 2 B(r, n − 1) α 2
(hij ) +
(h ) − n2 ||Ĥ ||2 + nC
n i,j=1
n − 1 i,j=1 ij
n−1
α 2 n2 + n(r − 1) − 2r n + r α 2
+
(hin ) + (hαni )2
(hii )
r
n
i=1
r
(n − 1)(n + r) (hαij )2 − 2
hαii hαjj + (hαnn )2
+
n
n
1≤i=j≤n−1
1≤i=j≤n
n−1
n2 + n(r − 1) − 2r α 2
r
(hii ) − 2
(hαii hαjj ) + (hαnn )2
r
n
i=1
1≤i=j≤n
Now, take into account F : Rn → R given by
F (hα11 , hα22 , . . . , hαnn ) =
n−1
i=1
1≤i=j≤n
n2 + n(r − 1) − 2r α 2
(hii ) − 2
r
r
(hαii hαjj ) + (hαnn )2
n
and optimization problem for invariant real constant K
min F subjected to F : hα11 + hα22 + · · · + hαnn = K
Now, the partial derivative of F for i ∈ {1, 2, . . . , n − 1} is given by
∂F
∂hαii
∂F
∂hαnn
2(n+r)(n−1) α
hii − 2 nk=1
r
α
= 2rn hαnn − 2 n−1
k=1 hkk ,
=
hαkk ,
(22)
Now, to get an extremum solution (hα11 , hα22 , . . . , hαnn ) of the problem P, the vector
gradF ∈ T ⊥ M at F, i.e., it is collinear with the vector (1,1, …, 1). From system of
Eq. (22), the critical point of the optimized problem outlined by
(hα11 , hα22 , . . . , hαnn ) = (
rλ
rλ
rλ
,
, ... ,
, λ)
n(n − 1) n(n − 1)
n(n − 1)
Since we have ni=1 hαii = K, which together with (23) implies that
Kn
λ = r+n
. Thus, finally we have
hαii =
rK
nK
and hαnn =
,
(r + n)(n − 1)
n+r
Now, using theorem 1 of [12], we have
(r+n)λ
n
for i = 1, 2, . . . , n − 1.
(23)
= K or
236
P. Bansal and M. H. Shahid
2(n + r)(n − 1)
− 2 for i = 1, 2, . . . , n − 1
r
2r
aij = −2 for i = j and ann = .
n
Hess(F) = (aij ) where aii =
Then, using the totally geodesic condition ofF in Rn and considering a vector
U = (U1 , U2 , . . . , Un ) tangent to F such that ni=1 Ui = 0, we derive
2(n2 − n + nr − 2r) 2 2r 2
A(U, U ) =
Ui + Un ≥ 0
r
n
i=1
n−1
Then, (23) asserts that the solution point (hα11 , hα22 , . . . , hαnn ) is the global minimum
point and F(hα11 , hα22 , . . . , hαnn ) = 0. Hence, P ≥ 0 which gives
B(r, n − 1)
C(L) + n2 + g(AN , N )2 − 1
n−1
+g((φS + Sφ)ei , φei ),
2τ̂ (p) ≤ rC +
Or equivalently, we get
ρ̂ ≤
B(r, n − 1)
n + 1 g(AN , N )2
r
C+
C(L) +
+
n(n − 1)
n(n − 1)
n
n(n − 1)
g((φS + Sφ)ei , φei )
+
.
n(n − 1)
(24)
So, both the inequalities (i) and (ii) of Theorem 4 follow from (24).
Also, we can see that both the equalities hold iff
hij = 0 for i = j ∈ {1, 2, ..., n},
n(n − 1)
n(n − 1)
n(n − 1)
h11 =
h22 = ... =
hn−1 n−1 .
hnn =
r
r
r
for eα = en+1 = N for α = n + 1. Hence, finally we get equalities in (i) and (ii) of
Theorem 4 iff the real hypersurface M is invariantly quasi-umbilical with trivial
normal connection in Qm where the matrix of the shape operator will be of the form
(19).
Proposition 1 Let M be a real hypersurface of Qm . Then, the normalized δ-Casorati
curvature δc (2m − 2) and δ̂c (2m − 2) holds
g(AN , N )2
g((φS + Sφ)ei , φei )
2m
+
+
2m − 1 (2m − 1)(2m − 2)
(2m − 1)(2m − 2)
2
g(AN , N )
g((φS + Sφ)ei , φei )
2m
+
+
.
(ii)ρ ≤ δ̂c (2m − 2) +
2m − 1 (2m − 1)(2m − 2)
(2m − 1)(2m − 2)
(i)ρ ≤ δc (2m − 2) +
Optimization Approach for Bounds Involving Generalized …
237
Moreover, both relations (i) and (ii) hold the equalities iff M is an invariantly quasiumbilical real hypersurface with flat normal connection in Qm : for some orthonormal
of Tp M and {e2m = N } of Tp⊥ M, the matrix of the shape operator SN
basis {ei }2m−1
1
has the form
SN =
M 0
0 2a
, where M is the scalar matrix of order 2m − 2 with entries a.
and using (12) (resp. (13)),
Proof From (16) (resp. (17)) by taking r = (2m−1)(2m−2)
2
we obtain our result for normalized δ-Casorati curvature of M in Qm .
References
1. Blair, D., Ledger, A.: Quasi-umbilical, minimal submanifolds of Euclidean space. Simon Stevin
51, 322. MR 0461304 (1977)
2. Blair, D.E.: Contact Manifolds in Riemannian Geometry. Lecture Notes in Math, vol. 509.
Springer, Berlin (1976)
3. Casorati, F.: Nuova definitione della curvatura delle superficie e suo confronto con quella di
Gauss. Rend. Inst. Matem. Accad. Lomb. 22, 18671868 (1889). (In Italian)
4. Chen, B.Y.: Some pinching and classification theorems for minimal submanifolds. Arch. Math.
60, 568–578 (1993)
5. Chen, B.Y.: An optimal inequality for CR-warped products in complex space forms involving
CR δ-invariants. internat. J. Math. 23(3), 1250045 (17 pages) (2012)
6. Chen, B.Y., Dillen, F., Van der Veken, J., Vrancken, L.: Curvature inequalities for Lagrangian
submanifolds: the final solution. Differ. Geom. Appl. 31(6), 808–819 (2013)
7. Ghisoiu, V.: Inequalities for the Casorati curvatures of slant submanifolds in complex space
forms, Riemannian geometry and applications. In: Proceedings RIGA 2011, pp. 145–150.
University of Bucuresti, Bucharest (2011)
8. Golab, S.: On semi-symmetric and quarter-symmetric linear connections. The Tensor Soc.
29(3), 293301 (1975)
9. Hayden, H.A.: Subspaces of a space with torsion. Proc. London Math. Soc. 34, 2750 (1932)
10. Lee, J.W., Vilcu, G.E.: Inequalities for generalized normalized δ-Casorati curvatures of slant
submanifolds in quaternionic space forms. Taiwan. J. Math. 19(3), 691–702 (2015)
11. Mondal, A.K., De, U.C.: Some properties of a quarter-symmetric metric connection on a
Sasakian manifold. Bull. Math. Anal. Appl. 1(3), 99108 (2009)
12. Oprea, T.: Optimization methods on Riemannian submanifolds. An. Univ. Bucur. Mat. 54(1),
127–136 (2005)
13. Rastogi, S.C.: On quarter-symmetric metric connection. Comptes Rendus de lAcademie Bulgare des Sciences 31(7), 811814 (1978)
14. Suh, Y.J.: Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. Math.
281, 886–905 (2015)
15. Suh, Y.J.: Real hypersurfaces in the complex quadric with Reeb parallel shape operator. Internat.
J. Math. 25, 1450059, 17 (2014)
16. Yano, K., Imai, T.: Quarter-symmetric metric connections and their curvature tensors. The
Tensor Soc. 38, 1318 (1982)
Particle Swarm Optimization
with Probabilistic Inertia Weight
Ankit Agrawal and Sarsij Tripathi
Abstract Particle swarm optimization (PSO) is a stochastic swarm-based algorithm
inspired by the intelligent collective behavior of some animals. There are very few
parameters to adjust in PSO which makes PSO easy to implement. One of the important parameter is inertia weight (ω) which balances the exploration and exploitation
properties of PSO in a search space. In this paper, a new variation of PSO has been
proposed, which utilizes a novel adaptive inertia weight strategy based on the binomial probability distribution for global optimization. This new technique improves
final accuracy and the convergence speed of PSO with better performance. This new
strategy has been tested against a set of ten benchmark functions and compared with
four other PSO variants. The result shows that this new strategy is better and very
competitive in most of the cases than other PSO variants.
Keywords Particle swarm optimization (PSO) · Inertia weight · Exploration and
exploitation · Convergence
1 Introduction
PSO is a swarm-based meta-heuristic algorithm which is inspired by group behavior
of some animals like fish schools or bird flocks. It was initially introduced by Kennedy
and Eberhart [1]. As compared to the other optimization techniques, PSO is simple
and easy to implement since it has very few parameters to tune. The inertia weight is
one such important parameter of PSO. Initially, the basic version of PSO presented
by Eberhart and Kennedy [1] has no inertia weight. Eberhart and Shi [2] introduced
the concept of inertia weight for the first time by presenting constant inertia weight.
They discovered that a small inertia weight facilitates a local search and a large inertia
A. Agrawal (B) · S. Tripathi
National Institute of Technology, Raipur 492010, Chhattisgarh, India
e-mail: aagrawal.phd2017.cse@nitrr.ac.in; ankitagrawal648@gmail.com
S. Tripathi
e-mail: stripathi.cs@nitrr.ac.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_24
239
240
A. Agrawal and S. Tripathi
weight encourages a global search. After this first introduction of the inertia weight,
many researchers introduced the dynamically adjusting inertia weight which can
enhance the searching capabilities of the PSO. Also, PSO is subjected to theoretical
[3, 4] and empirical [5, 6] investigations by many researchers since then.
The remainder of this paper is organized as follows: the background of PSO; the
proposed inertia weight technique; the benchmark problems used to test the proposed
technique, other inertia weight strategies used for comparison, evaluation criteria and
results; and the conclusion of the paper.
2 Background
Like other evolutionary methods, in PSO, the population of feasible solution often
called as the swarm. These feasible solutions are known as the particles. Following
the current optimum solutions, N particles move in the D-dimensional search space
modifying their individual velocity iteratively. This current optimum solution is comprised of best position found by particle personally (i.e., the personal best or the pbest
position) and global best position found by entire swarm (i.e., the global best or the
gbest position). The position of ith particle is expressed as xi (xi1 , xi2 , . . . , xi D ),
where xid ∈ [L d , Ud ], d ∈ [1, D] and Ud and L d are the upper and lower limit of
the dth dimension of the search space. The velocity of ith particle is represented as
vi (vi1 , vi2 , . . . , vi D ). At each time step t (or iteration t), each particle updates
their respective velocity and position using following equations:
vid (t) ω ∗ vid (t − 1) + c1 ∗ R1id ∗ pbestid − xid (t − 1) + c2
∗ R2id ∗ gbestd − xid (t − 1)
(1)
xid (t) xid (t − 1) + vid (t),
(2)
where ω is inertia weight, R1id and R2id are uniformly distributed random numbers in
the range (0, 1), c1 and c2 are cognitive and social learning factors, pbestid and gbestd
are the personal best and the global best position of the ith particle. The performance
of PSO depends on the ability of the inertia weight to perform proper exploration
and exploitation of the search space.
Some famous adaptive methods were proposed that adjusts the inertia weight using
the feedback provided by process to gain better knowledge and control over population diversity. Some of the feedback parameters include the best fitness achieved
[9], the number of updated best positions [10], the standard deviation in components
of all particles [11] and the distance between particles [12], etc.
Particle Swarm Optimization with Probabilistic Inertia Weight
241
3 Proposed Inertia Weight Law
For proper balancing between the global and the local search ability of PSO, a new
inertia weight strategy is introduced, i.e., P-PSO. This strategy is inspired by the
idea that all the particles of the swarm must follow the particles which have shown
improvement in their respective position in the search space. For this purpose, the
binomial probability is used as a feedback parameter.
For calculation of inertia weight with above technique, we need to figure out the
situation of the swarm at each time step. The improvement in the position of particle
i at time step t can be determined as
⎧
⎨ 0 if fit pbestt ≥ fit pbestt−1
i
i
(3)
s(i, t) ⎩ 1 if fit pbestt < fit pbestt−1
i
i
Then we determine the total number of particles that have shown improvement in
their position at time step t as
S(t) N
s(i, t)
(4)
i1
Each time step is considered as an experiment, in which every particle act as a
trial and the repeated trials are independent since every particle is independent of
each other. Each trial have two possible outcomes only, i.e., a success and the other,
a failure. When the position of particle is improved then it is considered as a success
otherwise it is a failure. Since the probability of a success or a failure is equiprobable,
the probability of the success p, for each trial (particle) is taken as 0.5 (constant).
The total number of trials (swarm size) is N in all the experiments. Here the binomial
random variable is S(t), i.e., the total number of particles with improved position in
N trials of the binomial experiment. Then the binomial probability is the probability
of obtaining S(t) particles with improved position in N experiments. It can be written
as
P(S) C SN p S q N −S ,
(5)
where S S(t). Now, inertia weight as a linear function of P(S) is given as
ω ωmin + ωmax − ωmin ∗ P(S)
(6)
The number of particles with improved position varies in each experiment which
results in the variation of binomial probability and provides the necessary feedback to
the PSO for the next experiment. The nature of the inertia weight can be seen in Fig. 1
where the P-PSO is applied on the sphere function of dimension 30. Here the range
of the inertia weight [6] is [0.4, 0.5]. Initial values of ω are large in first few iterations
242
A. Agrawal and S. Tripathi
Fig. 1 Variations of the
inertia weight as a function
of the step number t
since cumulative binomial probability observed are higher. This happened as more
number of particles are moving towards the gbest position which is considered as
successful movement or improvement in the position of the particles. This forces
the particles of the P-PSO to perform the initial exploration of the search space.
Afterwards, the inertia weight converges towards its minimum value which is a
suitable value for the sphere function and fluctuates around it.
4 Experimental Setup
4.1 Benchmark Functions
In order to evaluate the performance and compare our inertia weight method with
other inertia weight strategies, we have used ten well-known optimization test problems [13]. All the problems are minimization problems. Detailed description of
these problems is provided in Table 1. The first five functions ( f 1 − f 5 ) are unimodal functions while rest of the functions ( f 6 − f 10 ) are multimodal functions. All
the functions have symmetrical search space except for f 10 which has asymmetrical
search space.
4.2 Algorithms Compared
Experiments were carried out for the comparative performance study of the introduced algorithm, i.e., P-PSO with following as in [11]: (a) w-PSO [11], (b) AIWPSO
[10], (c) Sugeno [14], and (d) GPSO [6].
Particle Swarm Optimization with Probabilistic Inertia Weight
243
Table 1 Benchmark functions used in experiments
Function
Mathematical representation
D 2
Sphere function
f 1 i1
xi
D−1
2 − x 2 ) + (x − 1)2 ]
Rosenbrock function
f 2 i1 [100(xi+1
i
i
D i
2
Rotated hyper-ellipsoid function f 3 i1 j1 x j
D
Sum squares function
f 4 i1
i xi2
D
Dixon price function
f 5 (x1 − 1)2 + i2
i(2xi2 − xi−1 )2
Ackley function
Rastrigin function
D
f 6 −20 exp −0.2 D1 i1
cos(xi2 ) −
D
exp − D1 i1
cos(2π xi ) + 20 + exp(1)
f 7 10D +
D
Schwefel function
f 8 418.9829 −
Griewank function
f9 D
f 10 Powell function
− 10 cos(2π xi )]
√
i1 x i sin( |x i |)
2
i1 [x i
xi2
i1 4000
D
−
D
i1 cos
D/4 xi
√
i
+1
(x4i−3 + 10x4i−2 )2 + 5(x4i−1 − x4i )2
+(x4i−2 − 2x4i−1 )4 + 10(x4i−3 − x4i )4
i1
4.3 Evaluation and Comparison Criteria
For better judgment of the performance and comparison of the algorithms, we conducted two different set of experiments on MATLAB. We run each algorithm 50
times using the benchmark functions in dimensions D 10 and D 30. For both
the experiments, swarm size is 30 and we took, c1 c2 2 as it follows the law
c1 + c2 < 4(1 + ω) [15]. In first experiment, mean value and the standard deviation
of the best solution are recorded after 3 × 105 functional evaluations (FEs). It helps
in judging the accuracy of the PSO variants. In second experiment, algorithm stops
when either it has achieved the solution with specified accuracy or a fixed number
of FEs (105 for D 10 and 3 × 105 for D 30) have been carried. Number of
functional evaluations (FEs) required to achieve the solution with their respective
accuracy have been recorded for this set of experiment. Also, results are recorded
only when the algorithm are successful in finding solution at least 15 times out of 50
runs and only successful runs are considered for recording the solution. The lower
number of FEs corresponds to the faster algorithm.
244
A. Agrawal and S. Tripathi
4.4 Results and Discussion
On ten optimization test problems, four other inertia weight techniques are applied
and the result is compared with those of P-PSO. In both the experiments, the best
solutions were mostly found by the new algorithms. Considering first experiment
(Table 2), for dimension D 10 (Fig. 2), P-PSO outperformed other algorithms
for functions f 1 , f 3 , f 4 , f 10 and yield comparable result for functions f 2 , f 5 , f 6 , f 8
and f 9 . It performed worst for function f 7 . w-PSO was slightly better for functions
f 2 and f 5 . For dimension D 30 (Fig. 2), P-PSO outperformed others for functions f 1 , f 2 , f 3 , f 4 and yield comparable result for functions f 5 , f 8 , f 9 , f 10 and it
performed worst for function f 6 and f 7 . w-PSO performed better for functions f 5 ,
f 7 and f 8 , while Sugeno for function f 6 .
Considering the second experiment (Table 3), for functions f 2 , f 5 , f 7 , f 8 in both
dimensions and for function f 9 in dimension 10, no algorithms were able to find
solution following the criteria mentioned in Sect. 4.3. In rest of the cases, P-PSO
outperforms all the other algorithms. P-PSO was followed by AIWPSO. Other algorithms took considerable FEs to find the solution of specified accuracy.
Depending on the number of times the algorithm yield the best and competitive
outcomes in first experiment, all the algorithms are scored. The scores of P-PSO,
GPSO, Sugeno, AIWPSO, w-PSO are 17, 10, 11, 15, and 13, respectively. The
convergence curve of different algorithms gives insight of their searching behavior.
In logarithmic scale, for function f 1 , f 3 and f 4 (Fig. 2), P-PSO’s convergence graph
is straight line which indicates that its convergence rate remains constant during
complete course of run. Also the performance of P-PSO does nt deteriorate when the
dimension of search space in increased. However, this is not the case for the other
algorithms.
P-PSO converges towards the best solution very fast in comparison to other algorithms. This behavior can be seen more clearly in result of second experiment. P-PSO
is clear winner here as no other algorithms were able to locate solution of specified
accuracy taking less number of FEs than P-PSO. This experiment also proves that
the P-PSO has very less complexity in comparison to other algorithms.
The above result shows that the P-PSO is significantly better in most of the cases
then other algorithms and very competitive in nature. It has extraordinary capability
to converge quickly towards the solution.
5 Conclusions
In this paper, PSO algorithm (P-PSO) with novel adaptive inertia weight is proposed
for global optimization. The aim of the study is to balance the exploration and
exploitation effectively as the algorithm progress. The proposed adaptive inertia
weight is dynamic in nature and in order to improve the position of the particle, it
varies in range [0.4, 0.5] using particles’ best position as feedback. The movement
Particle Swarm Optimization with Probabilistic Inertia Weight
245
Table 2 Mean and standard deviation of the best solutions in 50 runs for 5 PSO variants
Fun Dim Mean best fitness (standard deviation)
f1
f2
f3
f4
f5
f6
f7
f8
f9
f 10
P-PSO
GPSO
10
0 (0)
7.07263e−247 0(0)
(0)
Sugeno
30
7.97375e−180 6.00000e+02 2.00000e+02 1.69582e−134 2.00001e+02
(0)
(2.39898e+03) (1.41421e+03) (7.27707e−134) (1.41421e+03)
10
1.02264e+04 1.51293e+04 3.51845e+04 1.50159e+04 5.10888e+03
(4.94510e+04) (5.99464e+04) (8.75565e+04) (5.99711e+04) (3.53472e+04)
30
2.52185e+04
(0)
7.55031e+04 6.57812e+04 4.51346e+04 5.56823e+04
(1.15414e+05) (1.10335e+05) (9.69703e+04) (1.04276e+05)
10
0(0)
8.58993e+01 8.58993e+01 0(0)
(6.07400e+02) (6.07400e+02)
30
2.66288e+03 1.65786e+04 1.20259e+04 5.23986e+03 6.78605e+03
(5.48036e+03) (1.56432e+04) (1.60703e+04) (1.01671e+04) (1.00853e+04)
10
0(0)
30
7.20000e+01 2.66000e+02 3.70000e+02 1.72000e+02 1.50000e+02
(1.40029e+02) (2.88281e+02) (4.08706e+02) (2.39080e+02) (2.67452e+02)
10
1.02289e+01 4.22467e+01 5.83770e+01 1.03293e+01 3.74667e+00
(4.76932e+01) (9.74518e+01) (1.11037e+02) (4.77097e+01) (1.59316e+01)
30
1.50943e+03 1.25790e+04 6.76249e+03 5.39596e+03 7.93700e+01
(1.02141e+04) (5.25425e+04) (2.34556e+04) (2.92704e+04) (1.20152e+02)
10
4.22773E−15 4.36984E−15 4.29878E−15 4.22773e−15 1.05878e−06
(8.52289e−16) (5.02430e−16) (7.03255e−16) (8.52289e−16) (1.66154e−06)
30
9.16059e−01 2.85606e−01 8.06466E−15 3.54019e−01 2.91506e−01
(9.89416e−01) (2.01954e+00) (1.13260e−15) (2.02777e+00) (2.02005e+00)
10
5.85036e+00 7.36270e−01 2.12921e+00 5.17520e+00 1.01486e+00
(3.57123e+00) (7.73570e−01) (1.15490e+00) (4.85731e+00) (1.07292e+00)
30
7.97187e+01 6.39798e+01 6.51141e+01 6.89191e+01 4.56611e+01
(2.96679e+01) (2.15895e+01) (2.33054e+01) (2.19724e+01) (2.35280e+01)
10
6.55026e+02 8.43125e+02 8.45524e+02 6.41692e+02 6.25049e+02
(2.66293e+02) (3.20994e+02) (2.72643e+02) (2.29994e+02) (2.72901e+02)
30
3.33331e+03 3.73524e+03 3.71089e+03 3.21875e+03 2.56594e+03
(6.51836e+02) (7.23630e+02) (6.32618e+02) (7.64587e+02) (7.74013e+02)
10
7.02596e−02 5.89321e−02 5.59861e−02 7.84152e−02 6.84178e−02
(3.15837e−02) (2.38919e−02) (2.64845e−02) (3.72866e−02) (2.96336e−02)
30
1.41369e−02 2.06286e−02 1.09784e−02 1.82514e+00 1.83800e+00
(1.92449e−02) (2.62049e−02) (1.17534e−02) (1.28294e+01) (1.27965e+01)
10
3.01875e+00 7.05632e+00 1.10996e+01 3.64936e+00 8.17172e+00
(1.22930e+01) (2.56088e+01) (2.75300e+01) (1.83486e+01) (2.40510e+01)
30
1.35398e+02 6.22940e+02 3.39737e+02 1.15016e+02 1.15934e+02
(3.37763e+02) (7.04093e+02) (4.14255e+02) (2.05672e+02) (1.11562e+02)
3.07229e−248 0(0)
(0)
AIWPSO
w-PSO
0(0)
4.33650e−12
(8.64629e−12)
2.69553e−11
(7.18060e−11)
2.00000e+00 1.59699e−12
(1.41421e+01) (5.01795e−12)
246
A. Agrawal and S. Tripathi
(a) Function f1 (D=10)
(b) Function f
(D=30)
(c) Function f
(D=10)
(d) Function f
(D=30)
(D=10)
(f) Function f
(D=30)
3
(e) Function f
9
1
3
9
Fig. 2 Mean of best fitness for 50 independent runs as a function of step number
of particle is determined by the probability of locating global optimum by particles
which have shown improvement in their position in comparison to their last position.
The new approach is experimented on ten common optimization test problems
and compared with other four inertia weight settings. The result shows that the PPSO converges very fast towards the best solution in comparison to other algorithms.
The performance of P-PSO is better and competitive to other algorithms in almost
all cases. The performance of P-PSO is intact in the higher dimensions, which is not
the case for other algorithms.
Particle Swarm Optimization with Probabilistic Inertia Weight
247
Table 3 Mean and standard deviation of FEs required out of 50 runs to find the solution with
specified accuracy for 5 PSO variants
Fun Dim Accuracy
Mean of functional evaluations (standard deviation)
f1
P-PSO
GPSO
Sugeno
AIWPSO
w-PSO
10
1.00E−03
2566.80
(264.83)
41275.20
(1825.47)
10981.20
(526.57)
3662.40
(276.54)
38011.80
(3680.79)
30
1.00E−02
8462.50
(545.04)
155344.90
(4328.88)
41190.63
(1416.61)
11388.37
(721.53)
117583.80
(16269.26)
f2
10
1.00E−03
0
0
0
0
0
f3
30
10
1.00E−02
1.00E−03
0
2915.51
(247.06)
0
42668.40
(1306.18)
0
11665.80
(576.07)
0
4043.40
(318.78)
0
43027.80
(5096.82)
30
1.00E−02
10556.67
(734.35)
0
0
14299.23
(1919.98)
180319.09
(31550.40)
10
1.00E−03
2428.80
(221.70)
40323.00
(1663.08)
10620.63
(505.36)
3454.80
(314.72)
34839.60
(5413.48)
30
1.00E−02
8690.67
(902.40)
0
0
11798.57
(793.98)
125583.64
(17569.97)
f5
10
1.00E−03
0
0
0
0
0
f6
30
10
1.00E−02
1.00E−03
0
4176.73
(468.23)
0
47483.40
(1060.37)
0
14255.40
(622.17)
0
5814.00
(437.44)
0
70581.60
(9584.57)
30
1.00E−02
13611.43
(2519.77)
168284.08
(3870.67)
49020.00
(2640.72)
17121.52
(2394.67)
204951.43
(44319.53)
f7
10
1.00E−03
0
0
0
0
0
f8
30
10
1.00E−02
1.00E−03
0
0
0
0
0
0
0
0
0
0
f9
30
10
1.00E−02
1.00E−03
0
0
0
0
0
0
0
0
0
0
30
1.00E−02
7116.21
(1223.67)
151290.00
(5439.28)
39906.25
(5025.36)
10216.45
(1931.14)
72480.00
(14652.83)
10
1.00E−03
2956.88
(619.81)
40539.47
(3592.61)
11514.32
(1112.56)
4044.38
(950.87)
18847.17
(12104.98)
30
1.00E−02
16750.59
(2294.39)
0
0
0
0
f4
f 10
References
1. Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of IEEE International
Conference on Neural Networks, vol. 4, pp. 1942–1948, Perth, Australia (1995)
2. Eberhart, R.C., Shi, Y.: A modified particle swarm optimizer. In: Proceedings of the IEEE
Congress on Evolutionary Computation (CEC’98), pp. 69–73, Anchorage, AK (1998)
3. Clerc, M., Kennedy, J.: The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Trans. Evol. Comput. 6(1), 58–73 (2002)
248
A. Agrawal and S. Tripathi
4. Kadirkamanathan, V., Selvarajah, K., Fleming, P.J.: Stability analysis of the particle dynamics
in particle swarm optimizer. IEEE Trans. Evol. Comput. 10(3), 245–255 (2006)
5. Kennedy, J.: Bare bones particle swarms. In: Proceedings of the 2003 IEEE Swarm Intelligence
Symposium, 2003, SIS’03, pp. 80–87 (2003)
6. Shi, Y., Eberhart, R.C.: Empirical study of particle swarm optimization. In: Proceedings of the
IEEE Congress, CEC 99, vol. 3, pp. 1945–1950, Washington, DC (1999)
7. Eberhart, R.C., Shi, Y.: Tracking and optimizing dynamic systems with particle swarms. In:
Proceedings of the IEEE Congress on Evolutionary Computation, vol. 1, pp. 94–100, Seoul,
South Korea (2001)
8. Xin, J., Chen, G., Hai, Y.: A particle swarm optimizer with multi-stage linearly-decreasing
inertia weight. In: International Joint Conference on Computational Sciences and Optimization,
vol. 1, pp. 505–508. IEEE, New York (2009)
9. Nikabadi, A., Ebadzadeh, M.: Particle swarm optimization algorithms with adaptive inertia
weight: a survey of the state of the art and a novel method. IEEE J. Evol. Comput. (2008)
10. Nickabadi, A., Ebadzadeh, M.M., Safabakhsh, R.: A novel particle swarm optimization algorithm with adaptive inertia weight. Appl. Soft Comput. 11, 3658–3670 (2011)
11. Kessentini, S., Barchiesi, D.: Particle swarm optimization with adaptive inertia weight. Int. J.
Mach. Learn. Comput. 5(5), 368–373 (2015)
12. Zhan, Z.-H., Zhang, J., Li, Y., Chung, H.S.-H.: Adaptive particle swarm optimization. IEEE
Trans. Syst. Man Cybern.-Part B: Cybern. 39, 1362–1381 (2009)
13. Surjanovic, S., Bingham, D.: Virtual Library of Simulation Experiments: Test Functions and
Datasets, http://www.sfu.ca/~ssurjano/ (2013)
14. Lei, K., Qiu, Y., He, Y.: A new adaptive well-chosen inertia weight strategy to automatically
harmonize global and local search ability in particle swarm optimization. In: 1st IEEE International Symposium on Systems and Control in Aerospace and Astronautics, pp. 977–980,
Harbin (2006)
15. Martínez, J.F., Gonzalo, E.G.: The PSO family: deduction, stochastic analysis and comparison.
Swarm Intell. 3(4), 245–273 (2009)
An Evolutionary Algorithm Based
Hyper-heuristic for the Job-Shop
Scheduling Problem with No-Wait
Constraint
Sachchida Nand Chaurasia , Shyam Sundar , Donghwi Jung ,
Ho Min Lee and Joong Hoon Kim
Abstract In this paper, we developed an evolutionary algorithm with guided mutation (EA/G) based hyper-heuristic for solving the job-shop scheduling problem with
no-wait constraint (JSPNW). The JSPNW is an extension of well-known job-shop
scheduling problem subject to the constraint that no waiting time is allowed between
operations for a given job. This problem is a typical N P-hard problem. The hyperheuristic algorithm comprises of two level frameworks. In the high-level, an evolutionary algorithm is employed to explore the search space. The low-level, which is
comprised of generic as well as problem-specific heuristics such as guided mutation,
multi-insert points and multi-swap. EA/G is a recent addition to the class of evolutionary algorithm that can be considered as a hybridization of genetic algorithms (GAs)
and estimation of distribution algorithms (EDAs), and which tries to overcome the
shortcomings of both. In GAs, the location information of the solutions found so far
is directly used to generate offspring. On the other hand, EDAs use global statistical
information to generate new offspring. In EDAs the global statistical information is
stored in the form probability vector, and a new offspring is generated by sampling
this probability vector. We have compared our approach with the state-of-the-art
approaches. The computational results show the effectiveness of our approach.
Keywords Scheduling · Job-shop · No-wait · Genetic algorithms · Constrained
optimization · Estimation of distribution algorithms · Guided mutation
Heuristic · Hyper-heuristic
S. N. Chaurasia · D. Jung · H. M. Lee
Research Center for Disaster Prevention Science and Technology,
Korea University, Seoul 136-713, South Korea
S. Sundar
Department of Computer Applications, National Institute of Technology,
Raipur, India
J. H. Kim (B)
School of Civil, Environmental and Architectural Engineering,
Korea University, Seoul 136-713, South Korea
e-mail: jaykim@korea.ac.kr
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_25
249
250
S. N. Chaurasia et al.
1 Introduction
This paper addresses the job-shop scheduling problem with no-wait constraint
(JSPNW) which is an extension of well-known job-shop scheduling problem. In
the JSPNW, an additional no-wait constraint is imposed, i.e., once a job starts its
operations on multiple machines, will continue without any interruption. In other
words, if a job starts its operations for a given sequence of operations, then there
will be no time gap or wait time between any consecutive operations. The imposed
no-wait constraint makes the JSPNW N P-hard in strong sense [1].
The JSPNW has many applications in real world optimization problems. Typical
examples of the application of JSPNW are metallurgical processing, scheduling
of perishable products whose decay rate is very high with time. The JSPNW has
applications in many other industries such as steel industry [2], chemical industries
[3], and so on. The applications of the JSPNW can increase the throughput of the
production in industries.
Mascis and Pacciarelli [4] mapped the JSPNW to an alternative graph, considered as a generalization of the disjunctive graph [5], and presented a branch & bound
and four greedy heuristics. Macchiaroli et al. [6] proposed a two-phase tabu search
algorithm and results show that the tabu search has better performance. A simple
encoding and decoding based genetic algorithm is proposed by Brizuela et al. [7].
Several approaches have been proposed to handle the two subproblems, sequencing
problem and timetabling problem, of JSPNW such as variable neighborhood search
method (VNS) [8], hybrid simulated annealing/genetic algorithms [8], complete local
search with memory (CLM) [9]. Further, a complete local search with limited memory (CLLM) [10], and modified complete local search with memory (MCLM) [11]
have also been proposed.
In this paper, we present an evolutionary algorithm with guided (EA/G) based
hyper-heuristic (HH) where EA/G is employed as a higher level in hyper-heuristic.
Hereafter, EA/G based HH will be referred as HH-EA/G. The rest of the paper is
structured as follows: Sect. 2 describes the JSPNW. Section 3 presents the overview
of hyper-heuristics (HH) and evolutionary algorithm with guided mutation. Section 4
dedicated to HH-EA/G for the JSPNW. Section 5 presents the computational results.
And finally, conclusions and future work are described in Sect. 6.
2 Problem Description
The JSPNW is a scheduling problem with an additional no-wait constraint in which
n jobs are processed on m machines in a predefined sequence of operations on
machines. A processing of a job on a machine is called operation. Any machine
can process only one job at any given time, and a job can be processed only once on
any given machine. Each job Ji has Ni operations. The notational representation of
the JSPNW is as follows:
An Evolutionary Algorithm Based Hyper-heuristic …
–
–
–
–
–
–
–
251
n ∈ N number of jobs
m ∈ M number of machines
J = {J1 , J2 , . . . , Jn } set of jobs
M = {M1 , M2 , . . . , Mm } set of machines
Ni is number of operations of job ji
Oik ∈ {Oi1 , Oi2 , . . . , Oi Ni } is the k th operation of job Ji
Pi jk is the processing time of operation Oi j on machine k
The JSPNW has been proven to be N P-hard even for two-machines no-wait job
shops [12]. The JSPNW is the composition of two subproblems: first is the sequencing problem in which an optimal processing sequence is searched, and the second
subproblem is the timetabling problem in which a feasible set of start times of the
jobs from the first sequencing problem is obtained to minimize makespan for the
processing. Both the subproblems are also proven to be N P-hard in strong sense
[13, 14]. The goal of the JSPNW is to minimizing the maximum completion time
among all the jobs.
3 Overview of Hyper-heuristic and EA/G
3.1 Hyper-heuristic
No free lunch theorem [15] states that no single algorithm is capable to perform
well overall problems. Even for the same problem, its performance can change with
variation in constraints, size of instances, and so on. As a result, for a new problem or
for the variants of the same problem metaheuristics need to modify or re(developed)
to get satisfactory solution. Modification in metaheuristic or re(development) of new
metaheuristic is not only time consuming but also needs parameters tuning which is
itself a challenging task. Apart from this, all the time need to do deep learning of the
structure of the instances and of the algorithm.
In 1997, Denzinger et al. [16] introduced the term hyper-heuristic (HH) to explain
a protocol, which combines several artificial intelligence techniques in the context of
automated theorem proving. Later, in 2000, Cowling et al. [17] used this term as an
independent term to explain a heuristic to select heuristic(s) for solving optimization
problems.
3.2 Estimation of Distribution Algorithms (EDAs) for
Scheduling Problems
There are many variants of EDAs [18, 19] that have been proposed for solving the
scheduling problems. In [18], Wang et al. proposed an EDA based approach to solve a
252
S. N. Chaurasia et al.
flexible job-shop scheduling problem. In [20], an effective EDA approach is proposed
to solve a stochastic job-shop scheduling problem with an uncertainty of processing
time, and the objective is to minimize the expected average makespan within a reasonable amount of calculation time. Inspired by the success of evolutionary algorithm
with guided mutation of [21–24], Chaurasia and Alok [25] proposed an extension of
evolutionary algorithm with guided mutation for single machine order acceptance
and scheduling problem and the computational results show the effectiveness of the
proposed approach.
4 EA/G Based Hyper-heuristic (HH-EA/G)
In this work, we have developed a new HH-EA/G approach. The EA/G is employed
as a higher level algorithm. At the higher level, we employed Guided heuristic (GH)
as a heuristic to select a heuristic from the pool of low-level heuristics which reside
at the lower level. The main components of HH-EA/G for the JSPNW are described
below:
4.1 Solution Encoding
Each solution is represented as a permutation of jobs. The sequence of jobs makes
sure that jobs will be processed in the given order. For example, suppose five jobs 1,
2, 3, 4, 5 are sequenced as {4, 5, 1, 3, 2}. That means the system will start processing
from job 4 then job 5, and so on.
4.2 Initial Solution
The initial population is generated randomly.
4.3 Higher Level Search Methodology
The proposed higher level of hyper-heuristic is consist of two steps. In the fist step, a
credit is assigned to each heuristic with help of Credit assignment procedure. And, the
second step is heuristic selection rule in which a guided heuristic, which is inspired
by the guided mutation operator of [25], is employed to select a heuristic from a
set of low-level heuristics. The following sections explain the credit assignment and
heuristic selection rule.
An Evolutionary Algorithm Based Hyper-heuristic …
253
1. Credit assignment:
A two-dimensional fitness matrix, say FMatri x , consists of W rows and H
columns where W and H represent the window size and number of heuristics, respectively. In matrix FMatri x , each value, say F j Hi , represents the fitness
returned by the heuristic Hi at stage j in the current generation. After each generation, the fitness matrix FMatri x is updated using first-in-first-out (FIFO). FIFO
means, when the (W + 1)th fitness is appended into the window, then the first
will be removed. The FMatri x is initialized using W × H number of solutions’s
fitness. Similar to the initialization of probability vector p of [25], the probability
vector, say δ H , for the heuristics is initialized. f j Hi is the fitness of Hi at stage j
and max( f k Hi ) is the best fitness returned by heuristic Hi for k = 1, 2, . . . , W .
Similar to the method of updating of the probability vector p of [25], after each
generation, the probability vector δ H is updated. A temporary window size, say Z
(1 ≤ Z ≤ W ) i.e, last Z rows in FMatri x is used for learning from the temporary
window.
2. Heuristic selection rule: A guided heuristic (GH) is employed to select a heuristic to generate a new offspring. Similar to the guided mutation of [25], the
guided heuristic (GH) uses the both the global information which is stored in
the form of probability and the location information about the fitness of the Z
solutions to generate new heuristic. A random value r1 is generated uniformly
in [0, 1] and if the r1 is less than the probability of heuristic Hi then the heuristic
Hi is returned by the guided heuristic, and then applied to generated an offspring. Otherwise, a heuristic is selected with the highest probability among all
the heuristics.
4.4 Lower Level Heuristics
The lower level contains four problem-specific heuristics viz. H1, H2, H3, and H4.
The detailed description about the heuristics is presented in Table 1.
5 Computational Results
The proposed approach has been coded in C language and executed on a Linux
based operating system with 3.30 GHz Intel Core i5-4590 processor and
4GB RAM. gcc 5.4.0 compiler with O3 flag has been used to compile the C
code. For the HH-EA/G, we set the following parameters values: Pop si ze = 250,
par ent si ze = 125, N umber o f generation = 200, λ = 0.65, β = 0.75. sw p = 3,
M p = 4 W = 400 and Z = W/4. The local search is applied for four consecutive
successful improvements. The local search is applied when the difference between
254
S. N. Chaurasia et al.
Table 1 Constructive low-level heuristics for the SPP
H1
H2
H3
H4
Guided mutation operator [25]: The proposed G M is inspired by the guided
mutation of [25]. A solution is generated in the same manner as the G M of [25]
generates new offspring
Multi-point insertion [26]: First two different solutions say s1 and s2 are selected
from the population. Then randomly M p number of positions are selected from
solution s1 and the jobs at these positions are copied at, exactly, the same positions in
the partial solution say s3 . After that remaining positions in s3 are filled by the
remaining jobs from the solution s2 in the same order as they appeared in s2
Multi-swap [26]: Two different positions of jobs are selected uniformly at random
and jobs at the selected positions are swapped. This process is repeated sw p of times
Local Search [26]: Improves the solution iteratively by interchanging the adjacent as
well as nonadjacent jobs
current and the global best solution is less than 50% of the global best solution.
HH-EA/G is executed 20 independent times with random seeds on each instance.
We compared our approach with the state-of-the-art approaches viz. CLLM [10],
MCLM [11] and HABC [26]. The computational results are presented in Table 2. In
Table 2, columns PRD is percentage deviation (PRD) from the best-known solution
(BKS), APRD is the average PRD of 20 independent runs and ATET is the average
computational time. In Table 2, the last row indicates the average of PRD, APRD and
ATET for each approach. From Table 2, it is confirmed that HH-EA/G has, overall,
better performance than CLLM [10], MCLM [11] and HABC [26] in terms of APRD
and ATET.
6 Conclusions
This paper presented an evolutionary algorithm with guided mutation based hyperheuristic for the job-shop scheduling problem with no-wait constraint (JSPNW).
The proposed approach, as far as the authors’s knowledge, is the first evolutionary
algorithm based hyper-heuristic for the JSPNW. In this paper, we considered only
small size instances and compared with the state-of-the-art approaches. The computational results show that our approach is able to achieve better or equal results in
lesser computational time.
As a future work, we would like to include many number of problem-specific
heuristics at the lower level and also improve the heuristic selection procedure at the
higher level.
(6, 6)
(10, 5)
(10, 5)
(10, 5)
(10, 5)
(10, 5)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
(10, 10)
Ft06
La01
La02
La03
La04
La05
Ft10
Orb01
Orb02
Orb03
Orb04
Orb05
Orb06
Orb08
Orb09
Orb10
La16
La17
La18
La19
La20
AVG
(n, m)
Instance
1526
1482
1417
1371
1575
1557
1445
1319
1555
1365
1653
1599
1485
1615
1607
777
887
820
937
971
73
BKS
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.15
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
PRD
0.27
0.13
0.14
1.13
0.22
0.00
0.00
0.83
0.00
0.00
0.15
0.42
0.00
1.89
0.00
0.00
0.50
0.00
0.00
0.21
0.10
0.00
APRD
CLLM
37.95
6.00
16.00
51.00
53.00
41.00
69.00
19.00
60.00
44.00
14.00
40.00
71.00
48.00
72.00
92.00
8.00
34.00
12.00
33.00
14.00
0.00
ATET
0.41
1.31
0.00
2.82
0.00
1.84
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2.16
0.00
0.00
0.51
0.00
0.00
0.00
0.00
0.00
PRD
0.47
1.31
0.61
2.82
0.12
1.84
0.00
0.00
0.00
0.00
0.00
0.12
0.00
2.16
0.00
0.00
0.90
0.00
0.00
0.00
0.02
0.00
APRD
MCLM
5.68
5.40
5.00
6.15
11.85
5.65
11.85
4.25
6.40
3.55
8.50
7.85
13.75
6.70
6.65
7.85
3.90
6.25
3.10
7.60
4.55
0.00
ATET
1526
1482
1417
1384
1575
1557
1445
1319
1555
1370
1653
1599
1485
1615
1607
781
887
820
961
975
73
ValueHABC
Table 2 Comparison of results obtained by CLLM [10], MCLM [11], HABC [26] and HH-EA/G
0.23
0.00
0.00
0.00
0.95
0.00
0.00
0.00
0.00
0.00
0.37
0.00
0.00
0.00
0.00
0.00
0.51
0.00
0.00
2.56
0.41
0.00
0.50
0.00
0.43
5.22
0.95
0.00
0.00
0.28
0.00
0.00
0.37
0.00
0.00
0.00
0.00
0.00
0.51
0.00
0.00
2.56
0.41
0.00
APRD
HABC
PRD
4.63
3.00
3.44
7.13
4.62
5.43
5.51
4.10
8.99
5.76
4.77
5.28
10.07
6.86
8.00
9.86
1.14
0.87
0.70
1.41
0.78
0.15
ATET
1526
1482
1417
1384
1557
1557
1445
1319
887
1370
1653
1599
1485
1615
1607
781
887
820
961
975
73
BEST
0.23
0.00
0.00
0.00
0.95
0.00
0.00
0.00
0.00
0.00
0.37
0.00
0.00
0.00
0.00
0.00
0.51
0.00
0.00
2.57
0.41
0.00
0.49
0.00
1.01
3.56
0.95
0.00
0.38
0.87
0.04
0.00
0.37
0.04
0.14
0.78
0.04
0.00
0.51
0.04
0.50
2.57
0.41
0.00
APRD
HH-EA/G
PRD
1.26
1.27
1.35
1.40
1.50
1.60
1.60
1.57
1.70
1.50
1.50
1.57
2.00
1.47
1.57
1.80
0.56
0.57
0.57
0.67
0.57
0.15
ATET
An Evolutionary Algorithm Based Hyper-heuristic …
255
256
S. N. Chaurasia et al.
Acknowledgements This work was supported by the grant [13AWMP-B066744-01] from the
Advanced Water Management Research Program funded by the Ministry of Land, Infrastructure,
and Transportation of the Korean government.
References
1. Lenstra, J.K., Rinnooy Kan, A.H.G., Brucker, P.: Complexity of machine scheduling problems.
Ann. Discret. Math. 1, 343–362 (1977)
2. Dorn, J., Shams, R.: Scheduling high-grade steelmaking. IEEE Expert 11, 28–35 (1996)
3. Hall, N.J., Sriskandarajah, C.: A survey on machine scheduling problems with blocking and
no-wait in process. Oper. Res. 44, 510–525 (1996)
4. Mascis, A., Pacciarelli, D.: Job-shop scheduling with blocking and no-wait constraints. Eur. J.
Oper. Res. 143, 498–517 (2002)
5. Roy, B., Sussmann, B.: Les problèmes d’ordonnancement avec constraintes disjonctives.
SEMA, Note D.S., No. 9, Paris (1964)
6. Macchiaroli, R., Molè, S., Riemma, S., Trifiletti, L.: Design and implementation of a Tabu
search algorithm to solve the no-wait job-shop scheduling problem. In: Proceeding of the
CESA96, pp. 467–472 (1996)
7. Brizuela, C.A., Zhao, Y., Sannomiya, N.: No-wait and blocking job-shops: challenging problems for GA’s. In: IEEE International Conference on Systems, man, and cybernetics, pp. 2349–
2354. IEEE, New York (2001)
8. Schuster, C.J., Framinan, J.M.: Approximate procedures for no-wait job shop scheduling. Oper.
Res. Lett. 31, 308–318 (2003)
9. Schuster, C.J.: No-wait job shop scheduling: Tabu search and complexity of subproblems.
Math. Methods Oper. Res. 63, 473–491 (2006)
10. Zhu, J., Li, X., Wang, Q.: Complete local search with limited memory algorithm for no-wait
job shops to minimize makespan. Eur. J. Oper. Res. 198, 378–386 (2009)
11. Zhu, J., Li, X.: An effective meta-heuristic for no-wait job shops to minimize makespan. IEEE
Trans. Autom. Sci. Eng. 9, 189–198 (2012)
12. Sahni, S., Cho, Y.: Complexity of scheduling shops with no wait in process. Math. Oper. Res.
4, 448–457 (1979)
13. Schuster, C.: No-wait job shop scheduling: complexity under local search. Ph.D. thesis, Dissertation Thesis, University of Duisburg-Essen, Germany (2003)
14. Sriskandarajah, C., Ladet, P.: Some no-wait shops scheduling problems: complexity aspect.
Eur. J. Oper. Res. 24(3), 424–438 (1986)
15. Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evol.
Comput. 1(1), 67–82 (1997)
16. Denzinger, J., Fuchs, M., Fuchs, M.: High performance ATP systems by combining several AI
methods. In: Proceedings of the 15th International Joint Conference on Artificial Intelligence
- Vol. 1, IJCAI’97, pp. 102–107. San Francisco, CA, USA. Morgan Kaufmann Publishers Inc.,
Burlington (1997)
17. Cowling, P., Kendall, G., Soubeiga, E.: A Hyperheuristic Approach to Scheduling a Sales
Summit, pp. 176–190. Springer, Berlin (2001)
18. Wang, S., Wang, L., Zhou, G., Xu, Y.: An Estimation of Distribution Algorithm for the Flexible
Job-Shop Scheduling Problem, pp. 9–16. Springer, Berlin (2012)
19. Salhi, A., Rodríguez, J.A.V., Zhang, Q.: An estimation of distribution algorithm with guided
mutation for a complex flow shop scheduling problem. In: Proceedings of the 2007 Genetic
and Evolutionary Computation Conference (GECCO 2007), pp. 570–576. ACM, New York
(2007)
20. He, X.-J., Zeng, J.-C., Xue, S.-D., Wang, L.-F.: An Efficient Estimation of Distribution Algorithm for Job Shop Scheduling Problem, pp. 656–663. Springer, Berlin (2010)
An Evolutionary Algorithm Based Hyper-heuristic …
257
21. Zhang, Q., Sun, J., Tsang, E.: An evolutionary algorithm with guided mutation for the maximum
clique problem. IEEE Trans. Evol. Comput. 9, 192–200 (2005)
22. Chaurasia, S.N., Singh, A.: A hybrid heuristic for dominating tree problem. Soft. Comput.
20(1), 377–397 (2016)
23. Chaurasia, S.N., Singh, A.: A hybrid evolutionary algorithm with guided mutation for minimum
weight dominating set. Applied Intelligence 43(3), 512–529 (2015)
24. Chaurasia, S.N., Sundar, S., Singh, A.: A hybrid evolutionary approach for set packing problem.
Opsearch 52(2), 271–284 (2015)
25. Chaurasia, S.N., Singh, A.: Hybrid evolutionary approaches for the single machine order acceptance and scheduling problem. Appl. Soft Comput. J. 52, 725–747 (2017)
26. Sundar, S., Suganthan, P.N., Jin, C.T., Xiang, C.T., Soon, C.C.: A hybrid artificial bee colony
algorithm for the job-shop scheduling problem with no-wait constraint. Soft. Comput. 21(5),
1193–1202 (2017)
An Evolutionary Algorithm Based
Hyper-heuristic for the Set Packing
Problem
Sachchida Nand Chaurasia , Donghwi Jung , Ho Min Lee
and Joong Hoon Kim
Abstract Utilizing knowledge of the problem of interest and lessons learned from
solving similar problems would help to find the final optimal solution of better quality.
A hyper-heuristic algorithm is to gain an advantage of such process. In this paper, we
present an evolutionary algorithm based hyper-heuristic framework for solving the set
packing problem (SPP). The SPP is a typical N P-hard problem. The hyper-heuristic
is comprising of high level and low level. The higher level is mainly engaged in generating or constructing a heuristic. An evolutionary algorithm with guided mutation
(EA/G) is employed at the high level. Whereas a set of problem-independent and
problem-specific heuristics, called low level heuristics, are employed at the low level
of hyper-heuristic. EA/G is recently added to the class of the evolutionary algorithms
that try to utilize the complementary characteristics of genetic algorithms (GAs) and
estimation of distribution algorithms (EDAs) to generate new offspring. In EA/G, the
guided mutation operator generates an offspring by sampling the probability vector.
The proposed approach is compared with the state-of-the-art approaches reported
in the literature. The computational results show the effectiveness of the proposed
approach.
Keywords Set packing problem · Constrained optimization · Genetic algorithm
Estimation of distribution algorithm · Guided mutation · Heuristic
Hyper-heuristic
S. N. Chaurasia · D. Jung · H. M. Lee
Research Center for Disaster Prevention Science and Technology,
Korea University, Seoul 136-713, South Korea
J. H. Kim (B)
School of Civil, Environmental and Architectural Engineering,
Korea University, Seoul 136-713, South Korea
e-mail: jaykim@korea.ac.kr
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_26
259
260
S. N. Chaurasia et al.
1 Introduction
The set packing problem (SPP) is a typical N P-hard combinatorial optimization
problem [1]. The SPP is formally defined as follows: I = {1, . . . , n} is a set of n
objects and T j , j ∈ J = {1, 2, . . . , m} a list of m subset of set I , a packing P ⊆ I
is a subset of set I such that |T j ∩ P| ≤ 1, ∀ j ∈ J , i.e., at most one object of the
set T j can be in packing P. Each subset T j , j ∈ J = {1, . . . , m} is considered as a
set of constraints between some objects of set I . A weight function assigns positive
weight, ci to each object i in the set I . The aim of the SPP is to make a subset P ⊆ I
that maximizes the sum of weights of objects in set P. We follow the same notation
to formulate the SPP as used in [2], and formally, formulated as:
Max z =
ci xi
(1)
i∈I
ti, j xi ≤ 1, ∀ j ∈ J
(2)
i∈I
xi ∈ {0, 1}, ∀i ∈ I
(3)
ti, j ∈ {0, 1}, ∀i ∈ I, ∀ j ∈ J
(4)
0, if i ∈ P
1, otherwise
– a vector c = (ci ) where ci = cost
of an object i
0, if i ∈ T j
– a vector t = (ti j ) where ti j =
1, otherwise
– a vector x = (xi ) where xi =
A polyhedral theory [3] based branch and cut method was proposed to solve the
SPP. Lusby et al. [4] transformed the problem of routing trains through junctions as
an SPP and solved it using a branch & price algorithm. The SPP has many practical
applications in real world optimization problems. Zwaneveld et al. [5] formulated
a real railway feasibility problem as an SPP and solved it using a branch & cut
method and reduction tests. Rönnqvist [6] formulated a cutting stock problem as an
SPP and solved it with the combination of Lagrangian relaxation and sub-gradient
optimization.
It has been proven that the SPP has N P-hard nature. And, the scope of the proposed exact method is limited to small size instances. To overcome this limitation,
an alternative approach, called metaheuristic, is used to find a satisfactory solution
in a reasonable amount computational time. However, metaheuristics do not give the
guarantee of optimality. Delorme et al. [2] proposed a greedy randomized adaptive
search procedure (GRASP) to solve the SPP. The GRASP is tested on random and
real railways problem instances. Gandibleux et al. [7] presented an ant colony optimization (ACO) approach and used random instances to test the proposed approach.
Further, two versions of ACO [8, 9] have proposed for the SPP and tested on real
railway problem instances. Recently, Chaurasia et al. [10] proposed an evolutionary
An Evolutionary Algorithm Based Hyper-heuristic …
261
algorithm with guided mutation (EA/G) for the SPP. The EA/G is hybridized with
problem-specific heuristic and local search to further improve the solution returned
by the EA/G. The literature [10, 11] can be referred for the detailed study.
Generally, the traditional metaheuristics which use problem-specific structures,
operators and parameters value to get a satisfactory solution. Although, configuring, manually, the metaheuristic such as adding or removing operators or tuning the
parameter’s value is a most difficult task. Therefore, there was a demand of a methodology which can remove the drawbacks of such approaches. On such methodology
to address these issues is hyper-heuristic. Unlike the traditional approaches, hyperheuristics work on a search space of heuristics rather than on the search space of
solutions as the traditional approaches do.
In this paper, we present evolutionary algorithm with guided mutation based
hyper-heuristic to solve the SPP. Hereafter, the proposed approach will be referred
as EA/G-HH. The proposed EA/G-HH approach is compared with the state-of-theart metaheuristic approaches. The computational results show the effectiveness of
EA/G-HH in comparison to the existing approaches for the SPP.
The remainder of this paper is organized as follows: Sect. 2 is focused on the
overview of hyper-heuristic and EA/G. Section 3 describes our EA/G-HH approach
for the SPP. Section 4 describe the computational results. Section 5 outlines some
concluding remarks.
2 Overview of Hyper-heuristic and EA/G
2.1 Hyper-heuristic
Hyper-heuristics are kind of automated design techniques inspired by the fact that different heuristics have different strength and limitation [12]. Recently, Edmund et al.
[13] given a definition of hyper-heuristic framework as “an automated methodology
for selecting or generating heuristics to solve hard computational search problem”.
Hyper-heuristics have similar nature to metaheuristics and can be applied to a variety of optimization problems. However, metaheuristics work directly on the solution
space of the problem. On the other hand, hyper-heuristics work level/layer wise [14,
15]. Generally, hyper-heuristics perform their tasks in two levels known as higher
level and lower level. The higher level works on search space and it is independent
from the problem domain knowledge. It is mainly engaged in constructing or generating a best possible heuristic from the set of heuristics. The heuristics, also called
low-level heuristics, set resides at the lower level of hyper-heuristic and directly
works on the solution space of the problem. Each low-level heuristic can search the
solution space, modify the solution and construct a new solution using the problem
domain knowledge [16].
262
S. N. Chaurasia et al.
2.2 Evolutionary Algorithm with Guided Mutation (EA/G)
In 2005, Zhang et al. [11] developed an evolutionary algorithm with guided mutation
(EA/G) for maximum clique problem (MCP). The EA/G is a relatively new member
in the class of evolutionary algorithms and is developed with the aim to overcome,
as far as possible, drawbacks of two other evolutionary algorithms, viz. genetic
algorithms (GAs) and estimation of distribution algorithms (EDAs). The success of
EA/G for the MCP motivated the researcher to extend it for other combinatorial
optimization problems. Chaurasia et al. [10, 17–19] investigated the extended and
modified versions of EA/G for several combinatorial optimization problems.
The EA/G combines the features of both GAs and EDAs and tries to overcome
the shortcomings of these approaches. Basically, GAs use genetic operators such as
crossover and mutation, which use the location information of the candidate solutions,
to generate new offspring . GAs do not keep the past information since the beginning
of the generation and failed to make the use of global information which can be
useful to generate new offspring. On the other hand, EDAs use only the global
information to generate new offspring. In EDAs, the global information is stored
in the form of probability vector which characterizes the distribution of promising
candidate solutions in the solution space since the beginning of the generation of the
algorithm. A new offspring is generated by sampling the probability vector.
Considering the complementary characteristics of GAs and EDAs, Zhang et al.
developed an ideal algorithm that uses the characteristics of GAs and EDAs while
generating new offspring and named this algorithm as evolutionary algorithm with
guided mutation. A mutation operator called guided mutation (GM) is used to generate new offspring. The GM operator utilizes the location information as well as
the global information about the search space to generate new offspring. For more
detailed studies literature [10, 11, 17–19] can be referred.
3 EA/G-HH for the SPP
In this work, we have developed a new EA/G-HH approach. The EA/G is employed
as a higher level algorithm. At the higher level, we employed Guided heuristic(GH)
as a heuristic to select a heuristic from the pool of low-level heuristics which reside
at the lower level. The main components of EA/G-HH for the SPP are described
below:
3.1 Solution Encoding
We adopted the similar solution encoding method of [10]. Subset encoding is used
to represent a solution. Each solution consists of a set of objects in a packing.
An Evolutionary Algorithm Based Hyper-heuristic …
263
3.2 Initial Solution
The initial solution generation method is a combination of randomness and greedy
approaches. Sixty percent of objects are selected with the help of greedy approach
and remaining 40% are included randomly. In greedy approach, a roulette wheel
selection method is used to include unselected object into the partial solution.
3.3 Higher Level Search Methodology
The higher level of hyper-heuristic consists of two steps. In the first step, a credit
is assigned to each heuristic with help of Credit assignment procedure. And, the
second step is heuristic selection rule in which a guided heuristic, which is inspired
by the guided mutation operator of [19], is employed to select a heuristic from a
set of low-level heuristics. The following sections explain the credit assignment and
heuristic selection rule.
1. Credit assignment:
A two-dimensional matrix of W rows and H columns where W and H represent
the size of window and number of heuristics, respectively. f Hi j represents the
fitness returned by the heuristic Hi at stage j in the current generation. After each
generation, the fitness matrix FMatri x is updated using first-in-first-out (FIFO).
FIFO means, when the (W + 1)th fitness is appended into the window, then the
first will be removed.
The FMatri x is initialized using W × H number of solution’s fitnesses. The probability δ Hi is initialized using Eq. (5). f j Hi is the solution fitness returned by
heuristic Hi at stage j and max( f j Hi ) is the best fitness returned by heuristic Hi
for i = 1, 2, . . . , H .
max( f j Hi )
,
δ Hi = f j Hi
j = 1, 2, . . . W, i = 1, 2, . . . H
(5)
The probability vector δ H is updated after each generation using the Eq. (6).
In Eq. (6), Z (1 ≤ Z ≤ W ) is the size of partial window, i.e., last Z rows in
FMatri x . For example, after each generation, the probability vector is updated
using Eq. (6).
max( f j Hi )
, i = 1, 2, . . . H, j = 1, 2, . . . W
δ Hi = (1 − ζ ) × δ Hi + ζ × W
f k Hi
k=(W −Z )
(6)
2. Heuristic selection rule: A guided heuristic (GH) is employed to select a heuristic from a set of low-level heuristics pool to generate a new offspring. Inspired
by the guided mutation of [10], the probability vector δ H characterizes the
distribution of promising heuristics in the heuristics search space. The guided
heuristic (GH) uses the both the global information which is stored in the form
264
S. N. Chaurasia et al.
Table 1 Constructive low-level heuristics for the SPP
H1
H2
H3
H4
H5
The Guided mutation (GM) operator has an advantage of using both the global and
location information to generate a new solution. The proposed G M operator is modified
version of the G M operator of [10], and it always generates feasible solution.
Improve operator [10] follows a greedy approach to improve the solution. In each
iteration, unselected object is added with the help of roulette selection method
1-1 exchange heuristic [10] based on 1-1 exchange strategy is adopted to improve the
solution fitness. An object in a packing P is tried to replace with an object not in the
packing without violating the feasibility constraint to improve the cost of the packing P
In 1-2 exchange heuristic [10], similar to 1-1 exchange heuristic, the first improvement
strategy is adopted. An object in packing P is tried to exchange exactly with two objects
which are not in the packing P without violating the feasibility constraint
In perturbation strategy, the aim is to maintain diversification in the solution population.
The current solution is replaced with random solution exactly as the solution is
generated in the initial population
of probability and the location information about the fitnesses returned by the
heuristics in the current window, and then a heuristic is chosen by sampling the
probability vector δ H .
3.4 Lower Level heuristics
The lower level consists of five heuristics. In Table 1, there are five heuristics viz.
H1, H2, H3, H4, and H5. Heuristics H1 to H4 are problem-specific heuristic and
heuristic H5 is used to maintain diversification in the solution population.
4 Computational Results
The proposed EA/G-HH approach has been implemented in C and executed on
Intel Core i5-4590 with 4 GB RAM under Linux based operating system
with gcc 5.4.0 compiler. For the EA/G-HH, we have used the population size
of (N p ) = 250 and the parent size (M) = 125 of the current population to update
the probability vector p. The values of parameters β, λ and ζ are set to 0.90, 0.90,
and 0.80 respectively. The larger value of λ and ζ indicates that the algorithm gives
weight to learn more from the current population. On the other hand, smaller value of
λ and ζ indicates that it learns from the past information which is stored in the form
of probability. The window size is fixed to 300 and Z =125. Like GRASP [2], ACO
[7] and EA/G [10], EA/G-HH is also executed sixteen independent times for each
instance. Parameter values are chosen empirically after a large number of experiment
trails. The proposed EA/G-HH is compared with the state-of-the-art approaches,
viz., GRASP [2], ACO [7] and EA/G [10]. The computational results are shown in
Table 2.
100
100
100
100
100
100
100
100
100
100
100
100
200
200
200
200
200
200
200
200
200
200
pb100rnd02
pb100rnd03
pb100rnd04
pb100rnd05
pb100rnd06
pb100rnd07
pb100rnd08
pb100rnd09
pb100rnd10
pb100rnd11
pb100rnd12
pb200rnd01
pb200rnd02
pb200rnd03
pb200rnd04
pb200rnd05
pb200rnd06
pb200rnd07
pb200rnd08
pb200rnd09
pb200rnd10
Var
pb100rnd01
Instance
200
200
200
200
1000
1000
1000
1000
1000
1000
300
300
300
300
100
100
100
100
500
500
500
500
Cnst
1.0
1.0
1.5
1.5
2.5
2.5
1.0
1.0
1.5
1.5
3.0
3.1
2.0
2.0
3.1
2.9
2.0
2.0
3.0
3.0
2.0
2.0
Density
(%)
2
2
4
4
8
8
2
2
4
4
4
4
2
2
4
4
2
2
4
4
2
2
Max
one
Characteristics
[1–1]
[1-20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
Weight
118
1324
83
1004
14
184
64
731
32
416
23
306
40
463
39
503
64
639
16
203
34
372
Opt
0.02
0.01
0.04
0.02
8068.20
1211.37
63970.91
5403.23
156109.36
8760.73
6.80
0.48
1.13
0.49
0.02
0.00
0.01
0.01
52.86
7.81
0.60
2.92
TET
0/1 Solution
118
1324
83
1002
14
184
63
726
32
416
23
306
40
463
39
503
64
639
16
203
34
372
Best
118.00
1324.00
82.87
1001.12
13.37
184.00
63.00
722.81
32.00
415.18
23.00
306.00
40.00
463.00
38.75
503.00
64.00
639.00
16.00
203.00
34.00
372.00
Avrg
GRASP
3.64
3.75
2.71
4.20
3.48
4.62
9.12
10.81
7.35
7.32
1.13
0.68
1.28
1.26
0.57
1.00
0.69
0.80
1.29
1.14
1.31
1.97
118
1324
83
1004
14
184
64
729
32
416
23
306
40
463
39
503
64
639
16
203
34
372
ATET Best
118.00
1324.00
82.75
1003.50
12.87
182.56
62.93
725.12
31.56
415.25
22.93
306.00
39.62
463.00
38.68
503.00
64.00
639.00
15.56
203.00
34.00
372.00
Avrg
ACO
4.00
7.33
2.67
6.33
4.00
16.00
24.33
44.33
14.67
27.33
0.33
1.67
1.00
1.67
0.67
1.00
1.00
1.67
0.67
2.00
2.00
3.33
118
1324
83
1004
14
184
63
731
32
416
23
306
40
463
39
503
64
639
16
203
34
372
ATET Best
Table 2 Comparison of EA/G-HH with GRASP [2] ACO [7] and EA/G [10] on instances with upto 200 variables
EA/G
118.00
1324.00
82.81
1003.94
13.50
184.00
62.75
727.00
32.00
416.00
23.00
306.00
39.88
463.00
38.81
503.00
64.00
639.00
15.69
203.00
34.00
372.00
Avrg
0.76
1.31
0.80
1.61
0.55
1.07
0.92
1.85
0.76
1.66
0.21
0.40
0.24
0.38
0.21
0.38
0.14
0.35
0.18
0.37
0.22
0.38
118
1317
83
1001
14
184
63
723
32
416
23
306
40
463
39
503
64
639
16
203
34
372
ATET Best
0.21
0.20
0.21
0.19
0.21
0.18
0.22
0.21
0.24
0.22
0.06
0.05
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.06
0.70
0.06
ATET
(continued)
118.00
1308.25
82.94
992.38
14.00
174.81
63.00
708.19
31.50
410.12
23.00
306.00
40.00
460.19
39.00
501.88
64.00
638.81
16.00
201.44
34.00
372.00
Avrg
EA/G-HH
An Evolutionary Algorithm Based Hyper-heuristic …
265
200
200
200
200
200
200
200
200
pb200rnd12
pb200rnd13
pb200rnd14
pb200rnd15
pb200rnd16
pb200rnd17
pb200rnd18
Var
pb200rnd11
Instance
600
600
600
600
600
600
200
200
Cnst
Table 2 (continued)
2.6
2.5
1.0
1.0
1.5
1.5
2.6
2.5
Density
(%)
8
8
2
2
4
4
8
8
Max
one
Characteristics
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
[1–1]
[1–20]
Weight
19
255
79
926
45
571
43
545
Opt
19285.06
741.52
14372.85
12.20
10066.91
830.39
1.70
0.33
TET
0/1 Solution
19
255
79
926
45
571
43
545
Best
GRASP
18.06
251.31
78.31
926.00
45.00
566.43
43.00
544.75
Avrg
2.35
3.61
6.80
4.22
3.92
6.01
1.01
2.36
19
255
79
926
45
571
43
545
ATET Best
ACO
18.12
253.25
78.37
926.00
44.43
568.50
43.00
545.00
Avrg
3.00
11.00
15.33
27.00
8.67
20.33
1.33
4.33
19
255
79
926
45
571
43
545
ATET Best
EA/G
18.19
254.19
78.12
926.00
44.62
570.75
43.00
545.00
Avrg
0.65
1.22
0.98
1.72
0.75
1.74
0.80
1.65
19
255
79
926
45
568
43
543
ATET Best
18.06
248.44
78.38
918.19
45.00
565.75
43.00
540.00
Avrg
EA/G-HH
0.21
0.21
0.21
0.20
0.21
0.21
0.21
0.19
ATET
266
S. N. Chaurasia et al.
An Evolutionary Algorithm Based Hyper-heuristic …
267
A total of 21 instances are divided into two categories: uni-cost and multicost for the SPP. In uni-cost instances, the weight of each object is fixed to 1
and for the multi-cost instances weight of each object is fixed in [1, 20]. For
detailed description about the instances [10] can be followed. From Table 2,
we can observe that the EA/G-HH is able to achieve the optimal solution for
all instances except pb200md04, pb200md07, pb200md09, pb200md11
and pb200md13. Generally, computational time is affected by many factors and
one of them is the system configuration. Therefore, it would not be fair to compare
the computational time directly. Therefore, we are doing rough comparison, and we
can say that HH-EA/G outperformed the ACO approach in terms of solution quality
as well as computational time. In comparison with EA/G, EA/G-HH has equivalent
performance in terms of computational time.
5 Conclusions
This paper presented a simple evolutionary algorithm based hyper-heuristic (EA/GHH) for the set packing problem (SPP). The proposed approach has tested over
the instances with the set size 100 and 200 for both uni-cost and multi-cost. The
proposed EA/G-HH approach is compared with the state-of-the-art approaches, viz.,
GRASP, ACO and EA/G for the SPP. The computational results show that proposed
approach also achieved the optimal solution on most of the instances. However, the
computational time taken by our approach is bit more than the GRASP but lesser than
the ACO. Whereas in comparison with EA/G, EA/G-HH taking almost equivalent
computational time.
The future aspect of the proposed approach is to enhance the capability of higher
level to explore the search space efficiently and then investigate their performance
on large size instances. And, also many heuristics can be added to the heuristics pool
at the lower level to improve the performance of higher level. Similar approach can
be developed for other similar types of combinatorial optimization problems.
Acknowledgements This work was supported by the grant [13AWMP-B066744-01] from the
Advanced Water Management Research Program funded by the Ministry of Land, Infrastructure,
and Transportation of the Korean government.
References
1. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of N P Completeness. Freeman, San Francisco (1979)
2. Delorme, X., Gandibleux, X., Rodriguez, J.: GRASP for set packing problems. Eur. J. Oper.
Res. 153, 564–580 (2004)
3. Padberg, M.W.: On the facial structure of set packing polyhedra. Math. Program. 5, 199–215
(1973)
268
S. N. Chaurasia et al.
4. Lusby, R., Larsen, J., Ryan, D., Ehrgott, M.: Routing trains through railway junctions: a new
set packing approach. Transp. Sci. 45, 228–245 (2011)
5. Zwaneveld, P.J., Kroon, L.G., Romeijn, H.E., Salomon, M., Dauzère-Pérès, S., Hoesel,
S.P.M.V., Ambergen, H.W.: Routing trains through railway stations: model formulation and
algorithms. Transp. Sci. 30, 181–194 (1996)
6. Rönnqvist, M.: A method for the cutting stock problem with different qualities. Eur. J. Oper.
Res. 83, 57–68 (1995)
7. Gandibleux, X., Delorme, X., TKindt, V.: An ant colony optimisation algorithm for the set
packing problem. In: ANTS 2004, vol. 3172, pp. 49–60. Berlin (2004)
8. Gandibleux, X., Angibaud, S., Delorme, X., Rodriguez, J.: An ant algorithm for measuring and
optimizing the capacity of a railway infrastructure. In: Monmarché, N., Guinand, F., Siarry, P.
(eds.) Artificial Ants, vol. 1. ISTE Ltd and Wiley, New York (2010)
9. Merel, A., Gandibleux, X., Demassey, S.: A collaborative combination between column generation and ant colony optimization for solving set packing problems. In: MIC 2011: The IX
Metaheuristics International Conference, pp. 25–28, Italy (2011)
10. Chaurasia, S.N., Sundar, S., Singh, A.: A hybrid evolutionary approach for set packing problem.
Opsearch 52(2), 271–284 (2015)
11. Zhang, Q., Sun, J., Tsang, E.: An evolutionary algorithm with guided mutation for the maximum
clique problem. IEEE Trans. Evol. Comput. 9, 192–200 (2005)
12. Sabar, N.R., Ayob, M., Kendall, G., Qu, R.: Automatic design of a hyper-heuristic framework
with gene expression programming for combinatorial optimization problems. IEEE Trans.
Evol. Comput. 19(3), 309–325 (2015)
13. Burke, E.K., Hyde, M., Kendall, G., Ochoa, G., Özcan, E., Woodward, J.R.: A Classification
of Hyper-heuristic Approaches, pp. 449–468. Springer US, Boston, MA (2010)
14. Koulinas, G., Kotsikas, L., Anagnostopoulos, K.: A particle swarm optimization based hyperheuristic algorithm for the classic resource constrained project scheduling problem. Inf. Sci.
277, 680–693 (2014)
15. Burke, E.K., Gendreau, M., Hyde, M., Kendall, G., Ochoa, G., Özcan, E., Qu, R.: Hyperheuristics: a survey of the state of the art. J. Oper. Res. Soc. 64(12), 1695–1724 (2013)
16. Burke, E.K., Kendall, G., Newall, J., Hart, E., Ross, P., Schulenburg, S.: Hyper-Heuristics: An
Emerging Direction in Modern Search Technology, pp. 457–474. Springer US, Boston, MA
(2003)
17. Chaurasia, S.N., Singh, A.: Hybrid evolutionary approaches for the single machine order acceptance and scheduling problem. Appl. Soft Comput. J. 52, 725–747 (2017)
18. Chaurasia, S.N., Singh, A.: A hybrid heuristic for dominating tree problem. Soft. Comput.
20(1), 377–397 (2016)
19. Chaurasia, S.N., Singh, A.: A hybrid evolutionary algorithm with guided mutation for minimum
weight dominating set. Appl. Intell. 43(3), 512–529 (2015)
Developing a Decision-Making Model
Using Interval-Valued Intuitionistic
Fuzzy Number
Syed Abou Iltaf Hussain , Uttam Kumar Mandal
and Sankar Prasad Mondal
Abstract A multi-criteria decision-making model is developed that considers the
nondeterministic nature of decision-maker along with the vagueness in the decision. The main objective of this model is to minimize the risk associated with each
alternative. For this reason, the ratings of alternatives versus criteria are assessed in
Parametric Interval-Valued Intuitionistic Fuzzy Number (PIVIFN). A defuzzification method is developed using the Riemann integral method. In addition, different
properties, theorems, and operators are redefined for PIVIFN. Finally, the model is
applied to solve a decision-making problem.
Keywords Priority index · Degree of vagueness · Parametric interval-valued
intuitionistic fuzzy number · Relative benefit matrix · Riemann integral method
1 Introduction
Over the time multi-criteria decision-making (MCDM) method has been evolved
as a significant tool of modern decision-making science. In MCDM method, the
most appropriate alternative is chosen from a group of identical alternatives on the
basis of some criteria. Some of its thriving application is pattern recognition [1],
material selection [2–6], supplier selection [7–11], site selection [12–14], and so on.
Due to the existence of vagueness or uncertainty [15] in the information as well as
impreciseness in the physical nature of the problems, the decision-makers face a lot of
complications during the process of decision-making. For obtaining a rational result,
many researchers have combined MCDM with fuzzy sets (FSs) [16–19], interval sets
S. A. I. Hussain (B) · U. K. Mandal
Department of Production Engineering, National Institute of Technology,
Agartala, Jirania 799046, Tripura, India
e-mail: syedaboui8@gmail.com
S. P. Mondal
Department of Mathematics, Midnapore College (Autonomous), West Midnapore
721101, West Bengal, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_27
269
270
S. A. I. Hussain et al.
(ISs) [20–23], gray relational analysis (GRA) [24, 25], and others [26–29]. Out of
all the methods, fuzzy-integrated MCDM approach is mostly been used for tackling
uncertainty-based decision-making problems.
Atanassov [30–32] generalized the concept of FSs, introduced by Zadeh [33] and
presented the intuitionistic fuzzy sets (IFSs). The interval-valued fuzzy sets (IVFSs)
[34, 35] and developed form of IFSs and vague sets (VSs) [36] were the extended
and developed form IFSs. Further, IFSs and IVFSs are integrated which lead to the
development of interval-valued intuitionistic fuzzy sets (IVIFSs) [37]. With the introduction of IVIFSs, it is coupled with MCDM problems [38–45] for finding the most
suitable alternatives in a scenario where decision variables are collected in the form
of interval-valued number. IVIFSs’ integrated MCDM techniques became a major
flare in decision-making because the information about criteria or attribute values
is usually uncertain or fuzzy due to the increasing intricacy of the socioeconomic
environment and the vagueness in psychological perspective of human [43].
An attempt is made in this paper to develop a model that takes into account the
nondeterministic nature of decision-maker along with the degree of vagueness in the
decision. For this reason, in this paper, a model is developed that considers the IVIFSs
in a parametric form called the parametric interval-valued intuitionistic fuzzy sets
(PIVIFSs). Above that, we have defined a defuzzification method based on Reimann
integral method.
2 Preliminaries
In this section, some of our preliminary research definitions, properties, and theorem
are summarized.
Definition 2.1 (Interval-valued fuzzy
number)
An interval-valued fuzzy number I
is represented by closed interval [Il , Iu ]; μ I˜ and defined as
I [Il , Iu ]; α x; μ I˜ (x) : Il ≤ x ≤ Iu , x ∈ R
The membership function for this interval can be written as
α, Il ≤ x ≤ Iu
μ Ii (x) 0, otherwise
where R is the set of real numbers; Il and Iu are the left and right limits of the interval
number, respectively, and μ I˜ (x) as mentioned in Definition 2.1.
Definition 2.2 (Interval-valued intuitionistic fuzzy number) An interval-valued intuitionistic fuzzy number Ii can be defined as [Il , Iu ]; α, β. The membership function
for this interval can be written as
Developing a Decision-Making Model Using Interval-Valued …
271
Ii [Il , Iu ]; α, β x; μ I˜ (x), ϑ Ii (x) : Il ≤ x ≤ Iu , x ∈ R
μ Ii (x) α, Il ≤ x ≤ Iu
0, otherwise
and the nonmembership function is written as
β, Il ≤ x ≤ Iu
ϑ Ii (x) 1, otherwise
where Il and Iu are the left and right limits of the interval number, respectively, and
μ I˜ (x) and ν I˜ (x) as mentioned in Definitions 2.1 and 2.2.
Definition 2.3 (Parametric interval-valued intuitionistic fuzzy number) Parametric intuitionistic interval-valued fuzzy number I ( p) is represented by
{ p ∗ Il + (1 − p) ∗ Iu }; α, β {Iu + p ∗ (Il − Iu )}; α, β and defined as
I ( p) {Iu + p ∗ (Il − Iu )}; α, β
x; μ I˜ (x), ν I˜ (x) : x {Iu + p ∗ (Il − Iu )}, p ∈ [0, 1]
The membership function for this interval can be written as
α, p ∈ [0, 1]
μ I˜ (x) 0, otherwise
and the nonmembership function is written as
β, p ∈ [0, 1]
ν I˜ (x) 1, otherwise
where Il and Iu are the left and right limits of the interval number, respectively. μ I˜ (x)
and ν I˜ (x) as mentioned in Definitions 2.1 and 2.2. p are called the priority index of
decision-maker and 0 ≤ p ≤ 1.
Definition 2.4 (Properties of PIVIFN) Let us consider the two parametric intuitionistic interval numbers I { p ∗ Il + (1 − p) ∗ Iu }; α1 , β1 and
u|
u|
J { p ∗ Jl + (1 − p) ∗ Ju }; α2 , β2 . If I |Il |+|I
and J |Jl |+|J
2
2
Addition:
I α1 + J α2 I β1 + J β2
,
A I + J { p ∗ (Il + Jl ) + (1 − p) ∗ (Iu + Ju )};
I + J I + J (1)
272
S. A. I. Hussain et al.
Subtraction:
I α1 + J α2 I β1 + J β2
,
S I − J { p ∗ (Il − Ju ) + (1 − p) ∗ (Iu − Jl )};
I + J I + J (2)
Multiplication:
M I ∗ J p ∗ min(Il Jl , Iu Ju , Il Ju , Iu Jl ) + (1 − p)
∗ max(Il Jl , Iu Ju , Il Ju , Iu Jl ); α1 α2 , β1 + β2 − β1 β2 (3)
Multiplication by constant:
When k > 0 and (Il & Iu ) > 0 or (Il < 0 & Iu > 0) or (Il & Iu ) < 0
Mc k I {kp ∗ Il + k(1 − p) ∗ Iu }; α1 , β1 (4)
When k < 0 and (Il & Iu ) > 0 or (Il < 0 & Iu > 0) or (Il & Iu ) < 0
Mc k I {kp ∗ Iu + k(1 − p) ∗ Il }; α1 , β1 (5)
Inverse:
When (Il & Iu ) > 0 or (Il & Iu ) < 0
1
(−1)
p∗
H (I )
+ (1 − p) ∗
Iu
1
Il
; α1 , β1
(6)
When (Il < 0 & Iu > 0)
(−1)
H (I )
p∗
1
Iu
+ (1 − p) ∗
1
Il
; α1 , β1
(7)
Division:
I
J
Il Iu Il Iu
+ (1 − p)
, , ,
Jl Ju Ju Jl
Il Iu Il Iu
∗ max
; α1 α2 , β1 + β2 − β1 β2
, , ,
Jl Ju Ju Jl
D
p ∗ min
(8)
Definition 2.5 (Defuzzification of PIVIFN) A parametric interval value intuitionistic
fuzzy number I { p ∗ Il + (1 − p) ∗ Iu }; α, β is converted into crisp number C
using Riemann integral which is as follows:
Developing a Decision-Making Model Using Interval-Valued …
p1
(1+α−β)
2
×
(1+α−β)
2
×
C
(1+α−β)
2
×
C
(1+α−β)
2
C
(1+α−β)
2
C
(1+α−β)×(Il +Iu )
4
C
C
273
{ p ∗ Il + (1 − p) ∗ Iu }d p
p0
p1
{ p ∗ (Il − Iu ) + Iu }d p
p0
p2
2
∗ (Il − Iu ) + p ∗ Iu
p1
1
p0
∗ (Il − Iu ) + 1 ∗ Iu − 0
1
× 2 ∗ (Il + Iu )
×
(9)
2
Theorem 2.1 If there are two PIVIFNs I 1 and I 2 , then the relation between them
is totally defined by their crisp number C 1 and C 2 as follows:
1. If C 1 C 2 then I 1 I 2 .
2. If C 1 C 2 then I 1 I 2 .
3. If C 1 ≺ C 2 then I 1 ≺ I 2 .
3 Model Establishment
Step 1: Creation of the decision matrix (D).
Decision from each of the decision-makers is taken into consideration, and
the decision matrix (D) is created using the IVIWAA operator having η
number of alternatives and γ number of criteria.
D d ji η×γ
Step 2: Converting the decision matrix into relative benefit matrix (R)
⎧
⎨ d ji − mini d ji , ∀ benefit criteria, i ∈ 1, γ
ji R r η×γ ⎩ maxi d ji − d ji , ∀ cost criteria, i ∈ 1, γ
Step 3: Calculation of the score of each alternative (σ )
σj γ
i1
ωi . r ji
η ji j1 r
Step 4: Defuzzification and ranking of alternatives
274
S. A. I. Hussain et al.
Table 1 Decision matrix
Criteria
A1
A2
C1
{ p ∗ 4.2 + (1 − p) ∗ 6}; 0.6, 0.4
{ p ∗ 5 + (1 − p) ∗ 6.7}; 0.5, 0.5
C2
{ p ∗ 8 + (1 − p) ∗ 9.2}; 0.7, 0.3
{ p ∗ 7.3 + (1 − p) ∗ 8.2}; 0.7, 0.3
C3
{ p ∗ 7.2 + (1 − p) ∗ 8}; 0.5, 0.5
{ p ∗ 6.2 + (1 − p) ∗ 7.8}; 0.6, 0.4
C4
{ p ∗ 8.2 + (1 − p) ∗ 8.5}; 0.6, 0.4
{ p ∗ 7.2 + (1 − p) ∗ 8}; 0.7, 0.3
Criteria
A3
A4
C1
{ p ∗ 7.3 + (1 − p) ∗ 8.5}; 0.7, 0.3
{ p ∗ 7.1 + (1 − p) ∗ 8.5}; 0.7, 0.3
C2
{ p ∗ 7.7 + (1 − p) ∗ 8.5}; 0.6, 0.4
{ p ∗ 6.6 + (1 − p) ∗ 7}; 0.5, 0.5
C3
{ p ∗ 6.5 + (1 − p) ∗ 7.5}; 0.5, 0.5
{ p ∗ 8.7 + (1 − p) ∗ 9.5}; 0.8, 0.2
C4
{ p ∗ 7 + (1 − p) ∗ 8.2}; 0.5, 0.5
{ p ∗ 7.6 + (1 − p) ∗ 8}; 0.6, 0.4
Table 2 Relative benefit matrix
Criteria
A1
A2
C1
{ p ∗ (−2.5) + (1 − p) ∗ 1.0}; 0.55, 0.45 { p ∗ (−2.5) + (1 − p) ∗ 1.0}; 0.55, 0.45
C2
{ p ∗ (−0.5) + (1 − p) ∗ 1.5}; 0.65, 0.35 { p ∗ (−0.5) + (1 − p) ∗ 1.5}; 0.65, 0.35
C3
{ p ∗ (−1.7) + (1 − p) ∗ 1.7}; 0.50, 0.50 { p ∗ (−1.7) + (1 − p) ∗ 1.7}; 0.50, 0.50
C4
{ p ∗ (−1.2) + (1 − p) ∗ 0.5}; 0.65, 0.35 { p ∗ (−1.2) + (1 − p) ∗ 0.5}; 0.65, 0.35
Criteria
A3
A4
C1
{ p ∗ (−1.8) + (1 − p) ∗ 0.6}; 0.55, 0.45
{ p ∗ (−1.3) + (1 − p) ∗ (−0.2)}; 0.65, 0.35
C2
{ p ∗ (−1.6) + (1 − p) ∗ 1.6}; 0.60, 0.40 { p ∗ (−0.8) + (1 − p) ∗ (0.8)}; 0.70, 0.30
C3
{ p ∗ (−1.3) + (1 − p) ∗ 1.3}; 0.55, 0.45 { p ∗ (−1.0) + (1 − p) ∗ (1.0)}; 0.60, 0.40
C4
{ p ∗ (−2.0) + (1 − p) ∗ 0.8}; 0.55, 0.45 { p ∗ (−2.0) + (1 − p) ∗ 0.8}; 0.55, 0.45
The σ j is converted into crisp value F j according to Eq. 9. The ranking of
alternatives is done in ascending order of the F j values.
4 Illustrative Example
The best alternative is to be computed among a set of four alternatives on the basis
of four criteria. The third and fourth criteria are a cost criterion,
and the remaining
are benefit criteria. The weightage of the criteria is ωi 0.3 0.25 0.35 0.1 . The
decision matrix, relative benefit matrix and ranking table is shown in Tables 1, 2 and
3 respectively.
The ordering of the alternatives is done in ascending order of their C j value.
A2
A4
A1
A3
Developing a Decision-Making Model Using Interval-Valued …
Table 3 Score and rank of alternatives
Alternatives Score in PIVIFN
275
Score in crisp value
Rank
A1
{ p ∗ (−0.404) + (1 − p) ∗ 0.48}; 0.35, 0.65
0.153
3
A2
{ p ∗ (−0.43) + (1 − p) ∗ 0.409}; 0.34, 0.66
0.143
1
A3
{ p ∗ (−0.55) + (1 − p) ∗ 0.39}; 0.35, 0.65
0.152
2
A4
{ p ∗ (−0.64) + (1 − p) ∗ 0.32}; 0.34, 0.66
0.161
4
Table 4 Comparison table
Alternatives
Rank
Proposed
model
MOORA
COPRAS
VIKOR
TOPSIS
A1
3
3
3
2
3
A2
1
1
4
1
4
A3
2
2
1
4
1
A4
4
4
2
3
2
5 Results and Discussions
5.1 Validation of Proposed Models
The result from the proposed algorithm is compared with the result from the established models like MOORA, COPRAS, VIKOR, and TOPSIS.
Some of the points observed during the study are as follows:
i. The values of Il & Iu are not constrained except Il < Iu .
ii. Nondeterministic nature of decision-maker is measured by priority index ( p)
and degree of vagueness in decision is the degree of confidence and the degree
of skepticism of decision-makers’ decision.
iii. The aggregated decision matrix is converted into relative benefit matrix which
implies the relative benefit of not selecting the alternative with least benefit and
highest cost for a benefit and cost criterion, respectively.
iv. Risk minimization is the basic idea for the conversion of aggregated decision
matrix into relative benefit matrix.
v. Defuzzification of PIVIFN is done using the Reimann integral approach for p
varying from 0 to 1.
vi. Two interval value intuitionistic numbers are said to be equal, lesser or greater
than each other if their crisp number are equal, lesser or greater, respectively.
vii. From Table 4, it is validated that the proposed algorithm could be used for the
decision-making.
276
S. A. I. Hussain et al.
6 Conclusions
In the present study, a robust MCDM model is developed that can be used in decisionmaking considering the nondeterministic nature of decision-maker along with the
vagueness in decision. The priority index is the measure of nondeterministic nature
of decision-maker, whereas the degree of confidence and skepticism is the measure
of degree of vagueness in decision. For this reason, the ratings of alternatives versus
criteria are assessed in PIVIFN. The PIVIFN defined in the paper is the generalized
form of IVIFSs. In the proposed model, the aggregated decision matrix is converted
into relative benefit matrix which implies the relative benefit of not selecting the
alternative with least benefit or highest cost. Risk minimization is the basic idea
for the conversion of the aggregated decision matrix. The relative weight of each
alternative is calculated which is in the form of PIVIFN, as a result of which the
relative weights are defuzzified using the Reimann integral approach for the value
of p varying from 0 to 1. The final ranking and comparisons of different alternatives
are done in descending order. Finally, the proposed model is used for solving a
numerical example. In order to study the stability of the model, result obtained from
the proposed algorithm is compared with other decision-making models and it is
noticed that 80% of the time returns the same best answer. Hence, we can accept
the model for ranking of alternatives when the decision matrix is created in PIVIFN
form.
References
1. Mahdipoor, H.R.: Flow pattern recognition in tray columns with MADM (multiple attribute
decision making) method. Comput. Chem. Eng. 30, 1197–1200 (2006)
2. Chauhan, A., Vaish, R.: Magnetic material selection using multiple attribute decision making
approach. Mater. Des. 36, 1–5 (2012)
3. Cicek, K., Celik, M.: Multiple attribute decision-making solution to material selection problem
based on modified fuzzy axiomatic design-model selection interface algorithm. Mater. Des. 31,
2129–2133 (2010)
4. Shanian, A., Savadogo, O.: A material selection model based on the concept of multiple attribute
decision making. Mater. Des. 27, 329–337 (2006)
5. Hussain, S.A.I., Mandal, U.K.: Entropy based MCDM approach for Selection of material. In:
National Level Conference on Engineering Problems and Application of mathematics (2016)
6. Mandal, U.K., Kulavi, S., Hussain, S.A.I.: A hybrid approach of AHP-cross entropy model
to select bearing material under MCDM environment. In: International Conference on Future
Trends and Challenges in Mechanical Engineering (FTCME) (2017)
7. Gurung, S., Phipon, R.: Multi-criteria decision making for supplier selection using AHP and
TOPSIS method. Int. J. Eng. Inventions 6(2), 13–17 (2016)
8. Ho, W., Xu, X., Dey, P.K.: Multi-criteria decision making approaches for supplier evaluation
and selection: a literature review. Eur. J. Oper. Res. 202, 16–24 (2010)
9. Weber, C.A., Current, J.R., Benton, W.C.: Vendor selection criteria and methods. Eur. J. Oper.
Res. 50(1), 2–18 (1991)
10. Degraeve, Z., Labro, E., Roodhooft, F.: An evaluation of supplier selection methods from a
total cost of ownership perspective. Eur. J. Oper. Res. 125(1), 34–58 (2000)
Developing a Decision-Making Model Using Interval-Valued …
277
11. Boer, L.D., Labro, E., Morlacchi, P.: A review of methods supporting supplier selection. Euro.
J. Purchasing Supply Manage. 7(2), 75–89 (2001)
12. Daim, T.U., Bhatla, A., Mansour, M.: Site selection for a data centre—a multi-criteria decisionmaking model. Int. J. Sustain. Eng. 6(1), 10–22 (2013)
13. Fadlalla, R., Elsheikh, A.: Multi-criteria decision making in hotel site selection. Int. J. Eng.
Sci. Invention 6(1), 15–18 (2017)
14. Bahrani, S., Ebadi, T., Ehsani, H., Yousefi, H., Maknoon, R.: Modeling landfill site selection by
multi-criteria decision making and fuzzy functions in GIS, case study: Shabestar. Iran. Environ
Earth Sci 75, 337 (2016)
15. Zadeh, L.A.: Toward a general theory of uncertainty (GTU)—an outline. Inf. Sci. 172, 1–40
(2005)
16. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, 141–164
(1970)
17. Kharat, M.G., Kamble, S.J., Raut, R.D., Kamble, S.S., Dhume, S.M.: Modeling landfill site
selection using an integrated fuzzy MCDM Approach. Model. Earth Syst. Environ. 2, 53 (2016)
18. Kuo, Y.C., Lu, S.T., Tzeng, G.H., Lina, Y.C., Huang, Y.S.: Using fuzzy integral approach
to enhance site selection assessment a case study of the optoelectronics industry. Procedia
Comput. Sci. 17, 306–313 (2013)
19. Chen, V.Y.C., Lien, H.P., Liu, C.H., Liou, J.J.H., Tzeng, G.H., Yang, L.S.: Fuzzy MCDM
approach for selecting the best environment-watershed plan. Appl. Soft Comput. 11, 265–275
(2011)
20. Bhunia, A.K., Samanta, S.S.: A study of interval metric and its application in multi-objective
optimization with interval objectives. Comput. Ind. Eng. 74, 169–178 (2014)
21. Karmakar, S., Mahato, S.K., Bhunia, A.K.: Interval oriented multi-section techniques for global
optimization. J. Comput. Appl. Math. 224, 476–491 (2008)
22. Rao, S.S., Berke, L.: Analysis of uncertain structural systems using interval analysis. J. Am.
Inst. Aeronaut. Astronaut. 35, 727–735 (1997)
23. Kulpa, Z., Pownuk, A., Skalna, I.: Analysis of linear mechanical structures with uncertainties
by means of interval methods. Comput. Assist. Mech. Eng. Sci. 5, 443–477 (1998)
24. Long, D.J.: Control problem of grey systems. Syst. Control Lett. 1, 288–294 (1982)
25. Wei, G.: Grey relational analysis model for dynamic hybrid multiple attribute decision making.
Knowl. Based Syst. 24, 672–679 (2011)
26. Wang, X., Chen, B., Qian, G., Ye, F.: On the optimization of fuzzy decision trees. Fuzzy Sets
Syst. 112(1), 117–125 (2000)
27. Wang, X., Hong, J.: On the handling of fuzziness for continuous-valued attributes in decision
tree generation. Fuzzy Sets Syst. 99(3), 283–290 (1998)
28. Wang, X., Hong, J.: Learning optimization in simplifying fuzzy rules. Fuzzy Sets Syst. 106(3),
349–356 (1999)
29. Wang, X.Z., Wang, Y.D., Xu, X.F., Ling, W.D., Yeung, D.S.: A new approach to fuzzy rule
generation: fuzzy extension matrix. Fuzzy Sets Syst. 123(3), 291–306 (2001)
30. Atanassov, K.: Intuitionistic fuzzy sets. VII ITKR’s session, Sofia, June (1983)
31. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)
32. Atanassov, K.: Operators over interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 64(2),
159–174 (1994)
33. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–356 (1965)
34. Türkşen, I.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 20, 191–210
(1986)
35. Gorzalczany, M.B.: A method of inference in approximate reasoning based on interval-valued
fuzzy sets. Fuzzy Sets Syst. 21, 1–17 (1987)
36. Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23(2), 610–614 (1993)
37. Atanassov, K., Gargov, G.: Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3),
343–349 (1989)
38. Xu, Z.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15(6), 1179–1187
(2007)
278
S. A. I. Hussain et al.
39. Xu, Z.: Methods for aggregating interval-valued intuitionistic fuzzy information and their
application to decision making. Control Decis. 22(2007), 215–219 (2007)
40. Xu, Z., Chen, J.: Approach to group decision making based on interval-valued intuitionistic
judgment matrices. Syst. Eng. Theor. Pract. 27(4), 126–133 (2007)
41. Xu, Z., Chen, J.: On geometric aggregation over interval-valued intuitionistic fuzzy information.
In: FSKD, 4th International Conference on Fuzzy Systems and Knowledge Discovery, vol. 2,
pp. 466–471 (2007)
42. Ye, J.: Multi criteria fuzzy decision-making method based on a novel accuracy function under
interval-valued intuitionistic fuzzy environment. Expert Syst. Appl. 36, 6899–6902 (2009)
43. Park, D.G., Kwun, Y.C., Park, J.H., Park, Y.: Correlation coefficient of interval-valued intuitionistic fuzzy sets and its application to multiple attribute group decision making problems.
Math. Comput. Model. 50, 1279–1293 (2009)
44. Wei, G., Wang, X.: Some geometric aggregation operators on interval-valued intuitionistic fuzzy sets and their application to group decision making. In: Proc. 2007 ICCIS, 2007,
pp. 495–499 (2007)
45. Zhao, X., Tang, S., Yang, S., Huang, K.: Extended VIKOR method based on cross-entropy
for interval-valued intuitionistic fuzzy multiple criteria group decision making. J. Intell. Fuzzy
Syst. 25(4), 1053–1066 (2013)
A Multi-start Iterated Local Search
Algorithm with Variable Degree
of Perturbation for the Covering
Salesman Problem
Pandiri Venkatesh, Gaurav Srivastava and Alok Singh
Abstract The covering salesman problem (CSP) is a variant of the well-known
traveling salesman problem (TSP), where there is no need to visit all the cities, but
every city must be either visited or within a predetermined distance from at least
one visited city in the tour. CSP, being a generalization of the TSP, is also NP-Hard.
CSP finds important applications in emergency planning, disaster management, and
rural healthcare. In this paper, we have proposed a multi-start iterated local search
algorithm for the CSP. We also incorporated a variable degree of perturbation strategy to further improve the solution obtained through our approach. Computational
results on a wide range of benchmark instances shows that our proposed approach
is competitive with other state-of-the-art approaches for solving the CSP.
Keywords Covering salesman problem · Iterated local search · Heuristic
Traveling salesman problem
1 Introduction
The covering salesman problem (CSP) is a variant of the traveling salesman problem
(TSP), where each city is either visited or covered by at least one of the visited cities
in the tour. Basically, a city is said to be covered if it is within a pre-specified distance
from a visited city in the tour. Given a set of n cities, where each city i ∈ {1, 2, . . . , n}
covers a set of cities within its predetermined distance, ci . The objective of the CSP
is to find a minimum-length Hamiltonian cycle over a subset of cities, which cover
P. Venkatesh · G. Srivastava · A. Singh (B)
School of Computer & Information Sciences, University of Hyderabad,
Hyderabad 500 046, India
e-mail: alokcs@uohyd.ernet.in
P. Venkatesh
e-mail: venkatesh78.p@gmail.com
G. Srivastava
e-mail: gauravsrignp@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_28
279
280
P. Venkatesh et al.
the remaining unvisited cities. The important applications of this problem arises in
emergency planning, disaster management, and rural healthcare delivery, where it
is not possible to visit all the zones and the objective is to visit a lesser number of
zones, and the people located in unvisited zones have to travel to the nearest visited
zone in order to receive the service [1].
The CSP was first introduced by Current and Schilling [1], who developed a
heuristic method to solve it. Their approach consists of two steps. As part of the first
step, they proposed to solve the associated set covering problem (SCP) to find the
minimum number of cities which cover all the cities. Later, in the second step, an
optimal tour is constructed over a subset of cities resulted after solving the SCP. If
there are multiple optimal solutions possible after solving the SCP, then the minimumlength tour is selected after successful application of TSP solver over all the optimal
solutions given by SCP.
Golden et al. [2] proposed two local search algorithms (L S1 & L S2) for the CSP by
incorporating different moves like swap, removal, and reinsertion, and perturbation to
improve the tour length (cost). The first algorithm L S1 uses removal and reinsertion
operations to improve the quality of solutions. Initially, the algorithm removes some
cities from the tour and later it substitutes them by a subset of unvisited cities to find a
feasible solution. The second algorithm L S2 uses more sophisticated moves, namely
removal-reassignment and perturbation procedures to improve the solution quality.
Initially, the algorithm uses removal-reassignment procedure to identify a better
subset of cities to be visited by the salesman, and then it uses Lin − K er nighan
heuristic [3] to find a better ordering of the cities to be visited by the salesman.
Salari et al. [4] proposed an integer linear programming based heuristic method
to solve the CSP. This method improves the given initial feasible solution by taking
the advantage of integer linear programming (ILP) techniques and heuristic search.
In this method, initially some cities are removed and reinserted into the tour, and
then it solves an ILP-based model to optimality to form a new feasible solution.
Salari et al. [5] proposed an hybrid heuristic algorithm for the CSP by combining
ant colony optimization (ACO) algorithm and dynamic programming (DP) technique.
The results show that the hybrid heuristic algorithm is more efficient than other
algorithms available in the literature for solving CSP.
Arkin and Hassin [6] proposed a geometric variation of the CSP. In this problem,
each neighborhood is a compact set in a plane and if that set is intersected, then all
the cities in that neighborhood will be covered. Here, the objective is to minimize
the length of the tour which starts and ends at the same city in a neighborhood set,
and intersects all the neighborhoods.
There are many other variants of the CSP, which have been studied in the literature
such as covering tour problem [7], multi-vehicle covering tour problem [8], ring star
problem [9], and generalized traveling salesman problem [10].
In this paper, we have proposed a simple and effective multi-start iterated local search (MS-ILS) algorithm for the CSP by incorporating a variable degree of
perturbation strategy.
A Multi-start Iterated Local Search Algorithm …
281
The remainder of the paper is organized as follows. Section 2 formally defines the
CSP. Our proposed MS-ILS approach for the CSP is described in Sect. 3. Section 4
presents the computational results on benchmark instances. Finally, Sect. 5 concludes
the paper by outlining the contributions made and gives directions for future research.
2 Problem Definition
Given a complete edge-weighted graph G = (V, E), where V = {1, 2, . . . , n} is the
set of cities, E = {(i, j)|i, j ∈ V } is the set of edges and a distance di j is associated
with each edge (i, j) ∈ E. All those cities which are within a prespecified distance
from a city i are said to be covered by city i. The objective of the CSP is to find a
minimum-length Hamiltonian cycle among all Hamiltonian cycles over subgraphs
induced by those subsets of cities where cities that do not belong to the subset are
covered by the cities of the subset. We will denote such a subset of vertices by V .
Further, cities in V are said to be visited, whereas the cities in V − V are said to be
covered. Let Q i denote the set of cities that can be covered by city i and Q i be the
set of cities that can cover the city i. By introducing binary variables, yi to indicate
whether city i is part of the subset (yi = 1) or not (yi = 0), and another binary
variable xi j to indicate whether edge (i, j) is part of Hamiltonian cycle (xi j = 1) or
not (xi j = 0), the mathematical model for the CSP may be represented as follows:
Minimize
di j xi j
(1)
i∈V j∈V
subject to:
yi +
y j ≥ 1, ∀i ∈ V
j∈Q i
xi j +
(i, j)∈E
(2)
x ji = 2yi ∀i ∈ V,
(3)
xi j ≤ |S| − 1, ∀S ⊂ V ⊂ V
(4)
( j,i)∈E
i∈S j∈S
xi j , yi ∈ {0, 1} ∀(i, j) ∈ E, i ∈ V.
(5)
Equation 1 is the objective function for the CSP and it minimizes the total distance.
Equation 2 enforces the coverage requirement of each city, i.e., a city is either visited
or covered. Equation 3 satisfies the constraints of indegree and outdegree of the
visited cities. Equation 4 represents the sub- tour elimination constraint. Equation 5
enforces the binary nature of the decision variables xi j , and yi .
282
P. Venkatesh et al.
Algorithm 1: Pseudocode of basic ILS
Input: Set of parameters for ILS Output: Best solution found
S := Generate_Initial_Solution();
S := Local_Search(S);
while Termination condition not satisfied do
S := Perturbation_Procedure(S );
S := Local_Search(S );
S := Acceptance_Criteria(S , S , histor y);
return best;
3 Multi-start Iterated Local Search Algorithm for the CSP
In this section, we give a brief introduction about the iterated local search algorithm
followed by the details about the proposed method for the CSP.
Iterated local search (ILS) is a single population (solution based) metaheuristic,
which iteratively improves the solution quality. According to [11] ILS has many of
the desirable features: simple, easy to implement, robust, and highly effective. The
ILS mainly consists of four components, viz., initial solution generation, local search,
perturbation procedure, and acceptance criteria. Starting from an initial solution, the
ILS finds a locally optimum solution through a local search algorithm, and then
an iterative process ensues. During each iteration, in a bid to improve the solution
further, a perturbation procedure is applied on the current solution and the local
search algorithm is applied on the resulting perturbed solution. Then depending on
the acceptance criteria, newly obtained solution may replace the current solution and
another iteration begins. The two commonly used acceptance criteria are replacing
the current solution with newly obtained solution if it is better than the current
solution and always replacing the current solution with newly obtained solution.
The former criteria lead to a first improvement type of strategy, whereas the latter
leads to random-walk sort of strategy. ILS has been successfully applied to many
optimization problems and has shown its effectiveness when compared with other
approaches [12]. The pseudocode of basic ILS is given in Algorithm 1.
3.1 Proposed Multi-start Iterated Local Search (MS-ILS)
Our proposed multi-start iterated local search (MS-ILS) for the CSP is an extension
of ILS, and restarts the ILS multiple times, each time starting with a new solution
generated by our initial solution generation procedure. We choose the multi-start
mechanism due to the fact that the unproductive iterations were consuming more
time, and restarting the search from a newly generated initial solution yielded better
solutions. The main components of our proposed approach are discussed in following
subsections.
A Multi-start Iterated Local Search Algorithm …
283
Initial Solution Generation: The initial solution generation procedure starts by
selecting a city uniformly at random and then an iterative process ensues. During
each iteration, a city which is neither visited nor covered is selected uniformly at
random and inserted into the best position in the salesman’s tour. To find this, one
has to check all the possible insertion positions and then the insertion is carried out at
the position which yields least increase in the cost. This procedure is repeated until
the feasibility condition is satisfied, i.e., every city is either visited or covered.
Perturbation and Local Search Procedure: The perturbation plays a vital role in
the ILS as it controls the divergent behavior of the search. Basically, an algorithm
performs well if it maintains an exquisite balance between convergent and divergent
behavior of the search. The goal of the perturbation is to escape from the present
local optimum solution by perturbing it, and providing a new starting solution to
the local search to move the search to unexplored points in the search space. The
success of our proposed approach (MS-ILS) lies in correctly estimating, how strong
the perturbations need to be.
If the degree of perturbation is extremely low, then the local search algorithm
may jump to the previously visited solutions, which eventually results in nothing,
but cycling around the same set of solutions. On the other hand, if the degree of
perturbation is extremely high, then the perturbation may produce almost random
solutions which may not retain any desirable features of the parent locally optimal
solutions, thereby making local search ineffective and hence only low-quality solutions are produced. We have utilized a variable degree of perturbation in our MS-ILS
approach. The degree of perturbation is directly controlled by parameter D p . The
main idea of this strategy is that the degree of perturbation needs to be high in the
initial iterations, and it has to be reduced as the algorithm progresses in order to get
good solutions. The parameter D p varies over iterations from maxdp to min dp . The
value of D p in an iteration iter is calculated as follows:
D p :=
maxdp − min dp
iter max
(iter max − iter ) + min dp
(6)
Here, iter max is maximum number of iterations up to which D p can be varied.
The value of D p cannot be decreased beyond a point, otherwise there will be no
perturbation and generated neighboring solution will be identical to the original
solution. The perturbation and local search procedures of the MS-ILS are interlaced
into a single procedure. This procedure consists of following three steps.
Step 1: Insertion of Cities
This step adds some cities to the solution under consideration. Suppose V and V represent the set of cities and set of visited cities in solution under consideration
respectively. A subset of cities of size D p × |V | is selected randomly from
V − V , and then these cities are inserted one-by-one, in the order in which they
are selected, into the best possible position in the solution under consideration.
Step 2: Removal of redundant Cities
284
P. Venkatesh et al.
Algorithm 2: Pseudocode of our proposed approach MS-ILS for the CSP
Input: Set of parameters for the MS-ILS and a CSP instance
Output: Best solution found
F(best) := ∞;
for st := 1 to Nr st do
S := Generate_Initial_Solution();
S := Local_Search(S);
while Termination condition not satisfied do
S := S;
S := Insert_Cities(S , D p × |V | );
S := Remove_Redundant_Cities(S );
if F(S ) < F(S) then
S := S ;
S := TSP_Neighboring(S);
if F(S ) < F(S) then
S := S ;
else if S is not improved over Limitnoimp trials then
S := S ;
if F(S) < F(best) then
best := S;
return best;
This step removes redundant cities from the solution under consideration. A redundant city is the one whose removal does not affect the feasibility of a CSP
tour. If there are multiple redundant cities, then a city is removed which yields
maximum decrease in the cost of CSP tour. The removal procedure continues until
there is no redundant city in the CSP tour. If the resulting solution after this step is
better than the input solution to perturbation and local search procedure then this
solution replaces the original input solution, otherwise the original input solution
is passed to step 3.
Step 3: TSP neighboring
This step removes some visited cities from the solution under consideration and
then reinserts these deleted cities only at their best positions to restore the feasibility. In this step, first, each visited city is removed with probability D p . All these
removed cities are added to a set, and then, one-by-one, a city is chosen randomly
from this set and inserted at best possible position in the solution.
Acceptance Criteria: Our acceptance criteria always compares the quality(fitness)
of the solution generated by perturbation and local search procedure with the solution
before applying this procedure. An improved solution is always accepted, and worse
solution is accepted only when there is no change in the solution continuously for a
Limit noimp number of iterations. The reason of accepting worst solutions is not to
waste the time in unworthy locally optimum solutions.
The pseudocode of our approach is given in Algorithm 2.
A Multi-start Iterated Local Search Algorithm …
285
4 Computational Results
Our approach, MS-ILS, is tested on the 48 CSP test instances of [5]. These instances
are actually derived from the TSPLIB.1 These instances have cities ranging from
51 to 200, and have a n × n distance matrix format. Like the approach of [5], our
approach, MS-ILS, is executed on each test instance 5 times independently, each
time with a different random seed.
Our approach MS-ILS is implemented in C and executed on a Linux-based 3.10
GHz Core-i5 system with 4 GB RAM. In all our experiments with the MS-ILS,
we used the following parameters—MS-ILS is restarted 100 times, i.e., Nr st = 100,
Limitnoimp = 50, I termax = 500, and D p is varied from 1 to 0.05, i.e., maxdp = 1
and min dp = 0.05. All these parameter values are chosen empirically. For benchmarking, our approach MS-ILS is compared with the state-of-the-art approaches
available in the literature, viz. LS2 [2], SN [4], CPLEX and Hybrid ACO [5].
Tables 1 and 2 report the performance of the MS-ILS along with aforementioned
approaches on instances with 51–100 cities and more than 100 cities respectively. In
Table 1, the columns of CPLEX are reported as follows. The column (best) reports
the best cost achieved by CPLEX. Rows with “–” represents the instances where
CPLEX was not able to find a feasible solution within the time limit. The column
(Rel-gap) reports the relative MIP gap by CPLEX, which is the gap between upper
bound and smallest of all lower bounds of open nodes in the search tree. The column
(gap) reports the gap between upper bound, found by CPLEX, and the best cost
overall the heuristics. In both of these Tables 1 and 2, the first column represents
the name of the instance containing some digits that means total number of cities it
contains. The second column (NC) shows the number of nearest cities that can be
covered by each city. The columns (Best cost & Avg. cost) reports the best and average
costs over five independent runs, respectively. The column (Deviation) reports the
deviation of the Avg.cost from the best cost over all heuristics. It is calculated as
, where best is the cost of best solution over all the heuristics. The
100× Avg.cost−best
best
column (TT) reports the average of computing times of five different runs. The last
two rows in both Tables 1 and 2, “Avg” reports the average values of each column,
whereas “NB” reports the number of best solutions achieved by each algorithm. The
best values are reported in bold for easy identification. These tables clearly show that
our approach is comparable with these state-of-the-art approaches, and whenever
our approach misses the best solution on an instance, it misses by small margin only
in comparison to other approaches. Further, our approach provides the best overall
average solution quality as can be seen in second last rows of these two tables.
1 http://elib.zib.de/pub/mp-testdata/tsp/tsplib/tsplib.html.
NC
7
9
11
berlin52 7
9
11
st70
7
9
11
eil76
7
9
11
pr76
7
9
11
rat99
7
9
11
kroA100 7
9
11
eil51
Instance
164
159
147
3887
3430
3262
288
259
250
219
198
177
50275
45387
44060
508
530
473
12762
10130
12843
CPLEX
Best
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
14.51
13.68
22.71
21.90
0.00
5.71
12.91
17.29
40.63
35.09
39.07
27.61
49.46
Rel-gap
149
220
681
140
212
255
490
1391
3600
3600
3600
3600
2488
3600
3600
3600
3600
3600
3600
3600
3600
Time
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.21
5.80
7.03
4.12
0.00
0.09
2.40
4.53
16.48
6.53
31.92
10.60
44.29
Gap
164
159
147
3887
3430
3262
288
259
247
207
186
170
50275
45348
43028
486
455
444
9674
9159
8901
164
159
147
3887
3430
3262
288
259
247
207
186
170
50275
45462.2
43028
486
455
444
9674
9159
8901
LS2
Best cost Avg.cost
Table 1 Comparison of various approaches on instances with 51–100 cities
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.54
0.00
0.00
0.25
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1
1
1
1
1
1
1
2
2
1
1
1
2
2
2
2
2
2
3
2
2
Deviation TT
164
159
147
3887
3430
3262
288
259
247
207
185
170
50275
45348
43028
486
455
444
9674
9159
8901
164
159
147
3887
3430
3262
288
259
247
207
185
170
50275
45348
43028
486
455
444
9674
9159
8901
SN
Best cost Avg.cost
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
(continued)
3
2
2
2
2
2
4
4
4
4
4
4
4
4
4
7
7
7
7
7
7
Deviation TT
286
P. Venkatesh et al.
NC
7
9
11
kroC100 7
9
11
kroD100 7
9
11
kroE100 7
9
11
rd100
7
9
11
Avg
NB
kroB100
Instance
Table 1 (continued)
–
10517
–
10477
10020
11226
10316
–
–
10609
10719
11879
3836
4049
3946
–
9
CPLEX
Best
–
36.97
–
22.34
30.76
47.91
18.25
–
–
14.39
26.10
53.50
24.00
52.00
49.00
21.12
Rel-gap
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
2867.40
Time
–
13.82
–
3.22
11.44
30.05
7.17
–
–
9.11
16.88
40.58
10.84
26.77
35.04
10.06
9
Gap
9537
9240
8842
9723
9171
8632
9626
8885
8725
10150
8991
8450
3461
3194
2922
8325.69
35
9537
9240
8842
9723
9171
8632
9626
8885
8725
10150
8991
8450
3485.6
3194
2922
8329.55
33
LS2
Best cost Avg.cost
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.71
0.00
0.00
0.04
3
3
3
3
2
2
2
3
3
2
2
2
2
2
2
1.92
Deviation TT
9537
9240
8842
9723
9171
8632
9626
8885
8725
10150
8991
8450
3461
3194
2922
8325.67
36
9537
9240
8842
9723
9171
8632
9626
8885
8725
10150
8991
8450
3493.8
3194
2922
8326.58
35
SN
Best cost Avg.cost
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.95
0.00
0.00
0.03
(continued)
7
7
7
7
7
7
6
7
7
6
7
7
6
6
6
5.31
Deviation TT
A Multi-start Iterated Local Search Algorithm …
287
7
9
11
7
9
11
7
9
11
7
9
11
7
9
11
7
9
11
7
9
11
eil51
kroA100
rat99
pr76
eil76
st70
berlin52
NC
Instance
Table 1 (continued)
164
159
147
3887
3430
3262
288
259
247
207
186
170
50275
45348
43028
486
455
444
9674
9159
8901
Hybrid ACO
Best cost
164
159
147
3887.4
3430
3262
288
259
247
207
186
170
50275
45348
43028
486
455
444
9674
9159
8901
Avg.cost
0.00
0.00
0.00
0.16
0.00
0.00
0.00
0.00
0.00
0.00
0.54
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Deviation
2
2
3
2
2
2
2
6
4
2
3
2
10
6
12
6
6
6
7
9
10
TT
164
159
147
3887
3430
3262
288
259
247
207
186
170
50275
45348
43028
486
455
444
9674
9159
8901
MS-ILS
Best cost
164.0
159.0
147.0
3887.0
3430.0
3262.0
288.0
259.0
247.0
207.0
186.0
170.0
50275.0
45348.0
43028.0
486.0
455.0
444.0
9674.0
9159.0
8901.0
Avg.cost
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.54
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
Deviation
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
TT
(continued)
288
P. Venkatesh et al.
7
9
11
7
9
11
7
9
11
7
9
11
7
9
11
kroB100
Avg
NB
rd100
kroE100
kroD100
kroC100
NC
Instance
Table 1 (continued)
9537
9240
8842
9723
9171
8632
9626
8885
8725
10150
8992
8450
3461
3194
2922
8325.72
34
Hybrid ACO
Best cost
9537
9240
8842
9724
9171
8632
9626
8885
8725
10150
8992
8451.2
3461
3194
2922
8325.96
31
Avg.cost
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.01
0.00
0.00
0.00
0.02
Deviation
4
9
8
5
10
5
4
5
11
7
5
10
4
6
6
5.71
TT
9537
9240
8842
9723
9171
8632
9626
8885
8725
10150
8992
8450
3461
3194
2922
8325.72
34
MS-ILS
Best cost
9537.0
9240.0
8842.0
9724.0
9171.0
8632.0
9626.0
8885.0
8725.6
10150.0
8992.0
8450.0
3461.0
3194.0
2922.0
8325.77
32
Avg.cost
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.01
0.00
0.01
0.00
0.00
0.00
0.00
0.02
Deviation
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1.00
TT
A Multi-start Iterated Local Search Algorithm …
289
7
9
11
7
9
11
7
9
11
7
9
11
kroA150
Avg
NB
kroB200
kroA200
kroB150
NC
Instance
11423
10056
9439
11457
10121
9611
13285
11708
10748
13100
11900
10676
11127.00
9
LS2
Best cost
11800.0
10062.4
9439.0
11491.2
10121.0
9611.0
13666.4
11716.8
10848.6
13511.6
11964.8
10809.6
11253.53
3
Avg.cost
3.30
0.06
0.00
0.30
0.00
0.00
2.87
0.08
0.94
3.53
0.85
1.56
1.12
Deviation
4
3
3
4
4
4
6
5
5
5
5
5
4.42
TT
Table 2 Comparison of various approaches on instances with more than 100 cities
11423
10056
9439
11457
10121
9611
13285
11708
10748
13051
11864
10644
11117.25
12
SN
Best cost
11423.0
10057.6
9439.0
11457.0
10121.0
9611.0
13327.0
11731.6
10865.6
13181.2
11878.4
10656.8
11145.77
5
Avg.cost
0.00
0.02
0.00
0.00
0.00
0.00
0.32
0.20
1.09
1.00
0.12
0.12
0.24
Deviation
(continued)
11
10
9
10
10
10
15
14
13
15
14
13
12.00
TT
290
P. Venkatesh et al.
7
9
11
7
9
11
7
9
11
7
9
11
kroA150
Avg
NB
kroB200
kroA200
kroB150
NC
Instance
Table 2 (continued)
11423
10056
9439
11457
10121
9611
13286
11710
10760
13051
11864
10644
11118.50
9
Hybrid ACO
Best cost
11423.0
10056.0
9439.0
11457.0
10121.0
9611.0
13286.0
11710.0
10764.2
13061.0
11871.2
10644.0
11120.73
12
Avg.cost
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.04
0.15
0.08
0.06
0.00
0.03
Deviation
6
10
8
9
6
11
12
12
15
8
12
9
10.29
TT
11423
10056
9439
11457
10121
9611
13286
11708
10775
13051
11864
10644
11119.58
10
MS-ILS
Best cost
11423.0
10056.0
9439.0
11457.0
10121.0
9611.0
13286.0
11709.2
10776.4
13051.0
11864.0
10644.0
11119.80
11
Avg.cost
0.00
0.00
0.00
0.00
0.00
0.00
0.02
0.01
0.26
0.00
0.00
0.00
0.02
Deviation
1
1
1
2
1
1
2
2
2
2
2
2
1.58
TT
A Multi-start Iterated Local Search Algorithm …
291
292
P. Venkatesh et al.
5 Conclusions
In this paper, we have proposed a simple and effective multi-start iterated local search
algorithm for the CSP. A key feature of our iterated local search algorithm is that the
perturbation and local search strategies are interlaced into a single procedure. We
have also incorporated a variable degree of perturbation strategy to further improve
the solution obtained through our approach. We have evaluated and compared our
proposed approach with the state-of-the-art approaches on the forty eight benchmark
instances of [5]. Computational results on these benchmark instances show that our
proposed approach is competitive with other state-of-the-art approaches.
We intend to extend our approach to generalized covering traveling salesman
problem. Similar approaches can be developed for other related problems such as
family traveling salesman problem, generalized traveling salesman problem. As a
future work, we would like to investigate the possibility of developing a population
based metaheuristic approach utilizing the components of the MS-ILS to improve
the results obtained through it.
Acknowledgements The first author acknowledges the financial support from Council of Scientific
& Industrial Research (CSIR), Government of India in the form of a Senior Research Fellowship.
The authors are also grateful to Dr. Majid Salari for providing the benchmark instances for the CSP.
References
1. Current, J.R., Schilling, D.A.: The covering salesman problem. Transp. Sci. 23(3), 208–213
(1989)
2. Golden, B., Naji-Azimi, Z., Raghavan, S., Salari, M., Toth, P.: The generalized covering salesman problem. INFORMS J. Comput. 24(4), 534–553 (2012)
3. Lin, S., Kernighan, B.W.: An effective heuristic algorithm for the traveling-salesman problem.
Oper. Res. 21(2), 498–516 (1973)
4. Salari, M., Naji-Azimi, Z.: An integer programming-based local search for the covering salesman problem. Comput. Oper. Res. 39(11), 2594–2602 (2012)
5. Salari, M., Reihaneh, M., Sabbagh, M.S.: Combining ant colony optimization algorithm and
dynamic programming technique for solving the covering salesman problem. Comput. Ind.
Eng. 83, 244–251 (2015)
6. Arkin, E.M., Hassin, R.: Approximation algorithms for the geometric covering salesman problem. Discrete Appl. Math. 55(3), 197–218 (1994)
7. Gendreau, M., Laporte, G., Semet, F.: The covering tour problem. Oper. Res. 45(4), 568–576
(1997)
8. Hachicha, M., Hodgson, M.J., Laporte, G., Semet, F.: Heuristics for the multi-vehicle covering
tour problem. Comput. Oper. Res. 27(1), 29–42 (2000)
9. Labbé, M., Laporte, G., Martín, I.R., González, J.J.S.: The ring star problem: polyhedral analysis and exact algorithm. Networks 43(3), 177–189 (2004)
10. Fischetti, M., Salazar González, J.J., Toth, P.: A branch-and-cut algorithm for the symmetric
generalized traveling salesman problem. Oper. Res. 45(3), 378–394 (1997)
11. Lourenço, H.R., Martin, O.C., Stutzle, T.: Iterated local search. Int. Ser. Oper. Res. Manage.
Sci. 321–354 (2003)
12. Lourenço, H.R., Martin, O.C., Stützle, T.: Iterated local search: Framework and applications.
In: Handbook of Metaheuristics, pp. 363–397. Springer, Berlin (2010)
A New Approach to Soft Hyperideals
in LA-Semihypergroups
Sabahat Ali Khan, M. Y. Abbasi and Aakif Fairooze Talee
Abstract In this paper, we introduce soft hyperideals in LA-semihypergroups
through new approach and investigate some useful results. Further, we characterize regular and intra-regular LA-semihypergroups using soft hyperideals.
Keywords LA-semihypergroups · Soft intersection hyperideals · Regular and
intra-regular LA-semihypergroups
1 Introduction and Preliminaries
Marty [10] introduced the notion of algebraic hyperstructures as a natural generalization of classical algebraic structures. In algebraic structures, the composition of
two elements is an element while in algebraic hyperstructures, the composition of
two elements is a nonempty set. Hasankhani [6] defined ideals in right(left) semihypergroups and discussed some hyperversions of Green’s relations.
The concept of LA-semigroup was given by Kazim and Naseeruddin [8]. Mushtaq and Yusuf [12] studied some properties of LA-semigroups. Hila and Dine
[7] defined LA-semihypergroups and studied several properties of hyperideals
in LA-semihypergroup. Yaqoob et al. [15] gave some characterizations of LAsemihypergroups using left and right hyperideals. Yousafzai and Corsini [16] characterized the class of an intra-regular LA-semihypergroup using one-sided hyperideals.
Molodsov [11] introduced a mathematical tool for dealing with hesitant, fuzzy,
unpredictable, and unsure articles known as soft set. Further, Maji et al. [9] defined
many applications in soft sets. Cagman and Aktas [1] introduced soft group theory
S. A. Khan (B) · M. Y. Abbasi · A. F. Talee
Department of Mathematics, Jamia Millia Islamia, New Delhi 110 025, India
e-mail: khansabahat361@gmail.com
M. Y. Abbasi
e-mail: yahya_alig@yahoo.co.in
A. F. Talee
e-mail: fuzzyaakif786.jmi@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_29
293
294
S. A. Khan et al.
and correlate soft sets with rough sets and fuzzy sets and gave [3] a new approach to
soft group called soft intersection group. Sezgin [13, 14] studied soft set theory in LAsemigroup with the concept of soft intersection LA-semigroups and soft intersection
LA-ideals.
We first recall the basic terms and definitions from the LA-semihypergroup theory
and soft set theory.
Definition 1 [4, 5] Let S be a nonempty set and let ℘ ∗ (S) be the set of all nonempty
subsets of S. A hyperoperation on S is a map o : S × S → ℘ ∗ (S) and (S, o) is called
a hypergroupoid.
If x ∈ S and A, B are nonempty subsets of S, then we denote
Ao B =
a o b, x o A = {x} o A and A o x = A o {x}.
a∈A,b∈B
Definition 2 [7] Let S be a nonempty set. A hypergroupoid S is called an LAsemihypergroup if for every x, y, z ∈ S, we have
(x ◦ y) ◦ z = (z ◦ y) ◦ x.
The law is called the left invertive law. Every LA-semihypergroup satisfies the following law:
(x ◦ y) ◦ (z ◦ w) = (x ◦ z) ◦ (y ◦ w),
for all x, y, z, w ∈ S. This law is known as medial law.
Definition 3 [15] Let S be an LA-semihypergroup, then an element e ∈ S is called
left identity (resp., pure left identity) if ∀ a ∈ S, a ∈ e ◦ a (resp., a = e ◦ a).
An LA-semihypergroup (S, ◦) with left identity satisfy the following laws,∀ x, y, z, w
∈ S.
(x ◦ y) ◦ (z ◦ w) = (w ◦ z) ◦ (y ◦ x),
called a paramedial law, and
x ◦ (y ◦ z) = y ◦ (x ◦ z),
Definition 4 [7] A nonempty subset T of an LA-semihypergroup S is called a subLA-semihypergroup of S if t1 ◦ t2 ⊆ T for every t1 , t2 ∈ T .
Definition 5 [7] A nonempty subset I of an LA-semihypergroup S is called a left
(resp., right) hyperideal of S if S ◦ I ⊆ I (resp., I ◦ S ⊆ I ) and is a hyperideal of S
if it is both a left and a right hyperideal.
A New Approach to Soft Hyperideals in LA-Semihypergroups
295
Definition 6 [15] An LA-semihypergroup S is called regular, if for each s ∈ S there
exists x ∈ S such that s ∈ (s ◦ x) ◦ s and intra-regular, if for each a ∈ S there exists
x, y ∈ S such that a ∈ (x ◦ a ◦ a) ◦ y.
Cagman and Enginoglu [2] gave the following concept of soft sets. Throughout
this paper, we represent:
S : an LA-semihypergroup, V : an initial universe, E : a set of parameters, F(S) : set
of all soft sets of S over V, P(V) : the powerset of V.
Definition 7 A soft set F A over V is a set defined by F A : E → P(V) such that
/ A.
F A (x) = ∅ if x ∈
Here, F A is also called an approximate function. A soft set over V can be represented
by the set of ordered pairs
F A = {(x, F A (x)) : x ∈ E, F A (x) ∈ P(V)}.
It is clear to see that a soft set is a parameterized family of subsets of the set V.
Definition 8 Let F A , F B ∈ F(S). Then, F A is called a soft subset of F B and denoted
F B , if F A (x) ⊆ F B (x) for all x ∈ E.
by F A
Definition 9 Let F A ,F B ∈ F(S). Then, the union of F A and F
B denoted by F A
F B , is defined as F A F B = F A∪ B , where F A∪ B (x) = F A (x) F B (x) for all x ∈
E.
Definition
10 Let F A , F B ∈F(S). Then, the intersection of F A and F
B denoted by
F A F B is defined as F A F B = F A∩ B , where F A∩ B (x) = F A (x) F B (x) for
all x ∈ E.
2 Soft Characteristic Function, Soft Intersection (S.I.)
Product, and Soft Intersection (S.I.) Hyperideals
In this section, we define soft characteristic function, soft intersection (S.I.) product,
and soft intersection (S.I.) hyperideals. Further, we study S.I. hyperideals with S.I.
product and some interesting results.
Definition 11 Let Y be a subset of S. We denote the soft characteristic function of
Y by SY and is defined as
SY (y) =
V,
∅,
if y ∈ Y
if y ∈
/ Y.
296
S. A. Khan et al.
In this paper, we denote an LA-semihypergroup S as a set of parameters.
Let S be an LA-semihypergroup. For x ∈ S, we define Sx = {(y, z) ∈ S × S | x ∈
y ◦ z}.
Definition 12 Let F S and G S be two soft sets of an LA-semihypergroup S over V.
Then, the soft product F S ˆ G S is a soft set of S over V, defined by
(F S ˆ G S )(x) =
⎧ ⎨
{F S (y) ∩ G S (z)} if Sx = ∅
(y,z)∈Sx
⎩
∅
if Sx = ∅
for all x ∈ S.
Theorem 1 Let X and Y be nonempty subsets of an LA-semihypergroup S. Then
(1) If X⊆ Y , then S X S
Y.
(2) S X SY = S X ∩Y , S X SY = S X ∪Y .
(3) S X ˆ SY = S X ◦Y .
Proof (1) and (2) are trivial.
(3). Let s ∈ S such that s ∈ X ◦ Y , then there exists x ∈ X and y ∈ Y such that s ∈
x ◦ y. Thus, (x, y) ∈ Ss . So, Ss is nonempty. Now,
(S X ˆ SY )(s) =
( p,q)∈Ss
{S X ( p) ∩ SY (q)} ⊇ S X (x) ∩ SY (y) = V.
It implies S X ˆ SY = V. Hence, S X ˆ SY = S X ◦Y .
Definition 13 A non-null soft set F S is said to be an S.I. sub-LA-semihypergroup
of S over V, if
F S (ϑ) ⊇ F S (x) ∩ F S (y)∀x, y ∈ S.
ϑ∈x◦y
Definition 14 A non-null soft set F S is said to be an S.I. left (resp., right) hyperideal
of S over V, if
F S (ϑ) ⊇ F S (y)(r esp.,
ϑ∈x◦y
F S (ϑ) ⊇ F S (x))∀x, y ∈ S.
ϑ∈x◦y
Definition 15 A non-null soft set F S is said to be an S.I. hyperideal of S over V if
it is both an S.I. left and an S.I. right hyperideal of S over V.
Example 1 Let (S, ◦) be an LA-semihypergroup, where S = {1, 2, 3} with a hyperoperation ◦ is given by following table:
Let V = {a, b, c}. Define a soft set F S : S → P(V) by
F S (1) = {a}, F S (2) = {a, b, c} and F S (3) = {a, b, c}.
A New Approach to Soft Hyperideals in LA-Semihypergroups
◦
1
2
3
1
{1, 3}
{2, 3}
{2, 3}
2
3
3
{2, 3}
297
3
{2, 3}
3
{2, 3}
Then, we can verify that ϑ∈x◦y F S (ϑ) ⊇ F S (y) and ϑ∈x◦y F S (ϑ) ⊇ F S (x) ∀
x, y ∈ S. Therefore, F S is an S.I. hyperideal of S over V.
Theorem 2 A non-null soft set F S is an S.I. left hyperideal of S over V if and only
if S S ˆ F S F S .
Proof Suppose F S is an S.I. left hyperideal of S over V. Then ϑ∈x◦y F S (ϑ) ⊇
F S (y) ∀ x, y ∈ S. Now, if Sx = ∅, then (S S ˆ F S )(x) = ∅. In this case, (S S ˆ F S )(x)
⊆ F S (x), therefore S S ˆ F S F S . If Sx = ∅, then there exists u, v ∈ S such that x
∈ u ◦ v. Thus, (u, v) ∈ Sx . So, Sx is nonempty. Then, we have
(S S ˆ F S )(x) =
S S (u)
V
F S (v)
F S (v) =
(u,v)∈S
(u,v)∈S
x
x
=
F S (v) ⊆
F S (ϑ) ⊆
F S (x)
x∈ p◦v
ϑ∈ p◦v
(u,v)∈S
(u,v)∈S
(u,v)∈S
x
x
x
⊆
F S (x) = F S (x).
(u,v)∈Sx
x∈u◦v
Therefore, S S ˆ F S F S .
Conversely, suppose that S S ˆ F S F S . Now, we have to show that F S is an S.I.
left hyperideal of S over V. Thus, we have
ϑ∈x◦y
=
⊇
ϑ∈x◦y
ϑ∈x◦y
ϑ∈x◦y
(u,v)∈Sϑ
(S S ˆ F S )(ϑ) =
ϑ∈x◦y
V
F S (v) =
ϑ∈x◦y
(u,v)∈Sϑ
F S (y) ⊇
F S (ϑ) ⊇
ϑ∈x◦y
(u,v)∈Sϑ
S S (u)
F S (v)
F S (v)
(u,v)∈Sϑ
F S (y) = F S (y).
(x,y)∈Sϑ
It follows that F S is an S.I. left hyperideal of S over V.
Theorem 3 A non-null soft set F S is an S.I. sub-LA-semihypergroup (resp., S.I. right
hyperideal) of S over V if and only if F S ˆ F S F S (resp., F S ˆ S S F S ).
Proof Proof is similar to the Theorem 2.
Theorem 4 Let S be an LA-semihypergroup and F(S) be the set of all soft sets of S
over V. Then (F(S), ˆ ) is an LA-semigroup.
298
S. A. Khan et al.
Proof Let us suppose that F S , G S and H S ∈ F(S) and x ∈ S such that x ∈ y ◦ z, y
∈ p ◦ q and t ∈ y ◦ p, where y, z, p, q, t ∈ S . Then,
F S ˆ G S (y) H S (z)
((F S ˆ G S ) ˆ H S )(x) =
(y,z)∈Sx
=
F S ( p) G S (q)
H S (z)
(y,z)∈Sx
( p,q)∈S y
F S ( p) G S (q)
H S (z)
=
x∈y◦z
y∈ p◦q
=
F S ( p) G S (q)
H S (z)
x∈( p◦q)◦z
H S (z) G S (q)
F S ( p)
=
x∈(z◦q)◦ p
H S (z) G S (q)
F S ( p)
=
x∈t◦ p
t∈z◦q
H S (z) G S (q)
F S ( p)
=
(t, p)∈Sx
(z,q)∈St
=
H S ˆ G S (t) F S ( p)
(t, p)∈Sx
= ((H S ˆ G S ) ˆ F S )(x).
Therefore, ((F S ˆ G S ) ˆ H S ) = ((H S ˆ G S ) ˆ F S ).
Theorem 5 If S is an LA-semihypergroup. Then medial law holds in F(S).
Proof Let us suppose that S is an LA-semihypergroup and F(S) is the set of all soft
sets of S over V and let F S , G S , H S , K S ∈ F(S), Then by applying invertive law, we
have
(F S ˆ G S ) ˆ (H S ˆ K S ) = ((H S ˆ K S ) ˆ G S ) ˆ F S = ((G S ˆ K S ) ˆ H S ) ˆ F S
= (F S ˆ H S ) ˆ (G S ˆ K S ).
Theorem 6 Let S be an LA-semihypergroup with left identity and F S , G S , H S ∈
F(S). Then following holds:
(i) F S ˆ (G S ˆ H S ) = G S ˆ (F S ˆ H S ).
(ii) (F S ˆ G S ) ˆ (H S ˆ K S ) = (K S ˆ H S ) ˆ (G S ˆ F S ).
Proof (i) Let us suppose that F S , G S and H S ∈ F(S) and x ∈ S such that x ∈ y ◦ z,
z ∈ p ◦ q and t ∈ y ◦ p, where y, z, p, q, t ∈ S. Then
A New Approach to Soft Hyperideals in LA-Semihypergroups
299
F S (y)
(F S ˆ (G S ˆ H S ))(x) =
G S ˆ H S (z)
(y,z)∈S
x
=
F S (y)
G S ( p) H S (q)
(y,z)∈S
( p,q)∈Sz
x
=
F S (y)
G S ( p) H S (q)
x∈y◦z
z∈ p◦q
F S (y)
=
G S ( p) H S (q)
x∈y◦( p◦q)
G S ( p) F S (y)
H S (q)
=
x∈ p◦(y◦q)
G S ( p)
F S (y) H S (q)
=
x∈ p◦t
t∈y◦q
G S ( p)
F S (y) H S (q)
=
( p,t)∈Sx
(y,q)∈St
G S ( p)
F S ˆ H S (t)
=
( p,t)∈Sx
= (G S ˆ (F S ˆ H S ))(x).
Therefore, F S ˆ (G S ˆ H S ) = G S ˆ (F S ˆ H S ).
(ii) Let us suppose that F S , G S , H S ∈ F(S) and x ∈ S such that x ∈ y ◦ z, y ∈ p
◦ q and z ∈ s ◦ t, where y, z, p, q, s, t ∈ S . Then
(F S ˆ G S )(y) (H S ˆ K S )(z)
((F S ˆ G S ) ˆ (H S ˆ K S ))(x) =
(y,z)∈Sx
F S ( p) ∩ G S (q)
H S (s) ∩ K S (t)
=
(y,z)∈Sx ( p,q)∈S y
(s,t)∈Sz
=
F S ( p) ∩ G S (q)
H S (s) ∩ K S (t)
x∈y◦z y∈ p◦q
z∈s◦t
F S ( p) ∩ G S (q) ∩ H S (s) ∩ K S (t)
=
x∈( p◦q)◦(s◦t)
=
K S (t) ∩ H S (s) ∩ G S (q) ∩ F S ( p)
x∈(t◦s)◦(q◦ p)
K S (t) ∩ H S (s)
G S (q) ∩ F S )( p)
=
x∈m◦n m∈t◦s
n∈q◦ p
K S (t) ∩ H S (s)
G S (q) ∩ F S )( p)
=
(m,n)∈Sx (t,s)∈Sm
(q, p)∈Sn
(K S ˆ H S )(m) (G S ˆ F S )(n) = ((K S ˆ H S ) ˆ (G S ˆ F S ))(x).
=
(m,n)∈Sx
Hence, (F S ˆ G S ) ˆ (H S ˆ K S ) = (K S ˆ H S ) ˆ (G S ˆ F S ).
Proposition 1 In an LA-semihypergroup S with left identity, for every S.I. left hyperideal F S of S over V, S S ˆ F S = F S .
Proof Let F S be an S.I. left hyperideal of S over V, then S S ˆ F S F S . We only
need to prove that F S S S ˆ F S . Since S is an LA-semihypergroup with left identity,
300
S. A. Khan et al.
then for any x ∈ S, x ∈ e ◦ x, where e is the left identity of S. Thus, (e, x) ∈ Sx .
So, Sx is nonempty. Now, we have
(S S ˆ F S )(x) =
(y,z)∈Sx
S S (y) ∩ F S (z)
⊇
S S (e) ∩ F S (x)
= F S (x).
Therefore, S S ˆ F S = F S .
Proposition 2 In an LA-semihypergroup S with left identity, for every S.I. right
hyperideal F S of S over V, F S ˆ S S = F S .
Proof Let F S be an S.I. right hyperideal of S over V, then F S ˆ S S F S . Now, it
is only remains to show that F S F S ˆ S S . Since S is an LA-semihypergroup with
left identity, thus for any x ∈ S, x ∈ e ◦ x ⊆ (e ◦ e) ◦ x =(x ◦ e) ◦ e, where
e is the left identity of S. Then, there exists y ∈ x ◦ e such that x ∈ y ◦ e. Thus,
(y, e) ∈ Sx . So, Sx is nonempty. Now, we have
(F S ˆ S S )(x) =
F S ( p) ∩ S S (q)
( p,q)∈Sx
⊇ F S (y) ∩ S S (e)
= F S (y)
(1)
As F S is an S.I. right hyperideal of S over V, we have ϑ∈x◦e F S (ϑ) ⊇ F S (x) ∀ x,
e ∈ S. Since y ∈ x ◦ e, it would imply that F S (y) ⊇ F S (x), therefore from (1)
(F S ˆ S S )(x) ⊇ F S (y) ⊇ F S (x).
It implies F S
F S ˆ S S . Hence, F S ˆ S S = F S .
Corollary 1 In an LA-semihypergroup S with left identity, S S ˆ S S = S S .
Proposition 3 Let S be an LA-semihypergroup with left identity. Then, every S.I.
right hyperideal of S over V is an S.I.hyperideal of S over V.
Proof Let F S be an S.I. right hyperideal of S over V. Then F S ˆ S S F S . Thus, we
have S S ˆ F S = (S S ˆ S S ) ˆ F S = (F S ˆ S S ) ˆ S S F S ˆ S S F S .
So, F S is an S.I. left hyperideal of S over V, hence an S.I. hyperideal of S over V.
Theorem 7 Let X be any nonempty subset of an LA-semihypergroup S. Then X
is a left (resp., right) hyperideal of S if and only if S X is an S.I. left (resp., right)
hyperideal of S over V.
Proof Let X be a left hyperideal of S. Then S ◦ X ⊆ X . Now S S ˆ S X = S(S◦X )
S X .This shows that S X is an S.I. left hyperideal of S over V.
Conversely, suppose that S X is an S.I. left hyperideal of S over V. Let x ∈ S ◦ X ,
then S X (x) ⊇ (S S ˆ S X )(x) = S(S◦X ) (x) = V. It implies x ∈ X . Hence, S ◦ X ⊆ X .
Therefore, X is a left hyperideal of S.
A New Approach to Soft Hyperideals in LA-Semihypergroups
301
Theorem 8 If F S is an S.I. right hyperideal of S over V and G S an S.I. left hyperideal
of S over V. Then
FS ˆ GS FS GS
Proof Proof is straight forward.
3 Characterizations of Regular and Intra-regular
LA-Semihypergroups
In this section, we characterize regular and intra-regular LA-semihypergroups using
S.I. hyperideals.
Theorem 9 For an LA-semihypergroup S, the following conditions are equivalent:
(1) S is regular;
(2) R ◦ L = R ∩ L for every right hyperideal R and every left hyperideal L of S;
(3) < a > R ∩ < b > L = (< a > R ◦ < b > L ) for all a, b ∈ S;
(4) < a > R ∩ < a > L = (< a > R ◦ < a > L ) for all a ∈ S.
Proof Proof is straightforward.
Theorem 10 For an LA-semihypergroup S, the following conditions are equivalent:
(1) S is regular; (2) F S ˆ G S = F S G S for every S.I. right hyperideal F S and every S.I. left hyperideal
G S of S over V.
Proof (1) =⇒ (2):
By Theorem 8,
Suppose that S is a regular LA-semihypergroup.
F S ˆ G S F S G S . Now, it remains to show that F S G S F S ˆ G S . To prove
this, let s ∈ S. As S is regular, there exists an element x ∈ S such that s ∈ (s ◦ x) ◦ s.
Now s ∈ (s ◦ x) ◦ s, it implies there exists b ∈ s ◦ x such that s ∈ b ◦ s. Thus, (b, s)
∈ Ss and hence Ss is nonempty. Then, we have
F S ˆ G S (s) =
{F S ( p) ∩ G S (q)}
( p,q)∈Ss
⊇ F S (b) ∩ G S (s)
(2)
As F S is an S.I. right hyperideal of S over V, we have ϑ∈s◦x F S (ϑ) ⊇ F S (s) ∀ s, x
∈ S. Since b ∈ s ◦ x, it would imply that F S (b) ⊇ F S (s). Now, we have from (2)
F S ˆ G S (s) ⊇ F S (b) ∩ G S (s) ⊇ F S (s) ∩ G S (s) = F S G S (s)
Therefore, F S ˆ G S = F S G S .
(2) =⇒ (1): Suppose that F S ˆ G S = F S G S . To show S is regular, we have to
prove that R ◦ L = R ∩ L, for every right hyperideal R and every left hyperideal
302
S. A. Khan et al.
L of S. Let R and L be right and left hyperideals of S, as we know that R ◦ L ⊆
R ∩ L. Thus, we will only show that R ∩ L ⊆ R ◦ L. Let a ∈ R ∩ L. By Theorem
7, characteristic soft function S R and S L of S are S.I. right hyperideal of S over V
and S.I. left hyperideal of S over V. Then, we have
S(R◦L) (a) = S R ˆ S L (a) = S R S L (a) = S(R∩L) (a) = V.
It follows that a ∈ R ◦ L and hence R ∩ L ⊆ R ◦ L. Therefore, (2) implies (1).
Theorem 11 If S is an intra-regular LA-semihypergroup, then every S.I. right hyperideal is an S.I. left hyperideal of S over V.
Proof Let F S be an S.I. right hyperideal of S over V, then ϑ∈a◦b F S (ϑ) ⊇ F S (a)
∀ a, b ∈ S. As S is an intra-regular LA-semihypergroup, thus for any a ∈ S, there
exists x, y ∈ S such that a ∈ (x ◦ a ◦ a) ◦ y. Now ϑ ∈ a ◦ b, it implies ϑ ∈ ((x ◦
a ◦ a) ◦ y) ◦ b, then there exists c ∈ (x ◦ a ◦ a) such that ϑ ∈ (c ◦ y) ◦ b. Since S
is an LA-semihypergroup, then ϑ ∈ (b ◦ y) ◦ c. So, there exists d ∈ b ◦ y such that
ϑ ∈ d ◦ c. As F S is an S.I. right hyperideal of S over V, we have F S (d)⊇ F S (b) and
F S (ϑ) ⊇ F S (d). It implies F S (ϑ) ⊇ F S (b) for all ϑ ∈ a ◦ b. Hence ϑ∈a◦b F S (ϑ)
⊇ F S (b) ∀ a, b ∈ S. Therefore, F S is an S.I. left hyperideal of S over V.
Theorem 12 For an LA-semihypergroup S, the following conditions are equivalent:
1.
2.
3.
4.
S is intra-regular;
L ∩ R ⊆ L ◦ R for every left hyperideal L and every right hyperideal R of S;
< a > L ∩ < b > R ⊆ < a > L ◦ < b > R for all a, b ∈ S;
< a > L ∩ < a > R ⊆ < a > L ◦ < a > R for all a ∈ S.
Proof Proof is straightforward.
Theorem 13 For an LA-semihypergroup S with left identity, the following conditions
are equivalent:
1. S is intra-regular;
2. F S ˆ G S F S G S for every S.I. left hyperideal F S and every S.I. right
hyperideal G S of S over V.
Proof (1) =⇒ (2): Let S be an intra-regular LA-semihypergroup with left identity
e , F S an S.I. left hyperideal of S over V and G S an S.I. right hyperideal of S
over V. Also, let a ∈ S. Since S is intra-regular, there exist x, y ∈ S such that a
∈ (x ◦ (a ◦ a)) ◦ y = (a ◦ (x ◦ a)) ◦ y = (y ◦ (x ◦ a)) ◦ a ⊆ (y ◦ (x ◦ a)) ◦ (e ◦ a) =
(y ◦ e) ◦ ((x ◦ a) ◦ a) = (x ◦ a) ◦ ((y ◦ e) ◦ a) = (x ◦ a) ◦ ((a ◦ e) ◦ y). Then, there
exists c ∈ x ◦ a, b ∈ a ◦ e and d ∈ b ◦ y such that a ∈ c ◦ d. Thus, (c, d) ∈ Sa . So,
Sa is nonempty. Then, we have
F S ˆ G S (a) =
{F S ( p) ∩ G S (q)}
( p,q)∈Sa
⊇ F S (c) ∩ G S (d).
(3)
A New Approach to Soft Hyperideals in LA-Semihypergroups
303
As F S is an S.I. left hyperideal of S over V, we have ϑ∈x◦a F S (ϑ) ⊇ F S (a) ∀
x, a ∈ S. Since c ∈ x ◦ a, it would
imply that F S (c) ⊇ F S (a). As G S is an S.I. right
hyperideal of S over V, we have ϑ∈a◦y G S (ϑ) ⊇ G S (a) ∀ y, a ∈ S. Since d ∈ b ◦ y
and b ∈ a ◦ e, it would imply that G S (d) ⊇ G S (b) and G S (b) ⊇ G S (a). So, G S (d) ⊇
G S (a). Now from (3), we have
F S ˆ G S (a) ⊇ F S (c) ∩ G S (d) ⊇ F S (a) ∩ G S (a) = F S G S (a).
Hence, F S ˆ G S F S G S .
(2) =⇒ (1): Suppose F S ˆ G S F S G S . We only need to show that L ∩ R ⊆
L ◦ R, for every right hyperideal R and every left hyperideal L of S. Let b ∈ L ∩ R,
where L and R are left and right hyperideals of S, respectively. By Theorem 7, the
characteristic soft functions S L and S R of S are S.I. left hyperideal of S over V and
S.I. right hyperideal of S over V. Now, we have
S(L◦R) (b) = S L ˆ S R (b) ⊇ S L S R (b) = S(L∩R) (b) = V.
It follows that b ∈ L ◦ R and hence L ∩ R ⊆ L ◦ R. Therefore, (2) implies (1).
References
1. Aktaş, H., Çağman, N.: Soft sets and soft groups. Inform. Sci. 177, 2726–2735 (2007)
2. Çağman, N., Enginoglu, S.: Soft set theory and uni-int decision making. Eur. J. Oper. Res. 207,
848–855 (2010)
3. Çağman, N., Çitak, F., Aktaş, H.: Soft int-group and its applications to group theory. Neural
Comput. Appl. 21, 151–158 (2012)
4. Corsini, P.: Prolegomena of Hypergroup Theory, Aviani (ed), 2nd ed. (1993)
5. Davvaz, B., Fotea, V.L.: Hyperring Theory and Applications, p. 115. International Academic
Press, Palm Harber (2007). (1993)
6. Hasankhani, A.: Ideals in a semihypergroup and Greens relations. Ratio Mathematica. 13,
29–36 (1999)
7. Hila, K., Dine, J.: On hyperideals in left almost semihypergroups. ISRN Algebra. Vol. 2011,
Article ID953124, 8 pages, (2011)
8. Kazim, M.A., Naseeruddin, M.: On almost semigroups. Alig. Bull. Math. 2, 1–7 (1972)
9. Maji, P.K., Biswas, R., Roy, A.R.: Soft set theory. Comput. Math. Appl. 45, 555–562 (2003)
10. Marty, F.: Sur une generalization de la notion de group. 8th Congres Math. Scandinaves Stockholm, pp. 45–49 (1934)
11. Molodtsov, D.: Soft set theory-first results. Comput. Math. Appl. 37, 19–31 (1999)
12. Mushtaq, Q., Yusuf, S.M.: On LA-semigroups. Alig. Bull. Math. 8, 65–70 (1978)
13. Sezgin, A.: A new approach to LA-semigroup theory via the soft sets. J. Intell. Fuzzy Syst. 26,
2483–2495 (2014)
14. Sezgin, A.: Certain characterizations of LA-semigroups by soft sets. J. Intell. Fuzzy Syst. 27(2),
1035–1046 (2014)
15. Yaqoob, N., Corsini, P., Yousafzai, F.: On intra-regular left almost semihypergroups with pure
left identity. J. Math. http://dx.doi.org/10.1155/2013/510790, 10 pages (2013)
16. Yousafzai, F., Corsini, P.: Some Characterization Problems in LA-semihypergroups. J. Algebra,
Number Theory: Adv. Appl. 10(1–2), 41–55 (2013)
Adjusted Artificial Bee Colony Algorithm
for the Minimum Weight Triangulation
Adis Alihodzic, Haris Smajlovic, Eva Tuba,
Romana Capor Hrosik and Milan Tuba
Abstract The minimum weight triangulation is a well-known NP-hard problem
often used for the construction of triangulated random network models of land contours. Since it is an intractable problem, the required computational time for an
exhaustive search algorithm grows exponentially with the number of points in 2D
space. Nature-inspired swarm intelligence algorithms are prominent and efficient
optimization techniques for solving that kind of problems. In this paper, we adjusted
the artificial bee colony algorithm for the minimum weight triangulation problem.
Our adjusted algorithm has been implemented and tested on several randomly generated instances of points in the plane. The performance of our proposed method was
compared to the performance of other stochastic optimization algorithms, as well as
with the exhaustive search for smaller instances. The simulation results show that
our proposed algorithm in almost all cases outperforms other compared algorithms.
Keywords Swarm intelligence algorithms · Artificial bee colony · ABC
Minimum weight triangulation
This research is supported by the Ministry of Education, Science and Technological Development
of Republic of Serbia, Grant No. III-44006.
A. Alihodzic · H. Smajlovic
Faculty of Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina
E. Tuba
Faculty of Informatics and Computing, Singidunum University, Belgrade, Serbia
R. Capor Hrosik
Maritime Department, University of Dubrovnik, Dubrovnik, Croatia
M. Tuba (B)
Faculty of Informatics and Computing, Singidunum University, Belgrade, Serbia
e-mail: tuba@ieee.org
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_30
305
306
A. Alihodzic et al.
1 Introduction
The minimum weight triangulation problem dates back to the 1970s and has been
called perhaps the most longstanding open problem in computational geometry. It
was introduced by Garey and Johnson in their open problems list [7]. Nearly 30
years after being enlisted, in 2008 Mulzer and Rote have proved that it was an
NP-hard problem [14]. In computational geometry, optimization problems related
to specific geometric configurations are interesting for research due to their use in
many fields of applications. One of such problems is planar triangulation which has
become very popular primarily because of their large set of applications such as
finite element methods, computer-aided geometric design, surface interpolation, for
calculations in numerical analysis, computer graphics, robotics, computer vision, and
image synthesis, visibility, ray shooting, kinetic collision detection, rigidity, guarding
as well as in mathematical and natural sciences. Also, other significant computational
geometry problems, like geometric searching and polyhedron intersection, use planar
triangulations as a preprocessing phase.
In computational geometry, there are many challenges which are intractable problems, or no polynomial algorithms are known. One of such challenges is the problem
of finding the minimum weight triangulation. It is based on searching the minimum
sum of the lengths of the edges over all possible triangulations for a given set of
n points in 2D space. Therefore, the required computational time for an exhaustive
search algorithm grows exponentially with the number of points in the plane. To
overcome this problem, some algorithms such as linear programming [25], nearest
neighbor graph [10], greedy search with local search [5], and others have already
been used. In this paper, we propose the use of prominent nature-inspired swarm
intelligence algorithm, artificial bee colony, to search for the suboptimal solutions
to reach fast convergence and reduce the CPU time. These metaheuristic algorithms
have a simple implementation and they can efficiently find approximate solutions for
NP-hard optimization problems [13]. The most popular nature-inspired algorithms
for optimization, with improvements, adjustments, and hybridizations, include ant
colony optimization [3, 4, 8], artificial bee colony algorithm (ABC) [9], firefly algorithm (FA) [16, 19, 22], bat algorithm (BA) [1, 2, 23], cuckoo search (CS) [18,
24], and fireworks algorithm [15, 17, 20].
The ABC algorithm was first proposed for unconstrained optimization problems
but was later also applied to constrained problems by extending the basic ABC
algorithm simply by adding a constraint handling technique into the selection step
of the ABC algorithm to prefer the feasible regions of the entire search space. In
this paper, an adjusted artificial bee colony (AABC) algorithm is presented applied
to the minimum weight triangulation problem. To show the power of this approach,
the AABC algorithm was applied to various randomly generated instances of points
in the plane, and the obtained results were compared with the results reached by
other well-known stochastic optimization algorithms, namely simulated annealing
(SA) and particle swarm intelligence (PSO). The experimental results show that
Adjusted Artificial Bee Colony Algorithm …
307
the AABC algorithm always produces better results compared to the PSO and SA
algorithms, considering both accuracy and, especially, convergence speed.
The rest of the paper is organized as follows. The minimum weight triangulation
(MWT) is described in Sect. 2, while the brief review of the artificial bee colony
(ABC) algorithm is presented in Sect. 3. Details of our adjusted ABC algorithm
are in Sect. 4. Experimental and comparative results of applying PSO, SA, and the
proposed adjusted ABC to the minimum weight triangulation are presented in Sect. 5.
Finally, conclusions and suggestion for future work are discussed in the last section
of the paper, Sect. 6.
2 Minimum Weight Triangulation Problem
Solving the polygon triangulation problem led to the discovery of Catalan numbers.
In fact, by trying to find the number of different ways to triangulate a convex polygon,
Euler was the first to discover, each and every time, that number takes a certain value
regarding some vertices. Even though Euler was the first to discover it, the number
became known as Catalan number later on, in honor of the Belgian mathematician.
The link between Catalan numbers Cn and the number of different ways to triangulate
convex polygon of n vertices is given by
Tn = C + n − 2, n ≥ 3,
(1)
where Tn stands for the number of possible triangulations of a convex polygon and
Cn is given by
(2n)!
.
(2)
Cn =
(n + 1)! n!
Considering the numerator in Eq. (2), it can be seen that finding all possible
triangulations of a given polygon, or, even more, of a given point set, is an exponential
combinatorial problem and NP-hard, as well as finding the one with the lowest sum
of edges’ weights [14].
Notations and common terms in this paper related to the triangulation are as
follows. Vertex set and edge set of a graph G are denoted as V (G) and E(G), respectively. Analogously, vertex and edge sets of triangulation T are denoted as V (T )
and E(T ), respectively. H (S) is the notation for the hull of a point set S. The problem
of triangulating point set S reduces to finding a planar graph G where V (G) = S
and every face of G, except the outer ones, is bounded by exactly three edges. Reference to the triangulation T is equivalent to a particular planar graph with attributes
mentioned above. The weight of triangulation T , denoted as wt (T ), presents a sum
of triangulation edges lengths. The problem of finding a minimum weight triangulation thus reduces to finding a triangulation with minimal sum of triangulation edges
lengths. Euclidean distance for calculating edge lengths is the most natural choice,
308
A. Alihodzic et al.
but often, including in this paper, for simpler calculations, squared Euclidean distance
is used.
In the past, several approaches to the minimum weight triangulation problem
were proposed. Numerous studies used modified and adjusted genetic algorithms.
In [21], quantum genetic algorithm was proposed for minimum weight triangulation
and it was proven to be better than the greedy method. Genetic algorithm combined
with annealing selection was proposed in [12]. Annealing selection prevented early
convergence, and thus performance of the proposed method was better compared
to the genetic algorithm. Besides genetic algorithms, swarm intelligence algorithms
were also used. In [6], ant colony optimization algorithm was applied to the minimum weight triangulation as well for the minimum weight pseudo-triangulation.
Ant colony optimization outperformed greedy and simulated annealing algorithms.
3 General Artificial Bee Colony Algorithm
In this section, a general artificial bee colony algorithm (ABC), adaptable to problems
with discrete search space, is described. The artificial bee colony algorithm is a swarm
intelligence algorithm proposed by Karaboga in 2005 [9]. It is another metaheuristic
inspired by some swarm behavior, particularly in this case, a swarm of bees. Four
main phases determine the ABC algorithm. These are the initial phase, employed
bees phase, onlooker bees phase and scout bees phase which are executed in the
following manner:
Algorithm 1 ABC Algorithm
1: Initial Phase
2: repeat
3: Employed bees phase
4: Onlooker bees phase
5: Scout bees phase
6: until some terminating condition is met
A. Initial phase: During this phase, a random solution set is generated within
the given search space. Since the ABC algorithm, just like any other populationbased algorithm, deals with a whole set or population of solutions all the time, this
initialization has to be performed.
B. Employed bees phase: Each solution in the working solution set is treated as a
food source and to each food source one employed bee is attached. Each employed
bee does a local search of its food source neighborhood and attaches to the food
source of better quality, if such one is found. It means that during this phase, for each
solution inside the current solutions set, some arbitrary local search heuristic with
the greedy selection is performed. It is recommended, though, for the local search
being used to be a rather randomized search, meaning that a neighbor solution is
Adjusted Artificial Bee Colony Algorithm …
309
randomly selected out of the available search space. Since this simple search method
with greedy selection is of significant importance, it can be treated as a separate
method. Namely, for a given solution S, its neighborhood is limited to an arbitrary
size and a random solution S is found out of it. If solution S is of better quality, S is
replaced by S.
C. Onlooker bees phase: For each employed bee, there exists one affiliated onlooker bee. With certain probability, typically calculated as
fitnessi
Pi = j fitness j
(3)
where fitnessi is the quality of the ith solution, each onlooker bee performs another
local search around the food source of the affiliated employed bee and keeps track
of new found food source with the better quality, or if no food source with better
quality is found, of the same food source of the affiliated employed bee. During
this phase, for each solution S in the current solution set, another local search of its
neighborhood is performed, in the same manner as in the previous phase, but with
the probability Pi from Eq. 3. The track is kept of the new found solution with the
better quality, or if no solution with better quality is found, of the solution S that is
now stored in another solution set, distinctive from the employed bees solution set.
Regarding the size of the neighborhoods, especially when the onlooker bees local
search is performed, the possibility of it being rather wide should be ensured. That
can be done by lowering the neighborhood limit while performing the search from
the mentioned method in the onlooker bee phase.
D. Scout bees phase: During the scout bees phase, each employed bee leaves its
food source if it did not change for some determined time and searches for another
free food source. This means that if some employed bee solution did not change for
some limiting number of iterations, it is immediately replaced with a completely new
random solution. During the whole process, one solution of the best quality is stored
and kept track of.
4 The Proposed AABC Approach for the MWT
In this section, we apply an adjusted ABC algorithm (AABC) to the MWT problem
through phases mentioned in the previous section. The initial phase requires a set
of random solutions to be found. The first problem is to generate a set of random
triangulations and even simpler question arises before as a nontrivial one, that being,
how to triangulate a given point set. Several algorithms for finding arbitrary triangulations are known so far, but for the purpose of this paper, we adopted the greedy
one from [11].
Probably, the most important remark for this phase is the fact that the initialization
of the starting population is done rather differently than the usual way. We start
off with all the bees attached to only one food source. We have chosen the same
310
A. Alihodzic et al.
greedy solution mentioned above to be the starting solution, which proved to be the
most efficient one for this purpose. After the initial population of bees, i.e., agents,
is established, the next phase of the algorithm, the employed bees phase, can be
performed.
Parallel, sequential, and stochastic hill climbing, based on edge flipping, characterizes this and partially the next phase of the algorithm as well. The procedure
for a single bee can be described as follows. First, the edges surrounded by convex
rectangles out of all inner edges in the solution are filtered. Among the newly filtered edges, only the ones with heavier weight than their counterpart diagonals in
surrounding rectangles are classified as improvable ones. After that, among the improvable edges, one is selected randomly and replaced with its counterpart diagonal.
This action, also known as edge flipping, provides a solution of lower weight after
each execution, that is, a stage where the stochastic hill climbing is applied. This
action of edge flipping is performed on each bee sequentially and thus the employed
bees phase search is simulated.
The onlooker bees phase does not differ much from the previous one. The only
difference is that it is allowed for the solution that the onlooker bees find to climb
to the local optimum immediately, with all the other bees waiting, that is, the hill
climbing is performed on only one bee/solution selected until it reaches the local
optimum. Also, here the track is kept of the best solutions found so far.
All employed bees will eventually reach their local optimum and once they do so,
they will not be able to improve their solution in the same manner as they did before.
In other words, they will be stuck in local optima, that is, where the scout bees phase
comes to act. During this phase, it is allowed for the bees that have reached the local
optimum to wander through the search space for a while. It is facilitated by letting
the agent choose any of the edges surrounded by the convex rectangle and not just the
improvable ones. That way the agent is allowed to move uphill as well as downhill
while conducting its search. For how long should an agent wander, that is, how many
edge flips should wandering bee perform, is determined by some previously defined
constant factor.
5 Experimental Study
In this experimental study, our proposed AABC algorithm was compared against
two other standard metaheuristic techniques, namely SA and PSO. Since there are
no standard collections of instances in the literature for the MWT problem, we
generated a collection of five instances where each of them consists of minimum 15
points and maximum 23 points. On these benchmarks, we performed a comparison
between the exhaustive search and our AABC approach. For the purpose of checking
the quality of obtained solutions as well as the stability of generated solutions, we
also randomly generated seven instances where each of them consists of 40 points.
Each instance is called RI-k-i, where k denotes the size of the ith instance, where
i = 1, . . . , 7.
Adjusted Artificial Bee Colony Algorithm …
311
The points are uniformly distributed and for each point (x, y), the coordinates are
x, y ∈ [−333, 333]. For experimental simulations, instances were preprocessed to
guarantee that the coincident points do not appear in the plane. Through the experimental evaluation, we assess the applicability of the adjusted ABC metaheuristic for
the MWT problem. The proposed AABC method has been implemented in Python
programming language. All tests were done on an Intel Core i7-3770K @3.5GHz
with 16GB of RAM running under the Windows 10 x64 operating system. To compare
the proposed AABC algorithm with the PSO, and the SA algorithms, the objective
function evaluation was computed N×G times, where is N the population size and
G is the maximum number of generations. Each algorithm was executed in 50 runs.
The population size for the AABC was set to N = 13 bees (agents), while for the
PSO it was set to N = 33 particles. The value of limit in the scout bees phase of
the AABC was equal to 7. The learning parameters of the PSO algorithm α and β
were initialized to 2.0 and 2.0, respectively. The velocity of particles was set to zero
at the beginning of the PSO algorithm. The number of generations for both AABC
and PSO was set to G = 33. For the purpose of this paper, in the case of simulated
annealing algorithm, the initial temperature T0 was set to 1.0 with its change α being
picked uniformly from the interval (0.85, 0.99). The SA algorithm terminates after
not changing the solution weight for a full cycle.
The optimal weights and the corresponding computational times found by the exhaustive search, together with the corresponding computational times for the AABC
algorithm, are presented in Table 1. These values can be compared to the calculated worst, mean, best, and standard deviations for 50 runs for three metaheuristic
algorithms presented in Table 2.
Only the rows (benchmark cases) where there was some difference in tested
algorithms are shown in Table 2. For all the rows that are not shown in Table 2, all
algorithms found best solution in all runs. As it can be seen by comparing Tables 1
Table 1 Optimal weights and time processing provided by the exhaustive search and the AABC
algorithm
Random
Exhaustive search method
AABC
Instances
Optimum weight
Time (s)
Mean time (s)
RI-15-1
RI-15-2
RI-15-3
RI-15-4
RI-15-5
RI-16-1
RI-16-2
RI-16-3
RI-16-4
RI-16-5
2111182
1909236
2049619
1549159
1995367
1546841
1702863
1700495
1463510
2055712
27.39
16.26
20.32
19.40
17.11
59.25
66.13
59.36
41.09
32.89
0.83
0.75
0.73
0.79
0.74
0.74
0.79
0.78
0.77
0.87
(continued)
312
Table 1 (continued)
Random
Instances
RI-17-1
RI-17-2
RI-17-3
RI-17-4
RI-17-5
RI-18-1
RI-18-2
RI-18-3
RI-18-4
RI-18-5
RI-19-1
RI-19-2
RI-19-3
RI-19-4
RI-19-5
RI-20-1
RI-20-2
RI-20-3
RI-20-4
RI-20-5
RI-21-1
RI-21-2
RI-21-3
RI-21-4
RI-21-5
RI-22-1
RI-22-2
RI-22-3
RI-22-4
RI-22-5
RI-23-1
RI-23-2
RI-23-3
RI-23-4
RI-23-5
A. Alihodzic et al.
Exhaustive search method
Optimum weight
Time (s)
AABC
Mean time (s)
15762801
1416507
2326401
1714096
1550093
2275043
2696259
2023911
1806271
1656804
2006489
1797201
1777187
1641380
1685899
2185726
1537246
1993018
1744241
2329993
2022009
2473327
2113566
1381651
2348732
2246770
1820994
1755667
2068914
2291740
1644823
1890912
2515459
2347326
2252239
0.85
0.87
0.90
0.87
0.81
1.07
1.12
1.07
1.03
1.02
0.96
1.14
1.09
1.04
1.06
1.09
0.94
1.21
1.14
1.21
1.27
1.25
1.51
1.25
1.26
1.25
1.38
1.33
1.37
1.37
1.44
1.34
1.46
1.36
1.46
101.26
206.64
181.81
131.35
122.44
240.71
326.56
408.3
290.33
147.41
614.85
401.56
360.84
263.18
685.72
709
892.15
1570.29
1039.4
1716.42
912.2
3226.62
6737.32
2281.67
5508.41
7446.92
3626.2
4961.5
7846.59
11868.93
13573.4
4254.22
33332.45
21125.1
8613.89
Mean
2114203
2092213
2034210
1559225
1712957
1440159
2332225
1732083
2703399
1756330
1804804
1781739
1645661
2248547
1756094
2343195
2357454
2081143
1652834
2285596
Rand.
Instan.
RI-15-1
RI-15-3
RI-15-5
RI-16-1
RI-16-3
RI-17-2
RI-17-3
RI-17-4
RI-18-2
RI-18-5
RI-19-2
RI-19-3
RI-19-4
RI-20-1
RI-20-4
RI-20-5
RI-21-5
RI-22-4
RI-23-1
RI-23-2
SA
2111182
2092213
1995367
1546841
1700495
1787137
2326401
1714096
2696259
1755344
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2068914
1644823
2252239
Best
1251
0
10308
2426
3191
4847
1732
3240
3271
491
2920
2156
2054
9824
1659
5468
2366
2672
3204
8243
SD
2111182
2056895
1995367
1546841
1701791
1416507
2326401
1714096
2696259
1656804
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2075512
1644823
2252239
Worst
2111182
2052665
1995367
1546841
1700598
1416507
2326401
1714096
2696259
1656804
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2074265
1644823
2252239
Mean
2111182
2049619
1995367
1546841
1700495
1416507
2326401
1714096
2696259
1656804
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2068914
1644823
2252239
PSO
Best
0
3580
0
0
351
0
0
0
0
0
0
0
0
0
0
0
0
2540
0
0
SD
2111182
2049619
1995367
1546841
1700495
1416507
2326401
1714096
2696259
1656804
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2075512
1644823
2252239
Worst
2111182
2049619
1995367
1546841
1700495
1416507
2326401
1714096
2696259
1656804
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2069309
1644823
2252239
Mean
2111182
2049619
1995367
1546841
1700495
1416507
2326401
1714096
2696259
1656804
1797201
1777187
1641380
2185726
1744241
2329993
2348732
2068914
1644823
2252239
AABC
Best
Table 2 Comparison of the mean values and standard deviations obtained for the SA, PSO, and AABC for five random instances over 50 runs
SD
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1566
0
0
Adjusted Artificial Bee Colony Algorithm …
313
RI-41-1
RI-42-2
RI-40-3
RI-41-5
RI-42-4
RI-43-6
RI-40-7
2849983
2272714
2385028
2390812
3256158
2887767
2505964
2848273
2271293
2384878
2384568
3235455
2887106
2501009
2847134
2271062
2384748
2382200
3234082
2886920
2500908
1395
573
139
2964
5232
350
707
2851232
2274818
2401029
2396415
3234082
2896889
2519285
2848588
2271969
2390623
2383177
3234082
2888788
2511475
2847134
2271062
2384748
2382200
3234082
2886920
2502858
1279
942
5666
2482
0
2922
4026
2847134
2271062
2384748
2382200
3234082
2886920
2500908
2847134
2271062
2384748
2382200
3234082
2886920
2500908
2847134
2271062
2384748
2382200
3234082
2886920
2500908
0
0
0
0
0
0
0
Table 3 Comparison of the worst, mean, best values, and standard deviations obtained in 50 runs for the SA, PSO, and AABC for seven randomly generated
instances
Rand.
SA
PSO
AABC
Instan.
Worst
Mean
Best
SD
Worst
Mean
Best
SD
Worst
Mean
Best
SD
314
A. Alihodzic et al.
Adjusted Artificial Bee Colony Algorithm …
315
and 2, the proposed AABC generated optimal weights for 44 out of 45 benchmarks
in only 429 evaluations.
Also, for all instances, AABC was able to generate optimal weights in the reasonable amount of time, around one second, compared to the exhaustive search method
which took for some instances over 9 hours. From Table 2, it can be observed that
the SA in some cases remained stuck in local optima, for example, RI-15-3, RI-17-2,
and RI-18-5. On the another hand, PSO and AABC produced satisfactory results
for each of 50 runs. However, AABC is by far the most stable one, even with the
fact that it used smaller number of evaluations. Since PSO used 1089 evaluations,
it implies that our proposed AABC is more than two times faster than the PSO algorithm. Therefore, our adjusted ABC algorithm in all cases outperforms PSO and
SA for all tested instances. Additionally, allowing ABC to use more agents, it can
be stabilized even more. Table 3 shows the testing on seven larger instances. Each
of them includes 40 points.
Our AABC with 33 agents remained completely stable (having its standard deviation equal to 0) for each run and all instances, while the PSO algorithm destabilized
heavily. It is important to highlight that PSO gets trapped into some local optima for
the instance RI-40-7 and therefore it is not capable of reaching the global optimum.
6 Conclusion
In this paper, we considered the design of approximation algorithm for solving the
minimum weight triangulation problem for sets of points in the 2D space. We adjusted
the ABC algorithm for this problem and compared it to other metaheuristics. Our
proposed adjusted ABC algorithm was tested on 35 smaller and 7 larger instances
of points in the plane. It proved to be superior to the PSO and especially to the SA
algorithm considering the quality of the solutions as well as the stability of obtained
solutions. This shows that our proposed algorithm is an excellent choice for the
MWT problem. Additional adjustments can be done in the future using larger sets
of points. Since this is the initial application of the artificial bee colony algorithm
to the minimum weight triangulation problem, the computational complexity of the
method is still subject to improvements. One point for possible improvement is
the triangulation algorithm. Using one with lower complexity could significantly
enhance the execution time quality of the AABC. Some hybridized of the ABC
algorithm is also a promising area for improvements. Finally, some other swarm
intelligence based metaheuristics applied to this problem could be investigated, since,
besides this one, only two others were researched.
316
A. Alihodzic et al.
References
1. Alihodzic, A., Tuba, M.: Improved bat algorithm applied to multilevel image thresholding.
Sci. World J. (special issue Computational Intelligence and Metaheuristic Algorithms with
Applications) 2014 Article ID 176718, 16 (2014)
2. Alihodzic, A., Tuba, M.: Improved hybridized bat algorithm for global numerical optimization. In: Proceedings of the 16th IEEE International Conference on Computer Modelling and
Simulation, UKSim-AMSS 2014, pp. 57–62 (2014)
3. Dorigo, M., Stutzle, T.: Ant Colony Optimization. MIT Press, Cambridge, MA (2004)
4. Dorzán, M.G., Gagliardi, E.O., Leguizamón, M.G., Penalver, G.H.: Using ACO metaheuristic
for MWT problem. In: 30th International Conference of the Chilean Computer Science Society
(SCCC), pp. 228–237. IEEE, New York (2011)
5. Dorzán, M.G., Leguizamón, M.G., Mezura-Montes, E., Hernández-Peñalver, G.: Approximated algorithms for the minimum dilation triangulation problem. J. Heuristics 20(2), 189–209
(2014)
6. Dorzan, M., Gagliardi, E., Leguizamon, M., Hernandez Pealver, G.: Approximations on minimum weight triangulations and minimum weight pseudo-triangulations using ant colony optimization metaheuristic. Fundam. Informaticae 119(1), 1–27 (2012)
7. Garey, M.R., Johnson, D.S.: A Guide to the Theory of NP-Completeness, p. 228. W. H. Freeman,
San Francisco, CA (1979) (Problem OPEN12)
8. Jahani, M., Bigham, B.S., Askari, A.: An ant colony algorithm for the minimum weight triangulation. In: International Conference on Computational Science and Its Applications (ICCSA),
pp. 81–85. IEEE, New York (2010)
9. Karaboga, D.: An idea based on honey bee swarm for numerical optimization. Technical Report
- TR06, Erciyes University, Engineering Faculty, Computer Engineering Department, pp. 1–10
(2005)
10. Kolingerová, I., Ferko, A., Vomáčka, T., Maňák, M.: Nearest neighbour graph and locally minimal triangulation. In: International Conference on Computational Science and Its Applications,
pp. 455–464. Springer, Berlin (2017)
11. Levcopoulos, C., Krznaric, D.: Quasi-greedy triangulations approximating the minimum
weight triangulation. J. Algorithms 27(2), 303–338 (1998)
12. Lin, Y.B., Wang, R.M., Li, Y.: Minimum weight triangulation based on improved genetic
algorithm with annealing selection. J. Inf. Comput. Sci. 1(2), 337–342 (2004)
13. Michalewicz, Z., Fogel, D.B.: How to Solve it: Modern Heuristics, p. 554. Springer Science
& Business Media, Berlin (2013)
14. Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM (JACM) 55(2), 1–45
(2008), Article A11
15. Tan, Y., Zhu, Y.: Fireworks algorithm for optimization. In: Tan, Y., Shi, Y., Tan, K.C. (eds.)
Advances in Swarm Intelligence: First International Conference (ICSI 2010), Proceedings, Part
I, pp. 355–364. Springer, Berlin (2010)
16. Tuba, E., Mrkela, L., Tuba, M.: Support vector machine parameter tuning using firefly algorithm. In: 26th International Conference Radioelektronika, pp. 413–418. IEEE, New York
(2016)
17. Tuba, E., Tuba, M., Dolicanin, E.: Adjusted fireworks algorithm applied to retinal image registration. Stud. Inf. Control 26(1), 33–42 (2017)
18. Tuba, M., Alihodzic, A., Bacanin, N.: Cuckoo search and bat algorithm applied to training
feed-forward neural networks. In: Yang, X.S. (ed.) Recent Advances in Swarm Intelligence
and Evolutionary Computation, Studies in Computational Intelligence, vol. 585, pp. 139–162.
Springer International Publishing, Basel (2015)
19. Tuba, M., Bacanin, N., Alihodzic, A.: Firefly algorithm for multi-objective RFID network
planning problem. In: 22nd Telecommunications Forum Telfor (TELFOR), pp. 95–98. IEEE,
New York (2014)
Adjusted Artificial Bee Colony Algorithm …
317
20. Tuba, M., Bacanin, N., Alihodzic, A.: Multilevel image thresholding by fireworks algorithm.
In: Proceedings of the 25th International Conference Radioelektronika, pp. 326–330 (April
2015)
21. Wang, R., Li, Y., Lin, Y., Luo, X., Zhang, X.: Minimum weight triangulation based on improved
genetic quantum algorithm. J. Comput. Inf. Syst. 1(2), 323–327 (2005)
22. Yang, X.S.: Firefly algorithm, levy flights and global optimization. Res. Dev. Intell. Syst. XXVI,
209–218 (2010)
23. Yang, X.S.: A new metaheuristic bat-inspired algorithm. Stud. Comput. Intell. 284, 65–74
(2010)
24. Yang, X.S., Deb, S.: Cuckoo search via Levy flights. In: World Congress on Nature Biologically
Inspired Computing, pp. 210–214 (December 2009)
25. Yousefi, A., Young, N.E.: On a linear program for minimum-weight triangulation. SIAM J.
Comput. 43(1), 25–51 (2014)
Decision-Making Proposition of Fuzzy
Information Measure with Collective
Restrictions
Anjali Munde
Abstract Information theory was founded by Shannon (A mathematical theory of
communication. Bell Syst Tech J 379–423, 623–659 (1948) [11]) who introduced the
concept of entropy in communication theory. Information theory is concerned with
communication systems and has applications in statistics, information processing,
and computing. The theory of fuzzy sets which was introduced by Zadeh (Fuzzy sets
as a basis for a theory of possibility. Fuzzy Sets Syst 3–28 (1978) [13]) gave fuzzy
entropy a measure of fuzzy information which was dependent on Shannon’s entropy.
A large amount of work is being done on characterization of various fuzzy entropies.
In this paper, a generalized measure of fuzzy information with multiple parameters
has been proposed and applications of fuzzy information measure in decision-making
have been discussed.
Keywords Fuzzy information measure · Fuzzy set theory · Decision-making
1 Introduction
1.1 Fuzzy Information Measures
One of the most prominent features of twentieth-century technology has been the
development and exploitation of new communication media concurrent with the
growth of devices of transmitting and processing information. The word information
and saying information is strength is very common and often they are encountered
in our daily life. Lot of information is transmitted through Human voice, Telephone,
Radio, Television, Books and Newspaper, etc. Nyquist [5, 6] and Hartley [3] studied
the qualitative nature of the measure of information. Shannon [11] was the founder
of information theory which was of the interest to communication engineers and
A. Munde (B)
Amity University, Noida, Uttar Pradesh, India
e-mail: anjalidhiman2006@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_31
319
320
A. Munde
proposed measure of entropy. Wiener [12] proved results similar to Shannon. Renyi
[10] generalized Shannon’s entropy and introduced entropy of order r.
Fuzzy sets were first proposed by Zadeh [13] in his paper entitled, “Fuzzy Sets”.
A fuzzy set A in X is characterized by a membership function μ A (x) which associates
with each point in X a real number in the interval [0, 1], with the values of μ A (x) at x
representing the “grade of membership” of x in A. With the ith element, it associates
a fuzzy uncertainty f (μ A (xi ), where f (x) function has the following four properties:
1.
2.
3.
4.
Fuzzy uncertainty becomes zero when x takes the value 0 or 1
Fuzzy uncertainty increases when x takes the value from 0 to 0.5
Fuzzy uncertainty decreases when x takes the value from 0.5 to 1.0
Fuzzy uncertainty remains same when the membership function changes from
μ A (x) to (1 − μ A (x))
De Luca and Termini [2] suggested that corresponding to Shannon’s probabilistic
entropy, the measure of fuzzy entropy should be
H (A) −
n 1
μ A (xi ) log μ A (xi ) + (1 − μ A (xi )) log(1 − μ A (xi ))
·
i1
n log 2
(1)
Deshmukh et al. [1] introduced parametric measure of fuzzy entropy of order α
and studied the monotonic character of the measures of fuzzy entropy.
Kumar et al. [4] introduced generalized measure of fuzzy entropy and proved that
the generalized measure is a valid measure of fuzzy entropy. Also, fuzzy entropy for
various values α was computed which showed that the fuzzy entropy is a concave
function. Further, an important property of maximality was being discussed.
Prakash and Gandhi [7] proposed two new generalized fuzzy entropies and proved
its validity. Also they showed that the maximum value of the generalized fuzzy
entropy is an increasing function and hence discussed the monotonic behavior of the
entropies.
Prakash and Gandhi [8] introduced two new fuzzy measures involving trigonometric functions and calculated the minimum parameter of the polygon of n sides
from the first measure and minimum area of the polygon of n sides from the second
measure.
Prakash et al. [9] introduced measure of entropy based upon χ 2 distribution, t
distribution; F distribution and proved that sampling distributions can be used to
develop new information measures.
A new parametric measure of Fuzzy information involving three parameters
A new generalized fuzzy information measure involving three parameters α, β
and γ has been suggested and their necessary and required properties are examined. Thereafter, its validity is also verified. Also, the monotonic behavior of fuzzy
information measure of order α, β and γ has been proved.
The generalized measure of fuzzy information involving three parameters α, β
and γ is given by
Decision-Making Proposition of Fuzzy …
Hα,β,γ (A) 321
n (α+β)μ (x )
1
A i
{μ A
+ (1 − μ A (xi ))(α+β)(1−μ A (xi )) }γ − 2γ ,
i1
(1 − α)
(2)
where α > 0, α 1, β 0, γ 0.
Properties of Hα,β,γ (A)
I have supposed that, 00.α 1, Further I have studied the following properties:
1. Hα,β,γ (A) ≥ 0 i.e., Hαβ (A) is nonnegative.
2. Hα,β,γ (A) is minimum iff A is a non-fuzzy set. For μ A (xi ) 0, it implies
Hα,β,γ (A) 0 and for μ A (xi ) 1, it has Hα,β,γ (A) 0.
3. Hα,β,γ (A∗) ≤ Hα,β,γ (A), where A* be sharpened version of A. When μ A (xi ) lies
between 0 and 21 then Hα,β,γ (A) is an increasing function whereas when μ A (xi )
lies between 21 and 1 then Hα,β,γ (A) is a decreasing function of μ A (xi ).
Let A* be sharpened version of A which means that if μ A (xi ) < 0.5 then μ A ∗
(xi ) ≤ μ A (xi ) and if μ A (xi ) > 0.5 then μ A ∗ (xi ) ≥ μ A (xi ) for all I 1, 2,…, n.
Since Hα,β,γ (A) is an increasing function of μ A (xi ) for 0 ≤ μ A (xi ) ≤ 21 and
decreasing function of μ A (xi ) for 21 ≤ μ A (xi ) ≤ 1, therefore μ A ∗ (xi ) ≤ μ A (xi )
implies that Hα,β,γ (A∗) ≤ Hα,β,γ (A) in [0, 0.5] and μ A ∗ (xi ) ≤ μ A (xi ) implies that
Hα,β,γ (A∗) ≤ Hα,β,γ (A) in [0.5, 1]. Hence Hα,β,γ (A∗) ≤ Hα,β,γ (A).
4. Hα,β,γ (A) Hα,β,γ A where A is the compliment of A i.e. μ A (xi ) 1−μ A (xi ).
Thus when μ A (xi ) is varied to 1 − μ A (xi ) the Hα,β,γ (A) does not change.
Under the above conditions, the generalized measure proposed in (2) is a valid
measure of fuzzy information measure.
Monotonic Behavior of fuzzy information measure
In this section, the study of monotonic behavior of the fuzzy information measure
has been discussed. For this, diverse values of Hα,β,γ (A) by assigning various values to α, β and γ has been calculated and further the generalized measure has been
presented graphically (Figs. 1, 2 and 3; Tables 1, 2 and 3).
Monotonic behavior of
( , , ) ( ) for = 2 = 2
=3
H(α,β, )A
10
8
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
μA(xi)
Fig. 1 Representing the monotonic behavior of Hα,β,γ (A) for α 2, β 2, γ 3
322
A. Munde
H(α,β, )A
Monotonic behavior of
( , , ) ( ) for = 3 = 2
=3
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
μA(xi)
Fig. 2 Representing the monotonic behavior of Hα,β,γ (A) for α 3, β 2, γ 3
Monotonic behavior of
( , , ) ( ) for = 3.5 = 2
=3
H(α,β, )A
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
1.2
μA(xi)
Fig. 3 Representing the monotonic behavior of Hα,β,γ (A) for α 3.5, β 2, γ 3
Table 1 Representing the values of Hα,β,γ (A) for α 2, β 2, γ 3
μ A (xi )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Hα,β,γ (A)
0.0
6.73
7.55
7.77
7.85
7.87
7.85
7.77
7.55
6.73 0.0
1.0
Table 2 Representing the values of Hα,β,γ (A) for α 3, β 2, γ 3
μ A (xi )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Hα,β,γ (A)
0.0
3.58
3.88
3.95
3.97
3.98
3.97
3.95
3.88
3.58 0.0
1.0
Table 3 Representing the values of Hα,β,γ (A) for α 3.5, β 2, γ 3
μ A (xi )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Hα,β,γ (A)
0.0
2.93
3.13
3.17
3.18
3.19
3.18
3.17
3.13
2.93 0.0
1.0
Numerical example based on fuzzy information measure
Example 1: Suppose i want to solve an evaluation of teaching quality problems in
which the alternatives are four young teachers to be evaluated according to their
teaching performances by the expert committee. The evaluation system includes
the four indexes: teaching altitude, teaching content, teaching method, and teaching
Decision-Making Proposition of Fuzzy …
Table 4 Representing the
membership values of the
teachers with respect to
teaching performance
323
(C 1 )
(C 2 )
(C 3 )
(C 4 )
(A1 )
0.80
0.75
0.80
0.85
(A2 )
0.65
0.85
0.80
0.75
(A3 )
0.75
0.75
0.85
0.70
(A4 )
0.80
0.70
0.75
0.75
result. For evaluating the preference, the decision makers formed four fuzzy sets as
(Table 4).
Fuzzy Information measure for each given option (A1 ), (A2 ), (A3 ), (A4 ) is given
as,
For α 2, β 2 and γ 3 it has,
H(A1 ) 30.0806
H(A2 ) 30.3572
H(A3 ) 30.1633
H(A4 ) 30.7244
Optimal Solution is with maximum entropy. So, the expert committee should
select A4 with preference order M 4 , M 2 , M 3, M 1.
Conclusion
I have introduced new generalized measure of fuzzy information and proved its validity. The development of this new fuzzy measure will definitely reduce uncertainty,
which will help to increase the efficiency and remove uncertainty for betterment. I
have also discussed the particular cases of α, β, γ and presented the fuzzy information
measure which clearly shows that fuzzy information measure is a concave function.
Further, the new fuzzy information measure has been applied to decision-making
problems.
References
1. Deshmukh, K.C., Khot, N.: Generalized measures of fuzzy entropy and their properties. World
Acad. Sci. Eng. Technol. pp. 994–998 (2011)
2. De Luca, A., Termini, S.: A definition of a non-probabilistic entropy in setting of fuzzy sets.
Inf. Control 20, 301–312 (1972)
3. Hartley, R.V.L.: Transmission of information. Bell Syst. Tech. J. 7, 535–563 (1928)
4. Kumar, A., Mahajan, S., Kumar, S.: Some generalized measures of fuzzy entropy. Int. J. Math.
Sci. Appl. 1, 821–829 (2011)
5. Nyquist, H.: Certain factors affecting telegraph speed. Bell Syst. Tech. pp. 324–325 (1924)
6. Nyquist, H.: Certain topics in telegraph transmission theory. AIEEE Trans. pp. 617–619 (1928)
7. Prakash, O., Gandhi, C.P. : New generalized measures of fuzzy entropy and their properties. J.
Inf. Math. Sci. 3, 1–9 (2011)
8. Prakash, O., Gandhi, C.P.: Applications of trigonometric measures of fuzzy entropy to geometry. Int. J. Math. Comput. Sci. 6, 76–79 (2010)
324
A. Munde
9. Prakash, O., Thukral, A.K., Gandhi, C.P.: Information measures based on sampling distributions. World Acad. Sci. Eng. Technol. pp. 1132–1136 (2011)
10. Renyi, A.: On measures of entropy and information. In: Proceedings of 4th Berkeley Symposium Mathematical Statistics and Problems. University of California Press, Berkeley,
pp. 547–561 (1961)
11. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. pp. 379–423,
623–659 (1948)
12. Wiener, N.: Cybernetics. MIT Press and Wiley, New York (1948)
13. Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. pp. 3–28(1978)
Exact Algorithm for L(2, 1) Labeling
of Cartesian Product Between Complete
Bipartite Graph and Cycle
Sumonta Ghosh, Prosanta Sarkar and Anita Pal
Abstract L(h, k) labeling is one kind of graph labeling where adjacent nodes get
the value differ by at least h and the nodes which are at 2 distance apart get value
differ by at least k, which has major application in radio frequency assignment,
where the assignment of frequency to each node of radio station in such a way that
adjacent station get frequency which does not create any interference. Robert in 1998
gives the direction to introduce L(2, 1) labeling. L(2, 1) labeling is a special case of
L(h, k) labeling, where the value of h is 2 and value of k is 1. In L(2, 1), labeling
difference of label is at least 2 for the vertices which are at distance one apart and label
difference is at least 1 for the vertices which are at distance two apart. The difference
between minimum and maximum label of L(2, 1) labeling of the graph G = (V, E)
is denoted by λ2,1 (G). In this paper, we propose a super-linear time algorithm to
label the graph obtained by the Cartesian product between complete bipartite graph
and cycle. We design the algorithm in such a way that gives exact labeling of the
graph G = (K m,n × Cr ) for the bound of m, n > 5 and which is λ2,1 (G) = m + n.
Finally, we have shown that L(2, 1) labeling of the above graph can be solved in
polynomial time for some bound.
Keywords Cartesian product · L(2, 1) labeling · Complete bipartite graph
Cycle
S. Ghosh · P. Sarkar (B) · A. Pal
National Institute of Technology Durgapur, Durgapur 713209, West Bengal, India
e-mail: prosantasarkar87@gmail.com
S. Ghosh
e-mail: mesumonta@gmail.com
A. Pal
e-mail: anita.buie@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_32
325
326
S. Ghosh et al.
1 Introduction
Graph labeling has become very useful in the domain of applied mathematics. So
in many applications like missile guidance code, design communication network
addressing system, etc., L(h, k) labeling problem has major application in radio frequency assignment, where assignment of frequency to each node of radio station is in
such a way that adjacent station get frequency which does not create any interference.
Robert in 1988 gives the idea of frequency assignment problem with the restriction
“close” and “very close”, where “close” node received frequency that is different and
“very close” node received frequency is two or more apart, which gives the direction
to introduce L(2, 1) labeling. The above idea of Robert was actually a vertex coloring
problem; here, color is just replaced by non-negative integer by which we can label the
vertex of a graph G = (V, E), where V and E represent the set of vertices and edges,
respectively. L(2, 1) labeling is also a vertex coloring problem where every color
is replaced by some non-negative integers with the restriction that if d(x, y) = 2,
∀x, y ∈ V , i.e., for “close” node frequency will be at least 1 apart and if d(x, y) = 1,
∀x, y ∈ V , i.e., for “very close” node frequency will be at least 2 apart.
Graph G = (V, E) has various bounds of λ2,1 (G) known in terms of , ω(G), and
χ (G). Maximum degree is denoted by , where ω(G) and χ (G) denote the size of the
maximum clique and chromatic number of the graph G, respectively. For the graph
K 1, , stable lower bound is + 1, Griggs and Yeh [2] gives the explanation that a
graph required 2 − span, and in 1992 they also prove that λ2,1 (G) ≤ 2 + 2.
In 2008, Gonclaves [5] improved the bound to λ2,1 (G) ≤ 2 + 2 −2 and later the
bound is improved to λ2,1 (G) ≤ 2 + 2 −3 for 3−connected graph. Then, Chang
and Kuo [6] improved the bound to λ2,1 (G) ≤ 2 + . The conjecture of Griggs
and Yeh [2] stable it for the graph of diameter 2. Then, Chang and Kuo improved the
bound to λ2,1 (G) ≤ 2 .
Conjecture 1 For any graph G = (V, E) with maximum degree ≥ 2, λ2,1 (G) ≤
2 .
The above conjecture of Griggs and Yeh [2] worked for the set of graphs like path
[2], wheel [2], cycle [2], trees [2, 6, 7], co-graphs [6], interval graphs [6], chordal
graphs [8], permutation graph [9, 10], etc. The bound λ2,1 (G) can be computed
systematically for some graphs like cycle, path, and tree [2, 6, 7]. For some graphs
G = (V, E) like path, cycle, complete bipartite graph, tree, star, bi-star, complete
graph λ2,1 (G) can be computed in polynomial time, but some other class of graphs
also there, namely interval graph [6], circular arc graph [11], chordal graph [8], etc.
λ2,1 (G) may not be computed in polynomial time, and complexity of such graphs is
either NP-complete or NP-hard.
These are the various results on Cartesian product between cycle and cycle, path
and cycle, and between complete graphs.
Exact L(2, 1) labeling problem is already done by many researchers on some
different kinds of graphs, and still it is a very interesting problem and challenging
work. For some simple graphs like path, cycle, complete bipartite graph, and tree,
Exact Algorithm for L(2, 1) Labeling of Cartesian …
327
it is manageable to find exact L(2, 1) labeling, but for complex graph structure like
Cartesian product between different graphs is not as smooth as simple graphs. In
this paper, we are trying to do exact L(2, 1) labeling of Cartesian product between
complete bipartite graph and cycle.
For digitalization of everything unknowingly, we enter into a wider, complex,
and hybridization network structure which leads to increase the number of node.
Increasing the number of node will also introduce collision of frequency, so need
to incorporate restriction. Here, we choose L(2, 1) labeling, but increasing of label
(frequency) may lead to high cost factor which may affect the feasibility of maintaining such complex network. So we partly introduce exact L(2, 1) labeling to remove
high cost factor and reduce failure to maintain reliability of the existing system.
The remaining part of the paper organized is as follows. Section 2 contains some
preliminaries and definition, and Sect. 3 presents our algorithms, analysis of algorithm, and lemma’s to study Griggs and Yeh [2] conjecture followed by conclusion.
2 Preliminaries
Definition 1 A graph G is called a complete bipartite graph if its vertices can be
partitioned into two subsets V1 and V2 such that no edges have both end points in
the same subset, and each vertex of V1 (V2 ) is connected with all vertices of V2 (V1 ).
Here, V1 = {X 11 , X 12 , . . . , X 1m } contains m vertices and V2 = {Y11 , Y12 , . . . , Y1n }
contains n vertices.
A complete bipartite graph with |V1 | = m and |V2 | = n is denoted by K m,n .
Definition 2 A cycle of a graph G = (V, E) is denoted by Cr , where
V = {v0 , v1 , . . . , vr } be the set of vertices and E = {e0 , e1 , . . . , er } be the set of
edges which form a cycle if every vertex say vi is adjacent to exactly two vertices.
Definition 3 Cartesian product is denoted by G × H , where G = (V, E) and H =
(V , E ) be the graph two graphs, which is defined by taking Cartesian product
between two sets of vertices V (G) × V (H ), where (u, u ) and (v, v ) are the order
pair of the Cartesian product that will be directly connected in G × H if and only if
either
1. u = v and u is directly connected with v in H , or
2. u = v and u is directly connected with v in G.
For Cartesian product between K m,n × Cr , we have to draw the graph K m,n , r
times. Here, each K m,n has two sets of vertices X ,Y where |X | = m and |Y | = n. Each
set of vertices of K m,n for r copies represented by (X 1 , Y1 ), (X 2 , Y2 ),(X 3 , Y3 ),…,
(X r , Yr ), where each X i = {xi1 , xi2 , xi3 , . . . , xim } and Yi = {yi1 , yi2 , yi3 , . . . , yin }.
328
S. Ghosh et al.
Fig. 1 The graph K m,n × Cr
Lemma 1 Let be the degree of the graph K m,n × Cr , then
=
m + 2 for m > n
n + 2 for n > m
(1)
Proof Let G = K m,n × Cr . If m > n, then the maximum degree of the graph K m,n
is m (Fig. 1). From Fig. 2 it is clear that every vertex is connected with previous copy
and next copy of K m,n . Therefore, only two degrees of each vertex will increase in
K m,n × Cr . Hence, the value of is m + 2.
The proofs of other cases are similar.
3 Labeling of Cartesian Product Between Complete
Bipartite Graph and Path
We already discussed various types of labeling of trivial graphs, intersection graphs,
and Cartesian product of some graphs with their bounds λ2,1 (G) in the form of and number of vertices. In this portion, we discussed the L(2, 1) labeling of cycle
Cr and exact algorithm of L(2, 1) labeling of Cartesian product between complete
bipartite graph and cycle followed by analysis of algorithm. We have shown that the
algorithm also follows the Griggs and Yeh conjecture.
Exact Algorithm for L(2, 1) Labeling of Cartesian …
329
Fig. 2 Exact L(2, 1)-labeling of the graph K 7,6 × C8
In this paper, we consider G = K m,n × Cr , where m, n > 5 for exact L(2, 1)
labeling. We give the name exact L(2, 1) labeling because number of label required
to label a complete bipartite graph K m,n , where m, n > 5 by L(2, 1) labeling is
equal to the number of label required to label Cartesian product between complete
bipartite graph and cycle by same labeling scheme. This is not bounded only for
L(2, 1) labeling but it is also verified for L(h, k) labeling. We consider the restriction
m, n > 5 because it always maintains the exact property discussed previously. For
m, n ≤ 5, we are able to label by L(2, 1) labeling but it is failed to attend the exact
labeling scheme.
3.1 Algorithm L21C
We use Algorithm 3.1 to label the cycle Cr of length r by L(2, 1) labeling. Let
v0 , v1 , v2 , . . . , vr −1 denote the vertices of the cycle Cr where vi is adjacent to vi+1
and v0 is adjacent to vr −1 . The rule of labeling of cycle Cr is given below. Maximum
label used to label a cycle of length r is λ2,1 (Cr ) = 4.
330
S. Ghosh et al.
Algorithm 1 Algorithm L21C
Input: The graph G = Cr .
Output: L(2, 1) labelled graph G = Cr
Step 1. If r ≡ 0(mod 3)
Step 2.
⎧
⎨ 0, i ≡ 0(mod 3);
f (vi ) = 2, i ≡ 1(mod 3);
⎩
4, i ≡ 2(mod 3)
Step 3. Else if n ≡ 1(mod 3), and cycle with multiple of 4 vertices only for vr −4 , vr −3 , . . . , vr −1
vertices, rest will follow the Step 1.
Step 4.
⎧
0, if i = r − 4;
⎪
⎪
⎨
3, if i = r − 3;
f (vi ) =
1, if i = r − 2;
⎪
⎪
⎩
4, if i = r − 1
Step 5. Else if r ≡ 2(mod 3), only for vr −2 , vr −1 , as follows.
Step 6.
1, if i = r − 2;
f (vi ) =
3, if i = r − 1
Stop.
3.2 Algorithm L21CBG
Algorithm 2 Algorithm L21CBG
Input: Complete bipartite graph G = K m,n .
Output: L(2, 1) labelled graph G = K m,n .
Initialize: c = 0.
Step 1 for i = 1 to m
Step 2 X [i] = c,c = c + 1.
end of loop i.
Step 3 c = c + 2
Step 4 for j = 1 to n
Step 5 Yk [ j] = c, c = c + 1.
end of loop j.
Stop.
Theorem 1 For G = K m,n , λ2,1 (K m,n ) = m + n.
Proof Let K m,n be the complete bipartite graph with two sets of verticesX =
{x1 , x2 , x3 , . . . , xm } and Y = {y1 , y2 , y3 , . . . , yn }, where |X | = m and |Y | = n. It
is clear that the vertices within a set are not connected, so each vertex in a particular set is at distance 2 whereas any two vertices from different sets are at distance 1. If we start label the vertex set X with 0, i.e., f (x1 ) = 0, we can increase
Exact Algorithm for L(2, 1) Labeling of Cartesian …
331
label by 1 for the next vertex because it is at distance 2 from the any vertex of
the set X , so we can continue with f (x2 ) = 1, f (x3 ) = 2, f (x4 ) = 3 similarly
f (xm ) = (m − 1). We can start label the set Y by the label (m − 1) + 2, i.e.,
m + 1. So f (y1 ) = m + 1, f (y2 ) = m + 2, f (y3 ) = m + 3, f (y4 ) = m + 4 similarly f (yn ) = m + n. So λ2,1 (K m,n ) = m + n.
3.3 Algorithm EL21LCC
The theme of the algorithm EL21LCC is that here we consider the graph Cartesian product between complete bipartite graph and cycle, i.e., G = K m,n × Cr . This
can also be incorporated in computer memory; for easy to understand, we use two
arrays for each copy of K m,n ; these are X k [i] and Yk [ j] for i = 1, 2, 3, . . . , m,
j = 1, 2, 3, . . . , n and k = 1, 2, 3, . . . , r . According to Fig. 2 we consider the cycle
form by the array elements X k [0], k = 1, 2, 3, . . . , r , which is the first vertex of set
X for each copy of K m,n . Another array Carr [r ] is considered for storing the L(2, 1)
labeling of cycle form by the first vertex of set X for each copy of K m,n , which is
labeled by the algorithm 3.1. For Algorithm 3, we consider two variables X max and
Ymax to store the maximum label use by X 1 [m] and Y1 [n], respectively.
Proof of correctness of Algorithm EL21LCC is given below.
Algorithm 3 Exact Algorithm for L(2, 1) Labeling of Cartesian Product Between
Complete Bipartite Graph and Cycle (EL21LCC)
Input: A Cartesian product between K m,n and Cr , i.e., G = (K m,n × Cr ).
Output:Exact L(2, 1) labeling of the graph G = (K m,n × Cr ).
Initialize: X max = m − 1, Ymax = m + n, Yk−1 [1] = X max + 2, X k−1 [1] = 0.
Step 1: For L(2, 1) labeling of the cycle Cr call Algorithm L21C
and store the value in the array Carr [r ].
Step 2: for k = 1 to r
Step 3: p = Carr [k]
Step 4: for i = 1 to m
Step 5: if ( p == X max ) then p = 0
end of if.
Step 6: X k [i] = p. p = p + 1
end of loop i.
Step 7: q = Yk−1 [1] + (X k [1] − X k−1 [1])
Step 8: for j = 1 to n
Step 9: if (q = Ymax ) then q = X max + 2
end of if.
Step 10: Yk [ j] = q, q = q + 1.
end of loop j.
end of loop k.
Stop.
Theorem 2 Algorithm EL21LCC exactly label the graph G = (K m,n × Cr ).
332
S. Ghosh et al.
Proof According to Theorem 1, we know that λ2,1 (K m,n ) = m + n. It is clear that
if we label complete bipartite graph K m,n starting by 0 then X (max) = m − 1 and
Y(max) = m + n. Now we consider the Cartesian product between complete bipartite
graph and cycle and we got the graph G = (K m,n × Cr ), where m, n > 5 (see Fig. 2).
From the graph G = K m,n × Cr , we get r copies of K m,n and we consider for each
copy of K m,n two arrays X and Y . We consider array Carr to label the cycle, which is
formed by the each first vertex of set X . In the array Carr [r ], the first element is the
label for the first vertex of vertex set X for the first copy of K m,n , i.e., X 1 [1] = Carr [1];
the second element is the label for the first vertex of vertex set X for the second copy
of K m,n , i.e., X 2 [1] = Carr [2]; and similarly, r th element is the label for the first
vertex of vertex set X for r th copy of K m,n , i.e., X r [1] = Carr [r ]. Now, we fetch the
first element of the array Carr [r ] and start labeling the first copy of K m,n according to
algorithm 2; next, fetch the second element from the array Carr and start labeling the
second copy of K m,n according to algorithm 2 just when it attends the value X (max)
and Y(max) ; we assign the label 0 and m + 1, respectively, to the very next vertex.
In a similar way, we can follow for the remaining steps. By shuffling the existing
label, we can achieve exact labeling for the graph G = (K m,n × Cr ) as the K m,n , i.e.,
λ2,1 (K m,n ) = λ2,1 (K m,n × Cr ) = m + n.
Analysis of Algorithm EL21LCC Let us consider the graph G = K 7,6 × C8 for
exact L(2, 1) labeling. For L(2, 1) labeling of the cycle C8 , call Algorithm L21C
and store the value in the array Carr [8] and the corresponding labels are Carr =
{0, 2, 4, 0, 2, 4, 1, 3}. Initially, X max = 7 − 1 = 6, Ymax = 7 + 6 = 13, Yk−1 [1] =
X max + 2 = 6 + 2 = 8 and X k−1 [1] = 0. For k = 1, we start labeling the first copy
of K 7,6 , and now initial value of p = Carr [1] = 0. For i = 1, first check p = 6
or not, clearly it is false then X 1 [1] = 0 and p = p + 1 = 1. For i = 2, checking condition false and X 1 [2] = 1, similarly for first copy of set X all the checking condition is false and last label of the last vertex is X 1 [7] = 6. Now q =
Yk−1 [1] + (X k [1] − X k−1 [1]) = 8 + (0 − 0) = 8, for j = 1 check whether q = 13
or not, which is false then Y1 [1] = 8, q = q + 1 = 9. For j = 2, checking conditions is false and Y1 [2] = 9, q = q + 1 = 10. Similarly, for the first copy set Y all
the checking condition is false and last label of the last vertex is Y1 [6] = 13. For
next iteration K = 2, we start labeling the second copy of K 7,6 , and now value of
p = Carr [2] = 2. For i = 1 condition, checking gives false and X 2 [1] = 1; similarly for i = 2, 3, 4, 5, corresponding label is X 2 [2] = 3, X 2 [3] = 4, X 2 [4] = 5,
X [5] = 6. But for i = 6 condition p = X max = 6, immediately assign p = 0 and
complete the remaining labeling and which is X 2 [6] = 0, X 2 [7] = 1. Now, q =
Yk−1 [1] + (X k [1] − X k−1 [1]) = 8 + (2 − 0) = 10, for j = 1 check whether q = 13
or not, which is false and then Y2 [1] = 10 , and similarly for the value of j = 2, 3, 4
corresponding label is Y2 [2] = 11, Y2 [3] = 12, Y2 [4] = 13. Now, for j = 5 condition q = Ymax = 13 the set q = X max + 2 = 8 + 2 = 10 and complete remaining
label Y2 [5] = 10, Y2 [6] = 11. Similarly, we can label rest of the copies of K 7,6 and
we carefully observe that label will not increase more that 13. Actually, we design
the algorithm in such a way where we shuffle the existing label of first copy of K 7,6 .
We also analyze the time complexity which ◦(nr ) if n > m otherwise ◦(mr ).
Exact Algorithm for L(2, 1) Labeling of Cartesian …
333
Analysis of Algorithm EL21LCC
Griggs and Yeh conjecture Algorithm 3 successfully follows the Griggs and Yeh
conjecture for m, n > 5.
Case 1: Griggs and Yeh conjecture satisfy for m > n
Theorem 3 For the graph G = (K m,n × Cr ), we already shown in Lemma 1 that
maximum degree is m + 2. Now from Theorem 1 we can conclude that λ2,1 (K m,n ×
Cr ) = m + n. As we know,
(m + 2)2 = m 2 + 4m + 4.
(m + 2)2 > m 2 + m.
(m + 2)2 > m + n as m > n.
Hence the proof.
Case 2: Griggs and Yeh conjecture satisfy for n > m
Theorem 4 For the graph G = (K m,n × Cr ) we already shown in Lemma 1 that
maximum degree is n + 2. Now from Theorem 1 we can conclude that λ2,1 (K m,n ×
Cr ) = m + n. As we know,
(n + 2)2 = n 2 + 4n + 4.
(n + 2)2 > n 2 + n.
(n + 2)2 > m + n as n > m.
Hence the proof.
4 Conclusion
L(2, 1)-labeling problem has wide application in real world and it has been already
applied to different kinds of graph structure. There exist a small number of graph for
which efficient algorithm is available. For any kind of graph may be simple or may be
complex, an exact labeling whose complexity can be measured by polynomial time
is always acceptable. In this paper, we design an exact algorithm with super-linear
time complexity to label a graph obtained by Cartesian product between complete
bipartite graph and cycle, i.e., G = (K m,n × Cr ). But our algorithm will work for a
certain bound that is for m, n > 5. Labeling technique acquires a broad area where
complex graph structure is always welcome and it is also a challenging work to label
these types of graphs.
Acknowledgements The work is supported by the Department of Science and Technology, New
Delhi, India, Ref. No. SB/S4/MS: 894/14.
334
S. Ghosh et al.
References
1. Hale, W.K.: Frequency assignment: theory and applications. Proc. IEEE 68, 1497–1514 (1980)
2. Griggs, J., Yeh, R.K.: Labeling graphs with a condition at distance two. SIAM J. Discret. Math.
5, 586–595 (1992)
3. Krl, D., Skrekovski, R.: A theory about channel assignment problem. SIAM J. Discret. Math.
16, 426–437 (2003)
4. Havet, F., Reed, B., Sereni, J.S.: L(2, 1)-labeling of graphs. In: Proceedings of the 19th Annual
ACMSIAM Symposium on Discrete Algorithms, SODA 2008, pp. 621–630. SIAM, Philadelphia, PA (2008)
5. Gonalves, D.: On the L(d, 1)-labellinng of graphs. Discret. Math. 308, 1405–1414 (2008)
6. Chang, G.J., Kuo, D.: The L(2, 1)-labeling on graphs. SIAM J. Discrete Math. 9, 309–316
(1996)
7. Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for L(2, 1)-labeling of trees.
Lect. Notes Comput. Sci. 5757, 35–46 (2009)
8. Sakai, D.: Labeling chordal graphs with a condition at distance two. SIAM J. Discret. Math. 7,
133–140 (1994)
9. Paul, S., Pal, M., Pal, A.: L(2, 1)-labeling of permutation and bipartite permutation graphs.
Math. Comput. Sci. https://doi.org/10.1007/s11786-014-0180-2
10. Bodlaender, H.L., Kloks, T., Tan, R.B., Leeuwen, J.V.: Approximations for -colorings of graphs.
Comput. J. 47(2), 193–204 (2004)
11. Calamoneri, T., Caminiti, S., Olariu, S., Petreschi, R.: On the L(h, k)-labeling of cocomparability graphs and circular-arc graphs. Networks 53(1), 27–34 (2009)
12. Georges, J.P., Mauro, D.W., Stein, M.I.: Labeling products of complete graphs with a condition
at distance two. SIAM J. Discrete Math. 14, 28–35 (2000)
13. Jha, P.K., Narayanan, A., Sood, P., Sundaram, K., Sunder, V.: On L(2, 1)-labelings of the
Cartesian product of a cycle and a path. Ars Combin. 55 (2000)
14. Klavar, S., Vesel, A.: Computing graph invariants on rotagraphs using dynamic algorithm
approach: the case of (2, 1)-colorings and independence numbers. Discrete Appl. Math. 129,
449–460 (2003)
15. Whittlesey, M.A., Georges, J.P., Mauro, D.W.: On the number of Qn and related graphs. SIAM
J. Discrete Math. 8, 499–506 (1995)
16. Schwarza, C., Troxellb, D.S.: L(2, 1)-labelings of Cartesian products of two cycles. Discrete
Appl. Math. 154, 1522–1540 (2006)
The Forgotten Topological Index
of Graphs Based on New Operations
Related to the Join of Graphs
Prosanta Sarkar, Nilanjan De and Anita Pal
Abstract The sum of degree cube of all the vertices of a graph is known as the
F-index or the “forgotten topological index” of that graph. In the present work, we
study the “forgotten topological index” of new operations of different subdivisionrelated graphs based on the join of graphs.
Keywords Forgotten topological index · F-join of Graphs, Graph operations
1 Introduction
A graph G is an ordered pair of two sets namely vertex set V (G) and edge set E(G),
respectively. The degree of a vertex v is the number of vertices in G which are connected to v by an edge and denoted by dG (v). A topological index is a graph invariant
which is a numerical parameter obtained from a graph and characterize its topology.
In chemical graph theory, there are different topological indices which have very
useful applications in chemistry, biochemistry, molecular biology, nanotechnology
for QSAR/QSPR investigation, isomer discrimination, pharmaceutical drug design,
and much more. The first and second Zagreb indices were introduced by Gutman
and Trinajestić in 1972 [1] and used it to the study of structure dependency of the
total π -electron energy(). These are, respectively, defined as
P. Sarkar · A. Pal
Department of Mathematics, National Institute of Technology, Durgapur, India
e-mail: prosantasarkar87@gmail.com
A. Pal
e-mail: anita.buie@gmail.com
N. De (B)
Department of Basic Sciences and Humanities (Mathematics),
Calcutta Institute of Engineering and Management, Kolkata, India
e-mail: de.nilanjan@rediffmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_33
335
336
P. Sarkar et al.
M1 (G) =
dG (v)2 =
v∈V (G)
[dG (u) + dG (v)]
uv∈E(G)
and
M2 (G) =
dG (u)dG (v).
uv∈E(G)
For further study on these indices, we refer the reader to [2, 3]. The “forgotten
topological index” was discovered in the same paper where Zagreb indices were
introduced. But in 2015, Furtula and Gutman [4] reinvestigated this index again and
named this index as the “forgotten topological index” or F-index. This index was
defined as
dG (v)3 =
[dG (u)2 + dG (v)2 ].
F(G) =
v∈V (G)
uv∈E(G)
We refer our reader to [5–7] for some recent study and application of this index. The
hyper Zagreb index was introduced by Shirrdel et al. in [8] and is defined as
HM (G) =
[dG (u) + dG (v)]2 .
uv∈E(G)
For different mathematical and chemical studies of this index, we refer our reader to
[9, 10]. One of the redefined versions of the Zagreb index is defined as
ReZM (G) =
dG (u)dG (v)[dG (u) + dG (v)].
uv∈E(G)
We refer the reader to [11, 12] for further study of this redefined Zagreb index. Note
that, as usual, Pn denotes a path with n vertices and (n − 1) edges, whereas Cn (n ≥ 3)
denotes a cycle graph with n vertices. In this paper all over, we deal with only simple
and connected graphs.
2 Preliminary
Let G 1 = (V (G 1 ), E(G 1 )) and G 2 = (V (G 2 ), E(G 2 )) be two connected graphs such
that |V (G 1 )| = n1 , |V (G 2 )| = n2 and |E(G 1 )| = m1 , |E(G 2 )| = m2 , respectively.
Different graph operations and derived graphs play an important role in graph theory.
In this paper, we consider one very important graph operation called join of graphs
and also some derived graphs such as different subdivision-related graphs. Thus, we
first define them.
S(G) is derived by putting a new vertex corresponding to every edge of G.
R(G) is derived by putting a new vertex into every edge of G, and then joining it
to the end vertices of their respective edge of G.
The Forgotten Topological Index of Graphs …
337
Q(G) is obtained by adding a new vertex into every edge of G, and then joining
those pairs of new vertices such that their respective edges are adjacent in G.
T(G) is derived by putting a new vertex into each edge of G, and then joining it
to the end vertices of the corresponding edge and joining with edges those pairs of
new vertices on adjacent edges of G.
Let F = {S, R, Q, T } and also let I (G) denote the set of vertices of F(G) which
are inserted into each edge of G, so that V (F(G)) = V (G) ∪ I (G). Here, we first
define vertex F-join and edge F-join of two connected graphs G 1 and G 2 , which
are defined as follows
Definition 1 [2] Let G 1 and G 2 be two simple graphs; the vertex F-join graph
of G 1 and G 2 is derived from F(G 1 ) and G 2 by connecting every vertex of
G 1 to all vertices of G 2 , so that the vertex and edge sets are V (F(G 1 ))∪V (G 2 )
and E(G 1 )∪E(G 2 )∪[uv : u ∈ V (G 1 ), v ∈ V (G 2 )], respectively, and is denoted by
˙ F G 2 . Replace F by S, R, Q, T , we get the vertex S-join, vertex R-join, vertex
G1∨
Q-join, and vertex T -join of graphs, respectively.
Definition 2 [2] The edge F-join graph of two simple graphs G 1 and G 2 is derived
from F(G 1 ) and G 2 by connecting all vertex of I (G 1 ) to every vertex of G 2 , so
that the edge and vertex sets are E(G 1 )∪E(G 2 )∪[uv : u ∈ I (G 1 ), v ∈ V (G 2 )] and
V (F(G 1 ))∪V (G 2 ), respectively, and is denoted by G 1 ∨F G 2 . Similarly, if we replace
F by S, R, Q, T , we get the edge S-join, edge R-join, edge Q-join, and edge T -join
of graphs, respectively.
In this paper, we will study the “forgotten topological index” of vertex and edge
F-join of graphs related to different subdivision graphs.
3 Main Results
In the following subsections, we consider the “forgotten topological index” of vertex
and edge F-join of graphs for different values of F as S, R, Q, T , respectively.
3.1 Vertex and Edge S-Join of Graphs
In this subsection, first we start with vertex and edge S-join of graphs. The figures
of vertex and edge S-join of P3 and P4 are given in Fig. 1.
Theorem 1 If G 1 and G 2 be two connected graph, then
˙ S G 2 ) = F(G 1 ) + F(G 2 ) + 3n2 M1 (G 1 ) + 3n1 M1 (G 2 ) + 6m1 n22 + 6m2 n21
F(G 1 ∨
+n1 n2 (n1 2 + n22 ) + 8m1 .
338
P. Sarkar et al.
P3 ∨˙ S P4
P3 ∨S P4
Fig. 1 The example of vertex S-join and edge S-join of graphs
Proof From definition of “forgotten topological index”, we have
˙ S G2) =
F(G 1 ∨
dG 1 ∨˙ S G 2 (v)3
˙ S G2)
v∈V (G 1 ∨
=
dG 1 ∨˙ S G 2 (v)3 +
v∈V (G 1 )
=
v∈V (G 2 )
(dG 1 (v) + n2 ) +
3
v∈V (G 1 )
=
dG 1 ∨˙ S G 2 (v)3 +
v∈I (G 1 )
(dG 2 (v) + n1 )3 +
v∈V (G 2 )
2
2
n32 ]
+
v∈V (G 1 )
+
dG 1 (v)3 + 3n2
v∈V (G 1 )
+n1 n32 +
+3n1
dG 2 (v)3 + 3n1
dG 2 (v) +
dG 1 (v)2 + 3n2 2
v∈V (G 1 )
v∈V (G 2 )
2
23
[dG 2 (v)3 + 3n1 dG 2 (v)2 + 3n1 2 dG 2 (v) + n31 ]
v∈V (G 2 )
=
v∈I (G 1 )
23
v∈I (G 1 )
[dG 1 (v) + 3n2 dG 1 (v) + 3n2 dG 1 (v) +
3
dG 1 ∨˙ S G 2 (v)3
dG 1 (v)
v∈V (G 1 )
dG 2 (v)2
v∈V (G 2 )
n2 n31
+ 8m1
v∈V (G 2 )
= F(G 1 ) + F(G 2 ) + 3n2 M1 (G 1 ) + 3n1 M1 (G 2 ) + 6m1 n22 + 6m2 n21
+n1 n2 (n1 2 + n22 ) + 8m1 .
Hence the result.
Example 1 Using Theorem 1, we get
˙ s Pm ) = (mn − 6)(m2 + n2 ) + 6mn(m + n) + 24mn − 10m − 2n − 36,
(i) F(Pn ∨
˙ s Cm ) = mn{(m2 + n2 ) + 6(m + n)} − 6m2 + 24mn − 10m + 16n − 22,
(ii) F(Pn ∨
˙ s Cm ) = mn{(m2 + n2 ) + 6(m + n)} − 6m2 + 24mn + 8m + 16n,
(iii) F(Cn ∨
˙ s Pm ) = mn{(m2 + n2 ) + 6(m + n)} − 6n2 + 24mn + 8m − 2n − 14.
(iv) F(Cn ∨
The Forgotten Topological Index of Graphs …
339
Theorem 2 If G 1 and G 2 be two connected graph, then
F(G 1 ∨S G 2 ) = F(G 1 ) + F(G 2 ) + 3m1 M1 (G 2 ) + 6m21 m2 + m1 (n2 + 2)3 + n2 m1 3 .
Proof Using the definition of “forgotten topological index”, we have
F(G 1 ∨S G 2 ) =
dG 1 ∨S G 2 (v)3
v∈V (G 1 ∨S G 2 )
=
v∈V (G 1 )
=
dG 1 (v) +
dG 1 (v) +
(dG 2 (v) + m1 ) +
(2 + n2 )
+3m1
(2 + n2 )3
3
[dG 2 (v)3 + 3m1 dG 2 (v)2 + 3m1 2 dG 2 (v) + m31 ]
dG 1 (v)3 + m1 (2 + n2 )3 +
v∈V (G 1 )
v∈I (G 1 )
v∈V (G 2 )
=
dG 1 ∨S G 2 (v)3
v∈I (G 1 )
v∈I (G 1 )
3
v∈V (G 2 )
3
v∈V (G 1 )
+
dG 1 ∨S G 2 (v)3 +
v∈V (G 2 )
3
v∈V (G 1 )
=
dG 1 ∨S G 2 (v)3 +
dG 2 (v)3
v∈V (G 2 )
dG 2 (v) + 3m1
v∈V (G 2 )
2
2
dG 2 (v) + n2 m31
v∈V (G 2 )
= F(G 1 ) + F(G 2 ) + 3m1 M1 (G 2 ) + 6m21 m2 + m1 (n2 + 2)3 + n2 m1 3 ,
which is the desired result.
Example 2 From Theorem 2, we get
(i) F(Pn ∨S Pm ) = (n − 1){(m + 2)3 + 6(m − 1)(n − 1) + m(n − 1)2 } + 12mn
−4m − 10n − 10,
(ii) F(Pn ∨S Cm ) = (n − 1){(m + 2)3 + 6m(n − 1) + m(n − 1)2 } + 12mn
−4m + 8n − 14,
(iii) F(Cn ∨S Cm ) = n{(m + 2)3 + 6mn + mn2 } + 12mn + 8m + 8n,
(iv) F(Cn ∨S Pm ) = n{(m + 2)3 + 6n(m − 1) + mn2 } + 12mn + 8m − 10n − 14.
3.2 Vertex and Edge R-Join of Graphs
In this subsection, we consider vertex and edge R-join of graphs. The figures of
vertex and edge R-join of P3 and P4 are given in Fig. 2.
Theorem 3 If G 1 and G 2 be two connected graph, then
340
P. Sarkar et al.
P3 ∨˙ R P4
P3 ∨R P4
Fig. 2 The example of vertex R-join and edge R-join of graphs
˙ R G 2 ) = 8F(G 1 ) + F(G 2 ) + 12n2 M1 (G 1 ) + 3n1 M1 (G 2 ) + 12m1 n22
F(G 1 ∨
+6m2 n21 + n1 n2 (n1 2 + n22 ) + 8m1 .
Proof We can prove this theorem similarly, as shown in Theorem 1.
Example 3 Using the result as in Theorem 3, we get
˙ R Pm ) = mn(m2 + n2 + 6n) + 72mn − 6n2 − 76m
(i) F(Pn ∨
+54n − 134, m, n ≥ 2,
˙
(ii) F(Pn ∨R Cm ) = mn(m2 + n2 + 6n) + m4 + 72mn − 84m
+72n − 120, m ≥ 3, n ≥ 2,
˙ R Cm ) = mn(m2 + n2 + 6n) + m4 + 8n4 + 72mn + 8n, m, n ≥ 3,
(iii) F(Cn ∨
˙ R Pm ) = mn(m2 + n2 ) + 6n2 (m − 1) + 72mn + 16m
(iv) F(Cn ∨
+46n − 22, m ≥ 3, n ≥ 2.
Theorem 4 If G 1 and G 2 be two connected graph
F(G 1 ∨R G 2 ) = 8F(G 1 ) + F(G 2 ) + 3m1 M1 (G 2 ) + 6m21 m2 + m1 (n2 + 2)3 + n2 m1 3 .
Proof The proof is same as shown in Theorem 2.
Example 4 Applying Theorem 4, we get
(i) F(Pn ∨R Pm ) = (n − 1)(m + 2)3 + m(n − 1)3 + 6(m − 1)(n − 1)2 + 12mn
−4m + 46n − 94, m, n ≥ 2,
(ii) F(Pn ∨R Cm ) = (n − 1)(m + 2)3 + m(n − 1)2 (n + 5) + 12mn − 4m
+64n − 112, m ≥ 3, n ≥ 2,
(iii) F(Cn ∨R Cm ) = n(m + 2)3 + mn3 + 6mn2 + 12mn + 8m + 64n, m, n ≥ 3,
(iv) F(Cn ∨R Pm ) = n(m + 2)3 + mn3 + 6(m − 1)n2 + 12mn + 8m
+46n − 14, m ≥ 2, n ≥ 3.
The Forgotten Topological Index of Graphs …
341
3.3 Vertex and Edge Q-Join of Graphs
Here, we consider vertex and edge Q-join of graphs. The figures of vertex and edge
Q-join of P3 and P4 are given in Fig. 3. Let us denote M4 (G) as follows:
M4 (G) =
dG (v)4 =
v∈V (G)
[dG (u)3 + dG (v)3 ]
uv∈E(G)
which is a particular case of general Zagreb index. In the following, we use this
topological index to represent vertex and edge Q-join of graphs.
Theorem 5 If G 1 and G 2 be two connected graph, then
˙ Q G 2 ) = F(G 1 ) + F(G 2 ) + 3n2 M1 (G 1 ) + 3n1 M1 (G 2 ) + M4 (G 1 )
F(G 1 ∨
+3ReZM (G 1 ) + 6m1 n22 + 6m2 n21 + n1 n2 (n1 2 + n22 ).
Proof Using the definition of “forgotten topological index”, we have
˙ Q G2) =
F(G 1 ∨
dG 1 ∨˙ Q G 2 (v)3
˙ Q G2)
v∈V (G 1 ∨
=
dG 1 ∨˙ Q G 2 (v)3 +
v∈V (G 1 )
=
v∈V (G 2 )
(dG 1 (v) + n2 ) +
3
v∈V (G 1 )
dG 1 ∨˙ Q G 2 (v)3
v∈I (G 1 )
(dG 2 (v) + n1 )3
v∈V (G 2 )
+
dG 1 ∨˙ Q G 2 (v)3 +
(dG 1 (u) + dG 1 (v))3
uv∈E(G 1 )
=
[dG 1 (v)3 + 3n2 dG 1 (v)2 + 3n2 2 dG 1 (v) + n32 ]
v∈V (G 1 )
+
[dG 2 (v)3 + 3n1 dG 2 (v)2 + 3n1 2 dG 2 (v) + n31 ]
v∈V (G 2 )
+
[dG 1 (u)3 + dG 1 (v)3 + 3dG 1 (u)dG 1 (v)(dG 1 (u) + dG 1 (v))]
uv∈E(G 1 )
=
dG 1 (v)3 + 3n2
v∈V (G 1 )
+n1 n32 +
v∈V (G 1 )
dG 2 (v)3 + 3n1
v∈V (G 2 )
+3n1
2
v∈V (G 2 )
+3
uv∈E(G 1 )
dG 2 (v) +
dG 1 (v)2 + 3n2 2
dG 1 (v)
v∈V (G 1 )
dG 2 (v)2
v∈V (G 2 )
n2 n31
+
(dG 1 (u)3 + dG 1 (v)3 )
uv∈E(G 1 )
dG 1 (u)dG 1 (v)(dG 1 (u) + dG 1 (v))
342
P. Sarkar et al.
P3 ∨˙ Q P4
P3 ∨Q P4
Fig. 3 The example of vertex Q-join and edge Q-join of graphs
= F(G 1 ) + F(G 2 ) + 3n2 M1 (G 1 ) + 3n1 M1 (G 2 ) + M4 (G 1 )
+3ReZM (G 1 ) + 6m1 n22 + 6m2 n21 + n1 n2 (n1 2 + n22 ),
which is the desired result.
Example 5 Using Theorem 5, we get
˙ Q Pm ) = mn(m2 + n2 ) + 6m2 (n − 1) + 6n2 (m − 1) + 24mn
(i) F(Pn ∨
−10m + 54n − 166, m, n ≥ 3,
˙
(ii) F(Pn ∨Q Cm ) = mn(m2 + n2 ) + 6m2 (n − 1) + 6n2 m + 24mn
−10m + 72n − 152, m, n ≥ 3,
˙ Q Cm ) = mn(m2 + n2 ) + 6mn(m + n) + 24mn + 8m + 72n, m, n ≥ 3,
(iii) F(Cn ∨
˙ Q Pm ) = mn(m2 + n2 ) + 6mn(m + n) − 6n2 + 24mn + 8m
(iv) F(Cn ∨
+54n − 14, m, n ≥ 3.
Theorem 6 If G 1 and G 2 be two connected graph, then
F(G 1 ∨Q G 2 ) = F(G 1 ) + F(G 2 ) + 3n2 2 M1 (G 1 ) + 3m1 M1 (G 2 ) + M4 (G 1 )
+3n2 HM (G 1 ) + 3ReZM (G 1 ) + m21 (6m2 + m1 n2 ) + m1 n2 3 .
Proof From the definition of “forgotten topological index”, we have
F(G 1 ∨Q G 2 ) =
dG 1 ∨Q G 2 (v)3
v∈V (G 1 ∨Q G 2 )
=
v∈V (G 1 )
=
dG 1 ∨Q G 2 (v)3 +
v∈V (G 2 )
dG 1 (v) +
3
v∈V (G 1 )
+
dG 1 ∨Q G 2 (v)3 +
(dG 2 (v) + m1 )
v∈V (G 2 )
(dG 1 (u) + dG 1 (v) + n2 )3
uv∈E(G 1 )
v∈I (G 1 )
3
dG 1 ∨Q G 2 (v)3
The Forgotten Topological Index of Graphs …
=
dG 1 (v)3 +
v∈V (G 1 )
343
[dG 2 (v)3 + 3m1 dG 2 (v)2
v∈V (G 2 )
+3m1 dG 2 (v) + m31 ] +
2
[(dG 1 (u) + dG 1 (v))3
uv∈E(G 1 )
+3n2 (dG 1 (u) + dG 1 (v))2 + 3n2 2 (dG 1 (u) + dG 1 (v)) + n2 3 ]
dG 1 (v)3 +
dG 2 (v)3 + 3m1
dG 2 (v)2
=
v∈V (G 1 )
+3m1
2
v∈V (G 2 )
dG 2 (v) +
n2 m31
+
v∈V (G 2 )
+3
v∈V (G 2 )
(dG 1 (u)3 + dG 1 (v)3 )
uv∈E(G 1 )
dG 1 (u)dG 1 (v)(dG 1 (u) + dG 1 (v)) + n2 3 m1
uv∈E(G 1 )
+3n2
(dG 1 (u) + dG 1 (v))2 + 3n2 2
uv∈E(G 1 )
(dG 1 (u) + dG 1 (v))
uv∈E(G 1 )
= F(G 1 ) + F(G 2 ) + 3n2 2 M1 (G 1 ) + 3m1 M1 (G 2 ) + M4 (G 1 )
+3n2 HM (G 1 ) + 3ReZM (G 1 ) + m21 (6m2 + m1 n2 ) + m1 n2 3 .
Hence the result.
Example 6 Applying our result as in Theorem 6, we get
(i) F(Pn ∨Q Pm ) = m(n − 1){(n − 1)2 + m2 } + 3m2 (4n − 6) + 60mn − 94m
+6(m − 1)(n − 1)2 + 54n − 148, m ≥ 3, n ≥ 4,
(ii) F(Pn ∨Q Cm ) = m(n − 1){(n − 1)2 + m2 } + 3m2 (4n − 6) + 6m(n − 1)2
+60mn − 94m + 72n − 152, m ≥ 3, n ≥ 4,
(iii) F(Cn ∨Q Cm ) = mn(m2 + n2 ) + 12m2 n + 6mn2 + 60mn + 8m
+72n, m ≥ 3, n ≥ 4,
(iv) F(Cn ∨Q Pm ) = mn(m2 + n2 ) + 12m2 n + 6mn2 − 6n2 + 60mn + 8m
+6n − 14, m ≥ 3, n ≥ 4.
3.4 Vertex and Edge T-Join of Graphs
Finally, we consider vertex and edge T-join of graphs. The figures of vertex and edge
T-join of P3 and P4 are given in Fig. 4.
Theorem 7 If G 1 and G 2 be two connected graph, then
˙ T G 2 ) = 8F(G 1 ) + F(G 2 ) + 12n2 M1 (G 1 ) + 3n1 M1 (G 2 ) + M4 (G 1 )
F(G 1 ∨
+3ReZM (G 1 ) + 12m1 n22 + 6m2 n21 + n1 n2 (n1 2 + n22 ).
344
P. Sarkar et al.
P3 ∨˙ T P4
P3 ∨T P4
Fig. 4 The example of vertex T-join and edge T-join of graphs
Proof We can prove this theorem in the same way as shown in Theorem 5.
Example 7 Using Theorem 7, we get
˙ T Pm ) = mn(m2 + n2 ) + 6n2 (m − 1) + 12m2 (n − 1) + 60mn
(i) F(Pn ∨
−64m + 110n − 264, m, n ≥ 3,
˙ T Cm ) = mn(m2 + n2 ) + 6n2 m + 12m2 (n − 1) + 60mn − 64m
(ii) F(Pn ∨
+128n − 250, m, n ≥ 3,
˙ T Cm ) = mn(m2 + n2 ) + 6n2 m + 12m2 n + 60mn + 8m
(iii) F(Cn ∨
+128n, m, n ≥ 3,
˙ T Pm ) = mn(m2 + n2 ) + 6n2 (m − 1) + 12m2 n + 60mn + 8m
(iv) F(Cn ∨
+110n − 14, m, n ≥ 3.
Theorem 8 If G 1 and G 2 be two connected graph, then
F(G 1 ∨T G 2 ) = 8F(G 1 ) + F(G 2 ) + 3n2 2 M1 (G 1 ) + 3m1 M1 (G 2 ) + M4 (G 1 )
+3n2 HM (G 1 ) + 3ReZM (G 1 ) + m21 (6m2 + m1 n2 ) + m1 n2 3 .
Proof Following the methods as in Theorem 6, we can similarly prove this theorem.
Example 8 From Theorem 8, we get
(i) F(Pn ∨T Pm ) = m3 (n − 1) + 3m2 (4n − 6) + m(n − 1)3 + 6(m − 1)(n − 1)2
+60mn − 94m + 110n − 246, m ≥ 2, n ≥ 3,
(ii) F(Pn ∨T Cm ) = m3 (n − 1) + 3m2 (4n − 6) + m(n − 1)3 + 6m(n − 1)2
+60mn − 94m + 128n − 250, m, n ≥ 3,
(iii) F(Cn ∨T Cm ) = m3 n + 12m2 n + mn2 (n + 6) + 60mn + 8m + 128n, m, n ≥ 3,
(iv) F(Cn ∨T Pm ) = m3 n + 12m2 n + mn3 + 6n2 (m − 1) + 60mn
+8m + 110n − 14, m, n ≥ 3.
The Forgotten Topological Index of Graphs …
345
4 Conclusions
In this work, we established some useful formula for the “forgotten topological index”
of graphs based on the vertex and edge F-join of graphs where F = {S, R, Q, T },
and hence we derived some useful examples for some known graphs. For future
study, some other topological indices for this graph operation can be computed.
References
1. Gutman, I., Trinajestić, N.: Graph theory and molecular orbitals total π -electron energy of
alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)
2. Sarkar, P., De, N., Pal, A.: The Zagreb Indices of graphs based on new operations related to
the join of graphs. J. Int. Math. Virtual Inst. 7, 181–209 (2017)
3. Mary, U., Arockiaraj, P.S., Albert, A.J.: The first and second Zagreb indices of corona graphs.
Int. J. Math. Archive 5(11), 80–89 (2014)
4. Furtula, B., Gutman, I.: A forgotten topological index. J. Math. Chem. 53, 1184–1190 (2015)
5. De, N.: F-index of graphs based on four operations related to the lexicographic product,
arXiv:1706.00464v1
6. De, N.: F-index and coindex of some derived graphs. Bull. Int. Math. Virtual Inst. 8, 81–88
(2018)
7. De, N.: F-index of total transformation graphs. Discrete Math. Algorithm. Appl. 9(3) (2017)
8. Shirrdel, G.H., Rezapour, H., Sayadi, A.M.: The hyper Zagreb index of graph operations. Iran.
J. Math. Chem. 4, 213–220 (2013)
9. De, N.: Hyper Zagreb index of bridge and chain graphs, arXiv:1703.08325v1
10. Gao, W., Jamil, M.K., Farahani, M.R.: The hyper-Zagreb index and some graph operations. J.
Appl. Math. Comput. 54, 263–275 (2017)
11. Kumar, R.P., Nandappa, D.S., Kanna, M.R.R.: Redefined Zagreb, randic, harmonic and GA
indices of graphene. Int. J. Math. Anal. 11, 493–502 (2017)
12. Farahani, M.R., Gao, W.: On multiplicative and redefined version of Zagreb indices of Vphenylenic nanotubes and nanotorus. British J. Math. Comput. Sci. 13(5), 1–8 (2016)
Clustering and Auction in Sequence:
A Two Fold Mechanism for Participatory
Sensing
Jaya Mukhopadhyay, Vikash Kumar Singh, Sajal Mukhopadhyay
and Anita Pal
Abstract Crowdsourcing with smart devices has gained a lot of popularity as a
research topic for the last several years. This is commonly known as participatory
sensing. In this paper, a double auction mechanism that also circumvents the position
(location) information of the participating agents is proposed.
Keywords Participatory sensing · Location information · Online double auction
1 Introduction
Participatory sensing [1, 7, 10, 11, 15] is a distributed problem-solving model in
which the task executers carrying smart devices (such as tablets, smartwatches, smartphones, etc.) may be engaged to accomplish the tasks or subtask posed by task
requesters through the third party (platform). Henceforth, task executers and task
requesters may also be termed as agents. In this work, we investigate a single task
execution problem (STEP), where there are multiple task requesters having a single
common task that is to be accomplished by the multiple task executers in an online
environment. By online environment, we mean that the agents arrive in the system
and depart from the system on a regular basis. The proposed model is shown in Fig. 1.
J. Mukhopadhyay (B) · A. Pal
Department of Mathematics, NIT, Durgapur 713209, West Bengal, India
e-mail: jayabesu@gmail.com
A. Pal
e-mail: anita.buie@gmail.com
V. K. Singh · S. Mukhopadhyay
Department of Computer Science and Engineering, NIT,
Durgapur 713209, West Bengal, India
e-mail: vikas.1688@gmail.com
S. Mukhopadhyay
e-mail: sajmure@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_34
347
348
Available
Agents
J. Mukhopadhyay et al.
Active
Cluster
Agents
Formation
at τi Platform
Implement DA
per
cluster
New
Active Buyers
Agents New
Sellers
Active
Agents
at τs
Fig. 1 System model
In our model, the location information of the agents are considered so that redundant data collection may be avoided by grouping the agents into clusters and then
putting them in the auction environment. The location-aware participatory sensing was first introduced in [9]. However, location-aware participatory sensing in
online double auction environment was not addressed in [9]. In this paper, we have
addressed the location-aware participatory sensing first time in an online double auction environment and have proposed truthful poly-time algorithm which also satisfies
individual rationality and budget balance property.
The remainder of the paper is structured as follows. Section 2 elucidates the preliminary concepts about participatory sensing. Section 3 describes our proposed model.
The proposed mechanisms is illustrated in Sect. 4. Conclusion and future directions
are given in Sect. 5.
2 Prior Works
In order to get a nice overview of the participatory sensing, the readers may refer
[4, 6, 7, 11, 17]. In past, for voluntary participation of the task executers, several
incentivizing schemes are discussed in literature. Reddy et al. [16] follow the fixed
price payment scheme, where the winning agents are paid a fixed price as their
payment. However, the fixed price-based incentive scheme may not satisfy the several
participating agents because the payment offered to them is much less as compared to
the effort made by them in the data collection process. Moreover, the incentive-based
schemes have got a special attention from the research community. Luo et al. [14]
address the incentive scheme under the reverse auction-based setting (single buyer
and multiple sellers). Several incentive schemes have been introduced in [8, 12, 20].
In [2, 18, 19], efforts have been made by the researchers to show the effect of quality
of data collected by the agents to the overall system by incorporating the quality
of data to the system in some sense. Some initial research has been carried out by
[5, 7, 13, 18] to preserve the privacy of the agents so that their private information
associated with the data are not revealed publicly. Recently, [9] provides the incentive
schemes under the location constraints. In their work, they have addressed locationaware participatory sensing in one buyer and multiple sellers’ environment. In our
model, we have explored more general multiple sellers–multiple buyers framework
in more challenging location-aware participatory sensing in online double auction
environment.
Clustering and Auction in Sequence: A Two Fold …
349
3 System Model
Let B = {B1 , B2 , . . . , Bm } be the set of task requesters and S = {S1 , S2 , . . . , Sn }
be the set of task executers such that m n. Each Si has a private valuation υie .
The set υ e denotes the set of valuations of all the task executers given as υ e =
{υ1e , υ2e , . . . , υne }. Similar to the task executers, we define υ r = {υ1r , υ2r , . . . , υmr } for
B. Each of the task executers and task requesters places their private information in
a sealed bid manner. It is to be noted that, due to the strategic nature of the agents,
they can misreport their respective private values. So, it is convenient to represent
the cost reported for performing the task by the task executer Si as υ̂ie and the value
of task requester Bi for buying the task as υ̂ir . υ̂ie = υie and υ̂ir = υir describe the fact
that Si and Bi are not deviating from their true valuations. In this model, there are
multiple task requesters (as buyers) and multiple task executers (as sellers). So, this
is a perfect setting to model the STEP as an online double auction problem (ODAP).
Due to online nature of the STEP, one of the realistic parameters that is perceived in
our proposed model is arrival and departure time of the agents. The arrival time of
each task executer Si and each task requester Bi are given as aie and air (misreporting
case is denoted as âie and âir ), respectively. The departure time of each task executer
Si and each task requester Bi are given as die and dir (misreporting case is denoted
as d̂ie and d̂ir ), respectively. In our proposed model, a day is termed as time horizon
T. The time horizon T is partitioned into several time slots (not necessarily of same
length) given as T = {τ1 , τ2 , . . . , τs }. For each time slot τi , a new set of active task
requesters R ⊂ B and a new set of active task executers U ⊂ S arrives in the auction
market. At each time slot τi , considering the newly active task executers U, a set of
clusters of task executers are formed and is given as £i = {£1i , £2i , . . . , £ki }, where £ij is
termed as the j th cluster for τi time slot. Over the T time horizon, the cluster vector
can be given as £ = {£1 , £2 , . . . , £s }. Once the clusters are formed, then for each
cluster £ij several independent double auctions will be performed. At each time slot
τi ∈ T and from each cluster £ij , the set of winning task executers-task requesters are
paired. At each time slot τi ∈ T, our proposed mechanism matches one task executer
to one task requester in a cluster. The payment of each task executer Si and each
task requester Bi is given as Pie and Pir , respectively. As the task executers and task
requesters are strategic in nature, they will try to maximize their utility. The utility
of any task executer is the payment received by the task executer minus the true
valuation of the task executer. More formally, the utility of Si is ϕie = Pie − υie , if
Si wins otherwise 0. Similarly, the utility of any task requester is defined as the true
valuation of the task requester minus the payment he pays. More formally, the utility
of Bi is ϕir = υir − Pir , if Bi wins 0 otherwise.
350
J. Mukhopadhyay et al.
4 STEM: Proposed Mechanism
4.1 Outline of STEM
In order to present the brief idea of the STEM to the readers, the outline of the
STEM is discussed before going into the detailed view. The outline of the STEM
can be thought of as a three-stage process: (1) For any auction round τi ∈ T, find
out the active task executers and task requesters (Defined formally in line no. 3 and
4 of Algorithm 1). (2) Cluster the active task executers based on k-means clustering
technique. (3) Run the online double auction separately for each cluster of task
executers. Task requesters will be the same for all the clusters.
4.2 Sketch of the STEM
The three-stage STEM can further be studied under four different sections: Main routine, cluster formation, payment, and allocation. First, the subpart of the proposed
mechanism, i.e., the Main routine phase is discussed and presented. The cluster
formation phase is addressed next. Next, the crucial part of the proposed mechanism, i.e., payment phase motivated by [3] is discussed and presented. Finally, the
allocation phase is addressed.
Main routine: The idea lies behind the construction of Main routine is to handle the
system partitioned into different time slots τi ∈ T. The input to the Main routine is
the set of task executers at τi time slot, i.e., Sτi , the set of available task requesters
at τi time slot, i.e., Bτi , the overall time horizon, i.e., T, the set of cost of execution
of all task executers, i.e., υ̂ e , and the set of value for buying the executed tasks by
all the task requesters, i.e., υ̂ r . The output is the set of allocation vector A.
Cluster formation: The idea behind the proposal of the Cluster formation phase is
to avoid the redundant data collection. The detailed process is shown in Algorithm 2.
Payment: For determining the payment of each agent, the valuation of the first losing
task executer and losing task requester is taken into consideration which is given
by I ∗j = argmin i {υ̂ir − υ̂ie < 0}. For defining the payment, we further require to
fetch the valuation of the task requester and the task executer at the index position
I ∗j . The valuation of the task requester at any index position is captured by the
bijective function ϒ r : Z → R≥0 , whereas the valuation of the task executer at any
index position is captured by the bijective function ϒ e : Z → R≥0 . Let us further
denote the valuation of the task requester at the index position I ∗j by ϒ r (I ∗j ) and the
valuation of the task executer at I ∗j by ϒ e (I ∗j ). For determining the payment of all
winning task executers and task requesters, we will take the help of the average of
the cost of the task executer at I ∗j and the value of the task requester at I ∗j given as
η=
ϒ r (I ∗j )+ϒ e (I ∗j )
.
2
Mathematically, the payment of i th task executer is given as
Clustering and Auction in Sequence: A Two Fold …
351
Algorithm 1 Main routine(Sτi , Bτi , T, υ̂ e , υ̂ r )
Output: A ← {A1 , A2 , . . . , Ak }
( j)
1: U ← φ, R ← φ, £e∗ ← φ, £r∗ ← φ, Uc ← φ, R ( j) ← φ
2: for all τi ∈ T do
3:
U ← active_T E (Sτi , τi )
∀Si ∈ U , âie ≤ τi ≤ d̂ie , and υ̂ie ≤ χie
4:
R ← active_T R (Bτi , τi )
∀Bi ∈ R, âir ≤ τi ≤ d̂ir , and υ̂ir ≥ χir
5:
£i ← Cluster formation (U , k)
6:
for each £ij ∈ £i do
7:
Uc ← Sor t_ascend(£ij , Si · υie )
Sorting based on υie ∈ υ e for all Si ∈ £ij
r
8:
R ← Sor t_descend(R, Bi · υi )
Sorting based on υir ∈ υ r for all Bi ∈ R
9:
Payment (Uc , R)
( j)
( j)
10:
Uc ← Uc ∪ Uc∗
11:
R ( j) ← R ( j) ∪ Rc
( j)
12:
£e∗ ← £e∗ ∪ Uc
∗
∗
13:
£r ← £r ∪ R ( j)
14:
Uc ← φ
15:
end for
16:
New task executers and task requesters comes.
17:
Sτi ← £e∗ ∪ {new task executer s}
18:
Bτi ← £r∗ ∪ {new task r equester s}
19: end forreturn A
Algorithm 2 Cluster formation (U, k)
1: C ← φ
k centroid determination
2: while |C | = k do
3:
x ∗ ← random(X )
Picking a random point X ∈ X
4:
C ← C ∪ {x ∗ }
5: end while
6: repeat
k cluster formation
7:
£i ← φ, £ij ← φ
8:
for each Sk ∈ U do
9:
for each X j ∈ C do
10:
D ← D ∪ {D(Sk , X j )}
Distance between Sk and X j
11:
end for
12:
j ∗ ← argmin j D
13:
£ij ∗ ← £ij ∗ ∪ {Sk }
14:
end for
15:
C←φ
16:
for j = 1 to k do
17:
£i ← £i ∪ £ij
18:
end for
19:
for each £ij ∈ £i do
20:
X j = 1i
x is the point i.e. a two dimensional vector in cluster £ij
x ∈£i x
|£ j |
j
21:
C ← C ∪ Xj
22:
end for
23: until change in cluster takes place return £i
352
J. Mukhopadhyay et al.
Algorithm 3 Payment (Uc , R)
1: Û ← φ, R̂ ← φ
2: for each Si ∈ Uc do
3:
if âie == τi then
4:
χie ← minρ e ∈[d̂ e −κ, τi ] {Pie (ρ e )}
i
5:
else
e
e
6:
χi ← min{Pi (τi − 1), Pie (τi )}
7:
end if
8:
if χie ≥ υ̂ie then
9:
Pe ← Pe ∪ {χie }
10:
Û ← Û ∪ {Si }
11:
else:
12:
Si is priced out.
13:
end if
14: end for
15: for each Bi ∈ R do
16:
if âir == τi then
17:
χir ← maxρ r ∈[d̂ r −κ, τi ] {Pir (ρ r )}
i
18:
else
r
r
19:
χi ← max{Pi (τi − 1), Pir (τi )}
20:
end if
21:
if χir ≤ υ̂ir then
22:
Pr ← Pr ∪ {χir }
23:
R̂ ← R̂ ∪ {Bi }
24:
else:
Bi is priced out.
25:
26:
end if
27: end for
28: Allocation(Û , R̂, Pe , Pr )
Pie (τi ) =
η,
ifϒ e (I ∗j ) ≥ ηandϒ r (I ∗j ) ≤ η
ϒ e (I ∗j ), otherwise
Fresh arrival
Still active
Fresh arrival
Still active
(1)
Similarly, the payment of the i th task requester is given as
Pir (τi )
=
η,
ifϒ e (I ∗j ) ≤ ηandϒ r (I ∗j ) ≥ η
∗
r
ϒ (I j ), otherwise
(2)
In this problem setup, for any particular time slot τi ∈ T, there might be two types
of agents: (a) Freshly arrived agents, (b) Still active agents. For freshly arrived
task executers and task requesters, the payment is calculated as shown below. More
formally, the payment of i th task requester is given as
Clustering and Auction in Sequence: A Two Fold …
353
⎧
maxρ r ∈[d̂ r −κ,..., τi ] {Pir (ρ r )}, if task requester is freshly
⎪
⎪
i
⎪
⎨
arrived
ζ r (τi ) =
r
r
⎪
if task requester are still
max{ζ (τi−1 ), Pi (τi )},
⎪
⎪
⎩
active
(3)
Here, κ is the maximum permitted gap between the arrival and departure of any
arbitrary agent i. The payment of i th task executer is given as
⎧
minρ e ∈[d̂ e −κ,..., τi ] {Pie (ρ e )}, if task executer is freshly
⎪
⎪
i
⎪
⎨
arrived
e
ζ (τi ) =
e
e
⎪
if task executers are still
min{ζ (τi−1 ), Pi (τi )},
⎪
⎪
⎩
active
(4)
Now, if after τi time slots if a task requester i is a winner, then the final payment
of that task requester will be given by Pir (τi ) = max{ζ r (τi−1 ), Pir (τi )} and similarly
if after τi time slots if a task executer i is a winner then the final payment of that task
executer will be given by Pie (τi ) = min{ζ e (τi−1 ), Pie (τi )}.
Allocation: The input to the allocation phase is the j th cluster in τi time slot, i.e.,
£ij , the set of task requester R, the payment vector of the task executers, i.e., Pe ,
and the payment vector of task requesters, i.e., Pr . The output is the set of task
requester–task executer winning pairs held in Ak .
Algorithm 4 Allocation (£ij , R, Pe , Pr )
1: Ak ← φ
2: Uc∗ ← Sor t_ascend(£ij , Si · χie )
3: Rc ← Sor t_descend(R, Bi · χir )
4: I j ← argmaxi {χir − χie ≥ 0}
5: for k = 1 to I j do
6:
Ůc ← Ůc ∪ {Sk ∈ Uc∗ }
7:
R̊ ← R̊ ∪ {Bk ∈ Rc }
8:
Ak ← Ak ∪ (Ůc , R̊)
9: end for
10: Uc∗ ← Uc∗ \ Ůc
11: Rc ← Rc \ R̊ return (Ak , Uc∗ , Rc )
Sorting based on χie ∈ Pe for all Si ∈ £ij
Sorting based on χir ∈ Pr for all Bi ∈ R
Lemma 1 Agent i cannot gain by misreporting their arrival time or departure time
or both.
Proof As the agents can misreport the arrival time or the departure time, the proof
can be illustrated into two parts considering both the cases separately.
– Case 1 (âie = aie ): Fix die , τi . Let us suppose an agent i reports the arrival time as
âie such that âie = aie or in more formal sense âie > aie . It means that an agent i will
354
J. Mukhopadhyay et al.
Fig. 2 An agent i
misreporting arrival time aie
τ1
τ2
τ3
τs
dei
aei
(dei − k)
âei
dei
t ∈ max[dei −k,...,aei ] {Pie } ≥ max[dei −k,...,âei ] {Pie }
Fig. 3 An agent i
misreporting departure
time die
τ1
(dˆei − k)
τ2
τ3
τs
dei
aei
(dei − k)
aei
dˆei
t ∈ max[dei −k,...,aei ] {Pie } ≥ max[dˆe −k,...,ae ] {Pie }
i
i
be aligned with more number of time slots before winning when reporting âie than
in the case when reporting truthfully, i.e., aie as shown in Fig. 2. Now, it is seen
from the construction of the payment function that the agent i will be paid less
than or equal to the payment he/she (henceforth he) is receiving when reporting
truthfully.
– Case 2 (d̂ie = die ): Fix aie , τi . Let us suppose an agent i reports the departure time
as d̂ie such that d̂ie = die or in more formal sense d̂ie < die . It means that an agent
i will be aligned with more number of time slots before becoming inactive when
reporting d̂ie than in the case when reporting truthfully, i.e., die as shown in Fig. 3.
Now, it is seen from the construction of the payment function is that the agent i
will be paid less or equal to the payment he is paid when reporting truthfully.
Considering the case 1 and case 2 above, it can be concluded that any agent i cannot
gain by misreporting arrival time or departure time. The proof is carried out by
considering the task executers, similar argument can be given for the task requesters.
This completes the proof.
Lemma 2 Agent i cannot gain by misreporting his/her bid value.
Proof Considering the case of task executers. Fix the time slot τi ∈ T and the cluster.
Case 1: Assuming that the i th task executer that lies in the winning set misreports
his bid value and is given as υ̂ie < υie . As the task executer was winning with υie , with
υ̂ie he would continue to win and his utility ϕ̂ie = ϕie . If instead he reports υ̂ie > υie .
Again two cases can happen. He may still lie in the winning set. Being in the winning
set, the utility will be ϕ̂ie = ϕie . By reporting υ̂ie , if he lies in the losing set, then his
utility will be ϕ̂ie = 0 < ϕie .
Case 2: Assuming that i th task executer lies in the losing set by reporting υie . Now,
let us check if he misreports his bid value, whether he will be able to gain or not. If the
reported bid value υ̂ie > υie , he would still lose and his utility ϕ̂ie = 0 = ϕie . If instead
Clustering and Auction in Sequence: A Two Fold …
355
he reports υ̂ie < υie , then two things can happen. If he still lies in the losing set, then
his utility ϕ̂ie = 0 = ϕie . But if he wins, then he had to bypass some valuation υ ej < υie
and hence υ̂ie < υ ej . Now as he wins his utility ϕ̂ie = Pie − υie = υ ej − υie < 0. Hence,
gain is not achieved.
Combining case 1 and case 2 above, we can say that any agent i cannot gain by
misreporting his bid value. The proof is carried out by considering the task executers,
and similar argument can be given for the task requesters. This completes the proof.
Lemma 3 STEM is weakly budget balanced.
Proof Budget balance means the sum of the payment of all the buyers minus sum
of the payments of all the sellers is greater than or equals to 0. To prove, this fixes
the time slot τi and cluster £ij . Now, by construction of our STEM, any task executer
and task requester are paired up only when Bi · Pir − Si · Pie ≥ 0. It means that,
for any task executer–task requester pair, there exist some surplus. In the similar
fashion,
in a particular
time slot τi and in a particular cluster considering all the
agents, i Bi · Pir − i Si · Pie ≥ 0. So, this is true for any arbitrary time slot τi .
This completes the proof.
Lemma 4 STEM is individual rational.
Proof Individual rationality means that an agent’s utility is non-negative. Fix the time
slot τi and cluster £ij . Considering the case of task requester, when the task requester
is winning then it is ensured that he has to pay an amount Pir such that υ̂ir ≥ Pir .
From this inequality, it is clear that the winning task requester has to pay amount
less than his bid value. So, in this case it can be concluded that ϕir = υ̂ir − Pir ≥ 0.
Moreover, if the task requester is losing in that case his utility is 0. So, this is true
for any arbitrary time slot τi and any cluster. Similar argument can be given for the
task executers. This completes the proof.
Theorem 1 STEM is truthful.
Proof Considering the case of task executers, fix the time slot τi and cluster £ij .
Truthful means that no participating agent can gain by misreporting their private
information(s). It is followed from Lemma 1 that the agents cannot gain by deviating
from their true arrival time or departure time or both. Considering the valuation of
the agents, it is clear from Lemma 2 that the agents cannot gain by deviating from
their true valuation. Similar argument can be given for the task requesters. This
completes the proof.
5 Conclusion and Future Works
An incentive compatible mechanism is given in this paper to circumvent the location
information in online double auction setting for the participatory sensing. In our
future work, we can investigate the quality consequence in this environment. Another
interesting direction is to find algorithms when the task requesters have some limited
budgets.
356
J. Mukhopadhyay et al.
References
1. Aflaki, S., Meratnia, N., Baratchi, M., Havinga, P.: Evaluation of incentives for body area
network-based healthcare systems. In: 8th International Conference on Intelligent Sensors,
Sensor Networks and Information Processing (ISSNP), pp. 515–520. IEEE, New York (2013)
2. Bhattacharjee, J., Pal, A., Mukhopadhyay, S., Bharasa, A.: Incentive and quality aware participatory sensing system. In: 12th International Conference on Dependable, Autonomic and
Secure Computing (DASC), pp. 382–387. IEEE Computer Society (2014)
3. Bredin, J., Parkes, D.C.: Models for truthful online double auctions. In: 21st International
Conference on Uncertainty in Artificial Intelligence (UAI), pp. 50–59. AUAI press (2005)
4. Chatzimilioudis, G., Konstantinidis, A., Laoudias, C., Zeinalipour-Yazti, D.: Crowdsourcing
with smartphones. IEEE Internet Comput. 16(5), 36–44 (2012)
5. Cristofaro, E.D., Soriente, C.: Participatory privacy: enabling privacy in participatory sensing.
IEEE Network Mag. 27(1), 32–36 (2013)
6. Ganti, R.K., Ye, F., Lei, H.: Mobile crowdsensing: current state and future challenges. IEEE
Commun. Mag. 49(11), 32–39 (2011)
7. Gao, H., Liu, C.H.: A survey of incentive mechanisms for participatory sensing. IEEE Commun.
Surv. Tutorials 17, 918–943 (2015)
8. Gao, Y., Chen, Y., Liu, K.J.R.: On cost-effective incentive mechanisms in microtask crowdsourcing. IEEE Trans. Comput. Intell. AI Games 7(1), 3–15 (2015)
9. Jaimes, L.G., Idalides, V.L., Labrador, M.A.: A location-based incentive mechanism for participatory sensing systems with budget. In: 16th International Conference on Pervasive Computing
and Communications (PerCom), pp. 103–108. IEEE, New York (2012)
10. Lan, K., Wang, H.: On providing incentives to collect road traffic information. In: 9th International Wireless Communications and Mobile Computing Conference (IWCMC). IEEE, New
York (2013)
11. Lane, N.D., Miluzzo, E., Lu, H., Peebles, D., Choudhury, T., Campbell, A.T.: A survey of
mobile phone sensing. IEEE Commun. Mag. 48(9), 140–150 (2010)
12. Lee, J.S., Hoh, B.: Dynamic pricing incentive for participatory sensing. Elsevier J. Pervasive
Mob. Comput. 6(6), 693–708 (2010)
13. Li, Q., Cao, G.: Providing privacy-aware incentives for mobile sensing. In: 34th International
Conference on Distributed Computing Systems (ICDCS), pp. 76–84. IEEE, New York (2014)
14. Luo, T., Tan, H.P., Xia, L.: Profit-maximizing incentive for participatory sensing. In: 33rd
Annual International conference on Computer Communications (INFOCOM), pp. 127–135.
IEEE, New York (2014)
15. Massung, E., Coyle, D., Cater, F.K., Jay, M., Preist, C.: Using crowdsourcing to support proenvironmental community activism. In: Proceedings of the SIGCHI Conference on Human
Factors in Computing Systems, pp. 371–380. ACM, New York (2013)
16. Reddy, S., Estrin, D., Hansen, M., Srivastava, M.: Examining micro-payments for participatory sensing data collections. In: 12th International Conference on Ubiquitous Computing
(UbiComp), pp. 33–36. ACM, New York (2010)
17. Restuccia, F., Das, S.K., Payton, J.: Incentive mechanisms for participatory sensing: survey
and research challenges. Trans. Sens. Netw. (TOSN). 12(2), 13:1–13:40 (2016). https://doi.
org/10.1145/2888398
18. Vergara-Laurens, I.J., Mendez, D., Labrador, M.A.: Privacy, quality of information, and energy
consumption in participatory sensing systems. In: 18th International Conference on Pervasive
Computing and Communications (PerCom). IEEE, New York (2014)
19. Yu, R., Liu, R., Wang, X., Cao, J.: Improving data quality with an accumulated reputation
model in participatory sensing systems. Sensors 14(3), 5573–5594 (2014)
20. Zhang, X., Yang, Z., Zhou, Z., Cai, H., Chen, L., Li, X.: Free market of crowdsourcing: incentive
mechanism design for mobile sensing. IEEE Trans. Parallel Distrib. Syst. 25(12), 3190–3200
(2014)
High-Order Compact Finite Difference
Scheme for Euler–Bernoulli Beam
Equation
Maheshwar Pathak and Pratibha Joshi
Abstract Structures such as buildings and bridges consist of a number of
components such as beams, columns, and foundations, all of which act together
to ensure that the loadings that the structure carries is safely transmitted to the supporting ground below. The study of the design and deflection of the beam under load
play an important role in the strength analysis of a structure. In the present paper,
we have applied high-order compact finite difference scheme using MATLAB to
approximate the solution of Euler–Bernoulli beam equation which determines the
deflection of the beam under the load acting on the beam.
Keywords Finite difference method · Beam equation · Differential equations
1 Introduction
In the construction of buildings, bridges etc., beams are used as the basis of supporting
structure. Every structure requires a safety and structural analysis with the knowledge
of beam theory. History of beam equations [1–35] came in existence by Leonardo
da Vinci (1452–1519), Galileo Galilee (1584–1642). Jacob Bernoulli (1654–1705)
first discovered that the curvature of an elastic beam at any point is proportional to
the bending moment at that point. Nephew of Jacob, Daniel Bernoulli (1700–1782)
formulated the differential equation of motion of a vibrating beam. Later, Leonhard
Euler (1707–1783) accepted Jacob Bernoulli’s theory in his investigation of the
shape of elastic beams under various loading conditions. There are so many real
life problems, where beam equations arise, some of them can be seen in [24–28].
M. Pathak (B) · P. Joshi
Department of Mathematics, College of Engineering Studies,
University of Petroleum and Energy Studies (UPES),
Energy Acres VPO Bidholi, PO Prem Nagar, Dehradun 248007, Uttarakhand, India
e-mail: mpathak81@gmail.com
P. Joshi
e-mail: pratibha.joshi@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_35
357
358
M. Pathak and P. Joshi
Normally, the horizontal beams can be made from steel, timber, or reinforced concrete
and have a cross-sectional shape that can be rectangular, T or I shape. The design of
such beams can be complex but is essentially intended to ensure that the beam can
safely carry the load it is intended to support. This will include its own self-weight,
the weight of the structure it is supporting and what is often referred to as “live load”
being the weight of people and furnishings in buildings or the weight of road or rail
traffic in bridges. In addition to the requirements for the beam to safely carry the
intended design loads there are other factors that have to be considered including
assessing the likely deflection of the beam under load (Fig. 1).
Finite difference methods are techniques to approximate the solution of an ordinary or partial differential equations. These methods require discretization of the
domain into a structured mesh of grid points, in which approximations are needed.
Then approximate the derivatives in the governing equations at the grid points by
Taylor series. Taking the value of desired function as an unknown at each grid points,
we get a system of linear algebraic equations, by solving the linear system of equations, required approximation obtain. Although, all times discretization of domain
is not an easy task. Nevertheless, there are so many problems where finite difference
methods are an easy and popular tool to solve them.
To achieve high-order accuracy, the general strategy is to expand the numerical
stencil which has the disadvantage of creating larger matrix bandwidths which complicates the numerical treatment near the boundaries and increases computational
cost. The high-order compact finite difference methods [1–17] give better accuracy
and the stencil points used in the methods come from compact stencil. Hence, matrix
bandwidth does not change and we achieve high accuracy without any above complications.
In the present work, we have taken the high-order compact scheme [18–23] which
increases the accuracy of the standard central difference approximation from O(h2 )
to O(h4 ) by including compact approximations to the leading truncation error terms
to solve the Euler–Bernoulli beam equation (static beam equation).
p(x)
Fig. 1 Deflection of simply supported beam
p(x)
High-Order Compact Finite Difference Scheme …
359
2 Euler–Bernoulli’s Beam Equation
The Euler–Bernoulli’s beam equation is given as
d2
d2 u
E
I
q(x)u + p(x), 0 ≤ x ≤ L
dx 2
dx 2
(1)
Here, u(x) is the deflection of the beam, E is the Young’s modulus, I is the
area moment of inertia of the beam cross section, q(x) is the coefficient of ground
elasticity, and p(x) is a load force normal to the beam at the distance x from one end
of the beam.
With boundary conditions
d2 u(0)
t0
dx 2
d2 u(L)
u(L) s L ,
tL
dx 2
u(0) s0 ,
Now, to apply the high-order compact method on the beam equation, we consider
the case when q(x) 0 and E I 1. So, under these circumstances the beam
equation can be written as:
d4 u
p(x), 0 ≤ x ≤ L
dx 4
(2)
with boundary conditions
d2 u(0)
t0
dx 2
d2 u(L)
u(L) s L ,
tL
dx 2
u(0) s0 ,
Consider the equivalent form of beam equation as given below
d2 v
p(x), 0 ≤ x ≤ L
dx 2
d2 u
v(x), 0 ≤ x ≤ L
dx 2
(3)
(4)
with boundary conditions
u(0) s0 , v(0) t0
(5)
u(L) s L , v(L) t L
(6)
360
M. Pathak and P. Joshi
3 High-Order Compact Finite Difference Scheme for Beam
Equation
In this section we apply high-order compact finite difference scheme on the equivalent
form of beam equation, (3) and (4) with boundary conditions (5) and (6).
Let us take n uniform subintervals for the whole length L of the beam, size of
each subinterval is h Ln , u i u(xi ) and xi i h, ∀i 0, 1, 2, . . . n. Now the
governing Eqs. (3) and (4) of the beam at node xi can be written as
d2 vi
pi
dx 2
d2 u i
vi
dx 2
(7)
(8)
With the Taylor series expansions of vi−1 , vi+1 and u i−1 , u i+1 , we get the standard
central difference approximations of (7) and (8) as given below
vi+1 − 2vi + vi−1
− τi1 pi
h2
u i+1 − 2u i + u i−1
− τi2 vi ,
h2
(9)
(10)
where τi1 and τi2 are the truncation error at the node i given by
h 2 iv
v + O(h 4 )
12 i
h 2 iv
τi2 u + O(h 4 )
12 i
τi1 (11)
(12)
To obtain a high-order compact formulation, we approximate the derivatives on
the right hand side of Eqs. (11) and (12) and include them in the truncation error τi1
and τi2 . Differentiating Eqs. (3) and (4) two times w.r.t. x, we get
d2 p
d4 v
4
dx
dx 2
4
d u
d2 v
dx 4
dx 2
(13)
(14)
With the Taylor series expansions of pi−1 , pi+1 and vi−1 , vi+1 , we get the standard
central difference approximations of (13) and (14) at the node i as given below
pi+1 − 2 pi + pi−1
h 2 iv
−
p − O(h 4 )
h2
12 i
vi+1 − 2vi + vi−1
h2
− viiv − O(h 4 )
2
h
12
viiv (15)
u iv
i
(16)
High-Order Compact Finite Difference Scheme …
361
Including the fourth derivatives of v and u at the node i from Eqs. (15) and (16)
in the Eq. (11) and (12) of truncation error τi1 and τi2 , we get
pi+1 − 2 pi + pi−1
− O(h 4 )
12
vi+1 − 2vi + vi−1
τi2 − O(h 4 )
12
τi1 (17)
(18)
Including the truncation error τi1 and τi2 form Eqs. (17) and (18) in Eqs. (9) and
(10), we get
vi+1 − 2vi + vi−1
pi+1 − 2 pi + pi−1
+ O(h 4 ) pi
−
h2
12
u i+1 − 2u i + u i−1
vi+1 − 2vi + vi−1
+ O(h 4 ) vi
−
2
h
12
(19)
(20)
Equations (19) and (20) provide compact approximations to (3) and (4) with
boundary conditions (5) and (6) with fourth-order asymptotic rate of convergence.
4 Numerical Solution
In this section, we have solved some examples of beam equation using high-order
compact finite difference scheme by developing MATLAB codes [see Appendix].
Example 1 Consider the beam equation
d4 u
16π 4 Sinπ xCosπ x, 0 ≤ x ≤ 1
dx 4
with the homogeneous boundary conditions
u(0) 0, u(1) 0
u (0) 0, u (1) 0
This equation has the exact solution
u(x) Sinπ xCosπ x
The grid refinement analysis of the above problem is described in Table 1, which
shows fourth-order accuracy of our proposed algorithm.
In Fig. 2 displays the numerical as well as exact solution of this problem in 40
subintervals. It can be seen that the numerical solution closely matches with the exact
solution.
362
M. Pathak and P. Joshi
Table 1 Grid refinement analysis of Problem 1
en ∞
N
10
20
40
80
160
6.276282295363811e−004
4.074746213433844e−005
2.539184200012201e−006
1.585822890781685e−007
9.909578491118509e−009
Ratio (r )
Order of
convergence ( p)
15.28
16.04
16.01
16.00
3.94
4.00
4.00
4.00
Fig. 2 Numerical and exact
solution of Problem 1 for 40
subintervals
L∞ = 4.07 × 10 −5
L∞ = 4.07 × 10
(a)
(b)
−5
Fig. 3 Error between numerical solution and exact solution of Problem 1 for a n 20 b n 80
In Fig. 3 the error between the obtained numerical solution and exact solution of
Problem 1 has been illustrated at n 20 and n 80.
High-Order Compact Finite Difference Scheme …
363
Figure 3 clearly displays that our algorithm gives high accuracy even at coarser
grid.
Example 2 To show high accuracy of our approach, we have taken the next beam
equation from Thankane and Styš [29], in which standard finite difference scheme
has been applied. Consider the following beam equation:
d2 d2 u
π 4 x sin π x − 4π 3 cos π x 0 ≤ x ≤ 1
dx 2 dx 2
which has the non-homogeneous boundary conditions:
u(0) u(1) 0
u (0) 2π, u (1) −2π
It has the exact solution u(x) x sin(π x). Since we want to compare our results
with the results of [29], we have first transformed the above equation in the following
beam equation with homogeneous boundary conditions
d4 u 0
π 4 x sin π x − 4π 3 cos π x + 96π (x − 1) + 12π (8x + 2(2x − 1))
dx
0 ≤ x ≤ 1,
(21)
where
u 0 (0) u 0 (1) 0
d2 0
d2 0
u
u (1) 0
(0)
dx 2
dx 2
by the transformation u(x) u 0 (x)+w(x) where w(x) −π(x − 1)2 x 2 (2x − 1) see
Ref. [29]. We applied high-order compact finite difference method in the boundary
value problem (21) and compared our results rounding off to four decimal places
with Thankane and Styš [29] in Table 2 for n 10.
Table 2 clearly illustrates that our approach gives higher accuracy compare to
approach of Thankane and Styš [29]. In [29] the maximum absolute error obtained
0.0143432 for n 10 whereas with our approach it is 0.000054908, which is a large
improvement. The graphs of our obtained numerical solution and the exact solution
are shown in Fig. 4 for 30 subintervals. It displays the similarity of both solutions.
364
M. Pathak and P. Joshi
Table 2 Comparison of our approach with approach of Thankane and Styš [29] for Problem 2 for
10 subintervals
X
Analytical
solution
u 0 (x)
u 0h (x) (in
Thankane and
Styš)
Abs. error (in
Thankane and
Styš)
u 0h (x) (in our
approach)
Abs. error (in
our approach)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0105
0.0693
0.1873
0.3442
0.5000
0.6068
0.6217
0.5185
0.2985
0.0069
0.0657
0.1864
0.3477
0.5083
0.6191
0.6361
0.5319
0.3072
0.0036
0.0036
0.0009
0.0035
0.0083
0.0123
0.0144
0.0134
0.0087
0.0105
0.0693
0.1872
0.3442
0.5000
0.6068
0.6217
0.5185
0.2985
0.0000
0.0000
0.0001
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Fig. 4 Numerical solution
and exact solution of
Problem 2 for n 30
5 Conclusion
In the present paper, we have obtained highly accurate numerical solution of
Euler–Bernoulli beam equation by developing MATLAB codes for high-order compact finite difference scheme which are mentioned in appendix. We have also compared our results with the results of Thankane and Styš [29] to show the better
approximation over standard finite difference method. It is clear from the results that
the approach mentioned in this paper increases accuracy of standard finite difference method without changing the bandwidth of the coefficient matrix or creating
complexity at the boundaries.
High-Order Compact Finite Difference Scheme …
365
Acknowledgements This paper is part of the project [No.-UCS&T/R&D/PHY SC.-10/1213/6180] funded by Uttarakhand State Council for Science and Technology, Dehradun, Uttarakhand,
India.
Appendix
Matlab Programs:
Example 1:
clc;
clear all;
t=0;
t=cputime;
format long;
n=input(‘enter the number of subintervals’);
x=linspace(0,1,n+1);
h=1/n;
%----------------------------------------------------f=[];
g=[];
fori=1:n+1
f=[f;(8*(piˆ4)*sin(2*pi*x(i)))];
end
%----------------------------------------------------%right side function g for v
fori=2:n
g=[g;(f(i-1)/12)+(f(i+1)/12)+(5*f(i)/6)];
end
%----------------------------------------------------%left boundary
lb=0;
%right boundary
rb=0;
%----------------------------------------------------%coefficient matrix A
A=sparse(n-1,n-1);
%diagonal elements of A
fori=1:n-1
A(i,i)=-2/(hˆ2);
end
%left elements of A
fori=2:n-1
A(i,i-1)=1/hˆ2;
366
M. Pathak and P. Joshi
end
%right elements of A
fori=1:n-2
A(i,i+1)=1/hˆ2;
end
%----------------------------------------------------%modified g due to left boundary
g(1)=g(1)-(lb/hˆ2);
%modified g due to right boundary
g(n-1)=g(n-1)-(rb/hˆ2);
%----------------------------------------------------v=[];
v=A\g;
v=[lb;v;rb];
%----------------------------------------------------%right side function q for u
q=[];
fori=2:n
q=[q;(v(i-1)/12)+(v(i+1)/12)+(5*v(i)/6)];
end
%----------------------------------------------------%modified q due to left boundary
q(1)=q(1)-0;
%modified q due to right boundary
q(n-1)=q(n-1)-0;
%----------------------------------------------------%calculate the value of u
u=[];
u=A\q;
%----------------------------------------------------%numerical solution
nums=[];
nums=[nums;0;u;0];
%----------------------------------------------------%exact solution
ex=[];
fori=1:n+1
ex=[ex;(sin(pi*(x(i)))*cos(pi*(x(i))))];
end
%----------------------------------------------------%absolute error
er=[];
fori=1:n+1
er=[er;abs(nums(i)-ex(i))];
end
High-Order Compact Finite Difference Scheme …
367
%----------------------------------------------------%maximum absolute error
maxer=max(er)
timing=cputime-t
%----------------------------------------------------% graph
plot(x,nums,’b’,x,ex,’*’);
plot(x,er);
Example 2:
format long;
clc;
n=input(‘enter the number of subintervals’);
x=linspace(0,1,n+1);
h=1/n;
%----------------------------------------------------f=[];
g=[];
fori=1:n+1
f=[f;((piˆ4)*x(i)*sin(pi*(x(i))))
-(4*(piˆ3)*cos(pi*(x(i))))];
end
%----------------------------------------------------%right side function g for v
fori=2:n
g=[g;(f(i-1)/12)+(f(i+1)/12)+(5*f(i)/6)];
end
%----------------------------------------------------%left boundary
lb=2*pi;
%right boundary
rb=-2*pi;
%----------------------------------------------------%coefficient matrix A
A=sparse(n-1,n-1);
%diagonal elements of A
fori=1:n-1
A(i,i)=-2/(hˆ2);
end
%left elements of A
fori=2:n-1
A(i,i-1)=1/hˆ2;
end
%right elements of A
368
M. Pathak and P. Joshi
fori=1:n-2
A(i,i+1)=1/hˆ2;
end
%----------------------------------------------------%modified g due to left boundary
g(1)=g(1)-(lb/hˆ2);
%modified g due to right boundary
g(n-1)=g(n-1)-(rb/hˆ2);
%----------------------------------------------------v=[];
v=A\g;
v=[lb;v;rb];
%----------------------------------------------------%right side function q for u
q=[];
fori=2:n
q=[q;(v(i-1)/12)+(v(i+1)/12)+(5*v(i)/6)];
end
%----------------------------------------------------%modified q due to left boundary
q(1)=q(1)-0;
%modified q due to right boundary
q(n-1)=q(n-1)-0;
%----------------------------------------------------%calculate the value of u
u=[];
u=A\q;
%----------------------------------------------------%numerical solution
nums=[];
nums=[nums;0;u;0];
%----------------------------------------------------%exact solution
ex=[];
w=[];
fori=1:n+1
ex=[ex;(x(i)*sin(pi*x(i)))];
end
%----------------------------------------------------%absolute error
er=[];
fori=1:n+1
er=[er;abs((nums(i)-ex(i)))];
end
%-----------------------------------------------------
High-Order Compact Finite Difference Scheme …
369
%maximum absolute error
maxer=max(er)
%----------------------------------------------------% graph
plot(x,nums,’b’,x,ex,’o’);
% plot(x,er);
References
1. Gupta Datta, A., Lake, L.W., Pope, G.A., Sepehrnoori, K.: High-resolution monotonic schemes
for reservoir fluid flow simulation. Situ 15, 289–317 (1991)
2. Leonard, B.P.: A stable and accurate convective modeling procedure based on quadratic
upstream interpolation. Comput. Methods Appl. Mech. Eng. 19, 59–98 (1979)
3. Carpenter, M.H., Gottlier, D., Abarbanel, S.: The stability of numerical boundary treatments
for compact high-order finite difference schemes. J. Comput. Phys. 108, 272–295 (1993)
4. Li, M., Tang, T., Fornberg, B.: A compact fourth-order finite difference scheme for the steady
incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 20, 1137–1151 (1995)
5. Gupta, M.M., Manohar, R.P., Stephenson, J.W.: A single cell high order scheme for the
convection-diffusion equation with variable coefficients. Int. J. Numer. Methods Fluids 4,
641–651 (1984)
6. MacKinnon, R.J., Carey, G.F.: Analysis of material interface discontinuities and superconvergent fluxes in finite difference theory. J. Comput. Phys. 75, 151–167 (1988)
7. MacKinnon, R.J., Carey, G.F.: Superconvergent derivatives: a Taylor series analysis. Int. J.
Numer. Methods Eng. 28, 489–509 (1989)
8. MacKinnon, R.J., Carey, G.F.: Nodal superconvergence and solution enhancement for a class
of finite element and finite difference methods. SIAM J. Sci. Stat. Comput. 11, 343–353 (1990)
9. MacKinnon, R.J., Johnson, R.W.: Differential equation based representation of truncation errors
for accurate numerical simulation. Int. J. Numer. Methods Fluids 13, 739–757 (1991)
10. Abarbanel, S., Kumar, A.: Compact high order schemes for the Euler equations. J. Sci. Comput.
3, 275–288 (1988)
11. Dennis, S.C.R., Hudson, J.D.: Compact h4 finite difference approximations to operators of
Navier-Stokes type. J. Comput. Phys. 85, 390–416 (1989)
12. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys.
103, 16–42 (1992)
13. Ananthakrishnaiah, U., Manohar, R., Stephenson, J.W.: High-order methods for elliptic equations with variable coefficients. Numer. Methods Partial Differ. Equ. 3, 219–227 (1987)
14. Spotz, W.F.: High Order Compact Finite Difference Schemes for Computational Mechanics.
Ph.D. thesis, University of Texas at Austin, Austin, TX (1995)
15. Spotz, W.F., Carey, G.F.: High order compact scheme for the stream-function vorticity equations. Int. J. Numer. Methods Eng. 38, 3497–3512 (1995)
16. Spotz, W.F., Carey, G.F.: High order compact formulation for the 3D Poisson equation. Numer.
Methods Partial Differen. Equ. 12, 235–243 (1996)
17. Jain, M.K., Jain, R.K., Mohanty, R.K.: Fourth order finite difference method for three dimensional elliptic equations with nonlinear first-derivative terms. Numer. Methods Partial Differen.
Equ. 8(6), 575–591 (1992)
18. Lax, P.D., Wendroff, B.: Difference schemes for hyperbolic equations with high order accuracy.
Commun. Pure Appl. Math. 17, 381–398 (1964)
19. MacKinnon, R.J., Langerman, M.A.: A compact high order finite element method for elliptic
transport problems with variable coefficients. Numer. Methods Partial Differen. Equ. 10, 1–19
(1994)
370
M. Pathak and P. Joshi
20. Dukowicz, J.K., Ramshaw, J.D.: Tensor viscosity method for convection in numerical fluid
dynamics. J. Comput. Phys. 32, 71–79 (1979)
21. Wong, H.H., Raithby, G.D.: Improved finite difference methods based on a critical evaluation
of the approximation errors. Numer. Heat Transf. 2, 139–163 (1979)
22. Dennis, S.C.R., Wing, Q.: Generalized finite differences for operators of Navier-Stokes type.
In: Proceedings of the 10th International Conference on Numerical Methods in Fluid Dynamics
vol. 264, pp. 222–228 (1986)
23. Spotz, W.F.: Superconvergent finite difference methods with applications to viscous flow. Master’s thesis, University of Texas at Austin, Austin (1991)
24. Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill (1993)
25. Gere, J.M.: Mechanics of Materials. Nelson Thomas Publishing (2002)
26. Timoshenko, S.P.: History of Strength of Materials. Dover Publications Inc., New York (1953)
27. Traill-Nash, R.W., Collar, A.R.: The effects of shear flexibility and rotatory inertia on the
bending vibrations of beams. Q. J. Mech. Appl. Mech. 6, 186–213 (1953)
28. Abbas, B.A.H., Thomas, J.: The second frequency spectrum of Timoshenko beams. J. Sound
Vib. 51, 123–137 (1977)
29. Thankane, K.S., Styš, T.: Finite difference method for beam equation with free ends using
mathematics. South. Afr. J. Pure Appl. Math. 4, 61–78 (2009)
30. Kumar, M., Joshi, P.: Some numerical techniques to solve elliptic interface problems. Numer.
Methods Partial Differen. Equ. 28, 94–114 (2012)
31. Joshi, P., Kumar, M.: Mathematical model and computer simulation of three dimensional thin
film elliptic interface problems. Comput. Math Appl. 63(1), 25–35 (2012)
32. Kumar, M., Joshi, P.: A mathematical model and numerical solution of a one dimensional steady
state heat conduction problem by using high order immersed interface method on non-uniform
mesh. Int. J. Nonlinear Sci. (IJNS) 14(1), 11–22 (2012)
33. Kumar, M., Joshi, P.: Mathematical modelling and computer simulation of steady state heat
conduction in anisotropic multi-layered bodies. Int. J. Comput. Sci. Math. 4(2), 85–119 (2013)
34. Pathak, M., Joshi, P.: High order numerical solution of an volterra integro-differential equation
arising in oscillating magnetic fields using variational iteration method. Int. J. Adv. Sci. Technol.
69, 47–56 (2014)
35. Pathak, M., Joshi, P.: A high order solution of three dimensional time dependent nonlinear
convective-diffusive problem using modified variational iteration method. Int. J. Sci. Eng.
8(1), 1–5 (2015)
Test Case Optimization
and Prioritization Based
on Multi-objective Genetic Algorithm
Deepti Bala Mishra, Rajashree Mishra, Arup Abhinna Acharya
and Kedar Nath Das
Abstract The validation of modified software depends on the success of Regression
testing. For this, test cases are selected in such a way that can detect a maximum
number of faults at the earliest stage of software development. The selection process
in which the most beneficial test case are executed first is known as test case prioritization which improves the performance of execution of test cases in a specific
or appropriate order. Many optimizing techniques like greedy algorithm, genetic
algorithm, and metaheuristic search techniques have been used by many researchers
for test case prioritization and optimization. This research paper presents a test case
prioritization and optimization method using genetic algorithm by taking different
factors of test cases like statement coverage data, requirements factors, risk exposure,
and execution time.
Keywords Regression testing · Test case prioritization · Risk exposure
Requirement factor · Genetic algorithm
D. B. Mishra · A. A. Acharya
School of Computer Engineering, KIIT University, Bhubaneswar 751024, India
e-mail: dbm2980@gmail.com
A. A. Acharya
e-mail: aacharyafcs@kiit.ac.in
R. Mishra (B)
School of Applied Sciences, KIIT University, Bhubaneswar 751024, India
e-mail: rajashreemishra011@gmail.com
K. N. Das
Department of Mathematics, NIT Silchar, Silchar, Assam, India
e-mail: kedar.iitr@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_36
371
372
D. B. Mishra et al.
1 Introduction
Nowadays, we are surrounded by many automated software. Qualitative, robust, and
trustworthy software are maintained by successful testing as it is the most important
phase of Software Development Life Cycle (SDLC) [1]. In the maintenance phase
when Software Under Test (SUT) is modified, we must ensure that the software
remains defect free and for this regression testing is required which is a process of
retesting of whole software and it is frequently performed on the altered version of
software for checking the validity. The same process needed more time, cost and
recourses during regression testing, so it is not always possible to run the whole test
cases again and again [2]. To avoid this problem, we need to prioritize the test cases
in such a manner that the most priority test cases are executed first than the lower one
and sometimes low priority test cases are need not be executed. The priority criteria
depend on the different factors of test cases [3].
The rate of risk can be identified during regression testing so that the debugging
process begins as soon as possible and high-risk faults are detected in testing life
cycle [4, 5]. Test case prioritization improves the cost-effectiveness of regression
testing by reordering the most important test cases to be executed very first. It also
increases the probability of running of the most beneficial test cases if the testing
process complete before the stipulated time [6, 7]. Several test case prioritization and
optimization techniques are developed by applying metaheuristic search techniques.
Genetic algorithm (GA) is one of the optimizing techniques used to solve different
optimization problem in software engineering field as it gives an exact fitness value
for each and every test case of a specific SUT. Based on the fitness value, the test
cases are prioritized [8, 9].
In this research paper, a test case prioritization method is developed based on
requirement priority, risk exposures, statement coverage, and execution time associated with test cases for a specific SUT within a given time constraint. The proposed
technique uses a multi criteria based GA for prioritization and optimization of test
cases by considering a stipulated time to execute test cases. The rest of the paper is
organized as: Sect. 2 describes test case prioritization for regression testing, Sect. 3
discusses related work on test case prioritization and minimization using GA. A brief
outline of GA is drawn in Sect. 4 and in Sect. 5 the proposed algorithm for prioritization is described. In Sect. 6, the proposed algorithm is implemented on a case study
and Sect. 7 discusses various factors taken for the proposed algorithm. Section 8
describes the experimental setup and result analysis for the case study taken. Finally,
the conclusion of the paper is drawn in Sect. 9 followed by some future works.
2 Test Case Prioritization for Regression Testing
Regression testing is done to make sure that any type of enhancement made to existing
software does not impact the previous functionality. It also ensures that the existing
Test Case Optimization and Prioritization Based …
373
Fig. 1 Various activities in regression testing
bug does not result in new bugs at the time of software modification [10]. During
regression testing all test cases are needed to run again with the new test cases created
for modified version. So it is very time-consuming and an expensive task to reexecute
all test cases again. To avoid this nonviable situation various activities are carried out
during Regression testing shown in Fig. 1 [11]. Test case selection and prioritization
provides the facility to execute a less numbers of test cases for maximum coverage
of the software, in such a manner that the most important test cases are executed
first than the others [8]. Through minimization, the redundant or obsolete test cases
are eliminated and hence the minimization techniques lower the cost of regression
testing by reducing a test suite to a minimal subset [5].
2.1 Problem Identification for Test Case Minimization
and Prioritization
Test Case Minimization Problem [12]:
Given: A test suite Ts and a set of test case requirements r 1 , r 2 , r 3 , …, r n that must
be satisfied to provide the testing coverage of a software. The subsets of Ts, T 1 , T 2 ,
T 3 , …, T n are associated with Traceability matrix, in such a way that each test case
T j belongs to T i can be used to test r i .
Problem: We have to find a representative set of test cases from Ts that satisfies
all the r i ’s, where the r i ’s can represent either all the requirements of the program or
the requirements related to the modified program.
Test case Prioritization problem [12]:
Given: A test suite T and T order refers to a number of ways that test cases are chosen.
Fitness f of T is calculated depending on some criteria to a real value.
Problem: We have to find T in such a way that T m T order for all T , where
T != T and f (T ) >= f (T ).
374
D. B. Mishra et al.
3 Related Work
Kaur and Goyal [13] proposed a new GA based method to prioritize the test cases by
taking the complete code coverage. They have used APCC metric to represent the
effectiveness of their result and analyzed different prioritized approaches. Konsaard
et al. [9] proposed total coverage based regression test case prioritization using GA.
Modified GA is used to simplify and it has the ability to change the population,
that supply a number of test cases used for prioritization process. Amr et al. [14],
presented an approach for automatic test case generation and prioritization using GA.
They have used multicriteria fitness function which evaluates the multiple control
flow data. They have taken different factors for prioritization such as coverage based
data, faults detected by test cases and their severity value. Their comparative result
showed superior to other similar work done previously. Prakash and Gomathi [1]
developed a multiple criteria coverage method for test case prioritization to improve
test efficiency by taking average information to prioritize test cases and found their
proposed method improves the performance of regression testing and the rate of
fault detection capacity for various SUT. Sharma and Sujata [15] defined an effective
model-based approach to generate and prioritize the effective test cases. They have
used GA to generate effective test paths based on the requirement and user view
analysis. They have taken cost factor for a specific model and estimate the overall
cost to test the functional behavior of the model. Kumar et al. [16] proposed a
prioritization technique based on requirement analysis such as requirement priority
and requirement factor with varying nature. Their proposed system improves the
testing process by ensuring the quality, cost, effort of the software and the user’s
satisfaction. Rhmann et al. [5] presented an approach for test came minimization and
prioritization by taking several factors of software projects under test. The selection
of test cases is based on given time constraints. They have used a novel approach of
0–1 integer programming to model the fitness.
4 Genetic Algorithm (GA)
GA is an evolutionary search technique used to solve many optimization problems.
It is inspired by the biological concept of evolution and based on “surviving the
fittest” [17]. The algorithm starts with the process of random generation of populations depending on the specified problem domain. Then the basic operators such as
selection, crossover, mutation, and elitism are applied on the initial population after
evaluation of fitness. The same process is repeated until reach an optimal solution. In
software engineering field, it is used to solve many complex and real-life problems
by producing high-quality test data automatically during testing phase [18].
In this research paper, permutation encoding [19] is used to create chromosomes
for the proposed method shown in Fig. 2. Average crossover [14] and insertion
mutation operators [20] are used to find the new offspring chromosome.
Test Case Optimization and Prioritization Based …
375
Fig. 2 Permutation
encoding
4.1 Average Crossover
Average crossover takes two parents to perform crossover and creates only one offspring by finding the average of two parents [14]. New offspring can be found by
averaging the genes of chromosomes of Fig. 2 and the resultant offspring is shown
in Fig. 3.
4.2 Insertion Mutation
In the proposed method, mutation is performed on a particular chromosome by
changing the gene position in that chromosome. The position of the gene is arbitrarily
changed by reinserting in a new position with a small mutation probability factor
[20]. An example of insertion mutation is shown in Fig. 4.
4.3 Fitness Function
The fitness function evaluates individual’s performance. Based on the fitness value,
the individuals with higher are selected to the next generation for better optimum
solution [21]. The fitness is defined in Eq. (1), which is based on total statement
coverage, total risk exposure, requirement priority values, and the time of execution
of test cases.
Fig. 3 New offspring after crossover
Fig. 4 Insertion mutation
376
D. B. Mishra et al.
5 Proposed Algorithm for Prioritization
This section describes the proposed algorithm to prioritize test cases and the factors
required for prioritization. To implement the proposed method, a case study as Accept
Saving Details for an Income Tax Calculator [11] is taken and GA is used to optimize
test cases and further, we prioritized those test cases based on their fitness. We have
taken some test case-oriented factors for prioritization such as requirement value,
statement coverage, and test case execution time during regression testing.
Prioritization and Minimization Algorithm:
Step 1. Create Initial Population i.e. Chromosomes (C1, C2, C3, . . . , Cn)
Step 2. Initialize the Population as Test Suites
Test Suite No. of Chromosomes
Step 3. Calculate Fitness
Step 4. Select Best Two Populations Based on Fitness
(Best 2 Chromosomes)
Step 5. Apply Cross Over Operator
Step 6. Apply Insertion Mutation on the new Chromosome
Step 7. Remove the Duplicates
Step 8. Check for the Multi objective fitness
If (Solution Feasible)
Print the Optimized Solution
Else
Go to Step 1
End
6 Case Study: Accept Saving Details for an Income Tax
Calculator
The proposed approach is implemented on a small java program as Accept Saving
for an Income Tax Calculator [11] shown in Fig. 5. The program takes 3 arguments as
Account number, Account type and the Amount to deposit in his/her saving account.
The correct values for different variables are given below. Test cases are generated
and listed in Table 1.
1. Account No—12345
2. Account type—“Saving”
3. Amount—Positive integer with 2 decimal points.
Test Case Optimization and Prioritization Based …
377
Fig. 5 Java program for Accept_saving
Table 1 Test cased generated for Accept_saving
Test case id
Input data
Output
Tl
(111, ‘Savin2’, 0)
Enter correct account number
T2
(12345, ‘Current’, 0)
Enter correct account type
T3
(12345, ‘Saving’, 0)
Enter correct amount
T4
(12345, ‘Saving’, 5000.00)
Print the balance
7 Factors Considered for Prioritization
The different factors like total statement coverage, requirement priority factor value,
and total risk exposure value are considered for prioritization. For minimization, the
total execution time has been taken as one of the proposed multi-objective constraint.
7.1 Requirement Priority Factor
From the Java coding shown in Fig. 5, the following requirements are needed:
378
D. B. Mishra et al.
Table 2 The priority factor value for requirements
Requirement
Manager
Developer
R1
R2
R3
R4
10
9
7
8
10
9
7
8
Customer
Total
10
10
9
9
30
28
23
25
• R1—If the user inputs wrong account no then the message should display to input
correct account number.
• R2—For wrong account type input the proper message should display.
• R3—If the user inputs a wrong amount the appropriate message should display to
input correct value.
• R4—Finally if accept saving is successful then the message should display indicating the operation success and display the total balance.
The requirement priority factor values are assigned by manager, developers, and
customers from 1 to 10 depending on their importance. It may be same or different
for different persons and the priority values of different requirements are recorded
in Table 2.
7.2 Risk Exposure
In software development life cycle, each module is tested by analyzing the potential
of risks. The testers use risk analysis to select most crucial test cases. So test cases are
prioritized by some potential problems occurs during software development. There
are mainly four types of risks which may occur during software development [5],
such as loss of power (LP), corrupt file data (CFD), unauthorized user access (UUA),
and slow throughput (ST). Risk exposure can be calculated using Eq. (1)
Risk Exposure UFP × RI
(1)
where UFP is the uncertainty factor of the potential problem with a scale of 1 (low)
to 10 (high) and RI is the risk impact values from 1 (low) to 10 (high). The risk
exposure for each requirement is computed by using Eq. 1 and shown in Table 3.
7.3 Total Statement Coverage
It counts the number of statements covered by each test case. In our case study, the
statement coverage of test cases T 1, T 2, T 3 and T 4 is (5, 7, 9, 12) respectively.
Table 4 shows the total number of statement covered and the time of execution of
Test Case Optimization and Prioritization Based …
379
Table 3 Total risk exposure of each requirement
Requirement Uncertainty Potential factor value
factor/risk
impact
R1
R2
R3
R4
UF
RI
UF
RI
UF
RI
UF
RI
Risk
exposure
LP
CFD
UUA
ST
4
9
8
7
6
3
6
9
5
8
9
3
4
7
7
6
7
10
6
9
8
9
9
5
10
8
3
6
5
9
7
10
Table 4 Statement coverage of different test cases
Test case
Total statement covered
T1
T2
T3
T4
5
7
9
12
226
155
163
211
Total execution time
15
21
27
36
each test case according to the java code for Accept_ Details ( ) shown in Fig. 5. The
execution time for each statement is taken as 3 milliseconds (ms).
8 Experimental Setup and Result Analysis
The initial population is created by using permutation encoding and the fitness function is designed keeping in mind the factors such as total statement coverage, requirement priority factor value and execution time, which is shown in Eq. (2). The test
case weight is divided by the order of the test case to decrease the weight when the
order of a particular test case increased. The test suite with high fitness value will be
selected for next generation. Next crossover and mutation operators are applied to
the selected chromosomes to generate new offspring (test suite). Finally, the number
of test cases is minimized by deleting the duplicates which gives the optimum result.
Maximize F(x) n
TCWi
i1
TCOi
Subject to: T 1 + T 2 + T 3 + T 4 ≤ 80
(2)
TCW Total Statement Covered + Requirement Priority Factor
+ Risk Exposure
(3)
380
D. B. Mishra et al.
Table 5 Prioritized test cases and their coverage
Test case factors Test case
(coverage)
% covered
T1
Total statement
covered
Requirement
covered
Risk covered
Total execution
time
T4
T3
5
12
9
30
25
23
226
15
211
36
163
27
93.33%
74%
79.47%
≤80 ms
where TCW represents the test case weight and it is calculated by using Eq. (3), n
represent the total number of test cases and TCOi indicates the test case order of
ith test case. Table 5 shows the resultant test case after minimization by considering
the time factor and we get the prioritization order of test cases as T 1, T 4 and T 3
according to the total factors covered by each test case.
9 Conclusion
In this paper, a new approach is used for test case prioritization by considering total
statement coverage, requirement priority factor value, risk exposure, and execution
time. The proposed method is implemented on a small case study namely Accept
Saving Details for an Income Tax Calculator. Further, the test cases are optimized
based on the time constraints.
In future, it is planned to implement the proposed algorithm to prioritize and
optimize the test cases for large and complex software.
References
1. Prakash, N., Gomathi, K.: Improving test efficiency through multiple criteria coverage based
test case prioritization. Int. J. Sci. Eng. Res. (2014)
2. Raju, S., Uma, G.V.: Factors oriented test case prioritization technique in regression testing
using genetic algorithm. Eur. J. Sci. Res. 74(3), 389–402 (2012)
3. Krishnamoorthi, R., Mary, S.S.A.: Factor oriented requirement coverage based system test case
prioritization of new and regression test cases. Inf. Softw. Technol. 51(4), 799–808 (2009)
4. Stallbaum, H., Metzger, A., Pohl, K.: An automated technique for risk-based test case generation and prioritization. In: Proceedings of the 3rd International Workshop on Automation of
Software test, pp. 67–70. ACM (2008)
5. Rhmann, W., Zaidi, T., Saxena, V.: Test cases minimization and prioritization based on requirement, coverage, risk factor and execution time
Test Case Optimization and Prioritization Based …
381
6. Jeyaprakash, S., Alagarsamy, K.: A distinctive genetic approach for test-suite optimization.
Procedia Comput. Sci. 62, 427–434 (2015)
7. Acharya, A.A., Mohapatra, D.P., Panda, N.: Model based test case prioritization for testing
component dependency in cbsd using uml sequence diagram. IJACSA Int. J. Adv. Comput.
Sci. Appl. 1(6), 108–113 (2010)
8. Sharma, N., Purohit, G.N.: Test case prioritization techniques “an empirical study”. In: 2014
International Conference on High Performance Computing and Applications (ICHPCA),
pp. 1–6. IEEE (2014)
9. Konsaard, P., Ramingwong, L.: Total coverage based regression test case prioritization
using genetic algorithm. In: 2015 12th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON),
pp. 1–6. IEEE (2015)
10. Kavitha, R., Sureshkumar, N.: Test case prioritization for regression testing based on severity
of fault. Int. J. Comput. Sci. Eng. 2(5), 1462–1466 (2010)
11. Ansari, A., Khan, A., Khan, A., Mukadam, K.: Optimized regression test using test case prioritization. Procedia Comput. Sci. 79, 152–160 (2016)
12. Chauhan, N.: Software Testing: Principles and Practices. Oxford University Press, Oxford
(2010)
13. Kaur, A., Goyal, S.: A genetic algorithm for regression test case prioritization using code
coverage. Int. J. Comput. Sci. Eng. 3(5), 1839–1847 (2011)
14. Umbarkar, A.J., Sheth, P.D.: Crossover operators in genetic algorithms: a review. ICTACT J.
Soft Comput. 6(1) (2015)
15. Sharma, N., Sujata, M.: Model based test case prioritization for cost reduction using genetic
algorithm
16. Kumar, A., Gupta, S., Reparia, H., Singh, H.: An approach for test case prioritization based
upon varying requirements. Int. J. Comput. Sci. Eng. Appl. 2(3), 99 (2012)
17. Goldberg, D.E.: Genetic Algorithms: In Search, Optimization and Machine Learning. Addison
Wesley, MA (1989)
18. Mishra, D.B., Mishra, R., Das, K.N., Acharya, A.A.: A systematic review of software testing using evolutionary techniques. In: Proceedings of Sixth International Conference on Soft
Computing for Problem Solving, pp. 174–184. Springer, Singapore (2017)
19. Ahmed, A.A., Shaheen, M., Kosba, E.: Software testing suite prioritization using multi-criteria
fitness function. In: 2012 22nd International Conference on Computer Theory and Applications
(ICCTA), pp. 160–166. IEEE (2012)
20. Boopathi, M., Sujatha, R., Kumar, C.S., Narasimman, S.: The mathematics of software testing
using genetic algorithm. In: 2014 3rd International Conference on Reliability, Infocom Technologies and Optimization (ICRITO) (Trends and Future Directions), pp. 1–6. IEEE (2014)
21. Mishra, D.B., Bilgaiyan, S., Mishra, R., Acharya, A.A., Mishra, S.: A review of random test
case generation using genetic algorithm. Indian J. Sci. Technol. 10(30) (2017)
PSO-SVM Approach in the Prediction
of Scour Depth Around Different Shapes
of Bridge Pier in Live Bed Scour
Condition
B. M. Sreedhara , Geetha Kuntoji, Manu and S. Mandal
Abstract Scour is one of the major factors which affects directly on the durability
and safety of the Bridge abutments. Based on the experimental data of Goswami in
2012, an effort is made to predict local scour by using a hybrid approach of Swarm
Intelligence based algorithms which is today one of the powerful tools of optimization techniques. In this work, an intelligent model based on support vector machine
in combination with the particle swarm optimization (PSO-SVM) technique is developed. The PSO-SVM models are developed with RBF, Polynomial and Linear kernel
functions. The circular, rectangular, round-nosed, and sharp-nosed shapes of piers
are considered in live bed scour condition. The scour depth around bridge piers is
predicted by considering Sediment size, flow velocity, and time of flow as input
parameters. Prediction accuracy of the models is evaluated using the model performance indicators such as Root Mean Square Error (RMSE, Correlation Coefficient
(CC), Nash Succlift Error (NSE), etc. The results obtained from the model are compared with the measured scour depth to validate the reliability of the hybrid model.
Based on the results, PSO based SVM model is found to be successful, reliable, and
efficient in predicting the scour depth around the bridge pier.
Keywords PSO-SVM · Kernel functions · Live bed scour · Bridge pier shapes
1 Introduction
Scour is one of the significant factors which affects the safety of the structure. It
occurs when the regular flow pattern disturbs due to the presence of structure across
the flow. When the uniform flow encounters a structure, there is a sudden change in
B. M. Sreedhara (B) · G. Kuntoji · Manu
Department of Applied Mechanics and Hydraulics, National Institute of Technology Karnataka,
Surathkal, Mangalore 575025, India
e-mail: shreedhar.am13f07@nitk.edu.in
S. Mandal
Department of Civil Engineering, PES University, Bangalore 560085, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_37
383
384
B. M. Sreedhara et al.
the flow pattern, due to which a large amount of eddy structure or system of vertices
develops at the base of the structure. The eddy structure usually is a combination of
horseshoe vortex and wake vortex. Horseshoe vortex develops at the upstream face
of the structure, and it plays a major role in the formation of scour hole. Wake vortex
appears at the downstream of the structure and its transport and deposits the eroded
materials from scour hole to the downstream.
Depending on the type and characteristics of the river bed, there are two types of
scour such as clear water scour and live bed scour. In the clear water scour condition,
there is no chance of refilling the scour hole by the sediment supply from the upstream
flow. Live bed scour occurs, when there is a continuous sediment supply from the
upstream flow. In this condition, the scour hole develops due to obstruction and
it refills by the upstream sediment supply. Here, the equilibrium state is reached
only when the rate of sediment erosion is equal to the rate of sediment supply to
the scour hole. The shape of the obstruction is one of the factors which effects on
the scour depth. To know the suitable shape for the construction across the flow
according to the scour condition, four types of shapes are considered in the study.
The different shapes namely, circular, rectangular, round-nosed, and sharp-nosed.
The present study concentrates on the prediction of live bed scour around different
shapes of bridge piers.
A large number of studies has been carried out to understand the mechanism
of the local scour and to predict the scour depth around the bridge pier and other
structures. Several researchers conducted an experimental study to analyze the scour
depth in live bed scour condition [1–3]. Those experimental data was further used for
numerical and soft computing studies. Later the soft computing techniques are used
as powerful tools to predict the scour depth using experimental data. The number
of Soft computing techniques are used form the researchers such as, artificial neural network (ANN), Genetic algorithm (GA), Fuzzy Logic, Group method of data
handling (GMDH), Linear regression (LR), Model tree approach, Neuro-AdaptiveInference-System (ANFIS), Support Vector Machine (SVM), etc. The various soft
computing models are developed to predict the scour depth around the bridge pier,
near spur dikes, below pipelines, around grade control structures, and others [4–6]. In
the present days, the efficiency of individual models is improved by hybridizing the
particular model with optimizing techniques. The common optimization techniques
are Particle swarm optimization (PSO), Ant Colony Optimization (ACO), Honey
Bee Search etc. It is observed from the literature that, the hybrid approaches like
ANFIS-ACO [7], ANFIS-LR [8], SVM-GA [9], GMDH-BP [10], PSO-ANN [11],
ANFIS-PSO [12, 13] and other models are developed to solve scour related and different problems. However, the application of PSO-SVM approach is not yet carried
out in the prediction of scour depth, and other scour related problems with live bed
scour condition. Therefore, in the current discussion, an effort is made to study and
estimate the scour depth around different shapes of bridge pier using PSO-SVM a
hybrid approach. Also, the study concentrates on suggesting the suitable shape of
the pier for live bed scour sites.
PSO-SVM Approach in the Prediction of Scour Depth …
385
2 Methodology
2.1 Data Analysis
The prediction of scour depth around the bridge pier using soft computing techniques
is based on the experimental data of Goswami Pankaj 2013. Experimental data are
generated using the 2D flume with a dimension of 1000 mm wide, 1300 mm depth,
and 19.25 m length. The size of bed material used in the study is uniformly graded
sand of d50 0.42 mm. The experiment conducted for live bed scour condition
with sediment quantity in the flow of 747.78 and 1066.67 ppm. The velocities of
flow considered in the study are 0.215 and 0.226 m/s. The data are collected from
0 to 4 h with 1-h interval. The circular, rectangular, round-nosed, and sharp-nosed
shaped piers are used in the experiment. Three input parameters, namely, sediment
quantity (ppm), velocity (U) and time (t) are used to estimate the depth of scour hole.
The experiments are conducted concerning different pier shapes such as circular,
rectangular, round-nosed, and sharp-nosed. The entire data set is randomly separated
and classified as training dataset (50%) and testing dataset (50%), based on the trial
and error technique for which there is no particular condition or criteria.
The statistical parameters are used to summarize the data sets; this defines the
distribution of data points, and their consistency in predicting the depth of scouring,
and the same are displayed in Table 1, regarding maxima, minima, mean, standard
deviation, and kurtosis for the entire data set of different pier shapes. The negative
value of kurtosis indicates that the distribution of data has lighter tails and flatter
peaks. The training data set is used to build the models to predict the scour depth.
Table 1 Statistical parameters
Statistical
Variables
parameters
Sediment
Velocity Time (h) Scour depth (mm)
quantity (pm) (m/s)
Max
Min
Mean
KD
Kurtosis
Max
Min
Mean
SD
Kurtosis
1066.67
747.78
907.225
159.45
−2.05
1066.67
747.78
907.225
159.45
−2.05
0.251
0.226
0.2385
0.0125
−2.05
0.251
0.226
0.2385
0.0125
−2.05
4
0
2
1.414
−1.31
4
0
2
1.414
−1.31
Circular Rectangular Round
nosed
Sharpnosed
98
71
83.575
7.69
−1.13
99
70
83.825
7.938
119
99
68
84.49
824
−1.034
98
68
85.24
8.95
−1.40
108
71
89.213
9.907
−1.16
106
73
89.633
10.024
−128
98
70
83.513
720
−1.02
97
68
83.35
7J394
−1.05
386
B. M. Sreedhara et al.
And then, the predicted values using testing data set are plotted against measured
values to analyze the model accuracy and efficiency in predicting the scour depth.
2.2 Development of PSO-SVM Model
Support Vector Machine (SVM) is a learning tool is derived from the past statistical
learning algorithms by Vapnik [14]. SVM acts as training algorithm and regression
tool for linear and nonlinear classification. In case of nonlinear data, the SVM can
map the data points of input space to the feature space of D-dimension by using
different kernel functions. As the kernel functions can convert nonlinear data points
them into linear ones. The SVM develops a different hyperplane margin between the
points in the feature space and amplifies edge between two informational indexes of
two input points. It made an effort of constructing a fit curve with a kernel function
and used on entire data points such that, data points should lie between two largest
marginal hyperplanes to minimize the error of regression [15, 16]. The predictive
capacity and classification error is dealt with learning some basic concept. First, the
hyperplane is separated, and then the process involves the selection of proper kernel
function and SVM between hard and soft margin.
Particle swarm optimization (PSO) is a population-based stochastic optimization
technique motivated by social behavior, such as bird flocking and fish schooling and it
was first proposed by Kennedy and Eberhart [17]. The particle swarm optimization
idea comprises of, at each time step, changing the speed of (accelerating) every
particle toward its pbest and gbest locations. Swarm intelligence concept began as a
simulation of improved social and simplified system. The first goal was to graphically
recreate the choreography of the bird of a bird block or fish school. In any case, it is
discovered that particle swarm model can be utilized as an optimizer. In PSO, every
single arrangement is a “bird” in the search space which is known as “particle.” Every
one of the particles has fitness values which are assessed by the function of fitness
to be optimized and have velocities which coordinate the flying of the particles. The
particles “y” through the issue space by taking the present ideal/optimum particles.
The methodology of the present study is illustrated in the flowchart given in Fig. 1.
2.3 Performance Analysis
The performances of the PSO-SVM models are analyzed using following statistical
parameters:
1. Normalized Root Mean Square Error (NRMSE)
PSO-SVM Approach in the Prediction of Scour Depth …
387
Fig. 1 Flowchart for the PSO-SVM model to predict scour depth
RMSE
× 100
NRMSE X
max −X min
N
2
i1 (X i −Yi )
where, RMSE N
2. Normalized Mean Bias (NMB)
(1)
388
B. M. Sreedhara et al.
NMB N Yi − X i
Ȳi
−1
Xi
X̄ i
i1
(2)
3. Nash–Sutcliffe coefficient (NSE)
N
NSE 1 −
i1
N
i−1
(X i − Yi )2
X i − X̄
(3)
2
4. Correlation Coefficient (CC)
N
CC N
i1
X i − X̄ · Yi − Ȳ
2 N
X i − X̄ · i1
Yi − Ȳ
i1
2
(4)
where,
X
Y
X̄
N
Observed/Measured value;
Predicted values;
Mean of actual data;
Total Number of Data Points.
3 Results and Discussion
The hybrid PSO-SVM models are developed with RBF, Polynomial, and Linear
kernel function to predict the scour depth around different shapes of bridge piers.
Circular, rectangular, round-nosed, and sharp-nosed shaped piers are considered in
the study. The experimental data used for training and testing the models are tabulated
in Table 1. The predicted results from the models are analyzed by using statistical
parameters as mentioned in the above section (Eqs. 1–4). The predicted results are
compared and plotted against experimentally measured scour depth. Figure 2 shows
the scatter plots of measured and predicted scour depth in the testing phase for all
four types of pier shapes. The model performances in case of both training and
testing are tabulated in Table 2. From the plots, it is clear that the PSO-SVM with
RBF and Polynomial kernel function performing better with higher CC, NSE, and
lower RMSE compared to a model with Linear kernel function. The models showing
good prediction for rectangular (CC 0.943, NSE 0.89) and sharp-nosed (CC 0.938, NSE 88) shapes compared to the circular (CC 0.920, NSE 0.845) and
round-nosed (CC 0.915, NSE 0.836) shapes. The negative NMB values show the
under-prediction, and positive NMB values show the overprediction of the models
as shown in Table 2. The box plots are plotted against the measured versus predicted
scour depth for all four shapes as shown in Fig. 3. It is observed from the box plot that,
PSO-SVM Approach in the Prediction of Scour Depth …
389
Fig. 2 Scatter plots of measured versus predicted scour depth from PSO-SVM model with different
kernel function for different pier shapes in testing phase
the spread of measured and predicted values are similar in the case of rectangular
and sharp-nosed shapes compared to other shapes.
4 Conclusion
The PSO-SVM model is applied to predict the scour depth around the different shapes
of the bridge pier. The RBF, Polynomial, and Linear kernel functions are used in the
study. The circular, rectangular, round-nosed, and sharp-nosed pier shapes with live
bed scour condition is considered. The predicted results are validated with experimental values. The PSO-SVM with RBF and Polynomial kernel function models
giving a good correlation for all the pier shapes compare to model with Linear kernel
function. The PSO-SVM model is well correlated in case of rectangular and sharpnosed shapes of piers. From the study, it can be concluded that PSO-SVM with RBF
390
B. M. Sreedhara et al.
Table 2 Performance analysis of PSO-SVM models
Pier shapes
Statistical PSO-SVM
indices
RBF
Polynomial
Circular
Rectangular
Round nosed
Sharp-nosed
Linear
Train
Test
Train
Test
Train
Test
CC
RMSE
NRMSE
NMB
NSE
CC
0.921
3.00
11.13
0.0013
0.847
0.966
0.908
3.33
11.50
−0.002
0.823
0.943
0.923
2.99
11.07
0.003
0.849
0.963
0.920
3.13
10.79
0.005
0.845
0.932
0.856
4.186
15.50
0.009
0.704
0.893
0.837
4.43
15.27
0.01
0.689
0.879
RMSE
NRMSE
NMB
NSE
CC
RMSE
NRMSE
NMB
NSE
CC
2.60
7.03
0.003
0.93
0.926
2.74
9.78
0.0008
0.855
0.918
3.38
10.24
−0.004
0.89
0.905
3.17
10.94
0.00
0.87
0.935
2.70
7.29
0.005
0.926
0.920
2.90
10.36
−0.001
0.84
0.92
3.66
11.09
0.0007
0.87
0.915
3.0
10.33
0.0016
0.836
0.938
4.50
12.19
0.0004
0.79
0.853
3.84
13.71
0.000
0.716
0.851
4.84
14.66
−0.007
0.77
0.852
3.91
13.48
−0.006
0.721
0.88
3.31
11.02
−0.007
0.86
3.44
11.09
0.01
0.825
3.15
10.52
−0.006
0.88
4.44
14.33
0.006
0.71
4.61
15.37
−0.018
0.73
RMSE
NRMSE
NMB
NSE
3.42
11.04
0.009
0.83
and Polynomial kernel function model could serve a better alternate for scour depth
prediction around bridge piers with live bed scour condition. The study also concludes that the rectangular and sharp-nosed shapes are suitable for live bed condition
comparing to circular and round-nosed pier.
Acknowledgements The authors would like to express their sincere gratitude to Dr. Goswami
Pankaj, Guwahati University for providing experimental data. Also, grateful to Director and Head
of the department, Applied Mechanics and Hydraulics, NITK, Surathkal for necessary support.
References
1. Sheppard, D.M., Miller Jr., W.: Live-bed local pier scour experiments. J. Hydraul. Eng. 132(7),
635–642 (2006)
2. Ballio, F., Radice, A., Dey, S.: Temporal scales for live-bed scour at abutments. J. Hydraul.
Eng. 136(7), 395–402 (2009)
PSO-SVM Approach in the Prediction of Scour Depth …
391
Fig. 3 Box plots of measured versus predicted scour depth for different pier shapes in testing phase
3. Ettmer, B., Orth, F., Link, O.: Live-bed scour at bridge piers in a lightweight polystyrene bed.
J. Hydraul. Eng. 141(9), 04015017 (2015)
4. Najafzadeh, M., Barani, G.A., Azamathulla, H.M.: Prediction of pipeline scour depth in clearwater and live-bed conditions using group method of data handling. Neural Comput. Appl.
24(3–4), 629–635 (2014)
5. Balouchi, B., Nikoo, M.R., Adamowski, J.: Development of expert systems for the prediction
of scour depth under live-bed conditions at river confluences: application of different types of
ANNs and the M5P model tree. Appl. Soft Comput. 34, 51–59 (2015)
6. Hong, J.H., Goyal, M.K., Chiew, Y.M., Chua, L.H.: Predicting time-dependent pier scour depth
with support vector regression. J. Hydrol. 468, 241–248 (2012)
7. Cus, F., Balic, J., Zuperl, U.: Hybrid ANFIS-ants system based optimisation of turning parameters. J. Achievements Mater. Manufact. Eng. 36(1), 79–86 (2009)
8. Akib, S., Mohammadhassani, M., Jahangirzadeh, A.: Application of ANFIS and LR in prediction of scour depth in bridges. Comput. Fluids 91, 77–86 (2014)
9. Chou, J.S., Pham, A.D.: Hybrid computational model for predicting bridge scour depth near
piers and abutments. Autom. Constr. 48, 88–96 (2014)
10. Najafzadeh, M., Barani, G.A.: Comparison of group method of data handling based genetic
programming and back propagation systems to predict scour depth around bridge piers. Sci.
Iranica 18(6), 1207–1213 (2011)
11. Hasanipanah, M., Noorian-Bidgoli, M., Armaghani, D.J., Khamesi, H.: Feasibility of PSOANN model for predicting surface settlement caused by tunneling. Eng. Comput. 32(4),
705–715 (2016)
12. Jannaty, M.H., Eghbalzadeh, A., Hosseini, S.A.: Hybrid ANFIS model for predicting scour
depth using particle swarm optimization. Indian J. Sci. Technol. 8(22) (2015)
13. Basser, H., Karami, H., Shamshirband, S., Akib, S., Amirmojahedi, M., Ahmad, R., Javidnia,
H.: Hybrid ANFIS–PSO approach for predicting optimum parameters of a protective spur dike.
Appl. Soft Comput. 30, 642–649 (2015)
392
B. M. Sreedhara et al.
14. Vapnik, V.: The Nature of Statistical Learning Theory. Springer Science & Business Media
(2013)
15. Mandal, S., Rao, S., Harish, N.: Damage level prediction of non-reshaped berm breakwater
using ANN, SVM and ANFIS models. Int. J. Naval Archit. Ocean Eng. 4(2), 112–122 (2012)
16. Mahjoobi, J., Mosabbeb, E.A.: Prediction of significant wave height using regressive support
vector machines. Ocean Eng. 36(5), 339–347 (2009)
17. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International
Conference on Neural Networks IV, vol. 1000 (1995, November)
Replenishment Policy for Deteriorating
Items Under Price Discount
Anubhav Namdeo and Uttam Kumar Khedlekar
Abstract This paper deals with a single-item inventory model with constant rate of
deterioration and shortage is not allowed. The stock-dependent and price-sensitive
demand is considered. Price discount is provided to uplift the market sell also the
preservation technology is applied to reduce deterioration. The objective of this paper
is to obtain the optimal replenishment time, preservation technology investment,
quantity, and profit. The model is illustrated by numerical examples and graphical
analysis. Discount policy could help to any business organization for smooth running
the business and obtain maximum profit.
Keywords Inventory · Stock dependent · Price-sensitive demand · Preservation
technology · Deterioration · Discount · Replenishment cycle
1 Introduction
Price is the economic value of something assigned by the manufacturer/dealer/retailer
or storekeeper to sold or offered for sale. Buyer convinces retailers to provide greater
pricing policy. Thus, retailers should have to decide the price of their product according to seasonality of demand, fashion, beyond the sell by date, the firms overall
objectives and reputation of the business. Retailers realize the importance of pricing strategy because the customers looking for good value, when they purchase the
products. To some customers, a good value means buying the product at a low price,
while the other getting their money’s worth in terms of product quality and service.
At the beginning of the inventory modeling, the inventory total cost calculated
by average method and there was no deterioration. The Harris–Wilson model for
obtaining the optimal quantity is the base model in the inventory modeling. Ford
Harris was the first one, who introduced the EOQ model published in 1915 [1]. A
A. Namdeo (B) · U. K. Khedlekar
Department of Mathematics and Statistics, Dr. Harisingh Gour Vishwavidyalaya
(A Central University), Sagar 470003, M.P., India
e-mail: namdeoanubhav@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_38
393
394
A. Namdeo and U. K. Khedlekar
frequent decadence in quality or quantity and market image of a product is called
deterioration, and such type of item is called deteriorating items. The first deteriorating inventory model studied by Whitin [2] on fashion apparels. Widyadana and Wee
[3] devised a deteriorating inventory model by considering price-dependent demand
and applied the concept of markdown policy. He found that there is a markdown time
and price that maximize the total profit.
For a long time, many authors kept thinking that the deterioration is a natural
phenomenon which is always present in the inventory. But the deterioration could be
control or at least reduce by applying some specific technics for a specific material
like there is cold storage and warehouse for vegetables, foods, fruits, and grains.
Hence, the cold storage is a kind of preservation technology which incurred a cost
or cost of preservation technology. Apart from this, packaging food items protect by
specific gas similarly, some chemical protects some other pharmaceutical substances.
In poultry form, hen and duck are kept in hot temperature because in low temperature
they are die. In this case, heater or thermowave is treated as preservation technology.
It is obvious that we have to invest some extra cost to preserve the items. This extra
cost is known as preservation cost that we have to bear.
In recent year, some authors are interested to include the preservation technology
in their deteriorating inventory model. Khedlekar et al. [7] presented an EPQ model
with disruption by incorporating time proportional demand and existence of shortage
after the completion of the production cycle also applied the preservation technology
to preserve the commodity from deterioration. In this study, they determined production time without disruption, production time with disruption, and if the disrupted
production system unable to fulfill the demand then they determined how much
and when we have to replenish from spot market. Weibull distribution deteriorating
inventory model with price-dependent demand and allowed the shortage developed
by Begum et al. [4]. Jagadeeswari and Chenniappan [5] devised an EOQ model for
deteriorating item with time quadratic demand in which partial back logging rate is
assumed as a decreasing function of waiting time for next replenishment.
Mishra et al. [10] developed an EOQ model for stock and price dependent demand
by considering complete and partial back ordering. Preservation technology is applied
to reduce deterioration and proved that the profit function is concave in price, time,
and preservation technology investment. Khedlekar et al. [6] designed an inventory
model for linearly declining demand in which some technic is used to preserve
the commodity. They have got the optimal replenishment time, optimal price, and
optimal preservation cost such that the total profit could be maximized. Khedlekar
et al. [8] extended his model [6] by taking exponential declining demand under the
cost of item preservation. Khedlekar and Namdeo [9] devised an inventory model
for stock and price dependent demand.
Some items have the property of completely perished or outdated in a specific
period of time, because they become harmful after the deadline or the new innovation
takes place of old one. Therefore, the manufacturer, retailer, and the storekeeper need
to completely sold out the stock before such time. For this, one can provide the price
discount as the basis of early to come and early to get profit. In view of this we have
Replenishment Policy for Deteriorating Items Under Price Discount
395
designed an inventory replenishment policy by simultaneously incorporating cost of
preservation technology, and discount policy.
2 Assumption and Notation
In this model, preservation cost and deterioration are co-related by λ(α) = λ0 e−μα ,
μ > 0. Here, we assume that the demand is price sensitive and has the constant
elasticity. The price-sensitive and stock-dependent demand at time t is assumed to
be D( p) = ψ(ξ p)−η . One time price discount is considered in one planning horizon.
The notations are as follows:
p
ch
cp
co
Io
ξ
ψ
η
I (t)
T1
T
r
I (t)
Q
λ(α)
α
TP(α, T)
Initial market price per unit,
The inventory holding cost unit per unit time,
The purchasing cost per unit,
Ordering cost per cycle,
Preservation cost,
Discount rate,
Stock-dependent parameter,
increase price rate,
Inventory level at time t,
Discount offering time,
The length of replenishment cycle,
Discount percentage in time,
The inventory level at time t,
The order quantity per cycle,
The deterioration rate,
Preservation technology investment cost per unit time to reduce the deterioration rate,
The total profit per unit time.
3 Mathematical Model
The inventory decrease due to demand and deterioration. The rate of change of
inventory could be presented in Fig. 1, and formulated by the differential equations:
∂ I (t)
+ λ(α)I (t) = −ψ(ξ p)−η , 0 ≤ t ≤ T
∂t
(1)
Since, there is no price discount in the interval t ∈ [0, T1 ], so ξ = 1 hence
∂ I (t)
+ λ(α)I (t) = −ψ p −η , 0 ≤ t ≤ T1
∂t
(2)
396
A. Namdeo and U. K. Khedlekar
Fig. 1 Graphical representation of inventory system
The boundary condition I (0) = Q, reveals
I (t) = Qe−λ(α)t +
ψ p −η −λ(α)t
e
− 1 , 0 ≤ t ≤ T1
λ(α)
(3)
Hence, the inventory level at point t = T1 ,
I (t) = Qe−λ(α)T1 +
ψ p −η −λ(α)T1
e
−1
λ(α)
(4)
Similarly, there is offered the price discount in the interval t ∈ [T1 , T ], so
∂ I (t)
+ λ(α)I (t) = −D( p), T1 ≤ t ≤ T
∂t
(5)
where, D( p) = ψ(ξ p)−η
By the Eq. (4), we have
I (t) = Qe−λ(α)t +
ψ p −η −λ(α)t
e
− e−λ(α)(t−T1 )
λ(α)
D( p) −λ(α)(t−T1 )
(e
− 1),
λ(α)
f or T1 ≤ t ≤ T
(6)
D( p) λ(α)T1
ψ p −η 1 − eλ(α)T1 −
(e
− eλ(α)T )
λ(α)
λ(α)
(7)
+
Boundary condition I (T )=0, reveals that
Q=−
Before calculating the total cost, we have to define the total inventory for 0 ≤ t ≤ T .
There are two different inventories, for 0 ≤ t ≤ T1 and T1 ≤ t ≤ T .
using the Eq. (3), the inventory for 0 ≤ t ≤ T1 is
Replenishment Policy for Deteriorating Items Under Price Discount
T1
I (t)dt = −
0
Q(e−λ(α)T1 − 1) ψ(e−λ(α)T1 + λ(α)T1 − 1) p −η
−
λ(α)
λ(α)2
397
(8)
Using the Eq. (6), the inventory for T1 ≤ t ≤ T is
T
Q
(e−λ(α)T − e−λ(α)T1 )
λ(α)
ψ p −η −λ(α)T
−
e
− e−λ(α)T1 − e−λ(α)(T −T1 ) + 1
λ(α)2
D( p) −λ(α)(T −T1 )
e
−
+ λ(α)(T − T1 ) − 1
2
λ(α)
I (t)dt = −
T1
(9)
The total profit of the season can be formulated as
TP(α, T ) = Sales Revenue (R) − Purchasing Cost (C p ) − Ordering Cost (Co )
− Inventory Holding Cost (C h ) − Preservation Cost (Io )
– Sales revenue: The total revenue consists of the both, before the price discount is
applied and after the price discount is applied. The total revenue could be formulated as
R1 = ψ p 1−η T1 , 0 ≤ t ≤ T1
where R1 is revenue for 0 ≤ t ≤ T1 and
R2 = ψ(ξ p)1−η (T − T1 ), T1 ≤ t ≤ T
where R2 is revenue for T1 ≤ t ≤ T
– Purchasing cost: According to Eq. (7), we know the order quantity Q. Therefore,
the formulation of total purchasing cost is
c p ψ p −η 1 − eλ(α)T1 + c p D( p)(eλ(α)T1 − eλ(α)T )
Cp = −
T λ(α)
– Ordering Cost:
Co =
co
T
– Inventory Holding cost: The formulation of the total inventory holding cost per
unit time is
T1
T
ch
I (t)dt +
I (t)dt
Ch =
T
T1
0
398
A. Namdeo and U. K. Khedlekar
Ch =
ch
Q(e−λ(α)T1 − 1) ψ(e−λ(α)T1 + λ(α)T1 − 1) p −η
−
−
T
λ(α)
λ(α)2
Q
ch
(e−λ(α)T + e−λ(α)T1 )
−
T λ(α)
ψ p −η −λ(α)T
−λ(α)T1
−λ(α)(T −T1 )
−
e
−
e
−
e
+
1
λ(α)2
ch D( p) −λ(α)(T −T1 )
e
+
λ(α)(T
−
T
)
−
1
−
1
T λ(α)2
– Preservation cost: The formulation of the total inventory preservation cost per
unit time is
T1
T
λ(α)c p
I (t)dt +
I (t)dt
Io =
T
0
T1
λ(α)c p
Q(e−λ(α)T1 − 1) ψ(e−λ(α)T1 + λ(α)T1 − 1) p −η
−
−
Io =
T
λ(α)
λ(α)2
λ(α)c p
Q
−
(e−λ(α)T + e−λ(α)T1 )
T
λ(α)
ψ p −η −λ(α)T
−λ(α)T1
−λ(α)(T −T1 )
−
e
−e
−e
+1
λ(α)2
λ(α)c p D( p) −λ(α)(T −T1 )
e
+ λ(α)(T − T1 ) − 1
−
T
λ(α)2
After taking T1 = r T , the total profit would have the form
co
T P(α, T ) = ψ p 1−η r + ψ(ξ p)1−η (1 − r ) −
T
ch + λ(α)c p −η
ψ p (1 − eλ(α)r T )
−
T λ(α)2
+D( p)(eλ(α)r T − eλ(α)T ) (e−λ(α)T − 1)
ch + λ(α)c p
ψ p −η e−λ(α)T − e−λ(α)r T
+
2
T λ(α)
−η 1 − eλ(α)r T + D( p)(eλ(α)r T − eλ(α)T )
cp ψ p
−
T λ(α)
ch + λ(α)c p
D( p) e−λ(α)(1−r )T + λ(α)(1 − r )T − 1
+
2
T λ(α)
ch + λ(α)c p
ψ p −η e−λ(α)r T + λ(α)r T − 1
+
2
T λ(α)
Replenishment Policy for Deteriorating Items Under Price Discount
399
Proposition 1 The total profit function TP(α, T ) is concave in T .
Proof Differentiate the total profit with respect of T and again double differentiate.
To obtain the value of T , equate the first derivative to zero. Now, we check the
optimality of total profit. To maximize the total profit, the second derivative must
be less than zero. By algebraically, It is very difficult to prove that the total profit is
concave function of T , for all positive parameters and r, ξ, λ(α) all are between zero
and one. So, we proved the concavity graphically. For this, all the parameters are
same as in Example 1 except that r = 0.5 and ξ = 0.7. Figure 2, indicates that the
second derivative have negative value between the feasible area T = 0.1 to 0.5. Hence,
the total profit is a concave function of the replenishment time T , i.e., 0.1 ≤ T ≤ 0.5
would maximize the total profit.
Proposition 2 The total profit function TP(α, T ) is concave in α.
Proof Differentiate the total profit in respect of α and again double differentiate. To
TP
obtain the value of α, equate ∂∂α
to zero. The preservation cost maximizes the total
profit if the second derivative less than zero. By algebraically, it is very difficult to
prove that the total profit is concave function of α, for all positive parameters and
r, ξ all are between zero and one. So, we proved the concavity graphically. For this,
all the parameters are the same as in Example 1 except that r = 0.5 and ξ = 0.7.
Figure 3, indicates that the second derivative has negative value for all value of α.
Hence, the total profit is a concave function of preservation cost α, i.e., α would
maximize the total profit.
Fig. 2 The graphical representation of d (T P) with respect to T
400
A. Namdeo and U. K. Khedlekar
Fig. 3 The graphical representation of d (T P) with respect to α
4 Numerical Example and Sensitivity Analysis
Example 1 In this paper, the preservation cost is defined by λ(α) = λ0 e−μα , μ >
0, λ0 =0.4, μ =0.9, and α = 1.3. The parametric value of the inventory system are
as follows: ch = 0.05, c p = 0.01, co = 100, ψ = 100000, η = 1.8. The value of r
is varying from 0.5 to 0.9 and ξ is also varying from 0.5 to 0.9. The computational
results are illustrated in Table 1.
Example 2 In this example, the parameters are the same as that in Example 1, except
for discount rate ξ . The value of r is varying from 0.5 to 0.9. For the given value of
ξ (= 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3), we find the corresponding optimal
value T, Q, and TP respectively. The computational results are shown in Table 2.
We observe from Table 2, that the increasing price discount is decreasing the total
profit. Also, the increasing price discount alleviates both the cycle time and order
Table 1 Computational results
ξ
T
Q
r = 0.5 0.5
0.7
0.9
r = 0.7 0.5
0.7
0.9
r = 0.9 0.5
0.7
0.9
0.5220
0.4961
0.4861
0.4946
0.4855
0.4818
0.4602
0.4682
0.4729
183
120
95
117
97
89
56
74
83
TP
S Rev
OC
PC
HC
Pre C
7615
6809
6332
6644
6375
6216
5672
5941
6101
8044
7233
6755
7069
6798
6639
6093
6364
6523
192
202
206
202
206
208
217
214
211
3.50
2.41
1.95
2.34
2.00
1.85
1.23
1.59
1.75
4.69
2.75
1.96
2.80
2.10
1.80
0.55
1.27
1.60
0.11
0.06
0.04
0.06
0.05
0.04
0.01
0.03
0.04
Replenishment Policy for Deteriorating Items Under Price Discount
Table 2 Sensitive analysis with respect to ξ
ξ
r = 0.5
r = 0.7
T
Q
TP
T
Q
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
0.5536
0.5216
0.5056
0.4961
0.4901
0.4861
0.4834
0.4815
0.4802
0.4793
260
183
143
120
105
94
87
81
77
74
8281
7615
7152
7809
6543
6332
6159
6014
5891
5786
0.5030
0.4946
0.4891
0.4855
0.4830
0.4813
0.4801
0.4793
0.4787
0.4783
139
117
105
97
92
89
86
84
83
82
401
TP
r = 0.9
T
Q
TP
6867
6644
6490
6375
6287
6216
6159
6110
6070
6034
0.4543
0.4602
0.4647
0.4682
0.4708
0.4727
0.4740
0.4750
0.4758
0.4763
5449
5672
5827
5942
6030
6101
6159
6207
6248
6283
38
56
67
74
79
83
85
87
89
90
quantity. Therefore, there could exist shortage due to large discount.Also, it is clear
that if we provide price discount later this will lead us to a loss. So, we need to
provide less price discount early in the cycle.
Example 3 Similarly, in this example, the parameters are the same as that in Example 1, except for preservation cost α. The value of r is varying from 0.5 to 0.9. For
the given value of α (= 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3), we find the
corresponding optimal value T, Q, and TP respectively. The computational results
are shown in Table 3.
Table 3 shows that successive investment on preservation technology provides
an increasing profit. Therefore, if we invest more in preservation then we have got
Table 3 Sensitive analysis with respect to α
α
r = 0.5
r = 0.7
T
Q
TP
T
Q
0.3
0.7
1.4
1.6
1.8
2.0
2.3
2.5
3.0
3.5
0.5199
0.5208
0.5218
0.5219
0.5220
0.5221
0.5222
0.5223
0.5225
0.5226
191
187
182
181
180
180
179
179
178
177
7615.13
7615.31
7615.50
7615.54
7615.57
7615.60
7615.63
7615.64
7615.67
7615.68
0.4942
0.4944
0.4946
0.4949
0.4953
0.4952
0.4951
0.4950
0.4949
0.4948
122
119
117
116
116
115
115
115
114
114
TP
r = 0.9
T
Q
TP
6644.20
6644.35
6644.50
6644.51
6644.52
6644.55
6644.57
6644.59
6644.62
6644.64
0.4605
0.4603
0.4602
0.4601
0.46006
0.46005
0.46004
0.46003
0.46002
0.46001
5672.42
5672.43
5672.45
5672.45
5672.45
5672.45
5672.45
5672.46
5672.46
5672.46
57
57
56
56
56
56
56
56
56
56
402
A. Namdeo and U. K. Khedlekar
Fig. 4 The graphical representation of profit TP in respect to price discount ξ
more profit. Also, Figs. 4 indicate that the profit will be more if discount offering
time occurs earlier. So, preservation technology investment beneficiary to inventory
management and this should apply from the beginning of the Enterprise. Also, more
investment in preservation technology slightly expand the replenishment time and
reducing the ordering quantity. This means that the items could keep longer time and
we have not to order extra quantity.
5 Conclusion
The optimal replenishment time and optimal ordering quantity have been derived and
two examples are provided to illustrate the model. The outcome reasserts to maximize
the total profit we have to apply the discount as early as possible. Also, the successive
investment in preservation technology is beneficial to inventory management. Also,
there exists an optimal time of replenishment cycle and an optimal cost for investing
in preservation technology. It is advised to retailer to keep less price discount and
the discount offering time should according to market need.
In future research time-dependent demand rate, variable holding cost, different
types of deteriorating function can be used to extend the model. Also, one can formulate the model in fuzzy enlivenment.
Replenishment Policy for Deteriorating Items Under Price Discount
403
References
1. Harris, F.W.: Operations and Cost. A.W. Shaw Company, Chicago (1915)
2. Whitin, T.M.: The Theory of Inventory Management, 2nd edn. Princeton University Press,
Princeton (1957)
3. Widyadana, G.A., Wee, H.M.: A replenishment policy for item with price dependent demand
and deteriorating under markdown policy. Jurnal Teknik Industri 9(2), 75–84 (2007)
4. Begum, R., Sahoo, R.R., Sahu, S.K., Mishra, M.: An EOQ model for varying items with Weibull
distribution deterioration and price dependent demand. J. Sci. Res. 2(1), 24–36 (2010)
5. Jagadeeswari, J., Chenniappan, P.K.: An order level inventory model for deteriorating items
with time quadratic demand and partial backlogging. J. Bus. Manage. Sci. 2(3), 79–82 (2014)
6. Khedlekar, U.K., Shukla, D., Namdeo, A.: Pricing policy for declining demand using item
preservation technology. SpringerPlus 5(1957), 1–11 (2016)
7. Khedlekar, U.K., Namdeo, A., Nigwal, A.: Production inventory model with disruption considering shortage and time proportional demand. Yugoslav J. Oper. Res. 27 (2017). https://doi.
org/10.2298/YJOR1611
8. Khedlekar, U.K., Namdeo, A., Chandel, R.P.S.: Pricing strategy with exponential declining
demand using preservation cost. Int. J. Oper. Res. 14(1), 1–10 (2017)
9. Khedlekar, U.K., Namdeo, A.: An inventory model with stock and price dependent demand.
Bull. Allahabad Math. Soc. 30(2), 253–267 (2015)
10. Mishra, U., Cardenas-Barron, L.E., Tiwari, S., Shaik, A.K., Garza, G.T.: An inventory model
under price and stock dependent demand for controllable deterioration rate with shortages and
preservation technology investment. Ann. Oper. Res. (2017). https://doi.org/10.1007/s10479017-2419-1
Performance Emission Characterization
of a LPG-Diesel Dual Fuel Operation:
A Gene Expression Programming
Approach
Amitav Chakraborty, Sumit Roy and Rahul Banerjee
Abstract The envisaged work attempts to explore the inherent capability of LPG
as a potent alternative fuel, in diesel dual fuel paradigms in order to address the
omni-present BTE-NOx-SOOT trade-off perspectives of an existing diesel engine.
Furthermore, considering the prohibitive costs of computational time of present day
3D CFD platforms in multi-objective calibration challenges in I.C. engine domains,
a unique gene expression programming (GEP) model has been proposed, to act as a
robust and computationally rational system identification tool (SIT) in the LPG-diesel
dual fuel platform. For the developed model, load, LPG energy share and injection
duration were the chosen input variables, whereas BSFCEQ , BTE, NOx, SOOT, and
HC were the corresponding output responses. Subsequent to GEP modeling, it was
revealed that developed GEP model was competent enough to map the experimental
engine output parameters with higher and commendable ranges of accuracy. The
obtained results of coefficient of correlation were in the ranges of 0.99262–0.99769,
while the error metrics of mean absolute percentage error values were in the ranges
of 1.03–3.08% and very low values root mean square errors, respectively.
Keywords Gene expression programming · Dual fuel · LPG · Diesel
Performance emission
1 Introduction
Since the commencement, diesel engine technology has undergone an archetype
swing in its belvederes to meet the desired directives of the increasingly stringent
A. Chakraborty (B) · R. Banerjee
Department of Mechanical Engineering, NIT Agartala, Tripura 799046, India
e-mail: amitavchakraborty.me@gmail.com
R. Banerjee
e-mail: iamrahul.ju@gmail.com
S. Roy
Department of Mechanical Engineering, BML Munjal University, Gurgoan, India
e-mail: samroy4u@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_39
405
406
A. Chakraborty et al.
emission dictates on one hand and consumer expectations on the other. In contrast to
the increasing energy insecurity, depletion of fossil reserves and air pollution which
collectively tends to destabilize the future energy sustainability issue, alternative
fuels plays a prolific role in extenuating these alarming challenges [15]. Among all
the alternative fuels LPG due to its inherent exploits, higher future energy security
index and innate capability of plummeting emission footprints hallmarks itself as a
promising counterpart [5].
Various in-depth research studies have been concentrated for the development
of an appropriate computationally cost-effective method to act as a robust system
identification tool, which in turn unfolded the vast scope of AI-based meta-modeling
techniques in the contemporary I.C. engine domain. But as ANN is mainly based
on black box methodology of computation, the evolved models always lacked the
closed analytical association between the desired input and output responses. Gene
expression programming (GEP) on the other hand has the capability to overcome this
limitation and to be hallmarked as an appropriate SIT to bridge the gap in contemporary meta-modeling era. According to [7], GEP provides an advantageous closed
form of analytical expression which provides an opportunity for explicit parametric
evaluation and analysis. Due to the inherent search structure characteristics of GEP,
it is independent of the choice of topology and other iteration algorithms as faced
by ANN. Furthermore, the output modeled responses are not being camouflaged by
complex weight matrices as like ANN, rather they are explicit analytical functions
of simpler mathematical operators.
1.1 Motivation of the Present Study
In line with the deliberations set forth in sec (1) on the proliferation of AI metamodeling paradigms in I.C. engine domains, the present study envisages an endeavor
to encapsulate the performance-emission trade-off characteristics of an existing
diesel engine made to operate in LPG dual fuel mode through a unique GEP metamodel. The perusal for such a study can be perceived to be motivated by the increasing
need of computationally cost-effective offline meta-model based calibration necessities for addressing the ever constrictive emission-performance trade-off legislative
requirements of the modern day.
2 Experimental Setup and Methodology
2.1 Experimental Setup
The present experiment was conducted on a single cylinder, water cooled, vertical,
direct injection existing diesel engine (Kirloskarmake) bench which is coupled to
Performance Emission Characterization of a LPG-Diesel …
407
an air cooled, eddy current dynamometer (Powermag make). And all the required
details of the setup and methodology are similar to works of same authors [5].
Experimental Results:
On preliminary observation of the performance-emission parameters, it was revealed
that in dual fuel mode of operation, an increased injection duration resulted in higher
values of BTE, compared to base line diesel operation. Such gains in the performance
characteristics were simultaneously accompanied by commendable reductions in the
NOx-SOOT footprint. The LPG was inducted into the inlet manifold under timed
manifold injection (TMI) sequence, the TMI has been projected on to the default
engine valve timing diagram as depicted in [4] for a ready reference. The LPG
induction duration was successively increased at each load stepping in increments
of 25%. The LPG injection duration was continued till 15,000 µS, beyond which
engine stability was compromised due to considerable misfire in the engine. The
increasing injection duration manifested a proportional increase in the LPG energy
share at a given load step of operation. Thus, the LPG-diesel dual fuel operation sets
forth a motivation to peruse a true multi-objective calibration problem to improve
upon the inherent BTE-NOx-SOOT tradeoff footprint of base line diesel operation
[5].
3 Evaluation of GEP as System Identification Tool
in Engineering Paradigms
GEP may be defined as a kind of genetic algorithm more or less like GA and GP
which uses population of individuals, next the individuals are selected according to
their fitness value and further presents genetic variation by the use of various genetic
operators. The vital difference among GEP, GA, and GP resides in the characteristics of the entities such as in GAs, the entities are of pre-determined static length
linear strings (chromosome); whereas in GP, the individuals are of different size and
shape (parse trees) of nonlinear entities, as far GEP is concerned first of all there is
assumption of encoding of individuals into linear strings of unaltered length (genome
or chromosome), followed by conversion to nonlinear entities of various shape and
size. Some of the main advantages of GEP compared to GA and GP is such that
the chromosomes are of uncomplicated entity: linear, small, compacted and easier
to manipulate genetically. Also the expression trees (ET) are purely the expressions
of their chromosomes and on these chromosomes the process of selection functions
and based on their fitness value they are nominated for reproduction.
One of the utmost advantage of using GEP in contrast to other data-oriented techniques like ANN, ANFIS, etc., is the innate capability of GEP in expressing explicit
formulations of the relationships which in turn dictates the physical phenomenon.
408
A. Chakraborty et al.
4 Evaluation of the Developed GEP Model
For the present study, the basic arithmetic operators (+, −, *, /) and elementary
mathematical functions (Pow, Sqrt, Cube root, Exp, Log, 1/x, x2, x3) have been
extensively used to develop the desired GEP model. The main objective of the present
envisaged study is to develop an explicit formulation of BSFCEQ , NOx, HC and SOOT
as a function in terms of load, LPG share % and injection duration [15]. Further, an
explicit formulation of the output parameters in terms of experimental factors can be
given by equation.
BSFCEQ , NOx, HC and SOOT f (Load, LPG Share%, Injection duration) (1)
The GEP developed explicit equations for BSFCEQ , NOx, SOOT, HC, and CO
is given by Eqs. (2)–(5), in the given equations the input term LES and INJ.DUR
signifies LPG energy share % and injection duration respectively.
LES
BSFCEQ (e)(((LOAD) )∗(INJ.DUR))−(INJ.DUR)
+ ((LOAD − LES) ∗ LES) ÷ ((LES) + (−2.512))2 − (LOAD)
+ LOAD − 2 (LOAD) ∗ ((3.720) − (INJ.DUR))
−((INJ.DUR) − (−1.488))]
(2)
3
3
NOx (LES) ∗ (LES + (−0.586)) ∗ (INJ.DUR) − (1.632 − 0.4258)
+ (((LOAD) ∗ 5.754) ∗ (LES − LOAD)) + (LOAD)2
+ (LOAD) − (LOAD)LES ∗ ((LOAD ÷ 1.485) ∗ (LES) − (LOAD))
(3)
HC [(1 ÷ (((12.574) ∗ (−3.981)) − ((−42.804) ∗ (INJ.DUR))))
+ (e)(12.574∗LOAD) + 1 − (LES)(1.571) ∗ (LOAD ∗ LES)
(L E S)
2
+ (−4.750)
− (1.005)(−13.235) + (e)(1.571)
+((INJ.DUR) ∗ LES)]
(e)(LOAD)(INJ.DUR)
+ (LES − 0.591)
SOOT (e)(LOAD)
3
(0.210)
(10.098)
LES
− (LES)
+
(−7.243 − LOAD)
+
2
(5.175 ∗ 5.175)(LOAD)
(LOAD∗(−0.423)∗(LOAD))2
(4)
(5)
Performance Emission Characterization of a LPG-Diesel …
409
Similarly, the expression trees (ET) for all the output parameters are represented
in Figs. 1, 2, 3 and 4 the symbols in expression trees d0, d1, and d2 denote input
parameters of injection duration, load and LPG energy share % respectively.
Fig. 1 Expression tree for BSFCEQ
Fig. 2 Expression tree for NOx
410
A. Chakraborty et al.
Fig. 3 Expression tree for SOOT
Fig. 4 Expression tree for HC
4.1 Evaluation of Statistical Error Metrics
For evaluation of the capability of predicting performance of the developed GEP
model in the present envisaged study, coefficient of correlation (R) and coefficient
of variation (R2 ) have been considered given by equations later on, which is in
Performance Emission Characterization of a LPG-Diesel …
411
accordance with metrics considered in similar works [1, 8–13, 16, 18] Root Mean
Square Error (RMSE), Mean Squared Error (MSE) and Mean Absolute Percentage
Error (MAPE). The mentioned error measures are given by the following equations:
n
2
i1 (ti − oi )
(6)
R2 1 −
n
2
i1 (oi )
n
1 RMSE (7)
(Ti − Oi )2
n i1
n
(Ti − Oi )2
M S E i1
(8)
n
n 1 (Ti − Oi ) (9)
M AP E X 100
n i1 Ti
5 Results and Discussion
From the present envisaged study, a GEP model has been established for forecast of
the engine performance and emission parameters of BSFCEQ , NOx, SOOT, and HC.
For predicting the performance and emission metrics, the input parameters chosen
were load, LPG injection duration, and LPG energy share. It was further revealed
that the implementation of GEP model to predict the experimental engine output
parameters represented excellent statistical correlation metrics. The same fact is
highlighted by Figs. 8–12 as given in Annexure 1.
5.1 Performance Parameters
Performance parameter of (BSFCEQ ) is represented in Fig. 5 also the GEP predicted
data of (BSFCEQ ) when compared with experimentally measured data revealed a
coefficient of determination (R2 ) value of 0.992646, Root mean square (RMSE)
value of 0.02261 kg/kW-h, Mean absolute percentage error (MAPE) of 1.5662%.
Emission parameters:
Emission parameter of NOx is cited in Fig. 6 which represents the plotting of
predicted values against observed values with (R2 ) value of 0.99538, (RMSE) of
0.049420 ppm, (MAPE) of 2.06%. From Fig. 7 the characteristics of HC emission
can be revealed which yielded a (R2 ) value of 0.98529, (RMSE) of 0.04971 ppm,
(MAPE) value of 2.52%. Finally, Fig. 8 represents the emission profile of SOOT
with corresponding (R2 ) value of 0.98948, (RMSE) of 0.04892 Mg/m3 and MAPE
of 1.03%.
412
A. Chakraborty et al.
Fig. 5 BSFCEQ GEP predicted versus EXP
Fig. 6 NOx GEP predicted versus EXP
Fig. 7 HC GEP predicted HC versus EXP
6 Conclusion
The envisaged study tries to develop a first of a kind GEP model for characterizing the
performance and emission indices of a LPG-diesel dual fuel investigation. Further in
the study, through explicit closed form of analytical expression correlating the input
parameters of choice model performance and robustness were evaluated against the
Performance Emission Characterization of a LPG-Diesel …
413
Fig. 8 SOOT GEP predicted versus EXP
statistical metrics of coefficient of correlation (R) and mean absolute percentage error
(MAPE). On evaluation of the results of developed GEP model it is revealed that
the predicted results were in excellent and higher ranges of accuracy with the actual
observed experimental results, the corresponding coefficient of correlation (R) values
ranged from 0.99262 to 0.99769, while the error metrics of mean absolute percentage
error values were in the ranges of 1.03–3.08% and very low values root mean square
errors respectively. To this end, it can thus be concluded that the developed GEP model
possess the inherent capability to emulate the chosen engine responses with ranges
of commendable accuracy and robustness throughout the entire range of engine
operation. Thus, the developed GEP model can be hallmarked as a robust and accurate
system identification tool (SIT) in the LPG-diesel dual fuel operational paradigms
which is in accordance with several other AI techniques.
References
1. Abdi, H.: The Kendall Rank Correlation Coefficient Encyclopedia of Measurement and Statistics, pp. 508–510. Sage, Thousand Oaks, CA (2007)
2. Armstrong, J.S., Collopy, F.: Error measures for generalizing about forecasting methods: Empirical comparisons. Int. J. Forecast. 8, 69–80 (1992)
3. Banerjee, R., Bose, P.: Development of a neuro genetic algorithm based virtual sensing platform
for the simultaneous prediction of NOx, opacity and BSFC in a diesel engine operated in dual
fuel mode with hydrogen under varying EGR conditions. SAE Int. J. Engines 5, 119–140
(2012). https://doi.org/10.4271/2011-01-2472
4. Banerjee, R., Debbarma, B., Roy, S., Chakraborti, P., Bose, P.K.: An experimental investigation
on the potential of hydrogen–biohol synergy in the performance-emission trade-off paradigm
of a diesel engine. Int. J. Hydrogen Energy 41, 3712–3739 (2016)
5. Chakraborty, A., Roy, S., Banerjee, R.: An experimental based ANN approach in mapping
performance-emission characteristics of a diesel engine operating in dual-fuel mode with LPG.
Journal of Natural Gas Science and Engineering 28, 15–30 (2016)
6. Çiftçi, O.N., Fadıloğlu, S., Göğüş, F., Güven, A.: Genetic programming approach to predict a
model acidolysis system. Eng. Appl. Artif. Intell. 22, 759–766 (2009). https://doi.org/10.101
6/j.engappai.2009.01.010
414
A. Chakraborty et al.
7. Ferreira, C.: Gene expression programming in problem solving. In: 6th Online World Conference on Soft Computing in Industrial Applications (invited tutorial) (2001)
8. Fisher, R.A.: On the probable error of a coefficient of correlation deduced from a small sample.
Metron 1, 3–32 (1921)
9. Gumbel, E.J.: Bivariate exponential distributions. J. Am. Stat. Assoc. 55, 698–707 (1960)
10. Hine, J., Wetherill, G.: Coefficient of correlation. In: A Programmed Text in Statistics Book 4:
Tests on Variance and Regression. Springer, pp. 32–41 (1975)
11. Najafi, G., Ghobadian, B., Tavakoli, T., Buttsworth, D., Yusaf, T., Faizollahnejad, M.: Performance and exhaust emissions of a gasoline engine with ethanol blended gasoline fuels using
artificial neural network. Appl. Energy 86, 630–639 (2009)
12. Nazari, A.: Prediction performance of PEM fuel cells by gene expression programming. Int. J.
Hydrogen Energy 37(24), 18972–18980 (2012)
13. Pearson, K.: Determination of the coefficient of correlation. Science 30, 23–25 (1909)
14. Poli, A.A., Cirillo, M.C.: On the use of the normalized mean square error in evaluating dispersion model performance. Atmos. Environ. Part A General Top. 27, 2427–2434 (1993)
15. Roy, S., Ghosh, A., Das, A.K., Banerjee, R.: A comparative study of GEP and an ANN strategy
to model engine performance and emission characteristics of a CRDI assisted single cylinder
diesel engine under CNG dual-fuel operation. J. Nat. Gas Sci. Eng. 21, 814–828 (2014). https://
doi.org/10.1016/j.jngse.2014.10.024
16. Sayin, C., Ertunc, H.M., Hosoz, M., Kilicaslan, I., Canakci, M.: Performance and exhaust
emissions of a gasoline engine using artificial neural network. Appl. Therm. Eng. 27, 46–54
(2007)
17. Theil, H.: Studies in Mathematical and Managerial Economics, vol. 6. North-Holland Publishing Company (1966)
18. Togun, N.K., Baysec, S.: Prediction of torque and specific fuel consumption of a gasoline
engine by using artificial neural networks. Appl. Energy 87, 349–355 (2010)
Comprehensive Survey of OLAP Models
Harkiran Kaur and Gursimran Kaur
Abstract On-Line Analytical Processing (OLAP) is an expertise that offers data
discovery, including capabilities of complex analytical calculations, infinite report
viewing and predictive “what if” scenario (forecast) planning. OLAP finds its applications in various Business Intelligence (BI) applications. OLAP’s life blood is multidimensional data. OLAP differs from traditional databases in a way data is stored
and conceptualized. As OLAP tools are intended to perform enquiry, these systems
deliver fast answers for queries that aggregate large amount of detail data to find general trends in Data Warehouse (DW). A number of OLAP models are available, that
show various contradictory trends in Data Warehouse. This paper summarizes the
existing OLAP models utilized for extensive list of applications and further compares
these models based on different parameters.
Keywords OLAP · Data cube model · Fusion model · Integral model
1 Introduction
1.1 Basic Knowledge About Data-ware House and OLAP
In past decades, we have been using various technologies to answer various simple or complex queries by users [1]. The prominent use of this database technology
is where decision-making is priority than everyday transactions. Transactional processing systems can only access few tuple for database reads and writes, which is
the major hitch in the present era. Whereas, decision-making systems compare the
past and present data instantly and generates inferences by utilizing this knowledge.
H. Kaur (B) · G. Kaur
Department of Computer Science and Engineering, Thapar Institute of Engineering
and Technology (Deemed to be University), Patiala, India
e-mail: harkiran.kaur@thapar.edu
G. Kaur
e-mail: gkaur.me16@thapar.edu
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_40
415
416
H. Kaur and G. Kaur
Another drawback of transactional system is that, it does not store any past data or
historical data. Therefore, to handle gigantic past and present data and to support
decision-making, many enterprises are using an unmitigated database technology
called Data Warehouse. Data Warehouse stores bulk of data that includes historical
and present data. These warehouses are used to analyze business trend for which
whole of the information (that enables better decision-making) is extracted by the
analysts. This type of interactive decision-making is provided by On-Line Analytical Processing applications. Also, it handles and answers all the real-time complex
queries. In OLAP, data is stored in dimensional form rather than relational forms.
Various dimensional forms can be—one dimensional, two dimensional, three dimensional, n dimensional and so on, where one dimensional represents a procession, two
dimensional represents oblong, a square, a triangle, a polygon, etc., and three dimensional represents a canister, a or b, a cube, a pyramid, a prism, etc.
1.2 Advantages of OLAP
Various advantages of OLAP systems include: multidimensional representation of
the database, for consistency of information, presents the “what-if” scrutiny, fast
information processing, supports interactive and ad hoc exploration and provides
a distinct platform for all the information and business needs that is budgeting,
forecasting, coverage, and scrutiny.
1.3 Outline
The paper is structured as follows. Section 2 describes the definitions and terms used
in OLAP technology. Section 3 summarizes the various OLAP models. Section 4
gives a comparative scrutiny of various models. Section 5 presents conclusion.
2 Definitions and Terms Used in OLAP
(i) OLAP: OLAP is defined as “information system that enables the user to query
the system, conduct an analysis and so on. The result of which is generated in
seconds. It is the most influential and thrilling business technology available
for real-time decision making”.
(ii) Data Warehouse: According to W. H. Inmon, “a DW is a subject-oriented,
integrated, time-variant, and nonvolatile collection of data in support of management’s decision making process”.
(iii) Slicing: This operation designates the selection of data using single dimension
of a cube [2].
Comprehensive Survey of OLAP Models
417
(iv) Dicing: This function describes about filtering the data using two or more
dimensions [2].
(v) Roll-Up/Drill-Up: It is also known as aggregation where data is viewed at
higher level of hierarchy of a dimension [2].
(vi) Roll-Down/Drill-Down: It allows data navigation from higher level to lower
level data, which is, using this operation, data is viewed at lower level of
hierarchy of a dimension [2].
(vii) Granularity: It refers to the level of detail of the data stored in fact tables in
DW. Granularity can be lofty for minute level transactional data or it may be
stumpy for summarized data [3].
(viii) Dimension table: Dimension table consist of dimension attributes, which
describe the dimension elements to enhance information conception from the
available data. Dimension attributes are stagnant values containing textual
data or isolated numbers that work as text values [3].
(ix) STAR Schema: The STAR Schema is the simplest and most effective schema
for handling Data-Warehousing schema queries. It consists of large central
table known as fact table that contains no redundancies. Fact table refers a
number of facet tables using foreign key relationship with these tables [3].
(x) Snowflake Schema: Snowflake schema is an alternative of STAR Schema. The
centralized fact table is connected to multiple dimension tables. Furthermore,
in this schema, dimensions are present in a normalized form in multiple related
tables. This snow-flaking effect has its impact only on the dimension tables
and not on the fact table [4].
(xi) Pivot: This operation allows alternative representation of data, by rotating the
axis dimension wise [1].
3 Summarization of Various OLAP Models
Lee et al. in [5] proposed a data cube model (or Surv-Cube) for multidimensional
indexing and retrieval of the reconnaissance videos. The method proposed in this
paper provides multimedia warehouse for managing CCTV reconnaissance videos
from different locations in centralized manner and analyzing the videos according
to sequential view, measures and sites by the means of a data cube. Since, CCTV
video reconnaissance system has been developed for public and private safety; it
is used for monitoring interesting areas and recording huge number of videos for
prevention and investigation of criminal cases. Since it is very difficult to retrieve
important information from a huge dataset of reconnaissance videos, the author of
this paper proposed a data cube model. This model supports On-Line Analytical
Processing (OLAP) operations that further provides various functions to users such
as—(a) providing fine-grained retrieval of objects and events, (b) tracking the paths
of interesting objects and (c) summarizing the videos to the abstract level of time,
location and fascinating objects.
418
H. Kaur and G. Kaur
Boutkhoum et al. in [6] proposed a multidimensional model, which is an operational approach for integrating multi-criteria examination in OLAP systems [6]. Since
OLAP was used for archiving, organizing, scrutiny and multidimensional modeling,
it was restricted in consideration of multi-criteria and quality characteristic of decision system. So the main objective of this approach was to generate fusion sculpt
that is a combination of On-Line Analytical Processing (OLAP) and Multi-Criteria
Scrutiny (MCA), to expand the capabilities of OLAP. Multi-criteria scrutiny is an
inspection method to resolve decision problems according to multiple criterion where
relative judgment is done between various projects and diverse measures such that
final result generated is unique. The author in this paper also reported some of
the advantages of OLAP systems such as vigorous scrutiny of multidimensional
data, accessibility of data, its ability to present the data hierarchy and full facts of
calculations and speed of processing data. Further, this paper elaborates numerous
advantages of MCA approach including its ability to simplify complex situations, the
presence of qualitative feature during data scrutiny and utility of MCA as a trading
tool in discussions between users. As an outcome, the paper identified the fusion
model or a multidimensional model as the topmost qualitative approach and suggests it MCA as an appropriate tool to deal and handle the complication of decision
problems.
Han et al. in [7] addressed the technique of dimensional modeling based on Data
Warehouse, for creation of an OLAP-based farm products examination model. The
authors in this article suggested this model to assimilate characters of farm product
market in China and farm product examination as the subject of data scrutiny. This
paper presented the farm products examination multidimensional model (FPE MD).
This FPE MD model allows exploring farm product data at diverse dimension and
applying different operations onto them including examination of these products.
This new approach was mainly entitled for the safety of farm products.
Korobko et al. in [8] proposed an integral model which provides essential multidimensional conceptual view of information by developing methods of diverse data
integration without physical loading. Basic concept of integral OLAP model explains
that when data is assembled from diverse sources, either interior data or protracted
data, the cubes formed are large meagre cubes. Applying the method of Formal Conceptual Scrutiny, it helps to split meagre cube into compact cubes, such that, the set of
compact cubes can be ordered by partial ordering relation that is they form a lattice.
The authors proposed this method of constructing integral OLAP model of a domain
as a lattice of MD cube concepts based on integration of MD model and Formal
Concept Scrutiny. Integral OLAP model covers all possible analytical queries of the
domain. This paradigm does not deal with preprocessed consistent data but leans on
current snapshots of integrated data sources. Principally, this model can be used for
rapid assumption testing, support brilliant idea, and support of analytical surveys. It
Comprehensive Survey of OLAP Models
419
is close to the mode of human thinking and on other hand, it supports performance
improvement due to its simple structure, which allows the user to support adaptive
influence.
4 Comparative Scrutiny of Various Models
See Table 1.
Table 1 Comparative scrutiny of data cube model, fusion model, multidimensional model and
integral model
Parameters
Models
Data cube model Fusion model
Multidimensional Integral model
model
Algorithm used
Tracking
Ponderation
N.A.
Dimensional
algorithm
algorithm
algorithm
Schema
STAR Schema
Figure 1
Multidimensional Snowflake
STAR Schema
schema
Figure 2
N.A.
Relational
database schema
Figure 3
Structural view
Aggregation
Dimensions
N.A.
Yes
N.A.
N.A.
Time,
Location,
Event,
Object
Criteria,
Action,
Time
Time,
Product,
Location
Age,
City,
Name,
Address
Modules
Preprocessing,
data cube,
Scrutiny
OLAP analysis,
Multi-criteria
scrutiny,
Visualization
Design concept,
Decide
dimension and
facts, construct
logical model
Pivot, Roll-up,
Drill-down
Slice, Dice,
Roll-up,
Drill-down, Pivot
Creating catalog
of data sources,
Implementing
special ETL
procedures for
each resource to
integrate one into
Data Warehouse
N.A.
OLAP operations Slice, Dice,
Roll-up,
Drill-down
420
Fig. 1 SurvCube STAR Schema [5]
Fig. 2 Multidimensional STAR Schema [6]
H. Kaur and G. Kaur
Comprehensive Survey of OLAP Models
421
Fig. 3 Relational Database Schema [8]
5 Conclusion
In this paper, a comparative study on various models of On-Line Analytical Processing has been performed. It has been identified that, the OLAP operations summarize
large amounts of data in Data Warehouse. STAR Schema finds its usage in a variety
of applications as compared to snowflake schema and relational database schema.
Using one of the available schemas, various models can be built as discussed in
this paper. Snowflake schema finds its application in exploring farm product data at
different dimension due to the advantage of normalization. In addition, Relational
Database Schema finds its application in diverse data integration without physical
loading due to its ability to form a lattice of multidimensional cube based on integration of MD model.
But of all models, it has been concluded that Data Cube model and Fusion models
which use the capabilities of STAR Schema, have their usage in multidimensional
indexing, retrieval of the reconnaissance videos, vigorous scrutiny of multidimensional data, accessibility of data, its capability to present the data pecking order and
full details of calculations and speed of processing data.
Hence, in order to choose any one model, we need to identify our requirements.
Since, complexity is an issue, so preference is given to STAR Schema.
422
H. Kaur and G. Kaur
References
1. Acharya, S., Prasad, R.N.: Fundamentals of Business Analytics (2016)
2. Dhanasree, K., Shobabindu, C.: A survey on OLAP (2017)
3. Razat, M.S., Nayak, A.K.: A study on designing a Layered STAR Schema for data mining
optimization (2015)
4. Levene, M., Loizou, G.: Why is the snowflake schema a good Data Warehouse design? (2003)
5. Lee, H., Park, S., Yoo, J.-H.: A data cube model for surveillance video indexing and retrieval
(2013)
6. Boutkhoum, O., Hanine, M., Tikniouine, A., Agouti, T.: Integration approach of multicriteria
scrutiny to OLAP systems: multidimensional model (2013)
7. Han, M., Ju, C.: Research and application on OLAP-based farm products examination model
(2008)
8. Korobko, A., Nozhenkova, L.: Ordered multidimensional model construction of relational source
for integral OLAP-modeling (2016)
Energy Efficiency in Load Balancing
of Nodes Using Soft Computing
Approach in WBAN
Rakhee
and M. B. Srinivas
Abstract Wireless Body Area Network (WBAN) has become very prominent in
recent years for patient monitoring systems which offer the flexibility for mobility
of patients and the medical staff in indoor hospital environment. A large number of
patients monitoring at hospital, includes more data collection of various Body Area
Network Coordinator (BANC) at different data rates transmitting to base station for
further diagnosis. Traffic generated at base station due to large number of packets
transmitted by Medical Device Coordinators (MDC) makes the network congestion
and more energy consumption of the nodes present close to the base station. We
propose soft computing approach, i.e., Ant Colony based energy-efficient algorithm
using cluster based for monitoring the data packets while balancing the load at each
and every intermediate node by using Ant Colony probabilistic function for routing
the data from source to the base station. Our proposed algorithm helps to ensure
maximum network lifetime by using the modified cluster head rotation process during
route construction. In the current work, we implemented monitoring of the patients
using Ant colony based and done experimental result on OMNeT++ to prove the
proposed method can find better results than conventional methods.
Keywords WBAN · BANC · Ant Colony · OMNeT++
1 Introduction
Wireless Body Area Networks (WBANs) have evolved from Wireless Sensor Networks (WSN), where sensors are mounted on patients’ body and all the physicalrelated parameters are being measured by monitoring remotely either indoor or outdoor hospital environment. The aggregated data collected on patients is sent to base
Rakhee (B)
Department of Computers Science, VNRVJIET, Hyderabad, India
e-mail: asinrakhee@gmail.com
M. B. Srinivas
School of Engineering and Technology, BML Munjal University, Gurgaon, Haryana, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_41
423
424
Rakhee and M. B. Srinivas
station for further treatment. Network Lifetime, energy consumption, latency, and
throughput are important parameters of interest for WBAN. Ant Colony technique is
a meta-heuristic search algorithm for problem solving that takes the inspiration from
the behaviors of real ants. The basic idea of ant colony lies on communicating among
individuals based on pheromone deposits with other ants for route construction. It
has been well proven that, combinatorial optimization problem by Dorigo et al. [1] is
used for many applications including WSN. The nodes present in WBAN needs to use
the resources properly like limited memory and communication bandwidth for better
network efficiency. Hence there is a need for balancing the load at each and every
intermediate node participation for overall network efficiency. Hence energy-saving
mechanism is required by choosing soft computing technique for prolonged overall
network lifetime. Design of routing technique plays an important role in managing
the energy consumption in the WBAN.
Routing protocols need to be designed such a way that, they consume less energy
by route constructing with load balancing at each node with minimum number of hops
for prolong network lifetime. Cluster-based routing helps in achieving the desired
goal by data aggregation mechanism. In this mechanism, a cluster are formed with
equal number of nodes in it and elects a cluster head based on their energy levels
which is responsible for receiving and aggregating data from all other members
in the cluster group. Cluster mechanism achieves high data throughput, reduces
energy consumption, and increases network lifetime. During this process it reduces
the burden on the BANs while communicating. Hence, an energy-efficient routing
protocols are needed to assist the operation effectively in a better way while load
balancing at every node.
2 Motivation
WBANs have limited communication range, so it requires participation of intermediate nodes to transfer data from source to destination. Energy consumption and
balancing load on the nodes is the main concern when designing the routing protocols for prolonged network lifetime. Existing protocols of WBANs focus more on
conventional techniques which may not give desired results. In this work, we implemented cluster-based routing by using multi-hop construction for data aggregation
while focusing on energy and load at each and every intermediate node. In this work
as suggested Ant Colony technique [2] has been modified to include a cluster-based
approach by considering higher pheromone and energy residual by electing from
one level to another level which forms a tree-based protocol to enhance the performance, before its transmission takes place while balancing the load at every node.
The proposed protocol uses similar system model approach as discussed in ZEQoS
[3] and EPR routing protocols. The proposed work uses a minimum cost function
value from routing table to select the shortest path from source to destination. For
validating the performance of the proposed protocol, we have compared with traditional techniques like Anybody and Zkhan. In Anybody routing protocol, a group
Energy Efficiency in Load Balancing of Nodes …
425
of sensors are formed into a group on the patient’s body which may not give better
results when compared to our proposed algorithm where group of MDCs are grouped
into clusters.
3 Proposed Algorithm
3.1 Introduction
The proposed algorithm introduces a clusters formation with equal number of MDCs
within the network are formed and in each cluster MDC will become cluster head
within its clusters. The MDCs in each cluster send its information to the cluster head
MDC for further transmission to the next level cluster head. This way the clusters
forward only critical information to the destination rather sending continuous data to
the base station, i.e., NSC. During situations where critical data need to transfer, then
MDCs play a vital role in transmitting directly instead next cluster head. We propose
a modified probabilistic function to choose next cluster head level by level [4]. This
algorithm helps in improving energy conservation of the nodes and balancing the
load within the network by using minimum node participation to the base station.
3.2 Clustering Technique
In this proposed algorithm, clusters are formed with equal number of nodes, i.e.,
MDCs in it. In cluster-based routing protocol we use probabilistic function based
Ant Colony algorithm in choosing the next neighbor. It consists of two ant agents
called forward and backward ants. Forward ants help in route construction from
source node to the destination. During its route discovery, the ants generated at
the source node select the neighbor node based on its high pheromone and energy
values. They update the routing table while traversing from node to node regarding
the pheromone and energy values of each node. The next level cluster head is selected
based on the pheromone and energy level of the MDCs in each cluster.
3.3 Ant’s Structure
The data configuration of the ant’s structure used in its route discovery process is
defined below. It comprises the following fields in Table 1, where ant-ID is the ant’s
ID. ant-type gives information of the type of the ant in the route discovery process.
This field can be forward ant or a backward ant. ant-nodes is the nodes visited stack,
contains the IDs of nodes by which the ant passes. ant-hop count is the field where it
426
Rakhee and M. B. Srinivas
Table 1 Ant’s structure
Ant-ID
Ant-type
Ant-nodes
Ant-hop count
Ant-information
Table 2 Ant’s information structure
Energy residual
Queue delay
Packet loss
calculates the number of hops by which the ant passed from its source to intermediate
nodes till it reaches the destination. ant-info fields Table 2 involve information about
the route of the nodes.
3.4 Pheromone Structure
An ant pheromone table is a structure that stores pheromone trail information for
routing from node i to the destination via intermediate nodes j. The structure of
pheromone table is as shown in Table 3.
3.5 Route Discovery
When a BAN node needs to send data to the destination, it checks its routing table
to find an appropriate path for transmission. It checks out its pheromone table in
order to find any non-expired node information. If that information is expired if
the value associated to the time expiration field is inferior to the node clock. If all
the information in the pheromone table is expired, a new route phase is generated.
Number of forward ants is generated to send for route checking.
Researchers proposed many routing protocols on cluster based such as LEACH
[5], TEEN, HEED, and PEGASIS. LEACH routing protocol is very well known for
data aggregation where all clusters are self-organized. In each cluster, a cluster head
is elected to collect the data by the other sensor within that cluster. During the setup
phase, the ants generate a random number between 0 and 1 and it compares with the
threshold value T (n) as shown in Eq. 1. After the routing discovery process, data is
immediately sent to the destination.
Table 3 Pheromone structure
Neighbor
node
Energy
pheromone
value
Delay
pheromone
value
Packet loss
pheromone
value
Available
memory
Device type Time expire
Energy Efficiency in Load Balancing of Nodes …
T (n) ⎧
⎨
p
⎩ 1 − p r mod
1
p
427
∗
ph current E current
∗
ph initial
E initial
if n ∈ G
(1)
After MDC becomes a cluster head, it sends a Hello Packet to all other nodes to
join and send the data to it. Further, the MDC in one cluster choose next cluster head
based on the modified probabilistic function until the destination is reached. This
way it try to choose the cluster based on the highest pheromone and energy value.
Cluster Head Probability gives the probability of each node to be next cluster head
is as follows:
Di j ∗ α + Pi j (t) ∗ β ∗ τi
(2)
Cluster Head Probability (t) N
i0 Di j ∗ α + Pi j (t) ∗ β ∗ τi
Cluster Head Probability gives the probability of each node to be next cluster head
is as follows:
Pi j τi j
l∈U
α
∗ ni j
τi j
α
β
∗ ni j
β
(3)
3.6 Backward Ant’s
When a forward ant reaches the destination, i.e., NSC, the evaluation of the found
route is carried out. This information collected by forward ants compares with the
parameter values set by the application for each metric. For instance, the demand
routes with a packet loss value that is inferior to 1% and residual energy ratio superior
to 85%. The destination node evaluates this information and decides whether the route
is adequate.
3.7 Load Balancing Along the Route Construction
The main objective of this paper is to make sensor nodes network efficiently by
balancing the load among the sensor nodes. During the discovery of the route, this
algorithm helps in balancing the load at each intermediate node. In the proposed
algorithm, different factors help in calculating the weight of the node by using buffer
load, pheromone value, energy consumption, and hops at the base station. To mitigate
the occurrence of traffic congestion at the link or at the base station we calculate the
weight of each node in order to improve the packets delivery ratio and reliability
of the data sent. The load on a base station is a function of processing load ‘PLWi ’
428
Rakhee and M. B. Srinivas
and communication “CLWi ” due to sensors in the network is defined as shown in
equations below.
n
CLW i Ci j
(4)
i0
LW i f (PLW , CLW i )
(5)
Wi αi + β Bi + α Pcurrent + Hi
(6)
where Wi is the weight of each node i
•
•
•
•
•
E i denotes energy consumption of i node
Bi indicates buffer space of node i
Pcurrent signifies current pheromone value
H i represents the number of hops to the base station from node i
α, β are control parameters with the values lying between 0 and 1.
The below algorithm finds the load balancing at the intermediate nodes.
Algorithm I
Input: routing table of node i
Input: forward packet structure
Output: Update tables require to transmit data to the node
Output: Weight of each node pheromone value, buffer and residual energy
Packet p=buffer.next()
If an intermediate nodes is the neighbor node of source node and
If p.bufferspace<threshold value then
Forward ant is accepted from the source node else find an alternate path
If ant create a loop then discard it
Processing load on a base station is due to processing the data from all the sensors
(BANs and MDCs) in the network and energy consumption. Communication load,
“CLWi ” of a base station NSC is calculated to be the summation cost of MDCs in the
routing as shown in Eq. 1.
We set a threshold value to alternately select nodes with most ability to assist the
traffic of the network. This way it proposes a weight calculation at the neighboring
node information. The calculation of weight value of each node can be obtained by
Eq. 3.
The proposed algorithm tries to find the congestion of the traffic along the path.
If congestion occurs there is possibility of the loss of the packets. Hence during the
route discovery phase, whenever the source sends a packet to network, it checks it
buffer size, pheromone value, residual energy, and hop count in the network.
Energy Efficiency in Load Balancing of Nodes …
429
4 Control Flow of Proposed Algorithm
Step 1: Start
Step 2: Initialize the parameters, NSC broadcast the information to all the nodes
in the network
Step 3: BANs and MDCs calculates the distance from the NSC
Step 4: Generation of random number
Step 5: If it is less than T(n) then ants are generated at source node and select the
cluster head among MDCs based on high energy and pheromone value
Step 6: Broadcast the message regarding the cluster head election to all other
MDCs in the network
Step 7: MDCs choose the neighbor node based on the Ant colony technique based
on the probability of highest value
Step 8: It selects the shortest path in order to move from one cluster to another
cluster based on Ant colony technique.
Step 9: Store the nodes information regarding the visited nodes
Step 10: Calculate the load at each node based on the routes construction by the
forward and backward ants using the equations above.
Step 11: Choose the shortest path among the routes constructed whichever is the
minimum cost value.
Step 12: Choose an alternate path in the routing table if the route fails due to looping
or node failures or time expiry.
Step 13: Route maintenance is performed by updating the pheromones and energy
values of the nodes for every few seconds.
Step 14: End
5 Performance Evaluation
The OMNeT++ based simulator is used to perform the experiments of our proposed
algorithm using Ant colony for WBAN. The results of the simulation are shown in
the figure which shows the lifetime, throughput and energy of the different sets of
cluster heads in the WBAN. Here nearly 5% cluster heads of total network nodes
are more energy efficient and also throughput is good as compare with Anybody and
ZKhan (Tables 4 and 5).
6 Conclusions
Wireless Body Area Network is an emerging technology for next generation of
healthcare services. In this proposal we proposed a cluster-based energy-efficient
ANT Colony Optimization algorithm which is able to find an shortest way of choosing
430
Rakhee and M. B. Srinivas
Table 4 Comparison of various algorithms with proposed algorithm without clustering
Proposed algorithm
Zk-BAN
Anybody
Network lifetime (ms) 40
32
30
Energy consumption
(mJ)
180
350
400
Throughput
(Packets/ms)
33,000
24,000
29,000
Latency (ms)
0.7 ms
3 min
2 min
Table 5 Comparison of various algorithms with proposed algorithm with clustering
Proposed algorithm
Zk-BAN
Anybody
Network lifetime (ms) 35
30.033
25
Energy consumption
(mJ)
295
350
310
Throughput
(Packets/ms)
32,279
25,000
28,000
Latency (ms)
0.4 ms
1 min
0.9 ms
the next hop by clustering process using a modified Ant Colony modified probabilistic
function based upon the pheromones and residual energy in each node and also
load balancing the intermediate nodes while route construction so that there is no
congestion occurs at the base station. Hence, our proposed system monitors the
patient’s data continuously and sends to the base station via. Cluster Heads, i.e.,
through MDCs during emergency situations. We simulated our proposed technique
in OMNet++ simulator and have compared with existing system in terms of network
lifetime, energy and throughput and it has been observed that our proposed system
has better performance than conventional systems.
References
1. Dorigo, M., et al.: The Ant System: optimization by a colony of cooperating agents. IEEE Trans.
Syst. Man Cybern. Part B 26(1), 1–13 (1996) (Ding, W., Marchionini, G. 1997)
2. Ramesh Babu, B., et al.: Application of Hybrid ant colony optimization algorithm for solving
capacitated vehicle routing problem. Int. J. Comput. Sci. Inf. Technol. 3(2) (2012)
3. Khan, Z., et al.: QPRD: QoS-aware peering routing protocol for delay sensitive data in hospital
body area network communication. In: IEEE BWCCA-2012, pp. 178–185 (2012)
4. Chen, M., et al.: Energy-efficient differentiated directed diffusion (EDDD) in wireless sensor
networks. Comput. Commun. 29(2), 231–245
5. Xiangning, F., et al.: Improvement on LEACH protocol of Wireless Sensor Network. In: Sensor
Technologies and Applications. IEEE (2007)
Single Image Defogging Based on Local
Extrema and Relativity of Gaussian
R. Vignesh and Philomina Simon
Abstract Various atmospheric particles such as fog and haze alter the appearance
of a natural scene. Fog may afflict many real-life applications such as detecting target
objects, tracking, and visibility. The defogging method not only removes fog from
images but also causes an improvement in the increase the scene clarity, boost the
visual perception of the image, and preserve the structural features. In the proposed
work, an improved defogging method based on the local extrema and Relativity
of Gaussian is discussed. Here, we consider the model for atmospheric scattering
as the background for fog removal. The local extrema method is tailored in such
a way as to determine three pyramid levels to calculate atmospheric veil. Then, a
multi-scale detail enhancement with Relativity of Gaussian (RoG) is applied to the
restored results to produce the images with better appearance. Several experimental
analyses are performed on the proposed algorithm to prove that this method achieves
more color restoration and detail preservation which have a greater impact on scene
perception. This method also focuses on preserving the edges and structures.
Keywords Atmospheric scattering model · Local extrema · Atmospheric veil
Single image fog removal
1 Introduction
Reduced or bad visibility is one of the common issues that needs to be solved for most
of the outdoor vision applications. Fog or haze is a common natural phenomenon,
which is typically caused by suspended fragments within the air. They severely
degrade the image. So for obtaining proper visibility, fog removal has become a
necessity especially in real-time applications. Researchers should be able to develop
R. Vignesh (B) · P. Simon
Department of Computer Science, University of Kerala, Kariavattom, Thiruvananthapuram, India
e-mail: vigneshr144@gmail.com
P. Simon
e-mail: philomina.simon@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_42
431
432
R. Vignesh and P. Simon
efficient fog removal algorithms to be applied in real-time applications. There are
many issues to be tackled during the fog removal process such as the improving the
efficiency of defogging algorithms, automate the fog removal process, and proper
assessment measures for accurate fog estimation. The problem of distortion present
in the foggy image remains unresolved, especially handling with images with heavy
fog or haze. Another challenge is the nonavailability of efficient quality measures
for fog removal. In this paper, an improved single image fog removal method based
on estimating the three-level envelops in local extrema is computed followed by the
detail enhancement through Relativity of Gaussian is discussed. In this work, our
main aim is to find more details within an image and improve the visual visibility
over the method proposed by Hongyu Zhao et al. The proposed work achieves better
restoration for the color and details.
The paper is structured as follows. Section 2, describes the various defogging
techniques presented in the literature. In Sect. 3, we highlight the details of the proposed method. Section 4, deals with the various experimental quantitative analyses
and qualitative analyses. Section 5, brings an end to the paper after mentioning the
summary of the work.
2 Literature Review
Different approaches for fog removal include methods based on dark channel prior,
optical transmission method, method based on atmospheric scattering model, and
filter based methods. Generally, the image defogging techniques can be categorized
into single image defogging and multiple images defogging. Since multiple images
defogging are time-consuming, single image defogging method is simple and efficient one. In multiple image defoggings, several input images are captured under
different climatic conditions and they are processed to remove fog. The main downside of this technique is the obscurity in acquiring the numerous images. Here, we
have to acquire a large number of images for processing, thereby huge memory and
plenty of time for computation are required. Single image defogging takes only one
input image for defogging. Tan [1] proposed an optical model based method by
taking advantage of the increase in the amount of contrast present in the restored
image. This method achieves appealing results with improvement in contrast. Fattal
[2] proposed a work which assumes that scene reflectance is a constant vector in
local region and that the transmission is locally statistically uncorrelated. Zhao et al.
[3] proposed an atmospheric scattering model based defogging method based on
mathematical manipulation of the local extrema points. Local extrema is used for
image restoration, thereby estimating different levels of atmospheric veil and contrast
detail enhancement is performed at various scales with multi-scale tone manipulation
algorithm (Fig. 1).
Single Image Defogging Based …
433
Fig. 1 Block diagram of the proposed framework
3 Framework of the Proposed Method
In this work, we introduce an improved and enhanced method based on local extrema
and RoG to remove fog from a single image. Generally, local extrema is referred as
all the local maxima and local minima at the critical points in a function. The proposed method has four main modules: skylight estimation and color correction module, atmospheric veil computation module, image refinement through local extrema
module, and finally RoG-based detail enhancement module. This proposed method
performs accurate restoration of color and details of foggy image and also enhances
the visual perceptibility of image. Relativity of Gaussian-based detail enhancement
avoids the mild halo and noise that are sometimes visible in some of the existing
defogging algorithms.
3.1 Skylight Estimation and White Balance
The dark channel of input image [4] is given as follows and it is used to calculate the
skylight:
Idark (x, y) min(min
Ic (x, y))
(1)
(x,y)εϕ cε{R,G,B}
We choose dark channels 0.1% brightest pixels as ideal sky region and such pixels
are defined as I 0.1%. The initial skylight can be modeled by calculating the mean value
of I 0.1% in the foggy image, i.e.,
Amean mean(I0.1% )
(2)
The color of foggy image is influenced by variations in illumination. For example,
the variations in the climatic conditions such as foggy weather, sunny weather, and
cloudy weather may lead to the changes in the color of visual scene to partial color.
434
R. Vignesh and P. Simon
Fog should be considered as white. So in order to maintain that, a color correction
needs to be performed to make the skylight computation better. This condition can
be represented in the given equation.
A
Acmean
max ( Acmean )
(3)
cε R,G,B
where A is the perfect skylight computation result. Equation (3) initializes Amean to a
value of (1, 1, 1). The resultant image I obtained after the color correction module
which is white balanced can be represented as
I (x, y) I (x, y)
A
(4)
3.2 Computation of Atmospheric Veil
The calculation of atmospheric veil is a most crucial step to achieve restoration.
We need to consider two conditions for computing the atmospheric veil. The first
condition is that the value of atmospheric veil V (x, y) should lie between 0 and
1, and the next condition is that V (x, y) should not be higher than the minimum
intensity value obtained from the R, G, B channels of I (x, y). A rough computation
of the atmospheric veil is as follows: Ṽ
Ṽ (x, y) min (I (x, y))
cε{R,G,B}
(5)
where Ṽ is the matrix representation of atmospheric veil estimation matrix. Atmospheric veil coarse computation can be considered as a similar operation to the
minimum filtering, but in such filtering, blocking or halo artifacts can be observed in
images. So we need a better model and constraint estimation for further refinement.
3.3 Image Restoration by Local Extrema
Atmospheric veil is greatly influenced by the depth of objects rather than the scene
reflectance [5]. In this work, local extrema provides the base for edge-preserving
smoothing approach for defogging and it computes a better atmospheric veil. The
method local extrema is motivated from the work of Subr et al. [6], which uses edgeaware interpolation to compute envelopes. This method extracts detail information
at fine scales without considering the contrast. For the approximation of atmospheric
veil, a single mean layer is not sufficient. The atmospheric veil can be estimated by
executing three steps [3] in a non iterative manner.
Single Image Defogging Based …
435
Local extrema of Ṽ identification, inference of extremal envelopes, and RoG
detail enhancement.
In the initial step, the extrema Ṽ is identified and detected. We need to find whether
the pixel obtains a maxima or minima value based on some presumptions [3], when
local extrema is processed from the image details with a kernel of K × K size which
computes the wavelengths of at least K /2 pixels. In this work, we set K 5 as the
size of the extrema location kernel. After finding out the local minima and maxima
points, next step is to evaluate the minimal and maximal extremal envelopes. S is
considered as the set of pixels obtained in local extrema. Extremal envelope E is
computed using an interpolation technique for image colorization. The approach of
[7] is adopted to solve extremal envelope calculation. For that quadratic functional
is minimized using their weighted least squares formulation, which can be executed
at a faster rate. The equation for computing three-scale atmospheric veil is
Ri (x, y) I (x, y) − q Vi (x, y)
× A, where i 1, 2, 3
1 − Vi (x, y)
(6)
where the value of parameter q varies from 0 to 1, but it is set as 0.95 for tuning the fog
removal process. Here, the maximal envelope produces more information and details
more accurately in bright regions; the minimal envelope extracts information in the
dark regions and performs better in smoothening detail even with huge variations in
amplitude; the mean envelope is a combination of both the maximal and minimal
envelopes. We can observe that atmospheric veil is smooth at all regions other than
edges but it obtains higher intensity in dense haze area. We can infer from the above
discussion that minimal envelope is most apt envelope for restoring the atmospheric
veil, but the maximal envelope and mean layer can preserve details.
3.4 Detail Enhancement Using RoG
Edge-preserving smoothening is applied in various research areas in image processing. It divides an image into piecewise smooth base layer [8] and local volatile detail
layer which can be employed in detail enhancement applications. After computing the
restored results in three scales, we employ the RoG-based detail enhancement [8] to
improve the details within the image while highlighting the edges. Preserving image
smoothing comprises of applying a local filter and then performing an optimization
globally. Local filter may cause ringing effect on the edge. Global optimization concentrates on relatively suppressing the small variance. A local regularization called
Relativity of Gaussian (RoG) based on a local regularization technique is optimized
globally to highlight and recognize the edges irrespective of the scale. Inspired by
DoG, Relativity of Gaussian (RoG) is performed to selectively smooth the gradient [9]. In this work, we introduce an improved defogging method based on local
extrema and RoG to remove fog and highlight the edges. This method performs better
restoration of color and details of foggy image and also boost the visual perception
436
R. Vignesh and P. Simon
Fig. 2 Flow diagram of RoG
(a)
(b)
Fig. 3 a Input image. b Detail enhancement using RoG [8]
of the scene. The detail enhancement performed using RoG avoids the mild halos
and noise. RoG method should enhance more details in the image while preserving
the edges. Figure 2 demonstrates RoG. Figure 3 represents the detail enhancement
using RoG [8].
The algorithm for edge-preserving smoothing via RoG [8] is given as follows:
Input: Input image I, Scale parameter σ 1; 2,
Positive parameter λ and maximum iterations K.
Output: S, Smoothing result.
1:
2:
3:
4:
5:
Initialize S 0 ← I
For k 1 to K do
Compute weights wx; y from (7)
Update S k using equation from (8)
End for
These are the equations to be calculated to obtain the value of weights, w(x, y) and
Sk .
1
|(G σ 2 ∗ ∇x S)(G σ 2 ∗ ∇x S)|
2
2
k
2
S args minS − I 2 + λ(Wx ∇x S k−1 2 + w y ∇ y S k−1 2
Wx,y G σ 1/2 ∗
(7)
(8)
Single Image Defogging Based …
437
Table 1 Comparison of the performance of both methods using parameters e and r̄
Images
e values
r̄ values
Image 1
Hongyu Zhao
et al.
1.961
Proposed
2.372
Hongyu Zhao
et al.
1.471
Proposed
Image 2
5.560
6.112
2.233
3.128
Image 3
0.6920
0.7124
1.5745
2.204
Image 4
1.975
2.412
2.003
2.501
2.224
4 Experimental Results
In this defogging method, two evaluation metrics e and r̄ are used to evaluate the
performance of the proposed method. The concept of gradient rationing is employed
at edges to assess the visibility of the edges [10]. The parameter e represents the
ability of the method to restore the edges that are not observed in the input image.
The parameter r̄ expresses the average visibility effect enhanced by the restoration
method. We evaluate the proposed work on natural color images and synthetic images
to demonstrate our method can generate good results. Experimental analysis of the
values e and r̄ illustrates that the proposed work performs better compared to Hongyu
Zhao et al. in preserving detail and avoiding the halo effects. These values e, r̄ are
evaluated for Hongyu et al. [3] and our proposed algorithm and shown in Table 1.
From Table 1 (e values), we can deduce that the proposed work obtain more edges
than [3]. Table 1 (r̄ values) gives average visibility effect enhanced by the restoration
method. From the values, we can say that the proposed method achieves a better result.
Thus, we can conclude that the proposed work achieves more defogging results in
different environments while preserving image detail (Fig. 4).
From this table, we can infer that as the value of e increases, better restoration
results produced. We can also understand that the visibility of the image increased
based on the increase in r̄ values.
5 Conclusion
In this work, an improved defogging method based on local extrema and relativity of
Gaussian is proposed. After obtaining restored results, multi-scale detail enhancement with RoG edge-preserving smoothing approach is applied for detail enhancement. Image smoothing methods via RoG, effectively to remove textures while preserving other detail information of the image. RoG does not depend on any specific
definition of scale feature. This method can be effective for practical application in
real fog fields. Another advantage of this work is that the fog estimation is simple and
there is no requirement for the generation of depth map for processing this algorithm.
438
R. Vignesh and P. Simon
Fig. 4 a Input image. b Hongyu Zhao result. c Proposed work. d Edge map of Hongyu Zhao result.
e Edge map of proposed work
This method also undergoes limitation that it does not obtain better results in heavy
haze.
References
1. Tan, R.T.: Visibility in bad weather from a single image. In: Proceedings of the 2008 IEEE
Conference on Computer Vision and Pattern Recognition. Anchorage: IEEE Computer Society,
pp. 1–8. Anchorage (2008)
2. Fattal, R.: Single image dehazing. ACM Trans. Graph. (TOG) 27(3), 1–9 (2008)
3. Zhao, H., Xiao, C., Yu, J., Xu, X.: Single image fog removal based on local extrema. IEEE/CAA
J. Automaticasinica 2(2) (2015)
4. He, K.M., Sun, J., Tang, X.O.: Single image haze removal using dark channel prior. In: Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1956–963
(2009)
5. Nishino, K., Kratz, L., Lombardi, S.: Bayesian defogging. Int. J. Comput. Vision 98(3), 263–278
(2012)
6. Subr, K., Soler, C., Durand, F.: Edge-preserving multiscale image decomposition based on
local extrema. ACM Trans. Graph. 28(5) (Article No. 147) (2009)
7. Levin, A., Lischinski, D., Weiss, Y.: Colorization using optimization. ACM Trans. Graph. 23(3)
(2004)
8. Cai, B., Xing, X., Xu, X.: Edge/Structure preserving smoothing via relativity of gaussian. In:
IEEE International Conference on Image Processing (ICIP 2017), pp. 250–254 (2017)
9. Lowe, D.G.: Distinctive image features from scaleinvariant keypoints. Int. J. Comput. Vision
60(2), 91–110 (2004)
Single Image Defogging Based …
439
10. Hautiere, N., Tarel, J.P., Aubert, D., Dumont, E.: Blind contrast enhancement assessment by
gradient ratioing at visible edges. Image Anal. Stereol. 27(2), pp. 87–95 (2008)
Improved Edge-Preserving
Decomposition Based on Single Image
Dehazing
S. K. Anusuman and Philomina Simon
Abstract Haze is a common phenomenon often produced during bad weather that
obscure scenes, reduces visibility, and degrades colors. Haze removal poses a serious
challenge because of the difficulty to develop an exact mathematical model when a
single hazy image is given as the input. In this paper, we propose a single image haze
removal method to highlight the edges. A edge-aware constraint weighting scheme
based on first-order derivative from Gradient domain Guided Image filter is adopted
in this algorithm which helps to preserve the edges better. In this dehazing algorithm, simplified dark channel of the haze image can be separated as base layer and
detail layer by applying a filter, i.e., weighted guided image filter. The transmission
map is obtained from the base layer and thereby it is utilized to restore haze-free
image. Result analysis show that the proposed algorithm preserves and highlights
the edges without any color distortion. Experimental results are performed on natural haze images, aerial images, and under water images. In qualitative analysis, edge
map shows the better performance of the proposed method. Quantitative Analysis
have been carried out which demonstrates the better performance of the proposed
algorithm when compared with the existing method.
Keywords Haze removal · Weighted guided image filtering · Edge-aware
weighting function · Transmission map · Koschmieder’s law
1 Introduction
Haze is an atmospheric condition which is produced due to bad weather conditions
that hinders the visibility. Haze removal [1] has applications in outdoor surveillance,
computational photography, and aerial imaging. Outdoor scene images suffer from
S. K. Anusuman · P. Simon (B)
Department of Computer Science, University of Kerala, Kariavattom, Thiruvananthapuram, India
e-mail: philomina.simon@gmail.com
S. K. Anusuman
e-mail: skanusuman@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_43
441
442
S. K. Anusuman and P. Simon
bad weather conditions. During worst weather conditions, atmospheric particles also
got merged with reflected airlight from various directions [2]. This produces color
distortion and contrast of the captured object in the scene will be reduced. The images
thus obtained will be vague and suffers quality degradation. This is because atmospheric particles can absorb and dispersion of light. Generally, the absorption and
dispersion is modeled using direct attenuation and airlight. Two significant components present in the haze are direct light and attenuation. Haze removal can be defined
as the technique of removing haze from the input image to improve the visual appearance and contrast. Haze removal brings significant enhancement by improving the
visual scene both locally and globally. It also reduce the color degradation caused by
the airlight and generates information about the scene depth. Haze removal can be
performed using two approaches; using single hazy image and using multiple images
of single scene as the input. In this paper, we focus on the improving the edge details
in the dehazing method without using any prior information. Single image dehazing
is considered as challenging because of the difficulty in estimating the unknown
parameters in terms of distance.
Section 2 presents the different techniques used in single image haze removal algorithms. Section 3 introduces the proposed haze removal method. Section 4 presents
experimental analysis of the proposed method. Section 5 summarizes the paper.
2 Literature Review
The related works are organized as review of different guided image filtering techniques and different haze removal algorithms based on these filtering techniques. He
et al. [3] introduced the guided image filtering to overcome the problems in gradient
reversal artifacts. Guided image filter(GIF) [3] estimates the filtered output mage
by taking into account the structure and content of the guidance image. In weighted
guided image filtering (WGIF) proposed by Li et al. [4], an edge responsive weighting
is added with guided image filtering. Larger weights are given to the edge pixels than
the flat area pixels. In guidance image, local variance is computed in a 3 × 3 window.
Hence compute the edge-aware weighting. This weighting in WGIF helps to preserve
sharp edges and the halo artifacts can be reduced. Kou et al. [5] introduced Gradient
domain guided image filter which extends the weighted guided image filtering by
modifying the first-order edge-projected constraint. The new regularization term is
incorporated with an edge constraint term which is. In this filter, edges are preserved
much better. Tan [6] proposed a method is based on the view that day images which
are clear have more contrast than the hazy image and the airlight depends on the
distance computed between objects and the observer. Fattal [7] proposed a the work
which assume that scene reflectance is a constant vector in local region and that
the transmission is locally statistically uncorrelated. Wang et al. discussed about the
dehazing method based on the depth information. This method depends up on atmospheric scattering model where a dark channel prior is applied selected region of
Improved Edge-Preserving Decomposition Based …
443
interest (ROI) to estimate the atmospheric light. Zhao et al. [8] proposed defogging
method based on mathematical manipulation of the local extrema points.
3 Improved Edge-Preserving Decomposition Using
an Edge-Aware Weighting—Proposed Method
Improved Edge-preserving method is proposed to compute an efficient transmission
map thereby performing a better haze removal based on Koschmieder’s law [9]. The
algorithm will take into consideration the steps for efficiently computing minimal
color channel and simplified dark channel. Weighted guided image filter (WGIF) [4]
is used to segregate the simplified dark channel of the haze image into a base layer
and a detail layer. The block diagram of the proposed method is given in Fig. 1.
The edge-aware weighting function is taken as the first-order edge-aware constraint from Gradient domain guided image filtering [5]. This weighting function
helps in preserving the edges. This proposed method is inspired from Li et al. [10].
The major objective of this work is to adopt a better dehazing method with giving
emphasis to edge preservation and no color distortion occured in the result image.
The proposed method overcome the limitations of Li et al. where the edges are not
visually prominent when shown the results by analyzing the edge map.
3.1 Haze Modeling
Haze modeling is done by Koschmieder’s law [9] without the usage of any image
prior. The following equation represents the mathematical model for atmospheric
scattering where X c represents a haze image, where c ∈ {r, g, b} color channel
index, Z c denote haze-free image, Ac refer to the global atmospheric light, t is the
transmission medium where a part of non scattered light reach the camera. We have
to estimate Z c to remove the haze from X c . The parameters Ac and t are unknown.
According to this law, Haze is represented using the following equation.
X c ( p) Z c ( p)t( p) + Ac (1 − t( p))
Hazy
Image
Haze
modeling
Simplified
Dark Channel
Separation
Atmospheric
light
Estimation
Transmission
Map
Estimation
first-order edge-aware
constraint
Fig. 1 Block diagram of proposed method
Scene
Radiance
recovery
Haze
Free
image
444
S. K. Anusuman and P. Simon
Z c ( p)t( p) represents direct attenuation and scene radiance [10]. Ac (1 − t( p))
represents Airlight, where transmission map can modeled by the given equation
[10].
t( p) e−αd( p)
α denote scattering coefficient.
The scene radiance decreases exponentially with the depth of the scene d( p). The
contrast reduces exponentially with the scene depth.
3.2 Global Atmospheric Light Estimation
Generally, atmospheric light Ac can be is anticipated as the brightest color in a hazy
image which can be obtained using the quad-tree subdivision method. The value for
each region can be calculated based on the average pixel intensities and standard
deviation. The region with maximum value is chosen and that region is again divided
into four smaller blocks of rectangular size each. Process this algorithm until the
selected area becomes window size of 32 × 32. The final region to be chosen is done
by identifying the pixel which minimize the difference for each R, G, B channel and
thereby determining the global atmospheric light.
3.3 Decomposition of Simplified Dark Channel
The majority of the patches which are present in the non sky region, the minimum
intensity value in such a patch should obtain less value. We compute dark channel
for an image J by the following equation.
c dark
min J (y)
J (X ) min
c∈{r,g,b}
y∈Ω(X )
A new haze image model is derived by using the simplified dark channels of
the haze image X and the haze-free image Z. The minimal color components can
be estimated from [4]. The dark channel prior is introduced by He.et al. with the
assumption that For many haze-free outdoor images, the local regions which do not
contain the sky region with dark pixels contain low intensity at least in one of the RGB
color channels. Such intensity of these dark pixels are caused by the airlight. The
simplified dark channels of haze image X and haze-free image Z can be modeled as
in [10]. Dark channel of the haze image can be separated as a base layer and a detail
layer. The base layer is composed of the transmission map. To avoid introducing
artifacts to the dehazed image, the structure of the base layer should be made similar
to the structure of the haze image.
Improved Edge-Preserving Decomposition Based …
445
3.4 Transmission Map Estimation
In Gradient domain guided filtering (GDGIF), a weighting function which project
the edges can be obtained by solving a first-order edge-aware constraint. ΓˆG p . It
can be calculated by computing the local variances with 3 × 3 window size.
Edge projecting weighting function [5] can be computed as follows:
ΓˆG ( p ) N
1 χ ( p ) + ε
N p1 χ ( p) + ε
whereas (2ζ1 + 1)×(2ζ1 + 1) is the window
size 3 × 3 of the filter, the weighting p
ˆ
measures the importance of pixel ΓG p with respect to the whole guidance image.
With the introduction of edge weighting in GDGIF, a pixel is classified as edge
pixel only if the both of the two scale variances get a high value. [5]. but in the
weighting scheme of WGIF [4], less edges and details are detected.
This algorithm is based on the computation of minimal color channel and dark
channel [10] to provide a haze-free algorithm. The importance of the simplified dark
channel is to reduce the deviation of direct attenuation. Here, the weighted Guided
Image Filter (WGIF) in [4] is applied to decompose the simplified dark channel of
the haze image.
3.5 Scene Radiance Recovery
When the parameter values for global atmospheric light Ac and transmission map
t(p) is computed, the scene radiance Z(p) is restored from the given equation.
Z c ( p) 1
t ∗ ( p)
(X c ( p) − Ac ) + Ac
4 Results and Discussion
Matković et al. [12] proposed Global Contrast Factor (GCF) can perceive the contrast
as in human visual system. It manipulates contrast at different resolutions to compute
the total contrast in the image. It uses the contrast at the various resolution levels to
compute the overall contrast. This metric can be used to evaluate the effectiveness of
the proposed method to prove that the overall global contrast has improved. From the
446
S. K. Anusuman and P. Simon
Haze Image
Haze Free image using GDGIF
(Proposed Method)
Haze removal using WGIF
Edge Map
ꜛ
Ariel Image
Edge Map
Fig. 2 Qualitative results of the proposed method
Table 1 containing the GCF values, it can be inferred that the contrast is improved
when the new edge-aware weighting function from Gradient domain guided Image
filtering is used (Fig. 2; Table 1).
Improved Edge-Preserving Decomposition Based …
447
Fig. 2 (continued)
5 Conclusion
Haze images are produced in bad weather conditions that degrade the visibility of
the scene present in the image. In this paper, an improved edge-preserving single
dehazing is proposed based on simplified dark channel. In this work, a first-order
edge-aware constraint from Gradient domain Guided Image Filtering is introduced
as the edge weighting function in dehazing algorithm. Result analysis show that the
proposed algorithm preserves and highlights the edges without any color distortion
448
S. K. Anusuman and P. Simon
Table 1 Quantitative analysis of proposed method
Image
Original image
Dehazed image
(weight in WGIF)
Dehazed image
(weight in GGIF)
(proposed)
People
0.3851
0.7488
0.8845
Canon
Aerial
Road
Cones
Dolls
Under image 1
0.1849
0.1735
0.2278
0.1430
0.2172
0.2646
0.4457
0.3376
0.5966
0.4260
0.7538
0.6087
0.4566
0.3422
0.6372
0.4336
0.9065
0.6871
Under image 2
0.2068
0.3791
0.4016
than the existing method. Experimental results are performed on different cases of
hazy images. In qualitative analysis, edge map shows the better performance of the
proposed method.
References
1. Shwartz, S., Namer, E., Schechner, Y.Y.: Blind haze separation. In: Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2006, pp. 19841991
2. Narasimhan, S.G., Nayar, S.K.: Contrast restoration of weather degraded images. IEEE Trans.
Pattern Anal. Mach. Learn. 25(6), 713724 (2003)
3. He, K., Sun, J., Tang, X.: Guided image filtering. IEEE Trans. Pattern Anal. Mach. Intell. 35(6),
13971409 (2013)
4. Li, Z.G., Zheng, J.H., Zhu, Z.J., Yao, W., Wu, S.Q.: Weighted guided image filtering. IEEE
Trans. Image Process. 24(1), 120–129 (2015)
5. Kou, F., Chen, W., Wen, C., Li, Z.: Gradient domain guided image filtering. IEEE Trans. Image
Process. 24(11), 4528–4539 (2015)
6. Tan, R.T.: Visibility in bad weather from a single image. In: Proceedings of IEEE Conference
on Computer Vision and Pattern Recognition (CVPR), Anchorage, AK, USA, June 2008, pp. 18
7. Fattal, R.: Single image dehazing. In: Proceedings of SIGGRAPH, New York, NY, USA, June
2008, pp. 19
8. Zhao, H., Xiao, C., Yu, J., Xu, X.: Single image fog removal based on local extrema. IEEE/CAA
J. Autom. Sin. 2(2), 158–165 (2015)
9. Koschmieder, H.: Theorie der horizontalen sichtw eite. In: Proceedings of Beiträge zur Physik
der freien Atmosphäre (1924)
10. Li, Z., Zheng, J.: Edge-preserving decomposition-based single image haze removal. IEEE
Trans. Image Process. 24(12), 5432–5441 (2015)
11. He, K., Sun, J., Tang, X.: Single image haze removal using dark channel prior. IEEE Trans.
Pattern Anal. Mach. Intell. 33(12), 23412353 (2011)
12. Matković, K., Neumann, L., Neumann, A., Psik, T., Purgathofer, W.: Global contrast factor—a
new approach to image contrast. In: Proceedings of the First Eurographics conference on Computational Aesthetics in Graphics, Visualization and Imaging (Computational Aesthetics’05),
Switzerland, pp. 159–167 (2005)
Global and Local Neighborhood Based
Particle Swarm Optimization
Shakti Chourasia, Harish Sharma, Manoj Singh
and Jagdish Chand Bansal
Abstract The particle swarm optimization (PSO) is one of the popular and simple
to implement swarm intelligence based algorithms. To some extent, PSO dominates
other optimization algorithms but prematurely converging to local optima and stagnation in later generations are some pitfalls. The reason for these problems is the
unbalancing of the diversification and convergence abilities of the population during
the solution search process. In this paper, a novel position update process is developed
and incorporated in PSO by adopting the concept of the neighborhood topologies
for each particle. Statistical analysis over 15 complex benchmark functions shows
that performance of propounded PSO version is much better than standard PSO
(PSO 2011) algorithm while maintaining the cost-effectiveness in terms of function
evaluations.
Keywords Swarm intelligence based algorithm · Nature-inspired algorithm
Neighborhood topology · Optimization
1 Introduction
Kennedy and Eberhart in 1995 [4, 9] examined that the swarm intelligence is showed
by the flocking of birds and schooling of fishes, inspiring from which an optimization
technique was introduced by them which was called the particle swarm optimization
S. Chourasia · M. Singh
Gurukul Institute of Engineering & Technology, Kota, India
e-mail: shakti.engg85@gmail.com
M. Singh
e-mail: manojsinghq100@yahoo.com
H. Sharma (B)
Rajasthan Technical University, Kota, India
e-mail: harish.sharma0107@gmail.com; hsharma@rtu.ac.in
J. C. Bansal
South Asian University, New Delhi, India
e-mail: jcbansal@sau.ac.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_44
449
450
S. Chourasia et al.
(PSO). When we talk about the easiest and robust metaheuristic optimization algorithm, PSO comes first, and also it is very facile to understand and enact. Although
PSO can be utilized for solving non-convex, multi-model, complex, and nonlinear
optimization problems, it has also some drawbacks like other swarm intelligence
based algorithms such as getting stuck into local optima [12], computationally inefficient, when computed by the required number of function evaluations [2]. By
the effect of these inadequacies, the applicability of PSO is reduced [11]. In order
to compensate for these limitations, researchers are trying continuously to escalate
the convergence speed of PSO and avoiding the premature convergence, so as to
explore applicability of PSO. Due to this, so many variants of PSO algorithm [2, 6,
12, 16, 21, 22] have been propounded in this direction. However, it is very difficult to
attain both the goals concurrently, as in the comprehensive-learning PSO (CLPSO)
[12], which is propounded by Liang et al. It aimed to ignore the local optima but
also remained suffered from slow convergence. To bring stability betwixt the social
and cognitive components of initial and later stages of PSO, Ratnaweera et al. [16]
propounded time-varying acceleration factors. Also, Zhan et al. [21] attempted to acclimatize the increasing or decreasing acceleration factors, depending on disparate
exploring or exploiting search space stages. Another researcher, Zhang et al. [22],
examined these factors on position expectation and variance and set the values of
cognitive acceleration factor and the social acceleration factor as 1.85 and 2, respectively, which improved the system stability. A self-adaption technique was applied to
these cognitive and social factors by Gai-yun et al. [6]. To overcome the inefficiencies
of PSO, a new variant is propounded in which self-adaptive acceleration factors are
being involved. Remaining paper is organized as follows: Sect. 2 encompasses basic
PSO algorithm, and the propounded global and local neighborhood based particle
swarm optimization is explained in Sect. 3. Section 4 includes the performance analysis of propounded algorithm through various experiments, and finally, the paper is
concluded in Sect. 5.
2 Standard PSO (PSO 2011) Algorithm
The behavior of flocking of birds is being simulated in PSO optimization technique. In
this, population is taken from the active and interactive agents with very less inherent
intelligence. In this, each possible candidate’s solution is considered as particle or
candidate solution, whereas swarm is considered as whole population. In PSO first,
initialization process is carried out in which each particle is initialized randomly in the
search space. The information about its personal best position, global best position(in
current generation), and current velocity are stored in memory in pbest, gbest, and
V, respectively. According to these three values, the position of each particle is
updated. By the following collaborative trial-and-error method, the whole swarm
converges to single best known solution and thus move in a better direction. At time
stamp t, the position of ith particle in the D-dimensional search space is represented
as Xit = (xi1 , xi2 , . . . , xiD ). Another D-dimensional vector Vit = (vi1 , vi2 , . . . , viD )
Global and Local Neighborhood Based Particle Swarm Optimization
451
depicts the velocity of this particle. Pit = (pi1 , pi2 , . . . , piD ) saves the previously best
visited position of the ith particle. Two equations are used for movement in PSO
namely, velocity update equation (1) and position update equation (2) as shown
below:
(1)
vit+1 = vit + c1 r1 (pit − xit ) + c2 r2 (ptg − xit )
xit+1 = xit + vit+1
(2)
where g shows the best so far solution among all the solutions, i = 1, 2, . . . , SN
shows the index of solutions, SN is the total number of solutions in the swarm, and
c1 and c2 are randomly selected constants, named as cognitive and social factors,
respectively. r1 and r2 are uniformly generated random number such that r1 , r2 ∈
[0, 1].
The terms in velocity update equation (1) are explained further as follows: The
memory of the previous direction, which can be considered as a momentum and
which prevents the particles to change their directions suddenly, is vit . The term
which enables the particles to do local search in swarm and forces to come back
to the previous best situation is c1 r1 (pit − xit ), also called as cognitive component
or persistence. Lastly, the social component is defined as c2 r2 (ptg − xit ), which is
responsible for global search and also allows to compare themselves to others in
group. The pseudocode PSO algorithm is explained in Algorithm 1.
Algorithm 1 PSO Algorithm:
Initialization of the initial population, velocities, and parameters: c1 , c2 ;
Compute the objective function value of each solution;
Identify the global best (gbest) and previous best (pbest) solutions;
while termination criteria do
for every solution, Xi do
for every dimension, xi do
(i) Using equation (1), estimate the velocity vi ;
(ii) Using equation (2), estimate the position xi ;
end for
end for
For updated solutions, calculate the objective function value ;
Again update the position of global best (gbest) and previous best (pbest) solutions;
end while
Return the best so far solution as the global optima;
As mentioned in [17], the PSO can be termed as local PSO (PSOL) and global
PSO (PSOG) based on the based on the neighborhood size.
Since there were no defined boundaries in PSO, so the particles which are far
from the gbest perform larger step size, and hence get escape from the search space.
So to prevent that situation, velocity clamping technique is used. In this, the velocity
is set to its bounds whenever it exceeds its bounds. Another alternative of velocity
452
S. Chourasia et al.
clamping was introduced to maintain a stability between diversification and convergence abilities of the population. A popular velocity clamping component, namely,
inertia weight w [19], is shown in Eq. 3.
vit+1 = wvit + c1 r1 (pit − xit ) + c2 r2 (ptg − xit )
(3)
So it was obvious that fine tuning of these parameters (w, c1 , c2 ) were required to
find optimum values as done by so many researchers [5, 7, 8, 13, 15, 18, 19]. They
suggested that according to the nature of the problem, alternative values of these
parameters can be applied, and stability betwixt the exploration and exploitation
abilities is established.
3 Global and Local Neighborhood Based PSO Algorithm
As mentioned in the literature, the PSO algorithm is capable enough to get the optimal
solutions but suffers from the problem of slow convergence in later stage of the
search process [1]. Further, objective function’s characteristics based on fine tuning
of control parameters is another problem associated with it. As mentioned in Eq. 1,
the velocity update process depends on three terms: first is self-persistence, second
is cognitive learning, and third is global learning. The self-persistence component
helps in maintaining a minimum momentum in the position update process, the
cognitive learning component helps to attract the individual toward its previous best
position, while the global learning component diverts the individual toward global
best solution to improve the convergence speed. It is established by Kennedy and
Eberhart [9] that the high weight of cognitive learning will enhance the exploitation
in the vicinity of the individuals while high weight to global learning will enhance
the convergence. But a proper stability is required to avoid stagnation and premature
convergence situation in the population.
While analyzing the solution search process of PSO, the following modifications
in velocity update equation of PSO are propounded:
1. We propounded fitness-based acceleration coefficient in velocity update equation,
simply to explore or exploit the available environment based on particles current
fitness as shown in Eq. 4:
c1 = 1 − probi
c2 = probi
(4)
(5)
where probi is the fitness-based probability of ith solution and is calculated as
shown in Eq. 6.
fitnessi
probi =
(6)
maxfitness
Global and Local Neighborhood Based Particle Swarm Optimization
453
In Eq. 6, fitnessi is the fitness of ith solution which is calculated by objective
function value as explained in Eq. 7. maxfitness is the fitness of best so far solution
in the current generation of swarm. In Eq. 7, obji is the objective value of the ith
solution.
1
,
if obji (G) ≥ 0.
fitnessi (G) = 1+obji (G)
(7)
1+ | obji (G) |, otherwise.
2. We modified the existing velocity update equation by incorporating the local
neighborhood concept. For each individual, local neighborhood is selected from
the nearby solutions. Here, we adopted the local neighborhood size of 10% of
total solution’s size (SN ), i.e., 5% from the forward indexed solutions and 5%
from the backward indexed solutions. If SN = 60 say, then neighborhood of kth
solution includes 6 (10% of SN ) solutions. The indexed for these neighborhood
solutions will be k + 1, k + 2, k + 3 from forward and k − 1, k − 2, k − 3 from
backward side.
3. Moreover, we also incorporated position information of two random particles
from the swarm to avoid to trap in local optima.
The propounded modified velocity position update equation for ith solution (xi )
is as follows:
vit+1 = vit + r3 (pit − xit ) + c1 r1 (plt − xit ) + c2 r2 (ptg − xit ) + rand (g1t − g2t )
(8)
where r1 , r2 , and r3 ∈ (0, 1) are uniformly generated random numbers, c1 and c2
are the modified acceleration factors or the weights to local and global components,
respectively, as explained in Eq. 4. rand is the random number in (−1, 1), pl is the
local best solution in the neighborhood of ith solution, and g1 , g2 are the two random
particles in the whole swarm. pi and pg are the previous best and best so far solutions
of the swarm in the current generation.
Each term in propounded velocity update equation has its own importance. First
term on right side keeps the momentum of particle with which it was moving so
that it could not explore new area every time. Second term is the personal cognitive
component which forces the particle to move in its previously visited best position.
Third term is term which directs the particle toward the best particle in its local
neighborhood. Fourth term guides the particles toward global best particle in the
swarm to enhance the exploitation. Due to the involvement of best particles in local
and global neighborhoods in the update equation, particles are very much expected to
converge at local optima in early stages. So in order to avoid the chance of stagnation,
premature convergence, and to enhance the exploration, the last term which is the
difference betwixt two random particles in the swarm is added to this equation. r1 , r2 ,
and r3 are the random numbers betwixt 0 and 1 which control the impact of best
values on current momentum. c1 and c2 are the weights to local best component and
global best component, respectively, which are the function of fitness now, instead of
constant. For more fit solutions, c2 will be high and c1 will be low. In this way, more
fit solutions will give more weightage to global best to enhance the exploitation and
454
S. Chourasia et al.
low fit solutions will give less weightage to global best and hence concentrate on
exploration of the search space.
In this way, after each iteration, probability of each particle is calculated to decide acceleration factor’s value, and best particle in local and global neighborhoods
are identified for each particle to apply the new velocity update equation 8. Finally,
this modified velocity is added to particle’s previous position in order to move it
further in search space. Propounded scheme of velocity update equation is inspired
by neighborhood concept in the article [3] “Differential evolution using a neighborhood based mutation operator (DEGL)”. Propounded modified PSO is hereby named
as global and local neighborhood based PSO (GLNPSO). The pseudocode for propounded algorithm GLPSO is same as Algorithm 1 but only difference is velocity
update equation. The GLPSO uses the velocity update equation 8 instead of ordinary
velocity update equation 1.
4 Experiments and Results
To prove the validity of propounded GLNPSO algorithm, 15 benchmark continuous
test functions, fun1 to fun15 , are used, as shown in Table 1. The last column of the table
depicts the acceptable error, which is the error limit of the benchmarks. If algorithm
solves these functions with less error, then it is considered as successful algorithm.
4.1 Experimental setting
In the results, successful simulations over total simulations (SR), average function
evaluations (AFE), and average error (ME) are calculated of propounded GLNPSO
over 15 benchmark functions, and reported results are compared with Standard PSO
(2011). The experimental setting is mentioned as below:
–
–
–
–
Swarm size NP = 50 and Inertia weight w = 0.8,
PSO Cognitive and social coefficients c1 = c2 = 0.5 + log2 [10],
The number of simulations/run =100,
The termination condition is either maximum function evaluations that is set to
200,000 or the acceptable error as shown in Table 1.
4.2 Results Comparison
Tables 2, 3 and 4 show the SR, AFE, and ME for GLNPSO and PSO, respectively.
To limit the stochastic effect of the algorithm, AFE over 100 simulations are taken.
The SR, AFE, and ME actually reflect the reliability, efficiency, and accuracy of
GLNPSO over PSO, i.e., AFE and ME of GLNPSO are less than the PSO, while
Global and Local Neighborhood Based Particle Swarm Optimization
Table 1 Test functions, D: Dimension, EA: Error Acceptable
Test function
Search range
Optimum value
Sphere
De Jong f4
Griewank
Ackley
Alpine
Michalewicz
Cosine
Mixture
Zakharov
Axis parallel
hyper-ellipsoid
Inverted cosine
wave
Levy montalvo
Shifted Sphere
Shifted
Griewank
Shifted Ackley
Shubert
D
EA
[−5.12 5.12]
[−5.12 5.12]
[−600 600]
[−1 1]
[−10 10]
[0, π ]
[−1, 1]
f (0) = 0
f (0) = 0
f (0) = 0
f (0) = 0
f (0) = 0
fmin = −9.66015
f (0) = −3
30
30
30
30
30
10
30
1.0E−05
1.0E−05
1.0E−05
1.0E−05
1.0E−05
1.0E−05
1.0E−05
[−5.12 5.12]
[−5.12 5.12]
f (0) = 0
f (0) = 0
30
30
1.0E−02
1.0E−05
[−5 5]
f (0) = −9
10
1.0E−05
[−5, 5]
[−100,100]
[−600 600]
f (1) = 0
f (o) = fbias = −450
f (o) = fbias = −180
30
10
10
1.0E−05
1.0E−05
1.0E−05
[−32 32]
[−10, 10]
f (o) = fbias = −140
f (7.0835, 4.8580) =
−186.7309
10
2
1.0E−05
1.0E−05
Table 2 Success rate (SR) for 100 runs, TP: Test Functions
TP
GLNPSO
PSO
TP
fun1
fun2
fun3
fun4
fun5
fun6
fun7
fun8
455
100
100
90
100
100
78
97
100
100
100
69
100
100
3
86
31
fun9
fun10
fun11
fun12
fun13
fun14
fun15
GLNPSO
PSO
100
99
99
100
75
91
100
100
6
87
100
7
58
73
GLNPSO successfully solves the functions. Hence, it is proved that GLNPSO costs
less than the PSO algorithm. Another three statistical tests are also carried out in next
sections namely, the Mann Whitney U rank sum test, acceleration rate (AR) [14],
and boxplots analysis [20] to scrutinize the algorithm’s output more accurately.
456
S. Chourasia et al.
Table 3 AFEs based comparison for 100 runs, TF: Test Functions
TF
GLNPSO
PSO
TP
GLNPSO
fun1
fun2
fun3
fun4
fun5
fun6
fun7
fun8
9172
7426
33,890
17,640
30,499
89,049
15,156
128,708
38,102
32,597
113,503
77,352
93,047
198,326
63,044
196,434
Table 4 ME for 100 runs, TP: Test Functions
TP
GLNPSO
PSO
fun1
fun2
fun3
fun4
fun5
fun6
fun7
fun8
8.88E−06
8.16E−06
1.09E−03
9.38E−06
9.42E−06
9.91E−03
4.44E−03
8.82E−03
9.33E−06
9.03E−06
3.87E−03
9.69E−06
9.63E−06
3.12E−01
2.22E−02
2.20E−02
PSO
fun9
fun10
fun11
fun12
fun13
fun14
fun15
10,401
109,056
10,433
15,422
102,742
121,989
24,313
44,375
195,748
57,884
15,718
193,125
187,126
80,824
TP
GLNPSO
PSO
fun9
fun10
fun11
fun12
fun13
fun14
fun15
8.93E−06
8.30E−06
1.19E−04
8.75E−06
4.08E−03
4.06E−01
4.90E−06
9.33E−06
1.48E+00
1.44E−03
8.28E−06
4.17E−02
1.69E+00
9.54E−05
4.3 Results Analysis
The graphical representation of the data which is empirically distributed is carried
out in boxplots analyses. In Fig. 1a, b, c, the proposed GLNPSO is compared with
PSO in terms of SR, AFE, and ME, respectively. It can be easily seen from Fig. 1b,
c that the interquartile range and median are comparatively low for GLNPSO than
PSO which proves that the GLNPSO is cost-effective and more accurate than the
PSO. Further, it is clear from Fig. 1a that the interquartile range and median of the
GLNPSO is higher than the PSO which proves that the GLNPSO is more reliable
than the PSO.
Further, Mann–Whitney U rank sum test analysis for the AFEs of 100 simulations
is presented in Table 5. In this test, two datasets are considered and their significant difference is measured. For nonsignificant difference (i.e., the null hypothesis
accepted), “=” sign is used. If null hypothesis is rejected (significant difference observed), “+” and “−” are used for showing less or more AFEs taken by GLNPSO as
compared to PSO, respectively. In Table 5, “+” sign indicates that GLNPSO is significantly better, while “−” shows that GLNPSO is significantly not better than the
PSO. The “=” symbol shows that the performance of both algorithms is equivalent.
Global and Local Neighborhood Based Particle Swarm Optimization
100
Success Rate
80
60
40
20
0
GLNPSO
PSO
(a) SR
5
x 10
Average number of
function evaluations
2
1.5
1
0.5
0
GLNPSO
PSO
(b) AFE
0
10
Mean error
Fig. 1 Boxplots for
GLNPSO and PSO: a SR, b
AFE, c ME, calculated on
benchmarks fun1 − fun15
457
−2
10
−4
10
GLNPSO
PSO
(c) ME
458
S. Chourasia et al.
Table 5 Mann–Whitney U rank sum test on AFEs, TP: Test Function
TP
GLNPSO versus PSO TP
fun1
fun2
fun3
fun4
fun5
fun6
fun7
fun8
+
+
+
+
+
+
+
+
fun9
fun10
fun11
fun12
fun13
fun14
fun15
GLNPSO versus PSO
+
+
+
=
+
+
+
Table 6 AR of GLNPSO compared to the basic PSO, TP: Test Function
TP
GLNPSO versus PSO TP
GLNPSO versus PSO
fun1
fun2
fun3
fun4
fun5
fun6
fun7
fun8
4.15416485
4.389577161
3.349159044
4.385034014
3.050821338
2.227155836
4.159672737
1.526198838
fun9
fun10
fun11
fun12
fun13
fun14
fun15
4.266416691
1.794931045
5.548164478
1.01919336
1.879708396
1.53395798
3.324312096
Table 5 has 14 “+” signs out of 15 comparisons, which means the performance of
GLNPSO is significantly cost-effective than the PSO algorithm.
Moreover, acceleration rate (AR) is calculated to compare the convergence speed
of the considered algorithm by means of AFEs. The AR calculation for two algorithms
PSO and GLNPSO is carried out through the following equation:
AR =
AFE PSO
.
AFE GLNPSO
(9)
The comparison for all considered functions in terms of AR is depicted in Table 6.
Here, AR > 1 which means GLNPSO is faster than PSO. It is easily observed from
Table 6 that the GLNPSO is more efficient than the PSO over considered benchmarks
functions to reach the acceptable solutions.
Global and Local Neighborhood Based Particle Swarm Optimization
459
5 Conclusion
To establish an efficient equilibrium betwixt diversification and convergence abilities of the swarm in the solution search process, a new position update strategy is
propounded and incorporated in PSO. The propounded position update strategy is
based on the social learning phenomena from the solutions exist in the local vicinity
as well as global search space. The propounded variant of PSO is named as global
and local neighborhood based particle swarm optimization (GLNPSO) algorithm. In
the propounded GLNPSO, every solution is get updated by taking inspiration from
its previous best position, neighborhood solution, and the global best solution. A difference vector of randomly selected solutions is also incorporated in the propounded
search strategy to avoid stagnation of the particles in the search space. The GLNPSO
has simulated over 15 complex benchmark functions, and the reported results are
compared with the standard PSO 2011. Some statistical analyses like boxplots, acceleration rate, and Mann–Whitney U rank sum test are also carried out to establish
the competitiveness of the GLNPSO over the standard PSO algorithm. It is observed
through the intensive analyses that the GLNPSO is an efficient variant of the PSO
algorithm.
References
1. Angeline, P.: Evolutionary optimization versus particle swarm optimization: philosophy and
performance differences. In: Evolutionary Programming VII, pp. 601–610. Springer (1998)
2. Ciuprina, G., Ioan, D., Munteanu, I.: Use of intelligent-particle swarm optimization in electromagnetics. IEEE Trans. Magn. 38(2), 1037–1040 (2002)
3. Das, S., Abraham, A., Chakraborty, U.K., Konar, A.: Differential evolution using a
neighborhood-based mutation operator. IEEE Trans. Evol. Comput. 13(3), 526–553 (2009)
4. Eberhart, R., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the
Sixth International Symposium on Micro Machine and Human Science, 1995. MHS’95, pp.
39–43. IEEE (1995)
5. Eberhart, R.C., Shi, Y.: Tracking and optimizing dynamic systems with particle swarms. In:
Proceedings of the 2001 Congress on Evolutionary Computation, 2001, vol. 1, pp. 94–100.
IEEE (2001)
6. Gai-yun, W., Dong-xue, H.: Particle swarm optimization based on self-adaptive acceleration factors. In: 3rd International Conference on Genetic and Evolutionary Computing, 2009.
WGEC’09, pp. 637–640. IEEE (2009)
7. Gupta, S., Sharma, K., Sharma, H., Singh, M., Chhamunya, V.: L’evy flight particle swarm
optimization (LFPSO). In: 2016 International Conference on Computing, Communication and
Automation (ICCCA), pp. 252–256. IEEE (2016)
8. Jadon, S.S., Sharma, H., Bansal, J.C., Tiwari, R.: Self adaptive acceleration factor in particle
swarm optimization. In: Proceedings of Seventh International Conference on Bio-Inspired
Computing: Theories and Applications (BIC-TA 2012), pp. 325–340. Springer (2013)
9. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, 1995, vol. 4, pp. 1942–1948. IEEE (1995)
10. Kim, J.J., Park, S.Y., Lee, J.J.: Experience repository based particle swarm optimization for
evolutionary robotics. In: ICCAS-SICE, 2009, pp. 2540–2544. IEEE (2009)
460
S. Chourasia et al.
11. Li, X.D., Engelbrecht, A.P.: Particle swarm optimization: an introduction and its recent developments. In: Genetic and Evolutionary Computation Conference, pp. 3391–3414 (2007)
12. Liang, J.J., Qin, A.K., Suganthan, P.N., Baskar, S.: Comprehensive learning particle swarm
optimizer for global optimization of multimodal functions. IEEE Trans. Evol. Comput. 10(3),
281–295 (2006)
13. Malik, R.F., Rahman, T.A., Hashim, S.Z.M., Ngah, R.: New particle swarm optimizer with
sigmoid increasing inertia weight. Int. J. Comput. Sci. Secur. (IJCSS) 1(2), 35 (2007)
14. Rahnamayan, S., Tizhoosh, H.R., Salama, M.M.A.: Opposition-based differential evolution.
IEEE Trans. Evol. Comput. 12(1), 64–79 (2008)
15. Rathore, A., Sharma, H.: Review on inertia weight strategies for particle swarm optimization.
In: Proceedings of Sixth International Conference on Soft Computing for Problem Solving,
pp. 76–86. Springer (2017)
16. Ratnaweera, A., Halgamuge, S.K., Watson, H.C.: Self-organizing hierarchical particle swarm
optimizer with time-varying acceleration coefficients. IEEE Trans. Evol. Comput. 8(3), 240–
255 (2004)
17. Kennedy, J., Eberhart, R.C.: A new optimizer using particle swarm theory. In: Proceedings of
6th Symposium Micro Machine and Human Science, Nagoya, Japan, pp. 39–43 (1995)
18. Sharma, K., Chhamunya, V., Gupta, P.C., Sharma, H., Bansal, J.C.: Fitness based particle
swarm optimization. Int. J. Syst. Assur. Eng. Manage. 6(3), 319–329 (2015)
19. Shi, Y., Eberhart, R.: A modified particle swarm optimizer. In: The 1998 IEEE International
Conference on Evolutionary Computation Proceedings, 1998. IEEE World Congress on Computational Intelligence, pp. 69–73. IEEE (1998)
20. Williamson, D.F., Parker, R.A., Kendrick, J.S.: The box plot: a simple visual method to interpret
data. Ann. Intern. Med. 110(11), 916 (1989)
21. Zhan, Z.H., Zhang, J., Li, Y., Chung, H.S.H.: Adaptive particle swarm optimization. IEEE
Trans. Syst. Man Cybern. Part B Cybern. 39(6), 1362–1381 (2009)
22. Zhang, W., Li, H., Zhang, Z., Wang, H.: The selection of acceleration factors for improving stability of particle swarm optimization. In: Fourth International Conference on Natural
Computation, 2008. ICNC’08, vol. 1, pp. 376–380. IEEE (2008)
Rough Set Theoretic and Logical Study
of Some Approximation Pairs Due
to Pomykala
Pulak Samanta
Abstract The paper inquires about the relationships among the rough set theoretic
and modal properties based on lower and upper approximation operators corresponding to approximation pairs introduced by Pomykala viz. P2 and P3 . It also explores
Rough Modus Ponens Rules with respect to Systems corresponding to P2 and P3
and discusses about Rough Logics based on these Rough Modus Ponens rules.
Keywords Rough sets · Covering · Lower approximation · Upper
approximation · Modal logic · Rough Modus Ponens rule · Rough logic
1 Introduction
Professor Z. Pawlak introduced the theory of Rough set in 1982 [3]. The theory begins
by taking an approximation space < U, R > where U = φ and R is an equivalence
relation on U . The relation R is generated usually from an information system or
attribute-value system [3] defined on the universe. Thus, R partitioned the universe
into disjoint equivalence classes. In case of any subset S of U , the lower and upper
approximations S and S are defined by S = {s| [s] R ⊆ S} and S = {s| [s] R ∩ X = φ}.
[s] R , the equivalence class with respect to the equivalence relation R to which s
belongs is termed as granule at s. A major part of research in rough set theory is
based on studies of these approximations S , S of a set S ⊆ U . One can immediately
observe that the following properties of lower and upper approximations hold:
(1a) U = U (Co-normality)
(2a) φ = φ (Normality)
(3a) S ⊆ S (Contraction)
(4a) S ∩ T = S ∩ T (Multiplication)
(5a) (S) = S (Idempotency)
(1b) U = U (Co-normality)
(2b) φ = φ (Normality)
(3b) S ⊆ S (Extension)
(4b) S ∪ T = S ∪ T (Addition)
(5b) (S) = S (Idempotency)
P. Samanta (B)
Department of Mathematics, Katwa College, Katwa, Burdwan 713130, West Bengal, India
e-mail: pulak_samanta06@yahoo.co.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_45
461
462
P. Samanta
(6) (∼ S) =∼ (S), (∼ S) =∼ (S) (Duality)
(7a) S ⊆ T ⇒ S ⊆ T (Monotone)
(7b) S ⊆ T ⇒ S ⊆ T (Monotone)
(8b) (S) ⊆ S
(8a) S ⊆ (S)
A detailed study of the theory is available [4]. Almost immediately after the
publication of Pawlak’s initial papers, a natural generalization had been proposed by
Pomykala [5] by taking a general covering on U in place of a partition. A covering
on U is a collection C of non-empty subsets of U such that ∪C = U , i.e., if C = {Ci }
then ∪ C = ∪{Ci } = U . Obviously a partition is a special kind of covering. Pomykala
introduced four different pairs of approximation operators based on a covering one
of which (the fourth one) was due to Pawlak and the properties of first one already
studied [9] and we have studied the second and third of them here. The definitions
of them will be given in Sect. 2.
Modal logical properties [2] with the help of propositional logic and modal operators L (necessitation) and M (abbreviation of ∼ L ∼) based on rough set theoretic
properties are discussed here.
A detailed study of Rough Logics based on Rough Modus Ponens Rules are
available in [1, 8].
Section 2 lists properties of P2 and P3 . Section 3 deals with system in respect
to P2 and P3 . The Sects. 4 and 5, discuss about Rough Modus Ponens (RMP) rules
based on the system and rough consequence logic, respectively. Some final remarks
have been made in Sect. 6.
2 Structure and Properties of P2 and P3
2.1 Structure of P2 and P3
Different types of granules in defining approximation pairs due to Pomykala [5] are
as follows:
Consider C = {Ci }, a covering of S and the following sets.
NsC = ∪{Ci : s ∈ Ci }
PsC = {t ∈ S : ∀Ci (s ∈ Ci ⇔ t ∈ Ci )}.
Four lower and upper approximation pairs of of a set S given by Pomykala [5] are
listed below. As mentioned in the Introduction Pawlak introduced the fourth one.
P1 (S) = {x : NsC ⊆ S}
P1 (S) = ∪{Ci : Ci ∩ S = φ}
P2 (S) = ∪{NsC : NsC ⊆ S}
P2 (S) = {s : ∀t (s ∈ NtC ⇒ NtC ∩ S = φ)}
P3 (S) = ∪{Ci : Ci ⊆ S}
P3 (S) = {s : ∀Ci (s ∈ Ci ⇒ Ci ∩ S = φ)}
Rough Set Theoretic and Logical Study of Some Approximation …
463
P4 (S) = ∪{PsC : PsC ⊆ S}
P4 (S) = ∪{PsC : PsC ∩ S = φ}
2.2 Properties of P2 and P3
A detailed Rough Set Theoretic study of the abovementioned approximation pairs is
available in [6, 7]. The Rough Set Theoretic properties hold in cases of P2 and P3
and corresponding modal properties are listed in Table 1.
Table 1 Properties based on Rough Set and analogous Modal features
Properties based on Rough Set Analogous Modal features
1
2
Duality of S , S
S=S
Duality of L and M
p
L p , N -rule
3
4
5
6
7
8
9
10
11
12
S∩T ⊆ S∩T
S∩T ⊆ S∩Y
S∪T ⊆ S∪T
S∪T ⊆ S∪T
S ⊆ T implies S ⊆ T
S ⊆ T implies S ⊆ T
S⊆S
S⊆S
S⊆S
S ⊆ (S)
L(s ∧ t) → (Ls ∧ Lt)
(Ls ∧ Lt) → L(s ∧ t)
M(s ∨ t) → (Ms ∨ Mt)
(Ms ∨ Mt) → M(s ∨ t)
13
14
(S) ⊆ S
(S C ∪ T ) ⊆ (S)C ∪ T
M Ms → Ms
L(s → t) → (Ls → Lt)
s→t
Ls→Lt
s→t
Ms→Mt
Ls → s
s → Ms
Ls → Ms
Ls → L Ls
3 Modal System Corresponding to P2 and P3
The system corresponding to P2 and P3 is given by
Definition 1 Axioms
(Propositional Axioms)
1. p → (q → p)
2. ( p → (q → r )) → (( p → q) → ( p → r ))
3. (∼ q →∼ p) → ( p → q)
464
P. Samanta
(Modal Axioms)
4. L( p → q) → (L p → Lq)
5. L p → p
6. L p → L L p
7. p → M p
Rules
Let Λ, Ω be sets of wffs and p, q be wffs. Then the rules are:
1. p is an axiom ⇒
p
(Ax)
2. p p
(id)
3. Λ p ⇒ Λ, Ω p
(wk)
4. p ⇒ L p
(N)
5. Λ p, Λ p → q ⇒ Λ q
(MP),
i.e., rules are: axioms, identity, weakening, necessitation, and modus ponens, respectively.
The following observations and theorems hold:
Observation 1 It is easy to derive Overlap rule from the identity (id) and weakening
(wk) rules above:
(Overlap) p ∈ Λ ⇒ Λ p (ov).
Theorem 1 1. Λ p ⇒ Λ, Ω p (Monotonicity).
2. Λ p ⇒ ∃r1 , r2 , ...rn ∈ Ω such that r1 , r2 , ...rn
3. Λ, p q and Ω p then Λ, Ω q (Cut).
Λ (Compactness).
Theorem 2 If p is a sub-formula of q and q1 is obtained from q by replacing zero or
more occurrences of p by q then L( p ↔ q) → (q ↔ q1 ) (The rule of substitution
of equivalence).
Theorem 3 Λ, p
q ⇒ Λ
p → q (Deduction Theorem).
The following are valid in the system corresponding to P2 and P3 :
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
( p → q) ⇒ (L p → Lq) (DR 1)
Mp ⇒ p
L( p ∧ q) → (L p ∧ q)
(L p ∧ Lq) → L( p ∧ q)
L( p ∧ q) ↔ (L p ∧ Lq)
(L p ∨ Lq) → L( p ∨ q)
( p ↔ q) ⇒ (L p ↔ Lq) (DR 2)
p→p
L p ↔∼ M ∼ p
( p → q) ⇒ (M p → Mq) (DR 3)
M( p ∨ Mq) ↔ (M p ∨ Mq)
M( p → q) ↔ (L p → Mq)
M( p ∧ q) → (M p ∧ Mq)
Rough Set Theoretic and Logical Study of Some Approximation …
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
465
L( p ∨ q) → (L p ∨ Mq)
Lp → Mp
M( p → L p)
M( p → p)
M Mp → Mp
Lp ↔ L Lp
Mp ↔ M Mp
MLMp → Mp
LMp → LMLMp
LMp ↔ LMLMp
M Lp ↔ M L M Lp
4 Rough Modus Ponens (RMP) Rules
Let p and q be wffs. Using the connectives L, M we can form the following implications:
List of wffs
(i) L p → Lq
(iv) p → Lq
(vii) M p → Lq
(x) M( p → q)
(ii) L p → q
(v) p → q
(viii) M p → q
(xi) L( p → q)
(iii) L p → Mq (vi) p → Mq
(ix) M p → Mq
We define a relation F1 iff F2 , where F1 , F2 are any of the listed formulas. Then, it
can be easily verified that it is an equivalence relation.
Proof of equivalences is given below.
1. L p → Lq (Assumption)
2. Lq → q (Axiom)
3. L p → q (HS 1, 2)
So, (i) implies (ii).
1. L p → q (Assumption)
2. L L p → Lq (DR 1)
3. L p → L L p (Theorem)
4. L p → Lq (HS 3, 2)
So, (ii) implies (i).
As a result (i) iff (ii).
1. L p → Mq (Assumption)
2. (L p → Mq) → (M( p → q)) (Theorem)
3. M( p → q) (MP 1, 2)
So, (iii) implies (x).
466
P. Samanta
1. M( p → q) (Assumption)
2. M( p → q) → (L p → Mq) (Theorem)
3. L p → Mq (MP 1, 2)
So, (x) implies (iii).
As a result (iii) iff (x).
1. p → q (Assumption)
2. L( p → q) (Rule N)
So, (v) implies (xi).
1. L( p → q) (Assumption)
2. L( p → q) → ( p → q) (Axiom)
3. p → q (DT 1, 2)
So, (ix) implies (v).
As a result (v) iff (ix).
1. p → Mq (Assumption)
2. M p → M Mq (DR 3)
3. M M p → Mq (Theorem)
4. M p → Mq (HS 2, 3)
So, (vi) implies (ix).
1. M p → Mq (Assumption)
2. p → M p (Axiom)
3. p → Mq (HS 2, 1)
So, (ix) implies (vi).
As a result (vi) iff (ix).
Others are singletons.
In this case, the group of similar implications is:
{(i), (ii)}, {(iii), (x)}, {(v), (xi)}, {(vi), (ix)}, {(iv)}, {(vii)}, {(viii)}.
So, the number of Rough Modus Ponens (RMP) Rules will be seven.
The corresponding RMP rules are given below:
RMP1
RMP2
RMP3
RMP4
RMP5
RMP6
RMP7
Λ|∼ p, Λ|∼q→r,
Λ|∼r
Λ|∼ p, Λ|∼q→r,
Λ|∼r
Λ|∼ p, Λ|∼q→r,
Λ|∼r
Λ|∼ p, Λ|∼q→r,
Λ|∼r
Λ|∼ p, Λ|∼q→r,
Λ|∼r
Λ|∼ p, Λ|∼q→r,
Λ|∼r
Λ|∼ p, Λ|∼q→r,
Λ|∼r
L p→Lq
L p→Mq
p→Lq
p→q
M p→Mq
M p→Lq
M p→q
Rough Set Theoretic and Logical Study of Some Approximation …
467
5 Rough Consequence Logics
Let T be a modal system with consequence relation T . Based on T two other
systems, L r and L r+ are defined axiomatically by using Rough consequence relation
|∼ as follows:
Lr :
(i) T p implies Λ |∼ p
(ii) { p} |∼ p
(iii) Λ |∼ p implies Λ ∪ Ω |∼ p
(iv) RMP may be used.
L+
r :
(i), (ii), (iii) are exactly the same as in L r and
(iv)+ : rule RMP+ may be used.
It is therefore essential to produce rules RMP+ .
There is a cluster of rules inside the class RMP viz.
Λ |∼ p, Λ |∼ q → r,
Λ |∼ r
T
ℵ( p, q)
where ℵ( p, q) may be chosen from any one of the list of wffs in section 4.
The rules under the cluster RMP+ have only a variation in the third component, viz.,
Λ T ℵ( p, q) in place of T ℵ( p, q).
The following Observations 2, 3, 4, 5, 6 are valid for all the logical systems L r and
L r+ .
Observation 2 Overlap rule viz.
p ∈ ΛimpliesΛ |∼ p
follows from (ii) and (iii).
Proof Assume (ii) and (iii). Let p ∈ Λ.
Then { p} |∼ p by (ii).
Then { p} ∪ (Λ \ { p}) |∼ p by (iii).
i.e., Λ |∼ p.
Conversely, assume overlap.
Then (ii) holds immediately.
For (iii) we need induction on Λ |∼ p.
Cases:
(a) Consider p. Then Λ ∪ Ω |∼ p since this holds for any premise.
(b) Consider p ∈ Λ. Then p ∈ Λ ∪ Ω. So, Λ ∪ Ω |∼ p.
(c) Λ |∼ p is obtained by RMP.
Λ|∼q Λ|∼s→ p
ℵ(q,s)
.
Λ|∼ p
Therefore,
Λ∪Ω|∼q
Λ∪Ω|∼s→ p
Λ∪Ω|∼ p
ℵ(q,s)
468
P. Samanta
Observation 3 Λ |∼ p implies s1 , s2 , ...sn |∼ p for some s1 , s2 , ...sn ∈ Λ.
(Compactness).
We can prove the case using induction on the length of derivation of Λ |∼ p.
Observation 4 Λ, p |∼ q and Ω |∼ p imply Λ ∪ Ω |∼ q (Cut).
In this case one can derive the proof by induction on the length of derivation of
Λ, p |∼ q.
Observation 5 (i)
(ii)
Λ,q|∼r
ℵ( p,q)
Λ, p|∼r
Λ|∼ p
ℵ( p,q)
Λ|∼q
Proof (i) This follows from q → q and consequently, Λ |∼ q → q.
Then any of the RMP rules applies.
(ii) Proof by induction on Λ, q |∼ r .
Cases:
(a) If r , then automatically, Λ, p |∼ r .
(b) If r ∈ Λ ∪ {q} then r ∈ Λ, then automatically Λ |∼ r and by monotonicity,
Λ, p |∼ r .
(c) r = q. So, we need to derive Λ, p |∼ q.
Now, Λ, p |∼ p and ℵ( p, q).
So, by (i) Λ, p |∼ q.
(d) Λ, q |∼ r be obtained by some RMP rule.
ℵ(s,t)
.
i.e., by Λ,q|∼s Λ,q|∼t→r
Λ,q|∼r
So, by induction hypothesis,
Λ,q|∼s ℵ( p,q)
Λ,q|∼t→s ℵ( p,q)
and
.
Λ, p|∼s
Λ, p|∼t→r
So, we have,
Λ, p|∼s
ℵ(s,t)
Λ, p|∼t
and hence,
Λ, p|∼t Λ, p|∼t→r
.
Λ, p|∼r
Observation 6 Ordinary Modus Ponens rule for |∼ viz.
Λ |∼ p, Λ |∼ p → r
Λ |∼ r
may be obtained as a particular case of all the RMP rules for that case
holds.
For hierarchy of RMP rules we have the following derivations.
1. M p → Lq (Assumption)
2. Lq → q Axiom T
3. M p → q H.S. 1,2.
So, RMP7 = RMP4 .
T
ℵ( p, p)
Rough Set Theoretic and Logical Study of Some Approximation …
469
1. M p → Lq (Assumption)
2. p → M p Axiom
3. p → Lq H.S. 2,1.
So, RMP7 = RMP6 .
1. M p → q (Assumption)
2. p → M p Axiom
3. p → q H.S. 2,1.
So, RMP4 = RMP3 .
1. p → Lq (Assumption)
2. Lq → q Axiom T
3. p → q H.S. 1,2.
So, RMP6 = RMP3 .
1. p → q (Assumption)
2. L p → Lq DR1
So, RMP3 = RMP1 .
1. p → q (Assumption)
2. M p → Mq (DR3)
So, RMP3 = RMP2 .
1. p → Mq (Assumption)
2. L p → q (Axiom T)
3. L p → Mq H.S. 2,1.
So, RMP1 = RMP5 .
1. L p → q (Assumption)
2. q → Mq (Axiom)
3. L p → Mq H.S. 1,2.
So, RMP2 = RMP5 .
The relational arrangement of RMP rules is given by
RMP7 = RMP4 = RMP3 = RMP1 = RMP5 ,
RMP7 = RMP6 = RMP3 = RMP2 = RMP5 .
Clearly, the hierarchical order of corresponding rough logics will be in reverse order
and they are:
Lr5 Lr1 Lr3 Lr4 Lr7 ,
Lr5 Lr2 Lr3 Lr6 Lr7 .
Remark 1 The system we have considered is not a standard modal system like K ,
T , S4 , B and S5 . Example of such type of system corresponding to standard modal
systems like S5 , S4 and B has already been discussed [1, 8].
470
P. Samanta
6 Conclusion
Thus, various types of Rough Logics based on the systems P2 and P3 with the help
of RMP rules are presented in this paper. Various Rough Logics based on various
systems with the help of RMP rules including L r+ will be discussed in future.
Acknowledgements The author acknowledges the financial support from the University Grants
Commission, Government of India.
References
1. Bunder, M.W., Banerjee, M., Chakraborty, M.K.: Some rough consequence logics and their
interrelations. In: Peters, J.F., Skowron, A. (eds.) Transactions on Rough Sets VIII. LNCS, vol.
5084, pp. 1–20. Springer, Berlin (2008)
2. Hughes, G.E., Cresswell, M.J.: A New Introduction to Modal Logic. Routledge (1996)
3. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11(5), 341–356 (1982)
4. Pawlak, Z.: Rough sets—Theoritical Aspects of Reasoning About Data. Kluwer Academic
Publisher (1991)
5. Pomykala, J.A.: Approximation operations in approximation space. Bull. Pol. Acad. Sci. Math.
35(1987), 653–662 (1987)
6. Samanta, P., Chakraborty, M.K.: Covering based approaches to rough sets and implication lattices. In: Sakai, H., et al. (eds.) Proceedings of the 12th International Conference, RSFDGrC
2009, Delhi, India. LNAI, vol. 5908, pp. 127–134. Springer, Berlin (2009)
7. Samanta, P., Chakraborty, M.K.: Generalized rough sets and implication lattices. In: Peters, J.F.,
et al. (eds.) Transactions on Rough Sets XIV. LNCS, vol. 6600, pp. 183–201. Springer, Berlin
(2011)
8. Samanta, P. and Chakraborty, M.K.: Interface of rough set systems and modal logics: a survey.
In: Peters, J.F., Skowron, A., Slezak, D., Nguyen, H.S., Bazan, J.G. (eds.) Transactions on Rough
Sets XIX (2015), pp. 114–137. Springer, Berlin (2015)
9. Samanta, P.: A detailed Rough Set Theoretic and Logical Study of an approximation pair due
to Pomykala viz. P1 : Preprint
The Benefits of Carrier Collaboration
for Capacity Shortage Under Incomplete
Advance Demand Information
Arindam Debroy and S. P. Sarmah
Abstract Shippers usually provide information about the demand to the carrier in
advance which is termed as advance demand information (ADI). However, ADI is not
always helpful for the carrier in the proper allocation of the available capacity because
of its incomplete nature. At times, the carrier faces an acute shortage of capacity for
completing orders making it imperative for the carriers to arrange for additional
capacity. A collaborative practice through sharing of capacity at the individual level
has been used to overcome this problem. A case-based study has been carried out
to depict the benefits of collaboration. The findings suggest that sharing of capacity
at individual level resulted in 8.6% increase in profit when compared with the profit
earned by a standalone carrier.
Keywords Advance demand information · Strict control practice · Collaboration
1 Introduction
One of the major problems globally faced by the road transport industry is that
of managing a large number of customers with limited resource. There is a huge
mismatch between the resources available with individual carriers and the number of
people waiting to be served. In a developing country like India, there are around 5.6
million trucks [1] whereas the number of people involved in the trucking industry
is around 200 million [2]. Around 75% of the truck owners have only five or fewer
trucks [3]. This leads to a substantial gap between demand and supply available
with the individual carrier. Developed countries are also affected by this problem
of cut-throat competition due to mismatch of demand and supply. To overcome this
A. Debroy (B) · S. P. Sarmah
Department of Industrial and Systems Engineering, Indian Institute of Technology Kharagpur,
Kharagpur 721302, India
e-mail: debroyarindam1@gmail.com
S. P. Sarmah
e-mail: spsarmah@iem.iitkgp.ernet.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_46
471
472
A. Debroy and S. P. Sarmah
problem collaboration among the different members of the road transport industry
can be used as an economically viable solution. Collaboration among the entities
involved in road transport industry will not only improve their performance but of
the supply chain as a whole. At the very least collaboration allows smooth movement
of materials as well as information through the supply chain.
In the trucking industry, the flow of information normally takes place between the
shipper and the carrier. At times shipper provides the information about the load much
before the shipping date. This information is termed as advance demand information
(ADI). It enables the carriers to make better plans with the available resources to fulfill
the requirement in the best possible manner. The availability of ADI helps the carriers
to make better planning of the capacity allocation, but at times, even ADI, cannot
stop the carriers from falling short of the adequate amount of capacity required to
complete an order. In such situations, collaboration with other truck owners can be a
feasible solution to overcome this capacity shortage. In such situation, ADI provides
the carrier sufficient time to arrange for any additional capacity that is required to
meet the demand. Collaboration along with the availability of ADI can act as a boon
for the truckload carriers.
In a highly competitive business environment like the one discussed in this paper, if
the information of the customer demand is available in advance, it enables the supply
chain practitioners to take better decisions in managing their resources. Researchers
have shown the effectiveness of sharing demand information in increasing the efficiency of operation in the trucking industry [4–7]. However, mostly application on
ADI has been carried out in pick-up and delivery problem [8]. An optimization-based
computational study to quantify the relative benefits and cost of sharing advance
demand information was proposed Tjokroamidjojo et al. [4]. In another paper, [5]
suggested that if the trucking companies collaborate with their shippers and receive
advance demand information, it helps in improving the profit earned. However, any
business enterprises which are not bound by any contract or collaboration might have
reservations towards revealing their information. To get a better deal, shippers may
not be willing to reveal their demand information in advance to the carriers. However,
[6] in his work explored the potential advantages the shipper gets by revealing the
order in advance to the carrier. Nature of the advance demand information revealed
might not always be perfect. Imperfect or incomplete advance demand information
results in a rather precarious situation for the business managers and prevents them
from making efficient operational decisions as this information might not result in
the arrival of an actual demand. The enterprises with limited resources are affected
in the worst manner by this situation.
In this paper, we have made an attempt to show the advantage of ADI and how it
aids in horizontal collaboration between carriers. We have studied the following two
cases for the above described problem (i) when all the carriers individually try to
maximize their profit, (ii) all the carriers collaborate by sharing their unused capacity.
This practice allows the carriers to have full control over their capacity and decide
how much to share, therefore termed as strict control practice.
The Benefits of Carrier Collaboration for Capacity Shortage …
473
2 Modeling Framework
The primary goal of this research is to understand the impact of advance demand
information (ADI) on the operation of a truck owner. Although ADI might be available to the truck owners, the shipper does not provide any information regarding
the exact date on which the demand is due to arrive, which makes this information incomplete and less helpful in making allocation plans. This situation creates
a dilemma for the carrier as managing the capacity available with them is a serious
issue. As the capacity available with the carriers is limited collaboration has been
used as a technique to overcome a situation where the carrier might fall short of the
adequate amount of capacity required to meet the demand.
2.1 Key Assumptions and Notations
Assumptions:
The assumptions made in the development of the mathematical models in this
research work have been discussed as follows:
• Assumption 1: The incoming demand follows a Poisson distribution with mean
μ.
• Assumption 2: The incoming demands become due following Poisson distribution
with mean μ.
• Assumption 3: The time taken by the demands to become due follows an exponential distribution with mean λ.
• Assumption 4: The time taken by the truck to complete an order and return to the
depot follows a uniform distribution with limiting values a and b.
• Assumption 5: The order that becomes due is shipped the very next day so that
neither the carrier nor the shipper has to incur holding cost.
Notations shown in Table 1 have been used in formulating the models:
2.2 Mathematical Formulation
We have developed mathematical models considering two different situations: (a)
Non-cooperative system and (b) Strict control practice. The total number of incoming
demand which the carrier receives must have an upper limit and will depend on the
total truck capacity that the carrier has with him. The maximum quantity of the
incoming demand per day has been set to M, which is considered to be a multiple of
starting capacity, C. The maximum number of incoming order has been considered
to be K, which will vary depending upon the amount of incoming demand. This can
be represented mathematically by Eqs. (1) and (2) respectively.
474
A. Debroy and S. P. Sarmah
Table 1 Notations
Sets
Description
T
Total number of trucks, indexed by
l, t ∈ T {1, 2, . . . , T }
N
Total number of days in the planning horizon,
indexed by i ∈ N {1, 2, . . . ., N }, k ∈ N {2, 3, . . . , N }, r ∈ N {3, 4, . . . , N }
K
The maximum number of orders that can be
accepted by the carrier in a day, indexed by
j ∈ K {1, 2, . . . ., K }
O
Total number of orders that become due on ith
day, indexed by m ∈ O {1, 2, . . . , O}
Decision variables
Description
Qi j
Amount of jth order arriving on ith day
dim
Amount of mth order that becomes due on ith
day
kt
Binary decision variables
Xj
Binary decision variables
1 , 2 , 3 , 4
Binary decision variables
K
E Q ji ≤ M
j1
K ≤
M−
K
j1
c
E Q ji
(1)
+n
(2)
c is the capacity of each truck. The incoming demand has been assumed to follow
a Poisson distribution with mean, μ. As the upper limit of incoming demand is M, the
demand follows a truncated Poisson distribution. The carrier will be able to accept
demands only when Eq. (1) is satisfied.
The expected value of the accepted demand,
M
x
−μ
x0 μ ∗ e
(3)
E Q ji (F(M) − F(0))
The incoming demand becomes due depending on the confirmation given by the
shipper. The time between load dispatches is exponentially distributed with mean, λ.
The Benefits of Carrier Collaboration for Capacity Shortage …
2.2.1
475
Mathematical Formulation for a Standalone Carrier
Here all the carriers operate individually as standalone carriers. In case they fall short
of their required amount of capacity to meet customer demands, they lose the order
to other competitors.
The objective of the problem is to maximize the profit earned by an individual
carrier (qth carrier) can be written as,
Maximize
N O
E(dim ) − F
(4)
Pq Γ1 ∗ R ∗
i1 m1
⎧
O
⎪
⎨ 1, if E(C ) − E(dim ) ≥ 0
i
Γ1 m1
⎪
⎩
0, otherwise
(5)
R is the freight charged by the carriers and F is the fuel cost incurred by the
carrier. As carrier has a fixed capacity with him, so when a truck leaves the depot,
the capacity available to the carrier reduces and until the truck returns, the carrier
cannot use this truck. So, the capacity available with the carrier varies on a daily
basis based on the orders being executed. The expected available capacity on ith day
can be expressed using Eq. (6).
Expected capacity,
N −1 N
E(Ci ) C − E
nk + E
nr
(6)
kt r 3
k2
1, if E(tk − (tk−1 + Tk−1 )) ≥ 0
0, otherwise
(7)
Equation (6) depicts the daily variation in capacity over a certain planning horizon.
The first term in Eq. (6) shows the starting capacity available with the carrier. The
second term in the equation shows the expected value of the amount of capacity
which has left till kth day whereas third term shows the expected amount of capacity
that has arrived till rth day.
kt is a binary variable and the value is 1 if the truck that has left to deliver the
order has returned to the depot. For maximizing the profit of the carriers the following
constraints were considered.
Constraint 1: All the incoming demand will not become due demand
O
N −1 i1 m1
E(dim ) ≤
K
N −2 i1 j1
E(Q i j )
(8)
476
A. Debroy and S. P. Sarmah
Constraint 2: All due demand for a particular day cannot be accepted by the
carrier as it will depend on the available capacity
O
X j ∗ E(dim ) ≤ E(Ci )
(9)
m1
O
Xj ≥ 0
(10)
j1
Constraint 3: Time taken by the truck to complete an order and return to the
depot follows a particular distribution.
m
x
(11)
E(Tm ) x1 ,
g
where, g b − a + 1.
As stated in assumption 4, the time taken by the trucks to return to the depot after
completing the orders follows uniform distribution, which varies between b and a
day. Expected value of the time taken by the truck to carry out mth order is given by
Eq. (10).
2.2.2
Mathematical Formulation for Strict Control Practise
In this sub-section, collaboration between carriers at the individual level has been
considered. A strict control practice has been used to show the collaboration between
carriers. The carriers have been considered to share their additional capacity with
their partners in case the other carriers need it. This additional capacity can be bought
or sold depending upon the requirement. If falling short of the required capacity the
carrier buys the additional capacity from his partner in exchange for a price which
has been termed as capacity exchange price, C e . The new objective function which
depicts this scenario can be expressed as follows:
Maximize
Pq N O
i1 m1
R ∗ Γ2 ∗ E(dim ) + Γ3 ∗ Ce ∗
N
N
E Cis − Γ4 ∗ Ce ∗
E CiB − F
i1
i1
(12)
Γ2 , Γ3 and Γ4 are binary numbers. The carrier can either buy additional capacity
from other carriers or sell them. Both these situations cannot occur at the same time.
This condition can be ensured by Eq. (13).
The Benefits of Carrier Collaboration for Capacity Shortage …
477
Binary constraints
⎛
O
⎞
E(dmi ) ⎟
⎜ 0, if E(Ci ) Γ3 + Γ4 ⎝
⎠
m1
1, otherwise
⎧
O
⎪
⎨ 1, if E(C ) + E C B ∗ Γ ≥ E(dim )
i
4
i
Γ2 m1
⎪
⎩
0, otherwise
(13)
(14)
Equation (14) represents the constraint when the carrier will be able to complete
the order. The other constraints namely constraints 1–3 which have been stated in
Sect. 2.2.1 have to be satisfied for the carrier to complete an order.
In practice, the major operating cost incurred for running a truck is fuel cost,
which is almost 55% of the total operating cost (TCIL-IIMC report, 2012). Hence in
this paper, fuel cost has been considered to calculate the profit earned by the carrier.
The two key factors affecting the fuel consumed are the velocity of the truck and the
load carried by it. Yao et al. [9] in their paper have given the relationship between the
fuel consumed and the two key factors. In another work, the relationship between
the mileage for a heavy duty truck and the fuel consumed was given by Suzuki
[10]. Using these relationships and converting the values given for each parameter
in these papers the relation used to calculate the fuel consumed was developed given
by Eq. (15)
FC 2.2.3
1
4.2 − 0.063702 ∗
O
m1
E(dim )
+ 31.201 + 0.005 ∗ V 2 − 0.659 ∗ V (15)
Solution Procedure
The problem considered in this study is of capacity shortage, i.e., when the carrier
does not have sufficient capacity to carry out all the orders. So, it becomes imperative for the carrier to select the best combination of demand which will give the
maximum revenue. Generating combinations of the demands which satisfies all the
constraints as mentioned in the mathematical models makes it highly complex and
cannot be solved by using normal optimization software like CPLEX. For such problems, algorithms are developed which give feasible and best combinations of decision
variables, which could enable to provide an optimal solution with feasible points.
The proposed algorithms have been designed and solved using MATLAB 14.0 for
this scenario. We have described the steps followed to solve the problem of the three
different scenarios proposed in this paper.
478
A. Debroy and S. P. Sarmah
The steps followed to develop the algorithms have been described as follows:
The Benefits of Carrier Collaboration for Capacity Shortage …
479
480
A. Debroy and S. P. Sarmah
Table 2 Destination of all the carriers
XX
Imphal
Pashighat Tezpur (3)
(1)
(2)
YY
Imphal
(1)
ZZ
Imphal(1) Pasighat
(2)
Biswanath
Chariali (4)
Dibrugarh Muzaffarpur Kohima (4)
(2)
(3)
Tawang (3)
Nogaon
(5)
Durgapur Udaipur
(6)
(7)
Silchar
(5)
Darjeeling Durgapur
(6)
(7)
Dharmanagar Tura (5)
(4)
Udaipur
(6)
Biswanath
Chariali
(7)
3 Case Study
In order to illustrate the models, a case study of the Indian trucking industry was
conducted. Indian trucking industry is one of the largest trucking industry in the
world with around 7 million trucks moving across the country carrying 1325 billion
ton-km (http://www.iamwire.com). Around 200 million people are involved in this
industry. Around 75% of the industry comprises of carriers having less than 5 trucks
resulting in acute shortage of capacity and high level of competition among them.
This scenario makes this industry ideal for testing our proposed practices. All the
truck owners considered in this study are located in the northeastern part of India.
Three carriers XX, YY, and ZZ have been considered for this case study (Table 2).
The distance of each node from the depot and the speed at which the truck moves
between these nodes and the depot for every carrier have been shown in Tables 3
and 4. The speed range of Indian trucks on national highways varies from 25 to 60
kmph. So, for this case study, we considered speed within that range. Figure 1 shows
the zone of operation of the carriers. The routes followed by the trucks to carry out
the orders were represented by using ArcMAP Tool of ArcGIS 10.
3.1 Scenario I: Standalone Carriers
In this sub-section the carriers operate individually without collaborating with their
competitors. The solution procedure adopted to solve this problem has been explained
in the following sub-section.
Table 3 shows the details of performance indicators of the carriers XX, YY and
ZZ respectively.
3.2 Strict Control Model for the Carriers
We have considered collaboration between three carriers here. An attempt has been
made to demonstrate the benefits of collaboration in situations where complete
35
Speed
(km/h)
40
2
600
Nodes
1
Distance 480
(km)
Carrier XX
30
3
180
20
4
240
25
5
150
Table 3 Distance and speed of each depot for XX and YY
40
6
900
30
7
700
40
40
2
450
Carrier YY
1
480
35
3
800
20
4
350
25
5
300
30
6
500
40
7
900
The Benefits of Carrier Collaboration for Capacity Shortage …
481
482
A. Debroy and S. P. Sarmah
Table 4 Distance and speed of each depot for Carrier ZZ
Nodes
1
2
3
4
Distance
(km)
Speed
(km/h)
5
6
7
480
540
400
200
300
700
240
40
35
30
40
35
30
20
Fig. 1 The zone of operation of the carriers
Table 5 Details of performance indicators earned by three carriers
Carriers
Carrier XX
Carrier YY
Carrier ZZ
Revenue (INR)
915,600
582,400
705,600
Fuel cost (INR)
139,760
104,170
147,930
Profit (INR)
775,840
478,230
557,670
demand information from shipper is not available. All the carriers are considered
to collaborate through sharing of capacity. Using the proposed algorithm profit
earned by each carrier was calculated which has been shown in Table 4. The capacity exchange price used by the carriers to buy or sell the additional capacity was
considered to by 50% of the freight rate.
Comparing the results obtained in Tables 5 and 6, it can be concluded that all
the carriers make profit when there is sharing of unused capacity between carriers.
However, it is important to check how the profit varies when there is variation in
capacity exchange price. This variation in profit with capacity exchange price has
been shown in Fig. 2.
The Benefits of Carrier Collaboration for Capacity Shortage …
Table 6 Details of performance indicators earned by three carriers
Carriers
Carrier XX
Carrier YY
483
Carrier ZZ
Revenue (INR)
974,400
680,600
792,400
Fuel cost (INR)
181,580
127,330
169,210
Profit (INR)
792,820
553,270
623,190
Fig. 2 Variation in profit for XX, YY and ZZ with variation in capacity exchange price
4 Conclusion
Collaboration through strict control has been proposed as an economical solution for
the carriers. Fuel cost has been considered to calculate the major operational cost
the carrier incurs. An algorithm has been developed to find out the best possible
combination of orders to be accepted by the carrier in case of a capacity crunch. The
computational results show the benefit of collaboration in uncertainty. Only three
carriers have been considered in this case and it can be seen there is variation in the
gain for all these three carriers. It can also be seen that the change in profit varies
in a different way for each carrier with varying capacity exchange price. It will be
interesting to find out a fixed value of capacity exchange price for which the gain of
each carrier is optimum. Another important factor which can be considered in future
studies is the compatibility of items being transported when carriers share their truck
as this might at times stop the carrier from accepting additional capacity from his
partner.
484
A. Debroy and S. P. Sarmah
References
1. LNCS Homepage: https://inc42.com/startups/trucksuvidha/, last accessed 15 Aug 2017
2. LNCS Homepage: http://www.business-standard.com/article/economy-policy/demonetisatio
n-hits-transport-business-truckers-fail-to-pay-116111200946_1.html, last accessed 17 Oct
2017
3. IRADe: The Impacts of India’s Diesel Price Reforms on the Trucking Industry Integrated
Research and Action for Development, New Delhi, http://www.iisd.org/gsi/sites/default/files/
ffs_india_irade_trucking.pdf (2013)
4. Tjokroamidjojo, D., Kutanoglu, E., Taylor, G.: Quantifying the value of advance load information in truckload trucking. Transp. Res. Part E: Logistics Transp. Rev. 42(4), 340–357 (2006)
5. Zolfagharinia, H., Haughton, M.: The benefits of advance load information for truckload carriers. Transp. Res. Part E: Logistics Transp. Rev. 70, 34–54 (2014)
6. Scott, A.: The value of information sharing for truckload shippers. Transp. Res. Part E: Logistics
Transp. Rev. 81, 203–214 (2015)
7. Zolfagharinia, H., Haughton, M.: Effective truckload dispatch decision methods with incomplete advance load information. Eur. J. Oper. Res. 252(1), 103–121 (2016)
8. Zolfagharinia, H., Haughton, M.: Operational flexibility in the truckload trucking industry.
Transp. Res. Part B: Methodol. 104, 437–460 (2017)
9. Yao, E., Lang, Z., Yang, Y., Zhang, Y.: Vehicle routing problem solution considering minimizing
fuel consumption. IET Intel. Transport Syst. 9(5), 523–529 (2015)
10. Suzuki, Y.: A new truck-routing approach for reducing fuel consumption and pollutants emission. Transp. Res. Part D: Transport Environ. 16(1), 73–77 (2011)
Allocation of Bins in Urban Solid Waste
Logistics System
P. Rathore and S. P. Sarmah
Abstract Management of solid waste is a very crucial and difficult work for any
municipal corporation across the world. Mismanaged solid wastes have very negative
impact on the environment which is a serious issue for many developing countries
like India. Municipal solid waste management (MSWM) consists of multidisciplinary
activities which include forecasting generation, storage and collection, transportation, treatment, and waste disposal. Of all these activities, waste collection and transportation account for 50–70% of total cost of the system. At present, due to the
insufficient number of waste bins and their poor allocations, our society is having
a littering habit of waste disposal which is hazardous for the environment. India
generates 133,760 metric tons of MSW per day but due to poor waste collection
system only 91,152 metric ton per day gets collected and remaining waste goes to
low-lying urban area. It has been estimated that dumping of waste requires 212,752
cubic meter of space every day which is a very critical issue because there is limitation in availability of space and waste generation is increasing every year at 5%. In
this paper we have formulated a MILP model for the calculation of total number of
bin required in any site for different types of waste and we have proposed a method
for the allocation of bins within the site such that it able to capture the total waste of
the site. For solving the MILP model CPLEX software is used and for allocation of
bins ArcGIS software.
Keywords Solid waste · Collection system · Bin allocation · MILP · GIS
P. Rathore (B) · S. P. Sarmah
Department of Industrial and Systems Engineering, Indian Institute of Technology Kharagpur,
Kharagpur 721302, India
e-mail: pradeeprathore@iitkgp.ac.in; pradeeprathore076@gmail.com
S. P. Sarmah
e-mail: spsarmah@iem.iitkgp.ernet.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_47
485
486
P. Rathore and S. P. Sarmah
1 Introduction
One of the most important subjects that affect and worry today’s mankind is the
issue related to waste management. Municipal solid waste (MSW) also termed as
“garbage” or “trash” is an inevitable by-product of human activity. It is generated
by many sources (household, hospitals, shops, hotels, etc.) and are of mainly two
types organic (food, fruit, plant leafs, etc.) and inorganic (paper, plastic, glass, dust,
etc.) [1, 2]. At present due to population growth, industrialization, urbanization,
and economic growth, a trend of significant increase in MSW generation has been
recorded worldwide [3]. According to the report of World Bank’s Urban Development
department, 2012 the amount of MSW worldwide will raise from the current 1.3
billion tons per year to 2.2 billion tons per year by 2025 and the annual, global cost
will raise from the current $205 to $375 billion [4].
Municipal solid waste management (MSWM) consists of multidisciplinary activities which include forecasting, generation, storage and collection, transportation,
treatment, and waste disposal. Of all these activities, waste collection and transportation alone account for 50–70% of total cost of the system [5, 6]. The share
is even higher when landfilling (waste disposal method) is adopted, where waste
directly enters landfill without any treatment [7]. Collection of waste also accounts
for the emission of carbon in atmosphere which is harmful to the human life. Nguyen
et al. [8] estimated the fuel consumption of truck during the kerbside collection of
waste and found that more than 60% of the total fuel is consumed during collection
of waste. This type of situation is very common in developing nations, including
India, where municipal authorities are unable to upgrade or scale up the facilities
required for proper management of MSW [9].
For managing the present situation, an efficient collection system is needed which
requires many strategic, tactical, and operational decisions to be taken at many stages.
The MSWM is a cluster of numerous activities which can be categorized into three
main phases. The first phase is “analysis of demand” where through specific data
collection and dedicated forecast models; the generation of waste over a territory is
determined. Second phase is “Supply planning” which involves the selection of the
specific collection model, like door-to-door service or a collection based on large
kerbside waste bin used for different types of waste. In addition, the scheduling of
number of frequencies and capacity allocation to collection sites can be determined.
It is to be noted that the two decisions about frequency and capacity allocation are
strongly interrelated [10, 11].
In this article, an attempt has been made to propose an efficient bin allocation
method for multiple sources and different types of waste by using different types of
bins within a site.
Allocation of Bins in Urban Solid Waste Logistics System
487
2 Problem Description
Bin allocation is a very important activity as because whole collection system efficiency depends on it. Determination of number of bins required, for a particular site
is the most necessary task in any collection system. The objective of our model is to
optimize the number of waste bins or to determine the number of waste bins required
at minimum cost. For better understanding of the impact of various components of
bin number problem on overall solution, we have studied six different scenarios,
or versions, with increasing complexity. Each scenario is an extension of the previous scenario with additional complexity, i.e., scenario 5 contains aspect of all the
scenarios of 1–4 plus a new one. The five scenarios are as follows:
• Scenario 1—fundamental case. Only single source (residential) is considered.
Multiple types of bins and waste are not considered.
• Scenario 2—multiple sources. Apart from residential, other sources are considered
without any bin and waste type.
• Scenario 3—general case. Bin types considered.
• Scenario 4—space availability. Restriction of space availability considered.
• Scenario 5—multiple wastes. Type of wastes considered.
We have developed the mathematical models considering the above five different
scenarios. The notations used in the model are presented in Table 1 and it is followed
by the assumptions considered in the development of the model.
Assumptions:
(i) There is no limit on available number of bins.
(ii) There is no cap on budget allotted for bins installation.
Scenario 1—fundamental case: In this scenario, we have taken only one source
(household), one type of bin with bin cost and idling cost and one type of waste
(unsegregated). The objective is to determine the number of waste bins at minimum
cost. The mathematical formulation is
Minimize
n
cxi
(1)
i1
Subject to Qxi ≥ Pi qT (1 + f ) ∀i 1, . . . , n.
(2)
xi ≥ 0 and integer, ∀i 1, . . . , n.
(3)
Actual number of bins required xi − pi
(4)
The objective function (1) determines the number of bins at minimum cost. Constraint (2) ensures that the combined capacity of all bins should be greater than the
quantity of waste generated in a site in time period T . As the generation of solid waste
varies daily, and from source to source, it becomes complicated to know the exact
amount of waste going to generate. Therefore a safety factor f has been taken into
consideration to collect all waste without overfilling of bins. Equation (4) represents
488
P. Rathore and S. P. Sarmah
Table 1 The parameters of bin number model
Symbol
Description
Indices
I
J
Site number
Type of bins
K
Type of waste
Type
r
Potential points in site
a
Hospital at site
b
g
Farmers market at site
Gardens and parks at site
s
Commercial place at site
Parameters
n
Number of sites
Integer
m
Number of type of bins
Integer
K
Number of type of waste
Integer
A
Number of hospitals
Integer
B
Number of vegetable market
Integer
R
Number of gardens or park
Integer
S
Number of commercial
complexes or area
Integer
F
Number of potential points
Integer
T
Number of days between two Integer
consecutive trips for collection
f
Safety factor to avoid
overfilling
α
Threshold quantity of waste
Real
generation for consideration as
a source
Site data
Pi
qk
H ik
Real
Population in site i
Integer
Per capita waste generation
(kg per day) of type k, by the
residents of site i
Average quantity of waste (of
type k) generated per day at
hospital of site i
Real
Real
(continued)
Allocation of Bins in Urban Solid Waste Logistics System
Table 1 (continued)
Symbol
489
Description
Type
Gik
Average quantity of waste (of
type k) generated per day at
parks and gardens of site i
Real
C ik
Average quantity of waste (of
type k) generated per day at
commercial places in site i
Real
M ik
Average quantity of waste (of
type k) produce per day at
vegetable markets in site i
Real
Oir
Space available for bin
allocation within the site i.
Number of potential locations
within the site i. r 1, … F.
Real
Fi
pi , pi j , pi jk
Integer
Number of bins present in site Integer
i of type j for waste type k
Bin data
Uj
Space required by a type j bin
Real
Qj
Capacity of type j bin
Real
c, cj
Purchasing cost of type j bin
Integer
Binary number which takes
value 1 if hospital is selected;
otherwise 0
Binary number which takes
value 1 if farmers market is
selected; otherwise 0
Integer
Y
Binary number which takes
value 1 if park or garden is
selected; otherwise 0
Integer
L
Binary number which takes
value 1 if commercial place is
selected; otherwise 0
Integer
Decision variables
X
B
Integer
the actual number of bin required on the site which is a difference of the number of
bin calculated and the number of bins present in the site.
Scenario 2—multiple sources: This scenario is an extension of the previous one
with additional sources. Households are not the only sources of generation of waste in
any site. Therefore for estimating the total amount of waste generation, some other
primary sources like hospitals, farmers market, gardens or parks and commercial
places (school, college, administrative area, offices, shopping complexes, and malls)
are taken into consideration. Since the generation of waste by these sources varies on
490
P. Rathore and S. P. Sarmah
a daily basis, average generation over the time period T is considered. This scenario
does not affect the objective function, but constraint (2) changes to
⎛
A
B
R
⎝
Qxi ≥ Pi q +
Hia +
Mib +
G ig
a1
+
S
g1
b1
Cis T (1 + f ) ∀i 1, . . . , n
(5)
s1
Scenario 3—general case: In previous scenarios only one type of bin is considered but practically more than one type of bin is used, and there are some bins which
are already present on the site. Type of bins is based on sizes or carrying capacity,
and each bin has its own purchasing and idling cost. In this scenario, bin type is
considered which transforms the Eqs. (1), (3), (4) and (5) into
Minimize
Subject to
m
m
⎛
Q j xi j ≥ ⎝ Pi qi +
A
a1
j1
+
S
(c j + cwj )xi j .
(6)
j1
Hia +
B
b1
Mib +
R
G ig
g1
Cis T (1 + f )
∀i 1, . . . , n.
s1
(7)
xi j ≥ 0 and integer, ∀i 1, . . . , n; j 1, .., m
(8)
Actual number of bins required xi j − pi j
(9)
Scenario 4—space availability: Every waste bin has its space requirement based
on its size and every potential point in a site has fixed limited space. This constraint is
very crucial especially for urban areas where availability of space along the streets is
very less. The scenario 3 remains unchanged with the addition of the space availability
constraint
m
j1
xi j U j ≤
F
Oir ∀i 1, . . . , n.
(10)
r 1
Scenario 5—multiple wastes: This scenario can also be termed as segregation
of waste scenario. This scenario comes into play when different type of wastes is
collected separately by providing separate waste bins. The objective function and
constraints are changed due to incorporation of type of waste. More precisely, xi j
changes to xi jk . The per capita waste generation qi transforms to qik . Similarly, Hia ,
Allocation of Bins in Urban Solid Waste Logistics System
491
Mib , G ig and Cis changes to Hika , Mikb , G ikg and Ciks . Thus Eqs. (6), (7), (8), (9)
and (10) changes to
Minimize
n m K
(c j + cwj )xi jk
(11)
i1 j1 k1
Subject to
m
Q i xi jk ≥
Pi qik +
A
X Hika +
a1
j1
+
R
B Mikb
b1
Y G ikg +
g1
B
S
L Ciks T (1 + f ) ∀i 1, . . . , n; k 1, . . . , K .
s1
(12)
K
m j1 k1
xi jk U j ≤
F
Oir ∀i 1, . . . , n.
(13)
r 1
xi jk ≥ 0 and integer, ∀i 1, . . . , n; j 1, . . . , m; k 1, . . . , K .
Actual number of bins required xi jk − pi jk
(14)
(15)
Equations 11–15, represent the final model after including all the scenarios. Since
the generation of waste varies from source to source it is very difficult to consider
each and every source.
Therefore, binary decision variables X , B , Y and L were taken into account
to select the sources which generate waste more than or equal to a threshold quantity
(α) in time period T . Threshold value is not constant and it varies from situation to
situation. The binary decision variables are defined as
1, if Hika ≥ α
(6)
X 0, otherwise
1, if Mikb ≥ α
B (7)
0, otherwise
1, if G ikg ≥ α
(8)
Y 0, otherwise
1, if Ciks ≥ α
L (9)
0, otherwise
The resulting Mixed Integer Linear Programming (MILP) model can be solved
by using commercial solver CPLEX within relatively short time.
492
P. Rathore and S. P. Sarmah
Table 2 Characteristic of bins
Type
Size (m3 )
Capacity (kg)
Cost (c) (Rs)
Organic
Inorganic
1
1.7
493
544
27,000
2
3
870
960
48,500
Table 3 Generation of waste in Ward 20
Ward no. 20 (Population 3024)
Sources
Organic (kg/day)
Inorganic (kg/day)
Brihaspati Vegetable Market
(BVM)
75
25
Chhattisgarh School (C)
1.7
3
Mission Hospital
3
5
RSBH (H)
5
7
Vivekanand Garden
Fast Food Centre (C)
6
5
2
2
Santosh Restaurant(C)
3
1
Residential Centre (RC)
363
544
CIMS Hospital
115
143
Panjab National Bank (C)
0.7
2
3 Results and Discussion
In order to test our approach, we use, current situation of Bilaspur city, India. We
test the proposed mathematical model by using the available data of ward 20 which
is one of the 15 selected wards and solved it with the help of software ILOG CPLEX
12.2 to find out the required number of bins. The results are represented by using
ArcGIS 10 software for better understanding.
Presently, Bilaspur Municipal Corporation (BMC) is solely responsible for the
collection and disposal of wastes every week and due to poor collection system of
BMC, 60% of the population resorts to open dumping of waste which is unhygienic
and hazardous to environment and health. The average rate of per capita generation
of waste in Bilaspur is 350 (gm/day). The required data for testing the proposed
model has been shown in Tables 2 and 3 and the present scenario is represented in
Fig. 1.
The number of bins calculated for the ward 20 is bin1: 8 organic and 4 inorganic
while for bin2: 3 inorganic only. As the number of bin1 and bin2 already present
in the ward is 1 which we considered as inorganic bin. Therefore, now the actual
number of bins required are 10 organic and 3 inorganic of bin1 along with 2 inorganic
of bin2.
Allocation of Bins in Urban Solid Waste Logistics System
493
Legend
Open dumping
1.7m3 waste
3m3 waste
Fig. 1 Current collection points at ward 20
The location sources along with the potential points are presented in Fig. 2 and
the allotted bins are presented in Fig. 3.
4 Conclusion
In this study we have consider the determination of optimal number and bin allocation
problem of waste collection system. An MILP model is formulated to calculate the
number of waste collection bins required in a site while considering the multiple type
of waste and bins. The number of bins identified by minimizing the cost incurred in
allocation of bins while fulfilling the capacity requirement for the generated waste
without exceeding the availability of space within the site. The model is solved in
CPLEX software and allocation of bins is represented by using ArcGIS software. A
real-world instance (Bilaspur city) is considered for testing the proposed approach.
Total 15 wards were considered and the results show the effectiveness of the model
in terms of 15% reduction in collection points.
The proposed model is highly flexible and robust as it can be used for different
scenarios with different type of waste having different type of bins. It is applicable
494
P. Rathore and S. P. Sarmah
Legend
Commercial Places
Farmers market
Hospital
Gardens
Residential
Fig. 2 Sources and potential points of ward 20
Legends
Bin1 Organic
Bin1 Inorganic
Bin2 Inorganic
Inorganic centre
Organic centre
Fig. 3 Bin allocation in ward 20
to festival like occasions when generation of waste is very high compared to normal
Allocation of Bins in Urban Solid Waste Logistics System
495
scenarios, only thing to do is re run the optimization model considering higher values
of waste generation.
Acknowledgements The authors acknowledge the support of Bilaspur Municipal Corporation for
providing the necessary data and having discussions over present situation and feasibility of the
proposed model.
References
1. Hazra, T., Goel, S.: Solid waste management in Kolkata, India: practices and challenges. Waste
Manag. 29(1), 470–478 (2009)
2. Minoglou, M., Komilis, D.: Resources, conservation and recycling optimizing the treatment
and disposal of municipal solid wastes using mathematical programming—a case study in a
Greek region. Resour. Conserv. Recycl. 80, 46–57 (2013)
3. Ghiani, G., Manni, A., Manni, E., Toraldo, M.: The impact of an efficient collection sites location on the zoning phase in municipal solid waste management. Waste Manag. 34, 1949–1956
(2014)
4. Khan, D., Samadder, S.R.: Allocation of solid waste collection bins and route optimisation
using geographical information system: a case study of Dhanbad City, India. Waste Manag.
Res. 34(7), 666–676 (2016)
5. Tavares, G., Zsigraiova, Z., Semiao, V., Carvalho, M.G.: Optimisation of MSW collection routes
for minimum fuel consumption using 3D GIS modelling. Waste Manag. 29(3), 1176–1185
(2009)
6. Rada, E.C., Ragazzi, M., Fedrizzi, P.: Web-GIS oriented systems viability for municipal solid
waste selective collection optimization in developed and transient economies. Waste Manag.
33(4), 785–792 (2013)
7. Ghose, M.K.: A GIS based transportation model for solid waste disposal—A case study on
Asansol municipality. Waste Manag. 26, 1287–1293 (2006)
8. Nguyen, T.T.T., Wilson, B.G.: Fuel consumption estimation for kerbside municipal solid waste
( MSW ) collection activities. Waste Manage. Res. 28, 289–297 (2010)
9. Boskovic, G., Jovicic, N.: Fast methodology to design the optimal collection point locations
and number of waste bins: A case study. Waste Manage. Res. 33(12), 1094–1102 (2015)
10. Hemmelmayr, V.C., Doerner, K.F., Hartl, R.F., Vigo, D.: Management models and algorithms
for the integrated planning of bin allocation and vehicle routing in solid waste management.
Transp. Sci. 48(1), 103–120 (2014)
11. Ghiani, G., Laganà, D., Manni, E., Triki, C.: Capacitated location of collection sites in an urban
waste management system. Waste Manag. 32(7), 1291–1296 (2012)
Image Segmentation Through Fuzzy
Clustering: A Survey
Rashi Jain and Rama Shankar Sharma
Abstract In modern years, image processing is a vast area for research. Image
segmentation is the most popular part of image processing which divides the image
into number of segments to analyze the better quality of image. It is used to detect
objects and boundaries in images. Main goal of image segmentation is to change
the representation of image into the more meaningful regions. Image segmentation
results in a set of segments that covers the whole image or curves that are extracted
from the image. In this paper, different image segmentation techniques and algorithms
are presented, and clustering is one of the techniques that is used for segmentation.
Fuzzy c-means clustering (FCM) algorithm is presented in this paper for image
segmentation. On the basis of literature reviewed, several problems are analyzed in
previously FCM, and the problems have been overcome by modifying the objective
function of the previously FCM, and spatial information is incorporated in objective
function of FCM. Fuzzy c-mean clustering is also known as soft clustering. The
techniques that are explained in this survey are segmentation of the noisy medicinal
images along spatial probability, histogram-based FCM, improved version of fuzzy
c-means (IFCM), fuzzy possibilistic c-means (FPCM), possibilistic c-means (PCM),
and possibilistic fuzzy c-means (PFCM) algorithms are to be explained in further
sections on the basis of literature review. Moreover, several recent works on fuzzy
c-means using clustering till 2017 are presented in this survey.
Keywords Segmentation · Segmentation techniques · Fuzzy c-means clustering
(FCM) · PCM · FPCM · PFCM · Membership function
R. Jain (B) · R. S. Sharma
Rajasthan Technical University, Kota, Rajasthan, India
e-mail: 28jainrashi1994@gmail.com
R. S. Sharma
e-mail: rssharma@rtu.ac.in
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_48
497
498
R. Jain and R. S. Sharma
1 Introduction
Image segmentation means dividing or splitting a image into the different parts or segments that contains uniform characteristics like color, intenseness, and composition.
Segmentation is a preprocessing step. It helps in recognition of objects, capturing,
and analysis of an image. Segmentation of image results in the collection of segments
which combine to form a perfect image. There are many issues faced during image
segmentation and the main objectives are to minimize overall inconsistency, maximize connectivity, minimize the error rate, etc. In the modern years, now the area of
image processing has been provoked as a great importance and it is becoming now
expanded and necessary. Soft computing approaches consist of segmentation based
on fuzzy clustering, fuzzy logic techniques, and artificial neural networks. Clusters
formation can be controlled by using a similarity measure. It can be explored by
using two types of clustering approaches that are generally used: One is the hard
clustering (K-means) and the other is soft/fuzzy clustering (Fuzzy clustering). On
the basis of properties of an image, image segmentation can be categorized in the
two groups:
1. Discontinuity detection based approach: In this approach, basically image is
segmented into the regions on the basis of discontinuity or disruption. Edge
detection method approach comes under the category of discontinuity approach.
2. Similarity detection based approach: In this approach, the image is segmented
or divided into the parts/segments on the basis of some correlation. Thresholding,
region growing, region splitting, and merging methods come under similaritybased approach (Fig. 1).
This survey explores the image segmentation techniques that execute the distribution of tasks through fuzzy clustering. Section I explains the introduction of image
segmentation by using fuzzy clustering. Many authors are also interested to solve the
segmentation problem and also find out the solution in a fast, robust, and efficient
way. In Sect. 2, we discuss segmentation techniques; Sect. 3 explains comparative analysis of techniques; Sect. 4 explains segmentation applications; and Sect. 5
Fig. 1 A clustering method [16]
Image Segmentation Through Fuzzy Clustering: A Survey
499
explains the clustering algorithms. In Sect. 6, we discuss the overall background of
literature and various algorithms which are reviewed during survey are explained
in this section. And in further sections, we conclude our survey and provide some
future recommendations on the basis of survey.
2 Methods for Image Segmentation
There are different methods available for image segmentation. Some of these methods
are listed here which are introduced by many researchers.
1. Segmentation based on thresholding,
2. Segmentation-based regions,
– Segmentation using region growing
– Segmentation using region merging and splitting
3.
4.
5.
6.
Segmentation using edge-based method,
Segmentation using clustering,
Bayesian-based method for segmentation, and
Segmentation using classification.
2.1 Thresholding
This is the very simple image segmentation technique. It differentiates the image
between the background image and the image center. It uses the intensity histogram
such that by using the intensity histogram the intensity values determine that values are known as threshold values and these threshold values separate the desired
classes, where “T1 ” is the threshold value. The value of the u, v shows the coordinates
for the threshold value point. p(u,v), q(u,v) points show the pixels of the gray-level
image.
2.2 Region Growing
In this method, the regions of the image which are connected to each other and
which contains the group of pixels that are having same intensities are separated
using region growing technique. In this method, a point which is initially defined is
referred to as image point. Then after all the points which are connected with the
image point whose having the similar intensities and are matched with that image
point are to be selected and finally added to the growing regions. The process is
repeated till no more pixel is added to the region.
500
R. Jain and R. S. Sharma
2.3 Region Splitting
This method divides the image into independent regions and after that again merges
it on the basis of some action. The region splitting method consists of two phases—
first is the splitting phase and second is the merging phase. Region splitting method
generally used “Quad tree”.
2.4 Clustering
This is the method which is based on unsupervised image segmentation method in
which the dataset is not trained. It is the form of unstructured data analysis. Clustering
classifies the image into a fixed number of clusters, and the clusters are defined by
the user or it can be determined using an algorithm. No training phases are required,
and still train themselves using the present data.
2.5 Edge Detection
In this, edge or pixels between distinct regions are to be detected by using edge
detection method. The condition for distinct regions may be fast transformation
of intensity. So those pixels are extracted and combined together to form a folded
enclosure.
2.6 Bayesian Method
For classification purpose, Bayesian method is used and it works by including an
event into the image to construct the models on the basis of certain events which
is further put to use for the class distribution of pixels into the image. There are
numerous paths which are involved in this method like MRF (Markov Random
Field) and expectation elaboration.
2.7 Classification
Classification method is used to classify the data. This method uses the data that
are having their common labels to divide the image characteristic zone. Or we can
say that image classification is done by constructing a characteristic zone from the
image. Further, this characteristic zone or area is subdivided into distinct regions.
Image Segmentation Through Fuzzy Clustering: A Survey
Table 1 Comparison of different segmentation techniques
Technique
Method description
Merits
Threshold method
Edge-based detection
method
It is based on the
histogram analysis,
used to find out the
single threshold value
It is based on
discontinuity detection
approach
Region-based
segmentation method
It divides the image
into uniform regions
Segmentation using
clustering method
It divides the data
elements into the
uniform clusters
501
Demerits
Simple method,
previous information
is not needed
It is highly dependent
on the points. No
consideration of
spatial information
Advantageous for the It is not perfect for
images that are having inaccurate detection,
good contrast in the
more number of edges
objects
High resistant to the
Costly in terms of
noise, advantageous
memory and time
when similarity
measure is easy to
define useful
Fuzzy uses
Membership function
membership function is not easily find out,
and degrees so it is
and hence it is difficult
more favorable for
task
real-world issues
3 Comparison
See Table 1.
4 Image Segmentation Applications
The different real-world feasible applications of image segmentation are as follows:
– Improvement of image based on content;
– Automobile vision;
– Medical images, which includes volume rendered image from the measured
tomography and MRI images. These are
1.
2.
3.
4.
5.
6.
Locating tumors images and pathologies,
Tissues quantity measurement,
Diagnosing, study about anatomical structure,
Copy of patient surgery,
Virtual surgery copy, and
Eventually surgery exploration.
– Detection of objects;
1. Pedestrian detection methods,
2. Face detection methods,
502
R. Jain and R. S. Sharma
3. Obstacle light detection, and
4. Locating objects in the satellite pictures (Lanes, woodlands, crops, and field).
– Task recognition;
1. Face recognition,
2. Finger copy recognition, and
3. Recognition of iris.
– Traffic controlling schemes; and
– Surveillance for videos.
Numerous widespread clustering algorithms and methods have been developed
for image segmentation. These methods need commonly to combine with a rangespecific knowledge to efficiently solve the region-based image segmentation problems.
5 Fuzzy C-Means Clustering Algorithm
Fuzzy c-means clustering is a impressive unsupervised method used for the comparative analysis of data and models construction. In some phases, FCM is more
instinctive in comparison to hard clustering. The objects which are present on the
boundaries among the distinct classes or groups are not bound to fully associate to
one of the classes or a group, but rather the membership degrees are to be assigned
between the 0 and 1 which shows their partial membership. This algorithm is most
widely used in image segmentation and also can be used to remove the noise from
the images. Fuzzy c-means algorithm was first revealed in the literature review for a
consideration of special case for the (m = 2) by the author “Joe Dunn” in 1974 [14].
The generalized case (for any m greater than 1) was introduced and extended version
was developed by “Jim Bezdek” in his research of Phd at Cornell University in
1973. It can be further improved by Bezdek in 1981. Fuzzy separation used the FCM
algorithm in which data elements are associated to all the groups with individual
membership grades between 0 and 1. Objective function of the conventional fuzzy
C-means algorithm can be defined by the following equation [19]:
Ji =
ni C
δi j xi − c j 2
(1)
i=1 j=1
Here, “n i ” defines the number of data elements or points, “c” defines the number of
clusters, center point c j associates for the cluster “j”, δi j represents the membership
degree for the ith data point xi in cluster “j”, the model xi − c j 2 measures data
point closeness xi to the center point c j of the cluster “j”.
Image Segmentation Through Fuzzy Clustering: A Survey
503
Fuzziness coefficient (“m”): Fuzziness coefficient “m”, where 1 < m < ∞, measures the strength of required clustering. This value indicates that how much the
clusters can overlap to one another. The more the value of the “m”, the more will be
the clusters overlap with each other (Fig. 2).
Fig. 2 Algorithm flowchart for FCM (This figure is taken from https://www.researchgate.
net/publication/279224750/figure/download/fig3/AS:324182819786756@1454302615263/
flowchart-of-fuzzy-C-means-clustering-algorithm.png)
504
R. Jain and R. S. Sharma
Algorithm 1 Fuzzy C-means Algorithm
1. Input matrix U=[u i j ], U (0)
2. At step-y:compute, number of centers vectors C (y) =[c j ] with U (y)
n
m
j=1 u i j x j
ci = n
m
j=1 u i j
3. update U y , U y+1
4.
n
di j = (xi − ci )
(2)
(3)
i=1
1
u i j = c
di j
2
( k=1 ( dk
j )( m i −1 ))
(4)
5. if U(y+1) − U(y) < ∈
Here “m”is a real number of any type whose value should be >1, u i j represents the membership
level in the cluster j of xi , xi shows the ith of d-dimensional uniform data, c j is cluster center,
– Advantages and Disadvantages:
Unsupervised converges are the advantage of this algorithm, and not considering any
spatial information for noisy images into the fuzzy c-means is the main disadvantage.
To overthrown these imperfections of the FCM, many other algorithms were introduced such as improved FCM (IFCM), possibilistic fuzzy c-means algorithm (PCM),
fuzzy possibilistic c-means (FPCM), and fuzzy c-means based on histogram. Limitation is affectibility to the initial hypothesis (speed or local minima) and affectibility
to noise.
6 Related Work
See Table 2.
7 Gaps in Literature
– Problem of determining the clusters should be removed, and hence it should be
automatically be prior decided.
– In future, color image segmentation should be possible to do.
– Problem of fuzzification parameter “m” should be fixed so that overlapping of
clusters should be reduced. It should be fixed with some value not varies.
– Advanced clustering technique should be used to increase efficiency and accuracy.
Image Segmentation Through Fuzzy Clustering: A Survey
Table 2 FCM modifications
Paper details
Chaur-Heh Hsieh (IEEE),
1994 [4]
Technique used
505
Modification
It facilitates the neighborhood
system, the associated cliques,
potentials of GRF (Gibbs
random field)
Redefine objective function of
FCM clustering algorithm to
include the energy function
that is the sum of potentials,
and new membership equation
is derived
D. Zung L. Pham (Elsevier),
Used adaptive FCM algorithm Modify the objective function,
1999 [8]
for segmentation in the
include a multiplier field,
presence of in homogeneities which allows the centroids for
each class to vary across image
Young Won Lim (IEEE), 1990 A segmentation algorithm for Reduce the computational
[6]
color images based on the
burden required for the FCM.
thresholding and the fuzzy
Fine segmentation assigns the
c-means (FCM) techniques.
pixels, which remain
Scale-space filter is used for
unclassified after the roughly
analyzing the histograms of
segmentation, to the closest
three color components
class using the FCM
Yong yang (Springer), 2009
FCM with spatial
Improved FCM, formulated by
[17]
neighborhood information
incorporating the spatial
neighborhood information into
original FCM by a priori
probability and initialized by
histogram-based FCM
B. Zhang (IEEE), 2010 [18]
For image-based particle
Removes the noise and
characterization
high-frequency components,
multi-resolution fuzzy
and textures and features can
clustering approach was used be obtained
Kun Qin, Kai Xu (Elsevier),
Type-2 fuzzy sets consider the Handles uncertainty of
2011 [9]
fuzziness of the membership
membership function
degrees
A. Rajendra (Elsevier), 2012
Fuzzy clustering and
Method is more accurate and
[11]
deformable model based on
robust for brain tumor
the region were used for
segmentation
segmenting tumor region on
MRI images
Deepali Aneja (IEEE), 2013
Comparison of the three-image IFCM takes the less number of
[1]
segmentation methods based
iterations and gives the less
on fuzzy logic namely fuzzy
percentage of misclassification
c-means, intuitionistic fuzzy
error
c-means (IFCM), and type-II
fuzzy c-means was presented
Cunyong Qiu, Jian Xiao
Used enhanced type-2 fuzzy
Reduce shortfalls as FCM is
(Elsevier), 2014 [10]
c-means algorithm with
not quite efficient to handle the
improved initial center
uncertainties well
(continued)
506
Table 1 (continued)
Paper details
R. Jain and R. S. Sharma
Technique used
Modification
Omer Sakarya (IEEE), 2015
[12]
Proposed a fuzzy clustering
method to color image
segmentation
Benson (IEEE), 2016 [2]
Brain tumor segmentation
from MR brain images using
improved fuzzy c-means
clustering and watershed
algorithm
Automatic crack detection on
concrete images using
segmentation via fuzzy
c-means clustering
Different distance measures
were used: Euclidean,
Manhattan metrics, and two
versions of Gower coefficient
similarity measure
Avoiding the
over-segmentation problem
Yohwan Noh (IEEE), 2017(1)
[7]
Dr. T. Karthikeyan (Springer),
2017(2) [5]
Natacha Gueorguievaa
(Elsevier), 2017(3) [3]
Le Hoang Son, Tran Manh
Tuan (Elsevier), 2017(4) [15]
Sharmila Subudhi (Elsevier),
2017(5) [13]
Detect cracks that are
photographed at large
distances from the surface.
Detect 0.3 mm cracks in
photographs taken at a distance
of 1 m from the surface
Fuzzy c-means is applied in
In comparison to K-means,
microscopic image
fuzzy c-means gives higher
segmentation for leukemia
accuracy than k-means. Gabor
diagnosis
texture extraction method was
used to extract color features
from images and finally
extracted features are used for
classification. Fuzzy c-means
gives 90% accuracy, whereas
k-means gives 83% accuracy
M and MFCM: Fuzzy c-means Improving the initial choice of
clustering with Mahalanobis
cluster number and for
and Minkowski distance
visualization and analysis of
metrics
cluster results for labeled and
unlabeled datasets
Described a novel
Work has better accuracy than
semi-supervised fuzzy
the original semi-supervised
clustering algorithm with
fuzzy clustering and other
spatial constraints for dental
relevant methods
segmentation from X-ray
images
Used the optimized fuzzy
Efficacy of the proposed
c-means clustering and
system is illustrated by
supervised classifiers for
conducting several
automobile insurance fraud
experiments on a real-world
detection
automobile insurance datasets
Image Segmentation Through Fuzzy Clustering: A Survey
507
8 Conclusion and Future Perspectives
Fuzzy c-means, one of the unique algorithms, has been overworked in broad domain
of engineering and systematic development, for a time, medical imaging, pattern
detection, and data mining. On the basis of background, the previously designed FCM
makes use of the coincide pattern to determine the similarity among the patterns and
the data elements, it performs hardy only in the case of clustering circular clusters.
Moreover, a lot of algorithms are presented by various authors and researchers based
on the fuzzy c-means with the goal of clustering more general dataset. Throughout
the survey, we investigate some number of valuable points that can stand in need for
further improvement. Future scope is by using some advanced clustering technique
to achieve the more good accuracy in the results and the time taken for large dataset
and/or information retrieval from large datasets should be reduced. Scope for color
image segmentation can be used through FCM, number of clusters should automatically be decided, fuzzification parameter should be fixed, and overlapping of clusters
problem should be removed on the basis of fuzzification parameter.
References
1. Aneja, D., Rawat, T.K.: Fuzzy clustering algorithms for effective medical image segmentation.
Int. J. Intell. Syst. Appl. 5(11), 55 (2013)
2. Benson, C.C., Deepa, V., Lajish, V.L., Rajamani, K.: Brain tumor segmentation from MR brain
images using improved fuzzy c-means clustering and watershed algorithm. In: 2016 International Conference on Advances in Computing, Communications and Informatics (ICACCI),
pp. 187–192. IEEE (2016)
3. Gueorguieva, N., Valova, I., Georgiev, G.: M&mfcm: fuzzy c-means clustering with mahalanobis and minkowski distance metrics. Procedia Comput. Sci. 114, 224–233 (2017)
4. Hsieh, C.-H., Kuo, C.M., Chao, C.-W., Lu, P.-C.: Image segmentation based on fuzzy clustering
algorithm. In: MVA, pp. 460–463 (1994)
5. Karthikeyan, T., Poornima, N.: Microscopic image segmentation using fuzzy c means for
leukemia diagnosis. Leukemia 4(1) (2017)
6. Lim, Y.W., Lee, S.U.: On the color image segmentation algorithm based on the thresholding
and the fuzzy c-means techniques. Pattern Recogn. 23(9), 935–952 (1990)
7. Noh, Y., Koo, D., Kang, Y.-M., Park, D., Lee, D.: Automatic crack detection on concrete images
using segmentation via fuzzy c-means clustering. In: 2017 International Conference on Applied
System Innovation (ICASI), pp. 877–880. IEEE (2017)
8. Pham, D.L., Prince, J.L.: An adaptive fuzzy c-means algorithm for image segmentation in the
presence of intensity inhomogeneities. Pattern Recogn. Lett. 20(1), 57–68 (1999)
9. Qin, K., Kai, X., Liu, F., Li, D.: Image segmentation based on histogram analysis utilizing the
cloud model. Comput. Math. Appl. 62(7), 2824–2833 (2011)
10. Qiu, C., Xiao, J., Han, L., Iqbal, M.N.: Enhanced interval type-2 fuzzy c-means algorithm with
improved initial center. Pattern Recogn. Lett. 38, 86–92 (2014)
11. Rajendran, A., Dhanasekaran, R.: Fuzzy clustering and deformable model for tumor segmentation on mri brain image: a combined approach. Procedia Eng. 30, 327–333 (2012)
12. Sakarya, O.: Applying fuzzy clustering method to color image segmentation. In: 2015 Federated
Conference on Computer Science and Information Systems (FedCSIS), pp. 1049–1054. IEEE
(2015)
508
R. Jain and R. S. Sharma
13. Subudhi, S., Panigrahi, S.: Use of optimized fuzzy c-means clustering and supervised classifiers
for automobile insurance fraud detection. J. King Saud Univ. Comput. Inf. Sci. (2017)
14. Suganya, R., Shanthi, R.: Fuzzy c-means algorithm-a review. Int. J. Sci. Res. Publ. 2(11), 1
(2012)
15. Tuan, T.M., et al.: Dental segmentation from x-ray images using semi-supervised fuzzy clustering with spatial constraints. Eng. Appl. Artif. Intell. 59, 186–195 (2017)
16. Xu, R., Wunsch, D.: Survey of clustering algorithms. IEEE Trans. Neural Netw. 16(3), 645–678
(2005)
17. Yang, Y.: Image segmentation based on fuzzy clustering with neighborhood information. Optica
Applicata 39(1) (2009)
18. Zhang, B., Mukherjee, R., Abbas, A., Romagnoli, J.A.: Multi-resolution fuzzy clustering
approach for image-based particle characterization. IFAC Proc. Volumes 43(5), 153–158 (2010)
19. Zou, Y., Liu, B.: Survey on clustering-based image segmentation techniques. In: 2016 IEEE
20th International Conference on Computer Supported Cooperative Work in Design (CSCWD),
pp. 106–110. IEEE (2016)
Study of Various Technologies in Solar
Power Generation
Siddharth Gupta, Pratibha Tiwari and Komal Singh
Abstract Energy is an essential ingredient of socio-economic development and
economic growth. Countries such as Germany and other European countries in the
world have been developed specific regulatory mechanisms. These mechanisms are
developed to encourage its use either by government programmes or by financial
and/or tax incentives. In India, there is large existing solar potential but despite it the
encouragement to technology is still incipient. India is in a state of perennial energy
shortage. Here the demand supply gap is almost 12% of the total energy demand.
This trend is significant in the electricity segment. This segment is heavily dependent on coal and other non-renewable sources of energy. Renewable energy (RE)
sources contribute only very less amount of energy as compared to the total installed
power capacity of in India. In coming days, solar energy will play important role in
development of renewable energy sector in India. Solar energy can be harnessed by
solar PV, solar thermal and solar hybrid technology. This paper focused on study of
various technology of solar power energy generation.
Keywords Solar · Renewable energy · Thermal · Photovoltaic · Power · Hybrid
1 Renewable Energy
Renewable Energy is seen as pollution-free energy source and optimum use of these
resources helps to reduce the environmental impact and to develop sustainably in
accordance with the current social needs of the society. Renewable energy technologies provide a huge opportunity to reduce greenhouse gas emissions and reduce
global warming by replacing conventional energy. Renewable Energy will play significant role to create the pollution-free environment throughout the globe, it is a
S. Gupta (B) · P. Tiwari
Sam Higginbottom University of Agriculture, Technology and Sciences, Allahabad, India
e-mail: siddharthgupta.india@gmail.com
K. Singh
Galgotias University, Noida, Uttar Pradesh, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_49
509
510
S. Gupta et al.
Table 1 Year-wise targets (in MW) [6]
S. No.
Category
Rooftop solar
Ground
mounted solar
projects
Total
1
2
2015–16
2016–17
200
4800
1800
7200
2000
12,000
3
2017–18
5000
10,000
15,000
4
2018–19
6000
10,000
16,000
5
2019–20
7000
10,000
17,000
6
2020–21
8000
9500
17,500
7
2021–22
9000
8500
17,500
40,000
57,000
97,000
Total
reagent that can be used to generate energy with the help of solar, wind biomass,
etc. It generates huge energy with zero emission of pollution, by proper implementation it can reduce the pollution like PM2.5, PM10, etc. which are main sources of
air pollution. Renewable energy technologies are good source of energy to mitigate
greenhouse gas emission, for reducing the global warming. Energy production from
renewable energy sources is increasing drastically. Due to enhancement of pollution
in world, there is a need to give attention on development of projects, policy framing,
and operations in renewable energy technologies and their implementations. Solar
energy is a renewable source of energy with zero emission, which can be collected
using variety of technologies. Solar energy is the solution for the long-lasting energy
issues, which are currently a major problem for the world and countries like India
which are still developing. Solar energy can help us to improve energy security in
India. It can also help us to alleviate the concern regarding environment problems
and creating huge market for renewable energy in country. The paper focuses on
status of solar energy in India uses and its application.
According to data collected from the International Energy Agency and Key World
Energy Statistics [1], it is obtained that the energy demand of Brazil, Russia, India and
China is 32% of world energy demand. Among them, the highest one is China with
2417 million toe (tonnes of oil equivalent), which is 19% of the total world energy
demand. Russia comes after China with 701 million toe (6% of world demand).
India is also account for 692 million toe (5%) and Brazil is for 265 million toe (2%).
Table 1 shows years wise target of solar energy in India.
2 India’s Power Scenario
The Ministry of Non-Conventional Energy Sources has been implementing comprehensive programmes. These programmes are for the development and utilization
of various renewable energy sources in the country. As a result of efforts applied
Study of Various Technologies in Solar Power Generation
511
during the past quarter century, various kind of technologies and devices have been
developed and these are commercially available for use. These technologies and
devices consist: biogas plants, improved wood stoves, solar lanterns, solar cookers,
solar water heaters, solar thermal power generation, wind electric generators, pumps,
street lights, water-pumping wind mills, biomass gasifiers and small hydro-electric
generators, etc. Some energy technologies are being actively developed for the future
such as hydrogen, bio-fuels and fuel cells. India is involved in implementing one of
the world’s largest programme in renewable energy [2–5].
Use of fossil fuels and petroleum products at large extent is very dangerous. In
recent years due to over exploitation of such products the environment is having
problems both locally and globally. So considering the huge demand of energy for
all sectors of economy, the SPV is being viewed a substitute due to its abundant
availability on earth. SPV convert solar energy into useful energy forms by directly
absorbing solar photons particles. We are using nanotechnology in new solar design
to enhance their efficiency as compare to conventional solar cells. As the popularity of
SPV is increasing, but they have some drawback like less efficiency and cost problem.
This research gives some good result and covers more numbers of its problem.
The current electricity installed capacity of India is 135,401.63 MW. Presently
there is peak power shortage here which is about 10% and overall power shortage of
7.5% [2].
3 Solar Energy Potential in India
Even though there are these huge power plants (more than a Gigawatt in capacity) that
have come up in various parts of the country, there still is a big need and possibility
of it the country. Solar power is a fantastic for India, and this is how:-High number
of sunny days in majority of the country’s land: India is luckiest one to receive high
volumes of solar light and energy and this is all throughout the year. Tapping this
energy effectively will help to resolve energy crisis in many energy-deficient regions
of the country. Grids may fail to reach some place, but sun doesn’t.
India resides in its villages: Villages in India are a house to a huge chunk of the
1.2 billion populations that we have. And it has been really tough for the transmission
grids to reach in certain regions. Such regions can easily be made self-reliable with the
use of Solar-Powered systems. Many international and Indian banks are supporting
the initiative like these by the State Renewable energy boards, for example in the
state of Assam.
Solar contributes about 3.8% to the total installed utility capacity in India where
the majority of the power still comes from coal, contributing almost 60%. However,
in the last 3 years alone, India has been able to quadruple its solar generation capacity
to 12.2 GW in 2017. The Government of India has laid down a challenge of achieving
at least 8% of total utility power totalling up to 175 GW from renewable energy and
solar is set to contribute a vast majority in the same with a target of 100 GW. In
the last financial year alone, we have been able to add 5.5 GW of solar energy and
512
S. Gupta et al.
with the dropping costs and several government initiatives finally taking shape, solar
will be widely used as a source of power. In terms of pure potential, India has one
of the highest solar electricity production per watt installed owing to its position on
the globe and the 300 + sunny days that the country experiences. Therefore, while
solar is still a baby in the grander scheme of things, this baby is growing fast and we
definitely would expect it to grow up very soon and not just start running but also
shouldering the maximum responsibility of power sector in the country.
An ever-growing demand of power: With limited resources in terms of coal, and
also a comparatively small reserve of natural gas, India will have to find ways to
keep itself powered. Solar is one option that comes to mind. Even if we can sustain
the house holds and societies with their own generated power, we can resolve a
major issue of power crisis. Concepts of Smart-grid, Off Grid houses, Hybrid power
systems, etc. are catching the market.
State Governments are learning: After the successful Gujarat model, and the
growing Solar market, thus the fall of panel prices, are encouraging the states to push
for renewable energy sources. The Gujarat government is also trying to implement
the solar roof top policy, in Gandhinagar, to understand its applicability in the state.
Similar models are in progress in many states. Also the tariff rates are interesting,
and so is the accelerated depreciation, which makes it even more tempting for the
companies to set up a solar power plant.
Youth wants cleaner energy: The very buzz word of “Green Energy” is drawing
people, especially youth. The upcoming entrepreneurs and businessmen are seeking
Green Buildings, Solar roof top houses, etc. Also seeing the potential of the market,
many new entrepreneurs are working in the area of green energy.
There are various renewable energy resources in country but solar energy potential
is the highest among them. In most parts of India, clear sunny weather is experienced
approx 250–300 days a year. The received annual radiation varies between 1600 and
2200 kWh/m2 , which can be compared to the radiation received in the tropical and
sub-tropical regions. The equivalent energy potential is approximately 6000 million GWh of energy per year (Fig. 1). .
The National Action Plan on Climate Change also explained: “India is a tropical
country, where sunshine is available for longer hours per day and in great intensity.
Solar energy, therefore, has great potential as future energy source. It also has the
advantage of permitting the decentralized distribution of energy, thereby empowering
people at the grassroots level”. The tremendous amount of the total energy reaching
the earth from the sun has attracted the attention of many engineers and scientists to
consider it as a substitute for some of the present energy demand.
There are various solar energy conversion systems in which the low temperature
thermal converter is popular one, in which flat plate solar collector is an example.
However, this device is used normally only for small collection system. The collector
area range is at most few thousand square metres. Beyond this limit it is not feasible
economically and so for this purpose more appropriate system has to employ for
large applications. The various solar technology methods are shown in Fig. 2. The
idea of solar thermal power plant presents an attractive way of collecting solar energy
on a large scale to meet the energy demand for a variety of large-scale applications,
Study of Various Technologies in Solar Power Generation
513
Fig. 1 Annual mean daily global solar radiation in India [2]
such as electric power generation and industrial process heat. The attractiveness lies
in the fact that solar panels tracking the sun in two axes are used to concentrate
the incident solar radiation on a point where it is absorbed by a working fluid and
converted to thermal energy. The thermal energy can then be used directly in the
industrial process heat or transported to an energy conversion subsystem where it
converted to electricity. Solar thermal system generate medium to high temperature,
upwards of 2000 °C, for industrial process heater electric power generation. Thus
these installations can also be categorized as solar electric technologies in some
situations [6–9].
514
S. Gupta et al.
Fig. 2 Various solar technologies
4 Solar Photovoltaic
Photovoltaic is a technology that reliably converts solar radiation into electricity.
There are different types of modules depending on power ratings. Every module
has a number of solar cells. Solar cells are fabricated by means of semiconductors
such as silicon. Photovoltaic cells generate electricity in clean and reliable manner
which is the prime concern for today’s environment. Variation in temperature affects
the efficiency of solar module [9]. Due to these variations, photovoltaic technology
faces enormous challenges in its power quality performance [10]. Integration of
renewable energy is also a tedious process [11]. Photovoltaic is a kind of technology
which converts the sunlight directly into electricity. When bunch of light energy
(i.e. solar radiation) strikes the panel (which consist of number of cell) then, the
photons of sufficient energy dislodged the electrons from the atom’s cell as a result
free electrons starts moving through cell, which is creation of holes and filling of
holes in the cell. Due to this process (i.e. electrons and holes movement) it generates
electricity. Capacity of sun to supply energy is so huge that it can feed all energy
demand of the world. Generally, till now the conversion efficiency of solar energy
into useful form of energy (i.e. electrical energy) is between 15 and 20%. Due to
high investment cost needed in manufacturing process of the Si cells prevented them
from their widespread use. There are also few drawbacks of Si cells that it is toxic
in nature so, to eliminate these drawbacks a huge research and money is needed.
Luckily energy provided by sun is huge, which is 10,000 times more than that the
total energy needs, means converting 0.1% of the incident solar energy radiation with
10% efficiency can fulfil global energy needs. Concentrating photovoltaic is a new
method for production of electricity by harvesting the sun’s energy. To concentrate the
solar light at a particular, angle the varieties of solar concentrator (parabolic mirror)
are used which are mounted on solar tracker system so that the focal point remains
constant while sun changes its position across the sky. Recently the developments of
the two-axis tracking systems are become useful in concentrating photovoltaic. By
using this technology electrical output of the photovoltaic module can be improved
as shown in Fig. 3.
Study of Various Technologies in Solar Power Generation
515
Fig. 3 Block diagram SPV technology [12]
Fig. 4 Equivalent circuit of photovoltaic cell [12]
4.1 Solar Photovoltaic Technology Architecture
A solar cell, also known as a photodiode, may be modelled by a current source in
parallel with a diode. The diode in the model represents a real physical diode which
is created by the junction of P and N materials which form the solar cell. As photons
strike the cell’s surface, they excite electrons and move them across the PN junction
of the diode. Shunt and series resistances are added to obtain a better modelling of
the current-voltage characteristic. When the photovoltaic (PV) cell is illuminated and
connected to a load a potential difference (V) appears across the load and a current (I)
circulates. The cell functions as a generator as shown Fig. 4. The photons reaching
the interior of the cell with energy greater than the band gap generate electron–hole
pairs that may function as current carriers. Some of these carriers will find themselves
in or near the potential barrier and are accelerated as shown to form the photonic
current.
516
S. Gupta et al.
Fig. 5 Solar thermal conversion system [2]
5 Solar Thermal
Solar thermal technology is used to generate large amount of green energy (solar
energy) which helps to mitigate the pollution and consequently give a good living
condition for human. Solar thermal energy (shown in Fig. 5) also help to mitigate the
use of fossil fuels which is primarily factor responsible for enhancing the temperature
of atmosphere on earth. A solar thermal power plant produces electric power by
converting large amount of sunlight energy (photons) into the high-temperature heat
energy with the help of various mirrors configurations. Solar thermal power plant
plants are used to work efficiently over a 20-year period. India can have solar thermal
power plants of 5–6 GW capacity by 2020. Large amount of solar thermal power
plant output is consumed by various states in North-Eastern part of India.
6 Solar Hybrid
A PV-T hybrid is a mixture of photovoltaic and thermal technologies, hybrid technologies can not only produce electricity but can also produce high temperature
thermal heat as shown in Fig. 6. As we know that demand of electricity is increasing,
it is very important to develop such devices which can produce both solar electricity
and solar heat which can further be converted into electricity. When sunlight strike at
the Photovoltaic cells the parts of incident rays of light are used to produce the electric
energy and the rest is converted into heat energy. If temperature of the photovoltaic
module increases after a certain value, the efficiency of the photovoltaic module start
decreases. So, by developing methods to cool the module can improve overall efficiency of PV-T integration. PV-T system has better way to utilize solar energy as they
can provide higher overall efficiency than other solar power systems. Poly-crystalline
(pc-Si), mono-crystalline (c-Si), thin-film solar cells or multi-junction cells can be
used as a photovoltaic material. There are many researches and development work
has been carried out in this field and also there are many researches and development
is going on PV-T hybrid. Due to the dual characteristics of PV-T hybrid it has huge
Study of Various Technologies in Solar Power Generation
517
Fig. 6 Schematics of thermo-photo-voltaic generator [2]
scope in future. There are few features of the Photovoltaic-Thermal (PV-T) hybrid
systems are as follows:
• Double purpose: single device used to generate electricity as well as heat output.
• Efficiency and Flexibility: It is experimentally proved that the effectiveness of
PV-T hybrid is always higher than the device which operate on photovoltaic and
thermal technologies independently and hybrid can be used where space is limited.
• Wide applications: The heat energy output produced by PV-T hybrid can be used
for various purposes,
• Low cost and practical in nature: PV-T hybrid can easily be combined or integrated
with buildings and its cost is also affordable.
7 Principal Limitation
There are several limitations in the effective conversion of solar energy into electric
power. Some of them are
1. The main problem is that the efficiency of the collection system decreases as the
collection (operating) temperature increases while the efficiency of the engine
increase as the working fluid temperature increases.
2. The theoretical efficiency that can be obtained by any heat engine operating
between two temperatures is well understood and provides fixed fundamental
barriers.
3. A part of heat is lost from the working fluid during its passage from the collector.
518
S. Gupta et al.
4. Due to the intermittent nature of the solar energy some kind of energy storage
device is required to operate the heat engine continuously. The heat storage
material degrades with time.
5. Like many other fields the material of construction of a heat engine and a suitable
working fluid and their interaction cause a problem. The construction material
should withstand the high temperature and pressure.
6. In solar electric power generation both solar collectors and engine cause problems. Solar collectors are generally more expensive than the engine. They require
large area for installations.
8 Conclusion
In India energy shortage is a big problem, so it needs huge additions in energy capacities especially in renewable energy capacities to meet the surging energy demand.
By developing solar energy, we can improve energy security in India which can help
India to be energy independent, it can also help in reduction of the fuel prices. In
India, many of the undeveloped states have great potential for solar energy they can
develop solar power systems with the help of government to use solar energy. PV-T
solar hybrid system can fulfil the energy needs in India as they efficiently convert
incident solar energy into electrical and thermal energy and it can also help in slowing
down the increasing rate of pollution.
PV-T hybrid technology can be enhanced by improvisation in the design, area of
the design and development in the field of exergy output of the system. PV-T hybrid
technology can be used for industrial and personal applications, i.e. solar heat pump,
water purification, solar cooling and solar greenhouse. By removing some hindrance
in the field of social aspect such as the lack of information, public awareness, and
social acceptance of green energy technology, its future scope will be enhanced.
References
1. Skoplaki, E., Palyvos, J.A.: On the temperature dependence of photovoltaic module electrical
performance: a review of efficiency/power correlations, Sol Energy 83, 614–624 (2009)
2. Ministry of New and Renewable Energy source (MNRE), http://mnre.gov.in/filemanager/ann
ual-report/2016-2017/EN/pdf/1.pdf (Data retrieved 31 Dec 2016)
3. Kern, E.C., Jr., Russell, M.C., Combined photovoltaic and thermal hybrid collector system. In:
Proceedings of 13th IEEE Photovoltaic Specialist, pp. 1153–1157 (1978)
4. Timilsina, G.R., Kurdgelashvili, L., Narbel, P.A.: Solar energy: markets, economics and policies. Renew. Sustain. Energy Rev. 16(1), 449–465 (2012)
5. Parlak, K.S.: PV array reconfiguration method under partial shading conditions. Electr. Power
Energy Syst. 63, 713–721 (2014)
6. Patra, S., Kishor, N., Mohanty, S.R., Ray, P.K.: Power quality assessment in 3-U grid connected
PV system with single and dual stage circuits. Electr. Power Energy Syst. 75, 275–288 (2015)
Study of Various Technologies in Solar Power Generation
519
7. Pinto, S.J., Panda, G.: Performance evaluation of WPT based islanding detection for grid
connected PV systems. Electr. Power Energy Syst. 78, 537–546 (2014)
8. Aringhoff, R., Brakmann, G., Geyer, M., Teske, S.: Concentrated solar thermal power. Greenpeace International (2005)
9. Stoddard, L., Abiecunas, J., O’Connell, R.: Economic, energy, and environmental benefits of
concentrating solar power in California. National Renewable Energy Laboratory (2006)
10. Arif Hasan, M., Sumathy, K.: Photovoltaic thermal module concepts and their performance
analysis: a review. Renew. Sustain. Energy Rev. 14, 1845–1859 (2010)
11. Chow, T.T.: A review on photovoltaic/thermal hybrid solar technology. Appl. Energy 87,
365–379 (2010)
12. Garg, P.: Energy scenario and vision 2020 in India. J. Sustain. Energy Environ. 3(1), 7–17
(2012)
Reduction of Test Data Volume Using
DTESFF-Based Partial Enhanced Scan
Method
Ashok Kumar Suhag
Abstract Scan architecture is widely used method for testing of transition delay
faults (TDF). Launch-on-capture (LOC) and Launch-on-shift (LOS) are methods in
scan-based test. In scan-based test all the possible combinations of two pattern delay
tests cannot be applied to the circuit under test due to the structural constraints of
scan which results in poor test coverage. This problem is alleviated in enhanced scan
method as it supports random test vectors for delay test vector pairs at the cost of
significant area overhead. The area overhead for enhanced scan chain method can be
reduced by replacing the redundant flip-flop with the hold latch in enhanced scan flipflop. Hold latch based enhanced scan design needs a fast hold signal similar to scanenable signal in LOS testing. Delay Testable Enhanced Scan Flip-Flop (DTESFF)
implements the enhanced scan cell with the slow hold signal. In this work, DTESFFbased partial enhanced scan method is proposed for the reduction of test data volume.
Simulation results on ISCAS ’89 benchmark circuit displays reduction of test data
volume.
Keywords Scan test · Enhanced scan design LOC · TDF test and partial enhanced
scan method
1 Introduction
Delay fault appears when critical path delay exceeds clock time period and unable
to satisfy timing requirements. Process variation leads to physical defects like gate
oxide failure, via voids and resistive open and short, etc., are the major source of
timing defects. Timing failures are caused by delay defects and coming in picture
more frequently in deep submicron technologies, that is why incorporation of delay
fault with traditional stuck at fault testing becomes mandatory. Conventional functional at speed test suffers from high test-development cost particularly when size of
A. K. Suhag (B)
School of Engineering and Technology, BML Munjal University, Gurgaon 122413, India
e-mail: ashoksihag@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_50
521
522
A. K. Suhag
design is in millions of gates. Moreover SOC designs have limited access to internal
cores which makes functional at speed tests impractical. Both controllability and
observability of the internal signals in SOC’s are improved with scan based delay
tests.
In scan-based delay test two test vectors are needed for testing of transition delay
faults (TDF). First vector is called initialization pattern (V 1 ) while second vector is
called launch pattern (V 2 ). Architectural limitation of scan did not allow all combination of V 1 & V 2 that can be applied via scan test. The method by which vector V 2
is created in scan test is classified as Launch-on-Shift (LoS) [1, 2], or Launch-onCapture (LoC) [3]. In both methods vector V 2 is reliant on vector V 1 which limits the
test coverage. LoS test displays better coverage compared to LoC test but it needs
fast scan-enable signal which is not assisted by the most of designs [4]. Other scan
designs for elimination of performance penalty of output gating method are discussed
[5, 6]. Scan chain diagnosis also has now become very crucial [7].
In enhanced scan design one additional redundant flip-flop is provided with scan
flip-flop which eliminate the vector V 2 dependency on vector V 1 and results in good
test coverage at the expense of significant area overhead [8]. The cost of duplicating
all flip-flop is very high so alternatively hold latch-based enhanced scan design
saves some hardware but utilizes extra signal similar to LoS test which is expensive.
This problem is resolved in delay testable scan flip-flop by supporting slow hold
signal [9–11]. Enhanced scan method provides good test coverage at cost of high
area overhead. The benefits of full enhanced scan design can be obtained by partial
enhanced scan design [12]. In this paper, partial enhanced scan design using delay
testable enhanced scan flip-flop (DTESFF) is used to reduce the test data volume with
low area overhead compared to full enhanced scan design. Simulation is carried out
through commercial test generation tool on ISCAS-89 benchmark circuits. Results
demonstrate the significant reduction in test data volume by implementing few scan
flip-flops to DTESFF.
2 Review of DTESFF Design
The Structure of DTESFF discussed [10, 11] is shown in Fig. 1. Redundant flip-flop
of enhanced scan design is replaced with a hold latch to reduce the area overhead
and one extra AOI gate is used in DTESFF design for the alignment of hold signal.
In the DTESFF design timed controlled hold signal is generated from slow hold
signal with the help of AOI gate and generates a proper timing signal (Fig. 2) for the
holding and transmission of test pattern supported by enhanced scan method.
Reduction of Test Data Volume Using DTESFF-Based Partial …
Fig. 1 Delay testable
enhanced scan flip-flop
(DTESFF)
D I/P
Q
0
Scan I/P
523
D- Flip-flop
1
Hold
Latch
Scan enable
Clock
Hold Signal
V1
V2
Clock
Hold Signal
AOI Gate Controlled Signal
Fig. 2 Timing diagram of delay testable enhanced scan flip-flop (DTESFF) using slow hold signal
3 Selection Procedure of Scan Flip-Flop for Partial
Enhanced Scan
In partial enhanced scan method some of the scan flip-flops are upgraded to enhanced
scan flip-flop. Selection criterion behind the upgradation is based on the poor controllability of flip-flops. Two test vectors are required for the detection of delay faults
in scan test. Test vector V 1 is arbitrarily selected and scanned inside the scan chain
while vector V 2 is generated with the help of vector V 1 in scan test. Test vector
V 2 cannot be selected arbitrarily because of structural limitation of scan and due
to this the probability of second vector to be either 0 or 1 in scan test is not 50%
sometimes which results in very poor controllability. Some of the second test patterns bits are biased bits as they retain most of times either 0 or 1 which results in
524
A. K. Suhag
degraded test coverage. To enhance the test coverage these biased values is changed
to unbiased values simply by upgrading that scan flip-flop to DTESFF which offers
the flexibility in selection of second test vector. Combinational logic block output
signal probabilities for unbiased random inputs are computed through probabilistic
analysis.
4 Proposed Method
In this approach first of all untestable TDF are eliminated as untestable fault has
no effect on decrease of test data volume after that fault collapsing is leveraged
to reduce the size of fault set and helps in achieving more precise fault set. After
this signal probability is computed for the selection of scan flip-flops which needs
to be upgraded and finally after the upgradation of scan flip-flops to DTESFF the
transition delay fault coverage is computed with the help of commercial available
test generation tool.
5 Experimental Results
The efficacy of partial enhanced design using DTESFF is validated on ISCAS 89
benchmark circuits. The profile of ISCAS 89 benchmark circuits involves 600–1650
flip-flop in the design. Benchmark circuits having fewer flip-flops offer significant
results. Large benchmark circuits like S38417 having 1636 flip-flops demonstrates
good TDF coverage of 95.8% using pure LoC method gives a little room for the
improvement. Experimental results are shown in Table 1.
Test patterns for TDF are generated with the help of commercial available ATPG
tool. The selection criterion behind the upgradation of scan flip-flops to the DTESFF
is based on the analysis of signal probability of input signals. Scan flip-flops with poor
controllability are first augmented by DTESFF design. Scan flip-flops are sequenced
for each design is based on signal probability and flip-flop with poorest controllability
Table 1 Percentage of DTESFF versus fault coverage and number of test pattern
Circuit
LoC method
1% replaced
2% replaced
5% replaced
S13207
S15850
S38417
S38584
Fault
Test
coverage patterns
Test
No. of
coverage test
patterns
Test
No. of
coverage test
patterns
Test
No. of
coverage test
patterns
72.34
63.85
95.80
67.35
72.34
63.86
95.8
67.39
72.42
63.85
95.81
67.35
92.4
91.1
97.8
78.21
209
164
194
520
90
107
181
204
79
98
182
176
45
67
164
136
Reduction of Test Data Volume Using DTESFF-Based Partial …
525
600
400
200
S13207
S15850
LoC Method
1% Replaced
2% Replaced
No. of test paƩerns
Test coverage
No. of test paƩerns
Test coverage
No. of test paƩerns
Test coverage
Test paƩerns
Fault coverage
0
S38417
S38584
5% Replaced
Fig. 3 Test pattern and fault coverage comparison
value is on the top in priority list. After upgradation of scan flip-flop to DTESFF
transition delay fault coverage with number of test patterns is examined by enhancing
the number of DTESFF in each design.
The results of Table 1 and Fig. 3 demonstrates that by augmenting fewer scan
flip-flops DTESFF offers a great improvement in terms of the LoC test coverage as
well as in reduction of test patterns at the expense of little area overhead.
6 Conclusion
In this work few scan flip-flops were augmented to the DTESFF to decrease the
number of test coverage using partial enhanced test method. Furthermore improvement in test coverage is also observed. Probability distribution function is used for
selection of scan flip-flops which needs to be upgraded to DTESFF. The simulation
results on ISCAS 89 benchmark circuits demonstrated that the same fault coverage
can be achieved with reduced number of test patterns if we augment one percent
scan flip-flop to the DTESFF design as compared to the traditional LOC method.
Moreover the transition delay fault coverage is also enhanced at the expense of little
area overhead. Partial enhanced scan based on DTESFF design allowed using hold
latch with slower hold signal.
526
A. K. Suhag
References
1. Patil, S., Savir, J.: Skewed-load transition test: part II, coverage. In: Proceedings of International
Test Conference, p. 714 (1992)
2. Savir, J.: Skewed-load transition test: part I, calculus. In: Proceedings of International Test
Conference, p. 705 (1992)
3. Savir, J., Patil, S.: On broad-side delay test. Very Large Scale Integr. (VLSI) Syst., 2, p. 368
(1994)
4. Xu, G., Singh, A.D.: Low cost launch-on-shift delay test with slow scan enable. In: Proceedings
of European Test Symposium (2006)
5. Suhag, A.K., Ahlawat, S., Shrivastava, V. Choudhary, R.R.: Output gating performance overhead elimination for scan test. Int. J. Electron. 102(7), 1244–1252 (2015)
6. Suhag, A.K., Ahlawat, S., Shrivastava, V., Singh, N.: Elimination of output gating performance
overhead for critical paths in scan test. Int. J. Circ. Archit. Des. 1(1), 62–73 (2013)
7. Ahlawat, S., Vaghani, D., Tudu, J., Suhag, A.: A cost effective technique for diagnosis of
scan chain faults. In: Kaushik, B., Dasgupta, S., Singh, V. (eds.) VLSI Design and Test (VDAT
2017), Communications in Computer and Information Science, vol. 711, pp. 191–204. Springer,
Singapore (2017)
8. Bushnell, M.L., Agrawal, V.D.: Essentials of Electronic Testing for Digital, Memory and
Mixed-Signal VLSI Circuits. Springer (2000)
9. Suhag, A.K., Shrivastava, V., Singh, N.: Flip–flop selection for partial enhance scan chain
using DTESFF for high transition delay fault coverage. Int. J. Syst. Assur. Eng. Manag. 4(3),
303–311 (2013)
10. Suhag, A.K., Shrivastava, V.: Delay testable enhanced scan flip–flop: DFT for high fault coverage. In: Proceedings of International Symposium on Electronic System Design (ISED), pp:
129–133, (2011)
11. Suhag, A.K., Shrivastava, V.: Performance evaluation of delay testable enhanced scan flip–flop.
Int. J. Syst. Assur. Eng. Manag. 3(3), 169–174 (2012)
12. Pei, S., Li, H., Li, X.: Flip-Flop selection for partial enhanced scan to reduce transition test
data volume. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 20(12), 2157–2169 (2012)
Performance Analysis and Optimization
of Vapour Absorption Refrigeration
System Using Different Working Fluid
Pairs
Paras Kalura, Susheem Kashyap, Vishal Sharma, Geetanjali Raghav
and Jasmeet Kalra
Abstract In this paper, an attempt has been made to analyse the performance by
simulation of vapour absorption refrigeration systems (VARS) by changing their
working fluid pairs. The study has been done with the help of simulation of VARS
for ease of and accuracy in calculations. The temperature and enthalpy of the components of VARS has been observed in this paper to obtain the coefficient of performance (COP) for the different fluids. The refrigerant pairs chosen for the study are
NH3 –H2 O, NH3 –LiNO3 , NH3 –NaSCN and LiBr–H2 O. The results depict the performance comparison of all the fluid under same circumstances and also explained
the pair of working fluid having optimum parameters with maximum C.O.P. The
paper should be of interest to readers in the areas of energy and its applications.
Keywords VARS · Refrigerant · Absorber · Generator
1 Introduction
Refrigeration is the process of moving heat from one location to other in controlled
conditions [1]. The primary objective of refrigeration is to decrease the temperature of
a controlled area by removing heat from this area and releasing it in the surroundings.
French scientist Ferdinand Carré invented the first aqua-ammonia absorption system in 1858 [2]. A liquid pump was used to increase the pressure of strong solution.
In 1922, Balzarvon Platen and Carl Munters, two students at Royal Institute of Technology, Stockholm invented a three-fluid system that did not need a pump. In 1926,
Albert Einstein and his former student Leó Szilárd proposed an alternative design
known as Einstein refrigerator [3]. Since, the 1960s emphasis has been made on
the development of renewable energy based refrigeration systems. This system uses
P. Kalura (B) · S. Kashyap · V. Sharma · G. Raghav
University of Petroleum and Energy Studies, Dehradun 248006, India
e-mail: kush_kalura@yahoo.co.in
J. Kalra
Graphic Era Hill University, Dehradun 248001, India
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_51
527
528
P. Kalura et al.
heat energy instead of mechanical energy, as in Vapour Compression Refrigeration System, in parliamentary procedure to alter the state of refrigerant needed for
the refrigeration cycle. In the existing scenario the market is dominated by current
technology of VCRS due to its high performance [4]. But VCRS are a threat to
environment; VARS on the other hand are eco-friendly and uses renewable sources
of energy [4]. VARS also has many advantages over the VCRS in terms of cost
effectiveness, renewability, and maintenance and energy consumption.
The refrigerant commonly used in vapour absorption system is Ammonia [5].
The primary focus of our project is to compare the performance of existing VARS by
changing their working fluids and to identify the fluid pair using which the system
can be optimized for highest efficiency. Many important parameters for selecting
refrigerants are heat of vapourization, heat of solution, vapour pressure, solubility of
refrigerant, heat capacity of solution, viscosity of solution.
2 Methodology
2.1 Analysis of Fluid Pairs with NH3 as Refrigerant, i.e.
NH3 –H2 O, NH3 –LiNO3 and NH3 –NaSCN
In parliamentary law to utilize the equation, mass and energy conservation should
be specified for each element.
For the generator, the mass and energy balances yield
m 7 m 1 + m 8 (mass balance)
(1)
m 7 x7 m 1 + m 8 x8 (NH3 mass balance)
(2)
Q8 m1h1 + m8h8 − m7h7
(3)
From Eqs. (2) and (3), the flow rates of hard and weak solutions can be explained
m 8 (1 − x7 )(m 1 )/(x7 − x8 )
(4)
m 7 (1 − x8 )(m 1 )/(x7 − x8 )
(5)
Ultimately, energy balances for the absorber condenser and evaporator yield
Q i m 4 h 4 + m 10 h 10 − m 5 h 5
(6)
Q c m 1 (h 1 − h 2 )
(7)
Q e m 1 (h 4 − h 3 )
(8)
If the generator, condenser, absorber and evaporator temperatures and the refrigeration mass flow rate or the required refrigerating load is turned over, the above
equation can be acted out simultaneously to make the system functioning.
Performance Analysis and Optimization of Vapour Absorption …
529
Thermodynamic Properties
For NH3 –H2 O, NH3 –LiNO3 and NH3 –NaSCN absorption refrigeration cycles, NH3
are the refrigerant, H2 O, LiNO3 and NaSCN are absorbents. The thermodynamic
properties of states (1)–(4) in Fig. 1 are determined by NH3 , and other properties in states (5)–(10) can be calculated based on the binary mixture of NH3 –H2 O,
NH3 –LiNO3 or NH3 –NaSCN solutions.
NH3 –H2 O Pair
The relation between saturated pressure and temperature variables of an ammoniawater mixture can be explained as
log P A − (B/T )
(9)
where [6]
A 7.44 − 1.767x + 0.9823x 2 + 0.3627x 3
B 2013.8 − 2155.7x + 1540.9x − 194.7x
2
Fig. 1 Schematic diagram of vapour absorption refrigeration system
(9a)
3
(9b)
530
P. Kalura et al.
The relation between temperature, concentration and enthalpy is mentioned below,
with respective coefficients mentioned
16
h T, x ai 100((T /273.16) − 1)mi x ni ,
(10)
i0
where x is the ammonia mole fraction which can be explained as follows (Table 1):
x 18.015x/(18.015x + 17.03(1 − x))
(10a)
NH3 –LiNO3 Pair
The relation between saturation pressure and temperature of an ammonia-lithium
nitrate mixture is given as
ln P A + (B/T ),
(11)
where
A 16.29 + 3.859(1 − x)3
(11a)
B −2802 − 4192(1 − x)
(11b)
3
The relation between the temperature, enthalpy and concentration can be written
as
h(T, x) A + B(T − 273.15) + C(T − 273.15)2 + D(T − 273.15)3
(12)
where A, B, C and D are constants and are calculated as [6]
A −215 + 1570(0.54 − x)2 if x ≤ 0.54
Table 1 Coefficients for Eq. (10) [8]
i
mi
ni
ai
i
(12a)
mi
ni
ai
9
2
1
2.84179 × 100
1
0
1
−7.61080 × 100
2
0
4
2.56905 × 101
10
3
3
7.41609 × 100
11
5
3
8.91844 × 102
3
0
8
−2.47092 × 102
4
0
9
3.25952 × 102
12
5
4
−1.61309 × 103
13
5
5
6.22106 × 102
5
0
12
−1.58854 × 102
6
0
14
6.19084 × 101
14
6
2
−2.07588 × 102
15
6
4
−6.87393 × 100
16
8
0
3.50716 × 100
7
1
0
1.14314 × 101
8
1
1
1.18157 × 100
Performance Analysis and Optimization of Vapour Absorption …
531
A −215 + 689(x − 0.54)2 if x ≥ 0.54
(12b)
B 1.15125 + 3.382678x
(12c)
−3
C 10 (1.099 + 2.3965x)
(12d)
−5
D 10 (3.93333x)
(12e)
NH3 -NaSCN Pair
The relation between intensity and temperature of an ammonia-sodium thiocyanate
mixture is given by
ln P A + (B/T ),
(13)
where
A 15.7266 − 0.298628x
B −2548.65 − 2621.92(1 − x)
(13a)
3
(13b)
The relation between the temperature, enthalpy and concentration are [6]
h(T, x) A + B(T − 273.15) + C (T − 273.15)2 + D(T − 273.15)3 ,
(14)
where
A 79.72−1072x + 1287.9x 2 − 295.67x 3
B 2.4081 − 2.2814x + 7.9291x − 3.5137x
C 10−2 1.255x−4x 2 + 3.06x 3
D 10−5 −3.33x + 10x 2 − 3.33x 3
2
(14a)
3
(14b)
(14c)
(14d)
2.2 Analysis of LiBr–H2 O Fluid Pair
Let m be the mass flow rate of refrigerant in kg/s
mss and mws be the mass flow rate of strong solution and weak solution in kg/s
The range of the temperature for the generator Tg can be taken from 55 to 90 °C.
The range of the temperature for the Condenser Tc can be taken from 24 to 46 °C
The range of the temperature for the Absorber Ta can be taken from 16 to 32 °C
The range of the temperature for the Evaporator Te can be taken from 2.5 to 10 °C
While the operating temperatures for the different components can be taken as follows:
Generator Temperature (Tg) to be 64 °C
Condenser Temperature (Tc) to be 30 °C
532
P. Kalura et al.
Absorber Temperature (Ta) to be 20 °C
Temperature of evaporator (Te) to be 4 °C
Equations
Heat (Q) and Mass (m) balance for every Component [7]:
Evaporator: Applying the heat and mass balance
Q e (Refrigerating effect) m(h 4 −h 3 ) 5.25 kW
Cs (Circulation ratio) ws (ss − ws) m ∗ cs
(16)
m ws (1 + cs) ∗ m
(17)
(15)
Absorber: Applying the balance of energy
Qa mh4 + mss h10 − mws h5
(18)
Solution Heat Exchanger (HE):
m ws ∗ (h 7 − h 6 ) m ss ∗ (h 8 − h 9 )
(19)
Q G mh 1 + m ss h 8 − m ws h 7
(20)
Q c m(h 1 −h 2 )
(21)
COP Q E /Q G
(22)
Generator:
Condenser:
3 Results
MATLAB was used to solve the empirical relations and equations, in the previous
section, to find the best refrigerant pair for a VARS system. The results obtained
from these calculations are as follows (Tables 2, 3, 4, 5 and 6).
The following graphs indicate the MATLAB results of the equations and relations
in (Figs. 2, 3, 4 and 5).
Performance Analysis and Optimization of Vapour Absorption …
Table 2 Observations for NH3 –H2 O Fluid Pair
State
T (°C)
P (kPa)
x (%)
533
m (kg/min)
h (kJ/kg)
Generator exit 100
(1)
1166.92
100
1.00
1448.44
Condenser
exit (2)
30
1166.92
100
1.00
333.78
Evaporator
exit (4)
−5
354.42
100
1.00
1456.60
Absorber exit
(5)
25
354.42
52.24
3.56
−125.37
Generator
inlet (7)
67
1166.92
52.24
3.56
76.48
Generator exit 100
(8)
1166.92
33.55
2.56
284.66
Absorber inlet
(10)
354.42
33.55
2.56
19.98
Table 3 Observations for NH3 –LiNO3
State
T (°C)
P (kPa)
x (%)
m (kg/min)
h (kJ/kg)
40
Generator exit 100
(1)
1166.92
100
1.00
1448.44
Condenser
exit (2)
30
1166.92
100
1.00
333.78
Evaporator
exit (4)
−5
354.42
100
1.00
1456.60
Absorber exit
(5)
25
354.42
52.24
4.09
−139.11
Generator
inlet (7)
67
1166.92
52.24
4.09
−6.31
Generator exit 100
(8)
1166.92
33.55
3.09
103.44
Absorber inlet 40
(10)
354.42
33.55
3.09
−74.89
534
P. Kalura et al.
Table 4 Observations for NH3 –NaSCN
State
T (°C)
P (kPa)
x (%)
m (kg/min)
h (kJ/kg)
Generator exit 100
(1)
1166.92
100
1.00
1448.44
Condenser
exit (2)
30
1166.92
100
1.00
333.78
Evaporator
exit (4)
−5
354.42
100
1.00
1456.60
Absorber exit
(5)
25
354.42
52.24
5.35
−97.40
Generator
inlet (7)
67
1166.92
52.24
5.35
25.53
Generator exit 100
(8)
1166.92
33.55
4.35
115.78
Absorber inlet
(10)
354.42
33.55
4.35
−41.79
40
Table 5 Observations for LiBr–H2 O
State point
Temperature Pressure (bar) Enthalpy h
(°C)
(kJ/kg)
1
2
3
4
5
6
7
8
9
10
Table 6 Results
Energy (kW)
20
20
55
64
20
20
64
30
30
4.0
−180
−180
−115.7
−120
−195
−195
2600
125.7
126
2510
6.1
32
32
32
32
6.1
32
32
6.1
6.1
Concentration m (kg/s)
x 0.48
0.48
0.48
0.56
0.56
0.56
–
–
–
–
0.0154
0.0154
0.0154
0.0132
0.0132
0.0132
0.0022
0.0022
0.0022
0.0022
NH3 –H2 O
NH3 –LiNO3
NH3 –NaSCN
LiBr–H2 O
Generator (Qg )
1.7576
1.8979
1.843
6.9384
Condenser (Qc )
1.1023
1.1147
1.2009
4.6784
Absorber (Qa )
1.6546
1.7196
1.5869
4.63
Evaporator (Qe )
1.0878
1.1229
1.1927
4.25
COP
0.6189
0.5917
0.6472
0.6125
Performance Analysis and Optimization of Vapour Absorption …
Fig. 2 Graph for Circulation
Ratio with respect to
Condenser Temperature
Fig. 3 Graph for Circulation
Ratio with respect to
Evaporator Temperature
535
536
Fig. 4 Graph for COP with respect to Temperature of Generator
Fig. 5 Graph for COP with respect to Evaporator Temperature
P. Kalura et al.
Performance Analysis and Optimization of Vapour Absorption …
537
4 Conclusions
• Using MATLAB it was observed that the coefficient of performance of NH3 NaSCN fluid pair is better than NH3 –H2 O, LiBr–H2 O and NH3 –LiNO3.
• From the graph it could be seen that COP increases with increase in generator and
evaporator temperature.
• Also from the graph, the COP of NH3 –NaSCN is maximum.
• So it can be concluded that best refrigerant pair to be used in a VARS system is
NH3 –NaSCN.
References
1. Arora, C.P.: Refrigeration and Air Conditioning, 2nd edn, pp. 427–437. McGraw-Hill Publication (2000, 1981)
2. Granryd, E., Palm, B.: Refrigerating engineering, Stockholm Royal Institute of Technology
(2005), see Chap. 4-3
3. Einstein, A., Szilard, L.: US Patent 1781541, Refrigeration Filed Dec 16 1927 Patented Nov 11,
1930 United States Patent Office
4. Micallef, D., Micallef C.: Mathematical model of vapour absorption refrigeration unit (2010)
5. Raghuvanshi, S., Maheshwari, G.: Analysis of ammonia–water (NH3 -H2 O) vapor absorption
refrigeration system based on first law of thermodynamics, August (2011)
6. Sun, D.-W.: Comparison of the performances of NH3 -H2 O, NH3 -LiNO3 and NH3 -NaSCN
absorption refrigeration systems, Dublin, July (1996)
7. Sun, D.-W.: Thermodynamic design data and optimum design maps for absorption refrigeration
systems. Appl. Therm. Eng. 17(3), 211–221 (1997)
8. ASHRAE: ASHRAE Handbook, Fundamentals, Chapter 17, pp. 17.45 and 17.81. ASHRAE,
Atlanta (1993)
Vehicle Routing Problem with Time
Windows Using Meta-Heuristic
Algorithms: A Survey
Aditya Dixit, Apoorva Mishra
and Anupam Shukla
Abstract Meta-Heuristic Algorithms are one of the most widely used optimization
algorithms. Vehicle routing problem with time windows (VRPTW) is a famous NPhard combinatorial optimization problem that plays a key role in logistics systems.
In this paper, we review some of the recent advancements in the VRPTW using
meta-heuristic techniques. Many variants of the classical Vehicle Routing Problem
(VRP) are also presented. An extensive survey of the related research is presented
with a stress on the different approaches used to solve this problem. A review of
various evolutionary and swarm intelligence based algorithms like Particle Swarm
Optimization (PSO), Ant Colony Optimization (ACO), Artificial Bee Colony algorithm (ABC), etc., for solving VRPTW is also presented. Finally, the research gaps
inferred after analyzing the previous works are highlighted along with the future
prospects in this field of research.
Keywords Vehicle routing problem · Nature-inspired algorithms
Meta-heuristics · Evolutionary algorithms
1 Introduction
VRP is a complex combinatorial optimization problem. It involves designing of paths
for a group of vehicles for customers [1]. Each path begins and ends at the depot. Each
customer is visited only once by exactly one vehicle. When VRP is combined with
time window constraint, the problem is termed as VRPTW. In recent years, logistics
distribution is playing an important role and no doubt VRPTW plays a crucial role in
A. Dixit · A. Mishra (B) · A. Shukla
ABV-Indian Institute of Information Technology and Management, Gwalior, India
e-mail: apoorvamish1989@gmail.com
A. Dixit
e-mail: a.dixit93@gmail.com
A. Shukla
e-mail: dranupamshukla@gmail.com
© Springer Nature Singapore Pte Ltd. 2019
N. Yadav et al. (eds.), Harmony Search and Nature Inspired Optimization Algorithms,
Advances in Intelligent Systems and Computing 741,
https://doi.org/10.1007/978-981-13-0761-4_52
539
540
A. Dixit et al.
that. The aim of VRPTW involves the minimization of the number of vehicles (NV)
and the total travel distance (TD). A typical solution to the VRP is represented by
Fig. 1 as shown.
The rest of this paper is organized as follows. Section 2 presents the definition of
the VRP