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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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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. 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(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. 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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. 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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