METAHEURISTICS BASED ON
GENETIC ALGORITHM AND TABU SEARCH
FOR VEHICLE ROUTING PROBLEM WITH STOCHASTIC DEMANDS
IRHAMAH
UNIVERSITI TEKNOLOGI MALAYSIA
i
METAHEURISTICS BASED ON
GENETIC ALGORITHM AND TABU SEARCH
FOR VEHICLE ROUTING PROBLEM WITH STOCHASTIC DEMANDS
IRHAMAH
A thesis submitted in fulfilment of the
requirements for the award of the degree of
Doctor of Philosophy (Mathematics)
Faculty of Science
Universiti Teknologi Malaysia
NOVEMBER 2008
iii
To those that have affected my life
in the most wondrous way:
My husband Hendro Nurhadi,
My daughter Hana Azizah Nurhadi,
My mother and father Alm. Noor Hidayati and Ali Muhayin,
My mother and father in law Sri Wahyuni and Setiyono
iv
ACKNOWLEDGEMENTS
BismillahirRahmaanirRahiim. Alhamdulillaahi rabbil ‘aalamiin.
I wish to deeply express my gratitude to my supervisor, Prof. Dr. Zuhaimy
Ismail, for the great opportunity to join in this research.
His confidence,
understanding, supervision and support have been essential factors for me to work on
this study.
Next, I would like to thank Ministry of Science, Technology and Innovation
Malaysia for financial support; Department of Mathematics and Research
Management Centre, Universiti Teknologi Malaysia. I also thank my employer
Institut Teknologi Sepuluh Nopember (ITS) Surabaya – Indonesia for granting me
the study leave.
My special thanks also go to Dr. Zaitul Marlizawati Zainuddin for her advice,
support and lessons on heuristic methods. My sincere appreciation also extends to
lecturer at Department of Mathematics-UTM, all my colleagues in ITS and others
(including Indonesian people in UTM) who have supported me in various occasions.
My sincere thanks should be extended to the members of ISDAG, FS – UTM where I
joined at. I am very grateful also to Mr. Triyono and Mr. Komarudin Sayuti for great
discussion and their assistance in programming.
And the last but not the least, I wish to express my heartfelt thanks for my
family during my study: my husband Hendro Nurhadi, my daughter Hana Azizah
Nurhadi, my parents Ali Muhajin and almh. Noor Hidayati, my parents-in-law Sri
Wahyuni and Setiono, and all of my big familiy in Malang, Cepu, Surabaya and
Blitar.
Their understanding, patience, love, prayer, support and sacrifice are
unbounded.
v
ABSTRACT
This study considers a Vehicle Routing Problem with Stochastic Demands
(VRPSD) where the demands are unknown when the route plan is designed. The
VRPSD objective is to find an a priori route under preventive restocking that
minimizes the total expected cost, subject to the routing constraints, under the
stochastic demands setting. Various metaheuristics based on Genetic Algorithm
(GA) and Tabu Search (TS) were proposed to solve VRPSD. This study began with
investigating the effect of static and dynamic tabu list size in TS. The results showed
the advantage of dynamic tabu list size in significantly reducing the probability of
cycling. Further, Reactive Tabu Search (RTS) which has never been used in VRPSD
was introduced. This study showed that RTS give significant improvement to the
solution quality of TS. This study then explored the enhancement of GA for VRPSD
by proposing Adaptive GA (AGA), Breeder GA (BGA) and two types of Hybrid GA
with Tabu Search (HGATS).
Solutions generated using AGA were better than
solutions from fixed parameter setting, and the use of AGA reduce the amount of
time required in finding the appropriate mutation probability values of GA. The
BGA also gave an improvement to the solution quality of GA. Different schemes of
incorporating TS to GA lead to a significantly different performance of the HGATS
algorithms. Next, comparative studies between metaheuristics implemented in this
study were carried out including the comparison with previous research on GA for
VRPSD. The HGATS showed superiority in terms of solution quality compared to
other metaheuristics, followed by BGA and RTS in the second and third best
performance respectively. Furthermore, the proposed bi-objective Pareto BGA gave
better solution qualities compared to Pareto GA. Finally, the use of metaheuristics in
a case study of solid waste collection reduced significantly the company current
operation cost.
vi
ABSTRAK
Kajian ini mempertimbangkan suatu Masalah Perjalanan Kenderaan bersama
Permintaan Stokastik (VRPSD) di mana permintaan tidak diketahui ketika
perancangan laluan dibuat.
Objektif VRPSD adalah untuk menentukan suatu
perjalanan a priori di bawah pencegahan stok semula bagi mengurangkan jumlah kos
jangkaan, bergantung kepada kekangan perjalanan dengan permintaan stokastik.
Pelbagai metaheuristik berasaskan Algoritma Genetik (GA) dan Carian Tabu (TS)
telah dicadangkan untuk menyelesaikan masalah tersebut.
Kajian ini bermula
dengan menyelidik kesan statik dan dinamik saiz senarai tabu di dalam TS. Hasil
kajian menunjukkan kebaikan saiz senarai tabu dinamik telah berupaya
mengurangkan dengan signifikan kebarangkalian kitaran. Seterusnya Reactive Tabu
Search (RTS) yang belum pernah digunakan dalam VRPSD diperkenalkan di mana
ia menunjukkan RTS mempertingkatkan kualiti penyelesaian TS.
Kajian ini
seterusnya menerokai peningkatan GA untuk VRPSD dengan mencadangkan
Adaptive GA (AGA), Breeder GA (BGA) dan dua jenis Hybrid GA-Tabu Search
(HGATS).
Penyelesaian yang dihasilkan AGA adalah lebih baik berbanding
penggunaan parameter tetap dan dengan menggunakan AGA, jumlah masa yang
diperlukan untuk mencari kebarangkalian nilai mutasi GA yang sesuai juga boleh
dikurangkan. Kaedah BGA juga meningkatkan kualiti penyelesaian GA. Skema
berlainan dalam menggabungkan TS dengan GA membawa kepada perubahan
pencapaian algoritma HGATS.
Seterusnya kajian perbandingan di antara
metaheuristik yang dilaksanakan di dalam kajian ini dilakukan termasuk
perbandingan dengan kajian terdahulu tentang GA untuk VRPSD. HGATS yang
dicadangkan, menunjukan keunggulan berbanding yang lain, diikuti oleh BGA dan
RTS di tempat kedua dan ketiga pada tahap pencapaian. Seterusnya Pareto BGA
yang dicadangkan untuk menyelesaikan dua-objektif VRPSD telah menghasilkan
penyelesaian yang lebih baik berbanding Pareto GA.
Akhirnya, penggunaan
metaheuristik dalam satu kajian kes pengangkutan bahan buangan pepejal didapati
mengurangkan dengan signifikan kos operasi semasa syarikat.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
TITLE
i
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
xiii
LIST OF FIGURES
xvi
LIST OF APPENDICES
xx
INTRODUCTION
1.1 Introduction
1
1.2 Background of the Problem
3
1.3 Problem Statement of Research
8
1.4 Objectives of the Study
9
1.5 Scope of the Study
10
1.6 Significance of the Study
11
1.7 Organization of the Thesis
11
viii
2
REVIEWS OF THE LITERATURE
2.1 Introduction
14
2.2 The Vehicle Routing Problem with Stochastic Demands
(VRPSD)
14
2.2.1 Real World Applications
15
2.2.2 Solution Frameworks for VRPSD
16
2.2.3 Types of Recourse
18
2.2.4 Mathematical Formulation of Single VRPSD
20
2.2.5 Threshold and Expected Cost Evaluation
23
2.2.6 Multiple Homogeneous VRPSD
24
2.2.7 Previous Works on VRPSD
25
2.3 Heuristic Methods
31
2.3.1
Why Use a Heuristic Methods
31
2.3.2
Measuring the Quality of a Heuristic
33
2.4 Metaheuristics
34
2.4.1
Classification of Metaheuristics
36
2.4.2
Trajectory Methods
38
2.4.2.1
2.4.3
Basic Local Search: Iterative
Improvement
38
2.4.2.2
Simulated Annealing
38
2.4.2.3
Tabu Search
39
2.4.2.4
Explorative Local Search Methods
40
Population-based Methods
40
2.4.3.1
Evolutionary Computation
41
2.4.3.2
Ant Colony Optimization
43
2.5 Hybrid Metaheuristics
43
2.6 Applications of Genetic Algorithm in the Vehicle
Routing Problems
45
2.7 Applications of Tabu Search in the Vehicle Routing
Problems
49
2.7.1
49
Two Early Tabu Search Algorithms
2.7.2 Osman’s Tabu Search Algorithm
50
ix
2.7.3 Taburoute
50
2.7.4 Barbarasoglu and Ozgur’s Algorithm
51
2.7.5 Adaptive Memory Procedure of Rochat and
Taillard
52
2.7.6 Granular Tabu Search of Toth and Vigo
52
2.8 Review on Genetic Algorithms and Tabu Search for
VRPSD
53
2.9 Summary
3
56
RESEARCH METHODOLOGY
3.1
Introduction
57
3.2
Research Framework
57
3.3
Data Source
58
3.4
Data Generation
60
3.4.1
Single Vehicle
61
3.4.2
Multiple Vehicles
61
3.5
Demand’s Data Testing
62
3.5.1 Chi-Square Goodness of Fit Test
63
3.5.2
65
Kolmogorov-Smirnov Test
3.6
Paired-Samples Procedure
67
3.7
Genetic Algorithm
71
3.7.1 Selection Methods
72
3.7.2 Crossover
74
3.8
3.7.2.1
Partially-Mapped Crossover (PMX)
76
3.7.2.2
Cycle Crossover (CX)
77
3.7.2.3
Modified Crossover
77
3.7.2.4
Order Crossover
78
3.7.2.5
Order-Based Crossover
79
3.7.2.6
Position-Based Crossover
79
3.7.3 Mutation
80
3.7.4 Operator Settings in Genetic Algorithm
81
Tabu Search
82
x
3.9
The Hybrid Genetic Algorithm and Tabu Search
Scheme
88
3.10 Illustration of the Calculation of VRPSD Objective
Function
3.11 Summary
4
88
94
SOLUTION APPROACHES BASED ON TABU
SEARCH AND REACTIVE TABU SEARCH FOR
SINGLE VRPSD
4.1 Introduction
95
4.2 Tabu Search for Single VRPSD
96
4.2.1 Initial Solution
96
4.2.2 Neighbourhood Exploration
98
4.2.3 Tabu Moves
98
4.2.4 Aspiration Criterion
100
4.2.5 Stopping Criterion
101
4.3 Reactive Tabu Search
103
4.4 Computational Results
107
4.4.1 Static versus Dynamic tabu list size
107
4.4.2 Tabu Search versus Reactive Tabu Search
110
4.5 Conclusion
5
117
VARIOUS GENETIC ALGORITHM-BASED
METAHEURISTICS FOR SOLVING SINGLE VRPSD
5.1 Introduction
118
5.2 Genetic Algorithm for Single VRPSD
119
5.2.1 Chromosome Representation
120
5.2.2
Initialization
121
5.2.3
Evaluation
123
5.2.4
Roulette Wheel Selection with Elitism
123
5.2.5
Order Crossover
124
xi
5.2.6
Mutation
125
5.2.7
Stopping Criterion
126
5.3 The Development of Adaptive Genetic Algorithm
128
5.3.1. Mutation probability is random numbers in the
range of [pmmin, pmmax]
5.3.2. Adaptive mutation probability based on PDM
129
129
5.3.3. Adaptive mutation probability based on
Euclidean Distance
130
5.3.4. Adaptive mutation probability based on Lr
Distance
131
5.4 The Development of Hybrid GA and TS
131
5.5 The Development of Breeder Genetic Algorithm
133
5.6 Results and Discussion
137
5.6.1
Preliminary Study on the Choice of Population
Size
137
5.6.2 Adaptive GA versus Standard GA
140
5.6.3
142
Hybrid GA and TS (HGATS) versus GA and TS
5.6.4 Breeder GA versus Standard GA
150
5.6.5 Comparative Study between Tabu Search and
Genetic Algorithm – based Metaheuristics
5.7 Conclusion
6
153
159
SOLVING THE BI-OBJECTIVE VRPSD VIA
BREEDER GENETIC ALGORITHM
6.1 Introduction
160
6.2 Problem Formulation of Multiple VRPSD
162
6.3 Multi-objective Optimization
163
6.4 Breeder Genetic Algorithm for Bi-objective VRPSD
165
6.4.1
Chromosome Representation
166
6.4.2
Initialization
166
6.4.3
Improvement
168
6.4.4
Objective Vectors
168
xii
6.4.5
Pareto-rank
169
6.4.6
Selection
169
6.4.7
Crossover and Mutation
170
6.4.8
Stopping Criterion
173
6.4.9
Acceptance Criterion
173
6.5 Computational Results
6.5.1
Effect of different crossover types
6.5.2
Performance of Bi-objective Breeder GA for
VRPSD
6.6 Conclusion
7
176
178
180
CASE STUDY: SOLID WASTE COLLECTION
7.1 Introduction
181
7.2 Overview of Solid Waste Company
182
7.3 The Data Collection Process
183
7.4 Data Analysis
187
7.4.1
Daily Demand Characteristics
187
7.4.2
Route Construction
190
7.5 Conclusion
8
176
198
GENERAL CONCLUSION AND
RECOMMENDATION
8.1 Introduction
199
8.2 General Conclusion
199
8.3 Recommendations for Future Researches
201
REFERENCES
203
Appendices A - C
220
xiii
LIST OF TABLES
TABLE NO.
TITLE
PAGE
2.1
Previous works on VRPSD
26
3.1
Data display for chi-square goodness of fit test
64
3.2
Customer coordinates and demands
88
3.3
Distance matrix of customers
89
3.4
Summary of metaheuristic algorithms considered in this
94
study
4.1
The expected total cost of solutions obtained from TS and
111
RTS
4.2
Average computation time of TS and RTS (secs)
112
4.3
Descriptive statistics measures
115
4.4
Kolmogorov-Smirnov normality test results for differences
5.1
between TS and RTS
115
An illustration of initial population
122
xiv
5.2
Reviews on several population sizes used in Genetic
123
Algorithm for VRP
5.3
The performances of GA for different population sizes
138
5.4
Kolmogorov-Smirnov normality test results for differences 139
of GA solution qualities for different population sizes
5.5
Paired Samples Test Results of the Differences
140
5.6
The performance of fixed parameter versus adaptive GA
141
5.7
Kolmogorov-Smirnov normality test results of differences
of GA, TS and HGATS for 10-node problems
5.8
Kolmogorov-Smirnov normality test results of differences
of GA, TS and HGATS for 20-node problems
5.9
152
Descriptive statistics for the solutions of metaheuristics
implemented
5.12
149
Normality test results of the differences between GA and
BGA
5.11
146
Kolmogorov-Smirnov normality test results of differences
of GA, TS and HGATS for 50-node problems
5.10
143
154
Summary of paired test of metaheuristics solution quality
differences
157
5.13
Average computation time of the metaheuristics
158
6.1
Parameters setting in bi-objective VRPSD
175
xv
6.2
Descriptive statistics for the solutions from different
crossover types
6.3
176
Normality test results of the differences between three
types of crossover
177
6.4
Independent samples test for crossover performance
178
6.5
Comparison of total expected cost between BBGA and
Pareto GA
6.6
179
Comparison of vehicles number between BBGA and
Pareto GA
179
6.7
Comparison between multiplicative aggregation
180
7.1
Number of houses on route
186
7.2
Descriptive statistics of total daily collection
188
7.3
Descriptive statistics of demand data of nodes
192
7.4
Uniform distribution fitting of demand data
193
7.5
Route optimization in Taman Setia Indah
194
xvi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
Function of q
2.2
Algorithm for the computation of the VRPSD objective 24
23
function f 0 (Q)
2.3
Bianchi et al.’s Genetic Algorithm for the VRPSD
54
3.1
Research framework
58
3.2
Procedure of genetic algorithm
72
3.3
Illustration of roulette wheel selection
72
3.4
Situation before ranking (graph of fitnesses)
73
3.5
Situation after ranking (graph of order numbers)
73
3.6
One point crossover
75
3.7
Two point crossover
75
3.8
Uniform crossover
75
xvii
3.9
Partially-mapped crossover
76
3.10
Cycle crossover
77
3.11
Modified crossover
77
3.12
Order crossover
78
3.13
Order-based crossover
79
3.14
Position-based crossover
80
3.15
Basic tabu search algorithm
87
3.16
5 customers network in Table 3.1 (depot excluded)
89
3.17
Illustration of hj determination
92
4.1
Or-opt algorithm
98
4.2
Illustration of tabu move
100
4.3
Tabu search algorithm for single VRPSD
102
4.4
Initialization and parameters setting in RTS
105
4.5
The reactive tabu search mechanism
106
4.6
The escape phase
106
4.7
The comparison between static and dynamic tabu list size
implementation
109
xviii
4.8
Box and whisker plots of solutions produced by TS and
RTS
114
4.9
Wilcoxon signed-ranks test results for 10-node problems
116
4.10
Paired t-test results for 20-node and 50-node problems
117
5.1
Illustration of order representation
121
5.2
Structure of genetic algorithm and adaptive genetic
algorithm
127
5.3
Illustration of Hybrid GA – TS schemes
132
5.4
Structure of Breeder Genetic Algorithm
136
5.5
Box and whisker plots of solutions obtained from GA,
TS and HGATS for 10-node problems
5.6
Paired samples test results of GA, TS and HGATS for
problem size 10-node problems
5.7
147
Box and whisker plots of solutions obtained from GA,
TS and HGATS for 50-node problems
5.10
146
Paired samples test results of GA, TS and HGATS for
20-node problems
5.9
144
Box and whisker plots of solutions obtained from GA,
TS and HGATS for 20-node problems
5.8
143
148
Paired samples test results of GA, TS and HGATS for
50-node problems
150
xix
5.11
Box and whisker plots of solutions obtained from GA 151
and BGA
5.12
Paired-differences test results between GA and BGA 153
performance
6.1
The data structure of chromosome representation
167
6.2
Illustration of route-exchange crossover
172
6.3
Structure of Breeder Genetic Algorithm for bi-objective 174
optimization
7.1
Arc routing and node-to-node routing
184
7.2
Map of Taman Setia Indah
185
7.3
Portion of maps of Jalan Setia 6 and Jalan Setia 7
187
7.4
Box and whisker plots of total demand by day
189
7.5
One-way ANOVA of total demand versus day
190
7.6
Global route of the vehicle
192
7.7
Routes generated by TS, GA, HGATS-A, and Bianchi
195
7.8
Routes generated by RTS, BGA and HGATS-B
196
7.9
The current company route
197
xx
LIST OF APPENDICES
APPENDIX
TITLE
PAGE
A
List of Publications
220
B
The Kolmogorov-Smirnov Normality Tests
222
C
The Paired Samples Test
225
1
CHAPTER 1
INTRODUCTION
1.1
Introduction
Optimisation is a part of life. In our day to day lives, we make decisions that
we believe can maximize or minimize our objectives, such as taking a shortcut to
minimize the time or distance required to reach a particular destination, or finding a
lowest priced items in the supermarket. Most of these decisions are based on our
years of experience and knowledge about the system without resorting to any
systematic mathematical formulation.
However, as the system becomes more
complicated, further it is needed to formulate it into specific mathematical model,
and with the advent of computer it is possible to exploit optimisation theories to their
maximum extent.
Combinatorial optimisation is a branch of optimisation that arises everywhere
and certainly in applied mathematics and computer science, related to operations
research, algorithm theory and computational complexity theory that sit at the
intersection of several fields, including artificial intelligence, mathematics and
software engineering.
Combinatorial optimisation algorithms solve problem
instances that are believed to be hard in general, by exploring the usually large
solution space of these instances. Combinatorial optimisation algorithms achieve
this by reducing the effective size of the space and by exploring the space efficiently.
The Vehicle Routing Problems (VRP), Traveling Salesman Problem (TSP),
minimum spanning tree problem and knapsack problem are examples of
combinatorial optimisation problem.
2
Since late fifties, the Vehicle Routing Problem (VRP) has been and remains a
rich topic for researchers and practitioners. It becomes an area of importance to
operations research as well as its use for many real world applications. An integral
component of logistics is transportation, and a frequently arising situation in the
transportation and distribution of commodities has usually been modeled as a
Vehicle Routing Problem (VRP). Usually real world VRP arises with many site
constraints. VRP is a generalized problem of the Traveling Salesman Problem (TSP)
in that the VRP consists in determining m vehicle, where a route is tour that begins at
the depot. The task is to visit a set of customer in a given order and returns to the
depot. All customers must be visited exactly once and the total customer demand of
a route must not exceed the vehicle capacity.
Given a set of geographically
dispersed customers, each showing a positive demand for a given commodity, the
VRP consists of finding a set of tours of minimum length (or cost) for a fleet of
vehicles. According to Secomandi (2003), the class of VRPs is a difficult one, since
its elements are usually NP-hard problems and they are generally solved by heuristic
methods. Lenstra and Kan (1981) has shown that VRP is NP-hard problem.
The majority of these researches conducted on operations research are focus
on static and deterministic cases of vehicle routing in which all information is
determined before the time of planning the routes.
Whereas in this ICT age,
information is gathered in real time and in many cases they are changing. The
complexity of the problem increases as more information is unavailable at the time
of the planning or when the service has begun such as the time to begin service, the
location and the actual demand. In most real life application, stochastic or dynamic
information occurs parallel to the routes being carried out. Many of the vehicle
routing problems have inherent randomness, which is not considered in deterministic
models, probably travel times or demands are random variables with known
distributions.
Tillman (1969) started the works to explore cases on VRP with
stochastic demands. Since that, many theories and algorithms on VRPSD have been
proposed and or developed. In this research, we are interested in studying the
demand as the stochastic component.
3
This chapter presents the flow of the research proposal and it begins with the
background and problem statement of the research. It is important that an extensive
work has been carried out in order to present a case for this work and this is given in
Chapter 2.
Research objectives, the scope of this study and discussion on the
research contribution are also given. Finally, the brief of each chapter is outlined.
1.2
Background of the problem
The classical VRP models usually do not capture an important aspect of real
life transportation and distribution-logistic problems, namely the fact that several of
the problem parameters (demand, time, distance, city location, etc) are often
stochastic. Most existing VRP models oversimplify the actual system by assuming
system parameter (e.g. customer demands) as deterministic value, although in real
application, it may not be possible to know all information about customers before
designing routes. Stochastic information occurs and has major impact on how the
problem is formulated and how the solution is implemented.
Neglecting the
stochastic nature of the parameters in a vehicle routing model may generate sub
optimal or even infeasible routes (Yang et al., 2000).
As compared to the development in deterministic case, research in Stochastic
VRP is rather undeveloped. Gendreau et al. (1996) summarise the solution concepts
and literature available on different kinds of SVRP including the TSP with stochastic
customers, the TSP with stochastic travel times, the VRP with stochastic demands,
the VRP with stochastic customers and the VRP with stochastic customers and
demands. Stochastic VRP cannot be solved as VRP since properties and the optimal
VRP solution do not hold for the SVRP (Dror et al., 1989). Further, it calls for more
complex solution methodologies (Gendreau et al., 1995).
This study focus on VRP with Stochastic Demands (VRPSD) in which
demand at each location is unknown at the time when the route is designed, but is
follow a known probability distribution.
This situation arises in practice when
whenever a company, on any given day, is faced with the problem of collection/
4
deliveries from or to a set of customers, each has a random demand. In this study,
we deal with specific case on solid waste collection. It is hoped that optimisation
can take into account the stochasticity of the problem in obtaining better routes and
reducing cost.
In stochastic environment, due to its randomness in customers’ demands, a
vehicle capacity may be exceeded during service. A route failure is said to occur if
the demand exceeds capacity and a recourse action needs to be taken at extra cost.
Assuming that enough capacity is available at the depot, the vehicle may return to
the depot, replenish its load, and then resume service at the point where failure
occurred. Therefore the vehicle will always be able to satisfy all demands but the
length of the corresponding tour becomes a random quantity.
The recourse action could be the vehicle resumes service along the planned
route, namely a priori approach, or visiting the remaining customers possibly in an
order that differs from the planned sequence known as re-optimisation approach.
There are two common recourse policies for a priori optimisation. The first is the
simple recourse policy (Dror et al., 1989; Gendreau et al., 1995; Chepuri and
Homem-de-Mello, 2005) where a vehicle returns to the depot to restock when its
capacity becomes attained or exceeded. The second approach is the preventive
restocking (Bertsimas et al., 1995; Yang et al., 2000; Bianchi et al., 2004) where
preventive restocking is planned at strategic points preferably when the vehicle is
near to the depot and its capacity is almost empty, along the scheduled route instead
of waiting for route failure to occur. On the other hand, most recent computational
studies in re-optimisation approach were done by Secomandi (2000, 2001, 2003).
In this study, we use a priori approach since redesign the routes when actual
demand becomes known appears to be a problem for several reasons:
1. Resources might not be available.
2. Even if resources are available, it might be that redesign of routes is not
sufficiently important to justify the required effort, cost and time.
3. Redesigning the route might create confusing to the driver.
5
4. Regularity and personalization of service by having the same vehicle and
driver visit a particular customer every day is not guaranteed if one redesigns
the routes.
For our case in solid waste collection, familiarity of driver and waste
collector on the route visited every day is highly emphasized; usually the route is
substantially not changed.
Redesign the routes might cause a problem in the
situation when demand is highly variable, thus the routes could be different each day,
creating route instability which has consequences to system nervousness in material
requirements planning. Further, if demand is stable, it still requires the company
personnel to understand day by day task of gathering and inputting new demand to
algorithm that generate routes and then deliver the output information to driver.
Tillman (1969) was the first to propose algorithm for the VRPSD in the case
where there were multiple terminal deliveries and multiple vehicles. Since then,
many researchers have studied this problem in two frameworks, namely the chance
constrained and stochastic with recourse. In the chance constrained VRPSD, the
problem is to design a number of vehicle routes of least distance traveled, subject to
the constraint that the probability of route failure on any route is within an allowable
limit. In contrast, VRPSD with recourse try to minimize the total expected cost (or
distance), including the cost of travel as well as the cost of recourse action when a
route failure occurs. The VRPSD with recourse is considerably more difficult than
chance constrained VRPSD (Gendreau et al., 1995).
Various formulations and algorithms have been proposed and investigated,
including the properties and solution frameworks of VRPSD studied by
Dror et al. (1989), Bertsimas (1992) who proposed cyclic heuristic and found a priori
solution for single vehicle and Dror et al. (1993) who have examined a priori
VRPSD in the context of Stochastic Programming where there is only one vehicle
and the number of potential failures is small. Yang et al. (2000) developed optimal
restocking policy in conjunction with routing decisions for a priori VRPSD for single
and multiple vehicles.
Secomandi (2001, 2003) considered re-optimisation-type
routing policy by means of rollout policy for single vehicle.
6
Bianchi et al. (2004) considered basic implementation of five metaheuristics
for single vehicle: Iterated Local Search, Tabu Search, Simulated Annealing, Ant
Colony Optimisation and Evolutionary Algorithm (Genetic Algorithm) that found
better solution quality in respect to cyclic heuristic. Chepuri and Homem-de-Mello
(2005) proposed a new heuristic method based on the Cross-Entropy method for
single vehicle. Instead of the work of Bianchi et al. (2004) and Gendreau et al.
(1995), the work on the application of GA and TS for VRPSD are lacking in the
literature. The work of Bianchi et al. (2004) results that the performance of GA and
TS seem to be not significantly different, due to the fact that these algorithms find
solutions values which are not very different to each other, but can not compete ILS.
These facts have opened a new direction to conduct research on GA and TS
for solving VRPSD.
Moreover, it is widely known that GA has been proven
effective and successful in a wide variety of combinatorial optimisation problems,
including TSP and certain types of VRP, especially where time windows are
included. TS, the approach that dominates the list of successful algorithms, is known
also as a robust, efficient and effective approach to the general VRP family of
problem (Laporte, 1992; Osman, 1993).
TS often outperform other heuristic
techniques in terms of computational speed and solution quality (Osman, 1993).
The number of published work on the application of GA for solving basic
VRP, TSP, VRPTW, VRPB, and multi depot VRP has been growing. Different
approaches were also proposed based on different crossover operator, different
mutation operator, or replacement methods.
Although pure GA performs well,
mostly it does not equal TS in terms of solution quality, sometimes pure GA perform
inefficient on practical combinatorial optimisation.
To improve pure GA
performance, some algorithms are combined with the simple GA, yielding a hybrid
algorithm.
The statement about GA hybridization is noted by Coley (1999) that hybrid
algorithms, which combine a GA with more traditional algorithms, have been hinted
as a highly powerful combination for solving practical problem, also by Lacomme et
al. (2006) that it is well known that a standard GA must be hybridized with another
search procedure to be able to compete with metaheuristics like TS. Baker and
7
Ayechew (2003) showed that hybrid GA with neighbourhood search in the basic
VRP is competitive with TS and SA in terms of solution time and quality. Hybrid
GAs also have widespread application to VRPTW, including the work of Blanton
and Wainwright (1993), Thangiah (1995a and 1995b), Berger et al. (1998) and
Braysy et al. (2000).
Based on previous research on algorithms developed for VRPSD and the
knowledge of the basic structure of GA and TS, in this study we develop
metaheuristics based on GA and TS as the enhancement of basic GA and TS for
solving single and multiple VRPSD in minimizing single objective function. Our
reviews also found that hybrid GA has not been used to solve VRPSD and most of
researches in VRPSD with preventive restocking are dealing with single VRPSD and
little work has been done for multiple VRPSD, even though decisions relating to
routing fleets of vehicles are frequently taken into consideration in distribution and
logistics operations. This brings us to also develop the meta-heuristics based on
hybrid GA and TS to solve a priori VRPSD comprises single vehicle and multiple
vehicles.
The approach developed was inspired also by the emerging interest in hybrid
metaheuristics that has risen considerably among researchers in combinatorial
optimisation. The best results found for many practical or academic optimisation
problems are obtained by hybrid algorithms (Talbi, 2002). In this study, the GA will
be hybridized with TS. It is highly expected that this hybrid could combine the
advantage of GA as population-based method and the strength of TS as trajectory
method. As known, population-based methods are better in identifying promising
areas in the search space, whereas trajectory methods are better in exploring
promising areas in search space.
In this study, we also propose metaheuristics for solving bi-objective VRPSD
to minimize the total expected cost and the number of vehicles. This study is
motivated by fact that many real world design or decision making problems involve
simultaneous optimisation of multiple objectives (Srinivas and Deb, 1994). But most
of the existing literatures in VRPSD, except of multi-objective’s Tan et al. (2007),
use single-objective based heuristics. In a single-objective optimisation, the main
8
goal is to find the global optimum solution.
However, in a multi-criterion
optimisation problem, there are more than one objective function, each of which may
have different individual optimal solution. If there is sufficient difference in the
optimal solutions corresponding to different objectives, the objective functions are
often conflict each other.
Multi-criterion optimisation with such conflicting
objective functions gives rise to as set of optimal solutions, instead of one optimal
solution. The reason for the optimality of many solutions is that none can be
considered to be better than any other with respect to all objectives.
1.3
Problem Statement of Research
The VRPSD is defined on a complete graph G = (V, A, C), where
V = {0, 1,…, n} is a set of nodes with node 0 denotes the depot and nodes
1, 2, …, n correspond to the customers,
A = {(i, j ) : i, j V , i z j} is the set of arcs joining the nodes, and
C = ( cij : i, j  V , i z j} ) is a non-negative matrix (a matrix where all the
elements are equal to or above zero, C t 0, (i, j ), cij t 0 ) that denotes the travel
cost (distance) between node i and j.
The cost matrix C is symmetric and satisfies the triangular inequality. Customers
have stochastic demands [ i , i = 1,…, n, which are a non-negative discrete random
variable with known probability distribution pik
Pr ob([ i
k ) , k = 0, 1,…, K d Q.
Assume further that customers’ demands are independent and identical. Actual
demand is only known after the vehicle arrives at customer’s location. If there is a
route failure at any node, the recourse action has to be taken, the recourse action is to
travel back to the depot for replenish and then resume its journey as planned at the
node where failure occurred. The problem objective is to find a priori routes that
minimize the total expected cost, including travel cost and the expected recourse
cost, subject to the routing constraints, under the stochastic demands setting.
9
A stochastic vehicle routing problem arises when not all information relevant
to the planning of the routes is known by the planner when the routing process
begins and information can change after the initial routes have been constructed.
According to Secomandi (2003), for the class of deterministic VRPs, they are
generally solved by heuristic or metaheuristic methods; whereas algorithms for
stochastic VRP are considerably more intricate than deterministic and it calls for
efficient algorithm that is able to work in real-time since the immediate requests
should be served. These have made them an important candidate for solution using
metaheuristics.
This research tries to propose new metaheuristics based on GA and TS to
enhance the performance of basic GA and basic TS for solving VRPSD. In this
work, we initially consider a single-vehicle model and later expand the analysis to
incorporate multiple vehicles.
We also develop metaheuristics for solving bi-
objective VRPSD. The performances of the proposed algorithms were compared
each other and with other heuristics/ metaheuristics. The implementation of these
algorithms also will be done to solve real problem in optimizing solid waste
collection.
1.4
Objectives of the Study
The aim of this study is to develop various approaches to optimize VRPSD
solution, particularly in the use of metaheuristics approach. The objectives of this
study are to:
x
develop metaheuristics for solving single VRPSD that include:
a. Tabu Search and Reactive Tabu Search.
b. Genetic Algorithm and the enhanced GA, i.e. :
-
Adaptive Genetic Algorithm
-
Breeder Genetic Algorithm and
-
Hybrid GA and TS
10
x
develop GA-based metaheuristic for solving bi-objective multiple VRPSD.
x
conduct comparative evaluation on the performance among these
metaheuristics.
x
solving real problem data in optimizing solid waste collection by using the
VRPSD model and the developed metaheuristics.
1.5
Scope of the Study
1.
In this research, we confine the application of our algorithm to a priori
VRPSD under restocking policy.
2.
The problem data for the performance testing in simulation study are
randomly generated problem data following specific probability distributions.
3.
We confine our real problem solving in the case study of optimizing solid
waste collection.
4.
It is assumed that vehicles start and end at a single depot.
5.
It is assumed that every customer demands have the same probability
distribution but can have different parameters (for example: different mean
and variance for each customer that have normal distribution).
6.
The problem data were generated using Minitab version 14 and were tested
and analyzed using SPSS version 13 for Windows, while the implementation
of metaheuristics was done using Delphi version 6.0. Minitab has special
feature in generating random data following specific probability distribution
which SPSS does not have, while SPSS has advantage in better graphical
representation of the output of data analysis than Minitab.
7.
The stopping criteria in addressing convergence for all procedures presented
in this study will not using numerical analysis component/ complexity
analysis.
11
1.6
Significance of the Study
1.
From the view point of algorithm development: our algorithms are the first
implementation of
-
Reactive Tabu Search,
-
Adaptive GA,
-
Breeder GA,
-
and hybrid GA (especially hybrid GA with Tabu Search)
for solving single and multiple vehicles appeared in the VRPSD literature.
2.
We also propose the new adaptive mutation probability measure.
3.
We contribute on the first considering bi-objective VRPSD under restocking
policy and proposing new GA-based for solving bi-objective multiple
VRPSD.
4.
From application aspect, real problem in optimizing waste collection was
solved by using the proposed algorithm.
5.
We also develop software package for solving VRPSD.
6.
And the result of this study will be presented and published at the
international publications/ journal.
Along the recent increase in the demand for an efficient management system
for VRP and logistics and the advances in computer and information technology, the
importance of being able to effectively make use of the huge amount of information
has become important for a wide range of applications. Cost efficient routing of
vehicles play an important role to a wide range of industries. As indicated earlier,
our focus would be to work on the problem related to VRP for solid waste collection
in the Municipality of the Johor Bahru city.
1.7
Organisation of the Thesis
This thesis contains eight chapters. The first chapter is the introduction. This
chapter gives an introduction to the background of the problem, the statement of the
problem, objectives and scope of the study, and significance of the study.
12
Chapter two is the Literature Review. This chapter presents a literature
review about the Vehicle Routing Problem with Stochastic Demands, solution
techniques appeared in literature and also techniques which may be applied for
solving VRPSD.
Chapter three is the Research Methodology.
This chapter presents the
direction of the study and an overview of the methods used. It begins with the
general steps of research framework. A description of the data source for this study
and test related to it are also presented include case-study of a real-life application at
solid waste collection company (Perniagaan Zawiyah Sdn. Bhd). It follows with the
description of algorithm implemented with emphasis on Genetic Algorithm, Tabu
Search and the enhancement of these two metaheuristics.
In Chapter four, the development of metaheuristic methods based on Tabu
Search and Reactive Tabu Search for solving single VRPSD are explored. The
discussion of this chapter begins with the detail development of Tabu Search and
Reactive Tabu Search for single VRPSD, followed by the experimental results and
discussion.
Chapter five discusses the development of several Genetic Agorithm-based
metaheuristics for solving single VRPSD.
The standard Genetic Algorithm is
presented, followed by the enhancement of it through Adaptive Genetic Algorithm,
hybrid Genetic Algorithm with Tabu Search, and the Breeder Genetic Algorithm.
Chapter six considers multiple VRPSD which analog of the deterministic
VRP. The use of Breeder Genetic Algorithm is proposed to solve the bi-objective
VRPSD in minimizing the number of vehicles required and the total expected cost
concurrently, using Pareto-rank by modifying and extending the single-objective
Breeder Genetic Algorithm.
VRPSD finds its application on wide range of logistics and distribution sector
in cases where it is impossible to know the demand of the customers before the
vehicles arrives at customer’s location. In Chapter seven, the implementation of
13
VRPSD in the case of picking up of garbage done by solid waste company in Johor
Bahru namely Perniagaan Zawiyah Sdn. Bhd. is discussed.
Finally, Chapter eight concludes the relevant and important findings from this
research. Recommendations on area related to the findings and possible directions
for future research are presented.
14
CHAPTER 2
REVIEWS OF THE LITERATURE
2.1
Introduction
The process of solving Vehicle Routing Problem (VRP) takes place again and
again in many business operations. This has led many researchers in this field to
develop solution techniques to generate (near) optimal solution for solving VRP
which may also be applied to other variant of VRPs. This research as indicated in
the previous chapter focuses on developing solution methods for solving VRP with
Stochastic Demands (VRPSD). This chapter reviews available literature on the
VRPSD under a priori approach. It commences with the description of the VRPSD,
the preview on the solution frameworks, and mathematical model of VRPSD
including previous works on VRPSD. These are followed by a discussion on issues
related to the use of heuristic methods, the basic concept of metaheuristics and
hybrid metaheuristics relevant to this study, and some criteria for classification of
metaheuristics. Finally, the last section of this chapter discusses the taxonomy of
hybrid metaheuristics.
2.2
Vehicle Routing Problem with Stochastic Demands (VRPSD)
VRP and its variants are at the core of many industrial applications in
transportation logistics. In this study, a variant of VRP is studied where customer
demands are not deterministically known until the time when the vehicle arrives at
15
the customer location. To deal with this problem, the VRP is extended to cover the
more realistic case of uncertainty in customer demands by using VRP with
Stochastic Demands model. The customer demands are unknown but assumed to
follow specific probability distribution according to the past experience about
customer demands.
2.2.1
Real world applications
The VRPSD finds its applications in all those cases where it is impossible to
know the customers demand before the vehicles arrive at customer’s location. The
picking up of garbage is one of them. It is of course impossible to know a priori how
much garbage has to be collected at each customer location. It is possible that the
total quantity of garbage could exceed the vehicle capacity. In such situation, the
vehicle needs to go to the depot to replenish and then resume its route.
The delivery to petrol stations is another case subject to the stochasticity of
demand. When the customer (stations) issues the order, it is still unknown how
much he will sell in the time between the order and the new delivery. Another
related application is in the cash replenishment of automatic teller machines (ATM)
of a bank. The daily demand for cash at a specific machine is a random variable,
while the maximum amount of cash that might be carried by a vehicle at any time is
specified by a security policy. Not all machines may be able to be supplied on a
route from its designated vehicle, forcing a decision to be made as to how to supply
the machines when restocking demand may exceed vehicle capacity.
According to Yang et al. (2000), VRPSD also appear in cases such as
salespersons on peddle routes where salesperson stock their trucks with goods that
they anticipate can be sold on the route. Visits are made to predetermined customer
sites with the intent of replacing depleted stock. Because there is no advanced
reporting of the current inventory level or the size of a replenishment order, the
amount to be sold is not known until the site is visited. Examples of such peddle
routes are (1) beer distribution to retail outlets, (2) supply of baked goods at food
16
stores, (3) replenishment of liquid gas at research laboratories, and (4) stocking of
vending machines.
2.2.2
Solution Frameworks for VRPSD
Dror et al. (1989) mentioned that there are two solution frameworks for
Stochastic VRP (including VRPSD), namely stochastic programming and Markov
decision process. Further in this study, we focus on the discussion on stochastic
programming.
The mathematical programming problem is defined as follows.
Minimise Z MP
subject to:
f (x)
g i ( x) d 0 , (i = 0, 1, …, m),
x  S Ž Rn ,
xt0
where the real functions f, g i (i = 0, 1, …, m) are assumed to be deterministic and
known.
In general, the stochastic programming is a mathematical programming
where one or more parameters are random variables. Such cases seem typical of
real-life problems, where it may be difficult to determine the values of the
parameters with certainty. The stochastic programming can be described as
Minimise Z SP
subject to:
f ( x, [ )
g i ( x, [ ) d 0 , (i = 0, 1, …, m)
x  S Ž Rn ,
x t 0.
17
where [ is a vector of random variables, hence ( f ( x, [ ) , g i ( x, [ ) , i = 0, 1, …, m) is
a random vector.
Stochastic programming terminology distinguishes between wait and see and
here and now situations.
The wait and see leads to the so called distribution
problems by computing the decision variables (planned vehicle routes) after
observing the realization of [ (customer demands). Here and now is the situation
where the computation of x (route solution) has to precede the observation of [ .
The SVRP more naturally belongs to the here and now category since usually routes
have to be planned in anticipation of customer demands which become only known
only in the process of executing the planned routing sequence. Again, the problems
in this category are usually modeled in two ways: so-called chance constrained
stochastic programming (CCP) and stochastic programming with recourse (SPR)
(Dror et al., 1989).
The CCP models consists of designing a set of vehicle routes of least total
cost, subject to the constraint that the probability of route failure of any route is
within an allowable limit. Stewart and Golden (1983) and Laporte et al. (1989)
shown that under some restrictive assumptions on the statistics of demand, the
problem can be reduced to a deterministic VRP thus existing deterministic
algorithms can be readily applied. Although the CCP tries to control the probability
of roué failure, the cost of such failures is ignored. SPR models differ from the CCP
at two major points: First, it is assumed that every customer demand has to be fully
satisfied, thus at a route failure a recourse action always includes the filling of the
uncompleted demand. Secondly, the SPR explicitly model the route failure and the
cost of recourse action. Therefore, the result more closely reflects the real routing
situation. In general, the SPR models are more difficult to solve than CCP models
but their objective functions are meaningful (Gendreau et al., 1996).
Stewart and Golden (1983) provide a comparison of the two mentioned
frameworks for solving a multiple VRPSD. They concluded that if the route failure
penalty cost is known, SPR models produce lower costs than CCP. Laporte and
18
Loveaux (1989) in Dror et al. (1989) propose a stochastic model with recourse with
full service (i.e., no split deliveries) is assumed and two cases are considered:
1.
The SVRP with early information: all customer demands are known after the
route is constructed but before the vehicle leaves the depot and route breaks
are planned in advance so as to minimise their expected cost;
2.
The SVRP with late information: breaks and failures coincide.
In both cases, it is assumed that the planned routing sequence is unaltered by failures
or breaks. The two models differ only in their recourse function. The theoretical
work in Bastian and RinnooyKan (1992) extends the Stewart and Golden
formulations. They revised the SPR model with a fixed penalty for every unit of
unsatisfied demand to include the expected cost of transportation instead of the
transportation cost for a complete route.
2.2.3
Types of Recourse
A general SPR model can be written as
Minimise z
E[ f ( x, [ ) I ( x, [ )]
subject to x  S
where x is the route solution, [ is a vector of random variables, f ( x, [ ) is the
deterministic route cost, I ( x, [ ) is the recourse function which is the cost of recourse
action and the S is the set of all possible combinations of m routes (Yang et al.,
2000)
The first two recourse versions were provided by Stewart and Golden (1983).
The first version is to charge a fixed penalty O k every time a route k fails. The
second version is to charge a penalty for every unit of demand in excess of the
vehicle capacity Q on route k. Both versions assume that all customers must be
19
visited on the route and therefore the objective function includes the deterministic
total cost of the route plus the penalty term. Dror and Trudeau (1986) replaced the
fixed linear penalty cost in Stewart and Golden model by a non-linear recourse that
considers failure location.
Their recourse assumed that after a failure all the
remaining customers are served through individual deliveries. Assuming this nonoptimistic recourse, the deterministic cost in the objective function is replaced by the
expected cost.
Dror and Trudeau (1989) introduced a theoretically new recourse defined by
a return to the depot whenever the vehicle is unable to satisfy the demand (the
capacity becomes exceeded) and resume the service at the customer on the planned
route where route failure had occurred.
Savelsbergh and Goetschalckx (1995)
simplified the recourse to achieve computational efficiency assuming at most one
failure in a route. All subsequent publications (e.g. by Laporte and Loveaux, 1998;
Gendreau et al., 1996) have maintained the simple recourse policy defined in Dror et
al. (1989) with the exception of Bertsimas et al. (1995), Yang et al. (2000), Bianchi
et al. (2004), and Chepuri and Homem-de-Mello (2005).
The idea for a proactive recourse is introduced by Bertsimas et al. (1995),
they improve the expected cost of a priori solutions for single VRPSD using dynamic
programming to define routes with proactive returns to the depot for restocking.
Yang et al. (2000) also adopt a proactive recourse (preventive restocking) and design
routes for a multiple VRPSD. Bianchi et al. (2004) use the dynamic programming as
Yang et al. (2000), for solving single VRPSD.
The recourse in Chepuri and
Homem-de-Mello (2005) is to terminate the route and impose a penalty such as lost
revenue and/or emergency deliveries. This recourse is most practical when the time
frame within which demand must be met, is critical.
20
2.2.4
Mathematical Formulation of Single VRPSD
This section presents the mathematical formulation of the single VRPSD.
Definitions of some of the frequently used notations for the VRPSD are given as
follows:
(1). Customers and depot
V = {0, 1, ..., n} is a set of nodes with node 0 denotes the depot and nodes
1, 2, …, n correspond to the customers to be visited. We assume that all nodes,
including the depot, are fully interconnected.
(2). Demands
Customers have stochastic demands [ i , i = 1, ..., n which follows discrete
uniform probability distribution pik
Pr ob([ i
k ) , k = 0, 1, 2, …, K. Assume
further that customers’ demands are independent. Actual demand of each
customer is only known when the vehicle arrives at the customer location.
(3). Vehicle and capacity constraint
The vehicle has a capacity limit Q. If the total demand of customer exceeds the
vehicle capacity, route failure is said to have occurred.
(4). Routes
Every route must start at the depot, visit a number of customers and return to
the depot. A feasible solution to the VRPSD is a permutation of the customers
s = (s(1), s(2), . . . , s(n)) starting and ending at the depot (that is, s(1) = s(n) =
0), and it is called a priori tour.
(5)
Route failure and recourse action
Route failure is said to have occurred if the total demand exceeds the vehicle
capacity and the preventive restocking policy (Yang et al., 2000, Bianchi et al.,
2004) is applied.
21
(6). Cost and VRPSD objective function
A = {(i, j) : i, j  V, i z j} is the set of arcs joining the nodes and C = { cij : i, j
 V, i z j} is a non-negative matrix denotes the travel costs (distances)
between node i and j.
The cost matrix C is symmetric and satisfies the
triangular inequality. The cost matrix is a function of Euclidean distance;
where the Euclidean distance can be calculated using the following equation:
d ij
( xi x j ) 2 ( y i y j ) 2
Given a vehicle based at the depot, with capacity Q, VRPSD under
restocking policy requires finding vehicle routes and a restocking policy at each node
to determine whether or not to return to the depot for restocking before visiting the
next customer to minimise total expected cost. The costs under consideration are:
-
Cost of traveling from one customer to another as planned.
-
Restocking cost: the cost of traveling back to the depot for restocking.
-
The cost of returning to depot for restocking caused by the remaining stock in the
vehicle being insufficient to satisfy demand upon arrival at a customer location.
This route-failure cost is a fixed nonnegative cost b plus a cost of traveling to the
depot and back to the route.
VRPSD is defined as a problem of
Minimise f 0 (Q) = E[ f ( x, [ ) I ( x, [ )]
(2.1)
subject to x  S
where x is route solution, [ is a vector of customer demands, f ( x, [ ) is the
deterministic route cost, I ( x, [ ) is the recourse function which is the cost of recourse
action and the S is the set of all possible route solutions. The VRPSD objective
function f 0 (Q) is calculated using dynamic programming presented as follows:
22
A feasible solution to the VRPSD is a permutation of the customers s = (s(1),
s(2), . . . , s(n)) starting at the depot (that is, s(1) = 0), and it is called a priori tour.
Let 0 1 2 … j j+1 … n be a particular vehicle route. Upon the service
completion at customer j, suppose the vehicle has a remaining load q (or the residual
capacity of the vehicle after having serviced customer j), and let f j (q) denote the
total expected cost from node j onward. If L j represents the set of all possible loads
that a vehicle can have after servicing customer j, then, f j (q) for q  L j satisfies
f j (q )
minimum ( f jp (q ), f jr (q ))
(2.2)
where
p
f j (q ) c j , j 1 ¦ f j 1 (q k ) p j 1,k
k dq
¦ [b 2c j 1,0 f j 1 (q Q k )] p j 1,k
k !q
(2.3)
and
K
f jr (q) c j ,0 c 0, j 1 ¦ f j 1 (Q k ) p j 1,k
(2.4)
k 1
with the boundary condition
f n (q)
c n,0 , q  S n
(2.5)
In equations (2.3-2.5), f jp (q ) represents the expected cost of going directly to the
next node, whereas f jr (q ) represents the expected cost of the restocking action.
These equations are used to recursively determine the objective value of the planned
vehicle route and the optimal sequence of decisions after customers are served
(Bianchi et al., 2004). In principle, this procedure leads to a dynamic programming
23
since each time a customer demand is revealed, a decision has to be taken as to
where the vehicle should proceed.
2.2.5
Threshold and Expected Cost Evaluation
The expected cost-to-go in case of restocking, is constant for every q, since in
the case of restocking the vehicle will have full capacity Q before serving the next
customer, whatever the current capacity q is.
On the other hand, f jp (q ) is a
monotonically non-increasing function in q, for every fixed customer j. Therefore
there is a capacity threshold value h j such that, if the vehicle has more than this
value of residual goods, then the best policy is to proceed to the next planned
customer, otherwise it is better to go back to the depot for replenishment, as seen in
Figure 2.1 (Yang et al., 2000).
Figure 2.1 : Function of q
The algorithm in Figure 2.2 is an implementation of the above dynamic
programming recursion for the calculation of f 0 (Q ) and of the thresholds (Bianchi et
al., 2004).
24
for (q = Q down to 0) do
f n (q) = c n ,0
for (j = n-1 down to 1) do
compute f jr using f j 1 (.)
for (q = Q down to 0) do
compute f jp (q )
compare
f jr
and
f jp (q )
for finding the
threshold h j
compute f j (q ) using f j 1 (.)
end for
end for
end for
compute f 0 (Q )
return f 0 (Q )
Figure 2.2 : Algorithm for the computation of the VRPSD objective function f 0 (Q )
2.2.6
Multiple homogeneous VRPSD
Given a fleet of vehicles indexed by m  M, is based at a depot with each
capacities Q, the problem is to find a set of a priori routes (and the associated optimal
restocking policies), each to be served by a vehicle, such that customer demands on
the route are fully satisfied and the total expected cost is minimised. The total
demand of customers assigned to each route does not exceed the vehicle capacity of
the vehicle assigned to it, and the total cost is minimised.
In deterministic vehicle routing problems, a single route will always be
optimal if the vehicle has enough capacity to carry all customer demand. However,
multiple vehicles (routes) are needed when the total customer demand exceeds the
vehicle capacity. It is assumed that there are m vehicles for n customer nodes. Let
r
^r1 , r2 ,..., rm ` be a particular solution in which
ri is the sub tour dedicated to ith
vehicle. The function to be minimised is the sum of expected cost for all vehicles
whereas the function of each vehicle is similar with the single vehicle (Yang et al.,
2000).
25
2.2.7
Previous Works on the VRPSD
The summary of main contributions to solve VRPSD and similar problems is
given in Table 2.1.
[two depots]
(1969)
(1989)
Dror
(1986)
et
Trudeau
Dror
[multiple vehicles]
[single vehicle]
al. Theory
[multiple vehicles]
and Heuristic
Golden (1983)
and Heuristics
Heuristic
Tillman
Stewart
Methods
Authors
Data
multiple terminal deliveries
for the VRPSD in the case where there were small points
on a modification of Clarke and Wright algorithm terminals and 7 demand
the first to propose simple heuristic algorithm based Example data for 2
Description
mean
variables with finite
mean and variances
Markovian decision processes from theoretical
point.
random
independent
They also present a new solution framework using
recourse
non-negative
along with some properties of optimal solutions.
new
Discrete or continuous
demand
the
third
expected cost.
of
between zero and one-
generated
uniformly
reversing the direction of a designed route on the and Eilon
multiple VRPSD. They also study the impact of customers Christofides
were
deviation
each location, standard
recourse.
Modify the Clarke and Wright algorithm to solve Extensions of the 75-
the mean demand at
(GLM) for solving VRPSD under the fixed penalty Christofides and Eilon
demand
quantity was used as
Original
Poisson (μ)
Demand distribution
the superiority of generalized Lagrange multiplier 75-customers
SPR with Overview concepts and main issues in VRPSD No data
SPR
SPR
CCP and Adapt the Clarke and Wright algorithm and show Extensions of the 50 and
Model
Table 2.1 : Previous works on VRPSD
26
(1993)
et
[single vehicle]
al. Theory
[single vehicle]
(1992)
Dror
Theory
independent
identically distributed
occur at one of the last k customers of the planned
route
and
are
demands
problem constraints, where route failure can only
individual
distribution
where
Normal
b. General demand
a. Binary demand
failures per route is restricted either by data or the
Consider VRPSD in which the number of potential No data
are very close to re-Optimisation.
b. Gives analytical evidence that a priori strategies
customers is the same.
especially if the distribution of the demand of the
quite well from a worst case perspective
based on finding an a priori sequence, performs
a. Gives analytical evidence that the cyclic heuristics No data
demand)
behaviour.
Bertsimas
it does not have any
Optimisation strategies have on average very close
of
demand of one unit or
strategy
(n
Optimisation. It was shown that a priori and re-
the
demand
customers either has a
to
Binary
Demand distribution
re-
competitive
strategy
[single vehicle]
Data
(1990)
Description
Introduce the idea of a priori Optimisation as a No data
Methods
Bertsimas et al. Theory
Authors
Table 2.1 : Previous works on VRPSD (continued)
27
Methods
(1996)
Data
sets
of
were Vertices
continuous
potential moves. Comparisons with known optimal
solutions (exact algorithm) vary from 6 to 46 customers
[number of
vehicle = 2]
from optimality.
in 89.45% of cases, with an average deviation of 0.38%
indicate that the heuristic produces on optimal solution
with proxy for the objective function to evaluate
VRPSDC
Develop a tabu search algorithm called TABUSTOCH
2 vehicles.
distribution
- Largest instances can be solved is 46 customers and uniform
customers.
these routes are followed by skipping the absent according
vehicle = 2]
2
a
first
demand
were
to
The
uniform distribution
according to a discrete
the appropriate range
was then generated in
value of each demand
probabilities.
a or (10,20) in equal
square range of (1,9), (5,15)
stages: planned collection routes are designed in first generated in the assigned
vertices
unit square
- Propose an exact integer L-shaped algorithm in two n
posteriori solutions of the deterministic realizations
Demand distribution
n Discrete uniform
solutions by comparing to sample averages of a random location in
Computational study for assessing quality of a priori 10
Description
stage, when the set of present customers is known; [0,100]
SPR
SPR
SPR
Model
[number of
for VRPSDC
Gendreau et al. Tabu Search for
(1995)
Gendreau et al. Exact algorihm
(1995)
Bertsimas et al. Cyclic heuristics
Authors
Table 2.1 : Previous works on VRPSD (continued)
28
SPR
algorithm
with
two-phases:
solve
the
(2000)
et
objective
function
and
in the second phase
restocking
policy
vehicles]
1) as the eleven
(50,50)
instances up to 15 customers. They also adapt the
under dynamic programming of stochastic case.
OrOpt local search into the insertion and deletion costs
- Vehicle capacity=1
30, 40, 50, 60}
15}, larger size {20,
small size {10, 12,
- No of customers: possible values
(0, 0.1, 0.2,…, 0.9,
square, depot at
comparisons with branch and bound solutions to
with
distribution
in a 100x100
first-route second. Both algorithms seem to be efficient
and robust for small size instances, as shown by
discrete
triangular
A
are known
mean and variance
with
generated uniformly
with Investigate the route first-cluster second and the cluster - Customer locations
and multiple
al. Heuristics [single
stochastic
neighbourhood search methods to improve the solution
the
distributed
general customers Christofides normally
consider
Yang
distribution
Demand
Solve the problem using a computationally efficient Extensions of the 75- Independently and
Data
(1995)
[multiple vehicles]
and
SPR
Description
assignment model for the VRP in the first phase and and Eilon
Heuristics
Savelsbergh
Model
Goetschalckx
Methods
Authors
Table 2.1 : Previous works on VRPSD (continued)
29
Methods
[single vehicle]
Mello (2005)
[single vehicle]
and Heuristic
Homem-de-
Chepuri
(2004)
SPR
2
-
A Poisson or a Normal
Demand distribution
100 vertices
Cross-Entropy method for VRPSD.
75
have
50
Di  [50,100]
with equal
demand
customers
uniform Gamma ( Ki , E ), E identical
independent random numbers for all nodes
between 0 and 1
are
probability
customers and 5 have 200 Spread Si  [1,5] with equal
customers, 40 have 100 probability: 0.5
- Instances:
Propose a new heuristic method based on Distances
heuristic.
average
[ Di Si , Di Si ]
Di  [1,49] and
[0,99]
solution quality in respect to cyclic
around
and Genetic Algorithm that found better
distribution
Tabu Search, Simulated Annealing, ACO
normal interval
two where
cluster, clusters centers in
following
metaheuristics: Iterated Local Search,
f = 0.9
location Uniform distribution in the
instances between 25 and
uniform distribution
according to a continuous distribution
Consider basic implementation of five - Customers
minimise the expected solution cost.
vehicles]
SPR
Data
Propose an Integer L-Shaped algorithm - N vertices in the [0,100]
Description
for the VRPSD. The objective is to
SPR
Model
[between 2 and 4
Bianchi et al Metaheuristics
(2002)
Laporte et al. Exact algorithm
Authors
Table 2.1 : Previous works on VRPSD (continued)
30
31
2.3
Heuristic Methods
Silver et al. (1980) define heuristic method as a procedure for solving welldefined mathematical problem by intuitive approach in which the structure of the
problem can be interpreted and exploited intelligently to obtain a reasonable
solution.
2.3.1
Why Use a Heuristic Method?
The VRP is one of the classical problems of Combinatorial Optimisation
(CO). In simple terms, it can be defined as the problem of designing optimal routes
for a fleet of vehicles that has to perform a set of transportation tasks. A large
number of businesses and public sector agencies have to deal with the VRP on a
regular basis. For instance, in waste collection, a company has a fleet of vehicles
performing various transportation tasks, including domestic, industrial and
commercial waste collection, the transportation tasks problem that also faced by
various passenger transport services. Most CO problems, including the VRP, are
quite simple to define but extremely difficult to solve. In theory, such problems are
known as NP-hard, and most likely it can only be solved by an exhaustive search if
an exact optimal solution is required. All known exact algorithms for such problems
have super-polynomial time complexity. Despite great efforts, no polynomial-time
algorithm has been found for any NP-hard problem, nor has it been proved that such
an algorithm does not exist (Silver et al., 1980). According to Lenstra and Kan
(1981), the VRP is a well-known integer programming problem which falls into the
category of NP-hard problems.
Given an NP-hard CO problem, one has essentially the following three
options; firstly, to solve small instances of the problem using a super-polynomial
exact algorithm; secondly, to solve special cases of the problem using a specially
designed polynomial-time algorithm and the third to solve the problem
approximately, using a polynomial-time approximation algorithm, often called a
heuristic. Real-life instances of the VRP are usually sufficiently large to rule out the
32
first option.
Moreover, they usually contain various additional constraints that
would be difficult (or even impossible) to incorporate into the exact procedure and,
therefore, some solutions obtained can be in fact infeasible. Polynomially solvable
cases required by the second option are currently known only for some restricted
versions of the VRP, in particular for the TSP. So, typically, there is little hope that
a real-life vehicle routing problem would fit a known special case.
For these
reasons, option 3 often remains the only choice.
According to Silver et al. (1980), there are several possible reasons for using
heuristic methods. These include:
a. The mathematical problem is such a nature that an analytic (closed form) or
iterative solution procedure is unknown.
b. Although an exact analytic or iterative solution procedure may exist, it may
be computationally prohibitive to use or perhaps unrealistic in its data
requirements.
c. The heuristic method, by design, may be simpler for the decision maker to
understand, hence markedly increasing the chances of implementation.
d. For well-defined problem that can be solved optimally, a heuristic method
can be used for learning purposes, e.g., to develop an intuitive feeling as to
what variables are important. (This closely parallel with one of the primary
reasons for using simulation methods in operational research.)
e. A heuristic may be used as part of an iterative procedure that guarantees the
finding of an optimal solution. Two distinct possibilities exist:
i.
To easily obtain an initial feasible solution.
ii.
To make a decision at an intermediate step of an exact solution
procedure.
f. Heuristics can be used to give such ‘good’ starting solutions. In implicit
enumeration approaches to problem solving a good starting solution can give
a bound that drastically reduces the computational effort.
33
2.3.2
Measuring the Quality of a Heuristic
Silver et al. (1980) stated that a good heuristic should possess the following
four properties:
1.
Realistic computational effort to obtain solution.
2.
The solution should be close to the optimum on the average, i.e., we want
good performance on the average.
3.
The chance of a very poor solution (i.e., far from the optimum) should be
low.
4.
The heuristic should be as simple as possible for the user to understand,
preferably explainable in intuitive terms, particularly if it is to be used
manually.
Here are some of the ways for measuring the quality of heuristics:
1.
Comparison with the optimum solution.
2.
Problem relaxation – bounding.
3.
Extreme value statistical methods.
4.
Other comparisons.
i. Comparison with an enumerative method, requiring much more
computational effort that is terminated after a large amount of
computation, but likely without having found the optimal solution.
ii. Comparison with performance of the decision maker, either during an
earlier time frame or directly in parallel – there are compelling arguments
for this type of comparison. Identification of significant improvement
over existing procedures is probably much more important for
encouraging implementation than any proof of optimality, or nearness to
optimality.
iii. Comparison with other heuristic procedures – where other heuristic
solution methods have been proposed and/or used, one certainly can
compare ‘our’ heuristic against the others.
iv. Comparison with a ‘random’ decision rule – an extreme type of heuristic
is where one makes a decision completely at random.
v. Worst-case analysis, it determines the worst possible performance of the
heuristic (and the associated values of the problem parameters). However,
34
even when worst case performance is very poor, typical or average
performance can often be very good.
2.4
Metaheuristics
The term metaheuristic was firstly introduced by Glover (1977). It derives
from the composition of two Greek words. Heuristic derives from the verb
heuriskein which means “to find”, while the suffix meta means “beyond, in an upper
level”.
Before this term was widely adopted, metaheuristics were often called
modern heuristics (Reeves, 1993).
Examples of metaheuristics include-but not
limited to- Ant Colony Optimisation, Evolutionary Computation including Genetic
Algorithm, Iterated Local Search, Simulated Annealing and Tabu Search. Nowadays
metaheuristics are widely used to solve important practical combinatorial
Optimisation problems. However, due to the variety of techniques and concepts
comprised by metaheuristics, there is still no commonly agreed definition for
metaheuristics.
A metaheuristic is formally defined as an iterative generation process which
guides a sub-ordinate heuristic by combining intelligently different concepts for
exploring and exploiting the search space, learning strategies are used to structure
information in order to find efficiently near-optimal solutions (Osman and Laporte,
1996). Stutze (1999) defined metaheuristics as typically high-level strategies which
guide an underlying, more problem specific heuristic, to increase their performance.
The main goal is to avoid the disadvantages of iterative improvement and, in
particular, multiple descents by allowing the local search to escape from local
optima. This is achieved by either allowing new starting solutions for the local
search in a more ‘intelligent’ way than just providing random initial solutions. Many
of the methods can be interpreted as introducing a bias such that high quality
solutions are produced quickly. This bias can be of various forms and can be cast as
descent bias (based on the objective function), memory bias (based on previously
made decisions) or experience bias (based on prior performance). Many of the
metaheuristic approaches rely on probabilistic decision made during the search. But,
35
the main difference to pure random search is that in metaheuristic algorithm
randomness is not used blindly but in an intelligent biased form.
The definition used in the Metaheuristics Network (2000) is as follows.
A metaheuristic is a set of concepts that can be used to define heuristic methods that
can be applied to a wide range of problems. In other words, a metaheuristic can be
seen as a general algorithmic framework which can be applied to different
Optimisation problems with relatively few modifications to make them adapted to a
specific problem.
Blum and Roli (2003) summarised the outline fundamental
properties which characterize metaheuristics:
-
Metaheuristics are strategies that guide the search process.
-
The goal is to efficiently explore the search space in order to find (near)
optimal solutions.
-
Techniques which constitute metaheuristics algorithms range from simple
local search procedures to complex learning processes.
-
Metaheuristic algorithms are approximate and usually non-deterministic.
-
They may incorporate mechanism to avoid getting trapped in confined areas
of the search space.
-
Metaheuristics are not problem-specific.
-
Todays more advanced metaheuristics use search experience (embodied in
some form of memory) to guide the search.
According to Blum and Roli (2003), metaheuristics are high-level strategies
for exploring search spaces by using different methods. Of great importance hereby
is that a dynamic balance is given between diversification and intensification. The
term diversification generally refers to the exploration of the search space, whereas
the term intensification refers to the exploitation of the accumulated search
experience. These terms stem from the tabu search field and it is important to clarify
that the terms exploration and exploitation are sometimes used instead, for example
in the Evolutionary Computation field, with a more restricted meaning. In fact the
notions of exploration and exploitation often refer to rather short term strategies tied
to randomness, whereas intensification and diversification also refer to medium and
long term strategies based on the usage of memory.
The use of the terms
36
diversification and intensification in heir initial meaning becomes more and more
accepted by the whole field of metaheuristics.
According to Glover and Kochenberger (2003) in Raidl (2006), “… these
methods have over time also come to include any procedure for problem solving that
employs a strategy for overcoming the trap of local optimality in complex solution
spaces, especially those procedures that utilize one or more neighbourhood structures
as a means of defining admissible moves to transition from one solution to another,
or to build or destroy solutions in constructive and destructive processes.”
2.4.1
Classification of Metaheuristics
There are several ways of classifying and describing metaheuristic
algorithms. Blum and Roli (2003) summarised the classification of metaheuristics as
follows.
x
Nature-inspired vs. non-nature inspired.
Perhaps, the most intuitive way of classifying metaheuristics is based on the origins
of the algorithm. There are nature-inspired algorithms, like Genetic Algorithm (GA)
and Ant Colony Algorithm (ACO), and non nature-inspired ones such as Tabu
Search (TS) and Iterated Local Search (ILS). In our opinion this classification is not
very meaningful for the following two reasons. Firstly, many recent hybrid
algorithms do not fit either class (or, in a sense, they fit both at the same time).
Secondly, it is sometimes difficult to clearly attribute an algorithm to one of the two
classes. So, for example, one might ask the question if the use of memory in Tabu
Search is not nature-inspired as well.
x
Population-based vs. single point search.
Another characteristic that can be used for the classification of metaheuristics is the
number of solutions used at the same time: Does the algorithm work on a population
or on a single solution at any time? Algorithms that work on single solutions are
known as trajectory methods and encompass local search-based metaheuristics.
37
They all share the property of describing a trajectory in the search space during the
search process. Population-based metaheuristics, on the contrary, perform search
processes which describe the evolution of a set of points in the search space.
x
Dynamic vs. static objective function.
Metaheuristics can also be classified according to the way they make use of the
objective function. While some algorithms keep the objective function given in the
problem representation “as it is”, some others, like Guided Local Search (GLS),
modify it during the search. The idea behind this approach is to escape from local
minima by modifying the search landscape. Accordingly, during the search the
objective function is altered by trying to incorporate information collected during the
search process.
x
One vs. various neighborhood structures.
Most metaheuristic algorithms work on one single neighborhood structure. In other
words, the fitness landscape topology does not change in the course of the algorithm.
Other metaheuristics, such as Variable Neighborhood Search (VNS), use a set of
neighborhood structures which gives the possibility to diversify the search by
swapping between different fitness landscapes.
x
Memory usage vs. memory-less methods.
A very important feature of metaheuristics classification is the use of search history,
that is, whether they use memory or not. Memory-less algorithms perform a Markov
process, as the information used exclusively to determine the next action is the
current state of the search process. There are several ways of making use of memory.
Usually we differentiate between the use of short term and long term memory. The
first usually keeps track of recently performed moves or visited solutions. The
second is usually an accumulation of synthetic parameters about the search. The use
of memory is nowadays recognized as one of the fundamental elements of a
powerful metaheuristic.
In the rest of this study, we use the metaheuristics classification according to
the single point versus population-based search classification, which divides
38
metaheuristics into trajectory methods and population-based methods. This choice is
motivated by the fact that this categorization permits a clearer description of the
algorithms.
Moreover, a current trend is the hybridisation of methods in the
direction of the integration of single point search algorithms in population-based
ones.
2.4.2
Trajectory Methods
2.4.2.1 Basic Local Search: Iterative Improvement
The basic local search is usually called iterative improvement, since each
move is only performed if the resulting solution is better than the current solution.
The algorithm stops as soon as it finds a local minimum.
2.4.2.2 Simulated Annealing
Simulated Annealing (SA) is one of the oldest among the metaheuristics and
surely among the first algorithms that has an explicit strategy to escape from local
minima.
The origins of the algorithm are in statistical mechanics (Metropolis
algorithm) and it was first presented as a search algorithm for CO problem in
Kirkpatrick et al. (1983) (given some assumptions on the cooling schedule of the
temperature, etc.) that could be shown to converge to an optimal solution. The
fundamental idea is to allow moves resulting in solutions of worse quality than the
current solution (uphill moves) in order to escape from local minima.
The
probability of doing such a move is decreased during the search.
Simulated Annealing comes from the annealing process of solids. A solid is
heated until it melts, and then the temperature of the solid is slowly decreased
(according to annealing schedule) until the solid reaches the lowest energy state or
the ground state. If the initial temperature is decreased rapidly, the solid at the
39
ground state will have many defects or imperfections. An easy implementation of
the algorithm makes it very easy to adapt a local search method (e.g. best
improvement local search) to a simulated annealing algorithm, usually rendering the
local search with much better results. But although it is proven to converge to the
optimum, it converges in infinite time. Not only for this reason, but also since we
have to cool down slowly, the algorithm is usually not faster than its contemporaries.
2.4.2.3 Tabu Search
Tabu Search (TS) was proposed by Glover in 1986. A description of the
method and its concepts can be found in Glover and Laguna (1997). The basic
principle of TS is to pursue a best improvement Local Search whenever it encounters
a local minimum by allowing non-improving moves, cycling back to previously
visited solutions is prevented by the use of memories called tabu lists. These tabu
lists record the recent history of the search. According to Glover, TS is not seen as a
proper heuristic, but rather as a metaheuristic, i.e., a general strategy for guiding and
controlling “inner” heuristics specifically tailored to the problems at hand
(Gendreau, 2002).
TS is among the most cited and used metaheuristics for
combinatorial problems. Many computational experiments have shown that TS has
now become an established Optimisation technique which can compete with almost
all known techniques and which – by its flexibility – can beat many classical
procedures.
The word tabu (or taboo) comes from Tongan, a language of Polynesia,
where it was used by the aborigines of Tonga Island to indicate things that can not be
touched because they are sacred. According to Webster’s Dictionary, the word tabu
now also means “a prohibition imposed by social custom as a protective measure: or
of something “banned as constituting a risk”. These current more pragmatic senses
of the word accord well with the theme of tabu search. The risk to be avoided in this
case is that of following a counter-productive course, including one which may lead
to entrapment without hope of escape. On the other hand, as in the broader social
context where “protective prohibitions” are capable of being superseded when the
40
occasion demands, the “tabus” of tabu search are to be overruled when evidence of a
preferred alternative becomes compelling.
The most important association with traditional usage, however, stems from
the fact that tabus as normally conceived are transmitted by means of a social
memory which is subject to modification over time. This creates the fundamental
link to the meaning of “tabu” in TS. The forbidden elements of TS receive their
status by reliance on an evolving memory, which allows this status to shift according
to time and circumstance.
More particularly, TS is based on the premise that
problem solving, in order to qualify as intelligent, must incorporate adaptive memory
and responsive exploration.
The adaptive memory feature of TS allows the
implementation of procedures that implement a form of sampling (Glover and
Laguna, 1997).
2.4.2.4 Explorative Local Search Methods
These are the Greedy Randomised Adaptive Search Procedure (GRASP),
Variable Neighbourhood Search (VNS), Guided Local Search (GLS) and Iterated
Local Search (ILS).
2.4.3
Population-based Methods
Population-based methods deal in every iteration of the algorithm with a set
(i.e., a population) of solutions rather than with a single solution. As they deal with a
population of solutions, population-based algorithms provide a natural, intrinsic way
for the exploration of the search space. Yet the final performance depends strongly
on the way the population is manipulated.
41
2.4.3.1 Evolutionary Computation
Evolutionary Computation (EC) algorithms are inspired by nature’s
capability to evolve living beings well adapted to their environment. EC algorithms
can be succinctly characterized as computational models of evolutionary processes.
At each iteration a number of operators is applied to the individuals of the current
population to generate the individuals of the population of the next generation
(iteration). Usually, EC algorithms use operator called recombination or crossover
to recombine two or more individuals to produce new individuals. They also use
mutation or modification operators which cause a self-adaptation of individuals. The
driving force in evolutionary algorithms is the selection of individuals based on their
fitness (this can be the value of an objective function or the result of a simulation
experiment, or some other kind of quality measure). Individual with a higher fitness
have a higher probability to be chosen as members of the population of the next
iteration (or as parents for the generation of new individuals). This corresponds to
the principle of survival of the fittest in natural evolution. It is the capability of
nature to adapt itself to a changing environment, which gave the inspiration for EC
algorithms.
There has been a variety of slightly different EC algorithms proposed over
the years.
Basically they fall into three different categories which have been
developed independently from each other. These are Evolutionary Programming
(EP), Evolutionary Strategies and GA. EP arose from the desire to generate machine
intelligence.
While EP originally was proposed to operate on discrete
representations of finite state machines, most of the present variants are used for
continuous Optimisation problems. The latter also holds for most present variants of
E, whereas GAs are mainly applied to solve combinatorial Optimisation problems.
GAs are an effective search and Optimisation method that simulates the
process of natural selection or survival of the fittest. Holland in 1974 developed the
idea and concepts behind the GA and many authors have refined his initial approach
(Ortiz et al., 2004). The purpose of GAs is to provide satisfactory results for
Optimisation problems that are hard to solve using exhaustive techniques. The
42
researchers who used GAs to solve complex real-world problems report good results
from these techniques.
Genetic Algorithms differ from traditional search and Optimisation algorithm
in four ways:
1. GAs work with a coding of solution set, not the solution themselves.
2. GAs search from a population of solutions, not a single solution.
3. GAs use payoff information (fitness function), not derivatives or other
auxiliary knowledge.
4. GAs use probabilistic transition rules, not deterministic rules. (Goldberg,
1989)
Whereas Jung (2000) notes that there are two distinguishing characteristics of
the GAs that separate them from the general Optimisation techniques. The first
characteristic is that the GAs start with an initial set of random feasible solutions, not
a single solution. The GA generates many possible solutions to a given problem and
then let them compete and mate to generate improved offspring. In conventional
Optimisation techniques, a point to point approach is applied with a stepwise
procedure to get the optimum solution, but this approach has the danger of falling in
local optima (Gen and Cheng, 1997). However the GAs are population to population
approach, and can escape from local optima and are very effective in global search
with a multi directional search. The second characteristic is that the GAs do not
have any specific functional requirement for the mathematical relationship that
express a given problem. Therefore the GAs can handle any kind of objective
function and constraints. It is difficult to apply simple GAs to complex Optimisation
problem.
However with some modification, GAs can solve some particular
problems (Jung, 2000).
There are three major advantages when applying GA to Optimisation
problems (Gen and Cheng, 1997), they are
x
Due to their evolutionary nature, GAs will search for solutions without regard
to the specific inner workings of the problem. Gas can handle any kind of
objective functions and any kind of constraints (linear and non linear) defined
on discrete, continuous or mixed search spaces.
43
x
The ergodicity of evolution operators makes GA very effective at performing
global search (in probability).
The traditional approaches perform local
search by a convergent stepwise procedure, which compares the value of
nearby points and moves to the relative optimal points. Global optima can be
found only if the problem possesses certain convexity properties that
essentially guarantee that any local optimum is global optimum.
x
GA provides a great flexibility to hybridize with domain dependent heuristics
to make an efficient implementation for a specific problem.
2.4.3.2 Ant Colony Optimisation
Ant Colony Optimisation (ACO) is a metaheuristic approach proposed in
Dorigo (1992) and Dorigo et al. (1999). The inspiring source of ACO is the foraging
behavior of real ants. This behavior enables ants to find shortest paths between food
sources and their nest. While walking from food sources and the nest and vice versa,
ants deposit a substance called pheromone on the ground.
They decide which
direction to go by choosing the higher probability paths that are marked by stronger
pheromone concentrations.
This basic behavior is the basis for a cooperative
interaction which leads to the emergence of shortest paths.
2.5
Hybrid Metaheuristics
Over the last decade a large number of algorithms were reported that do not
purely follow the concepts of one single traditional metaheuristic, but they combine
various algorithmic ideas. These approaches are commonly referred to as hybrid
metaheuristics. As for metaheuristics in general, there exist various perceptions of
what a hybrid metaheuristic actually is. According to the May 2006’s issue of the
Merriam Webster dictionary, the meanings of hybrid are
44
x
something heterogeneous in origin or composition,
x
something (as a power plant, vehicle, or electronic circuit) that has two
different types of components performing essentially the same function,
while the current entry in Wiktionary defines this term as
x
offspring resulting from cross-breeding different entities, e.g. different
species,
x
something of mixed origin or composition.
Blum and Roli (2003) distinguish different form of metaheuristics
hybridisation.
The first one consists of including components from one
metaheuristic into anoher one.
The second form concerns systems that are
sometimes labeled as cooperative search.
The third form is the integration of
approximate and systematic (or complete) methods.
-
Component Exchange among Metaheuristics
One of the most popular ways to hybridisation concerns the use of trajectory
methods in population-based methods. Most of the successful applications of EC
and ACO make use of local search procedures.
The reason for that becomes
apparent when analysing the respective strengths of trajectory methods and
population-based methods. The power of population-based methods is certainly
based on concept of recombining solutions to obtain new ones. A solution resulting
from recombination in population-based methods is usually more different from the
parents. The strength of trajectory methods is rather to be found in the way they
explore a promising region in the search space. A promising area in the search space
is searched in a more structured way than in population-based methods. In this way,
the danger of being close to good solutions but missing them is not as high as in
population-based methods.
-
Cooperative Search
Cooperative search consists of a search performed by possibly different algorithms
that exchange information about states, models, entire sub-problems, solutions or
45
other search space characteristics. Typically, cooperative search algorithms consist
of the parallel execution of search algorithms with a varying level of communication.
The algorithms can be different or they can be instances of the same algorithm
working on different models or running with different parameters setting.
Cooperative search nowadays receives more attention, which is among other reasons
due to the increasing research on parallel implementations of metaheuristics. The
aim of research on parallelization of metaheuristics is twofold.
Firstly,
metaheuristics should be redesigned to make them suitable for parallel
implementation in order to exploit intrinsic parallelism.
Secondly, an effective
combination of metaheuristics has to be found, both to combine different
characteristics and strengths and to design efficient communication mechanism.
-
Integrating Metaheuristics and Systematic Methods
A very successful example of such integration is the combination of metaheuristics
and Constraint Programming (CP) (Focacci et al., 2002).
The hybridisation is
usually achieved by integrating concepts developed for metaheuristics (e.g.,
probabilistic choices, aspiration criteria, heuristic construction) into tree search
methods.
A typical application of this integration is the use of a probabilistic
backtracking, instead of a deterministic chorological backtracking. For instance, an
extreme case is the random choice of a backtracking move. The list of possible
backtracking moves can also be sorted by means of a dynamic heuristic or a
sampling of the leaves of the search tree. This sampling can be performed by a
metaheuristic: the result of each possible backtracking move is chosen as the starting
point for producing a complete solution by a metaheuristic (more than one solution
can be generated from each partial solution). Then, the quality of these complete
solutions is used as score for probabilistically selecting a backtracking move.
2.6
Applications of Genetic Algorithm in the Vehicle Routing Problems
For many years, GAs have been proven to be effective and successful in a
wide variety of search domain, even for non-linear, multi modality, high
dimensionality, discontinuity and noised mathematical models. Therefore GAs have
46
been widely applied in solving combinatorial optimization problems, including
certain types of VRP, especially where time windows are included and in TSP.
Blanton and Wainwright (1993) were the first to apply a genetic algorithm to
VRPTW. They hybridized a genetic algorithm with a greedy heuristic. Under this
scheme, the genetic algorithm searches for a good ordering of customers, while the
construction of the feasible solution is handled by the greedy heuristic. Thangiah
(1995a and 1995b) uses a genetic algorithm to find good clusters of customers within
a “cluster first, route second” problem-solving strategy. Thangiah et al. (1995) test
the same approach to solve vehicle routing problems with time deadlines. In the
algorithm proposed by Potvin and Bengio (1996) new offspring are created by
connecting two route segments from two parent solutions or by replacing the route of
the second parent-solution by the route of the first parent-solution. Mutation is then
used to reduce the number of routes and to locally optimize the solution.
Berger et al. (1998) present a hybrid genetic algorithm based on removing
certain customers from their routes and then rescheduling them with well-known
route-construction heuristics. The mutation operators are aimed at reducing the
number of routes by rescheduling some customers and at locally reordering
customers. Bräysy and Gendreau (2001a, 2001b) continues the study of Berger et al.
(1998) by performing a comprehensive sensitivity analysis, and by creating new
crossover and mutation operators. Bräysy et al. (2000) hybridize a genetic algorithm
with an evolutionary algorithm consisting of several route construction and
improvement heuristics. The recent genetic algorithm by Tan et al. (2001) is based
on Solomon’s insertion heuristic, -interchanges and the well-known PMX-crossover
operator. Berger et al. (2001) proposed GAs approach involves parallel co-evolution
of two populations and partial temporal constraint relaxation to improve solution
quality. The first population evolves individuals to minimize total traveled distance
while the second population focuses on minimizing temporal constraint violation to
generate a feasible solution. Results from a computational experiment showed that
Parallel Hybrid GA is cost-effective and very competitive, matching and even
outperforming the best-known VRPTW metaheuristics. The developed heuristic
improved the overall total number of tours while generating six new best-known
solutions for Solomon data sets.
47
GAs also appear to find good solutions for the TSP. However it depends
very much on the way the problem is encoded and which crossover and mutation
methods are used. It seems that the methods use heuristic information or encode the
edges of the tour (such as the matrix representation and crossover) perform the best
and give good indications for future work in this area. Overall, it seems that GAs
have proved suitable for solving the TSP. As yet, GAs have not found a better
solution to the TSP than is already known, but many of the already known best
solutions have been found by some genetic algorithm method also.
Jog et al. (1991) in Bryant (2000) found that partially modified crossover
(PMX) with the tour notation and no mutation operator gave a tour which length was
ten percent farther than the known optimum for the 33 city problem. For the 100
city problem, the result was 210 percent larger than the known optimum. The
algorithm using order crossover gives a better performance, giving a result of length
425 for the 30 city problem, while cycle crossover only gives a tour of length 517 for
the same problem. The best known solution for the 30 city problem is 420, so order
crossover seems to be the best so far (Homaifar, 1992 in Bryant, 2000). Homaifar
(1992) also found that two point matrix crossover method as well as inversion
performed well with 30, 50, 75, 100 and 318 city problems giving tours of lengths
420, 426, 535, 629 and 42154 respectively, which are all less then two percent above
the best known solution. So it seems that the idea of using edges rather than the
position of cities as our variable is promising. This makes sense, as it is the edge
which holds the costs and we want to pick which edges to use to connect all cities.
Note however, that the matrix representation takes more space to store and also more
computation time for the crossover and mutation processes than the integer
representation and basic crossovers.
Jog’s heuristic algorithms also performed well. The heuristic crossover,
when combined with the 2-opt and Or-opt mutation operators sometimes gives the
best known solution for that particular problem, and otherwise returns a solution
very close to that value. Heuristic algorithms also take up more space, since the 27
costs of each edge need to be stored. So GAs work best for the TSP when a matrix
representation with matrix crossover or a heuristic crossover is used. In both cases
the 2-opt mutation operator improves the solution.
48
Choi et al. (2003) proposed GAs for asymmetric TSP with extension on
search space by generating and including good quality infeasible solutions with
subtours through genetic operators: crossover and mutation.
For performance
comparison, benchmark problem set: 19 problems from TSPLIB and 8 additional
problems from Fischetti and Toth solved by this algorithm are compared to branch
and bound algorithm for asymmetric TSP. The solution quality of GAs obtained in
the experiment is comparable with, or in some cases surpasses, that of publicly and
readily available branch and bound algorithm. The GAs perform better when the
assignment relaxation problem provides a poor quality lower bound on the
asymmetric TSP.
Baker and Ayechew (2003) study the application of a GA to the basic VRP in
which costumers of known demand are supplied from a single depot.
The
experiment was conducted on 14 benchmark problems which can be downloaded
from the OR-library. Different crossover methods were compared; the best distances
for 1-point and uniform crossover were respectively 0.5% and 0.6% higher than
those for 2-point crossover. Whereas the best distances for worst fitness/unfitness
replacement were 1.3% higher than those for ranking replacement. Although the
pure GAs perform well, it does not equal Tabu Search in terms of solution quality. It
appears to be better to use hybrid GA, that is neighborhood search incorporating with
GA, produce a significant improvement, giving distances that are only 0.5% above
best known result with solution times that are not excessive. The statement about
Genetic Algorithm hybridization is also notes by Coley (1999) that hybrid
algorithms, which combine a GA with more traditional algorithms, have been hinted
as a highly powerful combination for solving practical problem, also by Lacomme et
al. (2006) that it is well known that a standard Genetic Algorithm must be hybridized
with another search procedure to be able to compete with metaheuristics like Tabu
Search.
Applications of GAs have also been reported for the VRP with backhauls in
Potvin et al. (1996b), for a multi-depot routing problem (Salhi et al., 1998), and a
school bus routing problem (Thangiah and Nygard, 1992). A hybrid approach to
vehicle routing using neural networks and GAs has also been reported (Potvin et al.,
1996a).
49
2.7
Applications of Tabu Search in the Vehicle Routing Problems
In Tabu Search, sequences of solutions are examined as in simulated
annealing, but the next move is made to the best neighbour of the current solution
xt . To avoid cycling, solutions that were recently examined are forbidden, or tabu,
for a number of iterations.
To alleviate time and memory requirements, it is
customary to record an attribute of tabu solutions rather than the solutions
themselves.
The basic tabu search mechanism can be enhanced by several
computational features, such as diversification and intensification strategies, as
described by Glover and Laguna (1993, 1997) and Hertz et al. (1997), for example.
Over the last fifteen years or so, tabu search has been applied to the VRP by
several authors.
Some of the first tabu search algorithms not yield impressive
results, but subsequent implementations were much more successful. These include
the work of Osman (1993), Taillard (1993), Gendreau, Hertz and Laporte (1994), Xu
and Kelly (1996), Rego and Roucairol (1996), Rego (1998), Barbarosoglu and Ozgur
(1999). In addition, Rochat and Taillard (1995) introduced a useful and powerful
concept, the adaptive memory, which can be used to enhance any tabu search-based
algorithm. In the same vein, Toth and Vigo (2003) introduced granular tabu search,
whose principles have far-reaching applicability (Toth and Vigo, 2002).
2.7.1
Two Early Tabu Search Algorithms
On of the first attempts to apply tabu search to the VRP is due to Willard
(1989). Here, the solution is first transformed into a giant tour by replication of the
depot, and neighbourhoods are defined as all feasible solutions that can be reached
from the current solution by means of 2-opt or 3-opt exchanges (Lin, 1965). The
next solution is determined by the best nontabu move. On three of the Christofides
et al. (1979) benchmark problems, the proposed algorithm does not appear to be
competitive with most known approximation algorithms. Pureza and Franca (1991)
defined the neighbours of a solution by moving a vertex to a different route or by
50
swapping vertices between two routes while preserving feasibility. As in Willard
(1989), the best nontabu feasible move is selected at each iteration. While better
than Willard’s algorithm, this implementation did not produce especially good
results. Further research has shown that more sophisticated search mechanisms are
required to make tabu search work.
2.7.2
Osman’s Tabu Search Algorithm
In Osman (1993), neighbourhoods are again defined by means of the O interchange generation mechanism, with O = 2. This includes a combination of
2-opt moves, vertex reassignments to different routes, and vertex interchanges
between two routes. In one version of he algorithm called BA (best admissible), the
whole neighbourhood is explored and the best nontabu feasible move is selected. In
the other version, FBA (first best admissible), the first admissible improving move is
selected if one exists; otherwise, the best admissible move is implemented.
2.7.3
Taburoute
With respect to the previous tabu search implementation, the Taburoute
algorithm of Gendreau et al. (1994) is rather involved and contains several
innovative features. The neighbourhood structure is defined by all solutions that can
be reached from the current solution by removing a vertex from its current route and
inserting it into another route containing one of its p nearest neighbours using GENI,
a Generalized Insertion procedure developed by Gendreau et al. (1992) for the
Traveling Salesman Problem (TSP). This may result in eliminating an existing route
or in creating a new one. A second important feature of Taburoute is that the search
process examines solutions that may be infeasible with respect to the capacity or
maximum route length constraints. More precisely, the objective function contains
two penalty terms, one measuring overcapacity, the other measuring over duration,
each weighted by a self-adjusting parameter: every 10 iterations, each parameter is
51
divided by 2 if all 10 previous solutions were feasible or multiplied by 2 if all were
infeasible.
This way of proceeding produces a mix of feasible and infeasible
solutions and lessens the likelihood of being trapped in a local minimum. At various
points during the search process, Taburoute re optimises the route in which a vertex
has just been inserted. This is achieved by using the Unstringing and Stringing (US)
TSP postoptimisation routine developed by Gendreau et al. (1992).
Taburoute does not actually use a tabu list but instead uses random tabu tags.
Whenever a vertex is moved from route r to route s at iteration t, its reinsertion into
route r is forbidden until iteration t+ T , where T is an integer randomly drawn from
the interval [5, 10]. Yet another feature of Taburoute is the use of a diversification
strategy, which consists of penalizing vertices that have been moved frequently in
order to increase the probability of considering slow-moving vertices. The objectie
function is artificially increased by adding to it term proportional to the absolute
frequency of movement of the vertex v currently being considered.
Finally,
Taburoute uses false starts. Initially, several solutions are generated and a limited
search is carried out on each of them. The best identified solution is then selected as
a starting point for the main search.
2.7.4
Barbarosoglu and Ozgur’s Algorithm
Barbarosoglu and Ozgur (1999) describe a rather simple tabu search
algorithm containing no diversification strategy and in which only feasible solutions
are examined.
Neighbour solutions are defined by means of a O -interchange
scheme that favors vertices relatively far from the centroid of their current route and
close to the centroid of the new route. Route reoptimisations are performed by
applying a 2-opt procedure. The method was applied to the six capacitated instances
of the Christofides et al. (1979) set and yielded interesting results.
52
2.7.5
Adaptive Memory Procedure of Rochat and Taillard
One of the most interesting developments to have occurred in the area of tabu
search in recent years is the concept of adaptive memory developed by Rochat and
Taillard (1995). It mostly used in TS, but its applicability is not limited to this type
of metaheuristic.
An adaptive memory is a pool of good solutions that is
dynamically updated throughout the search process. Periodically, some elements of
these solutions are extracted from the pool and combined differently to produce new
good solutions. In the VRP, vehicle routes selected from several solutions will be
used as a starting point. The extraction process gives a larger weight to those routes
belonging to the best solutions. When selecting these routes, care must be taken to
avoid including the same customer twice in a solution. This restriction means that
the selection process often will terminate with a partial solution that will have to be
completed using a constructive heuristic. Rochat and Taillard (1995) showed that
the application of an adaptive memory procedure can enhance a search strategy.
This has enabled them to obtain two new best solutions on the 14 standard VRP
benchmark instances.
2.7.6
Granular Tabu Search of Toth and Vigo
Granular Tabu Search (GTS) is yet another very promising concept. It was
introduced by Toth and Vigo (2003) and has yielded excellent results on the VRP.
The main idea behind GTS stems from the observation that the longer edges of a
graph have only a small likelihood of belonging to an optimal solution. Therefore,
by eliminating all edges whose length exceeds a granularity threshold, several
unpromising solutions will never be considered by the search process.
53
2.8
Review on GA and TS for VRPSD
As compared to the development of research in deterministic cases, research
on Stochastic Vehicle Routing Problem is rather undeveloped. So, the study of
SVRPs is a relatively growing research area which should gain in importance with
the spread of real time vehicle routing problems. The Vehicle Routing Problem with
Stochastic Demands (VRPSD) is without any doubt the most studied among all
SVRPs.
Many algorithms for solving VRPSD have been proposed e.g. exact
algorithm, heuristic, rollout policy and metaheuristics for single vehicle and multiple
homogeneous vehicles. Comprehensive survey on previous researches in VRPSD is
presented in Section 2.2.7. In this section we only focus on the details of GA and TS
applied to VRPSD, since our research deals with the development of metaheuristics
based on GA and TS for VRPSD.
As we know, GA has been proven to be effective and successful in a wide
variety of search domain and have been widely applied in solving combinatorial
Optimisation problems, including certain types of VRP, especially where time
windows are included and in TSP. TS was also known as a robust algorithm and
often outperforms other heuristics in terms of computational speed and solution
quality. But, the use of GA and TS in VRPSD is lacking in the literature.
Bianchi et al. (2004) have developed the basic implementation of five
metaheuristics: ILS, TS, SA, ACO and GA for single VRPSD that found better
solution quality respect to cyclic heuristic. For GA application, a small sample size
of 10 individuals is used. Initial solutions are built with the randomized farthest
insertion heuristic which picks the first customer at random and after the tour has
been completed, shifts the starting customer to the depot. Local search is then
applied to each member of the initial population. In the steady state evolution
process only one couple of parents reproduces at each generation. Tournament
selection of size 2 is used to select which parents are going to be given the chance to
reproduce.
The crossover used in the final implementation of Bianchi et al. (2004) is the
Edge Recombination (ER) readapted to always start at the depot. OX and PMX were
54
also used but ER seems to work better. It tries to build an offspring exclusively from
the edges present in both parents, as outlined in Figure 2.3. Swap based mutation
was used with swaps two adjacent alleles and never the depot. It is applied with a
mutation rate of 0.2.
Order based mutation that picks 2 loci at random and
exchanges their alleles, and inversion were also tried. Local search is again applied
at each generation to improve the quality of the offspring. The improved offspring is
then replaces the worst member of the population.
1. Create an edge list from both parents, that provides for each
customer all the other customers connected to it in at least
one of the parent,
2. start from the depot as current customer,
3. select the customer in the edge list of the current customer,
with the smallest number of customers left in its edge list,
4. if there is no customer left in the edge list of the current
customer then
5.
select a non yet visited customer,
6. end if
7. the selected customer becomes the current customer,
8. update the edge list by removing the current customer from
each adjacent customer list,
9. if the tour is complete then
10.
stop.
11. else
12.
go to step 3.
13. end if.
Figure 2.3 : Bianchi et al. (2004)’s Genetic Algorithm for the VRPSD
For TS implementation, Bianchi et al. (2004) uses the randomized farthest
insertion heuristic as the initial solution. The neighbourhood is derived from Or-Opt
local search as states in Yang et al. (2000). The strategy of selection of new
solutions is a descent-ascent method with the use of tabu list.
Their experimental results show that Iterated Local Search is the only which
is always among the best. At the opposite extreme is the SA, which is always among
the worst. The performance of the other metaheuristics seems to be not significantly
different from one to another. It has been hypothesized that this is due to the fact
that all algorithms find solutions which are very near to the optimal, and thus
solution values found are not very different from each other.
55
Gendreau et al. (1996) developed TS for a stochastic VRP that combines two
types of uncertainty: stochastic demand and stochastic customers (customers present
at locations with some probabilities).
They were the first to implement exact
algorithm for VRPSDC in order to find optimal solutions. These can be used to
assess the quality of solutions produced by another heuristics. The problem consists
of designing a first stage solution that minimises the expected cost of the second
stage solution, i.e., the cost of the first stage solution plus the expected cost of
recourse action. The neighbourhood of current solution is defined by removing in
turn one of q randomly selected customers and inserting each of them either
immediately before, or immediately after one of its p nearest neighbours. If a vertex
is moved at iteration v, its reinsertion and displacement is tabu until iteration v+.
TABUSTOCH algorithm proceeds as follows:
x
Initialization.
x
Neighbourhood Search.
x
Incumbent update.
x
Coefficient update: update the list of tabu move.
x
Intensification or terminate.
Results indicate that the running time of the proposed heuristic is reasonable,
and optimal solutions were obtained in over 89% of all instances for which an
optimum was known. According to Gendreau et al. (1996), the success of TS seems
largely due to the neighbourhood structure, to the fact that intermediate infeasible
solutions are allowed, and to the proxy function used to evaluate candidate moves.
This device is responsible for large time savings as t avoids repeated costly
computations of the objective function.
In GA and TS application of Bianchi et al. (2004), the fact that that GA and
TS seem to have the same result but they can not compete with the ILS, have opened
a new direction in developing GA and TS for solving VRPSD. Further, it is well
known that a standard GA must be hybridized with another search procedure to be
able to compete with other metaheuristics. So this study proposes the development
of GA and TS – based metaheuristics for solving VRPSD and the first
56
implementation of hybrid GA with TS for the VRPSD that combines the advantage
of GA and the strength of TS.
2.7
Summary
This chapter provides an in-depth understanding of the area of VRPSD and
solution techniques for it. It began by looking at the description of VRPSD and
some related works from researchers and heuristics (or metaheuristics) which can be
used to solve the VRPSD. These led to the development of enhanced metaheuristics
for solving VRPSD for improving the solution quality by considering Genetic and
Tabu Search – based algorithms.
57
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
This chapter presents the direction of the study and an overview of the
methods used. It begins with the general steps of the research framework. A
description of the data source for this study and test related to it are also presented
which include the case-study of a real-life application at solid waste collection
company (Perniagaan Zawiyah Sdn. Bhd). It is followed by the description of
algorithms implemented with emphasis on Genetic Algorithm, Tabu Search and
hybrid of these heuristics while the details of the proposed algorithms are given in
the following chapters.
We also provide a brief introduction to some of the
terminologies used when dealing with different types of applications within a
stochastic environment.
3.2
Research Framework
This framework of this research is divided into two main parts; these are the
development of algorithm for single vehicle and multiple vehicles. The general steps
of research framework are summarized in Figure 3.1 while the detail will be given in
the subsequent sections.
58
Figure 3.1 : Research framework
3.3
Data Source
There are two types of data that will be used to assess the performance of the
proposed algorithm:
59
a.
Primary Data
The primary data is a real life instance of VRPSD originating from the
municipal solid waste collection. The data was collected from Perniagaan Zawiyah,
Sdn. Bhd., Johor Bahru, one of sub contractor of Southern Waste Management Sdn.
Bhd. (SWM) – Johor Bahru branch office. SWM is a provider of a diverse range of
waste management services for municipal authorities, commercial and industrial
sectors for the Southern Region of Malaysia. We confine our study on residential
area in Johor Bahru under municipal authorities of Majlis Perbandaran Johor Bahru.
The problem of optimization of waste collection can be described as follows.
The solid waste is located in box or containers along the streets. On any given day,
the company faces the problem of collecting waste from a set of customer location
where the amount of waste disposal is a random variable, while each collecting truck
has a limited capacity. The problem is to design a set of solid waste collection
routes, each to be served by a truck such that the waste at each customer is fully
collected and the total expected cost is minimized. Each vehicle can service several
such sites before going to dumpsite to unload. Each vehicle starts from the depot,
visits a number of customers and ends at the depot. When a vehicle is full, it needs
to go to the closest available dumpsite to empty its load, and then resume their visit.
Each vehicle can make multiple disposal trips per day.
Detail of company profile and data collected are given in Chapter 7. The
variables to be measured in the data collection include:
b.
x
The distance or coordinates of customer locations.
x
The customer demands.
Simulation Data
Several sets of instances will be generated to simulate the real life instance of
VRPSD in solid waste collection as described in Section 3.4.
60
3.4
Data Generation
Since there is no commonly used benchmark for the VRPSD as found from
the literature review, we generate our own test bed. We consider several sets of
randomly generated instances that simulate real problem data from case study of
solid waste collection.
Based on experiments reported in Gendreau et al. (1995), three factors seem
to give impact to the difficulty of a given VRP instances: number of customers n,
number of vehicles m, and filling coefficient f. In a stochastic environment, the
filling coefficient can be defined as
n
f
¦
i 1
E ([ i )
mQ
(3.1)
where E( [ i ) is the expected demand of customer i and Q denotes the vehicle
capacity. This is the measure of the total amount of expected demand relative to
vehicle capacity and can be approximately interpreted as the expected number of
loads per vehicle needed to serve all customers.
Customer locations will be generated in the [0, 100] square area following a
discrete uniform distribution with the depot fixed at coordinate (50, 50). Each cij is
then defined as travel cost from i to j, as a function of distance traveled. Without loss
of generality, it is assumed that the travel cost is 1 unit cost per unit distance, thus the
cost function here is equal to distance function. The distance between customers
location i and j is calculated using Euclidean distance as follows:
d ij
( xi x j ) 2 ( yi y j ) 2
and it is assumed further that the distance is symmetric, that is dij = dji and dii = 0.
We consider discrete uniform probability distribution for the data of
demands. The problem data will be generated using Minitab 14 software package.
61
3.4.1
Single vehicle
For a single vehicle, the customers demands follow discrete uniform
distributions: U(1,5), U(6,10) and U(11,15) respectively. There are three different
problem size used here that are:
-
small problem size, that is 10 nodes
-
moderate problem size, that is 20 nodes
-
large problem size, that is 50 nodes
Twenty instances were generated on each of the problem size. Each algorithm was
tested on each instance for a number of iterations and executed for a number of
independent runs, thus the reported computational results will be the average values
over the number of independent run. When m = 1 then the filling coefficient in
Equation (3.1) can be defined as
n
f
¦
i 1
E ([ i )
Q
(3.2)
where E( [ i ) is the expected demand of customer i and Q denotes the vehicle
capacity.
3.4.2
Multiple vehicles
The second experiment considered in this study is the multiple homogeneous
instances. For the case of a fleet of vehicles, VRPSD with multiple vehicles analog
with the deterministic VRP are considered. The difference is that a priori restocking
policy is designed for each vehicle on the routes. The objective is to find a set of
routes, each of them to be served by one vehicle such that objective functions are
minimized.
In this experiment, the bi-objective function to minimize the total
expected cost and number of vehicles is considered with the restriction on length
constraint of each vehicle.
62
3.5
Demand’s Data Testing
After the data are collected or generated, hypothesis test for distributional
adequacy of demand data (whether a sample of data comes from specific
distribution) will be conducted. One of the primary purposes of a statistics is to
estimate unknown properties of the population that are necessarily numerical and
include items such as unknown proportions, means, probabilities, and so on. The
form of statistical inference that has received the most attention is called hypothesis
testing. Some terminologies on hypothesis testing are given as follows.
-
Hypothesis testing
: the process of inferring from a sample whether or
not to accept a certain statement about the
population.
-
Null hypothesis
: hypothesis to be tested, denoted by H0.
-
Alternative hypothesis : the negation of the null hypothesis, denoted by H1.
-
A test statistic
: a statistic used to help make the decision in a
hypothesis test
-
The critical region
: the set of all points in the sample space that result
in the decision to reject the null hypothesis
-
The significance level : the maximum probability of rejecting a true null
hypothesis, denoted by D
-
The critical level
: the smallest significance level at which the null
hypothesis would be rejected for the given
observation, denote by p-value.
Goodness-of-fit test is a test designed to compare the sample obtained with
the type of sample one would expect from the hypothesized distribution to see if the
hypothesized distribution function “fits” the data in the sample (Conover, 1980).
Over the years, many procedures for goodness of fit problems have been proposed,
and two best known of these are the chi square goodness of fit test and KolmogorovSmirnov test.
The chi-square goodness-of-fit test can be applied to discrete
distributions, while Kolmogorov-Smirnov can be applied to continuous distributions.
The choice of these statistical goodness-of-fit tests is dependent on the type of
variable being studied.
Observations can be classified according to a standard
63
measurement scale that goes from “weak” to “strong” depending on the amount of
precision of information available in the scale. There are four scales of measurement
that can be defined as follows (Freund and Wilson, 2003):
1. Nominal : identifies observed values by name or classification, for example: sex,
colors and brand names of the product.
2. Ordinal : distinguishes between measurements on the basis of the relative
amounts of some characteristic they possess. Usually the ordinal
scale refers to measurements that make only “greater”, “less”, or
“equal comparisons between consecutive measurements.
For
example, if a person were asked to taste five chocolate pies and rank
them according to taste, the result would be a set of observations in
the ordinal scale of measurement.
3. Interval : uses the concept of distance or measurement and requires a “zero”
point, but the definition of zero may be arbitrary. An example of
interval scale is the use of degrees Fahrenheit or Celsius to measure
temperature.
4. Ratio
: uses the concept of a unit of distance or measurement and requires a
unique definition of a zero value. It is appropriate for measurements
of heights, weights, birth rates, and so on.
3.5.1
Chi-Square Goodness of Fit Test
DATA
The data consist of N independent observations of a random variable X. These N
observations are grouped into c classes/ categories, and the numbers of observations
in each class are presented in the form of a 1 x c contingency table (Conover, 1980).
The data for analysis may be displayed in tabular form as shown in Table 3.1. The
categories may be either nominal or numerical. For example, a sample of individuals
may be placed into one of the two nominal categories: male and female. If age is the
variable of interest, these same individuals may be placed in one or the other of
several age (numerical) categories, such as less than 15 years, 15-24, 25-34, 35-44,
45-54, 55 and older (Daniel, 1990).
64
Table 3.1 : Data display for chi-square goodness of fit test
Class/ category
1
2
…
i
…
c
Total
Observed frequencies
O1
O2
…
Oi
…
Oc
N
ASSUMPTIONS
1. The data is a random sample.
2. The measurement scale is at least nominal.
HYPOTHESES
The chi-square test is defined for the hypothesis:
H0 : The sample has been drawn from a population that follows a specified
distribution
H1 : The sample has been drawn from a population that does not follow the
specified distribution
TEST STATISTIC
We expect random samples drawn from populations to reflect the characteristics of
those populations. Thus, if we have drawn a sample from a specified (hypothesized)
population, we expect a close “fit” between observed and expected frequencies
falling into the various categories. If the null hypothesis is true, then, we expect
close agreement between the observed and expected frequencies falling into the
categories. The test statistic for chi-square goodness of fit test is
F
2
c (Oi Ei ) 2
¦
Ei
i 1
where Oi is the observed frequency for class i, c is the number of class/ category and
Ei is the expected frequency for class i.
DECISION RULE
Reject H0 if F 2 ! F (21D , c 1) where F (21D , c 1) is the (1- D ) quantile of a chi-square
random variable with (c-1) degrees of freedom. Otherwise do not reject the null
65
hypothesis. When we have to estimate parameters of the sample data, we compute
the degrees of freedom differently. In addition to subtracting 1 from c, we also
subtract 1 for each parameter that we have to estimate. If g is the number of
parameters that we have to estimate from sample data, the degrees of freedom are cg-1. For uniform distribution, g = 3 (Daniel, 1990).
3.5.2
The Kolmogorov-Smirnov Test
The chi square test discussed in Section 3.4.1 was designed for use with
nominal data. The Kolmogorov-Smirnov was designed for testing the goodness of fit
for continuous data. Hence it can be used with data measured on at least an ordinal
scale (Daniel, 1990).
DATA
The data consist of a random sample X 1 , X 2 ,..., X n of size n associated with some
unknown distribution function, denoted by F (x) .
ASSUMPTION
The sample is a random sample.
HYPOTHESES
If we let F0 ( x) be the hypothesized distribution function (cumulative probability
function), then we can state the null hypotheses and the corresponding alternatives as
follows.
A. Two-sided Test
H0 : F (x) = F0 ( x) for all values of x
H1 : F (x) z F0 ( x) for at least one value of x
B. One-sided Test
H0 : F (x) t F0 ( x) for all values of x
66
H1 : F (x) < F0 ( x) for at least one value of x
C. One-sided Test
H0 : F (x) d F0 ( x) for all values of x
H1 : F (x) > F0 ( x) for at least one value of x
TEST STATISTICS
Let S (x) designate the sample (or empirical) distribution function. That is, S (x) is
the cumulative probability function computed from the sample data. Specifically
S (x)
the number of sample observations less than or equal to x
n
The test statistic depends on the hypothesis under consideration.
A. Two-sided Test
D
sup S ( x) F 0( x)
x
which is read “D equals the supremum, over all x, of the absolute value of the
difference S (x) and F0 ( x) that is S ( x) F 0( x)
B. One sided Test
D
sup[ F 0( x) S ( x)]
x
which is similar to D except that this statistics considers only the greatest
difference where the function F 0( x ) is above the function S (x) .
C. One-sided Test
D
sup[ S ( x) F 0( x)]
x
67
which is similar to D except that this statistics considers only the greatest
difference where the function S (x) is above the function F 0( x) .
DECISION RULE
Reject H0 at the significance level D if the statistic under consideration exceeds the
(1- D ) quantile w1D . If the sample data have been drawn from the hypothesized
distribution, the discrepancies between S (x) and F 0( x) for the observed values of x
should not be too large (Daniel, 1990).
3.6
Paired-Samples Procedure
A paired t-test is used to compare two population means where we have two
samples which observations in one sample can be paired with observations in other
sample. Examples of where this might occur are:
-
Before and after observations on the same subject (e.g. students’
diagnostic test results before and after a particular module or course)
-
A comparison of two different methods of measurement or two different
treatments where the measurements/ treatments are applied to the same
subjects (e.g. blood pressure measurements using a stethoscope and a
dynamap)
Suppose a sample of n subjects were given two different treatments. We
want to find out if, in general, the treatment A is different to treatment B. We can
use the results from a sample to draw conclusion about this. Let x = results from
treatment A, y = results from treatment B (Freund and Wilson, 2003).
HYPOTHESES
H0 : The mean difference is zero
H1 : The mean difference is not equal to zero
68
TEST STATISTIC
The procedure is as follows:
1. Calculate the difference ( d i
yi xi ) between the two observations on
each pair, making sure positive and negative differences are
distinguished.
_
2. Calculate the mean difference, d .
3. Calculate the standard deviation of the differences, s d , and use this to
_
calculate the standard error of the mean difference SE( d ) =
sd
n
.
4. Calculate the t-statistic, which is given by
_
T
d
_
SE (d )
Under the null hypothesis, T follows a t-distribution with n-1 degrees of
freedom (tn-1).
5. Reject H0 at the significance level D if the statistic under consideration
exceeds the tn-1 (t-distribution with n-1 degrees of freedom) (Freund and
Wilson, 2003).
When t-test is used, it is assumed that the population from which the sample
data have been drawn is normally distributed. Although modest departures from this
assumption do not seriously affect the conclusion when the assumption is grossly
violated, we have to seek an alternative method of analysis. One such alternative is a
nonparametric procedure. Several nonparametric are available for making inferences
about a location parameter. A characteristic of these nonparametric procedures is
that the median rather than the mean is the location parameter (Daniel, 1990). The
Wilcoxon matched-pairs signed-ranks test can be used as alternative of t-paired
samples test when the normal distribution assumption is violated. The Wilcoxon
matched-pairs signed-ranks test can be described as follows.
69
ASSUMPTION
1. The data for analysis consist of n values of the difference Di
Yi X i . Each
pair of measurement (Xi, Yi) is taken on the same subject or on subjects that
have been paired with respect to one or more variables. The sample of (Xi,
Yi) pairs is random.
2. The differences represent observations on a continuous random variable.
3. The differences are measured on at least an interval scale.
4. The differences are independent.
(Daniel, 1990)
HYPOTHESES
A. Two-sided
H0 : The median of the population of differences is zero.
H1 : The median of the population differences is not zero
B. One sided
H0 : The median of the population of differences is less than or equal to zero.
H1 : The median of the population differences is greater than zero
C. One-sided
H0 : The median of the population of differences is greater than or equal to
zero.
H1 : The median of the population differences is less than zero (Daniel, 1990)
TEST STATISTIC
To obtain the test statistic, the following procedure is used.
1. Obtain each of the sign differences
Di
Yi X i
2. Rank the absolute values of these differences from smallest to largest. That
is, rank
Di
Yi X i
70
3. Assign to each of the resulting ranks the sign of the difference whose absolute
value yielded that rank
4. Compute
T = the sum of the ranks with positive signs
T = the sum of the ranks with negative signs
There are two types of ties. The first type occurs when X i = Yi for a given
pair. We eliminate all pairs of observations having Di = 0 from the analysis and
reduce n accordingly. The other type of tie occurs when two or more values of Di
are equal. For ties of this type, the Di receive the average of the ranks that
otherwise would be assigned to them (Daniel, 1990).
DECISION RULE
A. Either a sufficiently small value of T or sufficiently small value of T will
cause rejection of H0. Therefore the test statistic in this case is T or T whichever is smaller. Reject H0 at significance level D if T is smaller than
or equal to d for n and D (two-sided).
B. Small values of T will cause rejection of H0. Reject H0 at significance
level D if T is less than or equal to d for n and D (one-sided).
C. Small values of T will cause rejection of H0. Reject H0 at significance
level D if T is less than or equal to d for n and D (one-sided).
Suppose, for example, that the observations are taken on subjects before receiving
some treatment. Rejecting H0 in A enables us to conclude that the treatment has an
effect. If the data result from an experiment designed to compare two treatments,
rejecting H0 enables us to conclude that the treatments have different effects.
Rejecting H0 in B and C lead to the conclusion that one treatment is more effective
than the other (Daniel, 1990).
71
3.7
Genetic Algorithm
Solutions to GA are represented by data structures, often a fixed length
vector, called chromosomes. Chromosomes are composed of genes which have
value called alleles, and the position of the gene in the chromosome is called locus
(Goldberg, 1989). Chromosomes have been represented by vectors, strings, arrays,
and tables. Alleles have been represented by a simple binary system of zeroes and
ones, integers, real number, and letter of alphabet, or other symbols (Gen and Cheng,
1997). Each chromosome has a fitness value which is indicative of the suitability of
the chromosome as a solution to the problem (Goldberg, 1989).
A simple genetic algorithm can be summarized as follows.
1. Representation: Encode the characteristics of each individual in the initial
population as a chromosome (typically, a chromosome is a bit string). Set the
current population to this initial population.
2. Evaluation: Evaluate fitness of each chromosome.
3. Selection: Select chromosomes of parent P from the current population via
proportional selection, a chromosome with high fitness is more likely to be
selected.
4. Recombination: Choose at random a pair of parents for mating. Generate two
offspring from the two parents by exchanging sub strings between parent
chromosomes (crossover).
5. Mutation: process each offspring by the mutation operator, and insert the
resulting offspring in the new population.
6. Repeat steps (4) and (5), until P offspring are created.
7. Replace the old population of chromosomes by the new one.
8. Evaluate the fitness of each chromosome in the new population.
9. Go back to Step 3 if the number of generations is less than a fixed number of
generation or until convergence to a population of similar individuals is obtained.
Then, the best chromosome generated during the search is decoded into the
corresponding individual (Potvin, 1996). The procedure of GA can be described in
Figure 3.2 (Gen and Cheng, 1997).
72
Procedure: Genetic Algorithms
begin
tÅ 0
initialize P(t);
evaluate P(t);
while
(not termination condition) do
recombine P(t) to yield C(t);
evaluate
C(t);
select P(t + 1) from P(t) and C(t);
t Å
t + 1;
end
end
Figure 3.2 : Procedure of genetic algorithm
3.7.1
Selection Methods
In selection process, chromosomes are selected from the population to be
parents for crossover. According to Darwin's theory of evolution the best ones
survive to create new offspring. There are many methods in selecting the best
chromosomes. Some of them will be described in following (Obitko, 1998).
a. Roulette Wheel Selection
Parents are selected according to their fitness. The better the chromosomes are, the
more chances for it to be selected.
Imagine a roulette wheel where all the
chromosomes in the population are placed. The size of the section in the roulette
wheel is proportional to the value of the fitness function of every chromosome: the
bigger the value is, the larger the section is. For an example, see Figure 3.3.
Figure 3.3 : Illustration of roulette wheel selection
73
A marble is thrown in the roulette wheel and the chromosome where it stops is
selected. Clearly, the chromosomes with bigger fitness value will be selected more
times. This process can be described by the following algorithm.
1. [Sum] Calculate the sum of the fitness values of all chromosomes in the
population - sum S.
2. [Select] Generate a random number between 0 and the sum of the fitness
value.
3. [Loop] Select the chromosome whose fitness value added to the sum of the
fitness values of the previous chromosomes is greater or equal to the random.
(Obitko, 1998)
b.
Rank Selection
The previous type of selection will have problems when they are big differences
between the fitness values. For example, if the best chromosome fitness is 90% of
the sum of all fitnesses then the other chromosomes will have very few chances to be
selected. Rank selection ranks the population first and then every chromosome
receives fitness value determined by this ranking. The worst will have the fitness 1,
the second worst will have the fitness 2 etc. and the best will have the fitness PopSize
(number of chromosomes in population). The following picture describes how the
situation changes after changing fitness to the numbers determined by the ranking.
Figure 3.4 : Situation before ranking (graph of fitnesses)
Figure 3.5 : Situation after ranking (graph of order numbers)
74
Now all the chromosomes have a chance to be selected. However this method can
lead to slower convergence, because the best chromosomes do not differ so much
from other ones (Obitko, 1998).
c. Steady-State Selection
This is not a particular method of selecting parents. The main idea of this type of
selection is that a big part of chromosomes can survive to next generation. The
steady-state selection GA works in the following way. In every generation a few
good (with higher fitness) chromosomes are selected for creating new offspring.
Then some bad (with lower fitness) chromosomes are removed and the new offspring
is placed in their place. The rest of population survives to new generation (Obitko,
1998).
d. Elitism
When creating a new population by crossover and mutation, we have a big chance,
that we will loose the best chromosome. Elitism is the name of the method that first
copies the best chromosome (or few best chromosomes) to the new population. The
rest of the population is constructed in ways described above. Elitism can rapidly
increase the performance of GA, because it prevents a loss of the best found solution
(Obitko, 1998).
3.7.2
Crossover
The simplest type of crossover is one point crossover. As illustrated in Loft
and Snow (2006) where chromosomes are strings of symbols (here, 0’s and 1’s),
there are two chromosomes; one from Parent A and one from Parent B, both the of
same length. A crossover point is selected randomly. The two chromosomes are cut
at this point, and new chromosomes are formed by using the chromosome from
Parent A before the crossover point and from Parent B after the crossover point and
vice versa. This is depicted in Figure 3.6.
75
Parent A:
0
1
1
0
1
0
1
1
segment of A
Parent B:
1
0
1
1
1
0
0
0
segment of B
Offspring 1
0
1
1
1
1
0
0
0
Offspring 2
1
0
1
0
1
0
1
1
Figure 3.6 : One point crossover
A first generalization of one point crossover is two point crossover. In two
point crossover, two crossover points are randomly chosen. Genetic material is
copied from A before the first crossover point.
Between the crossover points,
material is taken from Parent B. After the second crossover point, material is again
taken from A. This is depicted in Figure 3.7.
Parent A:
0
1
1
0
1
0
1
1
initial and final
segments of A
Parent A:
1
0
1
1
1
0
0
0
middle segment of B
Offspring
0
1
1
1
1
0
1
1
Figure 3.7 : Two point crossover
From two point crossover, one can imagine three point crossover, four point,
five point, and so on. The logical conclusion of this is uniform crossover. In
uniform crossover, each symbol in the offspring’s chromosome is chosen randomly
to be equal to the corresponding symbol of the chromosome of either Parent A or
Parent B as shown in Figure 3.8.
Parent A:
0
1
1
0
1
0
1
1
Parent A:
1
0
1
1
1
0
0
0
Offspring
0
0
1
0
1
0
0
1
Figure 3.8 : Uniform crossover
(the underlined symbols are
the ones chosen for the
offspring)
76
Loft and Snow (2006) observed that if a pattern appears in the chromosomes
of both parents, then recombination will preserve that pattern. For example, both
parents above have 1’s as the third and fifth symbol in their chromosomes and 0 as
the sixth then in one point, two point, and uniform crossover the offspring has the
same pattern. Other crossovers that usually used for TSP and VRP are Partially
Match Crossover (PMX), Cycle Crossover (CX), Order Crossover (OX), Modified
Crossover (MOX), Order-Based Crosover (OBX) and Position-Based Crossover
(PBX) (Potvin, 1996). They can be described in the next sub sections.
3.7.2.1 Partially-Mapped Crossover (PMX)
For PMX operator, firstly we randomly select two cut points on both parents.
In order to create offspring, the substring between the two cut points in the first
parent replaces the corresponding substring in the second parent. Then, the inverse
replacement is applied outside of the cut points, in order to eliminate duplicates and
recover all cities. In Figure 3.9, the offspring is created by first replacing the subset
{2, 3, 6} in parent 2 by the subset {5, 6, 4}. Hence, city 5 replaces city 2, city 6
replaces city 3, and city 4 replaces city 6 (step 1). Since cities 4 and 6 are now
duplicated in the offspring, the inverse replacement is applied outside of the cut
points. Namely, city 2 replaces city 5, and city 3 replaces city 4 (step 2). In the latter
case, city 6 first replaces city 4, but since city 6 is already found in the offspring at
position 4, city 3 finally replaces city 6. Multiple replacements at a given position
occur when a city is located between the two cut points on both parents, like city 6 in
this example.
Parent 1 : 1 2 5 6 4 3 8 7
Parent 2 : 1 4 2 3 6 5 7 8
Offspring:
Step 1
: 1 4 5 6 4 5 7 8
Step 2
: 1 3 5 6 4 2 7 8
Figure 3.9 : Partially-mapped crossover
77
3.7.2.2 Cycle Crossover (CX)
The cycle crossover focuses on subsets of cities that occupy the same subset
of positions in both parents. Then, these cities are copied from the first parent to the
offspring (at the same positions), and the remaining positions are filled with the cities
of the second parent. In this way, the position of each city is inherited from one of
the two parents. However, many edges can be broken in the process, because the
initial subset of cities is not necessarily located at consecutive positions in the parent
tours. In Figure 3.9, the subset of cities {3, 4, 6} occupies the subset of positions {2,
4 , 5} in both parents. Hence, an offspring is created by filling the positions 2, 4 and
5 with the cities found in parent 1, and by filling the remaining positions with the
cities found in parent 2.
Parent 1 :
1
3
5
6
4
2
8
7
Parent 2 :
1
4
2
3
6
5
7
8
Offspring:
1
3
2
6
4
5
7
8
Figure 3.10 : Cycle crossover
3.7.2.3 Modified Crossover
This crossover was proposed by Davis in 1985 as an extension of the onepoint crossover for permutation problems. A cut position is chosen at random on the
first parent chromosome. Then, an offspring is created by appending the second
parent chromosome to the initial part of the first parent (before the cut point), and by
eliminating the duplicates. An example is provided in Figure 3.11. The two parent
chromosomes are cut in the position between second and third positions.
An
offspring is formed by using the chromosome of Parent 1 before the cut point, that is
the subset {1, 2} and from second parent (after cut position) {2, 3, 6, 5, 7, 8}, since
the city 2 is already on the offspring, the city 2 is discarded and replaced by 4.
78
Parent 1 :
1
2
5
6
4
3
8
7
Parent 2 :
1
4
2
3
6
5
7
8
Offspring:
1
2
4
3
6
5
7
8
Figure 3.11 : Modified crossover
3.7.2.4 Order Crossover (OX)
This crossover extends the modified crossover of Davis (1985) by allowing
two cut points to be randomly chosen from the parent chromosomes. In order to
create an offspring, the string between the two cut points in the first parent is first
copied to the offspring. Then, the remaining positions are filled by considering the
sequence of cities in the second parent, starting after the second cut point (when the
end of the chromosome is reached, the sequence continues at position 1). In Figure
3.12, the subset {5, 6, 4} in parent 1 is first copied to the offspring (step 1). Then,
the remaining positions are filled one by one after the second cut point, by
considering the corresponding sequence of cities in parent 2, namely subset {5, 7, 8,
1, 4, 2, 3, 6} (step 2). Hence, city 5 is first considered to occupy position 6, but it is
discarded because it is already included in the offspring. City 7 is the next city to be
considered, and it is inserted at position 6. Then city 8 is inserted at position 7, city 1
is inserted at position 8, city 4 is discarded, city 2 is inserted at position 1, city 3 is
inserted at position 2, and city 6 is discarded.
Parent 1 :
1
2
5
6
4
3
8
7
Parent 2 :
1
4
2
3
6
5
7
8
Offspring (step 1):
-
-
5
6
4
-
-
-
Offspring (step 2):
2
3
5
6
4
7
8
1
Figure 3.12 : Order crossover
79
3.7.2.5 Order-Based Crossover (OBX)
This crossover focuses on the relative order of the cities on the parent
chromosomes. First, a subset of cities is selected from the first parent. In the
offspring, these cities appear in the same order as in the first parent, but at positions
taken from the second parent. Then, the remaining positions are filled with the cities
of the second parent. In Figure 3.13, cities 5, 4 and 3 are first selected in parent 1,
and must appear in this order in the offspring. These cities occupy positions 2, 4 and
6, respectively, in the offspring. The remaining positions are filled with the cities
found in parent 2.
Parent 1 :
1
2
5
6
4
3
8
7
Parent 2 :
1
4
2
3
6
5
7
8
Offspring:
1
5
2
4
6
3
7
8
Figure 3.13 : Order-based crossover
3.7.2.6 Position-Based Crossover
Here, a subset of positions is selected in the first parent. Then, the cities
found at these positions are copied to the offspring (at the same positions). The other
positions are filled with the remaining cities, in the same relative order as in the
second parent. The name of this operator is rather misleading, because it is the
relative order of the cities that is inherited from the parents (the absolute position of
the cities inherited from the second parent are rarely preserved). This operator can
be seen as an extension of the order crossover, where the cities inherited from the
first parent do not necessarily occupy the consecutive positions. In Figure 3.14,
position 3, 5 and 6 are first selected in parent 1. Cities 5, 4 and 3 are found at these
positions, and occupy the same positions in the offspring. The other positions are
filled one by one, starting at position 1, by inserting the remaining cities according to
their relative order in parent 2, namely the set {1, 2, 6, 7, 8}.
80
Parent 1 :
1
2
5
6
4
3
8
7
Parent 2 :
1
4
2
3
6
5
7
8
Offspring:
1
2
5
6
4
3
7
8
Figure 3.14 : Position-based crossover
3.7.3
Mutation
Mutation is a genetic operator that alters one or more gene values in a
chromosome from its initial state. This may produce entirely new gene values being
added to the gene pool. With these new gene values, the GA may be able to arrive at
a better solution than what was previously possible. Mutation is an important part of
the genetic search as helps to prevent the population from stagnating at any local
optima. Mutation occurs during evolution according to a user-definable mutation
probability rate. This probability rate should usually be set low. If it is set too high,
the search will turn into a primitive random search (Mitchell, 1996).
The primary purpose of mutation is to increase variation in a population.
Especially in the case where the initial population may be a small subset of all
possible solutions, mutation is the most important operator in GA. It is possible, for
example, that every instance of an essential bit might be zero in the initial
population. In such a case, crossover could never set that bit to one; mutation,
however, can set that bit to one. Mutation takes place after the crossover has been
performed. Mutation changes the new offspring randomly. For binary encoding, it
is done by switching a few randomly chosen bits from 1 to 0 or from 0 to 1 (Mitchell,
1996; Shen et al., 2001).
A variety of mutation methods are used in GAs, e.g. inversion, insertion,
displacement and reciprocal exchange mutation. Inversion mutation selects two
positions within a chromosome at random and then inverts the substring between
these two positions. Insertion is where individual is randomly selected and inserted
in a random position. Displacement is where a subtour is selected at random and
81
inserted in a random position. Whereas reciprocal exchange where we select two
positions at random and swap them (Gen and Cheng, 1997).
Another mutation operators are 2-opt, 3-opt and Or-opt.
In the 2-opt
operator, two edges (a, b) and (c, d) are randomly selected from the tour and check if
we can connect these four nodes in a different manner that will give a lower cost. To
do this check if cab + ccd > cac + cdb. If this is the case, replace the edges (a, b) and
(c, d) with the edges (a, c) and (d, b). Note that it is assumed that a, b, c and d appear
in that specific order in the tour even if b and c are not connected. A 3-opt operator
looks at three random edges instead of two. If there are edges (a, b), (c, d) and (e, f),
check if cab + ccd + cef > cac + cbe + cdf. If it is, replace (a, b), (c, d) and (e, f) with
the edges (a, c), (b, e) and (d, f).
The Or-opt operator is similar to the 3-opt. A set of connected nodes are
randomly chosen and checking is made if the string can be inserted between two
other connected nodes to give a reduced cost. We can calculate this by finding the
total cost of the edges being inserted and the total cost of the edges being removed.
If the cost of the edges being removed is greater than the cost of those being inserted
the switch is made (Bryant, 2000).
3.6.4. Operator Settings in Genetic Algorithm
It has long been acknowledged that the choice of operator settings has a
significant impact upon GA performance.
These settings come in the form of
components and parameters such as chromosome representation, population sizes,
crossover and mutation operators, crossover and mutation probability, replacement
techniques, etc. However, finding a good choice is somewhat of a black art. The
most commonly used method is empirically through hand tuning. This is where the
GA’s settings are determined by first experimenting with a selection of combinations
of settings and then selecting the combination that gives the best results.
82
The crossover and mutation operators are considered to be two of the main
conduits by which to control exploration and exploitation of the search space of GA.
This is partly why most of the work performed on finding good configurations for a
GA consists of keeping the other settings constant and just looking for good operator
rates. In one of earliest works of variation of settings in a GA, Davis (1989) suggests
the division of operators into two categories based on the roles they play. In the first
category are those operators that directly improve a chromosome, while in the second
category, operators that set the stage for such improvement.
Different forms of schemata exist in solutions for different problems
(Starkweather et al., 1991) and different crossover or mutation operators are better at
exploring different forms of schemata. Where a crossover operator successfully
exploits wanted traits from a particular solution space, for a solution space of a
different problem, it may be that it is able to exploit the wanted traits, but at a
different rate of efficiency (thus explaining the need for different operator rates) or
unable to do so at all.
While classic GAs employ the use of settings that remain static throughout its
application run, some existing work was found that indicated that by varying these
settings appropriately during the GA’s application, improvements can be made to
overall performance of the GA in terms of both solution quality and the amount of
time required to find it. The adaptive variation approach, also known as online
approach (Spears, 1995) involves the utilization of information gained about the state
of the GA during the run of the GA, which may consist of information about the
population as a whole, individuals in the population or components of the GA, to
adaptively vary its settings there.
3.8
Tabu Search
Tabu Search (TS) is among the most cited and used metaheuristics for CO
problems. TS basic ideas were first introduced in Glover (1986), based on earlier
ideas formulated by Glover in 1977 (Blum and Roli, 2003).
Glover (1977)
83
introduced the basic strategic oscillation approach and associated “strong
determination” and “consistency” notions involving frequency memory and
statistical analysis. Recency memory, sometimes considered a primary feature of
TS, was illustrated as a way to implement strategic oscillation.
According to Glover (1990), TS is founded on three primary themes:
1.
The use of flexible attribute-based memory structures designed to permit
evaluation criteria and historical search information to be exploited more
thoroughly than by rigid memory structures (as in branch and bound) or by
memoryless systems (as in SA and other randomized approach).
2.
An associated mechanism of control – for employing the memory structures –
based on the interplay between conditions that constrain and free the search
process (embodied in tabu restriction and aspiration criteria).
3.
The incorporation of memory functions of different time spans, from short term
to long term, to implement strategies for intensifying and diversifying the
search. (Intensification strategies reinforce move combinations and solution
features historically found good, while diversification strategies drive the
search into new regions).
In the following are some terminologies used in Tabu Search:
x
A move
: a transition from a solution s to another solution s’ in
N (s)
x
An attribute
: the elements that constitute the move
x
Tabu list
: a list of moves that are currently tabu
x
Tabu list size : the number of iterations for which a recently
accepted move is not allowed to be reversed
x
Tabu tenure
: an integer number telling for how long a given move
will remain tabu
The TS features are presented as follows:
1.
Neighbourhood structures. The search starts with initial solution and defines a
subset, N (s), of possible solutions that are neighbours of s (solution) under N.
Each neighbour is evaluated against the objective function.
84
2.
Short term memory. The core of TS is embedded in its short term memory
process (Glover, 1990). Tabu is used to prevent cycling when moving away
from local optima through non-improving moves, it is most common to prohibit
reverse moves for certain iterations. Tabu is also useful to help the search
move away from previously visited portions of the search space and thus
perform more extensive exploration (Cordeau and Laporte, 2002).
The
neighbourhood of the current solution is restricted to the solutions that do not
belong to the tabu list. At each iteration the best solution from the allowed set
is chosen as the new current solution. Additionally, this solution is added to the
tabu list and one of the solutions that were already in tabu list is removed
(usually in FIFO order). The algorithm stops when a terminate condition is
met. It might also terminate if the allowed set is empty, that is, if all the
solution in N (s) are forbidden by the tabu list. The use of tabu list prevents
from returning to recently visited solutions; therefore it prevents from endless
cycling and forces the search to accept even uphill moves. The length of tabu
list (i.e., the tabu tenure) controls the memory of the search process. With
small tabu tenures the search will concentrate on small area of the search
spaces. On the opposite, a large tabu tenure forces the search process to
explore larger regions, because it forbids revisiting a higher number of
solutions (Blum and Roli, 2003).
3.
Aspiration criteria.
Tabu is sometimes too powerful, they may prohibit
attractive moves, even when there is no danger of cycling, or they may lead to
an overall stagnation of the searching process. It is thus necessary to use
algorithmic devices that will allow one to revoke (cancel) tabus. These are
called aspiration criteria. The simplest and most commonly used aspiration
criteria consist in allowing a move even if it is tabu, if it results in a solution
with an objective value better than of the current best-known solution (since the
new solution has obviously not been previously visited (Gendreau, 2002).
4.
Stopping criteria. In theory, the search could go on forever, unless the optimal
value of the problem at hand is known beforehand. In practice, obviously, the
search has to be stopped at some point. The most commonly used stopping
criteria in TS are:
-
after a fixed number of iterations (or a fixed amount of CPU time)
85
-
after some number of iterations without an improvement in the
objective function value (the criterion used in most implementations)
-
when the objective reaches a pre-specified threshold value
In complex tabu schemes, the search is usually stopped after completing a
sequence of phases, the duration of each phase being determined by one of the
above criteria (Gendreau, 2002).
5.
Intermediate and long-term memory as a basis of strategies for Intensification
and Diversification. In fact, the fundamental elements of intensification and
diversification strategies are already present in the short-term memory
component of TS, since a short-term memory tabu list has an intensification
role by temporarily locking in certain locally attractive attributes (those
belonging to moves recently evaluated to be good), while it also has a
diversification role by compelling new choices to introduce (or exclude)
attributes that are not among those recently discarded (or incorporated)
(Glover, 1990).
There are several types of tabu list:
x
Strict Tabu
: Forbid all previous solutions.
x
Fixed Tabu
:
x
ƒ
The size of tabu list is fixed
ƒ
The possibility of cycles remain
Reactive Tabu
: Adapt list size to properties of a problem
dynamically
According to Crainic et al. (2005), two key concepts in TS are those of search
intensification and search diversification. The idea behind search intensification is
that regions of the search space that appear “promising” (usually because good
solutions have been encountered close by) should be explored more thoroughly in
order to make sure that the best solutions in these regions are found. Intensification
is usually based on some type of intermediate-term memory, such as a recency
memory, in which one would record the number of consecutive iterations that
various “solution components” have been present without interruption in the current
solution. Intensification is typically performed periodically restarting the search from
the best currently known solution and by “freezing” (fixing) in this solution the
86
components that seem more attractive. It is also often involves switching to a more
powerful neighborhood operator for a short period of time.
Search diversification addresses the complementary need to perform a broad
exploration of the search space to make sure that the search trajectory has not been
confined to regions containing only mediocre solutions. It is thus a mechanism that
tries to force the search trajectory into previously unexplored regions of the search
space. Diversification is usually based on some form of long-term memory, such as
frequency memory, in which one would record the total number of iterations (since
the beginning of the search) that various “solution components” have been present in
the current solution or have been involved in the selected moves (Crainic et al.,
2005). The basic structure of TS (Glover, 1990) is depicted at Figure 3.15.
87
Begin with Starting Current Solution
Create a Candidate List of Moves
(each move would generate a new solution from the
current solution)
Evaluate each Candidate Move
Does the move yield a higher evaluation than any other move
found admissible so far from the current candidate list?
Yes
Check Tabu Status
Is the candidate tabu?
Tabu
Yes
Not Tabu
Check Aspiration Level
Does move satisfy aspiration
criteria?
No
Yes
No
Move is Admissible
Designate as best admissible
candidate
Candidate List Check
Is there a “good probability” of better moves left,
or should candidate list be extended
No
Make the Chosen Best Admissible Move
Stopping
Criterion
Continue
Stop
Terminate Globally or Transfer
A transfer initiates an intensification or
diversification phase embodied in an
intermediate or long term memory component
Figure 3.15 : Basic tabu search algorithm
Update Admissibility Conditions
Update Tabu Restrictions and
Aspiration Criteria
88
3.9
The Hybrid Genetic Algorithm and Tabu Search Scheme
One of the developments of metaheuristics for VRPSD was based on the
hybrid Genetic Algorithm with Tabu Search. The Tabu Search was incorporated to
the body of GA. There were two types of Hybrid GA and TS as described further in
Chapter 5.
The proposed hybrid algorithm of GA and TS consists of several
properties such as the order or permutation representation; small population in which
all individuals are distinct; inclusion of good simple heuristic solutions in the initial
population; Roulette Wheel Selection with elitism; Order Crossover and the Tabu
Search to be incorporated to the GA.
The algorithm is implemented in the Visual C++ language under the operating
windows XP. The data were analyzed using Minitab version 14 and SPSS version 13
for Windows.
3.10
Illustration of the Calculation of VRPSD Objective Function
Consider a case of 5 customers where customers’ coordinates and demands,
and its distance matrix are shown at Tables 3.2 and 3.3, respectively. Set route to be
followed is 0 Æ 2 Æ 4 Æ 5 Æ 3 Æ 1 Æ 0 as described in Figure 3.16.
Table 3.2 : Customer coordinates and demands
i
X
Y
Demand
0
50
50
0
1
82
23
U(1,3)
2
21
63
U(2,6)
3
79
46
U(0,2)
4
6
80
U(0,2)
5
35
33
U(1,3)
89
Table 3.3 : Distance matrix of customers
Node
0
1
2
3
4
5
0
0
42
32
29
53
23
1
42
0
73
23
95
48
2
32
73
0
60
23
33
3
29
23
60
0
81
46
4
53
95
23
81
0
55
5
23
48
33
46
55
0
The filling coefficient f defined in Equation (3.2) is the measure of the total
amount of expected demand relative to vehicle capacity and can be approximately
interpreted as the expected number of loads per vehicle needed to serve all
customers. If f is equal to 1 then the load to be serviced is the same with the vehicle
capacity. If f is less than 1 then the load to be serviced is less than the capacity,
while if f is bigger than 1 than the load to be serviced exceeds the vehicle capacity.
For the illustration of VRPSD objective function, let f = 1.1. Then the quantity of
garbage is
¦ E ([ )
i
Q
n
f
=
(5)([2 4 1] / 3)
= 11, vehicle’s quantity Q = 11
1.1
Route: 0 – 2 – 4 – 5 – 3 – 1 – 0
4 (j = 2)
2 (j = 1)
0
3 (j = 4)
5 (j = 3)
1 (j = 5)
Figure 3.16: 5 customers network in Table 3.2 (depot excluded)
From Figure 3.16, j represents the order/ sequence to be visited.
st
For
example: j = 1 means 1 location to be visited, that is node 2. j = 2 means 2nd
location visited, that is node 4, and so on.
90
Iteration 1 : j = 5
Set f j 5 (q)
cj
5, 0
= 42
note that j = 5 doesn’t mean that node 5, but the customer no 5th visited, that
is node 1. So f 5 represents travel cost from node 1 and node 0 that is 42.
Iteration 2 : j = 4
j = 5-1 = 4 Æ 4th visited Æ node 3,
a. Calculate f 4r (q) based on Eq. (2.4).
K
f jr (q)
c j ,0 c0, j 1 ¦ f j 1 (Q [ k ) p j 1,k
k 1
3
f 4r (q)
c 4, 0 c0, 41 ¦ f 41 (11 [ k ) p 41,k ,
k 1
where
x
c 4,0 Æ travel cost from node 4th visited (node3) to node 0 is 29
(see distance matrix in Table 3.3)
x
p5,k represents probability for node 5th visited (node 1). Node 1
follows U(1,3), so k = 1, 2, 3. p5,k =1/3, since there are 3 outcomes
(1,2 and 3)
for the 2nd iteration, set f 5 (q) = 42 for all q.
thus:
f 4r (q )
29 42 p 5,k [ f 5 (11 1) f 5 (11 2) f 5 (11 3)]
29 42 13 [ f 5 (10) f 5 (9) f 5 (8)]
29 42 13 [(3) x( f 5 )]
29 42 13 [(3) x(42)] = 113
The value f 4r (q ) is the same for all q, f 4r (q) = 113.
b. Calculate f 4p (q ) based on Eq. (2.3)
3
set b = 0, f 4p (11)
c 4,5 ¦ f 41 (11 k ) p 41,k
k 1
91
23 {[ f 5 (11 1) p 5 ,1 ] [ f 5 (11 2 ) p 5 , 21 ] [ f 5 (11 3) p 5 ,1 ]}
23 13 {[ f 5 (10)] [ f 5 (9)] [ f 5 (8)]}
23 ( 13 )(3)(42) = 65
f 4p (11)
f 4p (2)
f 4p (10)
...
f 4p (3)
65
2
3
k 1
k 3
c 4,5 ¦ f 5 (2 k ) p5,k ¦ 2c5,0 f 5 (2 11 k ) p5,k
= 23 {[ f 5 (11 1) p5,1 ] [ f 5 (11 2) p5, 2 ]} [(2).(42) f 5 (2 11 3) p5,3 ]
23 13 {[ f 5 (10)] [ f 5 (9)] 84 [ f 5 (10)]}
23 13 {42 42 84 42}
= 93
f 4p (1)
1
3
k 1
k 2
c 4 , 5 ¦ f 5 (1 k ) p 5 , k ¦ 2 c 5 , 0 f 5 (1 11 k ) p 5 , k
= 23 {[ f 5 (1 1) p5,1 ]} {[2c5,0 f 5 (12 2) p5, 2 ] [2c5, 0 f 5 (12 3) p5,3 ])}
23 13 {[ f 5 (0)] [(2)(42) f 5 (11) (2)(42) f 5 (10)]}
23 13 {42 [(2)(42) 42 (2)(42) 42]}
= 120.9
3
f 4p (0)
c 4,5 ¦ 2c5,0 f 5 (0 11 k ) p5,k
k 1
= 23 {[2c5,0 f 5 (11 1) p5,1 ] [2c5,0 f 5 (11 2) p5, 2 [2c5,0 f 5 (11 3) p5,3 ])}
= 23 13 {[ f 5 (10)] [(2)(42) f 5 (11) (2)(42) f 5 (10)]}
= 23 13 {42 [(2)(42) 42 (2)(42) 42]}
= 149
c. Compare f 4r (.) with f 4p (q) for finding threshold h4 and f 4 (q ) .
h4 is the last value of q when f 4r (.) d f 4p (q) , so h4 =2.
f 4 (q)
min{ f 4p (q), f 4r (q )}
thus f 4 (11)
f 4 (10)
...
f 4 (3)
65 , f 4 (2) =93, f 4 (1) =113, f 4 (0) =113
These values of f 4 (q ) will be used for the next iteration.
92
160
140
120
100
fp
80
fr
60
40
20
h4
0
0
2
4
6
8
10
12
Figure 3.17 : Illustration of hj determination
Iteration 3 : j = 3
j = 5-2 = 3 Æ 3rd visited Æ node 5,
a. Calculate f 3r (q)
2
r
3
f (q)
c3, 0 c0,31 ¦ f 31 (11 k ) p31,k ,
k 0
where
x
c3, 0 Æ travel cost from node 3th visited (node5) and node 0 is
23 (see distance matrix in Table 3.3)
x
p 4,k represents probability for node 4th visited (node 3).
Node 3 follows U(0,2), so k = 0,1, 2. p 4,k =1/3, since there are
3 outcomes (0,1,2)
thus:
f 3r (q )
23 29 p 4,k [ f 4 (11 0) f 4 (11 1) f 4 (11 2)]
23 29 13 [ f 4 (11) f 4 (10) f 4 (9)] , use result from 2nd iteration for
f j (q) values
52 13 [65 65 65]
= 117
The value f 3r (q ) is the same for all q, f 3r (q) = 117.
93
b. Calculate f 3p (q )
2
c3, 4 ¦ f 31 (11 k ) p31,k
f 3p (11)
k 0
46 {[ f 4 (11 0) p 4,1 ] [ f 4 (11 1) p 4, 2 ] [ f 4 (11 2) p 4,3 ]}
46 13 {[ f 4 (11)] [ f 4 (10)] [ f 4 (9)]}
46 ( 13 )(65 65 65) = 111
f 3p (11)
f 3p (1)
f 3p (10)
...
f 3p (2) 111
1
2
k 0
k 2
c3, 4 ¦ f 4 (1 k ) p 4,k ¦ 2c 4,0 f 4 (1 11 k ) p1,k
46 13 {[ f 4 (1) f 4 (0)] [(2)(29) f 4 (10)]}
46 13 {113 113 58 65}
= 162.33
0
2
k 0
k 1
c 3 , 4 ¦ f 4 ( 0 k ) p 4 , k ¦ 2 c 4 , 0 f 4 ( 0 11 k ) p 4 , k
f 3p (0)
46 13 {[ f 4 (0)] [(2)(2)(29) f 4 (10) f 4 (11)]}
46 13 {113 116 65 65}
= 165.67
c. Compare f 3r (.) with f 3p (q) for finding threshold h3 and f 3 (q) .
h3 is the last value of q when f 3r (.) d f 3p (q) , h3 =2.
f 3 (q)
min{ f 3p (q), f 3r (q)}
so f 3 (11)
f 3 (10)
...
f 3 (2) 111 , f 3 (1) =162.33, f 3 (0) =165.67
These values of f 3 (q ) will be used in the next iteration.
These procedures are repeated until f 0 (Q) is found. The f 0 (Q) = 221 and the
values of threshold are h4 = 2, h3 = 2, h2 = 2 and h1 = 1.
94
3.11
Summary
In this study, several metaheuristics based on Tabu Search and Genetic
Algorithm are developed. Two main parts of the development are considered; these
are the development of metaheuristics for single-objective single vehicle and
bi-objective multiple vehicles.
There are two kinds of data used to assess the
performance of the proposed algorithm, the first is real data from case study of solid
waste collection and the second is generated data simulates the real data from solid
waste collection problem. The performance of algorithms is compared each other
and with previous work. The summary of the metaheuristic algorithms developed is
given in Table 3.4.
Table 3.4: Summary of the metaheuristic algorithms considered in this study
Single vehicle – single objective
Multiple vehicles – bi objective
1. Genetic Algorithm (GA)
1. Pareto Genetic Algorithm
2. Tabu Search (TS)
2. Pareto Breeder Genetic Algorithm
3. Adaptive Genetic Algorithm (AGA)
4. Hybrid Genetic Algorithm and Tabu
Search (HGATS)
5. Breeder Genetic Algorithm (BGA)
6. GA of Bianchi (previous research in
single VRPSD)
95
CHAPTER 4
SOLUTION APPROACHES BASED ON
TABU SEARCH AND REACTIVE TABU SEARCH
FOR SINGLE VRPSD
4.1
Introduction
Tabu Search has been used for solving many complex combinatorial
problems but its potential for solving single vehicle routing problem with stochastic
demands (VRPSD) has not been fully explored. The aim of this chapter is to explore
the development of heuristic method based on Tabu Search for solving single
VRPSD followed by the development of Reactive Tabu Search. One of the ways our
method is different from previous researches in VRPSD is the use of reactive tabu
search (RTS). From our reviews, this is the first time that RTS has been applied to
the VRPSD. The initial development of Reactive Tabu Search was proposed by
Battiti and Tecchiolli (1994). This approach maintains the basic ideas of TS with the
following additional features:
-
A fully automated way of adapting the size to the problem and to the current
evolution of the search,
-
An escape strategy for diversifying the search when the first mechanism is
not sufficient.
The RTS scheme uses an analogy with the theory of dynamical systems, where the
tabu list size depends on the repetition of solutions and consequently tabu list size is
determined dynamically, as opposed to the standard version of TS where tabu list
96
size is fixed. RTS seems to be robust enough and there may be a little effect of
parameter changing (Wassan et al., 2007). The discussion of this chapter begins
with the detail development of TS and RTS for single VRPSD, followed by the
computational results and discussion.
4.2
Tabu Search for Single VRPSD
In the context of TS related to the VRPSD, a decision has to be made
regarding the most appropriate solution procedure to solve this problem.
This
involves the choice of the initial feasible solution, the neighbourhood structure, the
tabu criterion, the fitness function, the aspiration function and the stopping criterion.
All these decisions are related to the body of solution procedure which proved to be
the most challenging.
For the problem under consideration, the primary requirement of an initial
feasible solution is that each customer is fully served while maintaining that each
collecting truck has a limited capacity. At the same time the company faces the
problem of not knowing the exact amount of demand (for example the demand is
waste disposal to be picked up). The task here is to design a route such that the
demand of each customer is fully collected and the total expected cost is minimized.
4.2.1
Initial Solution
It is necessary to generate a number of solutions to initialize the main search
process. The choice of the initial solution is known to be important. Perttunen
(1994) investigated whether the use of a constructive heuristic leads to better results
than using random initial solutions.
He found that the use of initial solutions
generated by applying a constructive heuristic, instead of a random initial solution,
typically improve the performance of the algorithm. Derigs and Kaiser (2007) stated
that starting from a set of randomly generated solution takes significantly more CPU-
97
time/ iterations than applying constructive heuristics. Adrabinski and Syslo (1983)
in Helsgaun (2000) found that the Farthest Insertion construction heuristic was
capable of producing good initial solutions for the Lin Kernighan heuristic. In this
study, two simple and fast heuristic methods were used to generate the initial
solution; the methods were Nearest Neighbour (NN) and Farthest Insertion (FI)
heuristics. The best among them was chosen as the starting solution to the main
body of the search itself, from which heuristic method which yields lower VRPSD
objective value (the total expected cost).
a.
Nearest Neighbour
The nearest neighbour is a fast and simple heuristic for constructing a TSP tour
(Hurkens and Woeginger, 2004), but the solution generated is usually not the optimal
one. It can sometimes miss shorter routes which are easily noticed with human
insight, due to its "greedy" nature. These are the steps of the algorithm:
1.
Start from the depot.
2.
Scan remaining nodes for the node nearest to last node added in the tour.
Add this node to the existing tour.
3.
b.
Repeat step 2 until all nodes are added. Close path to form a tour.
Farthest Insertion
Farthest Insertion was firstly introduced by Golden and Stewart (1985). These are
the steps of the algorithm:
1.
Start with a subtour made of the depot.
2.
Select a node k farthest from depot. Form subtour depot-k-depot.
3.
Scan remaining nodes for the node k farthest from any node in tour.
4.
Find edge (i,j) that minimize cik + cjk - cij where i and j are in the tour. Insert
k between i and j.
5.
Repeat step 3 and 4 until all cities are in the tour.
In both Randomized NN and Randomized FI, the first node to be visited is randomly
chosen and after the tour has been completed, shifts the starting customer to the
depot.
98
4.2.2
Neighbourhood Exploration
The neighbourhood of a solution can be obtained by applying Or-opt local
search methods. The Or-opt procedure, well known as node exchange heuristic, was
first introduced by Or (Bard et al., 1998). Marinakis and Migdalas (2002) describe
the procedure of Or-opt algorithm.
Or-opt removes a sequence of up-to-three
adjacent nodes and inserts it at another location within the same route. Or-opt can be
considered as a special case of 3-opt (three arcs exchanges) where three arcs are
removed and substituted by three other arcs. When removing a chain of consecutive
nodes in or-opt, two arcs are deleted, and the third is eliminated when inserting the
chain back into the route. However, the number of possible or-exchanges is far less
than that of possible 3-exchanges.
Or-opt is also shown to produce improved
solutions of comparable quality to those produced by 3-opt, while requiring
significantly less computational time.
The procedure of Or-opt is described in
Figure 4.1.
Consider an initial tour and set t = 1 and s = 3.
Remove from the tour a chain of s consecutive vertices,
starting with the vertex in position t, and tentatively
insert it between all remaining pairs of consecutive
vertices on the tour
Step 2.1: If the tentative insertion decreases the cost
of the tour, implement it immediately, thus
defining a new tour
Step 2.2: If no tentative insertion decreases the cost
of the tour, set t = t+1. If t = n + 1 then
proceed to Step 3, otherwise repeat step 2.
Step 3: Set t = 1 and s = s - 1. If s > 0 then go to step 2,
otherwise stop.
Step 1:
Step 2:
Figure 4.1 : Or-opt algorithm
4.2.3
Tabu Moves
A move is a transition from a solution s to another solution s’ in N (s). Once
a move is made, revisiting this solution is prohibited (or declared tabu) for a number
of iterations. Tabu is used to prevent cycling when moving away from local optima
through non-improving moves, it is most common to prohibit reverse moves for
99
certain iterations. Tabu is also useful to help the search move away from previously
visited portions of the search space and thus perform more extensive exploration.
(Cordeau and Laporte, 2002) Tabu list is a list of moves that are currently tabu.
Tabu list is an ordered queue containing forbidden moves following FIFO (First In
First Out) order; whenever a move is made, it is inserted to the end of the tabu list
and the first element from the list is removed. For the first iteration, the tabu tenure
of all edges is set to be 0. If tabu list size is too small, then it may not be able to
prevent cycling but if tabu list size is too long then it will create too many
restrictions.
In this study, two types of tabu list size are used:
1. Static tabu list size (based on our preliminary experiment on the effect of
different static tabu list size to the Tabu Search performance, the tabu list size
set to be N/3 seems to be the best) and
2. Dynamic tabu list size. In dynamic tabu list size, if an edge is moved at
iteration v, its addition or deletion is tabu until iteration v + , where is
randomly selected value in the interval [tlsmin, tlsmax]. The idea of using
random tabu list size was introduced by Taillard (1991) in the context of the
quadratic assignment problem. In the application of Tabu Search for solving
VRPSDC, Gendreau et al. (1996) randomly select tabu list size in the interval
[n-5, n]. In this study, we use interval of tabu list size within [ N3 N
5
, N3 N
5
].
The best admissible move is then chosen as the highest evaluation move in
the neighbourhood of the current solution in terms of the objective function value
and the tabu restrictions. The whole neighbourhood is explored and the best nontabu move is selected; however, in this study we also consider the feasible tabu
improving move. An improving move is not accepted if it is in tabu list unless it
satisfies the aspiration criterion. After a consecutive s nodes has been extracted form
position i and re-inserted at position j, the concept of tabu move is:
1. It is tabu to re-insert subsequent node from s in the original tour to
immediately position after s.
100
2. It is tabu to re-insert preceding node from s in the original tour to
immediately position before s.
The illustration of tabu moves is given in Figure 4.2.
1 2 3 4 5 6 7 8 9 10
i
j
tabu
1 5 6 7 8 2 3 4 9 10
tabu
Figure 4.2 : Illustration of tabu moves
4.2.4
Aspiration Criterion
The aspiration criterion is a measure solely designed to override the tabu
status of a move if this move leads to a solution better than the best solution found by
the search so far. As a rule, the search process moves at each iteration from the
current solution to the best non tabu solution in neighbourhood solutions. However,
a tabu solution may be selected if the value of objective function decreased upon the
best value of a solution (aspiration criterion) so far encountered. We also used
aspiration by defaults namely:
x
free the “least tabu tenure” move when all the move is tabu.
x
if there is a tie of least tabu tenure, then free the one with smallest objective
value.
101
4.2.5
Stopping Criterion
In using the optimization algorithms, the goal is to search for the global
optimum. However, in general it is not clear when the goal is achieved especially in
our application in the real world situation. Therefore, it is sometimes not that easy to
decide when execution of an optimization algorithm should be terminated. There are
many different mechanisms can be used for the detection of an appropriate time for
ending an optimization run. The most obvious way is simply to stop after a pre
specified number of iterations. But perhaps it would be better to stop when there is
very little change between iterations, indicating that the process has reached a
convergence (Tong et al., 2004).
In our implementation, the TS procedure is
repeated until there is no improvement in aspiration cost for 500 successive
iterations.
The whole processes of Tabu Search Algorithm described in Section 4.2
above were summarized in Figure 4.3.
102
BEGIN
Initial Solution
using NN and FI
Calculate cost using VRPSD
objective function
Set iteration count = 1, aspiration level =
= min cost of Initial Solutions
,
Create a Candidate List of Moves
(Move: 2-opt and Lin and Kerghan,
Find neighbourhood and determine cost)
v=v+1
Yes
Anymore
neighborhood?
No
Choose the minimum cost of
neighbourhood
Consider next lower cost
neighbourhood solution
No
Cost <
No
?
No
Is move (that
result min cost) is
Tabu?
Yes
Is all
move Tabu?
Yes
No
Yes
Accept as current solution
Update
and Tabu list
Accept as current solution
Update Tabu list
Freeing strategy:
aspiration by default
Is stopping
criterion met?
Yes
Set optimal solution
as Final Solution
END
Figure 4.3 : Tabu search algorithm for single VRPSD
103
4.3
Reactive Tabu Search
The structure of our RTS-VRPSD algorithm is generally the same as
described in previous section about the development of TS for single VRPSD, while
its “reactive” aspects are discussed furthermore.
Step 1: Initial Solution
Generate initial solutions using two simple and fast constructive heuristics
Nearest Neighbour (NN) and Farthest Insertion (FI) heuristics. The best among them
was chosen as the starting solution S to the main body of the search itself, based on
which heuristic that yields lower cost from the calculation of VRPSD objective
function.
Step 2: Initialization
-
Initialize the tabu search data structures
-
Set number of iterations
-
Initialize the RTS parameters values
Step 3: Neighbourhood Exploration
-
Evaluate the neighbourhood of a solution defines by the Or-opt local search
methods and choose the best neighbouring solution, S’.
-
The best admissible move is then chosen as the highest evaluation move in the
neighbourhood of the current solution in terms of the objective function value
and the tabu restrictions. The whole neighbourhood is explored and the best
non-tabu move is selected; however, in this study we also considering the
feasible tabu improving move. An improving move is not accepted if it is in
the tabu list unless it satisfies the aspiration criterion.
-
A tabu solution may be selected if the value of objective function decreased
upon the best value of a solution (aspiration criterion) so far encountered.
Step 4: Stopping Criterion
Perform Step 3 until there is no improvement in aspiration cost for 500
successive iterations.
104
The RTS employs two mechanisms, both mechanisms react to the repetitions.
The first mechanism of RTS is used to produce a balanced tabu list size and consists
of two reactions that were fast and slow reactions. The fast reaction increases the
tabu list size when solutions are repeated. The slow reaction reduces the tabu list
size for those regions of the search space that do not need large list lengths. In a fast
reaction, after some reactions of repetition, the size is increased and becomes large
enough to break any limit cycle and explore new regions. While in slow reaction,
decrease tabu list size:
-
if a number of iterations is greater than a moving average
-
when all movements are tabu and none is better than the aspiration criterion
The second mechanism of RTS provides a systematic way to diversify the
search when it is only confined to one portion of the solution space. Sometimes tabu
(prohibition) is not sufficient to avoid long cycles. A "chaotic trapping" of the
trajectory in a limited portion of the search space is still possible (the analogy is with
chaotic attractors of dynamical systems, where the trajectory is confined in a limited
portion of the space, although a limit cycle is not present). In order to increase the
robustness of the algorithm, an escape diversity strategy is needed. The escape phase
is triggered when too many configurations are repeated too often, in other words
when the number of “chaotic” (the repetition of a solution is more than a specific
times) is greater than the threshold then reset the tabu (escape). Battiti and Tecchiolli
(1994) performed random repositioning of customers for this purpose. In our case,
the escape mechanism is modified, instead of random repositioning of customers the
diversification is obtained by generating Or-opt of randomized farthest insertion.
In developing RTS, we define the following variables:
-
tls
: tabu list size
-
repetition
: the number of times the solution was accepted.
When repetition > 1, it is an often-repeated solution
-
Chaotic
: the number of often repeated solutions
-
MovAvg
: an average of the detected repetitions
105
-
LastChange : the number of iterations executed after the last change of tabu
list size
-
GapRep
: Gap between two consecutive repetitions (number of iterations
between this and previous occurrence of current solution)
The initialization and parameter settings used in the RTS are depicted in
Figure 4.4.
Initialization:
tls = 1
Chaotic = 0
MovAvg = 0
GapRep = 0
LastChange = 0
Iter = 0
Repetition = 0
Iteration number when last
noticed: LastTimeRep = 0
time
an
identical
solution
was
Iteration number of the most recent repetition: CurTimeRep = 0
When the number of repetitions for a solution is greater than
REP, it is added to the set of often-repeated solutions: REP = 3
Maximum number for the set of often-repeated solutions allowed:
Chaos = 6
Percentage increase for the tabu tenure value: Increase = 1.1
Percentage decrease for the tabu tenure value: Decrease = 0.9
Constant used to compare with GapRep to get the moving average:
GapMax = 50
Figure 4.4 : Initialization and parameters setting in RTS
The constant values for the above parameters are the same as in the standard
RTS scheme of Battiti and Tecchioli (1994) except for the value of Chaos where it is
the same as implemented by Wassan et al. (2007).
The RTS mechanism, called in Step 3 of the RTS, is given in Figure 4.5
while the Escape phase is presented in Figure 4.6.
106
(1).
Search for repetition of S, if found then
GapRep = CurTimeRep – LastTimeRep, and go to (2);
Else go to (4).
(2).
If (repetition of S) > REP, then set Chaotic = Chaotic + 1;
Else go to (3).
If Chaotic > Chaos, then set Chaotic = 0, clear and rebuild
tabu search data structures and go to Step 3 of the RTS
algorithm;
Else go to (3).
(3).
If GapRep < GapMax, then
tls = tls x Increase, set LastChange = 0,
MovAvg = (0.1 x GapRep) + (0.9 x MovAvg); and go to (5);
Else go to (4).
(4).
If LastChange > MovAvg, then
tls = tls x Decrease, and set LastChange = 0;
Else set Last Change = LastChange + 1, store the solution S
and go to (5).
(5).
Stop the RTS mechanism.
Figure 4.5 : The RTS mechanism
x
Set the current and best escape solutions to S (i.e., Sec =
S and Seb = S )
x
Set escape iterations
x
Perform Or-opt neighbourhood moves
x
Scan for the best solution, S’ec, from neighbouring
solutions of Sec
x
If C(S’ec) < C(Seb), then update Seb = S’ec and Sec = S’ec
x
Else Sec = S’ec
x
Repeat until there is no improvement for 100 iterations
x
Return to the main algorithm by setting its current
solution S to Seb (if found) or to Sec
Figure 4.6 : The escape phase
107
4.4
Computational Results
The metaheuristics were tested on a set of instances generated from procedure
described in Section 3.5. For each instance, N nodes were generated in [0,100]2
according to a uniform distribution. The demand of each node was generated as
follows. Nodes were first assigned a demand range of [1,5], [6,10], and [11,15] in
equal probability. The value of each [ i (customer demands) was then generated in
the appropriate range according to a discrete uniform distribution. In this study,
twenty instances were generated on each of the problem size that are small problem
size (number of nodes is equal to 10), moderate problem size (number of nodes is
equal to 20) and large problem size (number of nodes is equal to 50). Actually the
algorithm and software presented here can solve larger number of nodes, but in this
study, the computational experiments were confined on these three problem sizes.
Each algorithm was tested on each instance until there was no improvement in the
aspiration cost for 500 successive iterations.
Here is how the TS works. In the search process, TS allows the search to
accept non improving move (move that obtain larger VRPSD objective function than
the aspiration cost) as long as it is not tabu, and on the other side it also allows tabu
move as long it gives an improvement to the aspiration cost. Accepting the non
improving move can be seen in upper graph of Figure 4.7. In iteration 2, the cost of
269, that larger than the aspiration cost of 217, was accepted as the current solution
since it was the best non tabu move.
4.4.1
Static versus Dynamic Tabu List Size
As described in Section 4.2.3, two types of tabu list size are used. The first is
static tabu list size which is set to be N/3 and the second is dynamic tabu list size
which is set to be a random number within [ N3 N
5
, N3 N
5
]. Static tabu list size
means that the tabu list size is fixed from iteration to iteration; while dynamic tabu
list size means that the tabu list size is dynamically changed from iteration to
108
iteration depend on the random number generated within the specific range. For
example for 30-node problems, using static tabu list size gives a tabu list size equal
to 10 for every iteration, while by using dynamic tabu list size, the tabu list size in
every iteration may be different each other according to the random number
generated.
Firstly, the effect of static and dynamic tabu list size to the search process
were analyzed. For the illustration, lets consider instance consist of 5 nodes with
demand ranges are in [0,2], [1,3] and [2,6]. The TS was executed until the number of
iteration is equal to 10. Although the final best result from both types of tabu list
size were the same, it was quite different in the search process refers to the current
solution values as depicted in Figure 4.7.
109
Figure 4.7 : The comparison between static and dynamic tabu list size
implementation
Cycling is a condition during the search when the move finds solutions that
have been visited. Here the cycling found in the search process of both types of tabu
list size was given:
-
Using static tabu list size:
-
the same solution configuration found at iteration 1 and 3 (current cost
value = 217)
110
-
the same solution configuration found at iteration 2 and 4 (current cost
value = 269)
-
the same solution configuration found at iteration 6 and 8 (current cost
value = 324)
-
the same solution configuration found at iteration 7 and 9 (current cost
value = 263)
-
Using dynamic tabu list size
-
the same solution configuration found at iteration 2 and 6 (current cost
value = 269)
-
the same solution configuration found at iteration 3 and 7 (current cost
value = 255)
The same current cost also occurred between iteration 1 and 5, and between iteration
8 and 9, but these were not cycle since the solutions were not in the same order of
node visited (the solution configurations were different) although the VRPSD
objective function computed were equal. Furthermore we can see that static tabu list
size shows weakness since it can still find solutions that have been reached from a
few last iterations. By using dynamic tabu list size, the probability of cycling can be
reduced significantly. Thus, for the rest of study, we prefer to use dynamic tabu list
size rather than the static one.
4.4.2
Tabu Search versus Reactive Tabu Search
In this section, performances of TS and RTS were investigated and evaluated.
Most of the many heuristics developed for large optimization problems are evaluated
empirically—by applying the procedures to a collection of specific instances and
comparing the observed solution quality and computational burden. In this study,
twenty instances were generated on each of the 10-node problems, 20-node problems
and 50-node problems where the depot is excluded. Each algorithm was tested on
each instance until there is no improvement in the aspiration cost for 500 iterations.
The computational experiment results based on the total expected cost obtained were
reported in Table 4.1.
Rows correspond to problem instances and columns
111
correspond to algorithms being tested for each problem size. Each cell represents the
running of a particular algorithm on a specific instance.
Table 4.1 : The total expected cost of solutions obtained from TS and RTS
PROBLEM
INSTANCE
ALGORITHM
Number of
Number of
Number of
nodes = 10
nodes = 20
nodes = 50
TS
RTS
TS
RTS
TS
RTS
Instance 1
357
343
411
409
628
610
Instance 2
353
348
447
447
676
654
Instance 3
253
248
467
456
648
623
Instance 4
267
257
376
375
612
582
Instance 5
300
300
399
393
599
590
Instance 6
310
311
343
343
612
580
Instance 7
319
319
449
432
609
601
Instance 8
295
295
367
368
629
625
Instance 9
287
280
403
404
592
577
Instance 10
352
344
415
399
578
557
Instance 11
351
351
425
422
619
619
Instance 12
308
303
447
408
710
654
Instance 13
296
296
372
340
622
604
Instance 14
269
266
395
396
607
606
Instance 15
242
242
327
327
585
578
Instance 16
338
338
414
413
622
618
Instance 17
288
288
389
396
606
572
Instance 18
331
331
445
445
560
548
Instance 19
261
261
345
337
588
586
Instance 20
328
326
384
373
570
565
AVERAGE
305.25 302.35
401 394.15 613.6 597.45
112
Table 4.2 presented the average computation time of the TS and RTS for each
number of nodes. Computation times in this section were based on laptop machine
of Intel Centrino CoreTM 2 Duo processor with 2.0 GB of RAM. The average
computation time for 10 nodes is 32.4 seconds using TS, and we need more than
24,354.3 seconds for the 50-node problems. On the other hand, on average, we can
solve the same problems in 27.4 and 3,039.5 seconds respectively with the RTS. The
RTS reduced computation time about 16.7% for 10-node problems and the
differences in computation time of TS and RTS grows considerably as problem size
increases. It seems that RTS was superior to TS in both of solution quality and
computation time since RTS returned to less total expected cost and less computation
time than TS.
Table 4.2 : Average computation time of TS and RTS (secs)
Number of nodes
TS
RTS
10
32.4
27.4
20
577.2
205.6
50
24,354.3
3,039.5
The data in Table 4.1 was analyzed further using statistical software package
SPSS Version 13. In order to get information from the data, usually the first step in
data analysis is to conduct descriptive statistics to describe the basic features of the
data gathered from computational experiments. It consists of simple summaries
about the distribution, the central tendency and the dispersion. There are various
techniques that are commonly used in descriptive statistics, the first is in graphical
displays of the data which graphs summarize the data or facilitate comparisons. The
second is tabular description in which tables of numbers summarize the data and the
third is summary statistics (single numbers) which summarize the data.
Figure 4.8 shows the graphical summary by means of a box and whisker plot
to describe the distribution of solutions obtained by the TS and the RTS for each
problem size. The solutions were grouped by algorithm. A box and whisker plot can
be used to show the shape of the distribution, its central value, and variability. The
picture produced consists of the most extreme values in the data set (maximum and
113
minimum values), the lower and upper quartiles and the median. Box and whisker
plots are very useful when large numbers of observations are involved and when two
or more data sets are being compared. As it can be observed from the box and
whisker plots, it was worth noticing that the relative performance of the algorithms
was similar across different problem size: the solution quality produced by the RTS
clearly outperforms the results of TS since the RTS can yield lower total expected
cost for every problem size.
The second descriptive statistics technique namely summary statistics were
reported in Table 4.3. It can be seen that the larger number of nodes, the higher the
total expected cost.
Central tendency measures (mean and median) and the
dispersion measures (range and standard deviation) of solution qualities generated
from RTS are smaller than solution qualities obtained by TS for all problem size (all
number of nodes examined in this study). Smaller mean value refer to higher
solution quality that is a lower total expected cost.
114
800
12
700
600
500
400
300
200
TS_10
RTS_10
TS_20
RTS_20
TS_50
RTS_50
Figure 4.8 : Box and whisker plots of solutions produced by TS and RTS
To verify whether the differences between solutions found by the two
algorithms are statistically significant or not, we performed paired samples test
between TS and RTS results for every problem size (every number of nodes
examined in this study). The paired-samples t-test procedure compares the means of
two variables for a single group. It computes the differences between values of the
two variables for each case and tests whether the average differs from 0. Before the
paired t-test was conducted, the test of normal distribution assumption of the
difference between two variables must be conducted. The results of KolmogorovSmirnov test for normal distribution test were given in Table 4.4. We reported the
p-value for the null hypothesis “The data follow normal distribution” where the
significance level which the null hypothesis rejected is 0.05. The p-value that smaller
than 0.05 is sufficient to reject the null hypothesis.
115
Table 4.3 : Descriptive statistics measures
Statistics
Number of independent
runs
Valid
Missing
TS_10
20
RTS_10
20
TS_20
20
RTS_20
20
0
0
TS_50
20
RTS_50
20
0
0
Mean
305.25
302.35
401.00
394.15
613.60
0
597.45
0
Std. Deviation
35.559
35.042
39.030
37.788
35.024
29.280
Range
115
109
140
129
150
106
Minimum
242
242
327
327
560
548
Maximum
357
351
467
456
710
654
25
273.50
269.50
373.00
369.25
589.00
577.25
50
304.00
301.50
401.00
397.50
610.50
595.50
75
336.25
336.25
440.00
419.75
626.50
618.75
Percentiles
In the Table 4.4, the p-values of normality test for 10-node problems, 20-node
problems and 50-node problems were 0.046, 0.241 and 0.809, respectively. Since
the p-values of normality test for 20-node problems and 50-node problems were
larger than = 0.05, thus we have no evidence to reject the null hypothesis. We can
conclude that the difference values for 20-node problems and 50-node problems
follow normal distribution and the paired t-test can be used for this data set.
Otherwise for 10-node problems, the p-value is less than = 0.05, thus we must
reject null hypothesis and conclude that the difference was not follow normal
probability distribution. Since the normal distribution assumption was violated, thus
the non parametric Wilcoxon signed rank test was used for this problem size (number
of nodes).
Table 4.4 : Kolmogorov-Smirnov normality test results for differences
between TS and RTS
One-Sample Kolmogorov-Smirnov Test
Number of independent runs
Normal Parametersa,b
Mean
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
diff_TS_
RTS_10
20
2.9000
4.16628
.307
.307
-.193
1.372
.046
diff_TS_
RTS_20
20
6.8500
11.60887
.230
.230
-.199
1.028
.241
diff_TS_
RTS_50
20
16.15
14.184
.143
.143
-.127
.639
.809
116
We were interested in comparing the performance of TS and RTS for each
problem size. In paired t-test, the hypothesis related to this comparison is stated as
follows.
H0 : P d
0
H1: P d z 0
whereas the null hypothesis in Wilcoxon signed-rank test is different from the null
hypothesis in paired t-test, that is: the median difference between pairs of
observations is zero. The result of Wilcoxon signed-ranks test for number of nodes
10 and paired t-test for 20-node problems and 50-node problems were presented in
Figure 4.9 and Figure 4.10, respectively. The p-value of difference between TS and
RTS performance is 0.007, 0.016 and 0.000 for 10-node problems, 20-node problems
and 50-node problems, respectively. Since all p-values are less than = 0.05, the
null hypothesis was rejected and it can be concluded that the two compared groups
tend to yield different result. There were statistically significant differences between
the performance of TS and RTS in term of solution quality for solving VRPSD. The
extent of this difference was shown clearly by the confidence interval of the
difference in Figure 4.10 which not encompasses zero.
Ranks
RTS_10 – TS_10
Negative Ranks
Positive Ranks
Ties
Total
Number of runs
9 a
Mean Rank
Sum of Ranks
6.00
54.00
1 b
1.00
1.00
10 c
20
a. RTS_10 < TS_10
b. RTS_10 > TS_10
Test Statistics
b
c. RTS_10 = TS_10
Z
Asymp. Sig. (2-tailed)
RTS_10 TS_10
-2.708a
.007
a. Based on positive ranks.
b. Wilcoxon Signed Ranks Test
Figure 4.9 : Wilcoxon signed-ranks test results for 10-node problems
117
Furthermore, it can also be shown that the RTS can produce less total
expected cost. As we can see from Figure 4.8, the number of negative ranks refer to
(RTS < TS) is higher than the number of positive ranks refer to (RTS > TS). The
same information can also be obtained from Figure 4.10, where the mean values of
the differences between performance of TS and RTS for 20-node problems and 50node problems were positive, means that the values of total expected cost resulted
from TS were higher than the cost produced from TS.
a. Paired Samples
for 20-node
problems
Paired Test
Samples
Test
Mean
6.850
TS_20 - RTS_20
Paired Differences
95% Confidence
Std.
Interval of the
Error
Std.
Difference
Mean
Deviation
Lower
Upper
11.609
2.596
1.417
12.283
t
Sig.
(2-tailed)
df
2.639
19
.016
b. Paired Samples
for 50-node
problems
Paired Test
Samples
Test
Paired Differences
Mean
TS_50 - RTS_50
16.150
Std.
Deviation
Std. Error
Mean
14.184
3.172
95% Confidence
Interval of the
Difference
Lower Upper
9.512
22.788
t
5.092
df
Sig.
(2-tailed)
19
.000
Figure 4.10 : Paired t-test results for 20-node and 50-node problems
4.5
Conclusion
In this chapter, the effect of static and tabu list size was analyzed and the
solution approaches based on Tabu Search and Reactive Tabu Search for single
VRPSD were developed. From the computational experiments, the result shows the
advantage of using dynamic tabu list size in TS. By using dynamic tabu list size, the
probability of cycling was reduced significantly. Furthermore, the RTS is developed
for solving the VRPSD and comparative study between TS and RTS is conducted.
The result shows that there are statistically significant differences between the
performances of the two algorithms; further, the RTS is better than TS in both of
solution quality (based on the total expected cost) and computation time.
118
CHAPTER 5
VARIOUS GENETIC ALGORITHM-BASED METAHEURISICS
FOR SOLVING SINGLE VRPSD
5.1
Introduction
Genetic Algorithm (GA) is a population based search method based on
principles of natural selection and genetics. A set of points is generated, promising
points are selected, and new points are generated using the genetic operators’
recombination/crossover and mutation (Muhlenbein et al., 1999). GAs categorized
as a global search heuristics, classed under Evolutionary Algorithm that use
technique inspired by evolutionary biology. In these early days, the application
domains were mostly learning tasks and optimization of numerical functions. In
application to solve combinatorial optimization problem like TSP and VRP, the
original GA has to be modified.
In this chapter, we explore the dynamism of GA in solving the VRPSD by
developing various GA-based metaheuristics for solving single VRPSD.
The
VRPSD is encoded in the form of chromosomal manner and the fitness values are
established to define good or bad solutions based on its objective function. Further
we examined the selection procedure, reproduction system and mutation with giving
more emphasis on the choice of probability of mutation. This chapter begins with
the development of GA followed by the development of the Adaptive GA for solving
single VRPSD. In this study, we propose a new adaptive mutation probability
scheme based on diversity measure.
The use of fixed and adaptive mutation
probability was also compared. Furthermore, two enhanced GAs were proposed, the
119
first is the Hybrid GA-TS and second is the Breeder GA. Brief description of GAbased metaheuristics that were implemented was given, followed by the experimental
results and the discussion.
5.2
Genetic Algorithm for Single VRPSD
GA starts with an initial set of solutions, called population. Each solution in
the population is called a chromosome, which represents a point in the search space.
The chromosomes evolve through successive iterations, called generations. During
each generation, the chromosomes are evaluated using some measures of fitness.
The fitter the chromosomes, the higher the probabilities of being selected to perform
the genetic operations, including crossover and mutation. In the crossover phase, the
GA attempts to exchange portions of two parents, that is, two chromosomes in the
population to generate an offspring. The crossover operation speeds up the process
to reach better solutions. In the mutation phase, the mutation operation maintains the
diversity in the population to avoid being trapped in a local optimum. A new
generation is formed by selecting some parents and some offspring according to their
fitness values, and by rejecting others to keep the population size constant. After the
predetermined number of generations is performed, the algorithm converges to the
best chromosome, which hopefully represents the optimal solution or may be a nearoptimal solution of the problem.
The main characteristics in GA for solving single VRPSD were given as
follows. These include the chromosome representation, selection, recombination and
mutation.
Step 0. [Define] Define operator settings of GA suitable with the problem which is
VRPSD.
Step 1. [Initialization] Create an initial population P of PopSize chromosomes that
consists of constructive heuristics solutions and randomly mutation of it
where all individuals are distinct or clones are forbidden.
120
Step 2. [Fitness] Evaluate the fitness f(Ci) of each chromosome Ci in the
population. The fitness is the VRPSD objective function (the total expected
cost). High fitness denoted by low VRPSD objective function.
Step 3. [Selection] Apply Roulette Wheel Selection. This gives the set of Mating
Population M with size PopSize.
Step 4. [Crossover] Pair all the chromosomes in M at random forming PopSize /2
pairs. Apply crossover with probability pc to each pair and form PopSize
chromosomes of offspring, if random number t pc then offspring is the
exact copy of parents.
Step 5. [Mutation] With a mutation probability pm mutate the offspring.
Step 6. [Replace] Evaluate fitness of parents and new offspring. Choose the best
PopSize chromosomes. Replace the old population with newly generated
population.
Step 7. [Test] If the stopping criterion is met then stop, and return to the best
solution in current population, else go to Step 2.
5.2.1
Chromosome Representation
In the early days, most Genetic Algorithms used binary encodings and wellstudied crossover and mutation operators.
More recently, alternate encoding
schemes have been considered in order to account for problem-specific information
(Grefenstette, 1987).
Poon and Carter (1995) have classified different vector
schemes to represent a tour of cities in the traveling salesman problem: a path
representation, an adjacency representation, an ordinal representation, a position
listing representation, and an adjacency listing representation. Among them, the path
representation in particular has been frequently utilized to develop many genetic
algorithms for the traveling salesman problem because of its simplicity (Choi et al.,
2003). In order to represent enlarged search space effectively, an appropriate stringcoding scheme is needed to describe efficiently an infeasible solution with subtours.
The path representation of a chromosome is a natural coding scheme for the traveling
salesman problem and vehicle routing problem (Potvin, 1996; Schmitt and Amini,
1998).
121
Davis (1991) proposed a hybrid Genetic Algorithm for the graph coloring
problem in which individuals are represented as permutations of the nodes. In
developing the algorithm, the permutation representation or the path representation
or order representation is used since the typical approach using binary strings will
simply make coding more difficult. Order representation is perhaps the most natural
and useful representation of a VRP tour, where customers are listed in the order in
which they are visited (Davis, 1991). A chromosome represents a route and a gene
represents a customer and the values of genes are called alleles. The search space for
this representation is the set of permutations of the customers; every chromosome is
a string of numbers that represent a position in a sequence. Works in GA for solving
VRP using path representation were including the work of Schmitt and Amini
(1998), Machado et al. (2002), Tong et al. (2004), Lacomme et al. (2006). Order
representation can be described as follows.
Chromosome A
3 2 5 4 7 1 6 9 8
Chromosome B
1 5 3 2 6 4 7 9 8
Figure 5.1 : Illustration of order representation
5.2.2
Initialization and Population Size
Term population used in genetic algorithm is different with term population
in statistics. In statistics, population is the set of all possible values of a random
variable under observation where a sample is a subset of the population. In genetic
algorithm, population is a set of candidate solutions to be selected as parents. In the
case of vehicle routing problem, genetic algorithm does not work with the number of
whole possible chromosomes (which is the number of possible permutation of n
nodes = n!) as the population but rather works with a subset of it. The number of
candidate solutions in genetic algorithm is called population size.
Usually the initial population of candidate solutions in genetic algorithm is
generated randomly across the search space. However, other information can be
122
easily incorporated to yield better results. The inclusion of good heuristic in initial
solution is stated in Prins (2004) by using Clarke and Wright, Mole and Jameson and
Gillett and Miller heuristics for solving distance-constrained VRP (DVRP) instances.
In this study, the Nearest Neighbour (NN), Farthest Insertion (FI) and mutation
results of NN and FI were applied to generate the initial solution. The population is
an array P of PopSize (population size) chromosomes. Each chromosome Pk is
initialized as a permutation of customers. Clones (identical solutions) are forbidden
in P to ensure a better dispersal of solutions and to diminish the risk of premature
convergence. An example of initial population is depicted in Table 5.1.
Table 5.1 : An illustration of initial population
Chromosome
PopSize is even number
PopSize is odd number
Generated by
Chromosome 1
P1
P1
NN
Chromosome 2
P2
P2
FI
Chromosome 3
P3
P3
Mutation of NN
…
…
…
…
P
P
2
( PopSize
2
2)
P
3
( PopSize
2
2)
P
2
( PopSize
2
3)
Mutation of NN
3
( PopSize
2
3)
Mutation of FI
…
…
…
…
Chromosome PopSize
PPopSize
PPopSize
Mutation of FI
The population size is one of the factors affecting the performance of genetic
algorithm. Small population size might lead to a premature convergence. On the
other hand, large population size may lead to unnecessary expenditure of valuable
computational time.
In practice, the population size is often determined by
experimenter. Reviews on several population sizes used in previous researches on
Genetic Algorithm for VRP were presented in Table 5.2.
123
Table 5.2 : Reviews on several population sizes used in Genetic Algorithm for VRP
AUTHORS
APPLICATION POPULATION SIZE
Whitley (1989)
TSP
Thangiah (1995a) VRPTW
Baker and
VRP
Ayechew (2003)
50
50
30
50
Jaszkiewicz and VRP
Kominek (2003)
Bianchi et al.
VRPSD
(2004)
Prins (2004)
VRP
40
Lacomme et al.
(2006)
Ho et al. (2008)
5.2.3
CARP
VRP
NO. OF NODES/
CUSTOMERS
100 customers
50 customers
Larger sizes of
problem (75, 100,
150, 199)
100 sectors
10
50, 100 and 200
customers
Population size < 25 or Ranging from 50 to
483 customers
> 50 will give
moderate degradation
of the average solution
and found that
population size equal
to 30 performs best.
60
7-27 nodes and 1155 edges
25
50 and 100
customers
Evaluation
Once the population is initialized or an offspring population is created, the
fitness values of candidate solutions are evaluated. The fitness value is the function
of VRPSD objective function (the total expected cost). High fitness denoted by low
VRPSD objective function.
5.2.4
Roulette Wheel Selection with elitism
This research employs the combination of two types of GA selection
techniques, namely roulette wheel and elitism. The roulette wheel selection works
by selecting the chromosome by looking at their proportional fitness rank. This is
124
where the evolution concept ‘survival of the fittest’ comes into plays.
Some
researchers found that this technique will lead to only the best chromosome been
selected in the population. It is because the fittest chromosome rank is bigger
compared to the less fit chromosome and in probability of course chromosome with
the highest rate will have a big chance to be selected. The elitism technique is a
simple selection operator, which reserved the best found chromosome in the current
population to rebirth for the next generation.
Mitchell (1996) said that elitism could increase rapidly the performance of
genetic algorithm, because it prevents losing the best-found solution. When creating
a new generation using the reproduction operator, we could have a chance of losing
the best-found chromosome in the current population. This is where elitism plays
their role in preventing the lost of this chromosome.
In this example, after
calculating the probability of each selection, the elitism operator will automatically
reserved the chromosome that produce the lowest expected cost.
5.2.5
Order Crossover
The standard mutation and crossover operators designed for bit-strings and
parameter lists produce illegal offspring when applied to permutation lists.
To
combat this problem, a number of special operators have been developed which
produce legal permutation lists for offspring. Many of these operators were first
developed and tested on the Traveling Salesman Problem (TSP), where the
permutation represents a (circular) list of cities in a TSP tour. A very large number
of permutation crossovers exist in the literature, many of which have been developed
specifically for the TSP, which remains the most popular of the test bed problems
involving permutations (for example see Whitley et al. (1989) (Valenzuela, 2000).
A recent review of permutation crossovers can be found in Whitley (1997).
In Oliver et al. (1987), Partially Match Crossover (PMX), Cycle Crossover (CX) and
Order Crossover (OX) were applied to the 30-city problem of Hopfield and Tank.
They found that the best tour generated with OX was 11% shorter than the best PMX
125
tour, and 15% shorter than the best CX tour. Starkweather et al. (1991) tested six
different crossover operators on the same problem.
Thirty different runs were
performed with each operator. In this experiment, OX found the optimum 25 times
(out of 30), while PMX found the optimum only once, and CX never found the
optimum solution.
offspring.
Thus in this study, OX was employed in the generation of
OX crossover operator extends the modified crossover of Davis by
allowing two cut points to be randomly chosen on the parent chromosomes. In order
to create an offspring, the string between the two cut points in the first parent is first
copied to the offspring. Then, the remaining positions are filled by considering the
sequence of cities in the second parent, starting after the second cut point (when the
end of the chromosome is reached, the sequence continues at position 1). The detail
of how OX crossover works was given in Chapter 3.
5.2.6
Mutation
Mutation alters one or more genes (positions) of a selected chromosome
(solution) to reintroduce lost genetic material and introduce some extra variability
into the population. The role of mutation in GAs has been that of restoring lost or
unexplored genetic material into the population to prevent premature convergence of
the GA sub optimal solutions (Laoufi et al., 2006). In this study, we implement a
swap mutation. In swap mutation, two customer locations are swapped, and their
positions are exchanged. This mutation operator is the closest in philosophy to the
original mutation operator, because it only slightly modifies the original tour. For
example, choose two random positions, i.e. position 2 and 7 and swap entries from
tour
7,4,0,3,2,1,5,6
and the tour becomes
7,5,0,3,2,1,4,6
In standard Genetic algorithm, the mutation probability is set to be fixed or
remains the same from generation to the next generation. Here we employ small
126
values of pm (0.001 until 0.05) as it was commonly used in GA practice as stated by
Davis (1991), Fogarty (1989) and Hesser and Manner in Laoufi et al. (2006).
5.2.7
Stopping criterion
Stopping the genetic algorithm process can be done in several ways,
depending on the problem.
The most obvious way is simply to stop after
a pre-specified number of generations. But perhaps it would be better to stop when
there is a little change between generations, indicating that the genetic algorithm
process has reached a convergence (Tong et al., 2004). In our implementation, the
GA procedure is repeated until there were non improving moves of the best solution
for 500 successive generations.
The structure of Genetic Algorithm is depicted in Figure 5.2 where the
mutation operator is fixed during generations.
127
Figure 5.2 : Structure of genetic algorithm and adaptive genetic algorithm
128
5.3
The Development of Adaptive Genetic Algorithm
In majority of GA implementation, the operator settings are remaining static
during the GA run. However, some existing works found were showed that varying
GA settings during the GA application can lead to improvement of GA performance.
Herrera and Lozano (1996) have reviewed many aspects of the adaptation of GA
parameters settings such as mutation probability (pm), crossover probability (pc) and
population size (N).
The awareness of the importance of mutation is growing within the
evolutionary algorithm community and interest is increasing in a closer consideration
of its features (De Falco et al., 2002). Mutation operator may be considered to be an
important element in GA for solving the premature convergence problem, since it
serves to create random diversity in the population.
If the lack of population
diversity takes place too early, a premature stagnation of the search is caused. Under
these circumstances, the search is likely to be trapped in a local optimum before the
global optimum is found.
Most approaches adapt pm along with other GA parameters (see Lee and
Takagi (1993), Srinivas and Patnaik (1994), Sugisaka and Fan (2001), Mei-yi et al.
(2004), Xing et al. (2007), Law and Szeto (2007)) and so, the specific effects of the
adaptation of pm alone were not studied.
Lee and Takagi (1993) present
experimental study which was aimed at isolating the effects of the adaptation by
Fuzzy Logic Controllers of PopSize, pc and pm. It showed that the adaptation of pm
contributes most to high performance.
A time-dependency of pm was first suggested by Holland (1992) although he
did not give a detailed choice of the parameter for the time-dependent reduction of
pm. Later, the variation of pm was proposed by decreasing it exponentially during
the GA run by Fogarty (1989). The idea of sustaining genetic diversity through
mutation is then developed by measuring the Hamming distance between the two
parents during reproduction. One of it was done by Herrera and Lozano (1996) using
the adaptive control of pm by Fuzzy Logic Controllers. Thierens (2002) propose the
constant gain and the declining adaptive pm for binary genotypes. The constant gain
129
method obtains feedback of the search process by testing the results of increased and
decreased pm, and adapts the pm with a constant multiplicative gain. Liu and Feng
(2004) present a modified GA based on tuning of pm by the value of individual
fitness. The modified GA implements mutation first, after that crossover. Skinner et
al. (2004) presents a performance study of a parallel, coarse-grained, multiple deme
GA with adaptive mutation where the adaptive pm is a function of maximum number
of generations in a single epoch T, the problem dimensionality n, the gene bit length l
and generation step t.
In the following section, we compare the performance of several schemes of
adaptive mutation probability.
5.3.1
Mutation probability is random numbers in the range of [pmmin, pmmax].
Mutation probability is dynamically changed in the range of [0.001, 0.05]
from generation to other generation.
5.3.2
Adaptive mutation probability based on PDM
Lee and Takagi (1993) proposed two Phenotypic Diversity Measure (PDM)
performance measures:
PDM 1
f best
f
and PDM 2
f
f
worst
PDM1 and PDM2 belong to the interval [0,1]. If they are near to 1, convergence has
been reached, whereas if they are near to 0, the population shows a high level of
diversity.
130
5.3.3
Adaptive mutation probability based on Euclidean Distance
Herrera and Lozano (1996) propose genotypic diversity measure (GDM)
based on Euclidean distances of the chromosomes in the population from the best
one. The GDM is denoted by ED.
d d min
d max d min
ED
N
where d
1
N
¦ d (C
best
, C i ) , d max
max{d (C best , C i ) , d min
min{d (C best , C i ) , Ci is
1
chromosome in population. Euclidean distance between object x and y is defined as
n
th
2
¦ ( xi yi ) , where xi is the value of x at its i element and yi is the
1
d ( x, y )
value of y at its ith element, i = 1, …, n. The range of ED is [0,1]. If ED is low, most
chromosomes in the population are concentrated around the best chromosome and so
convergence is achieved.
In Section 5.3.1 and Section 5.3.2, Lee and Takagi (1993) and Herrera and
Lozano (1996) respectively proposed the use of Fuzzy Logic Controller to
dynamically control the mutation probability. In this study we prefer the use of
simple linear formula of pm = PDM1 or pm = PDM2 and pm = 1-ED. We use these
formula since the value of PDM1, PDM2, and ED are in the range of [0,1] as the
mutation probability. The nearer to the convergence is shown by higher PDM1,
higher PDM2 (nearer to 1) and lower ED (nearer to 0). Higher PDM1, PDM2 and ED
require higher pm. But since we attempt to compare performance of algorithms, we
have considered that these techniques should handle the same range of possible pm
values in the range of [0.001, 0.1]. So, a transformation was made from the interval
considered by these techniques into [0.001, 0.1].
Beside the above measures, we also propose the new adaptive mutation probability
as follows.
131
5.3.4
Adaptive mutation probability based on Lr Distance
We propose another diversity measure based on Lr distance (LD) that was
usually used to measure dissimilarity between two objects. We adopt the formula
proposed by Herrera and Lozano (1996). We substitute the ED with LD where Lr
distance between two objects is d ( x, y )
5.4
xi y i , thus pm = 1-LD.
The Development of Hybrid GA and TS
The proposed hybrid algorithm of GA and TS (HGATS) consists of several
properties such as the order or permutation representation; small population in which
all individuals are distinct; inclusion of good simple heuristic solutions in the initial
population; Roulette Wheel Selection with elitism; Order Crossover and the
incorporation of Tabu Search to the GA. It is hope that this hybrid could combine
the advantage of GA as population-based method and the strength of TS as trajectory
method. As known, population-based methods are better in identifying promising
areas in the search space, whereas trajectory methods are better in exploring
promising areas in search space.
We propose two schemes for the HGATS. The first is TS used as a powerful
mutation operator in GA, called HGATS type A. Different to the standard GA where
mutation is applied with probability of pm; in this HGATS, the Tabu Search is
applied for each offspring. This approach may be viewed as multistart Tabu Search,
with starting points obtained from the search space obtained from GA. The second
scheme is TS used as an improvement of the offspring after mutation is applied; we
called it HGATS type B. This approach can be regarded as multistart Tabu Search
where the starting points are obtained through a sampling of the search space of GA.
The illustration of development of metaheuristic for VRPSD based on HGATS is
described in Figure 5.3.
132
INITIAL POPULATION
Nearest Neighbour
Mutation of NN
SELECTION
Random
Mutation of FI
Mutation of NN
Mutation of FI
Farthest Insertion
TS
TS
RECOMBINATION
.
.
.
.
.
.
SELECTION
.
.
.
Best
individuals of
parents and
offspring
TS
Parent
Offspring
.
.
.
New population
MUTATION
(i). HGATS type A
Parent
INITIAL POPULATION
Nearest Neighbour
.
.
.
Mutation of NN
Mutation of FI
Mutation of FI
Mutation of NN
SELECTION
Farthest Insertion
.
Mutation of FI
RECOMBINATION
MUTATIO N
Parent
MUTATIO N
.
.
.
.
.
.
MUTATIO N
.
.
+
Offspring
IMPROVE WITH
TABU SEARCH
SELECTION
Best
individuals of
parents and
offspring
.
.
.
.
New population
Offspring
(ii). .HGATS type B
Figure 5.3 : Illustration of Hybrid GA – TS schemes
133
5.5
The Development of Breeder Genetic Algorithm
According to Muhlenbein (1994), there exist at least three kinds of theories in
science:
-
theories which are able to predict the outcome of experiments
-
theories which conceptually describe experimentally observed phenomenon
-
theories of a “third kind” which do not have any predictive capabilities
The popular theory of Genetic Algorithm is of the third kind while the theory of
Breeder Genetic Algorithm (BGA) is of the first kind. It models artificial selection
as performed by human breeders. The science of breeding is based on advanced
statistical methods.
In fact, quantitative genetics was the driving force behind
modern statistics.
The development started with Galton and Pearson in Muhlenbein (1994), who
invented the scatter diagram, regression and correlation in order to understand the
heritability of the height of sons in families living in England. The need to recast the
macroscopic Darwinian theory, as formalized by Pearson in the microscopic
Mendelian chance model, leads to the concept of decomposition of variance and
covariance. This theory was initiated by Fisher. This subsection proposes the BGA
for solving VRPSD, according to our knowledge, this is the first time BGA has
applied for solving VRPSD.
The BGA is inspired by the science of breeding animals. In this algorithm,
each one of a set of virtual breeder has the task to improve its own subpopulation.
Occasionally the breeder imports individuals from neighboring subpopulations.
According to Muhlenbein and Schlierkamp-Voosen (1993a), the BGA is a robust
global optimization method where selection, recombination and mutation are well
tuned and have a synergetic effect. Now we define the ingredients of the BGA. First,
a genetic encoding of the VRPSD, initialization and evaluation of fitness function are
made.
Then improvement by OrOpt local search, crossover and mutation are
presented. The proposed BGA for VRPSD is presented as follows.
134
Step 0. [Define] Define operator settings of GA suitable with
the problem which is VRPSD.
Step 1. [Initialization]
PopSize
Create
chromosomes
an
that
initial
population
consists
of
P
of
constructive
heuristics solutions and randomly mutation of it where
all individuals are distinct or clones are forbidden.
Step 2. [Fitness]
Evaluate
the
f(Ci)
fitness
chromosome Ci in the population.
of
each
The fitness is the
function of VRPSD objective function.
Step 3. [Improvement] Apply improvement method by using OrOpt
local search for each individual
Step 4. [Selection] Select T % best individual in population
to be parents for mating, set this set as S(t), in
this study T is in the range of 35 – 50%.
Step 5. [Crossover] Pair all the chromosomes in S(t) at random
Apply OX crossover with probability pc
forming pairs.
to
each
number
t
pair
pc
and
then
produce
offspring
offspring,
is
the
if
exact
random
copy
of
parents.
Step 6. [Mutation] With a mutation probability pm mutate the
offspring using swap mutation.
Step 7. [New Population] Insert offspring to the population.
Form new population P(t+1)
Step 8. [Stopping Criterion] If the stopping criterion is met
then stop, and return to the best solution in current
population, else go to Step 9.
The BGA procedure is
repeated until there were non improving moves of the
best solution for 500 successive generations.
Step 9. [Acceptance Criterion] Check whether new population is
better
than
acceptance
criterion,
if
yes,
go
to
Step 3, if no then go to Step 4.
In general, the properties of BGA are similar to the standard GA as described
in Section 5.2 except that there were some additional features to the standard GA: the
addition of Step 3, 4 and 9. Or-opt local search in Step 3 was described in Section
4.3. The detail of Step 4 and 9 are presented as follows.
135
Step 4: Selection
Let M(t) denote the mean fitness of the population at time t. The change in
fitness caused by the selection is given by
R(t )
M (t ) M (t 1)
and is called response to selection. R(t) measures the expected progress of the
population. Breeder measure the selection with the selected differential, which is
symbolized by S(t). It is defined by the difference between the mean fitness of the
selected individuals, Ms(t) and the population mean:
S (t )
M s (t ) M (t )
These two definitions are very important.
They quantify the most important
variables. In the process of artificial breeding, both R(t) and S(t), can be easily
computed. The breeder tries to predict R(t) from S(t). Breeder often uses truncation
selection. In truncation selection with threshold T, the T % best individuals will be
selected as parents. T is normally chosen in the range 10% to 50% (Muhlenbein and
Schlierkamp-Voosen, 1993b).
Step 9: Acceptance Criterion
The connection between R(t) and S(t) is given by the equation
R(t )
bt .S (t )
In quantitative genetics bt is called the realized heritability. It is normally assumed
that bt is constant for a number of generations. This leads to
R(t )
b.S (t )
136
In general, the progress due to a genetic search as long as the ratio R(t)/S(t) is greater
than 0.1 (Crisan and Muhlenbein, 1998).
The structure of Breeder Genetic Algorithm is depicted in Figure 5.4.
START
Generate Initial Population Pop(0)
Set iteration t = 1
Apply OrOpt for each individual, set as Pop(t)
Calculate mean fitness of population Pop(t), M(t)
TRUNCATE POPULATION
Select 35% to 50% best individuals of population as parent for mating
Calculate mean fitness of selected population TruncPop(t), Ms(t)
Calculate R(t) = M(t) - M(t-1)
Calculate S(t) = Ms(t) - M(t)
Crossover the TruncPop(t)
t = t+1
t = t+1
Mutation
No
Yes
R (t )
! 0.1 ?
S (t )
Insert new offspring to the unselected 50% - 65% of population.
Form new population Pop(t+1)
No
Stopping
criteria met?
Yes
Choose the best individual
END
Figure 5.4 : Structure of Breeder Genetic Algorithm
137
5.6
Results and Discussion
Genetic Algorithm – based metaheuristics described in previous section were
tested on a set of instances following data generation procedure in Chapter 3. There
is an extra consideration when testing on randomized procedures like genetic
algorithms.
Even for a single instance and a single algorithm implementation,
successive trials can produce quite different outcomes because of the impact of
random choices along the way. Several runs will usually need to be made, to get a
sense of the robustness of the procedure. In this chapter, twenty instances were
generated on each of the number of nodes 10, 20 and 50. Each algorithm was tested
on each instance for 100 generations and 10 runs. The value to be observed was the
average value of 10 runs.
In Section 5.6.1, a preliminary study on the effect of the population size to the
performance of GA was presented, while in Section 5.6.2, the relative performance
of Adaptive GA and standard GA was evaluated and compared. A comparative
study between the performances of two types of HGATS and between each type of
HGATS with both TS and GA alone were reported in Section 5.6.2. Section 5.6.3
demonstrated the advantage of Breeder GA whereas Section 5.6.4 attempted to study
the comparison between Tabu Search and Genetic Algorithm – based metaheuristics
described in this study.
5.6.1
Preliminary Study on the Choice of Population Size
In this study, a preliminary study on the effect of population size to the
performance of Genetic Algorithm on VRPSD was presented; the results were given
in Table 5.3. The preliminary study objective was to determine which population
size that can give relatively good performance of Genetic Algorithm.
All the
parameter setting of the Genetic Algorithm is fixed including the mutation
probability, crossover probability and stopping criterion, except for the population
size. The GA was implemented in fifteen (15) instances and five (5) independent
runs for each instance, thus each cell contains the average value over the runs. The
138
problem instances consist of four problem sizes that were 10 nodes, 30 nodes and 50
nodes.
Table 5.3 : The performances of GA for different population sizes
Instance
Number
Population size
of nodes
10
30
50
100
10_1
10
355.4 [17]
343 [54]
343 [91]
343 [189]
10_2
10
344.4 [21]
340 [56]
340.5 [100]
340 [247]
10_3
10
249.3 [17]
248 [52]
248 [87]
248 [188]
10_4
10
265.3 [17]
263 [53]
263 [89]
262 [226]
10_5
10
262 [17]
261 [50]
261 [88]
261 [198]
20_1
20
411.2 [85]
403.7 [218]
398.2 [355]
400.3 [725]
20_2
20
447 [60]
447 [186]
447 [310]
447 [618]
20_3
20
453.2 [88]
445 [260]
447.2 [392]
445 [742.5]
20_4
20
403.6 [81]
400.6 [248]
399.8 [329]
404.6 [639]
20_5
20
376 [57]
376 [176]
373 [322]
376 [738]
50_1
50
608.6 [471]
602.4 [1497]
604.8 [2497]
603.3 [4721]
50_2
50
625 [619]
611.8 [1528]
607.4 [2539]
599 [5154]
50_3
50
633 [351]
629 [1433]
629.5 [2011]
633 [3700]
50_4
50
675.7 [480]
657 [1882]
654 [3215]
659 [3800]
50_5
50
655 [573]
639 [1542]
639 [2346]
639 [4831]
443.69
444.01
Average
450.98
444.43
* the computation times are given in brackets (in seconds)
To verify whether the differences between solutions found by different
population sizes are statistically significant or not, we performed paired samples test
for each pair of algorithms compared, i.e. between solutions found using population
size of 10 and 30, between solutions found using population size of 10 and 50, etc.
The paired-samples t-test procedure compares the means of two variables for a single
group. It computes the differences between values of the two variables for each case
and tests whether the average differs from 0. Before the paired t-test was conducted,
the test of normal distribution assumption of the difference between two variables
must be conducted. The results of Kolmogorov-Smirnov test for normal distribution
test were given in Table 5.4. We reported the p-value for the null hypothesis “The
data follow normal distribution” where the significance level which the null
hypothesis rejected is 0.05. The p-value that smaller than 0.05 is sufficient to reject
the null hypothesis.
139
Table 5.4 : Kolmogorov-Smirnov normality test results for differences of GA
solution qualities for different population sizes
One-Sample Kolmogorov-Smirnov Test
N
Normal Parameters a,b
Most Extreme
Differences
Mean
Std. Deviation
Absolute
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
diff_10_30
15
6.5500
6.01365
.173
.173
-.138
.670
.760
diff_10_50
15
7.2900
6.93717
.287
.287
-.147
1.113
.168
diff_10_100
15
6.9700
7.95384
.183
.183
-.158
.709
.696
diff_30_50
15
.7400
2.25414
.295
.295
-.158
1.144
.146
diff_30_100
15
.4200
3.87818
.343
.343
-.190
1.329
.058
diff_50_100
15
-.3200
3.28486
.205
.156
-.205
.796
.551
a. Test distribution is Normal.
b. Calculated from data.
Since the p-values of normality test for all the differences were larger than
= 0.05, thus we have no evidence to reject the null hypothesis. We can conclude
that the difference values follow normal distribution and the paired t-test can be used
for this data set. The results of paired t-test were presented in Table 5.5. Since the
p-value of difference between the use of population size of 10 and 30, 10 and 50, 10
and 100 are less than = 0.05, the null hypothesis was rejected and it can be
concluded that the solution qualities generated by using population size of 10 is
different with the use of other population sizes. While the use of population size of
30, 50 and 100 did not give significant differences on the GA performance (solution
qualities) since the p-values were larger than = 0.05.
The preliminary study on the effect of different population size to the GA
performance discussed in this sub section suggest that the use of population size 30 is
sufficient to number of nodes up to 50, since there were no statistically significant
differences between the solution qualities resulted from the use of population size of
30 with population size of 50 and 100 but the use of population size of 30 give much
less computation time than the use of population size of 50 and 100. This result also
supported by literature reviews on research on population size used in GA for VRP
in Table 5.2 at Section 5.2.2 that Baker and Ayechew (2003) used population size of
30 for 50-node problems, while Prins (2004) stated that population size < 25 or > 50
will give moderate degradation of the average solution and found that population size
equal to 30 performs best.
140
Table 5.5 : Paired Samples Test Results of the Differences
Paired Samples Test
Paired Differences
Pair 1
Pair 2
Pair 3
Pair 4
Pair 5
Pair 6
5.6.2
10-30
10-50
10-100
30-50
30-100
50-100
Mean
6.5500
7.2900
6.9700
.7400
.4200
-.3200
Std. Deviation
6.0136
6.9371
7.9538
2.2541
3.8781
3.2848
Std. Error
Mean
1.5527
1.7911
2.0536
.5820
1.0013
.8481
95% Confidence
Interval of the
Difference
Lower
Upper
3.2197
9.8802
3.4483
11.1316
2.5653
11.3746
-.5083
1.9883
-1.7276
2.5676
-2.1390
1.4990
t
4.218
4.070
3.394
1.271
.419
-.377
df
14
14
14
14
14
14
Sig. (2-tailed)
.001
.001
.004
.224
.681
.712
Adaptive GA versus Standard GA
In this section, the performances of fixed and adaptive mutation probability
were compared. Table 5.6 presented the solution quality obtained for number of
nodes 10, 20 and 50, based on performance: average of the best fitness function (best
total expected cost) found at the end of each independent run. For each combination
of number of nodes and type of mutation probability, five instances were generated.
Each cell is the average value of 10 independent runs. The minimum total expected
cost under fixed setting is in bold and italic, while the best performance measure (the
minimum total expected cost) under all mutation probability types (both for fixed
and adaptive setting) is in asterisk.
Regarding to the GA versions under fixed pm values, the best performance
measures for each problem set are obtained by different pm values. Considering the
performance between fixed versus adaptive mutation probability, it is found that the
adaptive algorithms leads in better solution quality rather than fixed parameter, since
for each problem set, it returns results that better than all fixed pm values, or the
same as the ones of the most successful GAs with fixed pm settings. The adaptive
scheme plays a role in improving solution quality and possibly directing search to
unknown regions to avoid being trapped in local optimum. By using the adaptive
GA, the amount of time to spend in finding appropriate mutation probability values
of GA can be reduced.
141
Table 5.6 : The performance of fixed parameter versus adaptive GA
Number
of nodes
10
20
50
Types and values of
mutation probability
Instance
1
Instance
2
Instance
3
Instance
4
Instance
5
Fixed pm = 0.001
346.6
342.0
248.0
263.0
300
Fixed pm = 0.005
347.4
341.6
248.0
263.0
300
Fixed pm = 0.01
346.0
342.0
248.0
263.0
300
Fixed pm = 0.05
344.4
342.6
248.8
264.2
300
Fixed pm = 0.1
346.6
341.8
248.0
264.2
300
RANGE (0.001;0.1)
344.4
341.8
249.0
263.2
300
ED
345.2
342.0
248.0
263.2
300
LD
343.0*
342.0
248.0
263.6
300
PDM1
343.8
341.2*
248.0
261.8*
300
PDM2
345.0
341.8
248.0
263.6
300
Fixed pm = 0.001
404.8
447
454.0
376.0
406.6
Fixed pm = 0.005
408.6
447
455.0
375.0
404.0
Fixed pm = 0.01
407.6
447
452.4
376.0
404.6
Fixed pm = 0.05
408.8
447
460.2
376.0
403.8
Fixed pm = 0.1
410.4
447
453.0
374.6
405.4
RANGE (0.001;0.1)
410.0
447
454.0
376.0
404.0
PDM1
404.8
447
454.4
373.8*
409.0
PDM2
412.2
447
454.6
375.0
406.2
ED
412.8
447
457.2
376.0
404.6
LD
404.0*
447
451.0*
374.8
399.0*
Fixed pm = 0.001
633.0
592.6
610.2
631.2
722.8
Fixed pm = 0.005
632.8
593.0
608.6
631.0
719.6
Fixed pm = 0.01
632.6
590.2
608.4
621.6
721.8
Fixed pm = 0.05
633.0
591.6
607.6
621.0
724.0
Fixed pm = 0.1
633.0
593.0
610.2
628.0
721.0
RANGE (0.001;0.1)
632.8
591.6
608.4
625.2
717.0*
PDM1
632.8
591.2
608.0
624.4
721.0
PDM2
632.4*
591.6
608.8
624.6
722.0
ED
632.6
592.8
608.8
625.4
719.0
LD
633.0
589.2*
607.2*
619.0*
717.4
* denotes best total expected cost for all mutation probability settings
142
From the results in Table 5.6 it was shown that different AGAs are better
suited to different problem sets.
Compared to the other adaptive schemes, the
proposed LD showed superiority since the proposed LD measure was proven to yield
best performance measure in almost all instances, while if we compare it to fixed pm,
the lowest values of VRPSD objective function from LD were still lower than almost
overall that can be resulted from fixed pm setting, or the same with fixed pm results.
The next best adaptive mutation probability is PDM1.
5.6.3
Hybrid GA and TS (HGATS) versus GA and TS
In this section, comparative study between the performances of two types of
HGATS and between each type of HGATS with both TS and GA alone were
conducted. Twenty instances were generated on each of the number of nodes 10, 20
and 50. Each algorithm was tested on each instance for 100 iterations. Figure 5.5
shows the graphical summary by means of a box and whisker plot to describe the
distribution of solutions obtained by GA, TS and two types of HGATS for number of
nodes 10. The solutions were grouped by algorithm. As it can be observed from the
box and whisker plots, the relative performance of GA was better than TS, similar
with HGATS type A and worse than HGATS B. Compared to other algorithms, TS
seems to give worst performance since the expected costs from TS were higher than
others. Regarding to the Hybrid GA with TS, it is worth noticing that different
implementation of TS in GA can produce significant difference. As shown in Figure
5.5, the HGATS type B can obtain better relative performance than HGATS type A.
143
360
340
320
300
280
260
240
GA_10
TS_10
HGATS_10A
HGATS_10B
Figure 5.5 : Box and whisker plots of solutions obtained from GA, TS and HGATS
for 10-node problems
To verify whether the differences are statistically significant, the paired test
should be used. Firstly, the Kolmogorov-Smirnov was conducted to test the null
hypothesis “The data follow normal distribution”. The probability to reject the null
hypothesis was set to be 0.05.
Table 5.7presented the Kolmogorov-Smirnov
normality test results of differences of GA, TS and HGATS for number of nodes 10.
Table 5.7: Kolmogorov-Smirnov normality test results of differences of GA, TS and
HGATS for 10-node problems
One-Sample Kolmogorov-Smirnov Test
Number of runs
Normal Parameters
Mean
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal
b. Calculated from data.
DIFF_
GA_TS_
10
DIFF_
GA_
HGATSA
_10
20
-2.650
4.308
.331
.212
-.331
1.479
.025
20
-.580
1.811
.426
.183
-.426
1.903
.001
DIFF_
GA_
HGATSB_
10
20
1.200
2.211
.274
.274
-.244
1.227
.098
DIFF_
TS_
HGATSA
_10
DIFF_
TS_
HGATSB
_10
DIFF_
HGTSA_
HGATSB
_10
20
2.070
2.9829
.406
.406
-.244
1.816
.003
20
3.850
5.6407
.303
.303
-.247
1.353
.051
20
1.780
3.3181
.404
.404
-.296
1.808
.003
144
The p-values for difference between GA and TS, GA and HGATSA, GA and
HGATS type B, TS and HGATS type A, TS and HGATS type B, and HGATS type
A and B were 0.025, 0.001, 0.098, 0.003, 0.051, and 0.003, respectively. Thus the
difference between solution obtained from GA and HGATS type B follows normal
distribution, and also the difference between TS and HGATS type B since the pvalues were larger than D = 0.05; while the other differences do not follow normal
distribution. For the differences that satisfy normal distribution assumption, the
paired t-test can be used, in the contrary when the normal distribution assumption
was violated; the Wilcoxon paired-matched signed rank test was used. The summary
of paired samples test results were reported in Figure 5.6 whereas the detail of it was
given in Appendix B.
Paired Samples Statistics
Pair
1
Pair
2
GA_10
HGATS_10B
TS_10
HGATS_10B
N
20
20
20
20
Std. Deviation
34.05735
33.94795
35.55851
33.94795
Paired Differences
Std.
95% Confidence
Std.
Error
Interval of the
Deviation
Mean
Difference
Mean
Pair 1
GA_10 HGATS_10B
Pair 2
TS_10 HGATS_10B
Mean
302.6000
301.4000
305.2500
301.4000
Lower
Upper
Std. Error
Mean
7.61546
7.59099
7.95113
7.59099
t
df
Sig.
(2-tailed)
1.200
2.21193
.49460
.16479
2.23521
2.426
19
.025
3.850
5.64078
1.26132
1.21003
6.48997
3.052
19
.007
(a). Paired t-test results
Test Statisticsc
Z
Asymp. Sig. (2-tailed)
TS_10 HGATS_10A - HGATS_10A - HGATS_10B GA_10
GA_10
TS_10
HGATS_10A
-2.448a
-.844a
-2.375b
-2.207b
.014
.398
.018
.027
a. Based on negative ranks.
b. Based on positive ranks.
c. Wilcoxon Signed Ranks Test
(b). Wilcoxon signed rank test results
Figure 5.6 : Paired test results of GA, TS and HGATS for 10-node problems
145
As shown in Figure 5.6 (a), the mean differences of GA and HGATS type B
solution costs and the mean differences of TS and HGATS type B solution costs
were statistically significant, since the p-values were less than 0.05 and the 95%
confidence interval of the differences were not encompass zero. Furthermore we can
notice that the relative performance of HGATS type B was better than GA and TS
since the mean value of its differences were positive (in terms of GA – HGATS type
B and TS – HGATS type B).
From Figure 5.6 (b) we can also notice that the
relative performance of GA seems to be not significantly different from HGATS type
A, while the relative performance of TS and GA, TS and HGATS type A and
HGATS type A and B, were statistically significantly different from one to another.
The relative performance of GA is better than TS, not significantly different from
HGATS A and worse than HGATS type B. The TS seems to be the worst among the
others whereas the HGATS type B seems to give best performance relative to the
others.
The distribution of solutions obtained from GA, TS and HGATS for number
of nodes 20 were given in Figure 5.7. The relative performance of the algorithms
was similar across number of nodes 10 and 20 although the differences in number of
nodes 20 were more lightly to be seen. As it can be observed from the box and
whisker plots, the relative performance of GA was better than TS, similar with
HGATS type A and worse than HGATS B. Compared to other algorithms, TS seems
to give worst performance.
The last is the comparison between two types of
HGATS; it was shown that the HGATS type B can obtain better relative performance
than HGATS type A.
146
475
450
425
400
375
350
325
GA_20
TS_20
HGATS_20A
HGATS_20B
Figure 5.7 : Box and whisker plots of solutions obtained from GA, TS and HGATS
for 20-node problems
The next step is normality test based on Kolmogorov-Smirnov test against the
null hypothesis of “The data follows normal distribution” with the probability to
reject the null hypothesis is set to be 0.05. The results were reported in Table 5.8.
Since all the p-values were larger than 0.05 thus the null hypothesis can not be
rejected and it was concluded that the differences satisfied normal distribution
assumption in order to conduct the paired t-test. The paired t-test results were
presented in Figure 5.8.
Table 5.8: Kolmogorov-Smirnov normality test results of differences of GA, TS and
HGATS for 20-node problems
One-Sample Kolmogorov-Smirnov Test
DIFF_GA
Number of runs
20
a,b
Normal Parameters
Most Extreme
Differences
DIFF_GA_
DIFF_GA_
HGSTSA_20 HGATSB_20
TS _20
20
20
DIFF_
DIFF_TS_
DIFF_TS_
HGATSA_
HGATSA_20 HGATSB_20
HGATSB_20
20
20
20
Mean
-5.7200
1.4800
3.8800
7.2000
9.6000
2.4000
Std. Deviation
9.85290
3.96479
4.87416
9.31665
10.32320
4.13458
Absolute
.271
.248
.173
.224
.198
Positive
.191
.248
.173
.224
.198
.292
Negative
-.271
-.134
-.163
-.170
-.152
-.231
1.212
1.110
.773
1.001
.884
1.307
.106
.170
.588
.268
.416
.066
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
.292
147
From Figure 5.8 we can see that the only difference that was not statistically
significant was the difference between GA and HGATS type A (shown by the
p-value = 0.111 that was larger than D = 0.05, thus we can not reject H0 that there
was no difference between the performance of GA and HGATS type A), while the
other pairs result significant differences since the p-values were less than D = 0.05.
This condition also occurred in number of nodes 10. It seems that the inclusion of
TS as mutation operator in GA does not give significant impact to the performance
of GA for small and moderate number of nodes (number of nodes 10 and 20). It is
worth noticing that the relative performance of the algorithms is similar across
number of nodes 10 and 20.
From associated figure we can draw the same
conclusion for number of nodes 20 as in number of nodes 10, that the relative
performance of GA was better than TS, similar to HGATS type A, and worse than
HGATS type B. TS seems to give worst performance among others whereas the
HGATS type B seems to obtain the best performance relative to the others since the
mean value of difference between HGATS type B with other algorithms were always
positive.
Mean
GA_20 TS_20
GA_20 HGATS_20A
GA_20 HGATS_20B
TS_20 HGATS_20A
TS_20 HGATS_20B
HGATS_20A HGATS_20B
Paired Differences
95% Confidence
Interval of the
Std.
Difference
Error
Std.
Deviation
Mean
Lower
Upper
t
df
Sig.
(2tailed)
-5.72
9.85290
2.20317
-10.33130
-1.10870
-2.596
19
.018
1.48
3.96479
.88655
-.37558
3.33558
1.669
19
.111
3.88
4.87416
1.08990
1.59882
6.16118
3.560
19
.002
7.20
9.31665
2.08327
2.83967
3.456
19
.003
9.60
10.32320
2.30834
4.76859
4.159
19
.001
2.40
4.13458
.92452
.46496
2.596
19
.018
11.5603
3
14.4314
1
4.33504
Figure 5.8 : Paired samples test results of GA, TS and HGATS for 20-node
problems
148
Finally we compared the performance of GA, TS and HGATS for large
number of nodes that is 50. Figure 5.9 shows the graphical summary by means of a
box and whisker plot to describe the distribution of solutions obtained by GA, TS
and HGATS. The solutions were grouped by algorithm. As it can be observed from
the box and whisker plots, the HGATS type B seems to give best relative
performance since the mean value of solution given by HGATS type B was smallest
compared to other metaheuristics.
The HGATS type A was the second best,
followed by TS, while GA gave the worst solution quality since it obtained the
highest total expected cost. It can be seen that for 50-node problems, inner outliers
(denote by circles) were present in the box and whisker for solution cost values of
GA, TS and HGATS-A for problem instance 12, but not present at solution cost
values of HGATS-B. Outlier refers to an observation that is numerically distant
from the rest of the data, where inner outlier is an observation which over 1.5 box
lengths from upper/lower quartiles.
This phenomenon indicated the power of
HGATS-B which gave much lower cost than GA, TS and HGATS-A for the same
problem instance, i.e. problem instance 12.
720
12
12
12
690
660
630
600
570
540
GA 50
TS 50
HGATS 50A
HGATS 50B
Figure 5.9 : Box and whisker plots of solutions obtained from GA, TS and HGATS
for 50-node problems
149
The following step is normality test based on Kolmogorov-Smirnov test
against the null hypothesis of “The data follow normal distribution” with the
probability to reject the null hypothesis is set to be 0.05. The results were reported in
Table 5.9 Since all the p-values were larger than 0.05 thus we can not reject the null
hypothesis and we conclude that the differences satisfied normal distribution
assumption to conduct the paired t-test. The paired t-test results were presented in
Figure 5.10.
Table 5.9: Kolmogorov-Smirnov normality test results of differences of GA, TS and
HGATS for 50-node problems
One-Sample Kolmogorov-Smirnov Test
DIFF_
HGATSA_
HGSTSA_50 HGATSB_50 HGATSA_50 HGATSB_50 HGATSB_50
DIFF_GA_TS DIFF_GA_
50
N
20
a,b
Normal Parameters Mean
Std. Deviation
6.6200
20
9.1450
DIFF_GA_
20
25.5700
DIFF_TS_
20
2.5250
DIFF_TS_
20
18.9500
20
16.4250
8.1377
8.8924
16.1688
7.8312
13.7935
13.9776
Absolute
.171
.145
.110
.225
.143
.127
Positive
.171
.145
.110
.225
.143
.127
Negative
-.145
-.126
-.081
-.132
-.120
-.106
Kolmogorov-Smirnov Z
.763
.650
.494
1.008
.639
.568
Asymp. Sig. (2-tailed)
.606
.792
.968
.262
.809
.903
Most Extreme
Differences
a. Test distribution is Normal.
b. Calculated from data.
From Figure 5.10 we can see that the only difference that was not statistically
significant was the difference between TS and HGATS type A, while the other pairs
result significant differences. From associated figure we can draw conclusion for
that the relative performance of TS is better than GA, similar to the HGATS type A,
and worse than HGATS type B. The relative performance of HGATS type A is
better than GA. GA seems to give worst performance among others whereas the
HGATS type B seems to obtain the best performance relative to the others since the
mean value of difference between HGATS type B with other algorithms were always
positive.
150
Paired Samples Statistics
Pair
1
Pair
2
Pair
3
Pair
4
Pair
5
Pair
6
GA_50
TS_50
GA_50
HGATS_50A
GA_50
HGATS_50B
TS_50
HGATS_50A
TS_50
HGATS_50B
HGATS_50A
HGATS_50B
Mean
620.6700
614.0500
620.6700
611.5250
620.6700
595.1000
614.0500
611.5250
614.0500
595.1000
611.5250
595.1000
N
20
20
20
20
20
20
20
20
20
20
20
20
Std. Deviation
35.12470
34.88398
35.12470
33.92502
35.12470
28.91530
34.88398
33.92502
34.88398
28.91530
33.92502
28.91530
Std. Error
Mean
7.85412
7.80030
7.85412
7.58587
7.85412
6.46566
7.80030
7.58587
7.80030
6.46566
7.58587
6.46566
Paired Differences
Mean
GA_50 TS_50
GA_50 HGATS_50A
GA_50 HGATS_50B
TS_50 HGATS_50A
TS_50 HGATS_50B
HGATS_50A HGATS_50B
Std.
Deviation
Std. Error
Mean
95% Confidence
Interval of the
Difference
df
Lower
Upper
t
Sig.
(2tailed)
6.620
8.13774
1.81965
2.81142
10.42858
3.638
19
.002
9.145
8.89248
1.98842
4.98319
13.30681
4.599
19
.000
25.570
16.16888
3.61547
18.00273
33.13727
7.072
19
.000
2.525
7.83124
1.75112
-1.14013
6.19013
1.442
19
.166
18.950
13.79350
3.08432
12.49445
25.40555
6.144
19
.000
16.420
13.97768
3.12551
9.88324
22.96676
5.255
19
.000
Figure 5.10 : Paired samples test results of GA, TS and HGATS for 50-node
problems
5.6.4
Breeder GA versus Standard GA
The algorithms compared here are the Breeder GA (BGA) and the standard
GA. Twenty instances were generated on each of the number of nodes 10, 20 and 50.
Each algorithm was tested on each instance for 100 iterations. Figure 2 shows the
results of solution quality based on total expected cost obtained by the GA and the
BGA for each number of nodes. As it can be observed from the box and whisker
plots, it is worth noticing that the relative performance of the algorithms is similar
across different number of nodes: the solution quality produced by the BGA clearly
151
outperforms the results of standard GA since the BGA can yields lower total
expected cost for every number of nodes.
800
12
700
600
500
400
300
200
GA_10
BGA_10
GA_20
BGA_20
GA_50
BGA_50
Figure 5.11: Box and whisker plots of solutions obtained from GA and BGA
To verify that the differences between solutions found by the two algorithms
are statistically significant, we performed paired samples test between GA and BGA
results for every number of nodes. The paired-samples t-test procedure compares the
means of two variables for a single group. It computes the differences between
values of the two variables for each case and tests whether the average differs from
0. Before the paired t-test was conducted, the test of normal distribution assumption
of the difference between two variables must be conducted.
The results of
Kolmogorov-Smirnov test for normal distribution test were given in Table 5.10 We
reported the p-value for the null hypothesis “The data follow normal distribution”
where the significance level which the null hypothesis rejected is 0.95. The p-value
152
that smaller than 0.05 is sufficient to reject the null hypothesis.
In the table
associated, the p-values for number of nodes 10, 20 and 50 are 0.066, 0.363 and
0.832, respectively. Thus we can conclude that the difference values of GA and
BGA for all number of nodes follow normal distribution and the paired t-test can be
conducted for these data.
Table 5.10 Normality test results of the differences between GA and BGA
One-Sample Kolmogorov-Smirnov Test
N
Normal Parameters a,b
Most Extreme
Differences
Mean
Std. Deviation
Absolute
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
DIFF_BGA_
GA_10
20
-1.0500
2.09649
.292
.208
-.292
1.305
.066
DIFF_BGA_
GA_20
20
-2.7300
4.66579
.206
.116
-.206
.922
.363
DIFF_BGA_
GA_50
20
-22.6700
16.44823
.139
.109
-.139
.623
.832
a. Test distribution is Normal.
b. Calculated from data.
The p-value of paired difference test were presented in Figure 5.12, the
values were 0.037, 0.017 and 0.000 for number of nodes 10, 20 and 50, respectively.
These results show that the two algorithms tend to yield different result. There were
statistically significant differences between the performance of GA and BGA in term
of solution quality for solving VRPSD. The extent of this difference was shown well
by the confidence interval of the difference which not encompasses zero. From the
mean values of the differences on Figure 5.12, it also can be shown that on average,
the BGA produced less total expected cost than GA since the mean value of the
differences between BGA and GA (i.e. BGA - GA) were always negative. In the
other words, the mean values of total expected cost resulted from BGA were less
than the cost obtained from GA.
153
Paired Samples Statistics
Pair
1
Pair
2
Pair
3
BGA_10
GA_10
BGA_20
GA_20
BGA_50
GA_50
Mean
301.5500
302.6000
392.5500
395.2800
598.0000
620.6700
N
20
20
20
20
20
20
Std. Deviation
34.07341
34.05735
37.97572
38.62704
28.98094
35.12470
Std. Error
Mean
7.61905
7.61546
8.49163
8.63727
6.48033
7.85412
Paired Samples Correlations
N
Pair 1
Pair 2
Pair 3
BGA_10 & GA_10
BGA_20 & GA_20
BGA_50 & GA_50
20
20
20
Correlation
.998
.993
.886
Sig.
.000
.000
.000
Paired Samples Test
Paired Differences
Pair 1
Pair 2
Pair 3
Mean
BGA_10 - GA_10 -1.05000
BGA_20 - GA_20 -2.73000
BGA_50 - GA_50 -22.67000
Std. Deviation
2.09649
4.66579
16.44823
Std. Error
Mean
.46879
1.04330
3.67794
95% Confidence
Interval of the
Difference
Lower
Upper
-2.03119
-.06881
-4.91366
-.54634
-30.36801 -14.97199
t
-2.240
-2.617
-6.164
df
19
19
19
Sig. (2-tailed)
.037
.017
.000
Figure 5.12 : Paired-differences test results between GA and BGA performance
5.6.5
Comparative Study between Tabu Search and Genetic Algorithm –
based Metaheuristics
In Sections 5.6.1, 5.6.2 and 5.6.3 we gave comparative studies between the
performance of standard GA with each of the proposed AGA, HGATS and BGA. In
this section we presented the comparative study between all metaheuristic algorithms
described in this study. Furthermore, comparison was made with previous work
considering single VRPSD under restocking action. From our review, previous work
on metaheuristics for solving single VRPSD under restocking action was done only
by Bianchi et al. (2004). Thus in this study, the relative performance of Genetic
Algorithm-based metaheuristics were compared to the GA’s work of Bianchi et al.
(2004). Descriptive statistics for solutions expected cost of various metaheuristics
implemented in this study were reported in Table 5.11.
154
Table 5.11 : Descriptive statistics for the solutions of metaheuristics implemented
ALGORITHM
Number
of
nodes
Number of
Problem
Instance
Range
Minimum
Maximum
Mean
Std. Deviation
BGA_10
109.00
242.00
351.00
301.5500
34.07341
GA_10
109.00
242.00
351.00
302.6000
34.05735
BIANCHI_10
109.00
242.00
351.00
302.0700
34.69927
TS_10
10
20
115.00
242.00
357.00
305.2500
35.55851
RTS_10
109.00
242.00
351.00
302.3500
35.04174
HGATS_10A
109.00
242.00
351.00
303.1800
34.98414
HGATS_10B
109.00
242.00
351.00
301.4000
33.94795
BGA_20
GA_20
129.00
133.20
327.00
327.00
456.00
460.20
392.5500
395.2800
37.97572
38.62704
BIANCHI_20
131.20
327.00
458.20
394.0300
37.92726
TS_20
140.00
327.00
467.00
401.0000
39.03035
RTS_20
129.00
327.00
456.00
394.1500
37.78788
HGATS_20A
127.00
327.00
454.00
393.8000
38.54130
HGATS_20B
120.00
327.00
447.00
391.4000
37.69741
BGA_50
GA_50
103.00
151.60
554.00
564.60
657.00
716.20
598.0000
620.6700
28.98094
35.12470
110.20
553.80
664.00
602.5000
28.52954
150.00
560.00
710.00
614.0500
34.88398
RTS_50
106.00
548.00
654.00
596.9500
29.49839
HGATS_50A
134.00
560.00
694.00
611.5250
33.92502
HGATS_50B
106.00
548.00
654.00
595.1000
28.91530
20
20
BIANCHI_50
50
TS_50
20
From Table 5.11 the comparative study showed that based on solution
quality, HGATS type B relatively perform the best among others for all number of
nodess, since the mean values of solutions obtained from HGATS were the smallest
among others for all the number of nodes. On the other hand, TS showed weakness
in number of nodes 10 and 20 since the mean values of solutions obtained from TS
were the largest compared to other metaheuristics. GA also performed worst in large
number of nodes (N = 50) since it obtained highest total expected cost. Firstly, the
performance of the metaheuristics in number of nodes 10 was evaluated.
As
mentioned before, HGATS type B produced lowest mean value of cost, followed by
Breeder GA, Bianchi, RTS, GA, HGATS type A and TS which obtain the largest
total expected cost.
Secondly in larger number of nodes that was 20, the
performance of algorithms based on the total expected cost can be ranked from the
best to the worst as HGATS type B, Breeder GA, HGATS type A, Bianchi, RTS, GA
and TS. Finally for number of nodes 50, the HGATS type B still performed best,
155
followed by RTS, BGA, Bianchi, HGATS type A, TS and GA. To verify whether
the differences of solution quality obtained from the metaheuristics are statistically
different, further the paired samples test is used. Before the paired samples tests
were conducted, the assumption of normality test of each pair differences must be
conducted.
The results of Kolmogorov-Smirnov normality test were given in
Appendix A. Table 5.12 gives the summary of paired test of metaheuristics solutions
to test whether the differences of solution quality between two groups are statistically
different. The detail of the paired test results were given in Appendix B.
From Table 5.11, Table 5.12 and Appendix B it can be concluded that the
relative performance of GA is better than TS for number of nodes 10 and 20. The
relative performance of GA, HGATS type A, RTS, and Bianchi are similar for
number of nodes 10 and 20, but GA performs worst for large number of nodes (N =
50) compared to TS, HGATS type A, RTS, and Bianchi.
Among other
metaheuristics, TS seems to perform worst except for N = 50 where TS can perform
better than GA and similar to the HGATS type A. RTS is proven to improve the
basic TS application; it is showed by the relative performance of RTS that was
significantly better than TS. In number of nodes 10 and 20, RTS gives similar
relative performance to GA, HGATS type A, and Bianchi but RTS showed better
performance for large number of nodes (N = 50), since the mean of total expected
cost obtained from RTS was less than GA, HGATS type A and Bianchi. However
RTS can not compete with HGATS type B in terms of solution quality.
The
interesting things are that even the mean values of solutions obtained from RTS are
larger than BGA for number of nodes 10 and 20, and at the contrary for number of
nodes 50 where the mean value of solutions obtained from RTS are smaller than
BGA; the relative performances of RTS are not statistically different from BGA for
all number of nodes.
The BGA shows superiority over GA, TS, Bianchi and HGATS type A for
almost all number of nodess, except for N = 10 where the relative performance of
BGA is similar to Bianchi, and N = 20 where relative performance of BGA is similar
to HGATS type A although for these N = 10 and N = 20 the mean value of cost
obtained from BGA is less than the cost obtained from Bianchi and HGATS type A.
Previous algorithm developed that is GA of Bianchi shows similar performance with
156
GA, RTS and HGATS type A in number of nodes 10 and 20. It is worth noticing
that standard GA developed in this study (that does not involve local search), is able
to compete with GA of Bianchi that involve twice application of Or-opt local search
method that are in improving the initial population and offspring. Probably it caused
by the management of initial population in our standard GA that have more
chromosome number than GA’s Bianchi that is 30 in this study that larger than 10 in
GA’s Bianchi, generated from good heuristics and all individuals must be distinct,
thus with good heuristics solutions involved in our standard GA, the diversity of
initial population is larger than GA of Bianchi but the solution qualities were not
worse than GA’s Bianchi.
In this study, we develop two types of Hybrid GA and TS and it was found
that different implementation of TS in GA can produce significant difference in the
performance of algorithms. The inclusion of TS as mutation operator in GA with
mutation probability of 1 (HGATS type A) results better performance than TS for
10-node and 20-node problems, but it can not give high improvement to GA in
10-node and 20-node problems since the relative performance of HGATS type A,
GA and Bianchi are not significantly different. From Table 5.9, we can see that the
larger number of nodes, the larger improvement can be given by HGATS type A to
GA. In small number of nodes 10 the mean value of GA solutions cost is smaller
than HGATS type A, while in larger number of nodes: 20 and 50, the mean value of
HGATS solutions cost is smaller than solutions obtained from GA. Although in
general HGATS type A is better than TS and GA for all number of nodes and similar
to GA of Bianchi in 10-node problems and 20-node problems, HGATS type A still
can not compete the GA’s Bianchi in large number of nodes.
157
Table 5.12 : Summary of paired test of metaheuristics solution quality differences
ALGORITHM
TS
GA
N
10
10
z
20
type-A
20
z
10
type-B
20
10
z
z
z
10
10
20
50
20
50
10
20
50
z
z
=
z
z
=
z
=
z
=
=
z
=
=
=
z
z
z
z
=
z
50
z
z
=
z
=
z
=
=
z
z
20
z
z
z
50
10
50
=
=
z
20
BIANCHI
20
=
z
=
50
BGA
10
z
z
=
50
HGATS
50
BGA
z
=
50
10
20
HGATS - B
z
=
20
HGATS
10
HGATS - A
z
50
10
50
RTS
z
20
RTS
TS
=
z
=
=
z
(under significance level of 0.05, z refer to the difference of two algorithms performance is
statistically different )
On the other side, the implementation of TS as improvement to the offspring
after mutation applied; highly affect the performance of the algorithm for solving the
problem by giving significant improvement to the solution quality. The HGATS
seems to give best performance among metaheuristics used in this study for all
number of nodes. Even in the time when RTS, BGA and Bianchi can not go further,
the HGATS is still able to find solutions better than or equal to those metaheuristics.
The reason behind this is that in our GA and HGATS; we already start from good
initial solution and by using roulette wheel selection, where the probability of higher
fitness solution to be selected as mating population is larger, the crossover operator
can not give high impact to produce offspring better than their parent. Mutation
plays important role here by helping to escape from this condition. It can help to
z
z
158
prevent the population from stagnating in any local optimum by introducing some
extra variability into the population in order to the search to go further.
The average computation time of the metaheuristics was summarised in
Table 5.13. Computation times in this section were based on laptop machine of Intel
Centrino CoreTM 2 Duo processor with 2.0 GB of RAM. It can be seen that the
weakness of HGATS type B is its longest average computational time among all
metaheuristics. RTS was the fastest algorithm although the computation time grows
significantly for 50-node problems. Among the metaheuristics, RTS was the best
one in combining the advantage of both solution quality and computation time. For
10-node problems, the computation time of TS, GA, BGA and HGATS was not
significantly different, but as the number of nodes increases the differences between
these algorithms computation times was considerably large. For large number of
nodes, GA was the fastest, but as mentioned before, GA has inferior performance in
solution quality if compared to other metaheuristics. If the metaheuristics based on
GA were compared (including GA, BGA, HGATS-A, and HGATS-B), GA was the
fastest, except for 10-node problems.
Compared to HGATS-A, the use of GA
showed 1.4% and 27% average reduction in computational time with no significant
deterioration for problems with 10 and 20 nodes. For 10-node problems, BGA was
the fastest and but the computation time grows very significant as the number of
nodes increases. If computation is not a big concern, the BGA provide significantly
better solution quality than GA. The longest computation time was produced by
HGATS type B where the computation time grows dramatically as the number of
nodes increases. Even for small number of nodes, HGATS-B gives much longer
computation time compared to other metaheuristics. But, the fact that HGATS type
B provide the best quality solution is a promising result for research in VRPSD.
Table 5.13 : Average computation time of the metaheuristics (secs)
Number of nodes
ALGORITHM
10
20
50
TS
32.4
577.2 24,354.3
RTS
27.4
205.6 3,039.5
GA
53.7
247.2 1,635.9
BGA
48.6
373.3 25,945.7
HGATS-A
54.5
342.4 4,580.5
HGATS-B
811.8 17,999.9
*
*more than 8 hours
159
5.7
Conclusion
In this chapter, several metaheuristics based on Tabu Search and Genetic
Algorithm for single VRPSD were developed. It started with the development of
Adaptive GA. The computational result of solutions generated using Adaptive GA
was better than the solution quality generated using a method with fixed parameter.
For each problem set, it returns results that better than all fixed pm values, or the
same as the ones of the most successful GAs with fixed pm settings. By using the
adaptive GA, the amount of time required in finding the appropriate mutation
probability values of GA can be reduced. From the results it can also be deduced
that different Adaptive GAs were better suited to different problem sets. When
compared to other adaptive schemes, the proposed LD has shown its superiority.
Furthermore, two types of HGATS were developed and comparative study between
HGATS and TS and between HGATS and GA were conducted. The result showed
that the proposed HGATS did improve the relative performance of TS and GA alone.
Finally all metaheuristics described in this study were evaluated, and the proposed
HGATS showed superiority compared to other metaheuristics, followed by BGA and
RTS in the second and third best performance respectively. But HGATS type B
showed weakness in producing longest average computation time.
RTS
demonstrated an advantage in combining both of relatively good solution quality and
average computation time. In general, the proposed HGATS, BGA and RTS may
lead to a better performance in terms of solution quality than the previous research
on GA by Bianchi et al. (2004).
160
CHAPTER 6
SOLVING THE BI-OBJECTIVE MULTIPLE VRPSD
VIA BREEDER GENETIC ALGORITHM
6.2
Introduction
In the deterministic VRP, multiple vehicles (routes) are used because the
vehicles have finite capacity. In the stochastic case, Yang et al. (2000) stated that a
single route is proven to be the optimal solution, if there are no additional
constraints. However, in most real world situations, a single route may not be usable
due to several company regulations such as
a. a limit on the expected length (or time) of the route,
b. customers time windows for delivery or pickup,
c. limit on the probability of more than k route failures and or replenishment
trips to a pre-specified level, say .
In this research, multiple VRPSD with the first restriction is considered, that is, the
expected length of each route must not exceed a pre-specified value, say L.
According to Yang et al. (2000), for a single route problem even for moderately
sized problems (ten or more customers), they have to rely on heuristic methods.
Therefore, for the more complex case like solving multiple vehicles (routes) in this
chapter, we focused on the development of metaheuristics, namely genetic algorithm.
Many real world decision making problems involve simultaneous
optimization of multiple objectives (Srinivas and Deb, 1994). But most of the
existing literatures in VRPSD, except for the work of Tan et al. (2007), use singleobjective based heuristics. In a single-objective optimization, the main goal is to find
161
the global optimum solution. However, in a multi-criterion optimization problem,
there are more than one objective function, each of which may have different
individual optimal solution. If there is a sufficient difference in the optimal solutions
corresponding to different objectives, the objective functions are often conflict each
other. Multi-criterion optimization with such conflicting objective functions gives
rise to as set of optimal solutions, instead of a single optimal solution. The reason
for the optimality of many solutions is that none can be considered to be better than
the other with respect to all the objectives. In a typical multi-objective optimization
problem, there exists a set of solutions which are superior to the rest of solutions in
the search space in one or more objectives. These optimal solutions are known as
Pareto-optimal solutions or non-dominated solutions. The rest of the solutions are
dominated solutions.
Since none of the solutions in the non-dominated set is absolutely better than
any other, any solution is an acceptable solution. One way to solve multi-objective
problems is to scalarize the vector of objectives into one objective by averaging the
objectives using a weight vector.
This process allows a simpler optimization
algorithm to be used, but the obtained solution largely depends on the weight vector
used in the scalarization process.
Since genetic algorithm (GA) works with a
population of points, a number of Pareto-optimal solutions may be captured using
GAs.
In this chapter, we propose the use of Breeder Genetic Algorithm (BGA) to
solve the bi-objective VRPSD in minimizing the number of vehicles required and the
total expected cost concurrently, using Pareto-rank by modifying and extending the
single-objective BGA. This approach is motivated by the results in Chapter 5 that
shown superiority of BGA compared to other GA types. Extensive reviews also
found that this is the first time BGA has applied for solving bi-objective VRP
especially on VRP with Stochastic Demand. Section 6.2 introduces the description
of multiple VRPSD. Multiobjective optimization and Pareto definition are described
in Section 6.3 followed by the proposed Breeder GA in Section 6.4 and
computational results in Section 6.5. Finally, a short conclusion is presented.
162
6.3
Problem Formulation of Multiple VRPSD
The mathematical model for the multiple VRPSD is directed into bi-objective
function subject to constrain under the stochastic demands setting. The definitions of
some of the frequently used notations for the multiple VRPSD are given as follows.
x
Customers and depot
V = {0, 1, ..., n} is a set of nodes with node 0 denotes the depot and nodes
1, 2, …, n correspond to the customers to be visited.
x
Demands
Customers have stochastic demands [ i , i = 1, ..., n follow discrete uniform
probability distributions pik
Pr ob([ i
k ) , k = 0, 1, 2, …, K. Assume
further that customers’ demands are independent.
Actual individual
customer demand will only be known when the vehicle arrives at the
customer location.
x
Vehicles and capacity constraint
All vehicles are identical and each one has a capacity limit Q. If the total
customer demand exceeds the vehicle capacity, route failure said to be
occured.
x
Routes
Every vehicle must start at the depot, visit a number of customers and
return to the depot. A feasible solution to the VRPSD is a permutation of
the customers s = (s(1), s(2), . . . , s(n)) starting and ending at the depot (that
is, s(1) = s(n) = 0), and it is called a priori tour.
x
Length constraint
The length constraint is such that the expected length of each route should
not exceed a given limit L.
x
Route failure and recourse action
Route failure is said to be occurred if the total demand exceeds the vehicle
capacity and the preventive restocking policy (Yang et al., 2000, Bianchi et
al., 2004) is employed.
x
Cost and VRPSD bi-objective function
A non-negative matrix C = { cij : i, j  V, i z j} denotes the travel costs
(distances) between node i and j. The cost matrix C is symmetric and
163
satisfies the triangular inequality.
The cost matrix is a function of
Euclidean distance. Let 0 1 2 … j j+1 … n be a priori vehicle
route. The objective value of the planned vehicle route cost is calculated
using a dynamic programming as implemented by Yang et al. (2000) and
Bianchi et al. (2004). The multiple VRPSD, therefore, involves a solution
that minimizes the bi-objective of the total expected cost and number of
vehicles m.
6.4
Multi-objective Optimization
Multi-objective optimization (also known as multi-criteria optimization,
multi-performance or vector optimization) can be defined as:
A vector of decision variables which satisfies constraints and optimizes a
vector function whose elements represent the objective functions. These
functions form a mathematical description of performance criteria which are
usually in conflict each other. Hence, the term ‘optimize’ means finding such
a solution which would give the values of the entire objective functions
acceptable to the designer (Coello, 1999).
In minimization problem, a multi-objective optimization problem can be formulated
as follows:
Find the vector x = >x1 , x 2 ,..., x n @ which minimizes the vector function
T
f (x) = [f1(x),f2(x),..,fk(x)]T, x  S
(6.1)
where k is the number of criteria, x = >x1 , x 2 ,..., x n @ is the vector of decision
T
variables, f is objective vector and S is a set of feasible solutions.
In other words, we wish to determine among the set of all members, the particular set
x1 , x 2 ,..., x n which yields the optimum values of all the objective functions.
164
Zitzler et al. (2000) assumed, without loss of generality, the following
definition in a minimization problem.
Definition 1 (Pareto Dominance)
A vector a = ( a1 , a 2 ,..., a k ) is said to dominate b = ( b1 , b2 ,..., bk ), also written as
(a % b) if and only if a is partially less than b, i.e.,
i  {1,..., k} , f i (a)% f i (b) š j  {1,..., k} : f j (a) % f j (b)
Additionaly, a covers b ( a%b ) if and only if a % b or f(a) = f(b)
Definition 2 (Non-dominated and Pareto–optimal)
1. The vector a is said to be non-dominated regarding a set S’ Ž S if and only if
there is no vector in S’ which dominates a;
2. The vector a is Pareto-optimal if and only if a is non-dominated regarding S.
Pareto-optimal vectors cannot be improved in any objective without causing
degradation in at least one other objective; as they represent globally optimal
solutions. However, analogous to the single-objective optimization problems, there
may also be local optima which constitute a non-dominated set within a certain
neighbourhood.
Definition 3 (Local and Global Pareto-optimal set)
Consider a set of vectors S’ Ž S.
1. The set S’ is denoted as a local Pareto-optimal set if and only if
a’  S '
a % a’ š
there exist no solution a  S :
a-a’ < H š
f(a)-f(a’) < G
where . is a corresponding distance metric and H ! 0, G ! 0
2. The set S’ is called a global Pareto-optimal set if and only if
a ' S ': there exist no solution a  S : a % a’
165
Note that a global Pareto-optimal set does not necessarily contain all Pareto-optimal
solutions.
Term ‘Pareto-optimal set’ refer to the entirety of the Pareto-optimal
solutions and the corresponding set of objective vectors is denoted as ‘Pareto-optimal
front’.
6.5
Breeder Genetic Algorithm (BGA) for Bi-objective VRPSD
The BGA is inspired by the science of breeding animals. In this algorithm,
each one of a set of virtual breeder has the task to improve its own subpopulation.
Occasionally the breeder imports individuals from neighboring subpopulations. The
component of the BGA may be described as follows. Firstly, a genetic encoding of
the VRPSD, initialization and evaluation of fitness function are made. After that,
improve the solutions using inter and intra route exchange based on OrOpt local
search. Finally the crossover and mutation are performed in addition to improving
the offspring by local search.
The proposed BGA for bi-objective VRPSD is
presented in the following steps.
Step 0. [Define] Define operator settings of GA suitable with the problem which is
VRPSD.
Step 1. [Initialization] Create an initial population P of PopSize chromosomes that
consists of randomized farther insertion solutions where all chromosomes
(individuals) are distinct or clones are forbidden.
Step 2. [Improvement] Apply improvement method by using inter-route exchange
and intra-route exchange based on OrOpt local search for each individual.
Step 3. [Objective] Compute individual’s objective vector
Step 4. [Pareto-rank] Pareto-rank the population
Step 5. [Selection] Select T % best individual in population to be parents for
mating, set this set as TruncPop(t), in this study T is in the range of 35 –
50%.
Step 6. [Crossover] Pair all the chromosomes in TruncPop(t) at random forming
pairs.
Apply crossover with probability pc to each pair and produce
166
(T%).(PopSize) offspring, if random number t pc then offspring is the
exact copy of parents.
Step 7. [Mutation] With a mutation probability pm mutate the offspring.
Step 8. [Improvement] Apply improvement method by using local search to the
offspring.
Step 9. [Objective] Compute objective vectors of offspring
Step 10. [Pareto-rank] Pareto-rank the population including offspring
Step 11. [New Population] Remove the (T%).(PopSize) least fit individual based on
Pareto-rank to keep size PopSize. Form new population P(t+1)
Step 12. [Test/ Stopping Criterion] If the stopping criterion is met then stop, and
return to the best solution in current population, else go to Step 13.
Step 13. [Acceptance Criterion] Check whether new population is better than
acceptance criterion, if yes, go to Step 2, if no then go to Step 5.
6.4.1
Chromosome Representation
In the case of multiple vehicles, the genetic material of an individual contains
several routes (the number of necessary vehicles), each of them composed by an
ordered subset of the customers visited. In every route there must be at least one
customer.
The chromosome representation follows variable-length chromosome
representation proposed by Tan et al. (2007) as depicted in Figure 6.1.
Such
representation is efficient and allows the number of vehicles to be manipulated and
minimized directly for the bi-objective optimization for VRPSD.
6.4.2
Initialization
Firstly, the initial giant routes through all customers are generated from
Randomized Farthest Insertion, Randomized Nearest Neighbour and randomly
generated solutions. All individual must be distinct to ensure the better dispersal of
population. Next, partitions each giant route into smaller routes such that each route
167
satisfies the vehicle capacity. The chromosome is built such that the sum of the
mean values of the customer demands on each route does not exceed the vehicle
capacity. According to preliminary study on the effect of population size to the
performance of GA in VRPSD discussed in Section 5.6.1, we see that the use of
population size 30, 50 and 100 were not give significant differences to the
performance of GA in VRPSD for 10-node, 20-node and 50-node problems. In
Chapter 5, we consider several GA-based metaheuristics for single vehicle for 10node, 20-node and 50-node problems, while in this chapter we consider multiple
vehicles problems with larger number of nodes than ones used in Chapter 5. Baker
and Ayechew (2003) use population size of 30 for 50-node problems and population
size of 50 for larger sizes of problem. Thus in this Chapter, we set the population
size to be 30 for 50-node problems and the population size of 50 for 75-node
problems.
Individual
0
0
0
0
1
2
7
6
4
9
0
5
0
3
0
8
0
Figure 6.1 : The data structure of chromosome representation
168
6.4.3
Improvement Method
The improvement method consists of two parts: the Inter-route exchange and
the Intra-route exchange that are applied to each individual.
Generally these
improvement methods are based on Or-opt local search.
(i).
Inter-route exchange
The purpose of inter-route exchange is to improve the route by moving a
route segment, s, from one route to another route within one individual.
The
procedure pair wise improves the routes for all possible pairs. For each pair of
routes, say route A and route B, the process proceeds as follows:
1. Start from s = 3.
2. Make the most improving move by moving s successive nodes from one
to the other route (either from A to B or from B to A). Repeat the
improvement on the exchanged routes.
3. If no further improvement can be made, decrease s by 1, and repeat the
above step again. If s = 0, then stop.
(ii).
Intra-route exchange
The inter-route exchange method is followed by intra-route exchange. The
intra-route exchange improvement method is Or-opt.
6.4.4
Objective Vectors
The bi-objective vectors consist of minimizing two components in the
objective function, namely the number of vehicles and the total expected cost. The
fitness value for each component is the objective function. High fitness is denoted
by low number of vehicles or low total expected cost.
169
6.4.5
Pareto-rank
Fonseca and Fleming (1993) proposed a scheme in which the rank of a
certain individual corresponds to the number of chromosomes in the current
population by which it is dominated. Consider, for example, an individual xi at
generation t, which is dominated by pi
(t )
individuals in the current generation. Its
current position in the individuals rank can be given by
rank ( xi , t ) 1 pi
where pi
(t )
(t )
(6.2)
denotes the number of individuals that dominate an individual xi at
generation t.
The whole population is checked and all non-dominated individuals are
assigned rank ‘1’. Other individuals (dominated ones) are ranked by checking the
non-dominance of them with respect to the rest of the population in the following
way. For an individual point, the number of points that strictly dominate the point in
the population is first found. Thereafter, the rank of that individual is assigned to be
one more than that number. Therefore, at the end of this ranking procedure, there
could be some points having the same rank. The selection procedure then uses these
ranks to select or delete blocks of points to form the mating pool.
6.4.6
Selection
Let M(t) = ( M f 1 (t ), M f 2 (t ) ) denote the vector of bi-objective mean fitness of
the population at time t. The change in fitness caused by the selection is given by
R (t )
M (t ) M (t 1)
(6.3)
170
and is known as response to selection. R(t) measures the expected progress of the
population. Breeder measure the selection with the selected differential, which is
symbolised by S(t). It is defined by the difference between the mean fitness of the
selected individuals, Ms(t) and the population mean:
S (t )
M s (t ) M (t )
(6.4)
These two definitions (Equation 6.3 and Equation 6.4) are very important. They
quantify the most important measurement in the process of artificial breeding. Both
R(t) and S(t) can be easily computed. The breeder tries to predict R(t) from S(t).
Breeder often uses truncation selection. In truncation selection with threshold T, the
T % best individuals will be selected as parents.
6.4.7
Crossover and Mutation
In this study, three types of crossover were considered. In the first crossover,
we modified the route-exchange Crossover proposed by Tan et al. (2007), the second
crossover is PMX and the third crossover is OX. According to Tan et al. (2007), the
route-exchange crossover allows good sequences of routes in a chromosome to be
shared with other chromosomes. In the route-exchange crossover, only the best
routes of the selected chromosome are eligible for exchange. In the case where one
of the selected chromosomes has only one route, a segment of the route is randomly
selected to exchange with the other chromosome’s best route which will be inserted
as a new route in the first chromosome. To ensure the feasibility of chromosome
after the crossover, duplicated customers are deleted. These customers are deleted
from the original routes while the newly inserted route is left intact. Then a random
shuffling is applied to increase the diversity of chromosome. In the random shuffling
operation, with the exception of the newly inserted route, the order of customers in
each of the remaining routes will be shuffled with a probability equal to the shuffle
rate. The Route-exchange crossover can be described in Figure 6.2.
171
The route-exchange crossover is then followed by swap mutation. Swap
mutation operator reallocate two nodes, say i and j, which are currently on different
routes. Customer i is removed from its route and inserted to a position on the route
from which j is removed. Customer j is moved to a position on the route formerly
containing i. All pairs of customers are considered for this operation within iteration,
and the best move is executed.
The disadvantage of route-exchange proposed by Tan et al. (2007) is that the
offspring produced from crossover operation always have larger number of vehicles.
This condition can be seen in Figure 6.2 where the crossover operation builds one
more route from the parent. If the parents have each 3 and 4 routes, crossover
produces 4 and 5 routes for the offspring. Thus we modify the route exchange
crossover by adding decrease vehicle number operator. This operator works first
with randomly choose a route and each one node from that route is moved to the end
of other routes. The reason for this modification is that it gives us flexibility to
change the nodes to possibly decrease the number of routes. If all nodes from the
route can be moved, then the number of routes can be reduced. In this study, the
routes to be exchanged were randomly chosen. In mutation process, we randomly
choose pairs of customers rather than check all pairs of customer.
172
Figure 6.2. Illustration of route-exchange crossover (source: Tan et al., 2007)
173
6.4.8
Stopping Criterion
In using the optimization algorithms, the goal is to search for the global
optimum. However, in general it is not clear when the goal is achieved especially in
our application to the real world situation. Therefore, it is sometimes not that easy to
decide when execution of an optimization algorithm should be terminated. There are
many different mechanisms that may be used for the detection of an appropriate time
for ending an optimization run. One of the commonly used mechanisms is the
algorithm is executed until there is no improvement in fitness value for certain
generations. In this study, the BBGA is repeated until there is no improvement in
fitness value for 500 generations.
6.4.9
Acceptance Criterion
The relationship between R(t) and S(t) is given by the equation
R (t )
bt .S (t )
(6.5)
where bt in the quantitative genetics is known as the realized heritability. It is
normally assumed that bt is constant for a number of generations. This leads to
R(t )
b.S (t )
(6.6)
In general, the progress due to a genetic search as long as the ratio R(t)/S(t) is greater
than 0.1 (Crisan and Muhlenbein, 1998). In bi-objective, we extend this criteria in
such a way that for every objective, the ratio R(t)/S(t) is greater than or equal to 0.1
and there exist R(t)/S(t) > 0.1 for at least one objective.
The structure of Breeder Genetic Algorithm for bi-objective optimization is
depicted in Figure 6.3.
174
START
Generate Initial Population Pop(0) of size PopSize
Set iteration t = 1
Apply Inter-route and Intra-route exchange
based on OrOpt for each individual, set as Pop(t)
Calculate mean fitness of population Pop(t), M(t)
Compute individual’s objective vector
Pareto-rank the population
TRUNCATE POPULATION
Select 35% to 50% best individuals of population as parent for mating
Calculate mean fitness of selected population TruncPop(t), Ms(t)
Calculate R(t) = M(t) - M(t-1)
Calculate S(t) = Ms(t) - M(t)
Crossover the TruncPop(t)
Mutation
t = t+1
t = t+1
Improvement
Compute objective vector of offspring
No
Pareto-rank offspring and parent
Yes
Form new population Pop(t+1)
R(t )
! 0.1?
S (t )
No
Stopping criteria met?
Yes
Choose the best individual
END
Figure 6.3 : Structure of Breeder Genetic Algorithm for bi-objective optimization
175
6.6
Computational Results
The subsequent sections present the extensive simulation results and the data
analysis. Section 6.5.1 demonstrates how the three crossover types will affect the
performance of the algorithm. In Section 6.5.2 the effectiveness of the proposed
BGA is analyzed and then compared to the Pareto GA. Three types of Pareto
Genetic Algorithm – based metaheuristics were compared:
1. Pareto GA
: Genetic Algorithm
2. Pareto BGA1
: Breeder Genetic Algorithm without local search
improvement on offspring.
3. Pareto BGA2
: Breeder Genetic Algorithm with local search
improvement on offspring.
The relative performances of metaheuristics for bi-objective optimization problem in
VRPSD were tested on two problem size: 50 and 75. The metaheuristics were
executed 20 times on each instance. Each simulation run was carried out using the
parameters setting chosen after some preliminary experiments as shown in Table 6.1.
Table 6.1 : Paramaters setting in bi-objective VRPSD
Number
of
non
improving 500
successive generations
Population size
30 for 50-nodes and 50 for 75-nodes
Crossover rate
0.8
Mutation rate
0.05
Capacity of vehicle (Q)
100
Length limit (T)
200, 400
Truncation percentage
Randomly selected within interval 0.35 to 0.5
Initial population
Randomized NN + Randomized FI + random
solution
Additional cost for route failure
50
Crossover type
1. OX
2. Modified Route-exchange crossover
3. PMX
176
6.4.1
Effect of different crossover types
The performance of different crossover types was evaluated for the proposed
BGA type 2 on number of customer 50 and T = 400. From all runs of the algorithms
on this problem size and length limit setting, the number of vehicles required was the
same (i.e. equal to 3). Thus we only reported the performance of solution quality
based on the total expected cost. The summary of descriptive measures of the result
was reported in Table 6.2.
Table 6.2 : Descriptive statistics for the solutions from different crossover types
Descriptive Statistics
Number of
Runs
Minimum
Maximum
Mean
Std.
Deviation
OX
20
Statistic
783.49
Statistic
785.87
Statistic
784.6375
Std. Error
.26243
Mod_RE
20
783.49
792.27
787.8585
.66910
2.99229
PMX
20
783.49
792.27
785.2381
.48601
2.17352
1.17361
From Table 6.2, it shows that the choice of crossover can give different effect
to the relative performance of the algorithm. The OX crossover seems to give the
best relative performance compared to other crossovers, since OX can obtain
minimum average of total expected cost. It was followed by PMX and the worst
performance is produced by Modified Route Exchange (RE) crossover. Even though
we have enhanced the performance of Route Exchange crossover of Tan et al (2007)
by using decrease vehicle number operator as described in Section 6.4.7, the solution
quality of RE is less than solution obtained by OX and PMX crossovers. OX give
best solution quality in lowest total expected cost, but the difference between OX
and PMX implementation is relatively low.
To verify whether the differences of relative performance of crossovers are
statistically significant, the independent samples test was used. Firstly we tested the
normality assumption where the results of Kolmogorov-Smirnov normality test were
given in Table 6.3. Since all the p-values against the null hypothesis “The data
177
follow normal distribution” were bigger than significance level 0.05, that were 0.055,
0.174 and 0.076 OX, Modified RE and PMX respectively; thus we can not reject the
null hypothesis.
The data follows normal distribution and further we can use
independent samples t-test.
Table 6.3 : Normality test results of the differences between three types of
crossover
One-Sample Kolmogorov-Smirnov Test
OX
Number of runs
Normal Parameters
Mod_RE
20
PMX
20
784.6375
787.8585
785.2381
1.17361
2.99229
2.17352
.300
.247
.286
20
a,b
Mean
Std. Deviation
Most Extreme
Differences
Absolute
Positive
.296
.247
.286
Negative
-.300
-.150
-.211
1.341
1.105
1.279
.055
.174
.076
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
The results of independent t-test were presented in Table 6.4. There was a
statistically significant difference between the relative performance of Modified RE
with both PMX and OX since the p-values of the null hypothesis “There is no mean
difference between two groups” were less than 0.05, which were 0.003 and 0.003 for
pair RE-PMX and pair RE-OX. The extent of this difference was shown clearly by
the confidence interval of the difference which not encompasses zero.
From
associated Table it can also be drawn that although the OX gave better quality in less
total expected cost than PMX, the effect of PMX and OX was not statistically
significantly different.
178
Table 6.4 : Independent samples test for crossover performance
Levene's Test
for Equality of
Variances
F
OX_
ModRE
OX_
PMX
ModRE
_PMX
6.5.2
Equal
variances
assumed
Equal
variances
not
assumed
Equal
variances
assumed
Equal
variances
not
assumed
Equal
variances
assumed
Equal
variances
not
assumed
38.895
1.811
7.666
Sig.
.000
.186
.009
t-test for Equality of Means
t
Sig. (2tailed)
df
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-4.481
38
.000
-3.22094
.71872
-4.67591
-1.76597
-4.481
24.710
.000
-3.22094
.71872
-4.70205
-1.73983
-1.087
38
.284
-.60057
.55234
-1.71872
.51758
-1.087
29.211
.286
-.60057
.55234
-1.72987
.52873
3.169
38
.003
2.62037
.82698
.94623
4.29451
3.169
34.684
.003
2.62037
.82698
.94096
4.29978
Performance of Bi-objective Breeder GA for VRPSD
It was previously commented in Section 6.5.1 that OX crossover produces the
best solutions among the three crossover types. Thus the OX crossover was used in
the performance evaluation and comparison between the proposed BBGA with
Pareto GA. Different with the number of customer to be examined in single vehicle
case as investigated in previous chapter. In this performance comparison, larger
numbers of customer are used that are 50 and 75, since in this chapter the multiple
vehicles are used. In Table 6.5 and Table 6.6, the total expected cost and expected
number of vehicles reported were the respective cost and number of vehicles over the
non-dominated solutions in the population over the 20 simulation runs. As we can
see from Table 6.5, BBGA2 produces the best performance with the smallest product
of total expected cost. From Table 6.6, we can also see that Pareto GA can not
compete with BBGA1 and BBGA2 in terms of minimization of vehicles number
179
required, while the relative performance of BBGA1 and BBGA2 were similar. The
BBGA2, augmented with the additional improvement to the offspring, was capable
to improve the solution quality of BBGA1.
Table 6.5 : Comparison of total expected cost between BBGA and Pareto GA
Expected total cost
Number
of
nodes
Length
limit
Measure
50
200
Min
816.26
788.31
791.41
Mean
838.97
794.25
792.13
Max
856.54
798.67
792.26
Min
815.35
783.49
783.49
Mean
845.59
790.72
784.64
Max
879.50
806.29
785.87
Min
1137.74
1052.97
1044.14
Mean
1159.11
1089.42
1073.94
Max
1189.59
1109.97
1122.98
Min
1149.26
1090.39
1082.78
Mean
1166.88
1110.10
1094.77
Max
1169.80
1140.59
1109.81
400
75
200
400
Pareto GA
BBGA1
BBGA2
Table 6.6 : Comparison of vehicles number between BBGA and Pareto GA
Problem
Length
Measure
Expected number of vehicles
Pareto GA
BBGA1
BBGA2
size
limit
50
200
Min
5
5
5
Mean
5
5
5
Max
5
5
5
400
Min
3
3
3
Mean
3
3
3
Max
3
3
3
75
200
Min
7
6
6
Mean
7.00
6.60
6.75
Max
7
7
7
400
Min
4
3
3
Mean
4
3
3
Max
4
3
3
180
The comparison between the Breeder GA and Pareto GA were performed
using the multiplicative aggregation method (Van Veldhuizen and Lamont, 1998;
Tan et al., 2007) of the total expected cost and number of vehicles required.
Expected cost and expected number of vehicles are the respective average over the
non-dominated solutions in the archive population found at the termination of the
algorithm over the 20 runs. The multiplicative aggregation as the multiplication
product of the total expected cost and number of vehicles required were presented in
Table 6.7. Taking two objectives into consideration, Table 6.7 showed that BBGA
has distinctly lower multiplicative aggregate in comparison with Pareto GA, and the
BBGA2 seems to give best performance.
Table 6.7 : Comparison between multiplicative aggregation
Problem size
Length limit
50
200
400
200
400
75
6.7
Expected multiplicative aggregate
Pareto GA
BBGA1
BBGA2
4194.85
2536.77
8113.77
4667.52
3971.25
2372.16
7190.17
3330.3
3960.65
2353.92
7249.09
3284.31
Conclusion
A bi-objective optimization VRPSD, that involves the optimization of routes
for multiple vehicles to minimize bi-objectives of the total expected cost and the
number of vehicles required, has been considered. The bi-objective Breeder GA
(BBGA) was featured with good initial population, enhanced genetic operators,
progressive breeding process, Pareto fitness ranking, and improvement of the parents
and offspring. The effect of different crossover types to the relative performance of
the proposed BBGA has been also analyzed. Experimental results showed that OX
and PMX crossover were superior to Route Exchange crossover.
Furthermore,
a comparative study of several algorithms for solving bi-objective VRPSD is
conducted. The Breeder GA offers an improvement in the quality of solution in
comparison to Pareto GA.
181
CHAPTER 7
CASE STUDY: SOLID WASTE COLLECTION
7.1
Introduction
VRP with Stochastic Demands finds its application on wide range of logistics
and distribution sectors in cases where it is impossible to know the customers
demands before the vehicles arrives at customer’s location. This chapter discusses
the implementation solutions using VRPSD models and metaheuristics algorithms
described in previous chapters. A case study was conducted at Perniagaan Zawiyah
Sdn. Bhd. (PZSB), a company responsible for solid waste management for the
municipality of the City of Johor Bahru. The problem of solid waste collection is
modeled as VRPSD. It is of course impossible to know a priori the demands (how
much garbage has to be collected) at each customer location, but the parameters can
be estimated using appropriate probability distribution based on previous data
collected. It is highly expected that taking into account the stochasticity nature of a
real world situation may produce better routes thus will reduce cost. This chapter
begins with an overview of the practices of solid waste collection at PZSB given in
Section 7.2 followed by the data collection process in Section 7.3. The data analysis
is presented in Section 7.4. Finally a short conclusion is given in Section 7.5.
182
7.2
Overview of Solid Waste Company
Solid waste is an element of environmental pollution, which, in turn, is the
contamination of air, water and soil by materials that interfere with human health and
the quality of life and nature. In every country, most residents of course consider
solid waste collection as a required service, that’s why there are many solid waste
collection companies that compete based on price and quality service, this condition
also occurs in Malaysia.
One of waste collection company in Malaysia is the
Perniagaan Zawiyah Sdn. Bhd. which is located on:
-
Office
: 46-02, Padi Mas 4/1 Bandar Baru Uda
-
Workshop and depot
: Jalan Tahana, Tampoi Industrial Area
On their duties, PZSB is funded by the municipal authority of Johor Bahru
(Majlis Bandaraya Johor Bahru -MBJB) or became a sub contractor for other
companies under agreement for usually 1 year contract. Its operation covers area
within the boundary of the municipality, namely:
a.
Kawasan Pekan Nenas (under Southern Waste Management Sdn. Bhd. as
main contractor).
b.
Kawasan Baru CFO Zon Tengah (under Majlis Bandaraya Johor Bahru).
c.
Kawasan Kuarters Guru Jalan Noja, Kondominium Datin Halimah, Jalan
Nibong 14, Taman Setia Indah, Taman JP Perdana, Taman Mount Austin,
Taman Austin Perdana and Taman Austin Height (under Southern Waste
Management Sdn. Bhd. as main contractor).
d.
Kawasan Zon 1A Pasir Gudang: Taman Cendana, Kompleks Pusat Bandar,
Taman Mawar, Taman Bukit Dahlia and Kg. Pasir Gudang Baru (under Pihak
Berkuasa Tempatan Pasir Gudang).
The company is in charge of collecting waste in two major sectors:
1.
Commercial and Residential sectors
Commercial customers include restaurant and traditional market, while
residential customers are generally people living in private houses or
government quarters.
2.
Roll-on-roll-off/ Industrial
183
Industrial routes are different from residential and commercial sectors have
different size containers. Truck can only handle one industrial container at a
time and commonly haul this container to the dumpsite, empty it and have it
back to the original customer’s location.
7.3
The Data Collection Process
This research is confined to solid waste collection problem on residential
area. Each residential route may serve approximately 1000 unit of houses per day
per truck. Residents of each house store their garbage in city provided cart or box
which is usually placed in front of their house along the alley for collection. Each
truck is assigned a four-person crew: one (1) driver and three (3) solid waste
collectors (or garbage collectors). Usually all trucks start their operation at 7am
from the depot, after collecting the waste they empty the truck at the dumpsite and
return to the depot. The dumpsites are located at three different locations namely the
Seelong dumpsite, Pasir Gudang dumpsite and Pekan Nenas dumpsite. A driver and
collectors may visit the same customer multiple times in a week. Customers are
commonly serviced three times a week. The weekly schedule is fairly static; a stable
set of customers is served on a regular basis (three times a week) as most customers
seldom change the frequency of service.
At the start of each day, each truck is provided with customers list. Usually
the working period in Malaysia is generally 8am to 5pm hours, while at PZSB, the
workers start their activity at 7am. The trucks crews are obligated to depart at
approximately 7am and empty the truck load at dumpsite before returning to the
depot. The dumpsites are opened for 24 hours, but for subcontractor companies are
not the owner of the dumpsites, therefore the activity of emptying the waste must be
done before 7pm. If the waste collection’s trucks arrived at the dumpsite after 7pm,
they must fill in form of reason why they were late to empty the waste. There is no
time constraint such as: each truck crew is directed to collect solid waste for a fixed
amount of time before going to the dumpsite. Although there is no hard time
windows of collection time, every truck crew aware that they should finish the route
184
collection as fast as possible, since the company has a regulation that overtime cost is
only given at the starting day of the week for regular basis route (for example on
Monday for Monday-Wednesday-Friday collection days and Tuesday for TuesdayThursday-Saturday collection days) in flat rate a day for approximately RM 10 per
truck crew (according to the company information), not a multiplicative of the
number of overtime hour.
In some developed countries, certain government has specific regulation
about the way garbage is collected. In residential routes, the waste companies are
permitted to serve only customers on the right side of the street (Figure 7.1). This
condition requires arc-routing solutions. But the situation is different in developing
country like Malaysia, exceptions are granted for this regulation especially for alleys
and one-way streets. In an alley, wastes in right and left side of an alley are picked
up simultaneously. Thus, the way for solving the residential routing in this problem
is different, point-to-point or node-to-node solutions are more acceptable.
Street
a. Arc routing problem
Street
b. Node-to-node routing problem
Figure 7.1 : Arc routing and node-to-node routing
The PZSB Company divided the operation areas into several routes where
each route consists of several cluster. For example in Taman Setia Indah residential
area, it divided into 12 Precincts as presented in Figure 7.2. Due to our limitation,
this study only considers the area of observation on one vehicle route that commonly
covers three clusters namely Pasar Awam Setia Indah (Precint 1), Taman Setia Indah
6 (Precinct 5) and Taman Setia Indah 7 (Precint 7). On any given service day, there
is a possibility that the vehicle covers additional zone, for example on Kempas Indah
area, or in the other day the vehicle may visit another collection area, according to on
call request of the resident management to the company. This routing problem could
be regarded as Vehicle Routing Problem with Stochastic Customers, where each
185
customer is present with probability pi . In this study we only consider Vehicle
Routing Problem with Stochastic Demands where customer is deterministic but the
demand is stochastic. Thus the data analysis is conducted on commonly fixed visited
clusters namely Pasar Awam Setia Indah (Precinct 1), Taman Setia Indah 6 (Precinct
5) and Taman Setia Indah 7 (Precinct 7) as mentioned before.
Pasar
Awam
Figure 7.2 : Map of Taman Setia Indah
186
The collection regular days are Tuesday, Thursday and Saturday for every
week with the dumpsite location at Seelong. The data are collected for six (6) days,
twice for each regular day. These are 23 and 30 August 2007 for Thursday, 25
August and 1 September 2007 for Saturday and 28 August and 20 November 2007
for Tuesday. Variables to be observed are demand (amount of solid waste).
Taman Setia Indah 6 consists of 24 alleys whereas Taman Setia Indah 7
consists of 14 alleys. Detail of number of houses to be served at each alley is given
in Table 7.1.
Table 7.1 : Number of houses on route
No
Street
1
2
3
4
5
6
7
8
9
10
11
12
13
Setia 6/2
Setia 6/3
Setia 6/4
Setia 6/5
Setia 6/6
Setia 6/7
Setia 6/8
Setia 6/9
Setia 6/10
Setia 6/11
Setia 6/12
Setia 6/13
Setia 6/14
Total
No. of
houses
31
30
25
37
20
30
22
35
33
20
25
30
74
412
Street
Setia 6/18
Setia 6/19
Setia 6/20
Setia 6/21
Setia 6/22
Setia 6/23
Setia 6/24
Setia 6/27
Setia 6/28
Setia 6/29
Setia 6/30
Setia 7/2
Setia 7/3
No. of
houses
56
28
56
28
28
21
21
56
32
32
56
25
49
488
Street
Setia 7/4
Setia 7/5
Setia 7/6
Setia 7/7
Setia 7/8
Setia 7/9
Setia 7/10
Setia 7/11
Setia 7/12
Setia 7/15
Setia 7/16
Setia 7/17
Pasar Awam
No. of
houses
36
30
30
30
15
24
48
48
25
19
41
20
1
366
The portion of detail maps of the area at Taman Setia Indah 6 and Taman
Setia Indah 7 is given in Figure 7.3. Since the total of customer number is huge, we
consider the alleys as the nodes (customers) on a network rather than houses; the
demand is the total amount of houses waste in each alley. We model the demand
(weight of solid waste collected) as random variables whose parameters are
estimated from previous knowledge about the data. The objective of this statistical
model is to minimize the total expected cost.
187
Jalan Setia 6
Jalan Setia 7
Figure 7.3: Portion of maps of Jalan Setia 6 and Jalan Setia 7
7.4
Data Analysis
7.4.1
Daily Demand Characteristics
In the application of vehicle routing problem with Stochastic Demands in
case of solid waste collection, the objective is to find a priori route (route that has
been planned before the trip to the customers made but using previous knowledge
about the demand characteristics) that minimise the total expected total cost. The
demands (weight of solid waste collected) is unknown before the vehicles arrive at
customer location but the demands can be modeled as random variables whose
parameters are estimated from previous demands data.
The information about
demands characteristics including its parameter estimation is used to build the routes.
The objective of this statistical model is to minimize the total expected cost.
188
The total amount of waste collected by each truck can be obtained from
dumpsites. Each dumpsite stores detailed data regarding the arrival time of the truck
to the dumpsite, the weight of garbage deposited by the truck, and the time at which
the truck departs from the dumpsite. In this section we confine our discussion on the
demand data.
Usually the first step to be done in data analysis is computing
descriptive statistics as presented in Table 7.2.
Table 7.2 : Descriptive statistics of total daily collection
Descriptive Statistics
Tuesday
Frequency
2
Minimum
6579
Maximum
7529
Mean
7054.00
Std. Deviation
671.751
Thursday
2
4003
4537
4270.00
377.595
Saturday
2
4409
4723
4566.00
222.032
Valid N (listwise)
2
It was shown that the largest amount of demand (waste weight) is usually
occurred in Tuesday with the average of 7,054 kg. The average waste collected on
Tuesday is almost twice of the average demand during the rest of the week. This
phenomenon is reasonable since the waste to be collected on Tuesday is stacked for
three (3) days from Saturday after the waste is picked until Tuesday before the waste
is picked up again, while for the other days the waiting time of waste to be picked up
is two (2) days. Minimum amount of solid waste to be collected is usually occurred
on Thursday. The average solid waste load per day for the vehicle is approximately
5296.67 kg. Regarding to the truck capacity that can be planned is 6 tons, indicating
an average capacity utilization of 88.28%.
189
Boxplot of Demand by Day
8000
Demand
7000
6000
5000
4000
Saturday
Thursday
Day
Tuesday
Figure 7.4 : Box and whisker plots of total demand by day
Box and whisker plot of demand in Figure 7.4 showed the same
information. From box plot we can also see that although the mean of demand on
Saturday is higher than Thursday, the mean of these two days can be regarded as the
same since the boxes are still overlapped, whereas the mean of Tuesday demand is
different from the others. To convince more, one-way ANOVA was conducted in
order to test null hypothesis that the mean of demand on Tuesday, Thursday and
Saturday are the same. Testing the hypothesis at 95% confidence level, the p-value
(0.016) is less than D (= 0.05). Thus the null hypothesis is rejected. Note that
confidence interval for mean of demand on Saturday and Thursday are overlapping,
it means that the mean of demand on Saturday and Thursday is the same. Tuesday
demand is different with the others since it was not overlapped with other confidence
intervals as shown in Figure 7.5.
190
One-way ANOVA: Total Demand versus Day
Source
Day
Error
Total
DF
2
3
5
S = 463.0
Level
Saturday
Thursday
Tuesday
N
2
2
2
Mean
4566.0
4270.0
7054.0
SS
9352277
643126
9995403
MS
4676139
214375
R-Sq = 93.57%
StDev
222.0
377.6
671.8
F
21.81
P
0.016
R-Sq(adj) = 89.28%
Individual 95% CIs For Mean Based on
Pooled StDev
--------+---------+---------+---------+(------*------)
(-----*------)
(------*------)
--------+---------+---------+---------+4500
6000
7500
9000
Pooled StDev = 463.0
Figure 7.5 : One-way ANOVA of total demand versus day
7.4.2
Route Construction
In this section we reported the results obtained from applying the
methodology described in the Chapter 4 and Chapter 5 to solid waste collection area
given in Section 7.4.1. We also compare routes that are generated with the one that
were previously implemented by the company.
So far, the workers used a
combination of experience and intuition to do the task. In areas described above,
Setia Indah is divided into several clusters (Precincts) and each precinct is divided
into several blocks. The blocks are laid out on a grid. Thus a truck follows a long
snake-like route traveling from the alley of one block to the alley of an adjacent
block. The grid structure of the blocks makes it easy for the company to form routes
so that wasted distance and travel time between blocks is kept small.
The flow of route construction is as follows. The problem of waste collection
is transformed to a graph G = (V, A, C), where
V = {0, 1,…, n} is a set of nodes with node 0 denotes the depot and nodes
1, 2, …, n correspond to the nodes,
191
A = {(i, j ) : i, j V , i z j} is the set of arcs joining the nodes, and
C = ( cij : i, j  V , i z j} ) is a non-negative matrix (a matrix where all the
elements are equal to or above zero, C t 0, ij cij t 0 ) that denotes the travel cost
(distance) between nodes i and j.
The customer location coordinates then defined on the graph. Since the total of
customer number is huge, we consider the alleys as the nodes (customers) on a
network rather than houses; the demand is the total amount of houses waste in each
alley.
Firstly we optimized the route of clusters (Precincts) and then optimize the
route on each cluster. The result then compared to existing route implemented by the
company. Since the number of clusters to be optimized is small (equal to 3) thus the
performance of all metaheuristics namely TS, RTS, GA, BGA, HGATS type A and
HGAATS type B was similar. The route was Depot Æ Pasar Awam Æ Setia Indah
6 Æ Setia Indah 7 Æ Seelong Æ Depot. Since there is an additional constraint that
the workers are have breakfast at Pasar Awam first before collecting the solid waste
collection and get lunch in Pasar Awam also, thus Pasar Awam is visited twice, the
first is in the starting of route and the second is in the end of the route before
returning to the dumpsite. The waste from Pasar Awam was taken in the second
visit. The route becomes Depot Æ Pasar Awam Æ Setia Indah 6 Æ Setia Indah 7
Æ Pasar Awam Æ Seelong Æ Depot. The route was depicted in Figure 7.6.
After the cluster route has been built, the route optimisation of each cluster
was taken. Let’s consider a cluster that is Taman Setia Indah 7. In order to build the
route using VRPSD, the demand characteristics should be known.
characteristics of the demand data, descriptive statistics was used.
To get the
Statistics
summary of nodes demand data for six (6) days was given in Table 7.3. Jalan Setia
Indah 7/10 seems to give highest average value of waste to be collected per each
collection day, followed by Jalan Setia Indah 7/11; while the lowest amount of waste
to be collected is given by Jalan Setia 7/8. These statistics were reasonable since the
number of houses in Jalan Setia Indah 7/10 and Jalan Setia Indah 7/11 are larger
among other street in Zona 7 which are 48 houses, while the number of houses in
Jalan Setia Indah 7/8 were smaller in Zona 7 which is only 15 houses.
192
Figure 7.6 : Global route of the vehicle
Table 7.3 : Descriptive statistics of nodes demand data
No of
observation
Minimum
Maximum
Mean
Std. Deviation
Setia Indah 7/2
6
75.00
129.00
92.8333
26.13363
Setia Indah 7/3
6
114.00
239.00
156.0000
47.37510
Setia Indah 7/4
6
72.00
229.00
124.6667
56.21269
Setia Indah 7/5
6
71.00
198.00
117.8333
43.72375
Setia Indah 7/6
6
71.00
152.00
110.3333
35.35911
Setia Indah 7/7
6
83.00
180.00
117.5000
38.44346
Setia Indah 7/8
6
29.00
122.00
73.5000
31.09180
Setia Indah 7/9
6
58.00
93.00
76.3333
13.70645
Setia Indah 7/10
6
134.00
311.00
213.8333
73.32508
Setia Indah 7/11
6
135.00
255.00
187.5000
48.41797
Setia Indah 7/12
6
71.00
151.00
94.5000
29.61587
Setia Indah 7/15
6
66.00
140.00
96.1667
28.37193
Setia Indah 7/16
6
93.00
222.00
137.6667
47.09848
Setia Indah 7/17
6
51.00
157.00
82.6667
41.22459
193
Furthermore, uniform distribution fitting based on Chi-Square Test (as
described in Section 3.5.1) was conducted for demand data of each street in Taman
Setia Indah 7. The hypotheses were given below:
H0 : The demand data follows uniform distribution
H1 : The demand data does not follow uniform distribution
Minimum and maximum values of historical actual demand were set as lower
and upper bound of demand distribution follows discrete uniform distribution. The
results of uniform distribution fitting tests were given in Table 7.4. Since p-values of
distribution test for all streets were larger than 0.05 thus the null hypothesis that the
demand data follow uniform distribution can not be rejected. Each street demand
satisfies uniform probability distribution for specified bounds.
Table 7.4 : Uniform distribution fitting of demand data
Street
7/2
7/3
7/4
7/5
7/6
7/7
7/8
7/9
7/10
7/11
7/12
7/15
7/16
7/17
Demand (kg)
Minimum Maximum
75
129
114
239
72
229
71
198
71
152
83
180
29
122
58
93
134
311
135
255
71
151
66
140
93
222
51
157
p-value
0.206
0.926
0.429
0.928
0.882
0.901
0.897
0.799
0.956
0.922
0.881
0.873
0.929
0.910
Distribution test
result
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Taman Setia Indah 7 consists of 14 nodes with the demand satisfies uniform
distribution. Based on customer coordinate locations defined on the graph of map
and information about demand probability distribution, the route on Zone 7 was
194
built. The metaheuristics proposed in Chapter 4, 5 and 6 were implemented. The
computational results from the metaheuristics were given in Table 7.5 including the
routes and the total expected cost (unit cost) associated with. It was shown that RTS,
HGATS type B and BGA obtained the same route. On the other hand, the route
obtained from TS, GA, HGATS type A and Bianchi was the same. The route
obtained from RTS, HGATS type B and BGA gained lower expected cost than the
route obtained by TS, GA, HGATS type A and Bianchi. This result is similar with
the experimental result from Chapter 4 and Chapter 5. The route generated by TS,
GA, HGATS-A, and Bianchi was illustrated in Figure 7.7, the route generated by
RTS, HGATS-B and BGA was illustrated in Figure 7.8 and the current company
route was illustrated in Figure 7.9.
Table 7.5 : Route optimization in Taman Setia Indah 7
Algorithm
Cost (unit
Route
cost)
TS
888.27
7/13 - 7/12 – 7/11- 7/10 – 7/9 – 7/8 – 7/7 – 7/6 – 7/5 –
7/2 – 7/2 - 7/3 – 7/4 - 7/15 – 7/16 – 7/17 - 7/15
RTS
876.17
7/4 – 7/2 - 7/3 – 7/5 – 7/6 – 7/7 – 7/8 – 7/9 – 7/10 –
7/11 – 7/12 – 7/16 – 7/17 – 7/15
GA
888.27
7/13 - 7/12 – 7/11- 7/10 – 7/9 – 7/8 – 7/7 – 7/6 – 7/5 –
7/2 – 7/2 - 7/3 – 7/4 - 7/15 – 7/16 – 7/17 - 7/15
HGATS-A
888.27
7/13 - 7/12 – 7/11- 7/10 – 7/9 – 7/8 – 7/7 – 7/6 – 7/5 –
7/2 – 7/2 - 7/3 – 7/4 - 7/15 – 7/16 – 7/17 - 7/15
HGATS-B
876.17
7/4 – 7/2 - 7/3 – 7/5 – 7/6 – 7/7 – 7/8 – 7/9 – 7/10 –
7/11 – 7/12 – 7/16 – 7/17 – 7/15
BGA
876.17
7/4 – 7/2 - 7/3 – 7/5 – 7/6 – 7/7 – 7/8 – 7/9 – 7/10 –
7/11 – 7/12 – 7/16 – 7/17 – 7/15
Bianchi
888.27
7/13 - 7/12 – 7/11- 7/10 – 7/9 – 7/8 – 7/7 – 7/6 – 7/5 –
7/2 – 7/2 - 7/3 – 7/4 - 7/15 – 7/16 – 7/17 - 7/15
Company’s
current route
1000.18
7/4 – 7/3 – 7/2 – 7/5 – 7/8 – 7/7 – 7/6 – 7/5 – 7/9 – 7/10
– 7/11 – 7/12 – 7/15 – 7/16 – 7/17 - 7/15
195
Figure 7.7 : Routes generated by TS, GA, HGATS-A, and Bianchi
196
Figure 7.8 : Routes generated by RTS, BGA and HGATS-B
197
Figure 7.9 : The current company route
198
In general, the routes obtained from all metaheuristics implemented here
were better than the current route implemented by the PZSB Company. From results
given in Table 7.5, the route from RTS, HGATS type B and BGA can give
improvement in reducing cost up to 12.39% from current company route, while the
route from TS, GA, HGATS type A and Bianchi can reduce cost in approximately
11.18% from current company route. It is proven that the use of metaheuristics does
give improvement to the route optimization in searching for a better route. Better
route refers to less expected collection cost. In this study, the use of metaheuristics
give high effect in substantially reducing the total expected cost of a cluster, whereas
practically in a service day, the company uses multiple vehicles where one vehicle
serves multi clusters. By using metaheuristics, cost saving that can be gained by the
company in a service day is the summation of cost saving by a number of vehicles,
where cost saving per vehicle is the total cost saving of a number of clusters serviced
by a vehicle.
Regarding to the total amount of money spent on the transportation cost is
approximately 60 – 80 % of overall cost (according to company’s statement);
therefore the identification of a route using metaheuristics for solid waste collection
is critical. A small percentage of improvement in the collection operation result
a significant saving in the overall operation cost of the company.
7.5
Conclusion
A case study of a solid waste company was considered. The metaheuristics
studied in this research were implemented.
It was proven that the use of
metaheuristics does generate better routes by reducing the operation cost compared
to the current company routes.
199
CHAPTER 8
GENERAL CONCLUSION AND RECOMMENDATION
8.1
Introduction
This chapter provides a conclusion for this study based on the development of
metaheuristics namely Tabu Search (TS), Reactive Tabu Search (RTS), Genetic
Algorithm (GA), Adaptive GA, Breeder GA, and Hybrid GA with Tabu Search in
solving single Vehicle Routing Problem with Stochastic Demands (VRPSD), and also
the development of Breeder GA for solving bi-objective multiple VRPSD.
Some
recommendations for future work are included in this chapter.
8.2
General Conclusion
This thesis has presented a study on VRPSD, and several new metaheuristics
was successfully developed for the VRPSD. We first investigated the effect of static
and dynamic tabu list size in TS. By using dynamic tabu list size, the probability of
cycling can be reduced significantly. The dynamic tabu list size is later applied
within the TS. Reactive Tabu Search which has never been used in VRPSD was
introduced in this study. We found that there are statistically significant differences
between the performances of TS and RTS where the RTS can yield better solution
quality in terms of much less total expected cost.
200
Furthermore, Adaptive GA was developed and the performance of several
adaptive mutation probability schemes was compared. The computational results
generated for VRPSD using Adaptive GA was better than the solution quality
generated using the fixed parameter settings. By using the adaptive GA, the amount
of time required in finding the appropriate mutation probability values of GA can
also be reduced. Different Adaptive GAs were better suited to different problem
sets. This work makes more contribution by proposing a new mutation probability
scheme based on Lr distance. When compared to other adaptive schemes, the
proposed LD has shown its superiority.
To the best of our knowledge, hybrid GA with other metaheuristics has not
been done in VRPSD. This becomes a reason for proposing two types of Hybrid GA
with TS (HGATS) for solving VRPSD. Compared to TS and GA alone, the two
HGATS do improve the relative performance of the two algorithms and it is worth
noticing that different schemes of integrating TS to GA lead to a significantly
different performance. Another contribution is the development of Breeder GA for
solving VRPSD, it was shown that Breeder GA can give an improvement to solution
quality of GA.
Finally all metaheuristics described in this study were evaluated, and the
proposed HGATS do shows superiority in terms of solution quality compared to
other metaheuristics, followed by BGA and RTS in the second and third best
performance respectively.
But HGATS type B showed weakness in producing
longest average computation time. RTS demonstrated an advantage in combining
both of relatively good solution quality and average computation time. In general,
the proposed HGATS, BGA and RTS may lead to a better performance than the
previous GA developed by Bianchi et al. (2004).
While the metaheuristics described above deal with single objective function,
we also propose the Breeder GA for solving bi-objective multiple VRPSD in
minimizing the number of vehicles required and the total expected cost concurrently,
using Pareto-rank by modifying and extending the single-objective BGA.
The
computational results show that the proposed bi-objective Pareto BGA gave better
solution qualities compared to Pareto GA. Finally, the metaheuristics developed in
201
this study has been applied to solve a case study of solid waste collection in
municipality of Johor Bahru city, but confined on single vehicle. It was proven that
the use of metaheuristics can give improvement in obtaining better routes and
significantly reducing the cost compared to the current company routes.
8.3
Recommendations for Future Researches
The results of this study provide several directions for further research. In
this study, the problem data generated are following uniform distribution for the
customer locations and customer demands.
Future work can be done by
implementing various probability distributions for the customer demands, e. g.
Normal probability distribution, Poisson probability distribution, etc. Future work
can also include a complexity analysis of the algorithm, which has not been done in
this study.
In this study, we have developed TS and RTS using Or-opt. But, the TS can
not compete with RTS. Several other local searches can be explored with various
types of move implementation or adding a diversification in order to improve the
performance of TS. Diversification has not been applied in the RTS, therefore this
can be taken into consideration for further research.
Genetic Algorithms proposed in this study are in permutation or order
representation that is specifically designed for Traveling Salesman Problem and
Vehicle Routing Problems, future works can be done by using the binary
representation that commonly used for Genetic Algorithm, then a comparison
between the performances of these two chromosome representations to the VRPSD
can be investigated.
Adaptive GAs implemented and proposed here are depend on diversity
measure of population. One can conduct a research on adaptive GA that takes into
account other measures.
In this research, two Hybrid GA and TS for solving
202
VRPSD also have been proposed. Alternatively, one could hybridize GA with other
metaheuristics, or hybridize other metaheuristics beside GA.
Another work has been done in this research is the development of Breeder
GA for solving bi-objective multiple VRPSD. Future works could consider another
metaheuristics for bi-objective multiple VRPSD, for example TS, Simulated
Annealing or others. In this study, due to our limitation, we only consider single and
multiple homogeneous vehicles. One can extend these to the multiple heterogeneous
vehicles case.
The implementation of VRPSD model conducted in optimizing single vehicle
solid waste collection problem can be extent to the optimization of multiple vehicles.
Finally, the methods developed in this study may also be explored into several
application areas such as parcel delivery services, school bus routing and other
logistic and transportation problems.
203
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APPENDIX A
LIST OF PUBLICATIONS
-
INTERNATIONAL JOURNALS
1.
Zuhaimy Ismail and Irhamah. (2008). Adaptive Permutation-Based Genetic
Algorithm for Solving VRP with Stochastic Demands. Journal of Applied
Sciences. 8(18): 3228-3234.
2.
Zuhaimy Ismail and Irhamah. (2008). Solving the VRP with Stochastic
Demands via Hybrid Genetic Algorithm - Tabu Search. Journal of
Mathematics and Statistics (Accepted Manuscript).
3.
Zuhaimy Ismail and Irhamah. (2008). A Breeder Genetic Algorithm for
Vehicle Routing Problem with Stochastic Demands. Journal of Applied
Sciences Research (Accepted Manuscript).
4.
Zuhaimy Ismail, Irhamah and Zaitul Marlizawati Zainuddin. (2008).
Dynamic Tabu Search for Vehicle Routing Problem with Stochastic
Demands. American Journal of Applied Sciences (Accepted Manuscript).
5.
Zuhaimy Ismail and Irhamah. (2008). A Reactive Tabu Search for Vehicle
Routing Problem with Stochastic Demands. Submitted to International
Journal of Computational and Mathematical Sciences.
-
INTERNATIONAL/ NATIONAL CONFERENCES
1.
Zuhaimy Ismail, Irhamah and Khairil Asmani. (2005). The Vehicle Routing
Problems. Minggu Penyelidikan Sains & Matematik 2005, DSI - UTM
Skudai, 21–23 September 2005.
2.
Zuhaimy Ismail and Irhamah. (2006). Vehicle Routing Problem in
Optimizing Solid Waste Collection. Proceedings of National Seminar on
Science and Its Applications in Industry, Malacca - Malaysia, 14 – 15
February 2006.
221
3.
Zuhaimy Ismail and Irhamah. (2006). A Genetic Algorithm for the Vehicle
Routing Problem. Proceedings of Simposium Kebangsaan Sains Matematik
Ke-14, Kuala Lumpur – Malaysia, 6-8 June 2006.
4.
Irhamah and Zuhaimy Ismail. (2006). Vehicle Routing Problem with
Stochastic Demand. Proceedings of The First International Conference on
Mathematics and Statistics, Bandung – Indonesia, 19 – 21 June 2006.
5.
Irhamah and Zuhaimy Ismail. (2007). Tabu Search for Vehicle Routing
Problem with Stochastic Demands. Proceedings of The 3rd International
Conference on Research and Education in Mathematics 2007, Kuala Lumpur
– Malaysia, 10-12 April 2007.
6.
Zuhaimy Ismail and Irhamah. (2007). Genetic Algorithm for Vehicle Routing
Problem with Stochastic Demands. Proceedings of The 2nd International
Conference on Mathematics and Statistics, UTM Skudai – Malaysia, 28 – 29
May 2007.
7.
Zuhaimy Ismail, Irhamah and Loh Serlee. (2007). Look Ahead Heuristic For
Modeling Solid Waste Collection Problem. Proceedings of The 2nd
International Conference on Mathematics and Statistics, UTM Skudai –
Malaysia, 28 – 29 May 2007.
8.
Zuhaimy Ismail and Irhamah. (2007). Tabu Search Heuristics for SingleDepot Vehicle Routing Problem with Stochastic Demands. Proceedings of
Optimization 2007, Porto - Portugal, 22 – 25 July 2007.
9.
Zuhaimy Ismail and Irhamah. (2008). Dynamic Taboo Search for VRP with
Stochastic Demands. (2008). Proceedings of Regional Annual Fundamental
Science Seminar 2008, Institute of Ibnu Sina, UTM Skudai – Malaysia, 27 –
29 May 2008.
222
APPENDIX B
The Kolmogorov-Smirnov Normality Tests
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
DIFF_GA_TS_DIFF_GA_TS_DIFF_GA_TS_ DIFF_GA_
RTS_10
10
20
50
20
20
20
20
-2.6500
-5.7200
6.6200
.2500
4.30856
9.85290
8.13774
2.37874
.331
.271
.171
.258
.212
.191
.171
.242
-.331
-.271
-.145
-.258
1.479
1.212
.763
1.154
.025
.106
.606
.139
DIFF_GA_
RTS_20
20
1.1300
8.32479
.287
.156
-.287
1.281
.075
DIFF_GA_
RTS_50
20
23.7200
17.29489
.139
.139
-.109
.621
.835
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
DIFF_GA_
DIFF_GA_
DIFF_GA_
DIFF_GA_
DIFF_GA_
DIFF_GA_
HGSTSA_10 HGSTSA_20 HGSTSA_50 HGATSB_10 HGATSB_20 HGATSB_50
20
20
20
20
20
20
-.5800
1.4800
9.1450
1.2000
3.8800
25.5700
1.81096
3.96479
8.89248
2.21193
4.87416
16.16888
.426
.248
.145
.274
.173
.110
.183
.248
.145
.274
.173
.110
-.426
-.134
-.126
-.244
-.163
-.081
1.903
1.110
.650
1.227
.773
.494
.001
.170
.792
.098
.588
.968
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
DIFF_GA_
BGA_10
20
1.0500
2.09649
.292
.292
-.208
1.305
.066
DIFF_GA_
BGA_20
20
2.7300
4.66579
.206
.206
-.116
.922
.363
DIFF_GA_ DIFF_GA_ DIFF_GA_ DIFF_GA_
BGA_50 BIANCHI_10 BIANCHI_20 BIANCHI_50
20
20
20
20
22.6700
.5300
1.2500
18.1700
16.44823
2.57050
7.00192
14.55433
.139
.282
.235
.082
.139
.282
.169
.082
-.109
-.268
-.235
-.052
.623
1.260
1.050
.367
.832
.084
.220
.999
223
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
DIFF_TS_
RTS_10
20
2.9000
4.16628
.307
.307
-.193
1.372
.046
DIFF_TS_
RTS_20
20
6.8500
11.60887
.230
.230
-.199
1.028
.241
DIFF_TS_ DIFF_TS_ DIFF_TS_ DIFF_TS_
RTS_50 HGATSA_10 HGATSA_20 HGATSA_50
20
20
20
20
17.1000
2.0700
7.2000
2.5250
14.31562
2.98295
9.31665
7.83124
.138
.406
.224
.225
.138
.406
.224
.225
-.116
-.244
-.170
-.132
.615
1.816
1.001
1.008
.844
.003
.268
.262
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
DIFF_TS_
DIFF_TS_
DIFF_TS_
HGATSB_10 HGATSB_20 HGATSB_50
20
20
20
3.8500
9.6000
18.9500
5.64078
10.32320
13.79350
.303
.198
.143
.303
.198
.143
-.247
-.152
-.120
1.353
.884
.639
.051
.416
.809
DIFF_TS_
BGA_10
20
3.7000
5.46857
.301
.301
-.199
1.345
.054
DIFF_TS_
BGA_20
20
8.4500
11.11886
.198
.188
-.198
.884
.415
DIFF_TS_
BGA_50
20
16.0500
14.53661
.186
.186
-.117
.833
.492
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
DIFF_TS_
DIFF_TS_
DIFF_TS_ DIFF_RTS_ DIFF_RTS_ DIFF_RTS_
BIANCHI_10 BIANCHI_20 BIANCHI_50 BIANCHI_10 BIANCHI_20 BIANCHI_50
20
20
20
20
20
20
3.1800
6.9700
11.5500
.2800
.1200
-5.5500
4.67914
11.38573
13.10426
2.70158
1.77989
6.53537
.302
.243
.192
.391
.327
.163
.302
.243
.192
.391
.327
.139
-.198
-.242
-.109
-.359
-.203
-.163
1.349
1.086
.856
1.750
1.462
.729
.053
.189
.456
.004
.028
.662
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
DIFF_RTS_
BGA_10
20
.8000
2.48363
.526
.526
-.374
2.354
.000
DIFF_RTS_
BGA_20
20
1.6000
5.42315
.466
.466
-.384
2.084
.000
DIFF_RTS_ DIFF_RTS_ DIFF_RTS_ DIFF_RTS_
BGA_50
HGATSA_10 HGATSA_20 HGATSA_50
20
20
20
20
-1.0500
-.8300
.3500
-14.5750
6.39469
2.14944
9.18394
14.68644
.265
.400
.286
.110
.185
.250
.286
.093
-.265
-.400
-.258
-.110
1.186
1.790
1.281
.494
.120
.003
.075
.968
224
One-Sample Kolmogorov-Smirnov Test
DIFF_
DIFF_
DIFF_
DIFF_RTS_ DIFF_RTS_ DIFF_RTS_ HGATSA_
HGATSA_
HGATSA_
HGATSB_10 HGATSB_20 HGATSB_50 HGATSB_10 HGATSB_20 HGATSB_50
20
20
20
20
20
20
.9500
2.7500
1.8500
1.7800
2.4000
16.4250
2.78104
9.12414
3.64583
3.31815
4.13458
13.97768
.484
.276
.256
.404
.292
.127
.484
.276
.244
.404
.292
.127
-.366
-.232
-.256
-.296
-.231
-.106
2.163
1.235
1.145
1.808
1.307
.568
.000
.095
.146
.003
.066
.903
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
DIFF_BGA_ DIFF_BGA_ DIFF_BGA_ DIFF_BGA_ DIFF_BGA_ DIFF_BGA_
HGATSA_10 HGATSA_20 HGATSA_50 HGATSB_10 HGATSB_20 HGATSB_50
20
20
20
20
20
20
-1.6300
-1.2500
-13.5250
.1500
1.1500
2.9000
3.13236
5.88374
13.94772
.36635
5.83343
7.26853
.399
.283
.155
.509
.278
.245
.251
.201
.096
.509
.278
.205
-.399
-.283
-.155
-.341
-.272
-.245
1.783
1.266
.694
2.276
1.244
1.095
.003
.081
.721
.000
.091
.181
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
DIFF_
DIFF_
DIFF_
BIANCHI_
BIANCHI_
DIFF_BGA_ DIFF_BGA_ DIFF_BGA_ BIANCHI_
BIANCHI_10 BIANCHI_20 BIANCHI_50 HGATSA_10 HGATSA_20 HGATSA_50
20
20
20
20
20
20
-.5200
-1.4800
-4.5000
-1.1100
.2300
-9.0250
1.68635
3.93548
5.76541
2.72955
8.21943
12.45771
.421
.327
.182
.408
.255
.121
.329
.266
.157
.242
.255
.098
-.421
-.327
-.182
-.408
-.247
-.121
1.883
1.464
.815
1.824
1.142
.541
.002
.027
.520
.003
.147
.932
N
a,b Mean
Normal Parameters
Std. Deviation
Most Extreme
Absolute
Differences
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
One-Sample Kolmogorov-Smirnov Test
N
Normal Parameters a,b
Most Extreme
Differences
Mean
Std. Deviation
Absolute
Positive
Negative
Kolmogorov-Smirnov Z
Asymp. Sig. (2-tailed)
a. Test distribution is Normal.
b. Calculated from data.
DIFF_
BIANCHI_
HGATSB_50
20
7.4000
5.43749
.158
.158
-.112
.709
.697
DIFF_
BIANCHI_
HGATSB_20
20
2.6300
8.16908
.289
.289
-.274
1.291
.071
DIFF_
BIANCHI_
HGATSB_10
20
.6700
1.75712
.405
.405
-.351
1.813
.003
Asymp. Sig. (2-tailed)
Z
Asymp. Sig. (2-tailed)
Z
.028
HGATS_10A BGA_10
-2.197(a)
HGATS_1
0B RTS_10
-1.604(b)
.109
.018
-2.375(b)
HGATS_10A TS_10
.007
-2.708(b)
RTS_10 TS_10
a Based on negative ranks.
b Based on positive ranks.
c Wilcoxon Signed Ranks Test
.089
HGATS_10A RTS_10
-1.703(a)
RTS_20 BGA_20
-1.604(a)
.109
.398
-.844(a)
HGATS_10A GA_10
.014
-2.448(a)
TS_10 GA_10
Wilcoxon Signed Ranks Test
The Paired Samples Test
APPENDIX C
.083
HGATS_10B BGA_10
-1.732(b)
.588
-.542(a)
RTS_10 BIANCHI_10
.176
BIANCHI_10 BGA_10
-1.355(a)
.798
-.256(b)
RTS_20 BIANCHI_20
.026
BIANCHI_
20 BGA_20
-2.228(a)
.180
-1.342(a)
RTS_10 BGA_10
225
226
The Wilcoxon Signed Ranks Test
Ranks
N
TS_10 - GA_10
HGATS_10A - GA_10
RTS_10 - TS_10
HGATS_10A - TS_10
RTS_10 - BIANCHI_10
RTS_20 - BIANCHI_20
RTS_10 - BGA_10
RTS_20 - BGA_20
HGATS_10A - RTS_10
HGATS_10B - RTS_10
HGATS_10A - BGA_10
HGATS_10B - BGA_10
BIANCHI_10 - BGA_10
BIANCHI_20 - BGA_20
a TS 10 < GA 10
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
Negative Ranks
Positive Ranks
Ties
Total
2a
8b
10c
20
4d
4e
12f
20
9g
1h
10i
20
7j
0k
13l
20
2m
3n
15o
20
6p
4q
10r
20
0s
2t
18u
20
0v
3w
17x
20
2y
5z
13aa
20
3bb
0cc
17dd
20
1ee
6ff
13gg
20
3hh
0ii
17jj
20
1kk
4ll
15mm
20
2nn
9oo
9pp
20
Mean Rank
1.75
6.44
Sum of Ranks
3.50
51.50
3.00
6.00
12.00
24.00
6.00
1.00
54.00
1.00
4.00
.00
28.00
.00
2.75
3.17
5.50
9.50
5.00
6.25
30.00
25.00
.00
1.50
.00
3.00
.00
2.00
.00
6.00
2.00
4.80
4.00
24.00
2.00
.00
6.00
.00
1.00
4.50
1.00
27.00
2.00
.00
6.00
.00
2.50
3.13
2.50
12.50
4.00
6.44
8.00
58.00
227
Paired Samples Correlations
N
Pair 1
Pair 2
Pair 3
Pair 4
Pair 5
Pair 6
Pair 7
Pair 8
Pair 9
Pair 10
Pair 11
Pair 12
Pair 13
Pair 14
Pair 15
Pair 16
Pair 17
Pair 18
Pair 19
Pair 20
Pair 21
Pair 22
Pair 23
Pair 24
Pair 25
Pair 26
Pair 27
Pair 28
Pair 29
Pair 30
Pair 31
Pair 32
Pair 33
Pair 34
Pair 35
Pair 36
Pair 37
Pair 38
Pair 39
Pair 40
Pair 41
Pair 42
Pair 43
Pair 44
Pair 45
Pair 46
GA_20 & TS_20
GA_50 & TS_50
GA_10 & RTS_10
GA_20 & RTS_20
GA_50 & RTS_50
GA_20 & HGATS_20A
GA_50 & HGATS_50A
GA_10 & HGATS_10B
GA_20 & HGATS_20B
GA_50 & HGATS_50B
BGA_10 & GA_10
BGA_20 & GA_20
BGA_50 & GA_50
GA_10 & BIANCHI_10
GA_20 & BIANCHI_20
GA_50 & BIANCHI_50
TS_20 & RTS_20
TS_50 & RTS_50
TS_20 & HGATS_20A
TS_50 & HGATS_50A
TS_10 & HGATS_10B
TS_20 & HGATS_20B
TS_50 & HGATS_50B
BGA_10 & TS_10
BGA_20 & TS_20
BGA_50 & TS_50
BIANCHI_10 & TS_10
BIANCHI_20 & TS_20
BIANCHI_50 & TS_50
HGATS_20A & HGATS_
20B
HGATS_50A & HGATS_
50B
BIANCHI_20 &
HGATS_20A
BIANCHI_50 &
HGATS_50A
BIANCHI_20 &
HGATS_20B
BIANCHI_50 &
HGATS_50B
BIANCHI_50 & RTS_50
BGA_50 & RTS_50
RTS_20 & HGATS_20A
RTS_50 & HGATS_50A
RTS_20 & HGATS_20B
RTS_50 & HGATS_50B
BGA_20 & HGATS_20A
BGA_50 & HGATS_50A
BGA_10 & HGATS_10B
BGA_50 & HGATS_50B
BGA_50 & BIANCHI_50
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
Correlation
.968
.973
.998
.977
.871
.995
.967
.998
.992
.890
.998
.993
.886
.997
.983
.916
.955
.915
.971
.974
.988
.964
.923
.989
.959
.913
.991
.957
.934
Sig.
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
20
.994
.000
20
.913
.000
20
.977
.000
20
.935
.000
20
.977
.000
20
.982
.000
20
20
20
20
20
20
20
20
20
20
20
.975
.976
.971
.902
.971
.992
.988
.913
1.000
.968
.980
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
GA_20 - TS_20
GA_50 - TS_50
GA_10 - RTS_10
GA_20 - RTS_20
GA_50 - RTS_50
GA_20 - HGATS_20A
GA_50 - HGATS_50A
GA_10 - HGATS_10B
GA_20 - HGATS_20B
GA_50 - HGATS_50B
BGA_10 - GA_10
BGA_20 - GA_20
BGA_50 - GA_50
GA_10 - BIANCHI_10
GA_20 - BIANCHI_20
GA_50 - BIANCHI_50
TS_20 - RTS_20
TS_50 - RTS_50
TS_20 - HGATS_20A
TS_50 - HGATS_50A
TS_10 - HGATS_10B
TS_20 - HGATS_20B
TS_50 - HGATS_50B
Pair 1
Pair 2
Pair 3
Pair 4
Pair 5
Pair 6
Pair 7
Pair 8
Pair 9
Pair 10
Pair 11
Pair 12
Pair 13
Pair 14
Pair 15
Pair 16
Pair 17
Pair 18
Pair 19
Pair 20
Pair 21
Pair 22
Pair 23
18.95
9.60
3.85
2.52
7.20
17.10
6.85
18.10
1.25
.53
-22.670
-2.730
-1.050
25.570
3.880
1.200
9.145
1.480
23.720
1.130
.250
6.620
-5.720
Mean
13.79350
10.32320
5.64078
7.83124
9.31665
14.31562
11.60887
14.55433
7.00192
2.57050
16.44823
4.66579
2.09649
16.16888
4.87416
2.21193
8.89248
3.96479
17.29489
8.32479
2.37874
8.13774
3.08432
2.30834
1.26132
1.75112
2.08327
3.20107
2.59582
3.25445
1.56568
.57478
3.67794
1.04330
.46879
3.61547
1.08990
.49460
1.98842
.88655
3.86726
1.86148
.53190
1.81965
2.20317
12.49445
4.76859
1.21003
-1.14013
2.83967
10.40009
1.41688
11.35836
-2.02700
-.67303
-30.36801
-4.91366
-2.03119
18.00273
1.59882
.16479
4.98319
-.37558
15.62574
-2.76612
-.86329
2.81142
-10.33130
Lower
25.40555
14.43141
6.48997
6.19013
11.56033
23.79991
12.28312
24.98164
4.52700
1.73303
-14.97199
-.54634
-.06881
33.13727
6.16118
2.23521
13.30681
3.33558
31.81426
5.02612
1.36329
10.42858
-1.10870
Upper
Paired Differences
Std. Error 95% Confidence Interval
Mean
of the Difference
9.85290
Std.
Deviation
Paired Samples T-test
6.144
4.159
3.052
1.442
3.456
5.342
2.639
5.583
.798
.922
-6.164
-2.617
-2.240
7.072
3.560
2.426
4.599
1.669
6.134
.607
.470
3.638
-2.596
t
df
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
.000
.001
.007
.166
.003
.000
.016
.000
.435
.368
.000
.017
.037
.000
.002
.025
.000
.111
.000
.551
.644
.002
.018
Sig. (2-tailed)
228
BGA_10 - TS_10
BGA_20 - TS_20
BGA_50 - TS_50
BIANCHI_10 - TS_10
BIANCHI_20 - TS_20
BIANCHI_50 - TS_50
HGATS_20A - HGATS_20B
HGATS_50A - HGATS_50B
BIANCHI_20 - HGATS_20A
BIANCHI_50 - HGATS_50A
BIANCHI_20 - HGATS_20B
BIANCHI_50 - HGATS_50B
BIANCHI_50 - RTS_50
BGA_50 - RTS_50
RTS_20 - HGATS_20A
RTS_50 - HGATS_50A
RTS_20 - HGATS_20B
RTS_50 - HGATS_50B
BGA_20 - HGATS_20A
BGA_50 - HGATS_50A
BGA_10 - HGATS_10B
BGA_50 - HGATS_50B
BGA_50 - BIANCHI_50
Pair 24
Pair 25
Pair 26
Pair 27
Pair 28
Pair 29
Pair 30
Pair 31
Pair 32
Pair 33
Pair 34
Pair 35
Pair 36
Pair 37
Pair 38
Pair 39
Pair 40
Pair 41
Pair 42
Pair 43
Pair 44
Pair 45
Pair 46
-4.50
2.90
.15
-13.53
-1.25
1.85
2.75
-14.57
.35
1.05
5.55
7.40
2.63
-9.025
.23
16.425
2.40
-11.55
-6.97
-3.18
-16.05
-3.7
-8.45
Mean
5.76541
7.26853
.36635
13.94772
5.88374
3.64583
9.12414
14.68644
9.18394
6.39469
6.53537
5.43749
8.16908
12.45771
8.21943
13.97768
4.13458
13.10426
11.38573
4.67914
14.53661
1.28919
1.62529
.08192
3.11880
1.31564
.81523
2.04022
3.28399
2.05359
1.42990
1.46135
1.21586
1.82666
2.78563
1.83792
3.12551
.92452
2.93020
2.54593
1.04629
3.25049
1.22281
2.48625
-7.19830
-.50178
-.02146
-20.05273
-4.00368
.14370
-1.52023
-21.44846
-3.94822
-1.94281
2.49135
4.85518
-1.19324
-14.85539
-3.61681
9.88324
.46496
-17.68298
-12.29869
-5.36990
-22.85334
Lower
-6.25937
-13.65378
-1.80170
6.30178
.32146
-6.99727
1.50368
3.55630
7.02023
-7.70154
4.64822
4.04281
8.60865
9.94482
6.45324
-3.19461
4.07681
22.96676
4.33504
-5.41702
-1.64131
-.99010
-9.24666
Upper
-1.14063
-3.24622
Paired Differences
Std. Error 95% Confidence Interval
Mean
of the Difference
5.46857
11.11886
Std.
Deviation
-3.491
1.784
1.831
-4.337
-.950
2.269
1.348
-4.438
.170
.734
3.798
6.086
1.440
-3.240
.125
5.255
2.596
-3.942
-2.738
-3.039
-4.938
-3.026
-3.399
t
df
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
.002
.090
.083
.000
.354
.035
.194
.000
.866
.472
.001
.000
.166
.004
.902
.000
.018
.001
.013
.007
.000
.007
.003
Sig. (2-tailed)
229
230