a genetic simplified swarm algorithm for...
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A GENETIC SIMPLIFIED SWARM ALGORITHM
FOR OPTIMIZING
n- CITIES OPEN LOOP TRAVELLING SALESMAN PROBLEM
CHIENG HOCK HUNG
UNIVERSITI TUN HUSSEIN ONN MALAYSIA
A GENETIC SIMPLIFIED SWARM ALGORITHM
FOR OPTIMIZING
n- CITIES OPEN LOOP TRAVELLING SALESMAN PROBLEM
CHIENG HOCK HUNG
A thesis submitted in
fulfillment of the requirement for the award of the
Degree of Master of Information Technology
Faculty of Computer Science and Information Technology
Universiti Tun Hussein Onn Malaysia
JANUARY 2016
iii
DEDICATION
Dedicated to the Lord Almighty God;
Heavenly Father, Jesus Christ and Holy Spirit.
Papa, Mama, Lydia, Debrorah, and Hope’s family.
iv
ACKNOWLEDGEMENT
Firstly, I would like to give thanks to my best friend, Heavenly Father, for
His unconditional love and favors in my life. He is the source of my strength,
wisdom and knowledge. He has empowered and granted me strength to soar on wing
like eagle above the raging sea.
Secondly, I would like to thanks my supervisor, Dr Noorhaniza Wahid as
well. She was the direct contributor for this work. I would like to express my
thankfulness to her from the bottom of my heart. Her understanding, wisdom and
personal experiences have provided a good basis for this present thesis. She always
guided and pointing me to the right direction toward fulfillment of this project. Other
than that, she is also a kind person that always shares her life experiences to
encourage and inspire me in this journey of research.
I am grateful to my beloved parents and sisters for their support, provision,
care and love. They were always encouraged me and even prayed a lot for me along
my Master research’s journey.
In addition, I would like to thanks Szakif Enterprize for providing me the best
service in thesis proofreading.
Lastly, my special gratitude is due to my brothers and sisters in Hope Life
Group. I wish to express my deepest and sincere thanks to them who had
accompanied me with prayers, love, concern, support and advice over the past whole
two years.
This research project would not have been completed without the help of
many including the writing of others, who are acknowledged within the reference
section.
Thank you!
v
PUBLICATION
Chieng, H. H., and Wahid, N. (2014). A Performance Comparison of Genetic
Algorithm’s Mutation Operators in n-Cities Open Loop Travelling Salesman
Problem. Recent Advances on Soft Computing and Data Mining pp. 89-97. Springer
International Publishing. (Indexed by ISI, DBLP, El-Compendex, Scopus).
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ABSTRACT
Open Loop Travelling Salesman Problem (OTSP) is one of the extension of
Travelling Salesman Problem (TSP) that finding a shortest tour of a number of cities
by visiting each city exactly once and do not returning to the starting city. In the past,
TSP and OTSP has been applied in various vehicle routing systems to optimize the
route distance. However, in real-life scenario such as transportation problem does not
seem similar as pictured in OTSP whereby do not all cities are required to be visited
but simply restrain to several number of n cities. Therefore, a new problem called n-
Cities Open Loop Travelling Salesman Problem (nOTSP) is proposed. In the past,
Genetic Algorithm (GA) is a popular algorithm that used to solve TSPs. However,
GA often suffers from premature convergence due to the difficulty in preventing the
loss of genetic diversity in the population. Therefore, Genetic Simplified Swarm
Algorithm (GSSA) is proposed in this study to overcome the drawback of GA.
GSSA is an improved GA based algorithm with Simplified Swarm Optimization
(SSO) algorithm’s characteristic named Solution Update Mechanism (SUM). The
SUM is modified by embedding three GA mutation operators. Then, GSSA is used
to optimize nOTSP in terms of finding the shortest tour. Later, the performance of
GSSA is compared with GA without crossover operator (GA-XX) and GA with one-
point crossover operator (GA-1X). Performance of the proposed algorithm is
measured based on the shortest distance and average shortest distance found by the
algorithm. Meanwhile, an investigation on influence of population size towards
algorithm was also studied. The experiment results show that GSSA can discover
shorter tour than GA-XX and GA-1X. Nevertheless, the study also found that most
of the good solutions are discovered in the larger population sizes from 3000 to 5000.
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ABSTRAK
Open Loop Travelling Salesman Problem (OTSP) merupakan salah satu lanjutan
kepada Travelling Salesman Problem (TSP) yang mencari laluan tersingkat dengan
syarat hanya mengunjungi setiap bandar sekali sahaja dan tidak kembali ke bandar di
mana perjalanan bermula. Namun begitu, dalam situasi sebenar contohnya masalah
pengangkutan, ia tidak sama dengan OTSP. Kebanyakan kenderaan tidak
mengunjungi setiap bandar, tetapi hanya terhad kepada sejumlah bandar n sahaja.
Oleh itu, masalah baru yang dinamakan n-Cities Open Loop Travelling Salesman
Problem (nOTSP) dicadangkan di dalam kajian ini. Pada masa lalu, Genetic
Algorithm (GA) merupakan algoritma yang popular untuk menyelesaikan masalah
TSP. Namun, GA sering mengalami masalah penumpuan pra-matang yang
disebabkan oleh kesukaran untuk mencegah kehilangan kepelbagaian genetik. Oleh
itu, satu algoritma yang dinamakan Genetic Simplified Swarm Algorithm (GSSA)
dicadangkan untuk mengatasi kelemahan GA di dalam kajian ini. GSSA merupakan
penambahbaikan GA yang mengandungi ciri-ciri algoritma Simplified Swarm
Optimization (SSO). Ciri-ciri ini dikenali sebagai Solution Update Mechanism
(SUM). SUM diubahsuai terlebih dahulu dengan menerapkan tiga operator mutasi
GA. Seterusnya, GSSA dilaksanakan untuk mengoptimumkan nOTSP dari segi
pencarian laluan yang tersingkat. Di samping itu, prestasi GSSA akan dibandingkan
dengan prestasi GA tanpa operator crossover (GA-XX) dan GA yang mengandungi
operator one-point crossover (GA-1X). Prestasi algoritma dinilai melalui jarak
terpendek dan purata jarak terpendek yang diperolehi. Selain itu, kajian ke atas
pengaruh saiz populasi terhadap algoritma juga dikaji. Keputusan kajian jelas
menunjukkan bahawa GSSA berupaya untuk meneroka laluan yang lebih pendek
berbanding GA-XX dan GA-1X. Di samping itu, kajian ini juga mendapati bahawa
kebanyakan penyelesaian yang baik ditemui dalam saiz populasi yang lebih besar
iaitu dari 3000 hingga 5000.
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CONTENTS
TITLE i
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
PUBLICATION v
ABSTRACT vi
ABSTRAK vii
CONTENTS viii
LIST OF TABLES xii
LIST OF FIGURES xiii
LIST OF ALGORITHMS xv
LIST OF SYMBOLS AND ABREVIATIONS xvi
LIST OF APPENDICES xix
CHAPTER 1 INTRODUCTION 1
1.1 Background of Study 1
1.2 Problem Statement 3
1.3 Objectives of the Study 4
1.4 Scope of the Study 4
1.5 Thesis Outline 5
CHAPTER 2 LITERATURE REVIEW 6
2.1 Introduction 6
ix
2.2 The Concept of Optimization 6
2.2.1 Continuous Optimization 7
2.2.2 Combinatorial Optimization 8
2.3 Travelling Salesman Problem (TSP) 10
2.3.1 Variants of TSP 12
2.3.2 Open Loop Travelling Salesman
Problem (OTSP) 13
2.3.3 n-Cities Open Loop Travelling
Salesman Problem (nOTSP) 15
2.4 Genetic Algorithm 16
2.4.1 Genetic Operators 18
2.4.2 Mutation Operators 19
2.4.2.1 Inversion Mutation 20
2.4.2.2 Displacement Mutation 21
2.4.2.3 Pairwise Swap Mutation 21
2.4.3 Application of GAs on TSPs 22
2.5 Swarm Intelligence 24
2.6 Simplified Swarm Optimization 25
2.6.1 Algorithmic Structure and Flow 25
2.7 Chapter Summary 29
CHAPTER 3 RESEARCH METHODOLOGY 30
3.1 Introduction 30
3.2 Research Framework 30
3.3 nOSTP Topology Development 31
3.3.1 Nodes Generation 31
3.3.2 Nodes Mapping 33
3.3.3 Determining the Starting and Ending Points 34
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3.3.4 nOTSP Creation 36
3.4 Genetic Simplified Swarm Algorithm (GSSA) 38
3.4.1 GSSA Development 40
3.4.2 Process of GSSA 45
3.5 Experimental Setup 46
3.5.1 Pre-processing: Determining the MaxGen 47
3.5.2 Parameter Setting and Data Collection 49
3.6 Performance Evaluation 49
3.6.1 Optimal Solution and Average 50
3.6.2 Influence of Population Size on Algorithm 50
3.7 Chapter Summary 50
CHAPTER 4 ANALYSIS AND RESULT 52
4.1 Introduction 52
4.2 Performance Analysis and Results 52
4.2.1 Shortest Path and Average Distance 53
4.2.2 Discussion on Algorithms’ Performance 56
4.2.2.1 New Characteristic of the Algorithm 56
4.2.2.2 Adequate Genetic Diversity 56
4.2.2.3 Destructive Effect of Crossover
Operator 57
4.2.2.4 Crossing Path 58
4.3 The Influence of the Size Population on Algorithms 59
4.4 Chapter Summary 62
CHAPTER 5 CONCLUSION 63
5.1 Introduction 63
5.2 Research Contributions 63
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5.2.1 New Variant of TSP 63
5.2.2 Improved GA with SSO’s Characteristic 64
5.2.3 Performance of Proposed Algorithm-GSSA 64
5.2.4 Influence of the Population Size 65
5.3 Conclusion 65
REFERENCES 67
APPENDICES 81
VITA
xii
LIST OF TABLES
2.1 Related literatures on GAs for TSPs 22
3.1 Generated coordinates of each node 33
3.2 The symmetric distance-matrix 37
3.3 Fixed and adjusted parameters 47
3.4 Pre-processing results for n=10 48
3.6 Pre-processing results and MaxGen
for n=10, 20, 30 and 40 49
3.6 Characteristics of the algorithms 49
4.1 The shortest distances discovered from the executed
10 runs using GSSA, GA-XX and GA-1X 53
4.2 The shortest distances discovered by GSSA, GA-XX
and GA-1X for n=10, 20, 30 and 40. 54
4.3 Average distance over 10 runs using GSSA, GA-XX
and GA-1X 55
4.4 Influence of the population size on the algorithms 60
xiii
LIST OF FIGURES
2.1 Illustration of the vehicle routing problem 10
2.2 Illustration of the Travelling Salesman Problem 11
2.3 Variants of TSP 13
2.4 The difference between OTSP and nOTSP 16
2.5 Before and after the inversion mutation
was performed 20
2.6 Before and after the displacement mutation
was performed 21
2.7 Before and after the pairwise swap mutation
was performed 21
3.1 Research process 31
3.2 Nodes Mapped in Cartesian coordinate plane 34
3.3 The starting and ending points in the Cartesian
coordinate plane 35
3.4 Nodes position before and after the manipulation
of starting and ending points 35
3.5 New location of the starting and ending points 36
3.6 Population of chromosomes in matrix form 37
3.7 Arrangement of the first and second chromosomes
with number of gene n in a matrix 38
3.8 The flowchart of the SSO 39
3.9 The flowchart of the GA 40
3.10 SUM with three decision points 41
3.11 New SUM after modification was performed 41
3.12 Embedding process of the new SUM into the GA 43
3.13 The flowchart of Genetic Simplified Swarm
Algorithm 44
3.14 Fixed and adjusted parameters 47
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4.1 Cloning effect caused by crossover operator 58
4.2 Simulation of nOTSP’s paths 59
4.3 Influence of the population size on the GSSA 61
4.4 Influence of the population size on the GA-XX 61
4.5 Influence of the population size on the GA-1X 62
xv
LIST OF ALGORITHMS
2.1 Genetic Algorithm 17
2.2 Particle Swarm Optimization 26
2.3 Simplified Swarm Optimization 28
3.1 Genetic Simplified Swarm Algorithm 45
xvi
LIST OF SYMBOLS AND ABBREVIATIONS
d - Distance
𝐷 - Variable domain
E - Edge
𝑓(𝑥) - Function of x
G - Graph
𝑔𝑏𝑒𝑠𝑡 - Global best
gloFit - Global fitness
i - Location of the particle in PSO
Inf - Infinity
maxFit - Maximum fitness value
MaxGen - Maximum generation
m - Number of total given cities
min - Smallest elements in array (MATLAB
SYNTAX)
n - Number of visited cities
newFit - New fitness
p - Population size
𝑝𝑏𝑒𝑠𝑡 - Local best
randperm - Random permutation (MATLAB SYNTAX)
rand - Random (MATLAB SYNTAX)
𝑠 - Optimal solution
S - Global optimal solution
t - Time
V - Vertex
w - Inertial weight
zeros - Create array of all zeros (MATLAB SYNTAX)
α - Personal experience
β - Population experience
xvii
ℝ - Real number
𝑣𝑖𝑡 - Particle velocity in PSO
𝑥𝑖𝑡 - Particle position in PSO
𝑐𝑖 - Constraints
𝐶𝑤, 𝐶𝑝, 𝐶𝑔 - Predetermined constants
ACO - Ant Colony Optmization
ABC - Artificial Bee Colony
AI - Artificial Intelligent
AIS - Artificial Immune System
AS - Ant System
BF - Bacterial Foraging
CSO - Cat Swarm Optimization
EA - Evolutionary Algorithm
DPfGA - Distributed Parameter Free Genetic Algorithm
DPSO - Discrete Particle Swarm Optimization
GA - Genetic Algorithm
GA-1X - Genetic Algorithm with one-point crossover operator
GA-GSTM - Greedy Sub Tour Mutation operator
GA-XX - Genetic Algorithm without crossover operator
GNSS - Global Navigation Satellite System
GSO - Glowworm Swarm Optimization
GSSA - Genetic Simplified Swarm Algorithm
nOTSP - n-Cities Open Loop Travelling Salesman Problem
OTSP - Open Loop Travelling Salesman Problem
PSO - Particle Swarm Optimization
SA - Simulated Annealing
SUM - Solution Update Mechanism
TS - Tabu Search
TSP - Travelling Salesman Problem
xix
LIST OF APPENDICES
APPENDIX TITTLE PAGE
A Computational Results of GSSA, GA-XX
and GA-X1 81
B Pre-processing results for the predetermined
MaxGen on n=10, 20, 30 and 40 85
CHAPTER 1
INTRODUCTION
1.1 Background of Study
Road networks play a significant role for the economics growth and development of
a city. Road networks can be described as the equivalent of the veins in the human
body, and the vehicles are the blood cells that carry the nutrition from one part of the
body to the another part. In the reality of transportation, it is important for the
transportation system to identify the best route to navigate the drivers to their
destination. For example, an emergency evacuation unit’s transportation fleet such as
ambulances and firetrucks are required to reach the emergency site quickly by using
shortest tour for evacuation and emergency purposes. The other example such as the
Global Positioning System (GPS) that often used by the drivers in order to navigate
and show them the shortest route to an unfamiliar place. Until today, many
researches are still continuously improving the real-world vehicles routing system in
order to provide more effective and efficient route to travel (Toth & Vigo, 2014;
Gomez & Salhi, 2014; Royo et al., 2015). One of the common vehicles routing
problem in computer science is Travelling Salesman Problem (TSP) (El-Gharably et
al., 2013). This problem is used to simulate and solve routing problems.
TSP is a well-known and important combinatorial optimization problem
(Ausiello et al., 2012). It was first introduced by a mathematician from Ireland
named William Rowan Hamilton and a British mathematician, named Thomas
Penyngton Kirkman in the 1800s (Matai et al., 2010). Later, the problem was
formulated by Karl Menger in 1930 (Maredia, 2010). TSP is closely related to the
Hamiltonian path problem, where it devotes a path in an undirected or directed graph
that visits all the vertices exactly once (Abdoun & Abouchabaka, 2012). However,
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the idea of TSP is in regard to a salesman who supposed to travel by visiting all the
given cities exactly once and returns to the city he started, with the shortest route.
Despite TSP is devoted to a complete closed Hamilton path, TSP generally
can be divided into two categories, which are Closed Loop TSP and Open Loop TSP
(OTSP). Closed Loop TSP is similar with the ordinary TSP, while OTSP has a slight
difference when compared with TSP. The difference between OTSP and TSP is that
it has different starting and ending points. In other words, salesman in the OTSP
travels to each city exactly once by departing from one city but does not return to the
city where he departed.
However, in reality, today’s transportation issue is not exactly similar to what
has been described in the TSP and OTSP. In contrary, the numerous of transportation
issues are not related with “visit all the given cities”, in fact, simply visiting certain
number of cities rather than all the given cities which can lead to shortest tour
distance. For example, in the logistics of merchandise delivery services, the drivers
are required to plan the route by departing from the depot to the destination without
required to visit all the cities along the route, yet, they are restrained to a certain
numbers of cities for cost and time saving purposes.
Inspired from this issue, this research proposes a new extension of OTSP
called n-Cities Open Loop Travelling Salesman Problem (nOTSP). The nOTSP can
be illustrated through the scenario where a salesman is given a set of cities, but only
required to visit a certain number of cities rather than all the cities in a minimum tour.
Over the past decades, many algorithms have been successfully applied to a
wide range of combinatorial problems including TSPs. These algorithms include
Tabu Search (TS) (Pedro et al., 2013), Genetic Algorithm (GA) (Nagata & Soler,
2012) and Simulated Annealing (SA) (Wang et al., 2013). Among them, GA is one
of the most popular algorithms that used to solve permutation problems such as TSP
(Ahmed, 2010). In GA, it generates a set of possible solutions through permutation
of the genes. Hence, the solutions of TSP can also be easily represented as
permutation of genes in the GA.
Besides GA, swarm-based algorithms also have been used to solve high
complexity problems such TSPs. The example of swam-based algorithms have been
use to solve TSPs in the past are Particle Swarm Optimization (PSO) (Eberhart &
Kennedy, 1995), Ant Colony Optimization (ACO) (Colorni et al.,1992), Artificial
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Bee Colony algorithm (ABC) (Karaboga, 2005) and Bat algorithm (BA) (Yang,
2010).
Recently, another new swarm-based algorithm that has been proposed was
named Simplified Swarm Algorithm (SSO) (Chung & Wahid, 2012). SSO is the
variant of PSO. In the past, SSO is used to solve the classification problem and has
shown good performances. One of the reasons for its success is due to its special
characteristic, known as the comparison strategy. The purpose of this comparison
strategy is to update the global best (gbest) solution once a better solution is found.
Therefore this research proposes a GA based algorithm by adopting the
characteristic of SSO in GA. Hence, this algorithm is named as Genetic Simplified
Swarm Algorithm (GSSA). Later, the algorithm is used to optimize the nOTSP.
1.2 Problem Statement
TSP is a well-known combinatorial problem that is often used to model vehicles’
routing issues such as in transportation scenarios. However, it is realized that the
problems are not exactly similar as what has been pictured in TSP and OTSP.
Conversely, the vehicle may only travel from the starting point to the ending point by
visiting only a certain number of cities with minimum total travelling distance. For
example, a logistic services company which is in charge of delivering goods from the
depot to the destination by visiting only several numbers of cities, without passing
through all the cities along the route to keep the minimum distance. In order to tackle
this issue, this research models a variant of TSP called nOTSP.
In recently year, nature-inspired algorithms are commonly used and are
popular in the context of optimization. Among them, GA has been highlighted to
have good performances in solving many combinatorial problems such as TSP
(Ahmed, 2010; Dwivedi et al., 2012; Bahaabadi et al., 2012). However, GA often
suffers from the tendency to converge towards local optima or also known as
prematurely converge (Vashisht, 2013). Genetic diversity is often considered as the
primary reason for prematurely convergence in GA (Gupta & Ghafir, 2012). The
premature convergence is generally due to insufficiency of the diversity within the
population (Malik & Wadhwa, 2014). Therefore, to ensure the adequate of genetic
4
diversity is crucial for the algorithm to avoid them to be trapped at the local optima
(Gupta & Ghafir, 2012; Malik & Wadhwa, 2014).
To overcome the drawback, this research proposes an improved GA with
SSO’s characteristic in order to prevent the loss of genetic diversity and improve the
solutions.
1.3 Objectives of the Study
The objectives of this research are:
i. to propose a new extension of OTSP variant named n-Cities Open Loop
Travelling Salesman Problem (nOTSP),
ii. to propose an improved technique of Genetic Algorithm (GA) with
Simplified Swarm Optimization (SSO) algorithm’s characteristic to prevent
the loss of the genetic diversity in the population,
iii. to develop the propose technique in (2) for optimizing the nOTSP in term of
finding the shortest path, and
iv. to evaluate the performance of the proposed technique with other GA variants
in terms of shortest distance and population size.
1.4 Scope of the Study
This research focuses on single vehicle travels from a given starting point to an
ending point in nOTSP with n number of cities. The performance of the proposed
algorithm and the other GAs will be compared and analyzed in terms of the shortest
tour and the influence of population size toward the solutions. In this research, the
total number of cities m is set to be 50 as an experimental test case to represent the
real cities. Thus, a set of 50 cities are represented as nodes in this study and all the
nodes are generated randomly by computer. However, there is high possibility for
starting and ending points are appeared to be closed to each other. Therefore, the
manipulation of the starting and ending points will be conducted to ensure that they
are far apart as what has been pictured in real-world scenario. In addition, four
different data sets which represent the number of visited cities n are set to be 10, 20,
30 and 40 are employed. Moreover, each n is tested with 5 different population sizes
5
p, that are 1000, 2000, 3000, 4000 and 5000. During each experiment, each n is
executed 10 times on different p. Meanwhile, the performances of the proposed
algorithm are compared with other GA variants, which are GA without crossover
operator (GA-XX) and GA with one-point crossover (GA-1X). On the other hand,
computational time and iteration are not taken into account in this research.
1.5 Thesis Outline
The remaining of the chapters are structured as follows.
Chapter 2 provides the fundamental theories regarding the optimization, TSP,
GA, SSO and their applications. This is followed by reviews of the research made by
past researchers and scholars in the similar field. In addition, this chapter lays a
foundation for constructing the nOTSP and the proposed algorithm.
Then, Chapter 3 discusses the methodology used in this research. This details
how the problem is constructed, how the proposed algorithm is developed, how the
experiment is carried out systematically and how the results are recorded, calculated
and analyzed.
Furthermore, Chapter 4 presents the analysis and results of this research. In
this chapter, the proposed algorithm that has been developed in Chapter 3 is further
validated for its efficiency and accuracy based on the recorded experimental results.
The analysis and evaluations are carried out based on the computational results. Later,
the reasons are justified.
Lastly, Chapter 5 concludes the research and findings as well as summarizing
the contributions of the proposed algorithm are summarized followed by the
recommendations of future works.
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
Many techniques have been proposed to optimize the Travelling Salesman Problem
(TSP) or its variants in term of discovering the shortest route. Although many
literatures covered a wide variety of the theories, this review only focuses on four
dominant themes regarding the research topic. The four dominant themes are: the
concept of optimization and its category, the definition of the TSP and n-Cities Open
Loop Travelling Salesman Problem (nOTSP), the Genetic Algorithm (GA) as the
base of the proposed technique and its application on similar problems, and the
introduction of the Simplified Swarm Optimization (SSO) and its comparison
strategy called solution update mechanism (SUM). Although many literatures have
presented these themes through a variety of contexts, this study focuses on improved
technique and its application on nOTSP. Thus, this chapter lays the foundation for
the further development that is discussed in the next chapter.
2.2 The Concept of Optimization
Optimization happens everywhere and anytime, it ranges from simple problem to
complex problem in daily routines. In addition to the industrial and scientific worlds,
optimization also plays a significant role in controlling and maintaining the
performance in minimizing and maximizing an objective function. For instance,
business organizations have to maximize their profit and minimize the cost,
engineering design has to maximize the performances of the designed product while
of course minimizing the cost at the same time (Yang, 2008).
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The root of “optimization” is “optimal” that carries the meaning of “best”,
“better” or “good enough” (Keeton et al., 2007; Fletcher, 2013). In other words, the
phrase “optimal solution” can be explained as the “best solution”. Blum & Roli
(2003) described optimization as concern the choice of a “best” configuration of a set
of variables in order to achieve the goals. On the other hand, optimization can be
defined as choosing the best solution among a given set of solutions (Khajehzadeh et
al., 2011). Therefore, the optimization theory and methods are needed to deal with
the problems by selecting the best alternative based on the given objective function
(Chong & Zak, 2013).
The area of optimization has received numerous attentions in recent years
particularly in the field of computer science, including the development of user-
friendly software, high performance processors as well as applied in solving various
high complexity problems by providing efficient solution from all feasible solutions.
For instance, in the Global Positioning System (GPS), optimization plays the role in
guiding the driver to reach the destination by discovering and providing the best
possible route. However, in general, the area of optimization can be divided into two
main categories, which are continuous optimization and combinatorial optimization
(Blum & Roli, 2003). Both types of optimization will be discussed in details in the
next subsection.
2.2.1 Continuous Optimization
Continuous optimization is a branch of optimization in applied mathematics and it is
the opposite of the discrete optimization, or combinatorial optimization. In
continuous optimization, the variables are allowed to take on any values, which are
usually real numbers (Gould, 2006). On the other hand, continuous optimization can
be defined as finding the minimum or maximum value of a function of one or many
real variable which subject to constraints. The constraints are usually in the form of
equations or inequalities. This characteristic of continuous optimization shows it is
different from combinatorial optimization, in which the variables in combinatorial
problem may be binary, integer, or abstract objects that are drawn finitely from sets
of many elements.
8
According to Gould (2006) & Saleh (2014), an optimization problem is the
minimization or maximization of an objective function f over a vector of variables x.
This is subject to a vector of constraints c that the variables in x must satisfy. An
optimization problem can be derived as in equation 2.1:
min 𝑓(𝑥) subject to {𝑐𝑖(𝑥) = 0 1 ≤ 𝑖 ≤ 𝑘
𝑐𝑖(𝑥) ≤ 0 𝑘 < 𝑖 ≤ 𝑚 (2.1)
where x = (𝑥1, 𝑥2, . . . ,𝑥𝑛) is a vector of n variables of the problem, f : ℝ𝑛→ℝ which
is the objective function to be minimized, {𝑐𝑖 (x) = 0 |1 ≤ i ≤ k} are the equality
constraints over the variables in the vector x and {𝑐𝑖 (x) ≤ 0 |k < i ≤ m} are the
inequality constraints over the variables in the vector x. By using this convention, the
standard form defines a minimization problem. A maximization problem can be
treated by negating the objective function f to −f.
In real-world scenarios, the continuous optimization has been applied in
many areas, such as optimize the high-pressure gas network in The National Grid
Gas National Transmission System (NTS) at United Kingdom (UK) and optimize the
electrical-power scheduling by minimizing the cost through controlling the flow of
the current (Gould, 2006).
However, the optimization problem in this research does not belong to this
category; instead, it belongs to the combinatorial optimization problem. The
combinatorial optimization will be discussed in detail in the next section.
2.2.2 Combinatorial Optimization
The field of optimization is a rapidly growing research field that concerned with the
choice of optimal solutions for a set of variables into achieving the objectives. In the
field of applied mathematics and theoretical computer science, combinatorial
optimization is a topic that consists of finding an optimal from a finite set of objects
(Schrijver, 2003). Cook et al., (2009) described that combinatorial optimization is a
combination of both combinatorics linear programming and the theory of algorithms
for solving the optimization problem over discrete structure. Furthermore, Luke
(2012) mentioned that the combinatorial optimization problem is the solution that
𝑥 ∈ ℝ𝑛
9
consists of a combination of unique components selected from a typically finite or
called set. In short, the ultimate purpose and objective are to find the optimal
combinatorial of components.
According to Blum & Roli (2003), a combinatorial optimization problem
P=(S, f) can be defined by, a set of variables X = {𝑥1, 𝑥2…..𝑥𝑛}, variable domains
𝐷1, 𝐷2 ….. 𝐷𝑛 , constraints among variables and an objective function f to be
minimized, where f : 𝐷1 × 𝐷2 ×…..× 𝐷𝑛 → ℝ+. As maximize the problem, one can
simply negate the objective function f to –f. Hence, the set of all possible feasible
assignments is shown here:
𝑆 = {𝑠 = {(𝑥1, 𝑣1), … , (𝑥𝑛, 𝑣𝑛)} | 𝑣𝑖 ∈ 𝐷𝑖, 𝑠 𝑠𝑎𝑡𝑖𝑠𝑓𝑖𝑒𝑠 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡𝑠}, (2.2)
where, S denotes the search space or solution space. Each element of the set can be
treated as the possible solution to the problem. Once the optimal solution 𝑠∗ ∈ 𝑆 with
minimum objective function has been found, that is, 𝑓(𝑠∗) ≤ 𝑓(𝑠) ∀𝑠 ∈ 𝑆. Thus, 𝑠∗
is called a global optimal solution of (S, f) and 𝑆∗ ⊆ 𝑆 is the set of globally optimal
solution (Blum & Roli, 2003).
One example of combinatorial optimization problem is the knapsack problem.
The purpose of the knapsack problem is to fill the items into the knapsack with the
total highest value yet without overfilling the knapsack (Luke, 2012). Another
ubiquitous example of combinatorial optimization problem is the vehicle routing
problem (VRP) that was proposed by Dantzig & Ramser (1959). The aim of this
problem is to determine a set of path with least cost, where a vehicle will depart from
the depot to visit each city to serve the demands exactly once by exactly one vehicle
and all the routes will start and end at the same depot. While, the total demand on
each route does not exceed the vehicle capacity. The VRP is illustrated in Figure 2.1
Besides VRP, another common problem involving combinatorial
optimization is the travelling salesman problem (TSP). The literature of TSP is
covered in the next section.
10
Figures 2.1: Illustration of the vehicle routing problem.
(Source: http://neo.lcc.uma.es/)
2.3 Travelling Salesman Problem (TSP)
TSP is a NP-hard problem that has been widely studied in the field of combinatorial
optimization (Yan et al., 2012). It was first formulated by Karl Menger in 1930
(Maredia, 2010; Singh & Lodhi, 2013). The name “Travelling Salesman Problem”
was introduced by Hassler Whitney in Princeton University at 1934 (Alexander,
2005). Figure 2.2 shows the illustration of the TSP. TSP can be described as follow,
a salesman who desires to visit n cities, and supposed to find out the shortest
Hamilton tour through visiting all the cities only once and finally returning to the city
where he started. In 1954, TSP was derived as an integer program and solved by
using cutting plane method (Dantzig et al., 1954).
Later, TSP was revealed as an NP-hard problem due to its computational
complexity in the manner of finding the optimal tour (Karp, 1972). Because the
problem is computationally difficult, a large number of heuristic and exact methods
have been proposed to provide the optimal solutions (Applegate et al., 2011).
11
Figure 2.2: Illustration of the Travelling Salesman Problem.
(Source: http://www.pixbam.com/germany-map/file:blank-map-germany-
states./2389)
According to Matai et al., (2010) the feasible solutions of TSP is given as (n-
1)!/2 where n represents the number of cities. TSP can be presented on a complete
undirected graph G = (V, E), where V = {1,…,n} is denoted to the vertex node or city,
E = {(i, j) : i, j ∈ V, i < j} is an edge set and A = {(i, j): i, j ∈ V, i ≠ j} is an arc set. A
non-negative distance matrix D = (𝑑𝑖𝑗) is defined on E or on A. In particular, this is
the case of planar problems in which the vertices are points 𝑃𝑖 = (𝑋𝑖,𝑌𝑖) in the plane,
and 𝑑𝑖𝑗=√(𝑋𝑖 − 𝑋𝑗)2+ (𝑌𝑖 − 𝑌𝑗)2 is the Euclidean distance. The triangle inequality
is also satisfied if 𝑑𝑖𝑗 is the length of a shortest path from i to j on G.
TSP is not just applied in route planning issue, it is applied in many of
today’s industry. The TSP has several applications, such as in global navigation
satellite system (GNSS) surveying networks in order to determine the geographical
positions of unknown points on and above the earth by using satellite equipments
(Saleh & Chelouah, 2004). In addition, Wakabayashi et al., (2014) have also
employed the TSP to determine the optimum location of the central post office in
Bangkok. Moreover, TSP has been applied in logistic practice such as in the
12
distribution of food products from producers to shops, the distribution of fuel to
petrol stations and the distribution of various products from producers or distributors
to customers (Filip & Otakar, 2011).
2.3.1 Variants of TSP
The idea of traditional TSP consists of a complete closed loop which visits all
the given cities once and returns to its original point. Due to the different scenarios
that happen today, TSP has been modelled into different variants. These variants of
TSP can be classified into two main categories which are the closed loop travelling
salesman problem (TSP) and open loop travelling salesman problem (OTSP). Both
problems carry the same objective which is to find the minimum tour length. The
only difference between them is the starting and ending points. TSP has the same
starting and ending point, while OTSP has the different starting and ending point
(Wang et al., 2013).
Apart from that, TSP and OTSP can also be divided into two sub-categories,
which are, single depot multiple salesman (SDMS) and multiple depots multiple
salesman (MDMS) (Tang et al., 2000; Nallusamy, 2013; Wang et al., 2013). In TSP,
the subcategories are known as single depots multiple salesman-TSP (SDMS-TSP)
and multiple depots multiple salesman-TSP (MDMS-TSP). However, in OTSP, the
subcategories are known as single depots multiple salesman-OTSP (SDMS-OTSP)
and multiple depots multiple salesman-OTSP (MDMS-OTSP). Literally, both SDMS
and MDMS are containing more than one salesman to operate in the same time;
meanwhile, the difference between both SDMS and MDMS is the number of depot.
The variants of TSP are shown in Figure 2.3.
However, this study will only focuses on OTSP due to the research on OTSP
is still limited compared with the researches on traditional TSP (Wang & Hou, 2013;
Vashisht, 2013). Meanwhile, TSP may not be reflective of today’s real-life
transportation scenarios whereby there are many vehicles today that travel from
different location and end at another location. The further discussion on OTSP is
discussed in the next section.
13
Variants of TSP
Closed Loop TSP (TSP)
Open Loop TSP (OTSP)
Single Depot Multiple
Salesman -Travelling
Salesman Problem
(SDMS-TSP)
Multiple Depot
Multiple Salesman -
Travelling Salesman
Problem (MDMS-TSP)
Single Depot Multiple
Salesman - Open Loop
Travelling Salesman
Problem (SDMS-
OTSP)
Multiple Depot
Multiple Salesman -
Open Loop Travelling
Salesman Problem
(MDMS-OTSP)
Figure 2.3: Variants of TSP
2.3.2 Open Loop Travelling Salesman Problem (OTSP)
OTSP can be modelled according to the real-life scenario of today’s transportation
services. The purpose of OTSP is to find the minimum total distance of the vehicle
when travelling from a starting point to the ending point by visiting all the given
cities exactly once. According to Čičková et al. (2013), the OTSP can be defined by,
n which refers to a set of nodes, the indices i and j refer to customers and take values
Depot
Depot
14
between 2 and n, while index i = 1 refers to the depot, 𝑑𝑖𝑗 refers to distance between i
and j, where i, j = 1, 2, …n. The binary variables 𝑥𝑖𝑗, i, j = 1, 2, ... n with a following
notation: 𝑥𝑖𝑗= 1 if customer i precedes customer j in a route of the vehicle and 𝑥𝑖𝑗= 0
otherwise, and variables 𝑢𝑖 , i = 2, 3,…n that based on the well-known Tucker’s
formulation of the TSP (Miller et al., 1960). Then, the formula for OTSP is:
min Σ𝑖=1𝑛 Σ𝑗=1
𝑛 𝑑𝑖𝑗𝑥𝑖𝑗 (2.3)
subject to
Σ𝑖=1𝑛 𝑥𝑖𝑗 = 1 j= 2, 3,…n i ≠ j (2.4)
Σ𝑗=2𝑛 𝑥𝑖𝑗 ≤ 1 i=1, 2,...n i ≠ j (2.5)
Σ𝑗=2𝑛 𝑥1𝑗 = 1 i= 2, 3,...n (2.6)
𝑢𝑖 − 𝑢𝑗 + 𝑛𝑥𝑖𝑗 ≤ 𝑛 − 1 i, j= 2, 3,...n i ≠ j (2.7)
𝑥𝑖𝑗 ∈ {0,1} i, j = 1, 2,...n i ≠ j (2.8)
The objective function (2.3) expresses the minimization of the total distance
of vehicle route; (2.4) is the standard constraints that ensure the vehicle visits every
customer; (2.5) is a constraints that ensure the vehicle does not need to depart from
every costumer, because the route ends after serving the last person; constraints (2.6)
to ensure the vehicle starts its route exactly once, (2.7) is the sub-tour elimination
constraints and (2.8) is the integrality constraints.
Recently, Vashisht (2013) have implemented GA in OTSP, and showed that
GA has proved its suitability to solve OTSP. However, the author also claimed that
GA has its difficulty to maintain the optimal solution over many generations.
Furthermore, he was suggested that perhaps there is better crossover or mutation
operators can be found and implemented to generate better solutions. Meanwhile,
Wang & Hou (2013) had employed a Simple Model (SModel) in multi-depots OTSP
to determine the best numbers of salesman with nearly minimum total distance. From
the experimental results, the reported performance was excellent and this almost
generates the minimum total distances.
Despite TSP and several of its extensions (e.g. Time windows) have been
applied perfectly for the route planning problem of today, but, only limited number
of researches has considered applying the OTSP. Meanwhile, OTSP still can be
15
modified and applied in today’s vehicle routing problems to provide better solution
for single vehicle travelling between the given source and its destination, this is
especially beneficial for logistic transportation routing such as for
merchandise delivery. Therefore, this study proposes another new variant of OTSP.
2.3.3 n-Cities Open Loop Travelling Salesman Problem (nOTSP)
The new variant of OTSP, which is proposed in this study named as n-Cities Open
Loop Travelling Salesman Problem (nOTSP). In the nOTSP, the salesman departs
from the starting city to another city without requiring him to visit all the given m
cities. However, he is restrained to visit only n cities with the minimum total distance.
This problem was inspired and modelled on real-life transportation problems. For
example, in the logistic transportation routings of merchandise delivery, the delivery
starts from the depot to the destination without passing through all the cities. Hence,
only limited number of cities is required. Figure 2.4 illustrates the difference of
OTSP and nOTSP. In Figure 2.4 (a) the pathway of a vehicle travels from a starting
point to the destination by required to visit all the cities in OTSP is illustrated.
Meanwhile, Figure 2.4 (b) illustrates the pathway of a vehicle that travels from the
starting point to the destination without being required to visit all the cities in the
nOTSP.
The formulation of the nOTSP is nearly similar as the formulation in OTSP.
In OTSP, the number of given cities, m is equal to the number of visited cities, n by
the salesman. In the other word, this can be defined as n = m. But in the nOTSP, the
number of cities to be visited n is not exactly equal to the total number of cities m
that has been given to the salesman and therefore, n≠m.
In the past, nature-inspired metaheuristic algorithms were highlighted to be
effective and efficient in solving the combinatorial problem like TSP. Examples of
these algorithms include the Tabu Search (TS) (Pedro et al., 2013), Particle Swarm
Optimization (PSO) (Gao et al., 2012), Ant Colony Optimization (ACO) (Hlaing &
Khine, 2011), Genetic Algorithm (GA) (Nagata & Soler, 2012) and Simulated
Annealing (SA) (Wang et al., 2013), which have been applied in solving TSP and its
variant.
16
Although there are many algorithms that can applied in TSP and its variants,
one of the best metaheuristic algorithms is GA (Abound & Abouchabaka, 2012;
Vashisht, 2013; Singh & Lodhi, 2013). The major reason behind this is its flexibility,
robustness and versatility, which have been widely studied to solve combinatorial
and optimisation problems such as in Singh & Lodhi (2013) and Singh & Singh
(2014). In addition, the study proposed by Philip et al. (2011) stated that GA is a
very good local search algorithm for solving TSP through generating a present
number of random tours and then improving the population until its stop condition is
met. Moreover, Vashisht (2013) also stated that GA is suitable to solve TSP because
it does not need to explore every possible solution in the feasible region in order to
obtain a good result. Hence, the GA will be discussed in the next section.
(a) (b)
Figure 2.4: The difference between OTSP and nOTSP. (a) Classic Open Loop
Travelling Salesman Problem (OTSP), (b) n-Cities Open Loop Travelling Salesman
Problem (nOTSP).
2.4 Genetic Algorithm
GA is one of the population based metaheuristic algorithm which belongs to the
larger class of evolutionary algorithm (EA). It was invented in the 1970s by John
Holland (Holland, 1975). GA is a type of randomized search technique which is
17
based on the natural selection and survival of the fittest chromosomes (Albayrak &
Allahverdi, 2011; Bahaabadi et al., 2012). Each chromosome is formed by genes.
The set of chromosomes is known as “population”.
The GA process starts by generating a random population based on the
principles of natural selection. In the population, each chromosome is evaluated to
determine the potential chromosomes. The potential chromosomes are selected for a
recombination process to produce new chromosomes to replace the poorer
chromosomes (Sallabi & El-Haddad, 2009). In this way, the better chromosomes
produced each new generation. The process will continue for many generations until
the condition is met. The following pseudocode describes the processes of GA.
A Genetic Algorithm Pseudocode
Step 1: Choose an initial random population of individuals, p.
Step 2: Evaluate the fitness of the individuals, f.
Step 3: repeat
Step 4: Select the best individual to be used by the genetic operators.
Step 5: Generate new individuals using crossover and mutation
operators.
Step 6: Evaluate the fitness of the new individuals.
Step 7: Replace the worst individuals of the population by the best
new individuals.
Step 8: until some stop criteria is met.
Algorithm 2.1: Genetic Algorithm (Goldberg & Holland, 1988)
In the past decades, GA has been successfully applied to many areas. For
example, GA was applied in the area of classification for the identification of genes
of similar function from a gene expression time series (To & Vohradsky, 2007).
Moreover, GA also has been applied in Airline Revenue Management (ARM) to
maximize the revenue of airline (George et al., 2012). In the area of control
engineering, GA was applied to control the seismic vibration in nonlinear multi-
damper configuration (Patrascu, 2015).
18
However, GAs often suffered from premature convergence that caused by the
loss of genetic diversity in the population (Malik & Wadhwa, 2014). Gupta & Ghafir
(2012) have considered the insufficiency of genetic diversity as the major reason that
causes GA to prematurely converge. In GA, insufficiency of genetic diversity tends
to lead the solutions converge towards the local optima or even the arbitrary points
rather than toward the global optimum (Ghosh, 2012). This phenomenon occurs
when the genetic operators can no longer produce offspring with a better
performance than their parents. In other word, sufficient genetic diversity in the
population could allow the algorithm continues searching for the better solutions,
avoiding them to be trapped at the local optima and become stagnant (Gupta &
Ghafir, 2012). Hence, in order to avoid the premature convergence happen, the
action of preserving the genetic diversity is needed.
2.4.1 Genetic Operators
GA maintains the genetic diversity and combines the existing chromosomes
with others through some mechanisms called genetic operators, such as encoding,
selection, crossover and mutation. Each operator has its own purpose and
responsibility.
The first operator in GA is known as the encoding operator. The purpose of
this encoding operator is to transform the problem solution into chromosome or
called gene sequence. There are many encoding techniques such as binary encoding,
permutation encoding, value encoding and tree encoding that can be applied
according to the model of the problem (Malhotra et al., 2011). Since that every
chromosome is a string of numbers in a sequence, the permutation encoding is the
best encoding for ordering or queuing problems such as TSP (Malhotra et al., 2011).
The second operator in GA is the selection operator. The role of the selection
operator is to select some chromosomes from the population based on their fitness
(Geetha et al., 2009; Malhotra et al., 2011). Individuals which are nearer to the
solution will have a high chance to be selected. In addition, there are few types of
selection operators, such as roulette wheel selection, proportional selection, ranking
selection, tournament selection, range selection, gender-Specific selection (GACD)
and GR based selection (Sivaraj & Ravichandran, 2011). The individuals that have
19
been selected will be moved to the mating pool while the remaining unselected
individuals are eliminated.
The mating pool is a place where the selected chromosomes (parents) will
undergo the recombining (mating) process to produce a new child (new chromosome
or offspring) (Geetha et al., 2009). Crossover operator is applied in this stage to
expect the better offspring to be produced from the parents. The examples of
crossover techniques are single point crossover, two point crossover, multi-point
crossover, uniform crossover and three parent crossover (Geetha et al., 2009).
Lastly, the new chromosomes are brought to the mutation operator. Mutation
operator manipulates and reallocates the genes in the chromosome hope to produce
better chromosomes or solutions that are closer to the fitness. Holland (1975)
underlined that the roles of mutation is to provide a guarantee to the algorithm is not
trapped on a local optimal and at the same time it introducing diversity. This view
was also supported by Sallabi & El-Haddad (2009), and Negnevitsky (2011) from the
perspective of algorithmic functioning that the purpose of the mutation is to prevent
the algorithm from being trapped in a local minimum and to avoid the loss of genetic
diversity.
Therefore, implement the mutation operator in the algorithm is crucial in
order to prevent the loss of diversity and from being trapped in local optima. The
examples of mutation techniques such as, flipping mutation, interchanging mutation,
boundary mutation and reversing mutation. More details in regards to the mutation
operators are presented on the next section.
2.4.2 Mutation Operators
The quality of GA solution relies on two important operators which are crossover
and mutation operators. The purpose of crossover operator is to exploit the current
solution in order to find the better ones. Meanwhile, the role of mutation is to
maintain the genetic diversity in order to prevent the algorithms from being trapped
in a local optimal and preventing the population of chromosomes from becoming too
similar to each other (Sivanandam & Deepa, 2007).
There have been numerous debates among researchers on the “usefulness”
and relative roles of crossover and mutation (Senaratna, 2005). Their opinion is
20
divided over the importance of crossover versus mutation. On one hand, Holland
(1975) claimed that crossover operator is more important than mutation operator. On
the other hand, there were scholars who believe that the role of mutation is more
significant than crossover. Meanwhile, Fogel & Atmar (1990), and Fogel (1990,
1993 and 2006) made a strong claim that crossover has no general advantage over
mutation since mutation can also do what crossover does. Furthermore, Fogel also
stated that mutation alone can do everything and it is very useful in optimising the
function task (Fogel & Atmar, 1990 and Fogel, 1993; 2006). Later, Sivanandam &
Deepa (2007) revealed that applying crossover operator into GA to solve TSP does
not produce good solution for overall performances. In addition, Thibert-Plante &
Charbonneau (2007) also found that crossover was not particularly helpful in
producing better solution and Zheng et al., (2010) have discovered the importance of
mutation, where without mutation, GA tends to converge prematurely.
There are few common mutation operators such as inversion mutation,
displacement mutation and pairwise swap mutation are usually found to be
implemented in GA to solve the TSP. These three mutations were used in the work
of Albayrak & Allahverdi (2011) and Singh & Lodhi (2013) to optimize the TSP in
term of finding the shortest tour. Therefore, these three mutation operators will be
used in the algorithm. The details of these three mutation operators are explained in
the next section.
2.4.2.1 Inversion Mutation
The inversion mutation performs inversion of the substring between two selected
cities. Figure 2.5 explains the inversion mutation concept. Suppose two selected
cities, which are city 9 and city 2. Then the substring is (9 3 7 4 6 2). After the
inversion mutation is performed, the substring (9 3 7 4 6 2) was inverted and become
(2 6 4 7 3 9).
Before mutation 1 5 9 3 7 4 6 2 8 0
After mutation 1 5 2 6 4 7 3 9 8 0
Figure 2.5: Before and after the inversion mutation was performed.
21
2.4.2.2 Displacement Mutation
Displacement mutation pulls the first selected gene out of the set of string and
reinserts it into a different place then sliding the substring down to form a new set of
string. In this case, city 9 was taken out from the tour and placed behind city 2, at the
same time the substring (3 7 4 6 2) was slid down to fill the empty space. This is
shown in Figure 2.6.
Before mutation 1 5 9 3 7 4 6 2 8 0
After mutation 1 5 3 7 4 6 2 9 8 0
Figure 2.6: Before and after the displacement mutation was performed.
2.4.2.3 Pairwise Swap Mutation
In pairwise swap mutation, the residues at the randomly chosen two positions
swapped. Sometimes, this technique is also called interchange mutation or random
swap (Sallabi and El-Haddad, 2009). For this case, the location of city 9 and city 2
will be swapped. This is shown in Figure 2.7.
Before mutation 1 5 9 3 7 4 6 2 8 0
After mutation 1 5 2 3 7 4 6 9 8 0
Figure 2.7: Before and after the pairwise swap mutation was performed.
22
2.4.3 Application of GAs on TSPs
Many researches had proven that GA and its hybrid variant have the potentials to
solve TSPs. Some of the reviews of the related literature are detailed in Table 2.1.
Table 2.1: Related literatures on GAs for TSPs.
Authors,
year
Problem
domain
Techniqu
e
Operators Performance
measurements
Results
Sallabi & El-
Haddad,
2009
TSP Improve
d Genetic
Algorith
m (IGA)
Crossover
operator: Swapped
Inverted crossover
(SIC).
Mutation
operators: Multi
mutation
Shortest distance. IGA can be
effectively
solving the
TSP. The total
distance found
by IGA is near-
optimal.
Khan et
al., 2009
Symmetric
TSP (STSP)
and
asymmetric
TSP (ATSP)
GA Crossover
operator: OR
crossover.
Mutation
operators:
Inversion mutation
Shortest distance. GA can give
near-optimal
solutions for
both STSP and
ATSP.
Yang et
al.，2013
TSP GA Crossover
operator: 2-points
crossover
Mutation
operators: Pairwise
swap mutation.
Shortest distance. GA
successfully
discovers the
shortest tour
when compared
to other
algorithms such
as SA, ACO
and PSO.
Arya et
al., 2014
Multiple
Traveling
Salesmen
Problem
(MTSP)
GA Crossover
operator: One-point
crossover.
Mutation
operators:
Inversion mutation
and pairwise swap
mutation.
Shortest distance
and computational
time.
The proposed
GA produced
better results;
the run time
was also
optimized. This
GA is suitable
for large-size
problems.
Liu, 2014 TSP GA Crossover
operator: Edge-
Swapping (ES)
crossover.
Mutation
operators: None.
Shortest distance. The proposed
GA has found
the optimal or
best known
solutions for
most
benchmark
instances and
reduces
computational
cost.
23
Yao, 2014 TSP GA +
PSO Crossover
operator: Edge-
Swapping (ES)
crossover.
Mutation
operators:
Inversion mutation
and pairwise swap
mutation.
Shortest distance
and convergence
rate.
The proposed
algorithm
overcomes the
drawbacks as
low
convergence
rate and local
optimum when
using PSO.
Chen &
Chien,
2011a
TSP GA+SA+
ACS+PS
O
Crossover
operator: Bone-
crossover and two-
point crossover
Mutation
operators: simulated annealing
mutation and
pheromone-
mutation
Shortest distance
and convergence
rate.
The proposed
algorithm
generates better
average tour
lengths and
smaller
percentage
deviations
compared to
previous
studies.
Chen &
Chien,
2011b
TSP GA+AC
S Crossover
operator: Bone-
crossover
Mutation
operators: Route-
mutation and
pheromone-
mutation
Shortest distance. The proposed
algorithm
generates better
average tour
lengths
compared to
previous
studies.
Zhang &
Lu, 2012
TSP GA+AC
O Crossover
operator: single
point crossover
Mutation
operators: Reversal mutation.
Shortest distance
and convergence
rate.
Proposed
algorithm has
higher
converging
speed, stability
and global
optimization
ability.
Dong et
al., 2012
TSP GA+AS Crossover
operator: single
point crossover
Mutation
operators: Reversal mutation.
Shortest distance
and convergence
rate.
Proposed
algorithm has
superior
performance for
solving TSPs in
terms of
capability and
consistency of
achieving the
global optimal
solution, and
quality of
average optimal
solutions,
particularly for
small TSPs.
From the reviews stated above, GA are proved of their suitability in solving
the TSPs in term of finding the shortest path. Apart from the finding of the shortest
path in TSP, the reviews above also revealed that the researchers were integrated the
24
GA with the swarm based algorithms for the purpose of improving the convergent
rate of the algorithm. This can be clearly seen in the researches of Yao (2014), Chen
& Chien, (2011a; 2011b), Zhang & Lu (2012) and Dong et al. (2012) that the
approach of integrating the GA with swarm-based algorithms have the potential to
aid the problem of prematurely convergence in the algorithm. Therefore, this study
also has the intention to integrate the swarm-based algorithm into the GA to
overcome its drawback in term of insufficiency of the genetic diversity that causing
the prematurely convergence of the algorithm.
2.5 Swarm Intelligence
Swarm intelligence is a sub-field of evolutionary computing. “Swarm” is a term
often used to describe a huge number of homogeneous living creatures or organisms
moving without central controls (Ahmed & Glasgow, 2012). For examples, colonies
of ants and bees, flocks of birds or schools of fishes. In the recent years, swarm-based
algorithms have been chosen and successfully applied in many areas to solve high
complexity problems through producing a set of effective solutions (Blum & Merkle,
2008; Hiot, 2010).
The expression of “Swarm Intelligence” has been used since 1989, when it
was first introduced by G. Beni and J. Wang in the context of cellular robotic
systems (Beni & Wang, 1989). Swarm intelligence can be defined as an efficient
computational model in the artificial intelligence (AI) field which was inspired by
the collective behaviors of the swarm of homogeneous living such as self-
organization, decentralized control and communication (Blum & Merkle, 2008;
Mishra et al., 2013).
The first swarm intelligence model is the ACO, which was introduced by
Dorigo et al. (1991; 1992; 2006). Consequently, more models were developed, such
as Particle Swarm Optimization (PSO) (Eberhart & Kennedy, 1995), Artificial Bee
Colony algorithm (ABC) (Karaboga, 2005) and Bat algorithm (BA) (Yang, 2010).
All these algorithms are also has been applied to solve the TSPs in the past and
proved to have good performance (Goldbarg et al., 2008; Li et al., 2011; Ouaarab et
al., 2014).
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