A PROJECT REPORT

ON

Genetic Algorithm for Travelling Salesman Problem

A Project Work (MS-405)

Submitted to Assam University, Silchar in partial fulfillment of the requirements for

the award of degree of Master of Science in Computer Science

Under the Guidance of

Dr.(Mrs.) ShahinAra Begum

Associate Professor

Department of Computer Science,

Assam University, Silchar

Submitted by

BishakhaChakrabarty

M.Sc. (2 years) 4th

Semester

Roll 101614 No 22220379

Regn. No.: 01-100008467 of 2010-2011

(Paper Code: MS-405)

Department of Computer Science

Assam University, Silchar- 788011

May 2016

DECLARATION

I, BishakhaChakrabarty, student of 4th

semester(M.Sc. 2 years), Department of Computer

Science do hereby solemnly declare that I have duly worked on my project entitled

―Genetic Algorithm for Travelling Salesman Problem‖ under the supervision of Dr.(Mrs.)

ShahinAra Begum, Associate Professor, Department of Computer Science, Assam

University, Silchar. This project work is submitted in the partial fulfillment of the

requirements for the award of degree of Master of Science in Computer Science. The

results embodied in this thesis have not been submitted to any other university or

institution for the award of any degree or diploma.

Date: (BishakhaChakrabarty)

Place: Assam University, Silchar Semester- 4th

sem(Msc 2 years)

Roll: 101614 No: 22220379

Regn. No: 01-100008467 of 2010-2011

Assam University, Silchar

CERTIFICATE

This is to certify that Miss. BishakhaChakrabarty, student of 4th

semester(M.Sc. 2 years),

Department of Computer Science, Assam University, Silchar, bearing the Roll: 101614

No: 22220379 has carried out her project work entitled ―Genetic Algorithm for

Travelling Salesman Problem‖ under my guidance in the partial fulfillment of the

requirement for the award of degree of Master of Science in Computer Science during the

period January 2016 to May 2016.

The project is the result of her investigation and neither the report as a whole, nor any

part of it has been submitted to any other university or institution for any degree or

diploma.

Date:Dr.(Mrs.) ShahinAra Begum

Place: Associate Professor,

Department of Computer Science,

Assam University, Silchar.

CERTIFICATE

This is to certify that Miss. BishakhaChakrabarty, student of 4th

semester(M.Sc. 2 years),

Department of Computer Science, Assam University, Silchar, bearing the Roll: 101614

No: 22220379 has carried out her project work entitled ―Genetic Algorithm for

Travelling Salesman Problem‖ under the guidance of Dr.(Mrs.) ShahinAra Begum,

Associate Professor, Department of Computer Science, Assam University, Silchar, in the

partial fulfillment of the requirement for the award of degree of Master of Science in

Computer Science during the period January 2016 to May 2016.

The project is the result of her investigation and neither the report as a whole, nor any

part of it has been submitted to any other university or institution for any degree or

diploma.

Date:Dr.BipulShyamPurkayastha

Place: Head of Department,

Department of Computer Science,

Assam University, Silchar.

ACKNOWLEDGEMENT

I would like to take the opportunity to extend my heartfelt gratitude to my project guide

Dr.(Mrs.) ShahinAra Begum, Associate Professor, Department of Computer Science,

Assaam University, Silchar, for her valuable time and for being always supportive to my

work and giving me the freedom of thought always.

I wish to express my deep sense of gratitude and indebtedness to Prof.

BipulShyamPurkayastha, Head, Department of Computer Science, Assam University,

Silchar, for providing me the opportunity to utilize the facilities of the Department of

Computer Science.

I would like to thank all the teachers of the Department of Computer Science, Assam

University, Silchar, for their useful discussion and advice throughout my project.

I am grateful to my parents, for their support, concern and help as and when required. I

thank all my friends and classmates for their support.

(BishakhaChakrabarty)

CONTENTS

Page No.

Declaration…………………………………………………………………… i

Certificates……………………………………………………………………. ii

Acknowledgement……………………………………………………….. iv

List of figures………………………………………………………………… vii

List of tables………………………………………………………………….. viii

Abstract…………………………………………………………………………. ix

Chapter-1 Introduction....................................................................... 1-7

1.1 Aim of the project…………………………………………………. 2

1.2 Search Techniques…………………………………………………. 3-7

Chapter-2 Literature Survey…………………………………………………………..8-9

Chapter-3 Genetic Algorithm…………………………………………………………. 10-16

3.1 Introduction…………………………………………………………… 10

3.2 Parameters of Genetic Algorithm……………………………. 11

3.3 Crossover Methods………………………………………………… 12

3.4 Mutation……………………………………………………………….. 13

3.5 Selection……………………………………………………………….. 13-14

3.6 Algorithm……………………………………………………………… 14

3.7 Flowchart……………………………………………………………… 15

3.8 Advantages and Disadvantages of GA…………………….. 16

3.9 Applications of GA………………………………………………….. 16

Chapter-4 Travelling Salesman Problem…………………………………… 17-22

4.1 Introduction…………………………………………………………. 17

4.2 Mathematical Formulation of TSP………………………… 17-18

4.3 Methods to solve TSP…………………………………………… 18-21

4.4 Applications of TSP………………………………………………. 22

Chapter- 5 Implementation…………………………………………………….. 23-25

5.1 Solving Travelling Salesman Problem using GA…….. 23

Chapter- 6 Results and Discussions…………………………………………. 26-31

Chapter- 7 Conclusions and Future Scope……………………………….. 32

7.1 Conclusion…………………………………………………………… 32

7.2 Future Scope……………………………………………………….. 32

References……………………………………………………………......................... 33-34

Appendices……………………………………………………………………………………. 35-39

Appendix A: Snapshots of I/O Screen……………………………. 35-38

Appendix B: Development Platform……………………………… 39

LIST OF FIGURES

Fig 1 Different search techniques to solve optimization problems ……… 3

Fig 2 Flowchart of Genetic Algorithm…………………………………..15

Fig 3 Snapshots of I/O screen……………………………………………35

LIST OF TABLES

Table 6.1(a) Results based on Roulette wheel selection for Uniform crossover ..26

Table 6.1(b) Results based on Roulette wheel selection for One point crossover.27

Table 6.1(c) Results based on Roulette wheel selection for Two point crossover.27

Table 6.2(a) Results based on Rank selection for Uniform crossover…………...28

Table 6.2(b) Results based on Rank selection for One point crossover………….29

Table 6.2(c) Results based on Rank selection for Two point crossover………….29

Table 6.3(a) Results based on Tournament selection for Uniform crossover……30

Table 6.3(b) Results based on Tournament selection for One point crossover…..30

Table 6.3(c) Results based on Tournament selection for Two point crossover….31

ABSTRACT

Genetic Algorithm is a search technique used in computing to find the optimal solution to

a computational problem that maximizes or minimizes a particular function. Genetic

Algorithm is used to solve the Travelling Salesman Problem where one has to find the

shortest route among the cities from the origin.

Genetic Algorithm is computer algorithm that search for good solutions to a problem

from various possible solutions. The basic elements of Genetic Algorithm are

chromosomes, fitness function, selection, crossover and mutation.

The Travelling Salesman Problem (TSP) is well known in the field of combinatorial

optimization. Since it is an NP-complete problem, there is no efficient method to solve

this problem and give the best result. Many algorithms are used to solve travelling

salesman problem. Some algorithms give optimal solution, but some other algorithms

give the nearest optimal solution. The genetic algorithm is a heuristic method which is

used to improve the solution space for the Travelling Salesman Problem. The genetic

algorithm results in nearest optimal solution within a reasonable time. This project

mainly focuses on the comparative study of different selection methods and crossover

operators in genetic algorithm to solve Travelling Salesman Problem and finally report

the results.

CHAPTER- 1

INTRODUCTION

Genetic Algorithm is used to solve the Travelling Salesman Problem where one

has to find the shortest route among the cities from the origin.

Genetic Algorithms are computer algorithm that search for good solutions to a problem

from various possible solutions. The basic elements of Genetic Algorithm are

chromosomes, fitness function, selection, crossover and mutation.

The computational paradigms of GA were inspired by the mechanics of natural

evolution, including survival of the fittest, reproduction, and mutation. These mechanics

are well suited to resolve a variety of practical problems, including computational

problems, in many fields.Some applications of Genetic Algorithm are combinatorial

optimization, automatic programming, machine learning and economics.

In the present work, a comparative study among three selection methods has been

carried out. Fitness costs are used to conduct a comparison test of performance among the

three selection methods. The results obtained from the comparison test are tabulated and

the best performance is found with Roulette Wheel Selection.

The project report is organized as follows: Chapter-1 introduces the aim and objectives of

the project, followed by a brief description of various search techniques. Chapter-2

describes the literature survey of Genetic Algorithm and Travelling Salesman Problem.

Chapter-3 describes the Genetic Algorithm, followed by a brief description of its basic

parameters, operators, working, applications and flowchart. In Chapter-4 the basic

concept of Travelling Salesman Problem is described, followed by mathematical

formulation, methods to solve Travelling Salesman Problem and its applications.

Chapter-5 describes the implementation of Travelling Salesman Problem using Genetic

Algorithm and description of GUI implementation in Java followed by the results and

analysis of three selection methods based on fitness measure in Chapter-6. Chapter-7

describes the conclusion and future work.

1.1AIM OF THE PROJECT

The aim of this project is to solve the Travelling Salesman Problem using Genetic

Algorithm.

The present study has been undertaken with the following objectives:

To study genetic algorithm.

To study various parameters and operators of genetic algorithm.

To study travelling salesman problem.

To find the solution of travelling salesman problem using genetic algorithm and its

operators.

To make a comparative study of different selection methods and crossover

operators and finally report the results.

1.2 SEARCH TECHNIQUES

There are various search techniques for solving optimization problems. The different

search techniques are shown in the figure 1. Among the various search techniques I have

chosen genetic algorithm to solve the travelling salesman problem.

Fig 1: Different search techniques to solve optimization problems.

Soft Computing

Fuzzy Systems Neural NetworksEvolutionary Computation

Genetic Algorithm

Genetic Programming

Machine Learning

1.2.1 SOFT COMPUTING:-

It is the fusion of methodologies that were designed to model and enable solutions to real

world problems, which are not modeled, or too difficult to model, mathematically.It is

associated with fuzzy, complex and dynamic system with uncertain parameters.

Quality of Service Networking is an example of the class of complex, fuzzy and dynamic

systems with uncertain parameters, which soft computing is intended to model and

compute.

Current Applications of Soft Computing include:-

Handwriting Recognition

Image Processing and Data Compression

Automotive Systems and Manufacturing

Decision –Support Systems

Neurofuzzy Systems

Fuzzy Logic Control

1.2.2EVOLUTIONARY COMPUTATION:-

Evolutionary computation simulates evolution on a computer. The result of such a

simulation is a series of optimization algorithms, usually based on a simple set of

rules. Optimization iteratively improves the quality of solutions until an optimal, or at

least feasible, solution is found.

Evolutionary Computations can be studied under 2 categories:

Genetic Algorithm

Genetic programming

GENETIC ALGORITHM

Genetic Algorithm is a branch of Artificial Intelligence stochastic search technique that is

widely used in the field of optimization. GA‘s are computer algorithms that search for

good solutions to a problem from various possible solutions. This is well suited to resolve

variety of practical problems, computational problems.

It randomly generates a set of possible solutions to a problem, representing each as a

fixed length character string. It then test each possible solution against the problem using

a fitness function to evaluate each solution. Keep the best solutions, and use them to

generate new possible solutions. Repeat the previous two steps until either an acceptable

solution is found.

GENETIC PROGRAMMING

In artificial intelligence, genetic programming (GP) is an evolutionary algorithm-based

methodology inspired by biological evolution to find computer programs that perform a

user-defined task. Essentially GP is a set of instructions and a fitness function to measure

how well a computer has performed a task. It is a specialization of genetic algorithms

(GA) where each individual is a computer program. It is a machine learning technique

used to optimize a population of computer programs according to a fitness landscape

determined by a program's ability to perform a given computational task. GP evolves

computer programs, traditionally represented in memory as tree structures. Trees can be

easily evaluated in a recursive manner. Every tree node has an operator function and

every terminal node has an operand, making mathematical expressions easy to evolve and

evaluate.

FUZZY SYSTEM

A fuzzy control system is a control system based on fuzzy logic—a mathematical system

that analyzes analog input values in terms of logical variables that take on continuous

values between 0 and 1, in contrast to classical or digital logic, which operates on discrete

values of either 1 or 0 (true or false, respectively) The term "fuzzy" refers to the fact that

the logic involved can deal with concepts that cannot be expressed as the "true" or "false"

but rather as "partially true". Although alternative approaches such as genetic algorithms

and neural networks can perform just as well as fuzzy logic in many cases, fuzzy logic

has the advantage that the solution to the problem can be cast in terms that human

operators can understand, so that their experience can be used in the design of the

controller. This makes it easier to mechanize tasks that are already successfully

performed by humans.

NEURAL NETWORK

a neural network is a system of programs and data structures that approximates the

operation of the human brain. A neural network usually involves a large number of

processors operating in parallel, each with its own small sphere of knowledge and access

to data in its local memory. Typically, a neural network is initially "trained" or fed large

amounts of data and rules about data relationships (for example, "A grandfather is older

than a person's father"). A program can then tell the network how to behave in response

to an external stimulus (for example, to input from a computer user who is interacting

with the network) or can initiate activity on its own (within the limits of its access to the

external world). neural networks use several principles, including gradient-based training,

fuzzy logic, genetic algorithms, and Bayesian methods.

MACHINE LEARNING

Machine learning is a subfield of computer science that evolved from the study of pattern

recognition and computational learning theory in artificial intelligence. Machine learning

explores the study and construction of algorithms that can learn from and make

predictions on data. Such algorithms operate by building a model from example inputs in

order to make data-driven predictions or decisions, rather than following strictly static

program instructions.

Machine learning is closely related to computational statistics; a discipline that aims at

the design of algorithms for implementing statistical methods on computers. It has strong

ties to mathematical optimization, which delivers methods, theory and application

domains to the field. Machine learning is employed in a range of computing tasks where

designing and programming explicit algorithms is infeasible. Example applications

include spam filtering, optical character recognition (OCR), search engines and computer

vision.

In the present work I have chosen genetic algorithm to solve the travelling salesman

problem. In the next chapter a survey on solving travelling salesman problem has been

carried out.

CHAPTER – 2

LITERATURE SURVEY

Varshika et al.,(2012) proposed the work which uses TSP domain and it is solved

using genetic algorithm operators. The genetic algorithm is used for the purpose of

improving the solution space. The crossover is the important stage in the genetic

algorithm.

Naveen et al.,(2012) have done the survey on the travelling salesman problem

using various genetic algorithm operators. The proposed work solves the travelling

salesman problem using various genetic algorithm operators. The various methods for the

genetic algorithm operators like selection methods, crossover methods and mutation

methods are also mentioned in the paper.

Omar et al.,(2009) proposed an improved genetic algorithm where the new

crossover operation, population reformulation operation, multi-mutation operation, partial

local optimal mutation operation, and rearrangement operations are used to solve the

Traveling Salesman Problem.

Chetan et al.,(2011) solved the travelling salesman problem using genetic

algorithm operators. The paper also includes a comparative study on various parent

selection methods such as Roulette Wheel, Elitism and Tournament selection methods for

Travelling Salesman problem. This paper concludes that all the three selection methods

give similar solution when the population size is small but when the population size is

large Elitism method gives the better result.

A. Arananayakgi(2014) solvedthe travelling problem using genetic algorithm

operators to reduce the total distance and time. This is achieved by generating the fittest

criteria using selection, crossover and mutation operators. The main aim of the proposed

method is to produce the high quality solutions in reasonable time. So a new crossover

method, the Sequential Constructive Crossover method is used. This method will selects

the better edges from the parent chromosome and produce a new offspring which may

have same edges as the parents or it may have new edges which is not present in the

parent chromosomes.

Kasassbeh et al.,(2012) proposed a new crossover method which is used and it is

called as a Shared Crossover method. This method is simple and fast and the main aim is

to reduce the execution time. This new technique depends on passing as many as possible

of the shared paths between cities to the next generation with a guarantee that none of the

cities will appear in the child chromosome more than once after applying the crossover

method. The experimental results also show the greater reduction in the time

consumption.

In chapter-3, genetic algorithm, parameters of genetic algorithm and the various operators

of genetic algorithm has been discussed.

CHAPTER - 3

GENETIC ALGORITHM

3.1 INTRODUCTION

Genetic Algorithm is a branch of Artificial Intelligence stochastic search technique that is

widely used in the field of optimization. GA‘s are computer algorithms that search for

good solutions to a problem from various possible solutions. This is well suited to resolve

variety of practical problems, computational problems.

It randomly generates a set of possible solutions to a problem, representing each as a

fixed length character string. It then test each possible solution against the problem using

a fitness function to evaluate each solution. Keep the best solutions, and use them to

generate new possible solutions. Repeat the previous two steps until either an acceptable

solution is found.

GA was proposed by John Holland, his students and his colleagues at the University of

Michigan. Carrasco et al (2001) Genetic Algorithm (GA) model introduced by John

Holland in 1975 triggered a wide interest in the application of types of heuristic, which

mimic the natural evolutionary characteristic present in the biological species with the

purpose of solving optimization problem efficiently.

Randy et al (1998) GAs is a branch of artificial intelligence‘s stochastic search technique

that is widely used in the field of optimization.

According to Biesbroek (1999) the study of GA was originated from studies done on

cellular automata, which took place when Goldberg was at the University of Michigan.

John Koza(1992) has used genetic algorithm to evolve programs to perform certain tasks.

He called his method ―genetic programming‖ (GP).

3.2 PARAMETRS OF GENETIC ALGORITHM:-

CHROMOSOME:- A set of genes; chromosome contains the solution in the form of

genes.

GENE:- A part of chromosome; a gene contains a part of solution. It determines the

solution. For e.g- 10011 is a chromosome and 1,0,0,1,1 are its genes.

POPULATION SIZE:-It says how many chromosomes are in population.It depends on

the type of encoding and the problem. If there are only few chromosomes, then GA

would have a few possibilities to perform crossover and only a small part of search space

is explored. If there are many chromosomes then GA slows down.

REPRODUCTION:- It is usually the first operator applied on population. Chromosomes

are selected from the population of parents to cross over and produce offspring. It is

based on Darwin‘s evolution theory of ―Survival of the fittest‖. Therefore, this operator is

also known as ‗Selection Operator‘.

CROSS OVER:-After reproduction phase, population is enriched with better

individuals. It makes clones of good strings but does not create new ones. Cross over

operator is applied with a hope that it would create better strings.

MUTATION:-After cross over, the strings are subjected to mutation. Mutation of a bit

involves flipping it,changing 0 to 1 and vice-versa.

3.3 CROSSOVER METHODS:-

3.3.1 Single Point Crossover

A commonly used method for crossover is called single point crossover. In this method,

a single point crossover position is chosen at random and the parts of two parents after

the crossover position are exchanged to form two offspring.

Parent 1: 1 0 1 1 | 1 1 1

Parent 2: 0 1 0 1 | 0 0 1

Offspring 1: 1 0 1 1 1 0 0 1

Offspring 2: 0 1 0 1 0 1 1 1

3.3.2 Multipoint Crossover

It is a generalization of single point crossover, introducing a higher number of cut

points.

Parent 1: 1 0 0 |1 0 1| 1 1

Parent 2: 0 1 1 |1 0 0 | 0 1

Offspring 1: 1 0 0 1 0 0 1 1

Offspring 2: 0 1 1 1 0 1 0 1

3.3.3 Uniform Crossover

It doesn‘t use cut points, but simply uses a global parameter to indicate the

likelihood that each variable should be exchanged between two parents.

Parent 1: 1 0 1 0 0 0 1 1 1 0

Parent 2: 0 0 1 1 0 1 0 0 1 0

Offspring 1: 0 0 1 1 0 0 1 0 1 0

Offspring 2: 1 0 1 0 0 1 0 1 1 0

3.4 MUTATION:-

It represents a change in the gene. The mutation operator flips a randomly selected gene

in a chromosome. It changes the bit 0 to 1 and vice-versa.

Original Offspring: 1 0 1 1 0 1 1 1

Mutated Offspring: 1 0 1 1 0 0 1 1

3.5 SELECTION:-

It is the process of determining the number of times a particular individual is chosen for

reproduction and thus the number of offspring that an individual will produce.

3.5.1 Types of Selection Method

I. Roulette Wheel Selection

II. Tournament Selection

III. Half-Truncation Selection

Roulette Wheel Selection:-

The simplest selection scheme is roulette wheel selection, also called stochastic sampling

with replacement. This is a stochastic algorithm and involves the following technique:

The individuals are mapped to contiguous segments of a line, such that each individual‘s

segment is equal in size to its fitness. A random number is generated and the individual

whose segment spans the random number is selected. The process is repeated until the

desired number of individuals is obtained(called mating population).

Tournament Selection:-

In tournament selection a number Tour of individuals is chosen randomly from the

population and the best individual from this group is selected as parent. This process is

repeated as often as individuals to choose. These selected parents produce uniform at

random offspring. The parameter for tournament selection is the tournament size Tour.

Tour takes values ranging from 2- Nind(number of individuals in population).

Half-Truncation Selection:-

Compared to the previous selection methods modeling natural selection is an artificial

selection method. It is used by breeders for large population/mass selection. In truncation

selection individuals are sorted according to their fitness. Only the best individuals are

selected for parents. These selected parents produce uniform at random offspring. The

parameter for truncation selection is the truncation threshold Trunc. Trunc indicates the

proportion of the population to be selected as parents and takes values ranging from 50%-

100%. Individuals below the truncation threshold do not produce offspring.

3.6 PSEUDO CODE OF GENETIC ALGORITHM

Begin

INITIALIZE population with random candidate solutions;

EVALUATE each candidate;

Repeat

SELECT parents;

RECOMBINE pairs of parents;

MUTATE the resulting children;

EVALUATE children;

SELECT individuals for the next generation

Until TERMINATION-CONDITION is satisfied

End

3.7 FLOWCHART

Fig 2: Flowchart for Genetic Algorithm.

3.8ADVANTAGES AND DISADVANTAGES OF GA:-

3.8.1 ADVANTAGES

1. Easy to understand.

2. Unlike older AI systems, the GA‘s do not break easily even if the inputs changed

slightly.

3. GA can be employed for a wide variety of optimization problems.

4. GA performs very well for large scale optimization problems which may be very

difficult or impossible to solve by other traditional methods.

3.8.2 DISADVANTAGES

1. Sometimes have trouble finding the exact global optimum because there is no

guaranty to find best solution.

2. It requires large number of fitness function evaluations depending on the number

of individuals and the number of generations.

3. GA may take long time to evaluate the individuals.

3.9APPLICATIONS OF GENETIC ALGORITHM

I. Scheduling: Facility, Production, Job and Transport Scheduling.

II. Design: Circuit board layout, Communication Network design, Parametric

design in aircraft.

III. Control: Missile evasion, Gas pipeline control, Pole balancing.

IV. Machine Learning: Designing Neural Networks, Classifier Systems, Learning

rules.

V. Robotics: Trajectory Planning, Path Planning.

VI. Combinatorial Optimization: TSP, Bin Packing, Set Covering, Routing.

VII. Image Processing: Pattern recognition.

VIII. Medical: Studying health risks for a population exposed to toxins.

In chapter-4, Travelling Salesman problem(TSP), mathematical formulation of TSP and

various methods to solve TSP has been discussed.

CHAPTER- 4

TRAVELLING SALESMAN PROBLEM

4.1 INTRODUCTION

Travelling salesman problem(TSP) is a well- known classic problem from computer

science. Its mathematical formulation is simple, and one can state a simple strategy to

solve. Such a strategy is often impractical and as yet there is no efficient algorithm for

this problem that works consistently in all instances.TSP is a choice problem for

combinatorial optimization problems and one that has received the most attention.

Another thing to note is that one can‘t find for the Travelling Salesman Problem a

consistently efficient method of solution, as it belongs to a set of combinatorial

problem that are called NP-complete problems. NP-complete problems have the

distinction that there is no known algorithm that is efficient, practical and works in all

instances.

The Travelling Salesman Problem is easy to state: given a finite number of cities along

with the cost of travel between each pair of them, TSP is the search of the shortest tour

that visits the given cities exactly once.

4.2 MATHEMATICAL FORMULATION OF TSP

Mathematically, the goal of this problem is to find a tour, among all the possible tours,

that minimizes the total distance the salesman travels. The other criterion that the

problem has to satisfy is that each city should be visited once and only once, except that

he returns to the city from which the salesman starts.

TSP can be formulated as an integer linear program. Label the cities with the numbers

0,…,n. For n cities to visit, let xij be the variable that has a value 1 if the salesman goes

from city i to city j and a value 0 if the salesman does not go from city i to city j. Letdijbe

the distance from city i to city j.

Then the TSP can be stated as-

Minimize the linear objective function:

𝑍 = 𝑥𝑛𝑗≠𝑖 𝑗=0

𝑛𝑖=0 ijdij

0≤xij≤1 i, j= 0,…,n

Subject to the constraint,

𝑥𝑛𝑖=0,𝑖≠𝑗 ij=1 j=0,…n

𝑥𝑛𝑗=0,𝑗≠𝑖 ij=1 i=0,…,n

4.3 METHODS TO SOLVE TSP

4.3.1 Total enumeration method

In principle it is a combinatorial solution. The method rests in evaluation of all potential

routes (sequences) in the total number of (n – 1)!. Theadvantage is that a global optimum

is always found, however, it is not employable if higher numbers of visited places are

considered. With every added element (node) the amount of possible solutions grows

exponentially and not even nowadays do we have computers powerful enough for being

able to provide optimum solution within reasonable time.

4.3.2 Branch and Bound method

This method belongs to the oldest ones and the most often used algorithms for the TSP

solutions. The merit of the method rests in a gradual decomposition of a possible solution

set into a number of mutually disjunctive subsets labelled asbranches. In each step the

following is estimated:

The upper limit of the objective function that is most often the value of the objective

function zHwithout respecting limits and maximum lower bound of an objective function

zDof acceptable solutions which are known to us within the step.

Both the estimates can be employed for seeking non-prospective directions of further

procedures: if for any branch zH<zD, then the given direction canbe excluded. However,

this method is also, especially for higher n, too laborious and does not always guarantee

an optimum solution at the first attempt.

4.3.3 Clarke and Wright Algorithm

A significant progress in TSP solutions was provided by the Clarke‘s and Wright‘s

method. The initial situation assumes that each place is supplied individually and always

a return to the starting base follows. The essential idea is based on thecalculation of

economies achieved through integrating other places into the circular route. An

indisputable asset of this algorithm is its function to respect further restrictions often

generated by the practice, e.g. the need to optimize more orbitalroutes, to use more

vehicles while respecting their various capacities etc.

Guerra, Murino and Romano worked with thisalgorithm for optimize the routing phase in

a Location-Routing Problem (LRP) in. LRP can by assimilate to a Vehicle Routing

Problem (VRP) and after that they combine and balance VRP with TSP. Both problems

were solved with Clarke and Wright saving algorithm and the Branch and Bound model.

4.3.4 Ant Colony Optimization(ACO) Algorithm

ACO is one of the metaheuristic methods for solving TSP. Jalali, Afshar and Marino

described ACO as observation of real ants, and upon finding food return to their colony

while laying down pheromone trails. If other ants find such a path, they are likely not to

keep travelling at random, but to instead follow the trail, returning and reinforcing it if

they eventually find food. There is a higher probability that the trail with a higher

pheromone concentration will be chosen.

The pheromone trail allows ants to find their wayback to the food source and in the

opposite way.The trail is used by other ants to locate food sourcediscovered by any ant.

When a number of pathsavailable from the nest to a food source, a colony ofants may by

able to exploit the pheromone trail leftby individual members of the colony to

discoverthe shortest path from the nest to the food sourceand back. As more ants choose a

path to follow, thepheromone on the path builds up, making it moreattractive to be

followed by other ants.

To solve TSP, we keep the strength of pheromone trail _i,jfor each combination of two

points. The role of each ant is to find a valid solution, thuspossible routes. From the

starting point, the ant gradually repeats a move when it chooses a place where it has not

been yet while moving to it fromits current location. Once there are no more vacancies

left, the ant returns to the starting location. ACO is one of the Swarm Intelligence

systems which include many other algorithms such as Particle Swarm Optimization and

River formationdynamics.

4.3.5 Particle Swarm Optimization(PSO) Algorithm

PSO proceed from the social behavior of organisms such as bird flocking and fishing

schooling. Through cooperation between individuals, the group often can achieve their

goal efficiently and effectively. PSO simulates this social behavior asan optimization tool

to solve some optimization problems. Each particle flies in the search spacewith a

velocity that is dynamically adjusted basedon its own flying experience and its

companions‘flying experience. In other word, every particle willutilize both the present

best position information of its own (pbest) and the global best position information

(gbest) that swarm has searched up-tonowto change its velocity and thus arrives in the

new position.

4.3.6 Genetic algorithms(GA)

In recent years there have been attempts to use so called genetic algorithms for TSP

solutions. Simply stated, genetic algorithms transfer evolution principles in living

organisms into intelligent searching and model optimization in other fields.

Biological terminology is applied also to this very description of the algorithm. Genetic

algorithm, as well as nature, works with population of individuals (P) defined by one or

more mathematical genes – chromosomes (i.e. sequencesof numbers in binary notation.

The genetic algorithm (GA) uses the following steps:

1. Generate a population- The GA randomly samples values of the changing cells

between the lower and upper bounds to generate a set of (usually at least 50)

chromosomes. The initial set of chromosomes is called the population.

2. Create a new generation- In the new generation, chromosomes with a smaller

fitness function (in a minimization problem) have a greater chance of surviving to

the next generation. Crossover and mutation are used to generate chromosomes for

the next generation.

3. Stopping conditions- At each generation, the best value of the fitness function in

the generation is recorded, and the algorithm repeats step 2. If no improvement in the

best fitness value is observed after many consecutive generations GA terminates.

4.4 APPLICATIONS OF TSP:-

VEHICLE ROUTING:-

Suppose that in a city ‗n‘ mail boxes have to be emptied everyday within a certain period

of time, say one hour. The problem is to do the particular work with less number of

vehicles in minimum time.

SCHOOL BUS ROUTING PROBLEM:-

The objective of this problem is to obtain a bus loading pattern such that the number of

routes is minimized.The total distance travelled by all buses is kept at minimum.

MISSION PLANNING PROBLEM:-

The objective is to determine an optimal path for army men to accomplish the goals of

the mission in the minimum possible time.

PRINTING PRESS SCHEDULING PROBLEM:-

One of the major and primary applications of the mTSP arises in scheduling a printing

press for a periodical with multi-editions. Here, there exist five pairs of cylinders between

which the paper rolls and both sides of a page are printed simultaneously. There exist

three kind of forms, namely 4-, 6- and 8-page forms, which are used to print the editions.

The scheduling problem consists of deciding which form will be on which run and the

length of each run. In the mTSP vocabulary, the plate change costs are the inter-city

costs.

CREW SCHEDULING PROBLEM:-

An application for deposit carrying between different branch banks is reported by

Svestka&Huckfeldt (1973). Here, deposits need to be picked up at branch banks and

returned to the central office by a crew of messengers. The problem is to determine the

routes having a total minimum distance.

CHAPTER- 5

IMPLEMENTATION

5.1 Solving Travelling Salesman Problem using Genetic Algorithm

Finding a solution to the travelling salesman problem requires that we set up genetic

algorithm in a specialized way. For instance, a valid solution would need to represent a

route where every location is included at least once and only once. If a route contain a

single location more than once, or missed out a location completely then it wouldn't be

valid .To ensure the genetic algorithm does indeed meet this requirement special types of

mutation and crossover methods are needed.Firstly, the mutation method should only be

capable of shuffling the route, it shouldn't ever add or remove a location from the route,

otherwise it would create an invalid solution. One type of method we can use is swap

mutation.With swap mutation two location in the route are selected at random then their

positions are simply swapped. Because swap mutation is only swapping pre-existing

values, it will never create a list which has missing or duplicate values when compared to

the original, and that's exactly what we want for the Travelling Salesman Problem. One

crossover method that's able to produce a valid route is ordered crossover.

In this project locations are typically referred to as cities, and routes are referred to as

tours.In this crossover method we select a subset from the first parent, and then add that

subset to the offspring. Any missing values are then added to the offspring from the

second parent in the order they are found.

Parents:

Offspring:

Representation of the cities

In our project cities are represented with the help of X- coordinates and Y-coordinates of

a particular point, which looks as follows:

Cities:

0 1 2 3

20|30 5|10 10|20 40|50

Selection functions

In the implementation of this project, we tried three selection methods: Roulette wheel

selection, Tournament selection and Half truncation selection, where the fittest

chromosomes are copied without changes to the new population, so the best solutions

found will not be lost.

Fitness Cost

It is a type of function that is used to summarise how close a solution is from the desired

solution.

Fitness Cost= 1/ Distance

Complexity of the program

An exhausted search for a TSP has an exponential time complexity. Suppose that ‗n‘ is

the number of vertices. An exact solution takes O(n!) time, which is prohibitively long.

Therefore, polynomial-time heuristic search approaches are proposed. Heuristic or local

search algorithms have complexities ranging from O(n2)(e.g., nearest neighbour, double

ended nearest neighbour and nearest insertion), O(n2log(n)) (e.g.,shortest edge first).

The genetic algorithms with specialized operators have time complexity O(kmn2) where

‗k‘ is the generation number and ‗m‘ is the population size. The n2 factor is due to the fact

that all the repair procedures need to scan all the possible pairs of the vertices which is

O(n2).(P Merz& B Freisleben, 1997)

GUI Implementation in JAVA

The project was implemented in Java. The graphical user interface (GUI) was designed to

solve the Travelling Salesman Problem using Genetic Algorithm which has two popup

menu for choosing selection and crossover method. The provision for dynamic user input

was also made to provide basic elements of GA such as number of generations, number

of cities and mutation rate to find the new population from the randomly created initial

population. After selecting a method from both the selection and crossover methods and

on submitting the input values, the initial distance and the final distance are evaluated. It

also shows the X-coordinates and Y-coordinates of the cities. A city map is generated

with the help of X-coordinates and Y-coordinates of the cities. The city map shows the

shortest route taken by the salesman to complete his tour.

Source code Details

The source code that implemented on Java has two packages tspGA and TspGui. The

files included in these packages are as follows:

City.java models a city. TourManager.java holds the tour of the cities. Tour.java

stores a candidate tour. Population.java manages the population of candidate tours.

GA.java, it adds the cities and evolve a tour for travelling salesman

problem.TspGuidisplays the main window where all the operations are performed by the

user using graphical user interface.

CHAPTER- 6

RESULTS AND DISCUSSIONS

This chapter describes the performance of Genetic Algorithm for solving Travelling

Salesman Problem based on three Selection methods. The initial population is generated

randomly. A comparative analysis was done among the three selection methods.

Number of Cities : 10

Number of Generations: 100

Mutation Rate : 0.012

The table 6.1 (a-c) shows the performance based on Roulette wheel selection. After

mutation, the new population and their fitness cost are generated from the initial

population for uniform crossover, one point crossover and two point crossover method.

The results obtained are analyzed. The minimum fitness cost obtained in this selection

method is 0.00123 which is for two point crossover method.

Table 6.1(a): Results based on Roulette wheel Selection with Uniform crossover

City=10 X- coordinate Y- coordinate Fitness cost(uniform

crossover)

1 47 191 0.00216

2 49 193 0.00216

3 41 197 0.00328

4 43 137 0.00414

5 65 135 0.00460

6 67 117 0.00558

7 66 127 0.00558

8 42 123 0.00621

9 45 119 0.00621

10 55 141 0.00621

Table 6.1(b): Results based on Roulette wheel Selection with One point crossover

City=10 X- coordinate Y- coordinate Fitness-cost(one-

point crossover)

1 55 135 0.00127

2 30 108 0.00127

3 100 62 0.00135

4 132 55 0.00135

5 159 12 0.00135

6 152 120 0.00143

7 196 198 0.00143

8 158 216 0.00144

9 97 208 0.00144

10 29 248 0.00144

Table 6.1(c): Results based on Roulette wheel Selection with Two point crossover

City=10 X- coordinate Y- coordinate Fitness-cost(two

point crossover)

1 53 151 0.00123

2 73 130 0.00132

3 164 28 0.00135

4 103 1 0.00135

5 20 66 0.00140

6 33 110 0.00141

7 13 127 0.00141

8 18 192 0.00141

9 122 241 0.00141

10 89 167 0.00141

Table 6.2(a): Results based on Rank Selection with Uniform crossover

City=10 X- coordinate Y- coordinate Fitness

cost(uniform)

1 161 158 0.00152

2 161 158 0.00210

3 161 158 0.00210

4 153 171 0.00210

5 153 171 0.00324

6 153 171 0.00324

7 153 171 0.00354

8 88 230 0.00490

9 88 230 0.00490

10 153 171 0.00490

Table 6.2(b): Results based on Rank Selection with One point crossover

City=10 X- coordinate Y-coordinate Fitness-cost(one

point crossover)

1 180 211 0.00127

2 91 215 0.00134

3 87 246 0.00134

4 14 234 0.00147

5 4 225 0.00147

6 23 127 0.00149

7 16 94 0.00156

8 31 69 0.00156

9 78 71 0.00156

10 88 133 0.00167

Table 6.2(c): Results based on Rank Selection with Two point crossover

City=10 X- coordinate Y- coordinate Fitness-cost(two

point crossover)

1 99 76 0.00124

2 7 31 0.00125

3 28 136 0.00127

4 50 170 0.00127

5 107 203 0.00140

6 187 193 0.00140

7 151 135 0.00140

8 139 144 0.00142

9 160 91 0.00146

10 143 8 0.00146

The table 6.2 (a-c) shows the performance based on Rank selection. After mutation, the

new population and their fitness cost are generated from the initial population for uniform

crossover, one point crossover and two point crossover methods. The resultsobtained are

analyzed. The minimum fitness cost obtained in this selection method is 0.00124 which is

for two point crossover method.

Table 6.3(a): Results based on Tournament Selection with Uniform crossover

City=10 X- coordinate Y- coordinate Fitness

cost(uniform)

1 140 153 0.00131

2 140 153 0.00156

3 140 153 0.00194

4 187 97 0.00196

5 192 103 0.00219

6 192 103 0.00228

7 192 103 0.00387

8 192 103 0.00387

9 148 206 0.00408

10 148 206 0.00408

Table 6.3(b): Results based on Tournament Selection with One point crossover

City=10 X- coordinate Y- coordinate Fitness-cost(one

point crossover)

1 100 199 0.00142

2 73 231 0.00142

3 46 233 0.00153

4 28 167 0.00162

5 86 44 0.00167

6 75 20 0.00179

7 114 27 0.00179

8 150 33 0.00179

9 131 72 0.00183

10 98 163 0.00183

Table 6.3(c): Results based on Tournament Selection with Two point crossover

City=10 X- coordinate Y- coordinate Fitness- cost(two

point crossover)

1 180 202 0.00130

2 199 164 0.00137

3 194 135 0.00137

4 137 165 0.00137

5 137 138 0.00137

6 171 67 0.00137

7 119 11 0.00140

8 11 82 0.00140

9 30 103 0.00140

10 109 239 0.00140

The table 6.3(a-c) shows the performance based on Tournament selection. After

mutation, the new population and their fitness cost are generated from the initial

population for uniform crossover, one point crossover and two point crossover methods.

The results obtained are analyzed. The minimum fitness cost obtained in this selection

method is 0.00130 which is for two point crossover method.

Considering the overall performance, the fitness cost are evaluated for Roulette wheel

selection, Rank selection and Tournament selection methods. The minimum fitness cost

and hence the best solution was given by the Roulette wheel selection method where the

fitness cost is minimum i.e., 0.00123. Also, the best crossover method found is two point

crossover method.

CHAPTER- 7

CONCLUSION AND FUTURE SCOPE

7.1 CONCLUSION

In this project, the three selection methods viz. Roulette wheel selection, Rank selection

and Tournament selection methods and the three crossover methodsviz.uniformcrossover,

one point crossover and two point crossover methods are implemented. A city map is

drawn to show the cities travelled by the salesman. It also shows the route taken to visit

all the cities. The results obtained are analyzed and tabulated for Roulette wheel

selection, Rank selection and Tournament selection using Uniform crossover, One point

crossover and Two point crossover methods. The best selection method was found to be

Roulette wheel selection based on the analysis of fitness cost. The minimum fitness cost

found is 0.00123 which is for Two point crossover method.

7.2 FUTURE SCOPE

In this project, basic genetic algorithm is used to solve the Travelling Salesman Problem.

As a future work, it can also be extended for hybrid genetic algorithm and parallel genetic

algorithm. The performance between the two algorithms can also be analyzed based on

the selection methods and crossover operators.

REFERENCES

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APPENDIX A

SNAPSHOTS OF I/O SCREEN

Fig 3: Performing selection and crossover operation to get the optimal solution and the

city map.

APPENDIX B

DEVELOPMENT PLATFORM

B.1 Operating System:

Windows 7

B.2 Software Tools:

Platform used- Java Netbeans IDE 7.0.1

Testing- Java Netbeans IDE 7.0.1

Java is a set of computer software and specifications developed by Sun Microsystems,

which was later acquired by the Oracle Corporation, that provides a system for

developing application software and deploying it in a cross- platform computing

environment.

B.3 Hardware Configuration:

Processor: Intel(R) Core(TM)i3-2370M CPU @ 2.40GHz 2.40GHz

RAM: 4.00 GB

System Type: 32-bit operating system