Company

LOGO

Searching Solutions of C-TSP

Harbin Institute of Technology

Lecturer: HUA Dingguo

Tutor: YAN Jihong

Utilizing GA

Contents

GA Introduction2

Searching Result4

Definition of C-TSP31

Searching Process33

References5

1. Definition of C-TSP

What is C-TSP?

C-TSP: China Travelling Salesman Problem Cities: 31 cities including [1]

• Beijing, Shanghai, Tientsin, Shijiazhuang, Taiyuan, Hohhot, Shenyang, Changchun, Harbin, Sian, Lanzhou, Yinchuan, Xining, Urumqi, Jinan, Nanking, Hangzhou, Hefei, Nanchang, Foochow, Taipei, Chengchow, Wuhan, Changsha, Canton, Nanning, Haikou, Chengdu, Guiyang, Kunming, Lhasa

1. Definition of C-TSP

Starting point: Beijing

Destination: Back to Beijing

Constraint: Every city has to be visited Every city except Beijing can be visited for

ONLY ONCESearching Target:

The shortest travelling path

Straight-Line Path Only straight-line path is considered for the simplicity of the problem

Direct Arrival

Direct arrival can be realized between any 2

of the 31 cities

Assumptions

1. Definition of C-TSP

2. GA Introduction

InverseMutation

MutationMutation

Distancerelated

FitnessFitnessEvaluationEvaluation

Roulette

SelectionSelection

Oder Crossover

CrossoverCrossoverPopulation

Near-Optimal Solution

2.1 Population & Encode

Population: The scale of initial population is very

crucial to the performance of GA; If the scale is too small, the diversity is

not guaranteed; If the scale is too large, the computing is

hence time consuming; The scale is finally determined as 500

321(31 1)! 1.3263 10

2 Problem

Scale

Relatively Small

2.1 Population & Encode

Encode Since the cities can be denoted as integers

• 1-Beijing; 2-Shanghai; 3- Tientsin …

Every chromosome can be encoded in the form of integer string of 1 to 31 which is arranged in a random order

Example 1-23-7-4-17-12-31-8-29-18-45-9-6-30-22-26-28-27-20-16-2-24-

3-5-19-25-14-10-21-11-13

2.2 Fitness Evaluation

Distance is the major concern of C-TSP

the fitness value of one chromosome can be calculated as follows:

First, a pseudo fitness value f is obtained by Eq. 1

1

1

kk M

ii

df

d

Second, Fitness value F can be obtained through linear fitness scaling

f

F

average

Eq. 1

2.3 GA Operators

SelectionOperator one

Roulette

2.3 GA Operator

Crossover Order Crossover

Operator two

11- 3- 4- 5- 7-10- 6-15- 9- 1- 2

3- 5- 4- 7- 6-11- 1- 2- 9-15-10

4- 5- 7-10- 6

4- 7- 6-11- 1

2.3 GA Operator

Crossover Order Crossover

Operator two

x- 3- x- 5- x-10- x-15- 9- x- 2

3- x- x- x- x-11- 1- 2- 9-15-x

9-15- 4- 5- 7-10- 6- 3-11- 1- 2

9 - 2- 4- 7- 6-11- 1- 3- 5-10-15

2.3 GA Operator

Mutation Inverse Mutation

Operator three

11- 3- 4- 5- 7-10- 6-15- 9- 1- 2

11- 3- 6-10- 7- 5- 4-15- 9- 1- 2

3 Searching Process

The

100th

Generation

3 Searching Process

The

500th

Generation

3 Searching Process

The

1000th

Generation

3 Searching Process

EvolutionEvolution

G 1000

G 500

G 200

G 100

4 Searching Result

试验次数 最优旅行路线距离 /kilometer

获得代数

1 15655.0093 675

2 15965.1518 819

3 15860.9205 851

4 15896.8518 375

5 15908.5454 394

6 15468.3336 863

7 15665.0205 690

8 16612.9431 847

9 15849.3315 719

10 17015.8367 937

4 Searching Result

The Near Optimal Solution obtained by Hopfield Artificial Neural Networks is

15904 Kilometers [1]

GA found 6 better solutions ! In 10 experiments

The best is 15468 Kilometers !

4 Searching Result

Near-Optimal solution

obtained by

Hopfield ANN

4 Searching Result

Near-Optimal solution

obtained by

GA

References

[1] JIN Pan, FAN Junbo, TAN Yongdong. Neural Networks and Neural Computer: Theory · Application [M]. Chengdu: Southwest Jiaotong University Press, 1991: 375-376

Company

LOGO

Harbin Institute of Technology