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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
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Harbin Institute of Technology