Traveling-Salesman-Problem Algorithm Based on Simulated Annealing and Gene-Expression Programming
Abstract
:1. Introduction
- (1)
- To better improve the ability to global search, we applied a simulated annealing algorithm to the genetic operation of GEP, and propose an improved GEP algorithm based on simulated annealing (IGEP-SA).
- (2)
- In order to solve traveling-salesman problems like graph vertex traversal optimization, we present a traveling-salesman optimization algorithm on the basis of IGEP-SA.
- (3)
- Experimental results show that the proposed algorithm outperforms traditional algorithms in terms of the best solution, the worst solution, the running time of the algorithm, the rate of difference between the best solution and the known optimal solution, and the convergent speed of the algorithms.
2. Related Work
3. Improved GEP Algorithm Based on Simulated Annealing
3.1. Simulated Annealing
Algorithm 1: SA |
Input: initial temperature , minimum temperature , maximum number of iteration , |
probability of temperature drop ; |
Output: the optimal solution ; |
1. Generating an initial solution ; |
2. ; |
3. Computing the value of the objective function and ; |
4. ; |
5. while do |
6. ; |
7. if then |
8. ; |
9. end if |
10. if then |
11. ; |
12. if then |
13. ; |
14. else |
15. ; |
16. end if |
17. end if |
18. ; |
19. ; |
20. end while |
21. Return ; |
3.2. IGEP-SA
3.2.1. New Population Generation Based on Simulated Annealing
Algorithm 2: NPG-SA |
Input: current temperature T, number of population , fitness value of previous |
generation population , and fitness value of population after genetic operations ; |
Output: new population ; |
1. ; |
2. while do |
3. ; |
4. ; |
5. if then |
6. ; |
7. end if |
8. ; |
9. end while |
10. Return ; |
3.2.2. New Mutation Operator Based on Simulated Annealing
Algorithm 3: NMO-SA |
Input: current temperature T, number of population , fitness value of the previous |
generation population , and fitness value of population after genetic operations ; |
Output: new population ; |
1. ; |
2. ; |
3. while do |
4. ; |
5. ; |
6. initial mutation rate |
7. if then |
8. ; |
9. else |
10. ; |
11. end if; |
12. ; |
13. ; |
14. end while |
15. Return ; |
3.2.3. Description of IGEP-SA
Algorithm 4: IGEP-SA |
Input: current temperature T, number of population , fitness value of the previous |
generation population , fitness value of population after genetic operations , probability |
of temperature drop , , , and ; |
Output: new population ; |
1. ; |
2. ; |
3. while do |
4. ; |
5. ; |
6. ; |
7. ; |
8. ; |
9. ; |
10. ; |
11. ; |
12. ; |
13. end while |
14. ; |
15. ; |
4. Traveling-Salesman Problem Based on Simulated Annealing and GEP
4.1. TSP-SAGEP Code
4.2. Fitness Function of the TSP-SAGEP
4.3. Algorithm Description
5. Experiment and Analysis
5.1. Experimental Environment
5.2. Experimental Analysis
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Instance | Cities | Optimal Solution |
---|---|---|
fri26 | 26 | 937 |
st70 | 70 | 675 |
gr120 | 120 | 6942 |
pcb442 | 442 | 50,778 |
gr666 | 666 | 294,358 |
rl1304 | 1304 | 252,948 |
Parameter Item | Value |
---|---|
Initial temperature | 100 |
Minimum temperature | |
The probability of temperature drop | 0.98 |
Population size | 500 |
The number of runs | 10 |
The number of generations | 1000 |
IS Transposition rate | 0.3 |
RIS Transposition rate | 0.3 |
Gene Transposition rate | 0.1 |
One-point recombination rate | 0.3 |
Two-point recombination rate | 0.3 |
Gene recombination rate | 0.1 |
Instances | Optimal | Algorithm | Best | Worst | Average | Generations | Time (s) |
---|---|---|---|---|---|---|---|
fri26 | 937 | TSP-SAGEP | 937 | 971 | 941 | 150 | 7.0671 |
GEP | 943 | 1132 | 961 | 800 | 8.1629 | ||
st70 | 675 | TSP-SAGEP | 675 | 699 | 677 | 150 | 12.1135 |
GEP | 698 | 822 | 710 | 800 | 14.0931 | ||
gr120 | 6942 | TSP-SAGEP | 6942 | 7406 | 6995 | 200 | 19.4612 |
GEP | 7147 | 10,035 | 7624 | 900 | 22.3573 | ||
pcb442 | 50,778 | TSP-SAGEP | 50,811 | 52,147 | 50,878 | 300 | 51.0964 |
GEP | 51,092 | 54,113 | 51,676 | 900 | 57.1428 | ||
gr666 | 294,358 | TSP-SAGEP | 294,419 | 305,036 | 295,542 | 990 | 79.7853 |
GEP | 310,982 | 410,368 | 328,459 | 990 | 85.1204 | ||
rl1304 | 252,948 | TSP-SAGEP | 253,104 | 271,582 | 254,699 | 990 | 160.2458 |
GEP | 281,296 | 334,871 | 290,611 | 990 | 165.5572 |
Instances | Optimal | Algorithm | Best | Worst | Average | Rate | Time (s) |
---|---|---|---|---|---|---|---|
fri26 | 937 | TSP-SAGEP | 937 | 971 | 941 | 0 | 7.0671 |
Ezugwu2017 | 937 | 1204 | 951 | 0 | 8.5247 | ||
Mohsen2016 | 937 | 1299 | 955 | 0 | 8.5916 | ||
Wang2016 | 937 | 1151 | 948 | 0 | 8.3103 | ||
st70 | 675 | TSP-SAGEP | 675 | 699 | 677 | 0 | 12.1135 |
Ezugwu2017 | 675 | 846 | 701 | 0 | 14.9045 | ||
Mohsen2016 | 675 | 907 | 715 | 0 | 15.1037 | ||
Wang2016 | 675 | 831 | 695 | 0 | 14.0985 | ||
gr120 | 6942 | TSP-SAGEP | 6942 | 7406 | 6995 | 0 | 19.4612 |
Ezugwu2017 | 6942 | 11,237 | 7786 | 0 | 25.3581 | ||
Mohsen2016 | 6942 | 11,354 | 7801 | 0 | 27.0433 | ||
Wang2016 | 6942 | 11,008 | 7655 | 0 | 23.6709 | ||
pcb442 | 50,778 | TSP-SAGEP | 50,811 | 52,147 | 50,878 | 0.00065 | 51.0964 |
Ezugwu2017 | 51,107 | 55,723 | 51,958 | 0.00644 | 61.0876 | ||
Mohsen2016 | 51,143 | 56,098 | 52,044 | 0.00714 | 63.9011 | ||
Wang2016 | 51,097 | 55,006 | 51,828 | 0.00624 | 59.4571 | ||
gr666 | 294,358 | TSP-SAGEP | 294,419 | 305,036 | 295,542 | 0.00021 | 79.7853 |
Ezugwu2017 | 311,855 | 417,702 | 331,024 | 0.05611 | 90.0014 | ||
Mohsen2016 | 312,096 | 420,809 | 332,528 | 0.05684 | 92.0745 | ||
Wang2016 | 311,003 | 411,984 | 329,199 | 0.05352 | 88.0163 | ||
rl1304 | 252,948 | TSP-SAGEP | 253,104 | 271,582 | 25,4699 | 0.00062 | 160.2458 |
Ezugwu2017 | 288,013 | 340,015 | 297,813 | 0.12175 | 169.0031 | ||
Mohsen2016 | 290,057 | 341,298 | 299,305 | 0.12798 | 170.1165 | ||
Wang2016 | 286,799 | 335,990 | 294,637 | 0.11803 | 167.9084 |
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Zhou, A.-H.; Zhu, L.-P.; Hu, B.; Deng, S.; Song, Y.; Qiu, H.; Pan, S. Traveling-Salesman-Problem Algorithm Based on Simulated Annealing and Gene-Expression Programming. Information 2019, 10, 7. https://doi.org/10.3390/info10010007
Zhou A-H, Zhu L-P, Hu B, Deng S, Song Y, Qiu H, Pan S. Traveling-Salesman-Problem Algorithm Based on Simulated Annealing and Gene-Expression Programming. Information. 2019; 10(1):7. https://doi.org/10.3390/info10010007
Chicago/Turabian StyleZhou, Ai-Hua, Li-Peng Zhu, Bin Hu, Song Deng, Yan Song, Hongbin Qiu, and Sen Pan. 2019. "Traveling-Salesman-Problem Algorithm Based on Simulated Annealing and Gene-Expression Programming" Information 10, no. 1: 7. https://doi.org/10.3390/info10010007