skip to main content
research-article

A Hybrid Genetic Algorithm for the Bottleneck Traveling Salesman Problem

Published:01 January 2013Publication History
Skip Abstract Section

Abstract

The bottleneck traveling salesman problem is to find a Hamiltonian circuit that minimizes the largest cost of any of its arcs in a graph. A simple genetic algorithm (GA) using sequential constructive crossover has been developed to obtain heuristic solution to the problem. The hybrid GA incorporates 2-opt search, another proposed local search and immigration to the simple GA for obtaining better solution. The efficiency of our hybrid GA to the problem against two existing heuristic algorithms has been examined for some symmetric TSPLIB instances. The comparative study shows the effectiveness of our hybrid algorithm. Finally, we present solutions to the problem for asymmetric TSPLIB instances.

References

  1. Ahmed, Z. H. 2000. A sequential constructive sampling and related approaches to combinatorial optimization. Ph.D. thesis, Tezpur University, Assam, India.Google ScholarGoogle Scholar
  2. Ahmed, Z. H. 2010a. Genetic algorithm for the traveling salesman problem using sequential constructive crossover. Int. J. Biometrics Bioinformatics 3, 96--105.Google ScholarGoogle Scholar
  3. Ahmed, Z. H. 2010b. A hybrid sequential constructive sampling algorithm for the bottleneck traveling salesman problem. Int. J. Comput. Intell. Res. 6, 475--484.Google ScholarGoogle Scholar
  4. Ahmed, Z. H. 2010c. A lexisearch algorithm for the bottleneck traveling salesman problem. Int J. Comput. Sci. Secur. 3, 569--577.Google ScholarGoogle Scholar
  5. Ahmed, Z. H. 2011. A data-guided lexisearch algorithm for the bottleneck traveling salesman problem. Int. J. Oper. Res. 12, 1, 20--23.Google ScholarGoogle ScholarCross RefCross Ref
  6. Ahmed, Z. H., Pandit, S. N. N., and Borah, M. 1999. Genetic algorithms for the Min-Max travelling salesman problem. In Proceedings of the Annual Technical Session, Assam Science Society. 64--71.Google ScholarGoogle Scholar
  7. Burkard, R. E., Deinko, V. G., van Dal, R., van der Veen, J. A. A., and Woeginger, G. J. 1998. Well-solvable special cases of the traveling salesman problem: A survey. SIAM Rev. 40, 195--206. Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Carpaneto, G., Martello, S., and Toth, P. 1984. An algorithm for the bottleneck traveling salesman problems. Oper. Res. 32, 380--389.Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Deb, K. 1995. Optimization for Engineering Design: Algorithms and Examples. Prentice-Hall International, New Delhi, India.Google ScholarGoogle Scholar
  10. Gabovic, E., Ciz, A., and Jalas, A. 1971. The bottleneck traveling salesman problem. (Russian), Trudy Vy cisl. Centra Tartu. Gos. Univ. 22, 3--24.Google ScholarGoogle Scholar
  11. Garfinkel, R. S. and Gilbert, K. C. 1978. The bottleneck traveling salesman problem: Algorithms and probabilistic analysis. J. ACM 25, 435--448. Google ScholarGoogle ScholarDigital LibraryDigital Library
  12. Gilmore, P. C. and Gomory, R. E. 1964. Sequencing a one state-variable machine: A solvable case of the traveling salesman problem. Oper. Res. 12, 655--679.Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Goldberg, D. E. 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, New York. Google ScholarGoogle ScholarDigital LibraryDigital Library
  14. Johnson, D. S. 2006. Machine comparison site, http://public.research.att.com/~dsj/chtsp/speeds.html.Google ScholarGoogle Scholar
  15. Kabadi, S. and Punnen, A. P. 2002. The bottleneck TSP. In The Traveling Salesman Problem and Its Variants, G. Gutin and A. P. Punnen Eds., Kluwer Academic Publishers, Secaucus, NJ.Google ScholarGoogle Scholar
  16. Kao, M.-Y. and Sanghi, M. 2009. An approximation algorithm for a bottleneck traveling salesman problem. J. Discrete Algorithms 7, 315--326. Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. Li, L. and Zhang, Y. 2007. An improved genetic algorithm for the traveling salesman problem. In Communications in Computer and Information Science 2, D.-S. Huang, L. Heutte, and M. Loog Eds., Springer, 208--216.Google ScholarGoogle Scholar
  18. Liu, F. and Zeng, G. 2009. Study of genetic algorithm with reinforcement to solve the TSP. Expert Syst. Appl. 36, 6995--7001. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Parker, R. G. and Rardin, R. L 1984. Guaranteed performance heuristics for the bottleneck traveling salesman problem. Oper. Res. Lett. 2, 269--272. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Philips, J. M. Punnen, A. P., and Kabadi, S. N 1998. A linear time algorithm for the bottleneck traveling salesman problem on a Halin graph. Inf. Process. Lett. 67, 105--110. Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Radcliffe, N. J. and Surry, P. D. 1995. Fitness variance of formae and performance prediction. In Foundations of Genetic Algorithms 3, D. Whitely, and M. D. Vose Eds., Morgan Kaufmann, San Mateo, CA, 51--72.Google ScholarGoogle Scholar
  22. Ramakrishnan, R., Sharma, P., and Punnen, A.P. 2009. An efficient heuristic algorithm for the bottleneck traveling salesman problem. Opsearch 46, 275--288.Google ScholarGoogle ScholarCross RefCross Ref
  23. TSPLIB. 1995. http://www.iwr.uni-heidelberg.de/iwr/comopt/software/TSPLIB95/.Google ScholarGoogle Scholar
  24. Vairaktarakis, G. L. 2003. On Gilmore-Gomorys open question for the bottleneck TSP. Operations Research Letters 31, 483--491. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Wang, C.-X., Cui, D.-W., Wang, Z.-R., and Chen, D. 2005. A novel ant colony system based on minimum 1-tree and hybrid mutation for TSP. In Proceedings of the 2nd International Conference on Fuzzy Systems and Knowledge Discovery. Lecture Notes in Computer Science, vol. 3611, Springer, 1269--1278. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Watson, J., Ross, C., Eisele, V., Denton, J., Bins, J., Guerra, C., Whitely, D., and Howe, A. 1998. The traveling salesrep problem, edge assembly crossover, and 2-opt. In Parallel Problem Solving from Nature V. Lecture Notes in Computer Science, vol. 1498, Springer, 823--832. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Whitely, D., Starkweather, T., and Shaner, D. 1991. The traveling salesman and sequence scheduling: quality solutions using genetic edge recombination. In Handbook of Genetic Algorithms, L. Davis Ed., Van Nostrand Reinhold, New York, 350--372.Google ScholarGoogle Scholar

Index Terms

  1. A Hybrid Genetic Algorithm for the Bottleneck Traveling Salesman Problem

    Recommendations

    Reviews

    Jonathan P. E. Hodgson

    The bottleneck traveling salesman problem (BTSP) refers to the challenge of finding a Hamiltonian circuit that minimizes the largest cost of any of the arcs in the circuit. This paper describes a hybrid genetic algorithm (HGA) that efficiently produces high-quality solutions for BTSPs. The algorithm has been tested on the TSPLIB benchmarks. For its initial population, the algorithm uses a mixture of chromosomes selected in part through a sampling procedure that starts with chromosomes considered to be likely candidates for solutions and supplements that selection with randomly generated ones. A simple example of this procedure would help the exposition of the paper, as the explanation given in the paper is somewhat obscure. The author states that starting the algorithm entirely with chromosomes generated by the sampling procedure tends to get the algorithm stuck in local minima. The paper details the algorithm by giving the crossover operation and the mutation operation, which combines insertion, inversion, and reciprocal exchange. In addition, to avoid ending up in a local minimum, ten percent of the population is periodically randomly replaced using a sequential sampling algorithm described in the paper. A useful diagram displays the structure of this algorithm. The paper gives numerous results from the application of the proposed algorithm to various TSPLIB instances. These are compared with results for the binary search threshold (BST) heuristic and the hybrid sequential constructive sampling (HSCS) algorithm. On average, the proposed HGA algorithm is substantially faster than BST and somewhat faster than HSCS. Both practitioners and students will find in this work a stimulating example of how to use a number of techniques to enhance a genetic algorithm. The ideas for the creation of an initial population are particularly interesting. Online Computing Reviews Service

    Access critical reviews of Computing literature here

    Become a reviewer for Computing Reviews.

    Comments

    Login options

    Check if you have access through your login credentials or your institution to get full access on this article.

    Sign in

    Full Access

    PDF Format

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader
    About Cookies On This Site

    We use cookies to ensure that we give you the best experience on our website.

    Learn more

    Got it!