The determination of the optimal circular path has become widely known for its difficulty in producing a solution and for the numerous applications in the scope of organization and management of passenger and freight transport. It is a mathematical combinatorial optimization problem for which several deterministic and heuristic models have been developed in recent years, applicable to route organization issues, passenger and freight transport, storage and distribution of goods, waste collection, supply and control of terminals, as well as human resource management. Scope of the present paper is the development, with the use of fuzzy sets, of a practical, comprehensible and speedy heuristic algorithm for the improvement of the ability of the classical deterministic algorithms to identify optimum, symmetrical or non-symmetrical, circular route. The proposed fuzzy heuristic algorithm is compared to the corresponding deterministic ones, with regard to the deviation of the proposed solution from the best known solution and the complexity of the calculations needed to obtain this solution. It is shown that the use of fuzzy sets reduced up to 35% the deviation of the solution identified by the classical deterministic algorithms from the best known solution.
1. Applegate, D.L., Bixby, R.E., Chvátal, V. and Cook, W.J. (2007) The Traveling Salesman Problem: A Computational Study. Princeton (NJ): Princeton University Press.
2. Bai, J., Yang, G.K., Chenb, Y.W., Hu, L.S. and Pan, C.C. (2012) A model induced max-min ant colony optimization for asymmetric travelling salesman problem. Applied Soft Computing, 13(3), 1365–1375. DOI: doi:10.1016/j.asoc.2012.04.008.
3. Bohács, G., Frikker, I. and Kovács, G. (2013) Intermodal logistics processes supported by electronic freight and warehouse exchanges. Transport and Telecommunication, 14(3), 206–213. DOI:10.2478/ttj-2013-0017.
4. Botzoris, G. and Papadopoulos, B. (2015) Fuzzy Sets: Applications for the Design and Operation of Civil Engineering Projects (in Greek). Thessaloniki: Sofia Editions.
5. Chernov, V., Dorokhov, O. and Malyaretz, L. (2012) Construction of estimates in the choice of alternative solutions by using the fuzzy utilities. Transport and Telecommunication, 13(1), 11–17. DOI: 10.2478/v10244-012-0002-z.
6. Christofides, N. (1976) Worst-case Analysis of a New Heuristic for the Travelling Salesman Problem. Pittsburgh (PA): Carnegie Mellon University – Graduate School of Industrial Administration. (Technical Report 388, ADA025602).
7. Dantzig, G., Fulkerson, R. and Johnson, S. (1954) Solution of a large-scale travelling-salesman problem. Operations Research, 2(4), 393–410. DOI: 10.1287/opre.2.4.393.
8. Garey, M.R. and Johnson, D.S. (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness. New York (NY): W.H. Freeman & Co.
9. Genova, K. and Williamson, D.P. (2015) An experimental evaluation of the best-of-many Christofides algorithm for the traveling salesman problem. In: Algorithms - ESA 2015, (Eds.) N. Bansal, and I. Finocchi, Berlin, Heidelberg: Springer-Verlag, pp. 570-581.
10. Guh, Y.Y., Po, R.W. and Stanley Lee, E. (2008) The fuzzy weighted average within a generalized means function. Computers and Mathematics with Applications, 55(12), 2699–2706. DOI:10.1016/j.camwa.2007.09.009.
11. Gutin, G., Yeo, A. and Zverovich, A. (2002) Travelling salesman should not be greedy: Domination analysis of greedy-type heuristics for the TSP. Discrete Applied Mathematics, 117(1-3), 81–86. DOI:10.1016/S0166-218X(01)00195-0.
12. Hansen, M. (2000) Use of substitute scalarizing functions to guide a local search based heuristics: The case of MOTSP. Journal of Heuristics, 6(3), 419–431. DOI: 10.1023/A:1009690717521.
13. Hwang, C.P., Alidaee, B. and Johnson, J.D. (1999) A tour construction heuristic for the travelling salesman problem. The Journal of the Operational Research Society, 50(8), 797–809. DOI: 10.2307/3010339.
14. Johnson, D.S., Gutin, G., McGeoch, L.A., Yeo, A., Zhang, W. and Zverovitch, A. (2002) Experimental analysis of heuristics for the ATSP. In: The Travelling Salesman Problem and its Variations, (Eds.) G. Gutin and A.P. Punnen, Dordrecht: Kluwer Academic Publishers, pp. 485-488.
15. Junjie, P. and Dingwei, W. (2006) An ant colony optimization algorithm for multiple travelling salesman problem. In: Proceedings of the 1st International Conference on Innovative Computing, Information and Control, Beijing, August 2006, pp. 210-213.
16. Khuller, S. and Vishkin, U. (1994) Biconnectivity approximations and graph carvings. Journal of the Association for Computing Machinery, 41(2), 214–235. DOI:10.1145/174652.174654.
17. Laporte, G. (1992) The travelling salesman problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(2), 231–247. DOI:10.1016/0377-2217(92)90138-Y.
18. Lin, S. (1965) Computer solutions of the travelling salesman problem. The Bell System Technical Journal, 44(10), 2245–2269. DOI:10.1002/j.1538-7305.1965.tb04146.x.
19. Liu, W., Li, S., Zhao, F. and Zheng, A. (2009) An ant colony optimization algorithm for the multiple traveling salesmen problem. In: Proceeding of the 4th IEEE Conference on Industrial Electronics and Applications, Xi'an, May 2009, pp. 1533-1537.
20. Mattas, K., Botzoris, G. and Papadopoulos, B. (2015) Improvement of route optimization algorithmic methods with application of fuzzy sets. In: Proceeding of the 7th International Congress on Transportation Research, Athens, November 2015.
21. Padberg, M. and Rinaldi, G. (1991) A branch-and-cut algorithm for the resolution of large-scale symmetric travelling salesman problems. Society for Industrial and Applied Mathematics, 33(1), 60–100. DOI:10.1137/1033004.
22. Papadimitriou, C.H. (1977) The Euclidean travelling salesman problem is np-complete. Theoretical Computer Science, 4(3), 237–244. DOI:10.1016/0304-3975(77)90012-3.
23. Reinelt, G. (1991) TSPLIB – A travelling salesman problem library. ORSA Journal on Computing, 3(4), 376–384. DOI: 10.1287/ijoc.3.4.376.
24. Reinelt, G. (1994) The Traveling Salesman: Computational Solutions for TSP Applications. Berlin, Heidelberg: Springer-Verlag.
25. Schonlau, M., Welch, W.J. and Jones, D.R. (1998) Global versus local search in constrained optimization of computer models. Institute of Mathematical Statistics: Lecture Notes-Monograph Series, 34, 11–25. doi:10.1214/lnms/1215456182.
26. Stattenberger, G., Dankesreiter, M., Baumgartner, F. and Schneider, J. (2007) On the neighbourhood structure of the travelling salesman problem generated by local search moves. Journal of Statistical Physics, 129(4), 623–648. DOI: 10.1007/s10955-007-9382-1.
27. Xu, Z., Xu, L., and Rodrigues, B. (2011) An analysis of the extended Christofides heuristic for the kdepot TSP. Operations Research Letters, 39(3), 218–223. DOI:10.1016/j.orl.2011.03.002.