African Journal of Mathematics and Computer Science Research Vol. 4(11), pp. 339-349, October 2011 Available online at http://www.academicjournals.org/AJMCSR ISSN 2006-9731 ©2011 Academic Journals Full Length Research Paper Solving traveling salesman problem by using a fuzzy multi-objective linear programming Sepideh Fereidouni Department of Industrial Engineering, University of Yazd, P.O. Box 89195-741, Yazd, Iran. E-mail: sepideh.fereidouni@gmail.com. Fax: (+98) 351 6235002. Accepted 10 August, 2011 The traveling salesman problem (TSP) is one of the most intensively studied problems in computational mathematics. Information about real life systems is often available in the form of vague descriptions. Hence, fuzzy methods are designed to handle vague terms, and are most suited to finding optimal solutions to problems with vague parameters. This study develops a fuzzy multi-objective linear programming (FMOLP) model with piecewise linear membership function for solving a multi-objective TSP in order to simultaneously minimize the cost, distance and time. The proposed model yields a compromise solution and the decision maker’s overall levels of satisfaction with the determined objective values. The primary contribution of this paper is a fuzzy mathematical programming methodology for solving the TSP in uncertain environments. A numerical example is solved to show the effectiveness of the proposed approach. The performance of proposed model with Zimmerman and Hannan’s methods is compared. Computational results show that the proposed FMOLP model achieves higher satisfaction degrees. Key words: Traveling salesman problem, fuzzy multi-objective linear programming, decision maker. INTRODUCTION The traveling salesman problem (TSP) is one of the well- studied NP-hard combinatorial optimization problems which determines the closed route of the shortest length or of the minimum cost (or time) passing through a given set of cities where each city is visited exactly once (Majumdar and Bhunia, 2011). In other words, find a minimal Hamiltonian tour in a complete graph of N nodes. The first instance of the TSP was from Euler in 1759 whose problem was to move a knight to every position on a chess board exactly once (Michalewicz, 1994). The traveling salesman first gained fame in a book written by German salesman BF Voigt in 1832 on how to be a successful traveling salesman (Michalewicz, 1994). He mentions the TSP, although not by that name, by suggesting that to cover as many locations as possible without visiting any location twice is the most important Abbreviations: TSP, Traveling salesman problem; FMOLP, fuzzy multi-objective linear programming; FLP, fuzzy linear programming; MOLP, multi- objective linear programming; DM, decision maker; LP, linear programming. aspect of the scheduling of a tour. The origins of the TSP in mathematics are not really known - all we know for certain is that it happened around 1931 (Bryant, 2000). In TSP as a multi-objective combinatorial optimization problem, each objective function is represented in a distinct dimension. Of this form, to decide the multi objective TSP in the optimality means to determine the k- dimensional points that pertaining to the space of feasible solutions of the problem and that possess the minimum possible values according to all dimension. The permissible deviation from a specified value of a structural dimension is also considerable because traveling sales man can face a situation in which he is not able to achieve his objectives completely. There must be a set of alternatives from which he can select one that best meets his aspiration level. A conventional programming approach does not deal with this situation however some researchers have specifically treated the multi-objective TSP (Rehmat et al., 2007). Branch and Bound approach was used to solve TSP with two sum criteria (Chaudhuri and De, 2011) An E-constrained based algorithm for Bi-Objective TSP was suggested by Melamed and Sigal (1997) and Rehmat et al. (2007). A