Research Article A Mathematical Model for Fuzzy -Median Problem with Fuzzy Weights and Variables Fatemeh Taleshian and Jafar Fathali Department of Mathematics, Shahrood University of Technology, University Boulevard, Shahrood 3619995161, Iran Correspondence should be addressed to Jafar Fathali; fathali@shahroodut.ac.ir Received 2 November 2015; Revised 29 January 2016; Accepted 9 March 2016 Academic Editor: Imed Kacem Copyright © 2016 F. Taleshian and J. Fathali. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We investigate the -median problem with fuzzy variables and weights of vertices. Te fuzzy equalities and inequalities transform to crisp cases by using some technique used in fuzzy linear programming. We show that the fuzzy objective function also can be replaced by crisp functions. Terefore an auxiliary linear programming model is obtained for the fuzzy -median problem. Te results are compared with two previously proposed methods. 1. Introduction Location theory is an important topic in the felds of trans- portation and communication. Te -median problem is a classic problem in this line of investigation which consists of locating  facilities to cover the given demands such that the total transportation cost is minimized. In the graph version of -median problem it is shown that there exists an optimal solution that all the facilities are located at  vertices of the graph and the demand of each vertex will be totally covered by the nearest facility. Te - median problem for arbitrary  in general graphs is NP- hard [1]. For more information about location problems on networks see [2]. Tere are many situations in real world that can be mod- eled using -median problem. In actual cases the amounts of parameters are seldom determined precisely. Hence the parameters are determined with some degree of uncertainty. On the other hand fuzzy set theory is the best tool to illustrate this uncertainty. Tat is, the amounts of parameters are considered as fuzzy numbers. In the -median problem, the weight of each point represents the amounts of corresponding customers demand and the aim is to fnd the  best places for locating  facility center which provide customers demand. Terefore in the problem with ambiguous and uncertain demands, providing the exact amount of customer’s need by facility centers is far from reality. Terefore it is expected that the value of objective function and the amounts of variables be in fuzzy form. However in last researches the exact amounts for objective value and variables were yielded. Tus in this paper we overcome this shortcoming and consider the variables as fuzzy variables. Te concept of decision making in fuzzy environment is presented by Bellman and Zadeh [3]. Many authors applied this concept for solving fuzzy linear programming problems. Lai and Hwang [4] provided an auxiliary multiple objective linear programming model to solve a linear programming problem with fuzzy constraint coefcients of objective func- tions. Recently Allahviranloo et al. [5] solved a full fuzzy linear programming using ranking function and Lotf et al. [6] solved this kind of models by lexicography method and fuzzy approximate solution. Kumar et al. [7] proposed a new method for fnding the fuzzy optimal solution of full fuzzy linear programming with equality constraints. Nasseri et al. [8] considered the case that constraints are in inequality forms and presented a new fuzzy solution for solving full fuzzy linear programming. Many researchers consider the fuzzy location problems. Can´ os et al. [9] considered the fuzzy -median problem. Tey presented a fuzzy formulation to combine the standard Hindawi Publishing Corporation Advances in Operations Research Volume 2016, Article ID 7590492, 13 pages http://dx.doi.org/10.1155/2016/7590492