Advances in Operations Research

Volume 2016 (2016), Article ID 7590492, 13 pages

http://dx.doi.org/10.1155/2016/7590492

## A Mathematical Model for Fuzzy -Median Problem with Fuzzy Weights and Variables

Department of Mathematics, Shahrood University of Technology, University Boulevard, Shahrood 3619995161, Iran

Received 2 November 2015; Revised 29 January 2016; Accepted 9 March 2016

Academic Editor: Imed Kacem

Copyright © 2016 Fatemeh Taleshian and Jafar Fathali. This 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.

#### Abstract

We investigate the -median problem with fuzzy variables and weights of vertices. The 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. Therefore an auxiliary linear programming model is obtained for the fuzzy -median problem. The results are compared with two previously proposed methods.

#### 1. Introduction

Location theory is an important topic in the fields of transportation and communication. The -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. The -median problem for arbitrary in general graphs is NP-hard [1]. For more information about location problems on networks see [2].

There are many situations in real world that can be modeled 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. That 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 find the best places for locating facility center which provide customers demand. Therefore in the problem with ambiguous and uncertain demands, providing the exact amount of customer’s need by facility centers is far from reality. Therefore 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. Thus in this paper we overcome this shortcoming and consider the variables as fuzzy variables.

The 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 coefficients of objective functions. Recently Allahviranloo et al. [5] solved a full fuzzy linear programming using ranking function and Lotfi et al. [6] solved this kind of models by lexicography method and fuzzy approximate solution. Kumar et al. [7] proposed a new method for finding 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. Canós et al. [9] considered the fuzzy -median problem. They presented a fuzzy formulation to combine the standard minimization of transport costs with an acceptable reduction of the covered demand. Their algorithm considered only slight modifications of the total demand that should be covered and the optimal transport cost associated with it. In [10] the same method applied in a global sense of fuzzy -median problem. For this problem Canós et al. [11] introduced some marginal analysis techniques to study how solutions depend on membership functions. Moreno-Perez et al. [12] considered some location problems with fuzzy weights and lengths and presented methods to solve them. Kutangila-Mayoya and Verdegay [13] proposed a formulation to find optimal solution for the -median problem in a fuzzy environment when data related to the node demands and the edge distances are imprecise and uncertain. Many other fuzzy location problems are studied by authors (e.g., see [14, 15]).

In this paper we consider the -median problem where the demands of clients and variables are fuzzy numbers and fuzzy variables, respectively. We show the fuzzy model can be transformed to a crisp linear programming. Our method is the extended method of Lai and Hwang [4]. Their model was linear programming and we extend the method for mixed integer programming problem and apply it for the -median problem.

In what follows in this paper the model of crisp -median and some basic definitions and arithmetics between two triangular fuzzy numbers are reviewed in Section 2. In Section 3 the fuzzy model is converted to a linear programming model. To illustrate the proposed method, numerical examples are solved and the obtained results are discussed and compared with two other methods in Section 4.

#### 2. Preliminaries

In this section the crisp model of -median problem and some necessary notions of fuzzy set theory are reviewed.

##### 2.1. The Crisp -Median Model

Let be existing points. Each point has a nonnegative weight , usually called the demand at , and is the distance between points and . The -median problem asks to select facilities of these points such that the sum of the weighted distances of the existing points to the closest facility is minimized.

The first integer linear programming for the -median problem was presented by ReVelle and Swain [16]. A general integer programming for this problem can be written as follows.

Let be the demand of customer in point which provide by facility in andand then the model can be written as follows:

##### 2.2. Fuzzy Backgrounds and Notations

*Definition 1 (see [17]). *Let and be two triangular fuzzy numbers and then the fuzzy operators are defined as follows: (1).(2) if and only if , , and .(3) if and only if .

*Definition 2 (see [18]). *Let be a set of fuzzy numbers defined on set of real numbers. A ranking function is a function which maps each fuzzy number into the real line, where a natural order exists.

In this paper we use the following ranking function:

*Remark 3 (see [18]). *An inequality of fuzzy numbers can be transformed to equality, by adding fuzzy numbers and to the left and right sides of inequality, respectively; that is, where .

#### 3. Fuzzy Models

Let and be triangular fuzzy numbers corresponding to and , respectively. Then the fuzzy model of can be written as follows:

Let and . Then is the optimal solution of problem if it satisfies the following characteristics:(1) is a nonnegative fuzzy number.(2) and satisfy conditions (6).(3)If there exist any nonnegative fuzzy number and a vector such that they satisfy conditions (6), then

Model can be transformed to a crisp model by the following steps.

*Step 1. *Let and for and ; then model is converted to the following model:

*Step 2. *Using Remark 3, by adding fuzzy variables and for inequality (11) can be replaced by the following equalities:

*Step 3. *With definition of fuzzy operators, the fuzzy equalities can be replaced by crisp equalities. Then the model can be written as follows:

In the following steps we consider the objective function of model and convert model to a crisp model by using the method of Lai and Hwang [4].

*Step 4. *Consider the following objective function: We havein which is the most possible value and and are the least possible values. This fuzzy objective is fully defined by three corner points geometrically. Thus, minimizing the fuzzy objective can be obtained by pushing these three critical points in the direction of the left-hand side (see Figure 1). Therefore the objective function can be replaced by the following auxiliary functions:The three new objectives also guarantee the previous argument of pushing the triangular possibility distribution in the direction of the left-hand side.