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

We investigate the 𝑝 -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 𝑝 -median problem. The results are compared with two previously proposed methods.


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 NPhard [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 2 Advances in Operations Research 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.

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

The Crisp 𝑝-Median
Model.Let  = {V 1 , . . ., V  } be  existing points.Each point V  has a nonnegative weight   , usually called the demand at V  , and   is the distance between points V  and V  .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 V  which provide by facility in V  and and then the model can be written as follows: (2)
Definition 2 (see [18]).Let () be a set of fuzzy numbers defined on set of real numbers.A ranking function is a function R : () →  which maps each fuzzy number into the real line, where a natural order exists.
In this paper we use the following ranking function: R (, , ) = 1 4 ( + 2 + ) . ( Remark 3 (see [18]).An inequality of fuzzy numbers X ≲ Ã can be transformed to equality, by adding fuzzy numbers S and T to the left and right sides of inequality, respectively; that is, where R( S) − R( T) ≥ 0.

Fuzzy Models
Let x = ⟨ 1 ,  2 ,  3 ⟩ and w = ⟨ 1 ,  2 ,  3 ⟩ be triangular fuzzy numbers corresponding to   and   , respectively.Then the fuzzy model of ( 1 ) can be written as follows: Let X = [x  ] × and  = [  ] × .Then [ X, ] is the optimal solution of problem ( 2 ) if it satisfies the following characteristics: (1) X is a nonnegative fuzzy number.
(3) If there exist any nonnegative fuzzy number X and a vector   such that they satisfy conditions (6), then Model ( 2 ) can be transformed to a crisp model by the following steps.
Step 2. Using Remark 3, by adding fuzzy variables s = ⟨ 1 ,  2 ,  3 ⟩ and t = ⟨ 1 ,  2 ,  3 ⟩ for  = 1, . . .,   = 1, . . .,  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 ( 4 ) and convert model to a crisp model by using the method of Lai and Hwang [4].
Step 4. Consider the following objective function: We have in which ∑  =1 ∑  =1  1 is the most possible value and ∑  =1 ∑  =1  2 and ∑  =1 ∑  =1  3 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.
Step 5. Let  be the set of feasible solutions of model ( 4 ).Consider the Positive Ideal Solutions (PIS) and Negative Ideal Solutions (NIS) of the three objective functions as follows: Then the linear membership function of the objective functions can be computed as follows: and for  = 2, 3, Now we solve the following equivalent single-objective linear programming model introduced by Zimmermann [19]:

Comparing with Other Methods and Numerical Examples
In this section we compare our method with the methods of Nasseri et al. [8] and Moreno-Perez et al. [12] and use our method to solve some test problems.First to compare our method with Nasseri et al. [8], we should mention that they used the following ranking for defuzzification fuzzy constraints: Their method is not always able to presented correct ranking.It is clear that their method is not able to rank the following two fuzzy numbers:  = ⟨1, 2, 3⟩ and  = ⟨0.99,200, 201⟩.In this paper we deal with this problem.By adding fuzzy variable, each fuzzy inequality constraint reduces to equality form.Then each fuzzy equality constraint transforms to three constraints with constant coefficients as follows: where (s  ) − ( t ) ≤ 0.
Second to compare our method with method of Moreno-Perez et al. [12], in the following example, first we solve the crisp case and then explain our method to solve the problem with fuzzy weights and variables.Finally we solve the fuzzy model by method of Moreno-Perez et al. [12] and compare their results with those obtained by our method.
Example 1.Consider a problem of finding two medians of five existing points on the plane.The coordinates of existing facilities and corresponding demands are given in Table 1.
Using the Euclidean distance the distance matrix of these points is obtained as follows: Thus the problem can be formulated as follows: min  =

(29)
The objective function also can be transformed to the following tree crisp objective functions:   Let  be the set of feasible solutions for the recent model.The Positive Ideal Solution and the Negative Ideal Solution for each of these functions are obtained the same as Step 5, for  = 5.
By solving the obtained problems, we will find the following values: Then the linear membership functions of them are obtained as follows:  Using model ( 5 ) it follows that the minimum degree of satisfaction  for this example is equal to 0.904 and  2 =  5 = 1.So the facilities should be located at points 2 and 5.The fuzzy optimal solution X obtained by solving the model is as follows: (34) Now suppose that the above problem is solved using the method of Moreno-Perez et al. [12].Based on their method the fuzzy weights are transformed to crisp weights After solving the problem with these new weights the value of objective function and the amounts of variables will be obtained as follows:   ( As it is clear based on their method a lot of information of problem is lost.Also they obtained crisp amount for objective value and variables.It is very unrealistic that the optimal value of fuzzy problem is obtained as real constant amount.However based on our method the fuzzy objective value and fuzzy amount for variables are yielded.Obviously our method presented realistic and effectiveness results.Therefore the decision maker is able to get the best decision based on the possibilities and circumstances. Example 2. As the second example we consider the first test problem (-median instances) from the ORLIB library which is composed of a graph with 100 nodes and 200 edges; see Beasley [20].A number of medians are supposed to be  = 5 and the fuzzy demands of vertices are given in Table 3.
By applying our new proposed method it yields that  = 0.911 and  4 =  8 =  10 =  23 =  42 = 1, which means the vertices 4, 8, 10, 23, and 42 should be chosen as the location of facilities.Also the fuzzy objective is obtained as ⟨125944, 170330, 180999.56⟩.Tables 4-8 presented how the facility centers are provided the demands of customers.

Conclusion
One of the most important facility location problems which is used to model real situations is -median problem.However in real world problems it is not easy to determine the parameters and data precisely and in some cases it is not possible at all.Hence most of the times the parameters and data are considered by some degree of uncertainty.As fuzzy set theory is a very useful tool to illustrate this ambiguity and uncertainty, in this paper the fuzzy numbers are used to determining uncertain parameters.In this paper the -median problem with fuzzy data is considered.Since the parameters of the presented model are known as the demands of costumers and variables are the amounts of demands that each customer served by facility centers, it is far from reality that the facilities cover constant amounts of customer's demands.Thus in this paper the variables are considered as fuzzy variables.Using some technique in fuzzy linear programming the fuzzy model is transformed to a mathematical linear model.By solving the obtained linear programming, fuzzy objective value and fuzzy variables are obtained.Some numerical examples are solved to illustrate the proposed method.

Table 1 :
The coordinates and demands of existing facilities for Example 1.

Table 2 :
The fuzzy weights of demand points.

Table 8 :
Fuzzy numbers of assignment facility 42 to the customers for Example 2.