A heuristic approach for fuzzy fixed charge transportation problem

The fixed charge transportation problem is a nonlinear programming problem in which a fixed charge is incurred if a distribution variable assumes a nonzero value. Its structure is almost identical to that of a linear programming problem, and indeed it can be written as a mixed-integer linear program. In this paper, we consider the fixed charge transportation problem under uncertainty, particularly when the direct and fixed costs are the generalized trapezoidal fuzzy numbers. The first step it transforms into the classical fuzzy transportation problem. The next we obtain the best approximation fuzzy on the optimal value of the fuzzy fixed-charge transportation problem. This method obtains a lower and upper bound both on the fuzzy optimal value of the fuzzy fixed-charge transportation problem which can be easily obtained by using the approximation solution. Conclusion: we propose a new method as the best approximation method, with representation both of the transportation cost and the fixed cost, to find a fuzzy approximation solution close to the optimal solution for fuzzy fixed charge transportation problem.


Introduction
Transportation models have wide applications in the real world situations (Hasani and Moghimi, 2014). A special version of the transportation problem (TP) is the fixed-charge transportation problem (FCTP). In the FCTP, each route is associated with a fixed cost and a transportation cost per unit shipped. Since problems with fixed charge are usually NP-hard problems (Adlakha and Kowalski, 2010), the computational time to obtain exact solutions increases in the distinguished Class P of problems and very quickly become extremely long as the size of the problem increase (Adlakha and Kowalski, 2010). Thus, any method which provides a good solution should be considered useful (Kiyoumarsi and Asgharian, 2014).
According to the available literature, a wide range of different strategies are used in order to find an optimal solution for FCTPs. Generally, the solving methods of the FCTP can be classified as: exact, heuristic and metaheuristic methods.

Preliminaries
In this section, we briefly review some fundamental definitions and basic notation of the fuzzy set theory in which will be used in this paper.
Definition 1. (Kaufmann and Gupta, 1988): If X is a collection of objects denoted generically by x, then a fuzzy set in X is a set of ordered pairs, = {(x,A(x))|x X} A , where A(x) is called the membership function which associates with each  xX a number in [0,1] indicating to what degree x is a number. Definition 2. (Kaufmann and Gupta, 1988) (Kaufmann and Gupta, 1988): A fuzzy number Ã is said to be a trapezoidal fuzzy number (TFN) if its membership function is given by otherwise Definition 4. (Chen and Chen, 2007): A fuzzy set Ã , defined R, is said to be generalized fuzzy number if the following conditions hold: (a) Its membership function is piecewise continuous function. where  01 w .
Definition 5. (Chen and Chen, 2007): A fuzzy number = A (a,b,c,d;w) is said to be a generalized trapezoidal fuzzy number (GTFN) if its membership function is given by is called a TFN and denoted as Ã = (a, b, c, d). In this subsection, we reviewed arithmetic operations on GTFNs (Chen and Chen, 2007). Let = 1 1 1 1 1 A (a , , , ; ) b c d w and B = (a 2 , b 2 , c 2 , d 2 ; w 2 ) be two GTFNs. Define,

Ranking function
A ranking function is suited to compare the fuzzy numbers (Saeednamaghi and Zare, 2014). A ranking function is defined as, , where F(R) is a set of fuzzy numbers, that is, a mapping which maps each fuzzy number into the real line. Now, suppose that Ã and B be two GTFNs. Therefore, 1.

Fixed charge transportation problem
Consider a TP with m sources and n destinations. Each of the source i=1,2,…,m has S i units of supply, and each the destination j=1,2,…,n has a demand of D j units and also, each of the m source can ship to any of the n destinations at a shipping cost per unit cij plus a fixed cost f ij assumed for opening this route (i,j). Let ij x denote the number of units to be shipped from the source i to the destination j. We need to determine which routes are to be opened and the size of the shipment on those routes, so that the total cost of meeting demand, given the supply constraints, is minimized. Then, the FCTP is the following mixed integer programming problem (Balinski, 1961): .. where ij c and ij f are the real numbers. Without losing generality, we assume that the TP is balanced. Let TP be unbalanced, then by introducing a dummy source or a dummy destination it can be converted to a balanced TP. Despite of its similarity to the conventional TP, the FCTP is significantly harder to solve because of the discontinuity in the objective function introduced by the fixed costs.

Fuzzy fixed-charge transportation problem
Now, we assume that the transportation cost and the fixed cost to open a route (i,j) denote by ij c and ij f , respectively, which are not deterministic numbers, but they are the GTFNs, so, total transportation costs become fuzzy as well. The fuzzy fixed-charge transportation problem (FFCTP) is the following mathematical form: where, ij c and ij f are the GTFNs. Balinski (1961) proposed an approximation solution with heuristic method for the FCTP. This paper tries to develop the Balinski's heuristic method for the FFCTP. To do so, first, suppose that both of the transportation cost and the fixed cost are GTFNs as c ij , respectively, then the Balinski matrix is obtained by formulating a linear version of the FFCTP by relaxing the integer restriction on y ij in the objective function of model (2) as follows: So, the linear version of the FFCTP can be represented as follows: We call this "the Approximation Fuzzy Transportation Problem (AFTP)" in which the unit transportation cost is recalculated according to: The AFTP is the classical FTP with the fuzzy transportation costs.
Assume that, * {} ij x is the optimal solution of the AFTP. It can be easily modified into a feasible solution '' {x ,y } ij ij of (2) as follows: ij is an arbitrary feasible solution of the FFCTP. Then, the objective value of {x ̅' ij , y ̅' ij } of (2) provides an upper bound to the optimal value of (2).
Proof. Its Proof is straightforward.

Results
This section proposes a method as the best approximation method, to find an approximation solution to the optimal solution of the FFCTP. Its steps are as follow: Step 1. Convert the given the FFCTP into the FTP as the AFTP by using the following formula: Step 2. Apply one of the well-known methods, as such generalized north-west corner method, the generalized fuzzy least-cost method, or the generalized fuzzy Vogel's approximation method (Kaur and Kumar, 2012) to obtain an initial basic feasible solution of the AFTP.
Step 3. Apply fuzzy modified distribution method (Kaur and Kumar, 2012) to obtain a fuzzy optimal solution of the AFTP.
Step 4. Provide a lower bound ( * L Z ) on the optimal value of the FFCTP ( * FFCTP Z ) according to the theorem 1, by calculating the optimal value of the AFTP.
Step 5. Provide an upper bound ( * U Z ) on the optimal value of the FFCTP ( * FFCTP Z ) according to theorem 2, by calculating the objective value of an arbitrary feasible solution of the FFCTP.

Discussion
Suppose that a company has three factories in three different cities of 1, 2 and 3. The goods of these factories are assembled and sent to the major markets in the three other cities. The demand ( = , 1,2,3 i Si ), supply ( , 1,2,3) j Dj = ) for the cities and the transportation cost associated with each route (i,j) are given by the Table 1.
Step 1. The transportation Table with  Step 2. The initial solution of the AFTP with the generalized north-west corner method is shown in Table 3. Table 3. The initial solution of numerical example Step 3. We use the generalized fuzzy modified distribution method, to find the fuzzy optimal value of the AFTP.

Conclusions
This paper proposed a new method as the best approximation method, with representation both of the transportation cost and the fixed cost of the generalized trapezoidal fuzzy numbers. To this end, it found an approximation solution for the optimal solution to the fuzzy fixed-charge transportation problem. The lower and upper bounds on the fuzzy optimal value of the FFCTP can be easily obtained by using the best approximation method and this is the main advantage of the proposed method. The proposed method has been illustrated using a numerical example.