Fuzzy Programming with Quadratic Membership Functions for Multi-objective Transportation Problem

In the present paper, a fuzzy programming model with quadratic membership functions has been developed for the solution of a Multi-Objective Transportation problem. In literature, several fuzzy programming approaches exist with various types of membership functions such as linear, exponential, hyperbolic etc. These membership functions are defined, by taking the lower and upper values of the objective functions into account. In some cases, these methods fail to obtain an integer compromise optimal solution. In the present method, two coefficients of the quadratic membership functions are determined by the lower and upper values of the objective functions. The other coefficient is taken as a variable in the fuzzy programming approach. This means that the membership curve is fixed at the two end points and set free in between. Application of the method on numerical examples proved that the approach could generate integer compromise optimal solutions.


Introduction
In the classical transportation problem, unit quantities of a homogeneous product are to be transported from m sources to n destinations in such a way that the total transportation cost is a minimum. A variable x ij represents the unknown quantity to be transported from the i th source to the j th destination. In addition, there is a penalty c ij associated with transporting a unit of the product from the i th source to the j th destination. The penalty may be cost or delivery time or safety of delivery etc. In the real world situations, the transportation problem usually involves multiple, incommensurable and conflicting objectives. This kind of problem is called multi-objective transportation problem (MOTP). The mathematical model of the MOTP is written as follows. (1) x ij  0 and are integer, i = 1,2,…,m, j=1,2,…,n The concept of fuzzy set theory, first introduced by Zadeh [12], is used for solving different types of linear programming problems. Zimmermann [14] applied the fuzzy set theory with some suitable membership functions to solve linear programming problem with several objective functions. Efficient solutions, as well as an optimal compromise solution for MOTP was obtained by Bit.et.al [4] using fuzzy programming technique with linear membership functions. Bit.et.al [6], have also presented an additive fuzzy programming model that considers weights and priorities for all non-equivalent objectives for the MOTP problem. A fuzzy compromise programming method to obtain a non-dominated compromise solution to the MOTP was developed by Li and Lai [10]. Fuzzy programming techniques with hyperbolic and exponential membership functions to obtain optimal compromise solutions of the MOTP were introduced by Verma et.al [11]. A fuzzy programming approach with the linear membership functions to determine the optimal compromise solution of the MOTP was presented by Abd Elwahed [1]. An interactive fuzzy goal programming for multi-objective transportation problems was developed by Abd El-wahed and Lee [2].

Solution of the problem
Let L k and U k be the aspired level of achievement and the highest acceptable level of achievement for the k-th objective function, respectively.
We assume the membership function of the k-th objective function as Setting the membership values of the k-th objective function at the aspired level and highest acceptable level, 1 and 0 respectively, we have the following equations By solving the above system of linear equations, a k2 and a k3 are expressed in terms of a k1 . Thus, the membership function of the k-th objective function takes the following form We, now describe the proposed approach for the solution of the MOTP.
Adopting the fuzzy decision of Bellmann and Zadeh [3], together with the Quadratic membership function , a fuzzy optimization model of the MOTP can be written as follows x ij  0 and are integer, i = 1,2,…,m, j=1,2,…,n The above model can be simplified as x ij  0 and are integer, i = 1,2,…,m, j=1,2,…,n STEP 4. Solve (4) using a nonlinear programming technique to get an integer compromise optimal solution Verma et al [11] presented the following three fuzzy optimization models using hyperbolic, exponential and linear membership functions respectively x ij  0 for i=1,2,…,m, j=1,2,…,n and x mn+1  0 x ij  0 for i=1,2,…,m, j=1,2,…,n and  0 Where  k (x) = (Z k (x) -L k )/(U k -L k ) and s is a non-zero parameter prescribed by the decision maker.
We, now define the family of distance functions of a compromise solution from the ideal solution as  1 ,  2 ,…,  k are the weights attached to the objective functions. The power p represents a distance parameter 1  p  . Assuming that the sum of all the weights is unity, we define L 1 , L 2 and L as follows The approach which gives a compromise solution close to the ideal solution, is better than the other if Min L p (,k) is achieved for it's solution with respect to some p. In the next section, we consider three numerical examples, calculate the distance functions, and carry out a comparison of the proposed method with the three existing methods.

Analysis of the results
The distances L 1 , L 2 and L from the ideal solutions, for all the approaches are computed, for all the considered examples and are tabulated as follows. We have assumed equal weights to all the objective functions, in all the three examples.
For the first example, the hyperbolic and the proposed approaches produce the same compromise solution, which is nearer to the ideal solution with respect to all the distances L 1 , L 2 and L than the solution produced by the linear and the exponential approaches.
In case of the second example, the hyperbolic approach does not give an integer compromise optimal solution. It, however produces a non-integer compromise optimal solution. The distances L 1 , L 2 and L are calculated using the non-integer solution in this case. All the other methods produce integer compromise optimal solutions. It can be observed that the proposed method gives a compromise optimal solution, which is nearer to the ideal solution with respect to all the distances L 1 , L 2 and L than the solutions produced by all the other methods.
However, in the case of the third example, where there are a greater number of objective functions and constraints, the linear and the exponential approaches produce a compromise optimal solution nearer to the ideal solution, than the solutions produced by the hyperbolic and the proposed approaches.

Acknowledgement
The author expresses his sincere thanks to the authorities of GITAM UNIVERSITY for providing the necessary facilities for this Research.