Fuzzy Optimal Solution to Fuzzy Transportation Problem: A New Approach

In this paper we propose a new algorithm for the initial fuzzy feasible solution to a fully fuzzy transportation problem. Then by using fuzzy version of modified distribution method, we obtain the fuzzy optimal solution for the fully fuzzy transportation problem without converting to a classical transportation problem. A numerical example is provided to illustrate the proposed algorithm. It can be seen that the proposed algorithm gives a better fuzzy optimal solution to the given fuzzy transportation problem. Keywords-Fuzzy transportation problem; Triangular fuzzy number; Fuzzy arithmetic; Optimal solution. AMS classification: 90C08,90C90 INTRODUCTION Transportation problem is a special class of linear programming problem which deals with the distribution of single commodity from various sources of supply to various destinations of demand in such a manner that the total transportation cost is minimized. In order to solve a transportation problem, the decision parameters such as availability, requirement and the unit transportation cost of the model must be fixed at crisp values. But in real life applications supply, demand and unit transportation cost may be uncertain due to several factors. These imprecise data may be represented by fuzzy numbers. The idea of fuzzy set was introduced by Zadeh [17] in 1965. Bellmann and Zadeh [2] proposed the concept of decision making in fuzzy environment. After this pioneering work many authors have studied fuzzy linear programming problem techniques such as S.C. Fang [5], H. Rommelfanger [13] and H. Tanaka [15]etc. Fuzzy transportation problem is a transportation problem whose decision parameters are fuzzy numbers. The objective of the fuzzy transportation problem is to determine the transportation schedule that minimizes the total fuzzy transportation cost while satisfying the availability and requirement limits. Chanas et al [4] developed a method for solving fuzzy transportation problems by applying the parametric programming technique using the Bellman–Zadeh criterion [2]. Chanas and Kuchta [3] proposed a method for solving a fuzzy transportation problem by converting the given problem to a bicriterial transportation problem with crisp objective function which provides only crisp solution to the given problem. Liu and Kao [8] proposed a new method for the solution of the fuzzy transportation problem by using the Zadeh’s extension principle. Using parametric approach, Nagoorgani and Abdul Razak [10] obtained a fuzzy solution for a two stage Fuzzy Transportation problem with trapezoidal fuzzy numbers. Omar et. al [11] also proposed a parametric approach for solving transportation problem under fuzziness. Pandian and Natarajan [12] proposed a fuzzy zero point method to find the fuzzy optimal solution of fuzzy transportation problems. Since the fuzzy transportation problem is a special class of fuzzy linear programming problem, the straight forward method is to apply the existing fuzzy linear programming techniques to solve the fuzzy transportation problem. But most of the existing techniques provide only crisp solution for fuzzy transportation problem. In general, the authors have transformed the given fuzzy transportation problem in one or more crisp transportation problems and then obtained the crisp optimal solution. In this paper, we propose a new algorithm to find the initial fuzzy feasible S. Mohanaselvi et al. / International Journal on Computer Science and Engineering (IJCSE) ISSN : 0975-3397 Vol. 4 No. 03 March 2012 367 solution to a fuzzy transportation problem without converting to a classical transportation problem. It is important to note that the proposed algorithm avoid degeneracy and provides the fuzzy optimal solution quickly for the given fuzzy transportation problem. The rest of the paper is organized as follows. In section 2, we recall the basic concepts and the results of triangular fuzzy number and their arithmetic operations. In section 3, we introduce the fuzzy transportation problem with triangular fuzzy numbers and related results. In section 4, we propose a new algorithm to find the initial fuzzy feasible solution for the given fuzzy transportation problem and obtained the fuzzy optimal solution by applying the fuzzy version of MODI method. A numerical example is also provided to illustrate the theory developed in this paper.