World Applied Sciences Journal 8 (5): 543-549, 2010 ISSN 1818-4952 © IDOSI Publications, 2010 Corresponding Author: Dr. Nuran Güzel, Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Davutpasa, Istanbul, Turkey 543 Fuzzy Transportation Problem with the Fuzzy Amounts and the Fuzzy Costs Nuran Güzel Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Davutpasa, Istanbul, Turkey Abstract: In this paper, we investigated a fuzzy transportation problem with fuzzy quantities, in which are bounded fuzzy triangular numbers and fuzzy transportation costs per unit that are bounded upper fuzzy numbers. The problem is solved at two stages. In first stage is calculated maximum satisfactory level satisfying balance between fuzzy supplies and fuzzy demands. The given transportation problem is rearranged according to the maximum satisfactory level. In second stage, by considering the unit transportation costs, the arranged problem has investigated all of optimal solution connected with satisfactory level for quantities on all intervals that constituted from breaking points of the unit transportation costs, from zero to maximum satisfactory level. We derive two different satisfactory levels to the problem: one is breaking points γ p , for p = 1,2,…, of transportation costs C = C ( γ) and the other is α s values for s = 1,2,…, that are violated positive condition for optimal solution X * = X * (α) in intervals [γ p-1 , γ p ], for p = 1,2,…. Optimal solutions could be different by depending on the constructed [α s-1 , α s ] intervals for s = 1,2,… on the constructed [γ p-1 , γ p ] intervals for p = 1,2,…. We proposed a new technique in which find all different optimal solutions to the fuzzy transportation problem with respect to α in the [α s-1 , α s ] intervals for s = 1,2,… on the [γ p-1 , γ p ] intervals for p = 1,2,…. Key words : Fuzzy transportation problem • breaking points • membership functions INTRODUCTION The transportation is called the transportation problem that transports a homogenous product from m sources to n different destinations to minimize total transportation cost. There are numerous solution methods for transportation problem when prices and quantities are given as crisp numbers [1, 2]. Several variations of transportation methods have been using with the table methods such as the northwest- corner method, the shortcut method and Russel’s approximation method [1, 2]. Some others [1-3] have been using special techniques for linear programming (LP) problem because the classic single objective transportation problem is a special case of the linear programming problem. However, recently fuzzy programming approach started to use the optimal solutions of multi objective or single objective transportation problem [4-8]. For instance, Wahed [4] presented a fuzzy programming approach to determine the compromise solution of multi objective transportation problem (MOTP). Kikuchi [6] proposed a simple adjustment method that finds the most appropriate set of crisp numbers. Wahed et al. [8] presented an interactive fuzzy goal programming approach to determine the preferred compromise solution for MOTP. In reality, it is not possible to determine both quantities and transportation unit prices, but the fuzzy numbers gives best approximation of them. A model solving the transportation problem is given in [5] when quantities are fuzzy and prices are crisp. The model uses the table method for solution. Again, in [6] is given a method determining quantities that is satisfied the higher satisfactory level while quantities is only fuzzy. This method uses LP model in solution. OhEigeartaigh [1, 9] considered the case where the membership functions of the fuzzy demands are triangular forms for transportation problems and solved it using table method. Chanas and Kulej [1, 10] provided an approach to solve a fuzzy linear programming problem with triangular membership functions of fuzzy resources. Geetha and Nair [11] formulated a stochastic version of the time minimizing transportation problem and developed an algorithm based on parametric programming to solve it when transportation time is considered to be independent, positive normal random variables. Chanas et al . [7] are analyzed the transportation problem with fuzzy supply values of deliverers and with fuzzy demand values of