Solving fuzzy transportation problems based on extension principle

Transportation models play an important role in logistics and supply chain management for reducing cost and improving service. This paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem, in that the cost coefficients and the supply and demand quantities are fuzzy numbers. The idea is based on the extension principle. A pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level α. From different values of α, the membership function of the objective value is constructed. Two different types of the fuzzy transportation problem are discussed: one with inequality constraints and the other with equality constraints. It is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem. Since the objective value is expressed by a membership function rather than by a crisp value, more information is provided for making decisions.