A Goal Programming-Based Solution Method for Fuzzy Multi-Objective Transportation Problems

Transportation planning in practice is often a multi-objective problem with conflicting objectives (e.g., minimizing total transportation cost, delivery time, damage, or wastage during the transportation process); supply and demand parameters are only approximately known in practice. This paper proposes a solution method for fuzzy multi-objective transportation problem (FMOTP) in which the transportation coefficients (cost, time, supply and demand) are represented as triangular fuzzy numbers and derived by using the goal programming model. The proposed approach first defuzzifies the fuzzy parameters by the following graded mean integration representation technique, then builds objective-wise membership functions according to Zimmermann's fuzzy programming ideas and at last turns the obtained problem into a weighted goal programming problem that each membership goal tends to unity as closely as possible. Underachievement of each fuzzy goal is captured by the use of deviational variables, and the relative importance of cost, time and other criteria is reflected by the use of priority or preference weights. The computational steps of the method are illustrated numerically with three sources, four destinations and two goals (cost and time) and the compromise solution obtained is compared with Zimmermann's fuzzy programming approach and the weighted additive model. The results show that the proposed goal programming formulation provides a balanced compromise solution that achieves a more even distribution of achievement of the membership goals across the objectives than the comparison methods, and at the same time is computationally tractable with standard linear programming solvers. It is general enough to be expanded with other goals, fuzzy or crisp constraints, and the different priority structures of the decision makers, and thus is a practical tool for logistics and supply-chain planning in an uncertain environment.