A Compensatory Fuzzy Approach to Multi-Objective Linear Transportation Problem with Fuzzy Parameters

Abstract

In this paper, we focus on the solution procedure of the Multi-Objective Linear Transportation Problem (MOLTP) with fuzzy parameters, i.e. fuzzy cost coefficients, fuzzy supply quantities and fuzzy demand quantities. Because of several linear objectives and its fuzzy parameters, this transportation problem is very complicated and also due to the fuzziness in the costs this problem has non-linear structure. To overcome these difficulties, we gave an approach with three stages. By using linear solution techniques, our approach generates compromise solutions which are both compensatory and Pareto-optimal. In the first stage, the fuzziness in supply and demand quantities is eliminated by using Zimmermann's "min" operator to satisfy the balance condition. In the second stage, breaking points (i.e. the values of cost-satisfaction parameters that changed the optimal solution) and cost-satisfaction interval sets are obtained for each objective.  In the third stage, considering cost-satisfaction interval sets of all objectives, an overall cost-satisfaction interval set is determined. And then for each member of this set, our approach generates compensatory compromise Pareto-optimal solutions using Werner's $\mu_{and}$ operator. To our knowledge, combining compensatory ($\mu_{and}$) operator with MOLTP has not been published up to now. An illustrative numerical example is given to explain our approach.