A Hybrid Approach to Fuzzy Transportation Problems: Adaptive Defuzzification Combined with VAM and MODI Optimization

Abstract

Transportation planning under uncertain conditions is one of the most challenging issues that fall within the scope of operations research. The paper proposes a hybrid solution approach to a set of fuzzy transportation problems (FTPs) that are modelled using Triangular Fuzzy Numbers (TFN) for Transportation costs, supply , and demand. The adaptive defuzzification phase selects the best defuzzification−operator (centroid, arithmetic mean, geometric mean, or harmonic mean) based on the distribution and asymmetry of the triangle fuzzy number (TFN). A defuzzified problem is then solved using improved VAM (IVAM) for the first BFS and MODI method for verifying optimality. An optimal basic feasible solution (BFS) of Rs. 157 with m+n−1=6 allocations and no MODI pivots can be obtained from the classical Penalty Method applied to the crisp cost matrix as found in the recently introduced Module IV, confirming that it is a valid BFS. The 3×4 numerical example gives Z* = 195 with no MODI pivot. Compared to the centroid technique, the validation process shows that there is a 57% decrease in defuzzification error and a 65% reduction in calculation time for ten randomized examples.