An Interactive Fuzzy Goal Programming Approach for The Multi-Objective Solid Transportation Problem with Trapezoidal Fuzzy Parameters

Abstract

Abstract The solid transportation problem (STP) generalises the classical transportation problem by adding a third resource dimension — conveyances — alongside sources and destinations, making it a more realistic model for logistics planning where mode of transport (rail, road, air, sea) materially affects cost, time and risk. In real supply chains the unit transportation cost, transportation time, quantity of goods, and the supplies, demands and conveyance capacities are rarely known with certainty; they are more naturally expressed through imprecise, fuzzy information. This paper develops a multi-objective solid transportation problem (MOSTP) in which the unit transportation cost and unit transportation time coefficients are represented by trapezoidal fuzzy numbers. A graded mean integration representation is used to defuzzify the fuzzy coefficients, after which an interactive fuzzy goal programming (IFGP) procedure with exponential membership functions is proposed to generate a compromise (Pareto-efficient) solution that balances the conflicting objectives of minimising total transportation cost and total transportation time subject to supply, demand and conveyance-capacity constraints. The proposed method is illustrated on a three-source, three-destination, two-conveyance numerical instance, solved and compared against the conventional weighted-sum and global-criterion approaches. Results show that the exponential-membership IFGP model yields a compromise solution with a better balance of achievement levels across objectives (i.e., a smaller maximum deviation from the individual optima) than the weighted-sum method, while remaining computationally tractable. A brief sensitivity analysis on the shape parameter of the membership function and on conveyance capacity is also reported, together with directions for future research.