A Game Theory-based Approach to Fuzzy Linear Transportation Problem

Abstract

Background: Transport models have wide application areas in the real world and play an important role in reducing transportation costs, increasing service quality, etc. These models may have uncertain transportation costs and supply or demand capacities of the product. Hence, it would be effective to model the vagueness of customer demands, economic conditions, and technical or non-technical uncertainties because of uncontrollable factors. Therefore, we focus on developing a mathematical solution approach to the fuzzy transportation problems. Objective: In this paper, an integrated approach is proposed for the solution of the fuzzy linear transportation problem that has fuzzy cost coefficients in the objective function. Since transportation problem is encountered frequently in the national and international environment, it is considered that proposing a new solution method to this problem will be useful. Methods: Fuzzy cost coefficients are taken as trapezoidal fuzzy numbers due to their widespread use in the literature. Firstly, the fuzziness is removed by converting the original single objective fuzzy transportation problem into a crisp Multi-Objective Linear Programming Problem (MOLPP). After the classical payoff matrix is constructed, ratio matrices are obtained to scale the objectives. Then, an approach based on game theory is implemented to solve the MOLPP, which is handled as a zero-sum game. Results: Creating different ratio matrices in the game theory part of the approach can generate compromise solutions for the decision-makers. To demonstrate the effectiveness of the proposed approach, two numerical examples from the literature are solved. While the same solution is obtained in one of the examples, a different compromise solution set is generated, which could be presented to the decision-maker in the other example. Conclusion: In this paper, we developed a novel game theory-based approach to the fuzzy transportation problem. The proposed approach overcomes the non-linear structure due to the uncertainty in the cost coefficients. The greatest advantage of the proposed approach is that it can generate more than one optimal solution for the decision-maker.