A Transportation Problem Considering Fixed Charge and Fuzzy Shipping Costs

This paper considers a fixed-charge transportation problem with fuzzy shipping costs. Whereas the shipping costs of routes are fuzzy intervals, with increasing linear membership functions, fixed costs, supplies, and demands are deterministic numbers. By defining a membership function associated with the objective values and utilizing Bellman-Zadeh’s max-min criterion instead of the main problem, a new crisp nonlinear mixed-integer programming problem is formulated. To solve this non-linear programming problem, at first, we find optimality conditions for a feasible solution, then, using these optimality conditions, reformulate the non-linear programming problem as a linear mixed-integer fractional programming problem, and finally transform the fractional programming problem as a mixed-integer linear programming problem, which can be solved using the existence methods. An illustrative example is solved to explain the presented details.