On Modeling the Step Fixed-Charge Transportation Problem

Abstract

Fixed-charge transportation problem (FCTP) deals with determining optimal quantities ofgoods to be shipped and the routes to be used to satisfy the customers’ demands at minimaltotal cost. The total cost contains a fixed component which is incurred for every route that ispart of the solution along with the variable cost that is proportional to the amount shipped. Stepfixed-charge transportation problem (SFCTP) is a variant of the FCTP where the fixed costsfollow a step function. Staircase cost structure is very common in the shipping industry,national postal services and couriers, and materials management. In this work, we propose aMILP model for SFCTP. After explaining the mathematical model in sufficient detail, wedemonstrate its applicability on a small numerical example. Using extensive computationalexperiments, we conclude that the problem is a very hard problem with much “higher degree”of polynomial complexity.We also report that the number of steps in the fixed componentappears to be the dominant factor that significantly affects the computational time. Though theproposed MILP model is applicable for SFCTP, with minor modifications, it can be generalizedand used for other network optimization problems that warrant modeling of staircase coststructures.