On step fixed-charge transportation problem

The fixed-charge problem is a nonlinear programming problem of practical interest in business and industry. One of its variations is the fixed-charge transportation problem (FCTP) where fixed cost is incurred for every route that is used in the solution, along with the variable cost that is proportional to the amount shipped. That cost structure causes the value of the objective function Z to also behave like a step function. Each time we open or close a route the objective function jumps a step. The step fixed-charge transportation problem (SFCTP) is a variation of the FCTP where the fixed cost is in the form of a step function dependent on the load in a given route. While the value of the objective function Z in the FCTP is a step function, the introduction of the step fixed cost in the SFCTP results in the objective function Z being itself a step function with many more steps. Fixed-charge problems are usually solved using sophisticated analytical or computer software. This paper discusses the theory of SFCTP and presents a computationally simple heuristic algorithm for solving small SFCTPs.