A stochastic multi-stage fixed charge transportation problem: Worst-case analysis of the rolling horizon approach

We introduce the Stochastic multistage fixed charge transportation problem in which a producer has to ship an uncertain load to a customer within a deadline. At each time period, a fixed transportation price can be paid to buy a transportation capacity. If the transportation capacity is used, the supplier also pays an uncertain unit transportation price. A unit inventory cost is charged for the quantity that remains to be sent. The aim is to determine the transportation capacities to buy and the quantity to send at each time period in order to minimize the expected total cost. We prove that this problem is NP-hard, we propose a multistage stochastic optimization model formulation, and we determine optimal policies for particular cases, having deterministic unit transportation prices or load. Rolling horizon is a classical heuristic approach for solving multistage stochastic programming models. Our aim is to provide the worst–case analysis of this approach, applied to this NP-hard problem and to polynomially solvable particular cases. Numerical results are also provided.