The replenishment problem with multiple articles and an order threshold

Abstract This paper treats a replenishment problem where an order in a single period is to be determined. Demand is assumed to be known for multiple periods. The first periods’ demands are required to be satisfied by the order, satisfying remaining demand is optional. Furthermore, a threshold needs to be satisfied, e.g., a minimum total order volume is given and orders can be placed only if they reach (or exceed) this volume. As a consequence, articles cannot be treated independently and, thus, we have to determine order volumes for all articles integratedly. The replenishment problem with multiple articles and an order threshold is formalized and its computational complexity is determined. We develop mixed-integer programming models and suited optimization approaches and apply them in static and dynamic problem settings. We see that the deterministic problem setting can be solved in reasonable time and that a decision maker can benefit from employing it in a decision support tool in a dynamic environment with uncertainty.