Optimality of ( s,  S ) Inventory Policies under Renewal Demand and General Cost Structures

Abstract

We study a single‐stage, continuous‐time inventory model where unit‐sized demands arrive according to a renewal process and show that an (s, S) policy is optimal under minimal assumptions on the ordering/procurement and holding/backorder cost functions. To our knowledge, the derivation of almost all existing (s, S)‐optimality results for stochastic inventory models assume that the ordering cost is composed of a fixed setup cost and a proportional variable cost; in contrast, our formulation allows virtually any reasonable ordering‐cost structure. Thus, our paper demonstrates that (s, S)‐optimality actually holds in an important, primitive stochastic setting for all other practically interesting ordering cost structures such as well‐known quantity discount schemes (e.g., all‐units, incremental and truckload), multiple setup costs, supplier‐imposed size constraints (e.g., batch‐ordering and minimum‐order‐quantity), arbitrary increasing and concave cost, as well as any variants of these. It is noteworthy that our proof only relies on elementary arguments.