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

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.