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

Abstract

W e 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.

Optimal policies for a deterministic continuous-time inventory model with several suppliers

“…Our results differ from those in [12,15,16,26,27]. These papers deal with the long-term average cost criterion for continuous-time models, and indicate that under the respective circumstances an (s, S) policy is optimal.…”

“…Remarkably, this means that from the inventory level σ 2 an order can be placed with two suppliers that are otherwise excluded from the policy. When future costs are discounted, the problem of minimizing the long-term average cost per unit time has been studied in [15,16].…”

Log-Concavity of Compound Distributions With Applications in Operational and Actuarial Models

“…Rosling [43], Huh et al [31] and Perera et al [39] identify important implications of leadtime demand distribution having a log-concave pdf or cdf as discussed above and, in more detail, in Section 4.…”

“…When the demand process is an ordinary renewal process, Perera, Janakiraman and Shun-Chen [39] recently established that an (s, S) policy is optimal under a fully general order cost function and a general quasi-convex instantaneous inventory and shortage cost function. However, this optimality result was only obtained when orders arrive instantaneously, or when the leadtime is deterministic and the demand process Poisson.…”

A Model for Managing Logistics Costs Throughout a Product’s Life Cycle: A Case Study of a Multinational Manufacturing Company

“…Becker et al (2016) developed a method for analysing the potential of distribution logistics in terms of logistics costs, delivery time and delivery reliability. Perera et al (2018) studied 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. Rybakov (2017) developed a mathematical model for the optimisation of total costs in the logistics systems of the wholesale trading company.…”