Asymptotic Optimality of Order-Up-To Control for Stochastic Inventory Systems with Sequential Probabilistic Service Level Constraints

Abstract

Service level constraint is often used as a metric to directly control the quality of service (e.g., managing the probability of stock-out) in practice. Many inventory problems with service level constraints are often difficult to solve and are typically approximated by deterministic formulations. This raises an important question regarding the quality of such an approach. To shed light on this question, in this paper, we consider two simplified yet fundamental inventory models (with backorder and lost-sales) with independent demands, positive lead times and sequential probabilistic service level constraints, and study the performance of a natural order-up-to policy whose parameters can be calculated using the optimal solution of a deterministic approximation of the backorder inventory system. We show that it is asymptotically optimal for both the backorder and lost-sales systems in the setting with a high service level requirement, with a stronger performance bound for the backorder system. Our analysis for the lost-sales system involves a construction of an alternative backorder system whose expected total cost can be related to that of the analogous lost-sales system. Overall, our result contributes to the growing body of inventory literature that suggests the near-optimality of simple heuristic policies. Moreover, it also gives credence to the use of deterministic approximation for solving complex inventory problems in practice, at least for applications where the targeted service level is sufficiently high.