Asymptotic Optimality of Semi-Open-Loop Policies in Inventory Models with Stochastic Lead Times

Abstract

Inventory models with large and uncertain lead times are notoriously difficult to manage due to the curse of dimensionality. Recent works suggest that in inventory models with large deterministic lead times, semiopen-loop policies are asymptotically optimal. In this paper, we provide a theoretical foundation for the superior performance of semi-open-loop policies in inventory models where the lead times are not only large but also exhibit high variability. In the single-sourcing lost-sales inventory model with divisible products, we show that the optimality gap of constant-order policies decays exponentially fast as the lead time increases. In the single-sourcing lost-sales inventory model with indivisible products, under the assumption that the placed orders cannot cross in time, we propose a bracket policy, which alternates deterministically between two consecutive integer order quantities, and prove that the bracket policy is asymptotically optimal. In the dual-sourcing backlog inventory model with divisible products, we show that a semi-open-loop policy, which places a constant order from the regular supplier in each period, and implements a state-dependent modified base-stock policy from the emergency supplier, is asymptotically optimal, and we also extend our analysis to the joint pricing and inventory model. Finally, we provide a comprehensive numerical study to demonstrate the good performance of the proposed policies, and derive further managerial insights.

Keywords: open-loop policy, asymptotic analysis, inventory, stochastic lead time