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

Abstract

Inventory models with lost sales and large lead times are notoriously difficult to manage due to the curse of dimensionality. Recently, Goldberg et al. (2016) and Xin and Goldberg (2016) proved that in the lost-sales inventory model with divisible products, as the lead time grows large, a simple open-loop constant-order policy is asymptotically optimal. In this paper, we consider the lost-sales inventory model in which the lead time is not only large but also random. Under the assumption that the placed orders cannot cross in time, we establish the asymptotic optimality of constant-order policies as the lead time increases for the model with divisible products. For the model with indivisible products, we propose an open-loop bracket policy, which alternates deterministically between two consecutive integer order quantities. By employing the concept of multimodularity, we prove that the bracket policy is asymptotically optimal. Our results on divisible products also hold for the models with order crossover and random supply functions. As our main methodological contributions, we establish the convergence of the average cost incurred by the optimal open-loop policy in a finite-period problem, which serves as a lower bound of the optimal long-run average cost, to the long-run average cost generated by our proposed open-loop policies. Finally, we provide a numerical study to derive further insights, and find out that the proposed open-loop policies perform well even for short lead times when the ratio between the lost-sales penalty cost and holding cost is moderate.

Keywords: open-loop policy, asymptotic analysis, lead time, lost sales