Closing the Gap: A Learning Algorithm for Lost-Sales Inventory Systems with Lead Times

• Published in: Management science Vol. 66; no. 5; pp. 1962 - 1980
• Main Authors: Zhang, Huanan, Chao, Xiuli, Shi, Cong
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.05.2020; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; base-stock policy; censored demand; Convergence; Demand; Inventory; Inventory management; lead time; Learning; learning algorithms; lost sales; nonparametric; Regret; regret analysis; Sales; Stochastic models
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.2019.3288

We consider a periodic-review, single-product inventory system with lost sales and positive lead times under censored demand. In contrast to the classical inventory literature, we assume the firm does not know the demand distribution a priori and makes an adaptive inventory-ordering decision in each period based only on the past sales (censored demand) data. The standard performance measure is regret, which is the cost difference between a learning algorithm and the clairvoyant (full-information) benchmark. When the benchmark is chosen to be the (full-information) optimal base-stock policy, Huh et al. [Huh WT, Janakiraman G, Muckstadt JA, Rusmevichientong P (2009a) An adaptive algorithm for finding the optimal base-stock policy in lost sales inventory systems with censored demand. Math. Oper. Res. 34(2):397–416.] developed a nonparametric learning algorithm with a cubic-root convergence rate on regret. An important open question is whether there exists a nonparametric learning algorithm whose regret rate matches the theoretical lower bound of any learning algorithms. In this work, we provide an affirmative answer to this question. More precisely, we propose a new nonparametric algorithm termed the simulated cycle-update policy and establish a square-root convergence rate on regret, which is proven to be the lower bound of any learning algorithm. Our algorithm uses a random cycle-updating rule based on an auxiliary simulated system running in parallel and also involves two new concepts, namely the withheld on-hand inventory and the double-phase cycle gradient estimation . The techniques developed are effective for learning a stochastic system with complex system dynamics and lasting impact of decisions. This paper was accepted by Yinyu Ye, optimization.