### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1962-1980 |

Number of pages | 19 |

Journal | Management Science |

Volume | 66 |

Issue number | 5 |

DOIs | |

State | Published - May 2020 |

### All Science Journal Classification (ASJC) codes

- Strategy and Management
- Management Science and Operations Research

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## Cite this

*Management Science*,

*66*(5), 1962-1980. https://doi.org/10.1287/mnsc.2019.3288