arXiv 2207.045502022-07math.OCcs.LG

面向存在销售损失与不确定供应的库存系统的学习订货

Learning to Order for Inventory Systems with Lost Sales and Uncertain Supplies

College of Business Administration, University of Illinois at Chicago（伊利诺伊大学芝加哥分校商学院）；Stern School of Business, New York University（纽约大学斯特恩商学院）

Boxiao Chen, Jiashuo Jiang, Jiawei Zhang, Zhengyuan Zhou

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AI总结：

针对需求与供应分布未知且数据删失的销售损失库存系统，提出在线学习与主动剔除方法，实现相对最优策略的 O(L+√T) 遗憾界。

摘要：

我们考虑在规划期 T 内、提前期为 L 的随机销售损失库存控制系统。供应具有不确定性，并且是订货数量的函数（由随机产出/产能等导致）。我们的目标是最小化 T 期成本；众所周知，即使需求和供应分布已知，该问题在计算上也难以处理。本文假设需求与供应分布均未知，并开发了一种计算高效的在线学习算法。我们证明，当 L≥log(T) 时，该算法实现 O(L+√T) 的遗憾（即在 T 期内该算法成本与最优策略成本之间的性能差距）。为此，我们：1）证明对于任意 L≥0，与完全信息下的最优常量订货策略（一种著名且广泛使用的算法）相比，我们的算法成本至多高出 O(L+√T)；2）利用现有文献中关于该策略的已知性能保证。据我们所知，在在线库存控制文献中，此前尚无以最优策略为基准、有限样本下 O(√T)（且关于 L 为多项式）的遗憾界。该学习问题的一个关键挑战是需求与供应数据都可能被删失，因此只能观测到截断值。我们绕过这一挑战的方式是证明：在订货数量 q² 下生成的数据不仅使我们能够模拟 q² 的表现，还能模拟所有 q¹<q² 的 q¹ 的表现；这一关键观察使我们即使在数据删失下也能获得充分信息。通过建立高概率耦合论证，我们能够在有限时间范围内评估并比较不同订货策略在稳态下的表现。由于该问题缺乏凸性，我们开发了一种主动剔除方法，自适应地排除次优解。

英文摘要：

We consider a stochastic lost-sales inventory control system with a lead time $L$ over a planning horizon $T$. Supply is uncertain, and is a function of the order quantity (due to random yield/capacity, etc). We aim to minimize the $T$-period cost, a problem that is known to be computationally intractable even under known distributions of demand and supply. In this paper, we assume that both the demand and supply distributions are unknown and develop a computationally efficient online learning algorithm. We show that our algorithm achieves a regret (i.e. the performance gap between the cost of our algorithm and that of an optimal policy over $T$ periods) of $O(L+\sqrt{T})$ when $L\geq\log(T)$. We do so by 1) showing our algorithm cost is higher by at most $O(L+\sqrt{T})$ for any $L\geq 0$ compared to an optimal constant-order policy under complete information (a well-known and widely-used algorithm) and 2) leveraging its known performance guarantee from the existing literature. To the best of our knowledge, a finite-sample $O(\sqrt{T})$ (and polynomial in $L$) regret bound when benchmarked against an optimal policy is not known before in the online inventory control literature. A key challenge in this learning problem is that both demand and supply data can be censored; hence only truncated values are observable. We circumvent this challenge by showing that the data generated under an order quantity $q^2$ allows us to simulate the performance of not only $q^2$ but also $q^1$ for all $q^1<q^2$, a key observation to obtain sufficient information even under data censoring. By establishing a high probability coupling argument, we are able to evaluate and compare the performance of different order policies at their steady state within a finite time horizon. Since the problem lacks convexity, we develop an active elimination method that adaptively rules out suboptimal solutions.

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