[论文解读] Distributionally robust inventory control when demand is a martingale

Linwei Xin, David A. Goldberg|arXiv (Cornell University)|Nov 30, 2015

Energy, Environment, and Transportation Policies参考文献 128被引用 13

一句话总结

本文提出了一个时间一致的、分布鲁棒的多阶段报童模型，其中需求作为已知均值和支撑集的鞅过程演化。该文以闭式形式推导出最小最大最优策略和最坏情况分布，表明最坏情况对应于库存的随机过时，并在时间范围增大时建立了其极限过程的弱收敛性。

ABSTRACT

Demand forecasting plays an important role in many inventory control problems. To mitigate the potential harms of model misspecification, various forms of distributionally robust optimization have been applied. Although many of these methodologies suffer from the problem of time-inconsistency, the work of Klabjan et al. established a general time-consistent framework for such problems by connecting to the literature on robust Markov decision processes. Motivated by the fact that many forecasting models exhibit special structure, as well as a desire to understand the impact of positing different dependency structures, in this paper we formulate and solve a time-consistent distributionally robust multi-stage newsvendor model which naturally unifies and robustifies several inventory models with forecasting. In particular, many simple models of demand forecasting have the feature that demand evolves as a martingale. We consider a robust variant of such models, in which the sequence of future demands may be any martingale with given mean and support. Under such a model, past realizations of demand are naturally incorporated into the structure of the uncertainty set going forwards. We explicitly compute the minimax optimal policy (and worst-case distribution) in closed form, by combining ideas from convexity, probability, and dynamic programming. We prove that at optimality the worst-case demand distribution corresponds to the setting in which inventory may become obsolete, a scenario of practical interest. To gain further insight, we prove weak convergence (as the time horizon grows large) to a simple and intuitive process. We also compare to the analogous setting in which demand is independent across periods (analyzed previously by Shapiro), and identify interesting differences between these models, in the spirit of the price of correlations studied by Agrawal et al.

研究动机与目标

• 为解决在需求相关性下的库存控制模型误设问题，提出一种时间一致的鲁棒框架。
• 统一并强化现有假设需求服从鞅过程的库存模型。
• 将过去的需求实现纳入未来决策的不确定性集。
• 以闭式形式刻画最小最大最优策略和最坏情况需求分布。
• 将所提出的鲁棒鞅模型与独立需求情形进行比较，并分析相关性带来的代价。

提出的方法

• 构建一个分布鲁棒的多阶段报童问题，其中未来需求序列被约束为具有给定均值和支撑集的鞅过程。
• 利用动态规划与凸分析，递归推导最小最大最优策略。
• 采用非平凡的归纳论证，结合概率论与优化方法求解鲁棒动态规划。
• 识别出最坏情况分布对应于库存随时间随机过时。
• 证明当时间范围趋于无穷时，极限动态过程弱收敛于一个简单且直观的随机过程。
• 将所提出的鲁棒鞅模型与Shapiro（2011）分析的独立需求情形进行比较，突出其结构差异。

实验结果

研究问题

• RQ1 如何在时间一致的前提下将分布鲁棒优化应用于具有鞅需求的多阶段库存问题？
• RQ2 当需求为有界支撑的鞅过程时，最小最大最优策略与最坏情况分布的结构是什么？
• RQ3 在鞅假设下，最坏情况分布与独立需求下的最坏情况分布有何不同？
• RQ4 当时间范围趋于无穷时，最优策略的极限行为如何？
• RQ5 需求相关性对鲁棒库存控制有何影响？与独立情形相比有何差异？

主要发现

• 通过动态规划与凸分析，以闭式形式推导出最小最大最优策略与最坏情况分布。
• 在最优解下，最坏情况需求过程对应于库存随机地在某一时刻过时，该情形具有实际应用意义。
• 当时间范围趋于无穷时，最优策略动态弱收敛于一个具有简单且直观闭式表达的极限过程。
• 与独立需求情形相比，鲁棒鞅模型导致显著不同的最优策略与成本，凸显了相关性带来的非平凡代价。
• 该模型自然地将过去的需求实现纳入不确定性集，确保时间一致性并提升鲁棒性。
• 数值实验验证了所提模型在缓解模型误设影响方面的优势。