Inventory Control with Partial Observations and Inspections

本课题研究状态不直接可见的最优库存控制问题。相反，观察到的是称为信号的替代度量，用于表征状态变量的概率分布。我们的项目借鉴了非线性滤波和随机控制的有用概念，将问题分为两类。在第一类中，我们找到了一个有限维的充分统计量，它的演化完全描述了系统的行为，目的是得到一个最优解。这种情况发生在库存模型中，其中关于库存水平的信息是延迟的，并且只知道库存水平的条件分布（给定较早时期的库存水平）。在这些情况下，我们表明参考库存位置可以被构造为一个充分的统计量，并且根据该位置的基本库存策略是最优的。换句话说，存在一个向上有序的水平，当参考位置低于该水平时，对差进行排序是最优的，而当参考位置高于该水平时，则不进行任何排序。在第二类中，不存在有限维统计量。第二类的一个例子涉及销售损失，只有当库存水平为零时才能完全观察到库存水平。否则我们所知道的就是库存水平是正的。因此，任何时候的信号都显示库存是零还是正。当库存水平为正值时，只有库存水平的条件分布是已知的。因此，我们需要使用条件概率，它通常是一个无限维的实体。此外，条件概率以高度非线性的方式演化。我们开发了一种加权方案——类似于滤波文献中的Zakai方程——将条件概率转换为线性演化的非规范化概率。所得到的线性系统极大地方便了我们对库存问题和相关的动态规划方程的研究。它使我们能够得到最优反馈库存排序策略的存在性及其表征。它有助于开发实用的和近似的最优策略作为有限维函数，并计算它们的过程。我们还研究了具有重要特征的库存问题，例如审查需求（只观察到销售，而不是全部需求），随机损坏等。我们开发了新的方法来分析库存控制问题，特别是在部分观察下，以及在更广泛的管理和工程环境中应用不完整的信息。我们的多学科项目有助于在制造、供应链、机器维护、质量控制和财务等具有部分观察的各种系统中实现最佳决策和风险管理。这些系统中的部分观测结果通常与库存水平、客户需求、机床工具、生产产量和潜在分布的参数有关。我们应该注意到，虽然在这些领域中存在大量的文献，但其中大部分假设系统状态是完全观察到的。因此，在本项目中获得的政策和程序有可能改善该行业使用的库存控制方法，从而节省成本并改善客户服务。从我们的项目中产生的新的理解和数学方法应该是对运筹学、管理科学和金融文献的有价值的贡献，并且应该促进这些方法在这些领域的许多有趣问题的使用。

This project studies optimal inventory control problems whose states are not directly observed. What is observed instead are surrogate measures called signals, which are used to characterize the probability distribution of the state variables. Our project draws upon useful concepts developed for nonlinear filtering and stochastic control to group the problems into two classes. In the first class, we find a finite-dimensional sufficient statistic whose evolution describes the system behavior completely for the purpose of obtaining an optimal solution. Such cases occur for inventory models where the information about the inventory level is delayed, and only the conditional distribution of the inventory level, given the inventory level in an earlier period, is known. In these cases, we show that a reference inventory position can be constructed as a sufficient statistic and that a base-stock policy in terms of this position is optimal. In other words, there exists an order-up-to level such that it is optimal to order the difference when the reference position is below this level and not to order any when the reference position is above the level. In the second class, no finite-dimensional statistic exists. An example of the second class concerns lost sales, where the inventory level is fully observed only when it is zero. Otherwise all we know is that the inventory level is positive. Thus, the signal at any time reveals whether the inventory is zero or positive. When positive, only the conditional distribution of the inventory level given that it is positive is known. Thus, one needs to work with the conditional probability, which in general is an infinite-dimensional entity. In addition, the conditional probability evolves in a highly nonlinear fashion. We develop a weighting scheme--similar to the Zakai equation in the filtering literature--that converts the conditional probability into an unnormalized probability which evolves linearly. The resulting linear system facilitates considerably our study of the inventory problem and the associated dynamic programming equations. It enables us to obtain the existence of an optimal feedback inventory ordering policy and its characterization. It helps in developing practical and approximate optimal policies as finite-dimensional functions, and procedures to compute them. We also study inventory problems with important features, such as censored demand (only sales but not the full demand are observed), random spoilage, etc. We develop new approaches for the analysis of inventory control problems under partial observations in particular and of applications in broader management and engineering contexts with incomplete information in general. Our multi-disciplinary project contributes to optimal decision making and risk management in a variety of systems with partial observations, such as those found in manufacturing, supply chains, machine maintenance, quality control and finance. The partial observations in these systems are often associated with the inventory level, customer demand, machine-tool ware, production yield, and the parameters of the underlying distributions. We should note that while a substantial literature exists in each of these domains, much of it assumes that the system states are completely observed. Thus, the policies and procedures obtained in this project have the potential to improve the inventory control methodologies used in the industry, which would result in cost savings and better customer service. The new understanding and mathematical methods that stem from our project should be valuable contributions to the operations research, management science and finance literatures, and should facilitate the use of such methods for many interesting problems in these areas.