Inventory Control with Partial Observations and Inspections

本课题研究状态不可直接观测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，该度量用于表征状态变量的概率分布。我们的项目利用了为非线性滤波和随机控制开发的有用概念，将问题分成两类。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，目的是获得最优解。这种情况发生在库存模型中，其中关于库存水平的信息被延迟，并且在给定较早时期的库存水平的情况下，只知道库存水平的条件分布。在这些情况下，我们证明了参考库存头寸可以被构造为一个充分的统计量，并且根据该头寸制定的基础库存政策是最优的。换言之，存在向上排序级别，使得当基准位置低于该级别时对差值进行排序是最优的，而当基准位置高于该级别时则不对任何差值进行排序。在第二类中，不存在有限维统计量。第二类的一个例子是销售损失，只有当库存水平为零时才能完全观察到它。否则，我们所知道的就是库存水平为正。因此，任何时候的信号都会显示库存是零还是正。如果为正，则只知道库存水平为正的条件分布。因此，人们需要处理条件概率，条件概率通常是无限维实体。此外，条件概率以高度非线性的方式发展。我们开发了一种加权方案--类似于过滤文献中的Zakai方程--将条件概率转换为线性演化的非标准化概率。由此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订购策略的存在性及其刻画。它有助于开发实用的和近似的最优策略作为有限维函数，以及计算它们的过程。我们还研究了具有重要特征的库存问题，如截尾需求(只观察到销售而不是全部需求)，随机损坏等。我们发展了新的方法来分析部分观测下的库存控制问题，以及在一般信息不完全的更广泛的管理和工程背景下的应用。我们的多学科项目有助于在不完全观察的各种系统中进行最佳决策和风险管理，例如制造、供应链、机器维护、质量控制和金融中的系统。这些系统中的局部观察通常与库存水平、客户需求、机床设备、产量和基础分布的参数相关联。我们应该注意到，虽然在每个领域都有大量的文献，但其中大部分都假设系统状态是完全观察到的。因此，在该项目中获得的政策和程序有可能改进该行业使用的库存控制方法，从而节省成本和更好的客户服务。从我们的项目中产生的新的理解和数学方法应该对运筹学、管理科学和金融文献做出有价值的贡献，并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.