Analytical Solution to a Partially Observable Machine Maintenance Problem with Obvious Failures
研究一台按马尔可夫过程劣化直至故障的机器维护问题，推导出部分可观测马尔可夫决策过程的精确解析解，无需值迭代即可计算，并识别出六种最优解结构。
We study the maintenance of a machine that deteriorates according to a Markov process until it fails. When failure occurs (which is observable), corrective replacement is made. Otherwise, the machine can be in one of two unobservable working states, and the decision maker can choose production, inspection, or preventive replacement. The state is revealed upon inspection and is reset by corrective or preventive replacement. The objective is to minimize the expected total discounted cost over an infinite horizon. We derive an exact, analytical solution to this problem via a dual framework for partially observable Markov decision processes. The solution can be easily computed without value iteration. We identify six possible structures of the optimal solution, which are represented as graphs. Each graph contains an absorbing, cyclic subgraph that governs the steady-state behavior of the machine. The exact analytical solution facilitates comparative statics analysis, comprehensive numerical studies, and the generation of insights. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2014-04979]. Supplemental Material: The online appendix and data are available at https://doi.org/10.1287/mnsc.2022.4547 .