Analytical Solution to a Partially Observable Machine Maintenance Problem with Obvious Failures

We study the maintenance of a machine which 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 (POMDPs). 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