Optimal Inspection Policy under Imperfect Predictions: Structural Analysis and Insights

Abstract

We consider a critical component with a deterioration process, modeled as a three-state discrete-time Markov chain with a self-announcing failed state and two unobservable operational states: good and defective. Periodic monitoring by a defect-prediction model generates imperfect binary signals. To optimize the inspection decision and minimize the expected total discounted cost, we develop an infinite-horizon partially observable Markov decision process (POMDP). Our analysis demonstrates that the optimal policy exhibits a threshold structure. By introducing the novel concept of a chain-based threshold policy, we classify threshold policies into chain-based and non-chain-based categories. We provide sufficient conditions for a threshold policy to be chain-based and derive a set of equations to compute its value function exactly. For non-chain-based threshold policies, we present an easy-to-implement algorithm to identify the most impactful discontinuous beliefs and construct a modified policy accordingly. Implementing the modified policy allows us to approximate the value function and establish an error bound on the gap between the approximated and exact value functions. Furthermore, we establish criteria for determining the optimality of a threshold policy and propose methods for improving suboptimal policies.

Keywords: Maintenance Optimization, POMDP, Imperfect Signals

JEL Classification: C18, C61, O14