We consider a critical component that deteriorates according to a three-state discrete-time Markov chain with a self-announcing failed state and two unobservable operational states: good and defective. The component is periodically monitored by a defect-prediction model that generates binary signals, but the signals are imperfect. The problem is to decide how to use the signals to make the inspect-or-not decision with the objective of minimizing the expected total discounted cost. We build a partially observable Markov decision process to address the problem. We show that the structure of the optimal policy is threshold type, and the objective value at any given belief state is unimodal in the threshold value. By introducing a novel concept referred to as a chain-based threshold policy, we formalize specific properties of the belief space to explicitly link the optimal policy to a critical number of signals of different types coming from the prediction model. In this way, the optimal policy can be implemented in practice by simply counting the number of signals in a specific order.
We further provide a sufficient condition that confirms the optimal policy belongs to the class of chain-based threshold policies, and propose an approximate algorithm for the case when the optimal policy is not chain-based.