On an optimal maintenance policy for a Markovian deteriorating system with uncertain repair

This paper examines a system which is inspected at equally spaced points in time. We express the observed states of the system as a discrete time Markov chain with an absorbing state. It is assumed that the true state is certainly identified through inspection. After each inspection, one of three actions can be taken: Operation, repair, or replacement. We assume that the result of repair is uncertain. If repair is taken, we decide whether to inspect the system or not. When inspection is performed after completion of repair, we select an optimal action. After replacement, the system becomes new. We study the optimal maintenance policy which minimizes the expected total discounted cost for unbounded horizon. It is shown that, under reasonable conditions on the system's deterioration and repair laws and the cost structures, a control limit policy is optimal. We derive several valid properties for finding the optimal maintenance policy numerically. Furthermore, numerical analysis is conducted to show our theoretical results could hold under weaker conditions.