Optimal inspection and replacement policies for multi-state deteriorating systems

In this paper, a phase-type approach is proposed to derive optimal inspection and replacement policies for semi-Markovian deteriorating systems. In this approach, the general sojourn time distributions of a semi-Markovian maintenance model are approximated by acyclic phase-type distributions. Using the approximation, a semi-Markovian maintenance model can be transformed into a Markovian maintenance model such that the analytical tractability of Markov processes can be preserved. Based on the Markovian model, algorithms are provided to derive the optimal state-dependent and state-age-dependent inspection and replacement policies such that the expected long-run cost rate is minimized. Furthermore, procedures are developed to implement the optimal policies on semi-Markovian deteriorating systems. The implementation of the optimal policies is illustrated by numerical examples.