Optimal inspection and maintenance policy: integrating a continuous-time Markov Chain into a homing problem

The state of a machine is modeled as a controlled continuous-time Markov chain. Moreover, the machine is being serviced at random times. The aim is to maximize the time until the machine must be repaired, while taking the maintenance costs into account. The dynamic programming equation satisfied by the value function is derived, enabling optimal decision-making regarding inspection rates, and special problems are solved explicitly. This approach minimizes direct maintenance costs along with potential failure expenses, establishing a robust methodology for determining inspection frequencies in reliability-centered maintenance. The results contribute to the advancement of maintenance strategies and provide explicit solutions for particular cases, offering ideas for application in reliability engineering.