Optimal overhaul intervals with imperfect inspection and repair

This paper is motivated by the idea of a maintenance-free operating period whose objectives are to improve mission reliability and carry out as much maintenance as possible as a second-line activity. The system may be in one of three states (good, faulty, and failed), and expressions are developed for the average cost per unit time until failure. The system is periodically inspected, the inspection being imperfect in the sense that it can result in both false-positive and false-negative results. Simple faults can be fixed, but a repair is imperfect, in that there is a non-zero probability of a fault remaining after a repair. After a fixed number of inspections, the system is overhauled. If the system fails during operation, it is replaced at increased cost. The sojourn time in each state has non-constant failure rate, and discretization and supplementary variables are used to give a Markovian structure which allows easy computation of the average costs. Minimizing the average cost gives the optimal number of inspections before overhauling the system.