Optimal scheduling of non-perfect inspections

In this paper, we study the determination of optimal inspection policies when three types of inspections are available: partial, perfect and imperfect. Perfect inspections diagnose without error the system state. The system can fail because of three competing failure types: I, II and III. Partial inspections detect without error type I failures. Failures of type II can be detected by imperfect inspections which have non-zero probability of false positives. Partial and imperfect inspections are made at the same time. Type III failures are detectable only by perfect inspections. If the system is found failed in an inspection, a repair is made which renders the system in a good-as-new condition. The system is preventively maintained following an age-based policy. Preventive maintenance actions return the system to a good-as-new condition. We consider cost contributions of inspections, repairs, preventive maintenance and periods of unavailability. The model presented permits to determine the optimal (constant) inter-inspection period for partial, imperfect and perfect inspections and the optimal times of preventive maintenance actions.