Optimal inspection of a system under unrevealed minor failures and revealed catastrophic failures

This work deals with a maintenance model for a system that undergoes both revealed catastrophic failures and unrevealed minor ones. The system undergoes a perfect testing procedure at periodic times along with an imperfect repair whenever a minor failure is detected, while a perfect repair follows any revealed failure. The maintenance procedure is completed with a renewal of the system after the Nth unrevealed failure. Inspection are carried out so that expected cost is minimized through an infinite time span. We are concerned with the conditions under which the existence of an optimum policy is guaranteed.