Determination of inspection schedules of equipment by variational method

Maintained equipment can be subject to random failures. If failures are evident, they can be detected immediately when they occur. On the other hand, hidden failures that are not revealed during the performance of regular duties may be detected by inspection process. If concerned equipment has hidden failures, loss is assumed to incur from the moment its failure until the time when it is detected. Optimal inspection schedule should be considered when inspection is costly. In this study, optimal inspection schedules for equipment are derived by the variational method employing the inspection density function. For the case in which the time-to-failure distribution of equipment is given, a conditional equation that optimal inspection schedules should satisfy is derived. Furthermore, when the time-to-failure distribution is unknown, an ordinary differential equation that optimal inspection schedules should satisfy is obtained. Optimal inspection schedules in a closed form and numerical examples are shown for some potential loss rate functions.