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Mathematical Problems in Engineering
Volume 2006, Article ID 95843, 16 pages

Determination of inspection schedules of equipment by variational method

Department of Mechanical Systems Engineering, University of Shiga Prefecture, Hikone, Shiga 522-8533, Japan

Received 4 January 2006; Accepted 24 March 2006

Copyright © 2006 Susumu Okumura. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.