Determination of inspection schedules of equipment by variational method

Susumu Okumura

doi:10.1155/mpe/2006/95843

Abstract

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.

Highlights

Inspection is one of the means to confirm whether maintained equipment operates normally, or whether it works correctly when required for operation, and identify deterioration level of equipment
If equipment deteriorates gradually and the operational condition of equipment can be identified by inspection process based on condition monitoring techniques using measurement of vibration or wear, the deterioration trend plot can be useful for determining the optimal inspection schedule [10]
The optimal inspection schedules in a closed form are shown for some potential failure distributions and loss rate functions

Summary

Introduction

Inspection is one of the means to confirm whether maintained equipment operates normally, or whether it works correctly when required for operation, and identify deterioration level of equipment. Optimal inspection schedules for a single-unit system whose time to failure is described by a probability distribution were derived based on the variational method [5,6,7]. Okumura and Okino [8] studied optimal inspection schedules when failed equipment has a general type of loss rate function. A method for determining optimal inspection schedules is discussed, in which failures of equipment are detected only by inspection. The optimal inspection schedules in a closed form are shown for some potential failure distributions and loss rate functions. Optimal inspection density functions, optimal sequences of inspection times, and the conditions that inspection is effective are obtained explicitly for the case that failure distributions are unknown

Formulation and optimal inspection density function

Determination of inspection schedules

C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4

Some explicit results

Numerical examples of a virtual system

Conclusions