This paper is a state-of-the-art review of the literature related to optimal maintenance models of systems subject to failure. The emphasis is on work appearing since the 1976 survey, “A Survey of Maintenance Models: The Control and Surveillance of Deteriorating Systems,” by W.P. Pierskalla and J.A. Voelker, published in this journal.

Optimal maintenance models for systems subject to failure–A Review

Reliability analysis : Optimal inspection and maintenance schedules of failing systems

Non-informative and practical Bayesian confidence bounds for reliable life in the Weibull model

Reliability/availability estimation in the classical sense has been well developed and widely discussed. The parameters in the classical reliability/availability distributions are considered to be unknown constants to be determined. If there is information about these parameters, besides that from some current experiment, it can be used as the basis of Bayesian inference. This paper illustrates procedures for using the Bayesian approach for the study of reliability/availability problems. References on the subject are collected and classified. Specifically, included in this paper are: 1) the statement of the classical estimate and Bayesian estimate, 2) the structure of the prior distribution, 3) the reliability life testing, 4) the empirical Bayes approach in reliability, and 5) Bayesian availability. A Bayesian parameter estimate is generally independent of the sampling stopping rule and has smaller variability than the classical estimate; however, a Bayesian estimate is usually biased and associated with difficulties of choosing a prior. The empirical Bayes approach eliminates some of these difficulties, but frequently at the cost of mathematical tractability. The Bayesian approach to reliability/availability is part of the general trend toward using comprehensive probabilistic methods for dealing with the uncertainties associated with modern engineering problems. This trend should also move toward some system effectiveness measures other than simply reliability or availability. For a large system the Bayesian approach could apply, especially when the test data are scarce and the testing procedure is expensive.

Bayesian Reliability & Availability - A Review

Equipment mean time between failures (MTBF) is assumed to be a frequency random variable. The goodness of fit of the uniform prior as a probability model for the MTBF is compared to the goodness of fit of the inverted gamma prior for actual failure data. These distributions can both be adequately fitted to the same failure data when the method of moments is used to fit the distributions. A comparison of posterior producer and consumer risks in a Bayesian reliability demonstration test is made using the fitted inverted gamma and uniform distributions as the priors. There can be rather large differences in the values of the posterior risks even when the two priors fit the data equally well or equally poorly.

On the Behavior of Some Quantities Used in Bayesian Reliability Demonstration Tests

The rudiments of Bayesian philosophy are introduced, and the mathematics of its application are surveyed; there is no uniformity of thought concerning either. The more extreme Bayesian philosophy, which allows subjective probabilities, is a means of plausible reasoning, or of making inferences, through inductive logic. Because inferences concerning reliability concepts are important in the decision process, this philosophy has a place in the reliability field. The difficulties of interpreting the probability function and of assigning prior distributions restrict the presentation of a unified philosophy. Thus, only techniques for describing prior probabilities under various circumstances can be given. Application of the Bayesian approach depends on how the reliability decision is conceptualized.

Gary L. Crellin

Papers

The Philosophy and Mathematics of Bayes' Equation