Ronald V. Canfield

Cost Optimization of Periodic Preventive Maintenance

Most preventive maintenance(PM) models assume that the hazard function of a system after each PM occurrence is restored to like new or to some specified level. Thus, there is no provision for system degradation with time. Operation of many systems causes stress which results in system degradation and hence an increase in the level of the hazard function with time. PM is assumed to relieve stress temporarily and hence slow the rate of system degradation. However, this type of activity does not reverse degradation; so the hazard function is monotone. A hazard function is developed in this paper consistent with this concept of PM effect. The special case for which PM reduces the operational stress to that of a new system is considered in greater detail (eg, a new system begins operation with the full benefit of a PM activity; each subsequent PM completely restores the benefits). It is shown for this case that the hazard function under PM is approximately a 2-parameter Weibull with shape parameter 2 for systems with strictly increasing hazard without PM. Cost optimization of the PM intervention interval is obtained by determining the average cost-rate of system operation. When the hazard function without PM is unknown, optimization may be achieved through an iterative process. This avoids the necessity of estimating system failure characteristics without PM; such testing can be destructive or use expensive equipment.

A Bayesian Approach to Reliability Estimation Using a Loss Function

A Bayesian estimate of reliability for the exponential case is developed which utilizes the basic notion of loss in estimation theory. Since the loss associated with overestimation is usually greater than the loss associated with underestimation of reliability, the loss function can be a useful tool. The loss function and prior distribution of reliability presented are sufficiently flexible to be compatible with many situations in which reliability estimates are require. When no prior information is at hand and a symmetric loss is used, the resulting estimate is seen to be the minimum variance unbiased estimate of reliability. This agreement gives some credibility to the precision of the estimation approach.

Selecting the Prior Distribution in Bayesian Estimation

A major problem associated with Bayesian estimation is selecting the prior distribution. Fishers information measure is extended to cover prior distributions so that a comparative measure of the amount of information in the sample and in the prior is obtained. The amount of information is used as an intuitive measure of the relative value or weight of experimental data and prior information. By determining the relative weights of both types of information beforehand, it is possible to select a prior which has a known and controlled influence on the estimation process.

Comments on "Diagnostic-strategy selection for a series system" by J.A. Nachlas et al

The authors comment on the work of J.A. Nachlas et al. (see ibid., vol.39, no.3, p.273-80, 1990) that presents diagnostic strategies for series systems with both perfect and imperfect testing. Their comment concerns the results given for the perfect-test case. They hold that the proof of the theorem which provides a simple optimal test sequence is incomplete and give a necessary criterion for an optimal strategy. >

Conservative Bayes experimental design

A major difficulty in applying Bayes methods is the potential for lack of objectivity when quantifying prior information. Existing Bayes design theory for acceptance/demonstration tests is heuristically examined for sensitivity of the design to changes in the prior distribution. The changes considered are associated with location and shape (information content) of the prior. The concept of a risk-conservative prior is developed as a means of controlling unintentional introduction of subjective prejudice in the choice of prior. The intuitive notion of conservatism in Bayes estimation (a broad flat prior has less influence than a more peaked prior) does not carry into design. It is shown that definition of a conservative prior in experimental design requires an understanding of the effects of changing the prior location and shape on the resulting Bayes risk, and conditional consumer and producer risks. A modified minimax principle is used to simplify prior choice. >

BAYES MODIFIED MINIMAX EXPERIMENTAL DESIGN

Existing Bayes strategies for design have been known to be inadequate when the prior density does not describe a real random variable. The responses of the designs to conservative prior information are illogical in that the more conservative the prior density, the smaller the design sample size required. Thus prior ignorance is rewarded. A design strategy based upon a Bayesian modification of the minimax principle is developed for test of the form H0 :θ ɛ Θ0 vs.H1 : θ ɛ Θ1. The noninformative prior for hypothesis testing is defined and the Bayesian modified minimax test with noninformational prior is compared with the classical minimax test.

Nonparametric Estimation and Inference for Reliability

A nonparametric procedure for both point and interval estimation of percentiles, hazard rate, cummulative hazard and survival function is developed. The procedure is based upon the assumption that a time to failure random variable can be closely approximated by a polynomial in an exponential random variable. Accuracy of the approximation is demonstrated. Monte Carlo s-confidence bounds are derived for the estimated functions. Within the range of observed data, the estimated functions are useful for computations such as cost optimization of maintenance and replacement policies.