Determining Optimum Burn-In and Replacement Times Using Bayesian Decision Theory

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Abstract
An important problem facing a manufacturer is the determination of the amount of time to burn-in items (in order to eliminate early failures) and the age at which to replace items (to avoid failures due to wearout). This problem becomes difficult to solve if the time-to-failure distribution of an item is unknown and must be estimated from test and operational data. This paper describes a method of statistical data analysis which is readily applied to the solution of this decision problem under a realistic but general loss (or gain) function. The method is a multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss (or gain) resulting from a particular choice of burn-in time and item replacement age. This integration is performed by a Monte Carlo Procedure using importance sampling. An example demonstrates the flexibility of this method of analysis. The data are a mixture of ``point'' and truncated data, which often create difficulties when using conventional methods of decision analysis. In addition, since the method permits up to ten parameters for the family of time-to-failure distributions, a ``bathtub'' hazard rate function is used to generate the data for the example. The results are presented in the form of Bayesian confidence intervals for the true hazard rate function and a presentation of the expected loss as a function of burn-in time and age at replacement.

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