6. Three-Point Lifetime Distribution Elicitation for Maintenance Optimization in a Bayesian Context

A general three-point elicitation model is proposed for eliciting distributions from experts. Specifically, lower and upper quantile estimates and a most likely estimate in between these quantile estimates are to be elicited, which uniquely determine a member in a flexible family of distributions that is consistent with these estimates. Multiple expert elicited lifetime distributions in this manner are next used to arrive at the prior parameters of a Dirichlet Process (DP) describing uncertainty in a lifetime distribution. That lifetime distribution is needed in a preventive maintenance context to establish an optimal maintenance interval or a range thereof. In practical settings with an effective preventive maintenance policy, the statistical estimation of such a lifetime distribution is complicated due to a lack of failure time data despite a potential abundance of right-censored data, i.e., survival data up to the time the component was preventively maintained. Since the Bayesian paradigm is well suited to deal with scarcity of data, the formulated prior DP above is updated using all available failure time and right-censored maintenance data in a Bayesian fashion. Multiple posterior lifetime distribution estimates can be obtained from this DP update, including, e.g., its posterior expectation and median. A plausible range for the optimal time-based maintenance interval can be established graphically by plotting the long-term average cost per unit time of a block replacement model for multiple posterior lifetime distribution estimates as a function of the preventive maintenance frequency. An illustrative example is utilized throughout the paper to exemplify the proposed approach.