Generalisation and Bayesian Solution of the General Renewal Process for Modelling the Reliability Effects of Imperfect Inspection and Maintenance based on Imprecise Data

Common Stochastic Point Processes used in the analysis of Repairable Systems do not accurately represent the true life of a repairable component mainly due to the underlying repair assumption. Specifically, the Ordinary Renewal Process uses an as-good-as-new repair assumption while the Non-Homogenous Poisson Process uses an as-bad-as-old repair assumption. However, it is highly unlikely that any repairable system will readily fit into either repair assumption. Additionally, there is the possibility that any inspection or maintenance activity may actually worsen the system; worse-than-old. Regardless of the underlying repair assumptions and the limitation they impose on any solution, these point processes continue to be used to assist engineering and logistic decision making. While other solutions, mainly GRP based, have offered some resolution, no solution has sufficiently resolved the combined complexities of imperfect maintenance of multiple dependent failure modes, imperfect inspections and data uncertainty, specifically unknown times to failure. Accordingly, the solution offered here offers a model that can contend with all these factors through a Bayesian solution thereby allowing additional "soft-data" to be utilised during the analysis. The modelling scheme consisted of 10 cases divided into 2 main types; Type I with known failure times, and Type II with unknown failure times (data uncertainty). Each of the cases are incrementally modified through the addition of factors including imperfect maintenance of a single failure mode through to multiple dependent failure modes, and finally imperfect inspection. Generalisation of the GRP equations and Bayesian estimation models were developed for these cases. As a closed form solution to each of these cases is unavailable, numerical procedures were formulated. Specifically, an alternative Markov Chain sampling methods, Slice Sampling, was utilised to solve the Bayesian implementation of the needed extensions to the KIJIMA Type I GRP model with an underlying 2-parameter Weibull Time-To-Failure distribution. Based on a number of examples the resulting models have shown the ability to accurately predict future failure trends. Furthermore, the model provides a number of insights into the results including relative maintenance effectiveness and the merit of optimising imperfect maintenance or inspection to maximise availability