A generalized Brown-Proschan model for preventive and corrective maintenance

In the study of repairable systems reliability, the basic assumptions on repair efficiency are known as minimal repair or As Bad As Old (ABAO) and perfect repair or As Good As New (AGAN). Obviously, reality is between these two extreme cases. This is known as imperfect repair. One of the most famous imperfect repair model is the Brown-Proschan (1983)(BP) model in which the system is perfectly repaired with probability p and minimally repaired with probability (1-p). This random effect of repair actions can be represented with random variables B(i): B(i)=1 if the i th repair is AGAN, and B(i)=0 if the i th repair is ABAO. Authors have usually supposed that the effect of each repair, B(i), is known. But, in practical case, repair effects are unknown. To our knowledge, only four papers deal with BP model with unknown repair effects. Lim (1998) has estimated the parameters of the first time to failure distribution and repair efficiency with the Expectation-Maximization algorithm. Lim and Lie (2000) have used a SEM algorithm in order to estimate the parameters of a generalization of the BP model that allows first-order dependency between two consecutive repair effects. But they have assumed that only some repair effects are unknown. Lim, Lu and Park (1998) have proposed another method based on Bayesian analysis: they have assumed a prior beta distribution for the parameter p. Langseth and Lindqvist (2004) have generalized the BP model in the case of imperfect preventive maintenance. They have proposed to estimate the parameters of the model (including the repair efficiency one) with the likelihood function. All these papers propose a single estimation of the parameter p that represent the average efficiency of every repair or maintenance actions. Then, the drawback of all these methods, and more generally of all maintenance efficiency estimation methods (Shin Lim Lie (1996), Jack (1998), Yun Choung (1999), Karminskiy Krivstov (2000), Doyen Gaudoin (2004)), is that the efficiency of each maintenance is never individually assessed. In this paper we propose a model, with Brown-Proschan PM effect and ABAO CM effect. Several methods based on likelihood maximization are proposed to estimate the parameters of the first time to failure distribution and the PM efficiency parameter p. Thanks to this model and using a hidden reconstruction method, the efficiency of each preventive maintenance is individually assessed. Forecast reliability indicators are also derived. The framework proposed in this paper can take into account a left and right censorship of the failure process. Finally all the results are applied and validated on a real data set issued from the French electricity manufacturer EDF.