A fundamental but impossible to be addressed problem in repairable system modelling is how to estimate the system repair improvement (or damage) effects because of the large-sample requirements from the standard statistical inference theory. On the other hand, repairable system operating and maintenance data are often imprecise and vague and therefore Type I fuzzy sets defined by point-wise membership functions are often used for the modelling repairable systems. However, it is more logical and natural to argue that Type II fuzzy sets defined by interval-valued membership function, called interval-valued fuzzy sets (IVFS), should be used in characterizing the underlying mechanism of repairable system. In this paper, we explore a small-sample based GM(1,1) modelling approach rooted in the grey system theory to extract the system intrinsic functioning times from the seemly lawless functioning-failure time records and thus to estimate the repair improvement (damage) effects. We further explore the role of interval-valued fuzzy sets theory in the analysis of the system underlying mechanism. We develop a framework of the GM(1,1)-IVFS mixed reliability analysis and illustrate our idea by an industrial example.
R-fuzzy sets and grey system theory