Optimal generalized age-replacement model of a system with the generalized Polya process

This study developed an age-replacement model for a repairable but ‘worse than old’ system, and the corresponding optimal policy was derived and discussed. Most age-replacement models for repairable systems have been studied under the assumption that the failure process between two consecutive replacements follows the non-homogeneous Poisson process (NHPP), which implies that minimal repair rectifies each failure and returns the system to its original ‘as bad as old’ condition. However, this paper considers a different counting process, called the ‘generalized Polya process (GPP)’ (defined by Cha, 2014), instead of the NHPP. This means that after the repair (GPP repair), the system’s condition is worse than it was immediately before the failure. Such deficient maintenance was due to undertrained operators, low-quality spare parts, or possibly the negative effect caused by a system failure. Additionally, two types of failure were considered when the system failed, depending on its age. Type I failure can be rectified only through GPP repair, and type II failure can be eliminated only by replacing the entire system. The system is replaced either preventively at a planned time or correctively at the time of Type II failure, whichever occurs first. A model is developed for the average cost per unit of time based on the stochastic behavior of the assumed system. The minimum-cost policy time is described and discussed. Because the developed model and its analysis are general, several existing results were shown to be subsumed by this model.