A bivariate optimal imperfect preventive maintenance policy for a used system with two-type shocks

This article studies an optimal imperfect preventive maintenance policy based on a cumulative damage model for a used system with initial variable damage. The used system is subject to shocks occurring to a non-homogeneous Poisson process, and suffers one of two types of shocks with stochastic probability: type-I shock (minor) yields a random amount of additive damage of the system, or type-II shock (catastrophic) causes the system to fail. A bivariate preventive maintenance schedule (n, T) is presented in which the system undergoes preventive maintenance at a planned time T and the nth type-I shock, or corrective maintenance at any type-II shock and the total damage exceeds a threshold level, whichever occurs first. The optimal preventive maintenance schedule which minimizes the expected cost rate is derived analytically and discussed numerically.