A bivariate optimal replacement policy with cumulative repair cost limit under cumulative damage model

Abstract

In this paper, a bivariate replacement policy (n, T) for a cumulative shock damage process is presented that included the concept of cumulative repair cost limit. The arrival shocks can be divided into two kinds of shocks. Each type-I shock causes a random amount of damage and these damages are additive. When the total damage exceeds a failure level, the system goes into serious failure. Type-II shock causes the system into minor failure and such a failure can be corrected by minimal repair. When a minor failure occurs, the repair cost will be evaluated and minimal repair is executed if the accumulated repair cost is less than a predetermined limit L. The system is replaced at scheduled time T, at n-th minor failure, or at serious failure. The long-term expected cost per unit time is derived using the expected costs as the optimality criterion. The minimum-cost policy is derived, and existence and uniqueness of the optimal \( n^{*} \) and \( T^{*} \) are proved. This bivariate optimal replacement policy (n, T) is showed to be better than the optimal \( T^{*} \) and the optimal \( n^{*} \) policy.