Optimal replacement policy with replacement last under cumulative damage models

In this study, we discuss a preventive replacement policy under cumulative damage models that include the concept of replacement last. The system is subject to two types of shocks (represented by type 1 shock and type 2 shock) whose probabilities depend on age. Each type 1 shock leads to an amount of additive damage to the system and also causes a minor failure of the system, which can be rectified by a minimal repair. The system will have a catastrophic failure as long as the accumulated damage exceeds a failure level threshold L. Meanwhile, type 2 shock will cause catastrophic failure of the system. The system is replaced preventively at age tau or the n-th type 1 shock or the time instant when the accumulated damage exceeds a pre-specified level k (but less than L), whichever occurs last, and is replaced correctively at the first type 2 shock or the time which the accumulated damage exceeds L, whichever occurs first. The average cost rate is developed and the optimal replacement policy is discussed analytically and computed numerically.