Extended preventive replacement policy for a two-unit system subject to damage shocks

We consider a system consisting of two units (A and B), which is subject to two types of shocks (I and II) that occur according to a non-homogeneous Poisson process. The probabilities of these two shock types are age-dependent. Each type-I shock causes a minor failure of unit A, which can be corrected by a minimal repair. Meanwhile, this type of shock results in a certain amount of damage to unit B. These damages to unit B are accumulated to trigger a preventive replacement (PM) or a corrective replacement (CM) action. In addition, a minor failure for unit B with the cumulative damage of z will occur with probability πz$ \pi \left( z \right) $ at a type-I shock instant. Type-II shock is a major one that causes system replacement. We consider a two-dimensional PM policy, which prescribes that the system is preventively replaced at age T, or at the time when the total damage to unit B exceeds a prespecified level Z (but less than the failure level K where K > Z) or is replaced correctively at first type-II shock or when the total damage to unit B exceeds a failure level K, whichever occurs first. Thus, both PM and CM actions may be performed in our model. To minimise the expected cost per unit time, the optimal policy (T∗$ {T^*} $, Z∗$ {Z^*} $) is derived analytically and determined numerically. We also show that our model is a generalisation of many previous maintenance models in the literature.