Extended replacement policy for a system under shocks effect

We investigate a bivariate replacement policy for a system under shocks effect. A system including two units experiences to one of two types of shocks. Whenever a type I shock arrives, unit 1 has a minor failure which can be removed through a minimal repair; while a type II shock leads to a catastrophic failure of the system. The occurrence probabilities of shock types depend on the number of shocks since the last replacement. Whenever unit 1 suffers a minor failure, it causes some additive damage to unit 2. Once cumulative damage of unit 2 reaches a threshold level L, unit 2 will fail, which will also cause unit 1 to fail at the same time, causing a catastrophic failure of the system. Furthermore, unit 2 whose cumulative damage is x may fail minorly with probability π(x) at the instant of minor failure of unit 1, and requires minimal repair when it fails. It is assumed that the system implements preventive replacement as the system’s age reaches T, or the mth type I shock occurs, or corrective replacement as a type II shock occurs or the cumulative damage to unit 2 reaches a threshold level L, whichever comes first. For this model, we derive the formula for the mean cost rate and determine analytically and numerically the corresponding optimal policy. Finally, it is also shown that the policy can be regarded as a generalization of existing policies. This generalization makes our policy more flexible and applicable in real situations than existing policies.