Imperfect preventive maintenance model with two modes of failure

In this paper, an imperfect preventive maintenance model with two modes of failures is proposed and analyzed for the deteriorating system. We introduce the concept of two modes of failures: maintainable and non-maintainable. The deteriorating system experiences minimal repair when the system fails and experiences preventive maintenance at a periodic interval. The hazard rate of maintainable failures is reduced by preventive maintenance, but the hazard rate of non-maintainable failures isn't changed by preventive maintenance. Therefore, the preventive maintenance is imperfect. Generally, it is assumed that the dependence of two modes of failures is independent, but, in fact, the maintainable failure is influenced by the non-maintainable. The paper considers the dependence of two modes of failures and introduces the doubly stochastic Poisson in order to describe the dependence of two modes of failure. The optimal interval between the preventive maintenance and the optimal number of preventive maintenances are determined for minimizing the expected cost rate. Finally, the feasibility of model is illustrated by a numerical example.