Optimum maintenance policy for a system with three types of failures

Consider a system subject to three types of failures with different rate functions and different costs in which repairs and replacement. The type I and II failures arrive according to a non-homogeneous Poisson process where type I can be repair (k _ 1) times by a minimal repair policy. We suppose that type III of failure is catastrophic failure and the system should be replaced with a new one. We are interested in determining an optimal planned replacement time which minimizes the expected discounted costs.