A maintenance model with opportunities and interrupt replacement options

A component has time to failure X with density f(.). Opportunities arise as a Poisson process of rate lambda. At an opportunity the component may at choice, be replaced at cost ct. At failure the components can either be minimally repaired at cost c(1) or replaced at cost c(4). Finally the component may be replaced at any time at an 'interrupt' cost of c(3). We derive expressions for the long-ran expected cast rate, and give examples of numerical optimisation with respect to control limits on age at an opportunity, and at an interrupt replacement.