An optimal machine maintenance problem with probabilistic state constraints

We consider a machine that is maintained via two types of maintenance action: (i) continuous (minor) maintenance that curbs natural degradation of the machine; and (ii) overhaul (major) maintenance that takes place at certain discrete time points and significantly improves the condition of the machine. We introduce an impulsive stochastic differential equation to model the condition of the machine over the time horizon. The problem we investigate is to choose the continuous maintenance rate and the overhaul maintenance times to minimize the total cost of operating and maintaining the machine, where probabilistic state constraints are imposed to ensure that the machine’s state and output meet minimum acceptable levels with high probability. This impulsive stochastic optimal control problem is first transformed into a deterministic optimal control problem with state jumps and continuous inequality constraints. We then show that this equivalent problem can be solved using a combination of the control parameterization technique, the time-scaling transformation, and the constraint transcription method. Finally, we illustrate our approach by solving a numerical example.