Optimal maintenance policy and planned sale date for a machine subject to deterioration and random failure

The decision problem concerning the optimization of the maintenance policy and the selection of the sale date for a machine subject to deterioration and random failure is considered from a control-theoretic viewpoint. The originally stochastic optimal control problem is converted to a deterministic optimal control problem with the coefficients of the state and control variables modified in the performance index. The maximum principle is applied to derive the conditions for the optimal maintenance policy and for the optimal planned sale date. Economic interpretations of these conditions are presented in terms of marginal costs and revenues. An explicit solution is found analytically for the problem in the special case when the failure probability is independent of maintenance. The case of exponentially distributed life time for the machine is analyzed in full detail. Finally, the results are illustrated by an example.