Optimal Maintenance and Sale Age for a Machine Subject to Failure

A problem of optimal machine maintenance policy is studied. The value of the machine's output is independent of age, but the natural probability of machine failure increases with age. Preventive maintenance can be applied to reduce the probability of machine failure (but a failed machine can only be junked). The machine may be sold at any time providing it is still effective. The problem is to select an optimal preventive maintenance policy for the period of ownership and a planned sale date at which the machine will be sold providing it has not yet failed. The optimal policy calls for a continuous nonincreasing maintenance rate, although not necessarily a decreasing rate of maintenance expenditure. The necessary condition for machine age at sale is critical in the determination of optimal maintenance policy. The necessary conditions for an optimal maintenance policy are shown to be sufficient as well. The novelty of the approach lies in the explicit distinction between the machine's "natural" failure rate and the actual failure rate resulting from selection of preventive maintenance policy.