Temporal problem of subcontracting of manufacturing unit under joint production control and maintenance policy

In this paper, we consider a repairable production unit subject to random failures, which supplies input to a direct (main) customer. The production unit is submitted to a preventive maintenance action at instants kT (k=1,2,...). The corrective and preventive maintenance actions have random durations. In order to maximize the use of production capacity of the unit, it was decided to allocate it to perform periodic tasks of subcontracting for a contractor. A buffer stock is built up in order to guarantee a continuous supply of customer demand at a constant rate during repair and preventive maintenance actions, and subcontracting periods. We determine analytically the optimal instant to begin subcontracting which minimize the total average cost per time unit.