Optimal control of a dual service rate M/M/1 production–inventory model

We analyse a dual-source, production–inventory model in which the processing times at a primary manufacturing resource and a second, contingent resource are exponentially distributed. We interpret the contingent source to be a subcontractor, although it could also be overtime production. We treat the inventory and contingent sourcing policies as decision variables in an analytical study and, additionally, allow the primary manufacturing capacity to be a decision variable in a subsequent numerical study. Our goal is to gain insight into the use of subcontracting as a contingent source of goods and whether it can fulfill real-world managers' expectations for improved performance. We prove that a stationary, non-randomised inventory and subcontracting policy is optimal for our M/M/1 dual-source model and, moreover, that a dual base-stock policy is optimal. We then derive an exact closed-form expression for one of the optimal base stocks, which to our knowledge is the first closed-form solution for a dual-source model. We use that closed-form result to advantage in a numerical study from which we gain insight into how optimal capacity, subcontracting, and inventory policies are set, and how effectively a contingent source can reduce total cost, capacity cost, and inventory cost. We find that (i) the contingent source can reduce total cost effectively even when contingent sourcing is expensive and (ii) contingent sourcing reduces capacity cost more effectively than it does inventory cost.