A Brownian approximation of a production–inventory system with a manufacturer that subcontracts

This paper considers a production–inventory problem where a manufacturer fulfills stochastic, stationary demand for a single product from a finished-goods inventory. The inventory can be replenished by two production resources, in-house production and a subcontractor, which both have finite capacity. We construct a Brownian approximation of the optimal control problem, assuming that the manufacturer uses a ‘dual base-stock’ policy to control replenishment from the two sources and that her objective is to minimize average cost. A closed-form expression is obtained for one optimal base-stock policy and an analytical expression is derived from which the other optimal base stock can be computed numerically. We show conditions under which the objective is convex in capacity, and the unique globally optimal capacity can be computed numerically. We thus provide a tractable approximation to the two-source problem, which is generally intractable. We demonstrate the accuracy of this approximation for an M/M/1 model. We also draw managerial insight from the Brownian optimal base-stock results into how the optimal base-stock policies control the inventory distribution and under what conditions the contingent source is used to build inventory or to resolve backorders.