The authors consider the control of a manufacturing system producing one product. The demand for this product is random. The manager can product the product at a fixed rate r, or choose to stop the production. Each time the production is resumed, a setup cost must be paid. To prevent the backlog, which is not allowed, the manager can buy the product from outside vendors, paying a fixed order cost and a variable cost. There is also a linear inventory holding cost. The objective is to minimize the total expected discounted cost. Under heavy traffic conditions, i.e., when the production capacity is close to the average demand, this problem is approximated by an impulse control problem for Brownian motion, which is solved explicitly. The solution is then used to derive a control policy for the original system. The resulting control is characterized by three parameters 0<q0<Q0<S0 and is dubbed double band policy. When the inventory reaches S0 the machine is turned off and it is turned on again when inventory decreases to level Q0. An amount of q0 units is purchased from the outside vendor if the inventory level drops to zero. The authors prove that this policy is nearly optimal. That is, the relative difference between the cost under this policy and the optimal cost is small if traffic intensity is close to one.
