Diffusion approximation for controlled stochastic manufacturing system with setup costs

The authors consider the control of a manufacturing system producing one product. Under a heavy traffic condition, 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<S ⁰ and is called the 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. It is proved 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