Diffusion approximation for a controlled stochastic manufacturing system with average cost minimization

We consider the problem of production control in a single machine, single product, unreliable manufacturing system facing a constant demand d. The goal is to minimize the expected average (per unit time) inventory/backlog costs. Under heavy traffic condition, i.e., when the average production capacity is close to demand, the problem is approximated by a singular stochastic control problem. The approximation problem can be solved explicitly. The solution is then interpreted in terms of the actual manufacturing system and a control policy for this system is derived. We prove that the resulting policy is nearly optimal under the heavy traffic condition. This policy is characterized by a critical level z(0). That is, produce at maximal rate r when inventory is less than z(0) and at the demand rate d when the inventory is equal to z(0). The quality of the approximate control is also discussed by comparing it to known results, when they are available.