Production Control in a Failure-Prone Manufacturing System: Diffusion Approximation and Asymptotic Optimality

Abstract

We consider a problem of controlling the production rate of a single machine, single product, stochastic manufacturing system in order to minimize the total discounted inventory/backlog costs. The demand has two components: one is deterministic with constant rate $d$ and the other is stochastic with random demand batches. Under heavy loading (or heavy traffic) conditions, that is, when the average production capacity is close to the average demand, the control problem is approximated by a singular stochastic control problem. The approximate 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 single critical level $z_0$. The commodity should be produced only when inventory is less than or equal to $z_0$: The production rate is maximal if the inventory is less than $z_0$ and equal to the deterministic component $d$ of the demand rate if the inventory is equal to $z_0$.