Optimal discrete-flow control of a single-stage failure-prone manufacturing system

We consider a single-stage single-product and two-machine-state discrete-flow manufacturing system. The machine is subject to time-dependent failures. All the random variables including processing times, time to failure and time to repair are exponentially distributed. The objective is to determine, in this case, a production control, which minimizes an infinite horizon discounted backlog/surplus cost. This problem has been solved for the continuous-flow manufacturing system and it has been proved that the optimal production control is the hedging point policy. We prove in this paper, for the discrete-flow manufacturing system, that the optimal policy is of hedging point policy type. Based on the value iteration and the sample path analysis methods, we prove the convexity of cost function and the nonnegative value of hedging point.