Optimal production control of a failure-prone machine

We consider a problem of optimal production control of a single unreliable machine. The objective is to minimize a discounted convex inventory/backlog cost over an infinite horizon. Using the variational analysis methodology, we develop the necessary conditions of optimality in terms of the co-state dynamics. We show that an inventory-threshold control policy is optimal when the work and repair times are exponentially distributed, and demonstrate how to find the value of the threshold in this case. We consider also a class of distributions concentrated on finite intervals and prove properties of the optimal trajectories, as well as properties of an optimal inventory threshold that is time dependent in this case.