Abstract
We consider a monotone optimal policy for a discrete, time problem of controlling the arriving customers. At each period one customer arrives at a manufacturing factory to order a job distinguished by the reward and the service time with a constant delivery interval. The basic properties of optimal policies are obtained. It is shown that, contrary to intuition, from counterexamples an optimal policy cannot generally be monotone in such cases as finite-horizon problems with and without discounting and an infinite-horizon problem with discounting, while there exists a monotone optimal policy for infinite-horizon problems without discounting.
