An optimal production flow control problem with piecewise constant demand

We consider a production system consisting of one machine and producing one part type. The machine is reliable and has a finite capacity. We consider a finite planning horizon consisting of N intervals of different lengths. The demand is constant in each interval and may vary from one interval to another. Intervals with demand greater than the production capacity cause the production to deviate from the demand in the preceding and following intervals. In order to keep the production close to the demand there is a trade off between inventory build-up and customer waiting time. An optimal control problem is formulated to determine the optimal production strategy. Solution technique is developed via Pontryaginʹs Minimum Principle.