3. Hierarchical Controls under Discounted Cost 
Criterion

In this section, the problems of hierarchical production planning with the discounted cost is discussed. We present asymptotic results for hierarchical production planning in manufacturing systems with machines subject to breakdown and repair. We make use of the singular perturbation methods to reduce the original problems into simpler problems and then we describe a procedure to construct a control, derived from the solution to the simpler problems, for the original systems. The simpler problems turn out to be nothing but limiting problems obtained by averaging the given stochastic machine capacities and modifying the objective function in a reasonable way to account for the convexity of the cost function. Therefore, by showing that the associated value function for the original systems converge to the value functions of the limit systems, we can construct controls for the original systems from the optimal control of the limit systems. It turns out that the controls so constructed are asymptotically optimal as the fluctuation rate of the machine capacities goes to infinity. Furthermore, error estimates of the asymptotic optimality are provided in terms of their corresponding cost functions.

Here we will discuss hierarchical controls in single/parallel machine systems, flowshops, jobshops, and production-investment systems. Finally some computational results are given.