A 'bottleneck problem' in the sense that the level of economic activity will be determined by the resource in shortest supply is discussed. A typical problem of the bottleneck type involving the auto industry, the steel industry and the tool industry is presented. The problem is formulated mathematically employing a discrete approximation. After a short discussion of the difficulties of this approach, a continuous approximation is advanced. Finally, the functional equation approach is applied to the theory of dynamic programming.

The theory of dynamic programming treats problems involving multi- stage processes by means of a transformation of the problem from the space of decisions to the space of functions. This is accomplished by deriving a functional equation whose solution is equivalent to the solution of the original problem. To illustrate this approach most clearly, free of extraneous analytic details, a simple but nontrivial multi-stage investment problem is considered. How exact solutions may be obtained in some cases, approximate solutions in others, and how these approximate solutions may be used to obtain more accurate solutions in the general case is shown. Of particular importance is the decrease in the number of independent variables made possible by this approach. This is not only important from the theoretical standpoint, but is also of great value in reducing the cost in time and effort of numerical computation.

The description for this book, Adaptive Control Processes: A Guided Tour, will be forthcoming.