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Unformatted text preview: Chapter 19 Page 1 6/3/02 Dynamic Programming Models Many planning and control problems in manufacturing, telecommunications and capital budgeting call for a sequence of decisions to be made at fixed points in time. The initial decision is followed by a second, the second by a third, and so on perhaps infinitely. Because the word dynamic describes situations that occur over time and programming is a synonym for planning, the original definition of dynamic programming was planning over time. In a limited sense, our concern is with decisions that relate to and affect phenomena that are functions of time. This is in contrast to other forms of mathematical programming that often, but not always, describe static decision problems. As is true in many fields, the original definition has been broadened somewhat over the years to connote an analytic approach to problems involving decisions that are not necessarily sequential but can be viewed as such. In this expanded sense, dynamic programming (DP) has come to embrace a solution methodology in addition to a class of planning problems. It is put to the best advantage when the decision set is bounded and discrete, and the objective function is nonlinear. This chapter is primarily concerned with modeling of deterministic, discrete systems. Although it is possible to handle certain problems with continuous variables, either directly or indirectly by superimposing a grid on the decision space, such problems will not be pursued here because they are better suited for other methods. In any case, modeling requires definitions of states and decisions, as well as the specification of a measure of effectiveness. For the usual reasons, a reduction in complexity of the real problem is also necessary. From a practical point of view, it is rarely possible to identify and evaluate all the factors that are relevant to a realistic decision problem. Thus the analyst will inevitably leave out some more or less important descriptors of the situation. From a computational point of view, only problems with relatively simple state descriptions will be solvable by dynamic programming. Thus abstraction is necessary to arrive at a formulation that is computationally tractable. Often a particular problem may have several representations in terms of the state and decision variables. It is important that the analyst realize that the choice of formulation can greatly affect his or her ability to find solutions. Dynamic programming has been described as the most general of the optimization approaches because conceivably it can solve the broadest class of problems. In many instances, this promise is unfulfilled because of the attending computational requirements....
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This note was uploaded on 01/26/2011 for the course IEOR 4004 taught by Professor Sethuraman during the Fall '10 term at Columbia.
- Fall '10