Int. Journal of Math. Analysis, Vol. 2, 2008, no. 9, 425 - 431
Solution of a Dynamic Programming Model using Caratheodory Successive Approximation Method G. Panda Department of Mathematics Indian Institute of Technology Kharagpur - 721302, West Bengal, India
[email protected] Abstract A functional equation in dynamic programming is developed using multistage allocation processs. The existence of the solution of this equation is proved using caratheodory successive approximation process. Uniqueness of the solution is established.
Mathematics Subject Classification: 90C39, 91A20 Keywords: Dynamic programming, Caratheodory successive approximation method, Ascoli lemma
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Introduction
Dynamic programming is a multistage decision process of selecting an optimum allocation of resources, where the decisions are taken at several stages. The basic principle of this technique is based upon Bellman’s Principle of Optimality. In a multistage discrete deterministic process different types of functional equations arise depending upon different situations as a result of the transformation of the state and decision variables. R.Bellman [9] has contributed a lot for the formulation of such types of functional equations and studied the existence of their solution. P.C.Bhakta and S.Mitra [8] have discussed the nature of a particular type of functional equation arising in dynamic programming. They have used contraction principle and Brower fixed point theorem [1]. G.Panda and K.D.Senapati [3], Panda [4],[5] have formulated a two person zero sum continuous game theory model using dynamic programming approach and discussed the existence and uniqueness of its solution. In fact, depending upon the stage transformations, different types of multistage decision making models can be formulated. In this paper a dynamic programming model is developed from a real life situation and the existence of its solution is studied.
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Formulation of the model
The common feature of a multistage allocation process is that , at any stage, we have a physical system characterized by a small set of parameters known as the state variables. The effect of a decision is a transformation of the state variables yielding some return (may be profit). The objective of this multistage allocation process is to maximize the total return, which is a function of state variables and decision variables. Suppose a poultry farm has hens (chikens) and their prerogatives. At the end of each year it sends some of them to the market, retaining the other part for breeding purpose. Suppose x is the source (number of total hens) available at the beginning of the multistage allocation process, Φ(y) be the selling cost of y hens and x − y be retained for breeding purpose for next period, which yields a(x − y), a > 1, at the beginning of next year. The objective is to determine a breeding policy which maximizes the total return over an N period. The recurrence relation will take the form, Sup [Φ(y)] 0
