Dynamic Programming Sequential Scientific Management by A. Kaufmann and R. Cruon (Eds.)

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By A. Kaufmann and R. Cruon (Eds.)

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XJ) = X 2 ( . x3; in accordance with the theorem of optimality, the part (xo , x1 , x,} of an optimal subpolicy from xo to xg is optimal from xo to x 2 . T o find the optimal subpolicies from x, to x3 we need only consider those which are optimal from x, to x2 . T h e optimal value from xo to x3 will then be f0,3(x0 ? ” 3. MATHEMATICAL MODEL OF A DISCRETE DYNAMIC PROGRAM 21 and the optimal subpolicies between these points will be those which pass through the maximum values of x 2 . ). ) ~ A ,r i i x n .

A ,r i i x n . mO*,n-l(&l ? Xn*-I)> 4. 19) I n this way we obtain the optimal subpolicies from xo to every x, . ), in the same way that we have found all the optimal subpolicies in doing so. I n certain problems the value of xN is not imposed, and we then choose the one for which the optimal policy from xo to xN has the greatest value. I n such a case it is important to carry out the optimization as we have done for increasing values of n. Inversely, xN might have a fixed value while xo varies, and optimization would then have to begin at n = N and progress by decreasing values of n.

Nevertheless, this type of reasoning becomes much more subtle in a more complicated case, as we shall now show. Z6 = O 0 1 2 3 FIG. 1 4 5 6 6. A NUMERICAL EXAMPLE W H I C H IS NONLINEAR 47 6. A Numerical Example Which I s Nonlinear I n order to render the example which we studied in Sections 2 and 5 less elementary, we shall introduce two modifications, without claiming that even then we will entirely reproduce a practical case. Our aim, however, as already explained, is to familiarize the reader with the methods of dynamic programming and to point out the numerous pitfalls that may be encountered during the calculations.

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