# Dynamic Programming (Dover Books on Computer Science)

## Richard Bellman

Language: English

Pages: 384

ISBN: 0486428095

Format: PDF / Kindle (mobi) / ePub

An introduction to the mathematical theory of multistage decision processes, this text takes a "functional equation" approach to the discovery of optimum policies. Written by a leading developer of such policies, it presents a series of methods, uniqueness and existence theorems, and examples for solving the relevant equations. The text examines existence and uniqueness theorems, the optimal inventory equation, bottleneck problems in multistage production processes, a new formalism in the calculus of variation, strategies behind multistage games, and Markovian decision processes. Each chapter concludes with a problem set that Eric V. Denardo of Yale University, in his informative new introduction, calls "a rich lode of applications and research topics." 1957 edition. 37 figures.

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chose Ln, x1, x2 arbitrarily close to their respective upper bounds Fn — 1 + Fn — 2, Fn — 1 and Fn — 2 respectively. Then Fn = Fn — 1 + Fn — 2. This yields the proof of Theorem 11. Furthermore, it yields the optimal policy, since each xi is either discarded or is the optimal first choice for the remaining subinterval. The sequence {Fn} has as its first few terms with F20 > 10,000. Hence the maximum of a strictly unimodal function can always be located within 10–4 of the original interval

replaced by precisely the equation of (3). Since , this equation reduces to which possesses exactly one root under the assumption that φ(s) > 0. Observe that the limiting cases behave properly. If ap — k = 0, y = 0, if a = 1, y = ∞; if p = ∞, y = ∞. Having determined , we proceed to determine f(x) as follows. For we have and f′(x)= — k, or, Substituting (10) in (9), and setting x = 0, we obtain the following result for f(0)6, To determine f(x) for x ≥ 7 we have the equation

existence and uniqueness theorems and determine in which ways the theorems established above must be modified in order to remain valid. 13. In what ways is the problem of ordering for a military supply depot different from the problem of ordering items for a department store ? 14. Assume that there is no penalty for not being able to meet the demand, but that there is a return of b dollars for each item demanded and supplied. Suppose that this return can be used to increase the quantity

= 0 for t near 0. For 0 < t < t1 we let thus keeping x3 at zero level. At t1 we must distinguish different subcases : In case IA we can produce autos at capacity without running out of steel. Hence we let for t1 ≤ t ≤ t0; and for t0 < t ≤ T we let This solution for Case IA is optimal because it can be paired with our basic ω solution of Fig. 1. In Case IB we do not have enough steel to produce autos at capacity. Hence we continue to produce no autos for t > tl, i.e., We do this

functional subject to relations where x, y, c and g are n-dimensional column vectors, and F is a scalar function of x and y,7 we can proceed in a similar fashion. Setting the principle of optimality yields the functional equation The classical transversality conditions fall out as a special case in this equation, as might be expected on the basis of the duality between point and tangential coordinates which we have indicated above. Carrying through the calculations similar to those