@MISC{R(xn_chapter2, author = {Minimize E R(xn}, title = {Chapter 2 Dynamic Programming}, year = {} }

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Abstract

2.1 Closed-loop optimization of discrete-time systems: inventory control We consider the following inventory control problem: The problem is to minimize the expected cost of ordering quantities of a certain product in order to meet a stochastic demand for that product. The ordering is only possible at discrete time instants t0 < t1 < · · · < tN−1, N ∈ N. We denote by xk, uk, and wk, 0 ≤ k ≤ N − 1, the available stock, the order and the demand at tk. Here, wk, 0 ≤ k ≤ N − 1, are assumed to be independent random variables. Excess demand (giving rise to negative stock) is backlogged and filled as soon as additional inventory becomes available. Consequently, the stock evolves according to the following discretetime system (2.1) xk+1 = xk + uk − wk, 0 ≤ k ≤ N − 1. The cost at time tk invokes a penalty r(xk) for either holding cost (excess inventory) or shortage cost (unsatisfied demand) and a purchasing cost in terms of a cost c per ordered unit. Moreover, there is terminal cost R(xN) for left over inventory at the final time instant tN−1. Therefore, the total cost is given by (2.2) E