0752_FrA08.6 - FrA08.6 43rd IEEE Conference on Decision and...

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Optimality Inequalities for Average Cost MDPs and their Inventory Control Applications Eugene A. Feinberg Mark E. Lewis Abstract — We show that the assumptions guaranteeing ex- istence of a solution to the average cost optimality inequalities presented in Sch¨al [1] for compact action sets can be extended to include the non-compact action set case. Inventory and stochastic cash balance models (with Fxed costs) are natural candidates for the application of our results. Extension of the classic models for a general demand distribution are discussed in detail. I. INTRODUCTION In a discrete-time Markov decision process the usual method to study the average cost criterion is to ±nd a solution to the average cost optimality equations. A policy that achieves the minimum in this system of equations is then average cost optimal. When the state and action spaces are in±nite, one may be required to replace the equations with inequalities, yet the conclusions are the same; a policy that achieves the minimum in the inequalities is average cost optimal. Sch¨al [1] provides two sets of general conditions that imply the existence of a solution to the average cost optimality inequalities (ACOI). The ±rst, referred to as Assumptions (W) in Sch¨al [1], require weak continuity of the transition probabilities. The second group, Assumptions (S) , require setwise continuity of the transition probabilities. The purpose of this paper is to relax the assumptions in Sch¨al [1] so that the results can be applied directly to several problems in the literature; in particular to those related to inventory control. Recall the typical dynamic state equation for inventory control models x n +1 = x n + a n D n +1 ,n =0 , 1 , 2 ,..., (I.1) where x n is the inventory at the end of period n , a n is the decision how much should be ordered, and D n is the demand during period n . Let q ( dy | x, a ) be the transition probability for the control problem (I.1). Weak continuity of q means that E x k n ,a k n f ( x n +1 ) E x n ,a n f ( x n +1 ) for any sequence { ( x k n ,a k n ) ,k 0 } such that ( x k n k n ) Feinberg: Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794-3600 (efeinberg@notes.cc.sunysb.edu) Lewis: Department of Industrial and Operations Engineering, Uni- versity of Michigan, 1205 Beal Avenue, Ann Arbor, MI 48109-2117 (melewis@engin.umich.edu) ( x n n ) where f is any bounded, continuous function and the expectation is taken with respect to q . This holds in most inventory applications in light of (I.1) and Lebesgue’s dominated convergence theorem. On the other hand, set- wise continuity is too strong. Recall that this means that q ( B | x k n k n ) q ( B | x n n ) as ( x k n k n ) ( x n n ) for any Borel set B . For example, let D n =1 (deterministi- cally), a k n = a n + 1 k and x k n = x n . Then q ( B | x n n )=1 for B =( −∞ ,x n + a n 1] and q ( B | x n k n )=0 for all k , 2 ,...
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This note was uploaded on 12/06/2011 for the course MATH 101 taught by Professor Eugenea.feinberg during the Fall '11 term at State University of New York.

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0752_FrA08.6 - FrA08.6 43rd IEEE Conference on Decision and...

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