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A sample-path approach to the optimality of echelon order-up-to policies in serial inventory systems



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Operations Research Letters 36 2008 547 550 Contents lists available at ScienceDirect Operations Research Letters journal homepage www elsevier com locate orl A sample path approach to the optimality of echelon order up to policies in serial inventory systems Woonghee Tim Huh a Ganesh Janakiraman b a Department of IEOR Columbia University United States b Stern School of Business New York University United States article info Article history Received 22 January 2008 Accepted 8 May 2008 Available online 11 June 2008 a b s t r a c t We present a new proof of the optimality of echelon order up to policies in serial inventory systems first proved by Clark and Scarf Our proof is based on a sample path analysis as opposed to the original proof based on dynamic programming induction 2008 Elsevier B V All rights reserved Keywords Inventory Multi echelon Sample path analysis 1 Introduction In this paper we provide an alternate proof of the optimality of echelon order up to policies in serial inventory systems first proved in the seminal paper by Clark and Scarf 5 Our proof is based on a sample path analysis as opposed to the original proof based on dynamic programming induction The crux of our proof is the following claim for every policy that is not of the echelon order up to type we can construct a policy that dominates the original policy strictly We now highlight the weaknesses and strengths of our approach relative to the original In our analysis we assume just as Clark and Scarf do that the single period cost is a sum of N functions where N is the number of echelons of the individual echelon inventory positions While Clark and Scarf require these functions to be weakly convex we require strict convexity Although this is a stricter assumption than weak convexity note that any convex function can be approximated arbitrarily closely by a strictly convex function Furthermore their proof also provides an algorithm for computing the optimal order up to levels whereas our



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