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Duality and Existence of Optimal Policies in Generalized Joint Replenishment

Graduate School of Business, University of Chicago, Chicago, Illinois 60637
Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
dan.adelman{at}ChicagoGSB.edu
klabjan{at}uiuc.edu

We establish a duality theory for a broad class of deterministic inventory control problems on continuous spaces that includes the classical joint replenishment problem and inventory routing. Using this theory, we establish the existence of an optimal policy, which has been an open question. We show how a primal-dual pair of infinite dimensional linear programs encode both cyclic and noncyclic schedules, and provide various results regarding cyclic schedules, including an example showing that they need not be optimal.

Key Words: deterministic inventory theory; infinite linear programming duality; existence of optimal policies; semi-Markov decision process; cyclic schedule
History: Received: November 20, 2003; revision received: March 26, 2004;

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