Duality and Existence of Optimal Policies in Generalized Joint Replenishment

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 non-cyclic schedules, and provide various results regarding cyclic schedules including an example showing that they need not be optimal. Received November 20, 2003; Revised March 8, 2004; Revised March 26, 2004 MSC 2000 subject classification. Primary: 90B05, Secondary: 90C40, 90C90. OR/MS subject classification. Primary: Inventory/production: deterministic multi-item, Secondary: Dynamic programming/optimal control: deterministic semi-Markov, Programming: infinite dimensional