Formulations for dynamic lot sizing with service levels

In this article, we study deterministic dynamic lot‐sizing problems with a service‐level constraint on the total number of periods in which backlogs can occur over a finite planning horizon. We give a natural mixed integer programming formulation for the single item problem (LS‐SL‐I) and study the structure of its solution. We show that an optimal solution to this problem can be found in \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathcal O(n^2\kappa)\end{align*} \end{document} time, where n is the planning horizon and \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\kappa=\mathcal O(n)\end{align*} \end{document} is the maximum number of periods in which demand can be backlogged. Using the proposed shortest path algorithms, we develop alternative tight extended formulations for LS‐SL‐I and one of its relaxations, which we refer to as uncapacitated lot sizing with setups for stocks and backlogs. {We show that this relaxation also appears as a substructure in a lot‐sizing problem which limits the total amount of a period's demand met from a later period, across all periods.} We report computational results that compare the natural and extended formulations on multi‐item service‐level constrained instances. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013