Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems

NP{hard cases of the single{item capacitated lot{sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known which produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist, by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is rst developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models. Subject classi cation: Analysis of algorithms, suboptimal algorithms: fully polynomial approximation schemes. Dynamic programming/optimal control: lot{sizing models. Inventory/production: single{item capacitated lot{sizing. In the single{item capacitated economic lot{sizing problem we consider a production facility which manufactures a single product to satisfy known integer demands over a nite planning horizon of T periods. At each period, the production and holding{ backlogging cost functions are given, and the amount of production is subject to a Department of Quantitative Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands; e{mail: [email protected] Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands; e{mail: [email protected]