Efficient Approximation Algorithms for the Economic Lot-Sizing in Continuous Time

We consider a continuous-time variant of the classical Economic Lot-Sizing (ELS) problem. In this model, the setup cost is a continuous function with lower bound Kmin > 0, the demand and holding costs are integrable functions of time and the replenishment decisions are not restricted to be multiples of a base period. Starting from the assumption that certain operations involving the setup and holding cost functions can be carried out efficiently, we argue that this variant admits a simple approximation scheme based on dynamic programming: if the optimal cost of an instance is OPT, we can find a solution with cost at most (1 + �)OPT using no more than O � 1 �2 OPT Kmin log OPT Kmin � of these operations. We argue, however, that this algorithm could be improved on instances where the setup costs are “generally” very large compared with Kmin. This leads us to introduce a notion of input-size σ that is significantly smaller than OPT/Kmin on instances of this type, and then to define an approximation scheme that executes O � 1 �2σ 2 log 2 � OPT Kmin �� operations. Besides dynamic programming, this second approximation scheme builds on a novel algorithmic approach for Economic Lot Sizing problems. 1 Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium; e-mail: [email protected]. Supported by FSR Incoming Post-doctoral Fellowship of the Catholic University of Louvain (UCL), funded by the French Community of Belgium. 2 Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium; e-mail: [email protected] This text presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Ministers Office, Science Policy Programming. The scientific responsibility is assumed by the authors.