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LISCOS, Large Scale Integrated Supply Chain Optimization Software Based on Branch–and–Cut.pdf

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LISCOS, Large Scale Integrated Supply Chain Optimization Software Based on Branch–and–Cut

CORE DISCUSSION PAPER 2001/8 Discrete Lot–Sizing and Convex Integer Programming Andrew J. Miller 1 Laurence A. Wolsey 2 February 2001 Abstract We study the polyhedral structure of variants of the discrete lot–sizing problem viewed as special cases of convex integer programs. Our approach in studying convex integer programs is to develop results for simple mixed integer sets that can be used to model integer convex objective functions. These results allow us to define integral linear programming formulations for the discrete lot–sizing problem in which backlogging and/or safety stocks are present, and to give extended formulations for other cases. Our results help significantly to solve test cases arising from an industrial application motivating this research. 1CORE, Universite? Catholique de Louvain, Belgium 2CORE and INMA, Universite? Catholique de Louvain, Belgium This paper presents research results of the Belgian Program on Interuniversity Poles of Attrac- tion initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific responsibility is assumed by the authors. This research was also supported by the European Commission GROWTH Programme, Research Project LISCOS, Large Scale Integrated Supply Chain Optimization Software Based on Branch–and–Cut and Constraint Programming Methods, Contract No. GRDI–1999–10056. 1 1 Introduction This article is motivated by a production planning problem from a well-known multina- tional company encountered recently. The problem in question is a multi–item discrete lot–sizing problem with backlogging and with additional complicating constraints, so finding tight formulations for different variants of the discrete lot–sizing problem is a principal topic of this paper. The second topic arose from the observation that the different variants can also be viewed as convex integer programs (CIPs) with simple piecewise affine objective functions and simple linear constraints. This second viewpoint lead