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Keywords:

  • cutting stock;
  • integer decomposition;
  • linear relaxations

Abstract

We discuss two formulations of the pattern minimization problem: (1) introduced by Vanderbeck, and (2) obtained adding setup variables to the cutting stock formulation by Gilmore-Gomory. Let zmath image(u) be the bound given by the linear relaxation of (i) under a given vector u of parameters. We show that zmath image(u) ≥ zmath image(u) and provide a class of instances for which the inequality holds strict. We observe that the linear relaxation of both formulations can be solved by the same column generation procedure and discuss the critical role of parameter u. The article is completed by a numerical test comparing the lower bounds obtained through (1) and (2) for different values of u. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011