A note on Integer Programming Models for Cutting Stock Problems with Setup Costs

The 1-dimensional Cutting Stock Problem (1-CSP) seeks to fulfill with a minimum trim loss the demand of parts of various lengths by cutting them out from a given stock size. In the Pattern Minimization Problem (1-PMP) one wants an optimal 1-CSP solution that uses a minimum number of distinct cutting patterns. We discuss two formulations of 1-PMP: (1) introduced by Vanderbeck, and (2) obtained by introducing setup variables to the classical Gilmore-Gomory formulation for the 1-CSP. Let z i (u) be the bound given by the linear relaxation of (i) under a given vector u = (uk) of parameters. We show that z 2 (u) ≥ z 1 (u), and provide a case where the inequality holds strict.