New Stabilization Procedures for the Cutting Stock Problem

• Published in: INFORMS journal on computing 2011-09, Vol.23 (4), p.530-545
• Main Authors: Clautiaux, Francois, Alves, Claudio, Valerio, Jose, Rietz, Carvalho, Jurgen
• Format: Article
• Language: English
• Subjects: Algorithms; combinatorial optimization; Convergence (Mathematics); cutting stock; Cutting stock problem; Decomposition (Mathematics); Integer programming; Mathematical optimization; Optimization algorithms; Studies
• ISSN: 1091-9856; 1526-5528; 1091-9856
• DOI: 10.1287/ijoc.1100.0415

In this paper, we deal with a column generation-based algorithm for the classical cutting stock problem. This algorithm is known to have convergence issues, which are addressed in this paper. Our methods are based on the fact that there are interesting characterizations of the structure of the dual problem, and that a large number of dual solutions are known. First, we describe methods based on the concept of dual cuts , proposed by Valério de Carvalho [Valério de Carvalho, J. M. 2005. Using extra dual cuts to accelerate column generation. INFORMS J. Comput. 17 (2) 175-182]. We introduce a general framework for deriving cuts, and we describe a new type of dual cut that excludes solutions that are linear combinations of some other known solutions. We also explore new lower and upper bounds for the dual variables. Then we show how the prior knowledge of a good dual solution helps improve the results. It tightens the bounds around the dual values and makes the search converge faster if a solution is sought in its neighborhood first. A set of computational experiments on very hard instances is reported at the end of the paper; the results confirm the effectiveness of the methods proposed.