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Integer programming column generation strategies for the cutting stock problem and its variants

Abstract : This thesis gives a comprehensive view of the scope of formulations and
related solution approaches for the cutting stock problem (CSP) and its
variants. The focus is on branch-and-price approaches. Specialized
algorithms are developed for knapsack subproblems that arise in the
course of the algorithm. Thorough numerical tests are used to identify good strategies
for initialization, stabilization, branching, and producing
primal solutions. Industrial variants of the
problem are shown to be tractable for a branch-and-price approach.

The models studied are the following: the standard cutting stock and
bin packing problems, a variant in which the production levels lie in
a prescribed interval of tolerance, the multiple width cutting stock
problem in which stock pieces are of different size, a variant with
additional technical constraints that arise typically in industrial
applications, and a variant where the number of distinct cutting
patterns used is minimized given a waste threshold.

First, we consider various formulation of the Cutting Stock Problem
(CSP): different models of the knapsack subproblem can be exploited to
reformulate the CSP. Moreover, we introduce different ways of
modeling local exchanges in the solution (primal exchanges imply dual
constraints that stabilize the column generation procedure). Some
models are shown to be valid integer programming (IP) reformulations while others define
relaxations. The dual bounds defined by their LP solution are compared

Then, we study the variants of the knapsack subproblem that arise
in a column generation approach to the CSP. The branching constraints
used in the branch-and-price algorithm can result in class bound and
setup cost or the need for a binary decomposition in the subproblem.
We show how standard knapsack solvers (dynamic programming approach and specialized
branch-and-bound algorithm) can be extended to these variants of the
knapsack problem.

Next, we discuss some branch-and-price implementation strategies. We compare
different modes of initialization of the column generation procedure, we present our numerical study of various stabilization
strategies to accelerate convergence of the procedure. We compare in particular the impact of the various ways of introducing
local exchanges in our primal model and other stabilization techniques
such as dual solution smoothing techniques or penalization from a
stability center that prevent the fluctuation of the dual variables.
To generate the columns we study different strategies based on the use of heuristic columns or on a multiple generation of columns.
We also consider the use of heuristics based on column generation to find a primal bound. These are compared to a classic constructive heuristic. Then, we compare the different branching rules that are used in the branch-and-price procedure.

Finally, we present numerical results on two industrial applications that
correspond to the variant with technical restrictions for which we
minimize first the waste and then the number of setups.
Document type :
Complete list of metadatas
Contributor : Nancy Perrot <>
Submitted on : Tuesday, February 21, 2006 - 4:21:41 PM
Last modification on : Thursday, January 11, 2018 - 6:12:27 AM
Long-term archiving on: : Saturday, April 3, 2010 - 8:44:04 PM


  • HAL Id : tel-00011657, version 1




Nancy Perrot. Integer programming column generation strategies for the cutting stock problem and its variants. Mathematics [math]. Université Sciences et Technologies - Bordeaux I, 2005. English. ⟨tel-00011657⟩



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