This thesis gives a comprehensive view of the scope of formulations and<br />related solution approaches for the cutting stock problem (CSP) and its<br />variants. The focus is on branch-and-price approaches. Specialized<br />algorithms are developed for knapsack subproblems that arise in the<br />course of the algorithm. Thorough numerical tests are used to identify good strategies<br />for initialization, stabilization, branching, and producing<br />primal solutions. Industrial variants of the <br />problem are shown to be tractable for a branch-and-price approach.<br /><br /><br />The models studied are the following: the standard cutting stock and<br />bin packing problems, a variant in which the production levels lie in<br />a prescribed interval of tolerance, the multiple width cutting stock<br />problem in which stock pieces are of different size, a variant with<br />additional technical constraints that arise typically in industrial<br />applications, and a variant where the number of distinct cutting<br />patterns used is minimized given a waste threshold. <br /><br /><br />First, we consider various formulation of the Cutting Stock Problem<br />(CSP): different models of the knapsack subproblem can be exploited to<br />reformulate the CSP. Moreover, we introduce different ways of<br />modeling local exchanges in the solution (primal exchanges imply dual<br />constraints that stabilize the column generation procedure). Some<br />models are shown to be valid integer programming (IP) reformulations while others define<br />relaxations. The dual bounds defined by their LP solution are compared<br />theoretically.<br /><br />Then, we study the variants of the knapsack subproblem that arise<br />in a column generation approach to the CSP. The branching constraints<br />used in the branch-and-price algorithm can result in class bound and<br />setup cost or the need for a binary decomposition in the subproblem. <br />We show how standard knapsack solvers (dynamic programming approach and specialized<br />branch-and-bound algorithm) can be extended to these variants of the<br />knapsack problem.<br /><br />Next, we discuss some branch-and-price implementation strategies. We compare <br />different modes of initialization of the column generation procedure, we present our numerical study of various stabilization<br />strategies to accelerate convergence of the procedure. We compare in particular the impact of the various ways of introducing<br />local exchanges in our primal model and other stabilization techniques<br />such as dual solution smoothing techniques or penalization from a<br />stability center that prevent the fluctuation of the dual variables. <br />To generate the columns we study different strategies based on the use of heuristic columns or on a multiple generation of columns.<br />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. <br /><br />Finally, we present numerical results on two industrial applications that<br />correspond to the variant with technical restrictions for which we<br />minimize first the waste and then the number of setups.
| Identifer | oai:union.ndltd.org:CCSD/oai:tel.archives-ouvertes.fr:tel-00011657 |
| Date | 29 June 2005 |
| Creators | Perrot, Nancy |
| Publisher | Université Sciences et Technologies - Bordeaux I |
| Source Sets | CCSD theses-EN-ligne, France |
| Language | English |
| Detected Language | English |
| Type | PhD thesis |
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