This paper describes an attempt to solve the one‐dimensional cutting stock problem exactly, using column generation and branch‐and‐bound. A new formulation is introduced for the one‐dimensional cutting stock problem that uses general integer variables, not restricted to be binary. It is an arc flow formulation with side constraints, whose linear programming relaxation provides a strong lower bound. In this model, a cutting pattern, which corresponds to a path, is decomposed into single arc variables. The decomposition serves the purpose of showing that it is possible to combine the branch‐and‐bound method with variable generation. Computational times are reported for one‐dimensional cutting stock instances with a number of orders up to 30.
We explore an arc flow formulation with side constraints for the one-dimensional bin-packing problem. The model has a set of flow conservation constraints and a set of constraints that force the appropriate number of items to be included in the packing. The model is tightened by fixing some variables at zero level, to reduce the symmetry of the solution space, and by introducing valid inequalities. The model is solved exactly using a branch-and-price procedure that combines deferred variable generation and branch-andbound. At each iteration, the subproblem generates a set of columns, which altogether correspond to an attractive valid packing for a single bin. We describe this subproblem, and the way it is modified in the branch-and-bound phase, after the branching constraints are added to the model. We report the computational times obtained in the solution of the bin-packing problems from the OR-Library test data sets. The linear relaxation of this model provides a strong lower bound for the bin-packing problem and leads to tractable branch-and-bound trees for the instances under consideration.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.