Exact solution of bin‐packing problems using column generation and branch‐and‐bound

Abstract

We explore an arc flow formulation with side constraints for the one‐dimensionalbin‐packing problem. The model has a set of flow conservation constraints and a set ofconstraints that force the appropriate number of items to be included in the packing. Themodel is tightened by fixing some variables at zero level, to reduce the symmetry of thesolution space, and by introducing valid inequalities. The model is solved exactly using abranch‐and‐price procedure that combines deferred variable generation and branch‐and‐bound.At each iteration, the subproblem generates a set of columns, which altogether correspondto an attractive valid packing for a single bin. We describe this subproblem, and theway it is modified in the branch‐and‐bound phase, after the branching constraints are addedto the model. We report the computational times obtained in the solution of the bin‐packingproblems from the OR‐Library test data sets. The linear relaxation of this model provides astrong lower bound for the bin‐packing problem and leads to tractable branch‐and‐boundtrees for the instances under consideration.