Exact Solution of Cutting Stock Problems Using Column Generation and Branch‐and‐Bound

Abstract

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.