A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting

The one-dimensional cutting stock problem (1D-CSP) and the twodimensional two-stage guillotine constrained cutting problem (2D-2CP) are considered in this paper. The Gilmore-Gomory model of these problems has a very strong continuous relaxation which provides a good bound in an LP-based solution approach. In recent years, there have been several efforts to attack the one-dimensional problem by LP-based branch-and-bound with column generation (called branch-and-price) and by general-purpose CHVÁTAL-GOMORY cutting planes. When cutting planes are added to the LP relaxation, the pricing problem becomes very complex and often cannot be solved optimally in an acceptable time. Moreover, the modeling and implementation of its solution method as well as of the cutting plane apparatus for the chosen kind of cuts requires much effort. We develop a new upper bound for this pricing problem. We improve the numerical stability of the cutting plane algorithm and integrate mixed-integer GOMORY cuts. For 2D2CP we propose a pricing procedure which enables the search for strips of different widths without repetitions. Various branching strategies and tools such as pseudo-costs and reduced cost bounding are investigated. Tests show that, for 1D-CSP, general-purpose cuts are useful only in exceptional cases. However, for 2D-2CP their combination with branching is more effective than either approach alone and mostly better than other methods from the