A Class of Subpattern Formulations for One-Dimensional Stock Cutting

In the classical Gilmore-Gomory model for the one-dimensional cutting stock problem (1D-CSP) we have to deal implicitly with a huge number of variables representing all feasible patterns. The advantage is a strong relaxation and absence of symmetries. To reduce the number of variables, we can restrict them to subpatterns, i.e., partial patterns which are combined to produce whole patterns. Each subpattern can be a part of different resulting patterns; thus, all patterns have to be numbered, which brings much symmetry into the model. Moreover, the continuous relaxation may not only have non-integer pattern application intensities but also overcapacity patterns, resulting in a weak bound. However, it is possible to choose a small set of elementary subpatterns so that standard optimization software can be applied to the model.