In cutting stock problems, after an optimal (minimal stock usage)
cutting plan has been devised, one might want to further reduce the
operational costs by minimizing the number of setups. A setup
operation occurs each time a different cutting pattern begins to be
produced. The related optimization problem is known as the Pattern
Minimization Problem, and it is particularly hard to solve exactly.
In this paper, we present different techniques to strengthen a
formulation proposed in the literature. Dual feasible functions are
used for the first time to derive valid inequalities from different
constraints of the model, and from linear combinations of constraints. A new arc
flow formulation is also proposed. This formulation is used to
define the branching scheme of our branch-and-price-and-cut
algorithm, and it allows the generation of even stronger cuts by
combining the branching constraints with other constraints of the
model. The computational experiments conducted on instances from the
literature show that our algorithm finds optimal
integer solutions faster than other approaches. A set of computational
results on random instances is also reported.