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.