A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems

Abstract We consider the one-dimensional cutting stock problem which consists in determining the minimum number of given large stock rolls that has to be cut to satisfy the demands of certain smaller item lengths. Besides the standard pattern-based approach of Gilmore and Gomory, containing an exponential number of variables, several pseudo-polynomial formulations were proposed in the last decades. Much research has dealt with arcflow models, their relationship to the standard model, and possible reduction methods, whereas the one-cut approach has not attracted that much scientific interest yet. In this paper, we aim to compare both alternative formulations from a theoretical and numerical point of view. As a theoretical main contribution, we constructively prove the equivalence of the continuous relaxations of the one-cut model, the arcflow model, and the pattern-based model. In particular, the relationship between the one-cut model and the pattern-based model has remained an open question since the one-cut approach was proposed. Moreover, in order to make a computational comparison of the arcflow model and the one-cut model, we present how reduction methods, partly originating from arcflow considerations, can successfully be transferred to the one-cut context. Furthermore, we derive relations between the numbers of variables and constraints in both models, and investigate their influences in numerical simulations.