Using Benders decomposition to implicitly model tour scheduling

This paper introduces a new integrated model for the combined days-off and shift scheduling problem (the tour scheduling problem). The model generalizes the forward and backward constraints, previously introduced by Bechtold and Jacobs for the shift scheduling problem, to the tour scheduling problem. This results in a general and compact formulation that can handle several types of scheduling flexibility. We also provide a new proof of the correctness of forward and backward constraints based on Benders decomposition. The latter approach is interesting in itself because it can be used to solve the problem when extraordinary overlap of break windows or start-time bands is present. A discussion of model size for a set of hypothetical test problems is presented to show the merits of the new formulation.