On the Sum Constraint: Relaxation and Applications

The global constraint sum can be used as a tool to implement summations over sets of variables whose indices are not known in advance. This paper has two major contributions. On the theoretical side, we present the convex hull relaxation for the sum constraint in terms of linear inequalities, whose importance in the context of hybrid models is then justified. On the practical side, we demonstrate the applicability of the sum constraint in a scheduling problem that arises as part of the development of new products in the pharmaceutical and agrochemical industries. This problem can be modeled in two alternative ways: by using the sum constraint in a natural and straightforward manner, or by using the element constraint in a trickier fashion. With the convex hull relaxation developed earlier, we prove that the linear relaxation obtained from the former model is tighter than the one obtained from the latter. Moreover, our computational experiments indicate that the CP model based on the sum constraint is significantly more efficient as well.