Consider a graph G = (V,E), where the set of nodes V is partitioned into two sets, the set R of the nodes required to be in the ring and the set S of the nodes which may or may not be included in the ring. Besides the usual edge link costs, revenue profits between each pair of nodes are also considered. The objective is to find a ring including all the nodes in R plus some nodes in S such that link costs minus revenues is minimized. For a similar problem, Gendreau, Labbé and Laporte have presented a formulation with a quadratic term in the objective function. In this paper we follow a different strategy by pointing out that extra variables used in extended formulations for the Asymmetric Travelling Salesman Problem (ATSP) can be used to model the revenues in a linearized way. We adapt a recent formulation for the ATSP proposed by Gouveia and Pires and discuss several classes of valid inequalities. We also point out that the flow variables in a multicommodity flow model can also be used for modelling the revenues in a linearized way. Relationships between the linear programming relaxations of all the models presented in this paper are also discussed. Computational results will also be presented.
