Models for a Steiner multi-ring network design problem with revenues

The Steiner multi-ring network design problem with revenues consists of designing node-disjoint multiple rings connected by a specific node (hub) and passing through all the nodes with high priority of service and some of the nodes with low priority of service. The number of nodes in each ring has an upper bound to assure a certain level of service. Besides the usual arc link costs, we also consider revenues between each pair of nodes in the same ring, even when they are not connected by a direct link. The objective is to minimize the difference between the total connection cost and total revenue. The problem is a generalization of the problem studied in Gouveia and Pires (Eur J Oper Res 133:21–31, 2001a) and it can also be seen as a combination of variants of two NP-Hard problems, the vehicle routing problem and the maximum edge-weighted clique problem. We introduce and discuss two types of integer linear programming formulations and propose some valid inequalities to strengthen the linear programming relaxation. Computational results are presented to evaluate the quality of the linear programming relaxation bounds associated with these formulations as well as efficiency of the models to obtain the optimal integer solutions.