Formulations for the clustered traveling salesman problem with d-relaxed priority rule

In the classical Traveling Salesman Problem, the order in which the vertices are visited has no restrictions. The only condition imposed is that each vertex is visited only once. In some real situations this condition may not be sufficient to represent the problem, as there are cases in which the vertex visit order becomes extremely important. In other words, it is also necessary to consider a priority between the vertices. To deal with these situations, some formulations are proposed in the literature and are based on a rule called d-relaxed priority rule that captures the trade-off between total distance and vertex priorities. In this paper, the formulations from the literature are improved using valid inequalities and different formulations based on precedence variables are proposed. Computational results, based on data from the literature, are presented to demonstrate the competitiveness of the proposed approaches.