Mixed integer programming formulations for the generalized traveling salesman problem with time windows

The generalized traveling salesman problem with time windows (GTSPTW) is defined on a directed graph where the vertex set is partitioned into clusters. One cluster contains only the depot. Each vertex is associated with a time window, during which the visit must take place if the vertex is visited. The objective is to find a minimum cost tour starting and ending at the depot such that each cluster is visited exactly once and time constraints are respected, i.e., for each cluster, a single vertex is visited during its time window. In this paper, four mixed integer linear programming formulations for the GTSPTW are proposed and compared. They are based on different definitions of variables. All the formulations are compact, which means the number of decision variables and constraints is polynomial with respect to the size of the instance. Dominance relations between their linear relaxations are established theoretically. Computational experiments are conducted to compare the linear relaxations and branch-and-bound performances of the four formulations. The results show that two formulations are better than the other ones.