A time-based formulation and upper bounding scheme for the selective travelling salesperson problem

The selective travelling salesperson problem involves determining a tour of maximal value over a set of cities subject to a tour length constraint. The problem has the characteristic that not all cities have to be visited. An integer formulation which combines permutation variables with flow variables is presented. An upper bounding scheme for the number of nodes visited in the final solution is used to reduce the size of the integer program. The problem is solved using an off-the-shelf mixed-integer solver called CPLEX. Results demonstrating the usefulness of the bound are presented. The methodology is also applied to solve a 15-zone fisheries surveillance problem.