New mathematical formulations for the selective travelling salesman problem

Travelling Salesman Problem (TSP) is the basis of many real-life problems as vehicle routing, scheduling etc. The objective of TSP and its extensions is usually to minimize the total cost or the total amount of time spent or the total distance travelled. In recent years, researchers take an interest in problems which aim to maximize the profit such as revenue, income etc. by relaxation of the constraint which ensures to visit all nodes in the network, under the given budget or travel time constraint. In the literature, these problems are named as "The Selective TSP or Orienteering Problem ", "The Prize-collecting TSP " and "The Profitable Tour Problem". These problems are gathered under the common name of "TSP with Profits". Just as TSP and its extensions, it is clear that heuristic methods or special algorithms are primarily preferred as solution approaches of TSP with profits, therefore mathematical models don't attract enough attention. In the scope of this study, two new mathematical models (node-based and edge-based) have been presented for the Selective TSP which is the most popular problem of TSP with profits. Size of the new models is polynomial according to the numbers of its constraints and binary decision variables so that they can be solved by an integer-programming solver. Some of Orienteering Problem benchmark instances are solved with new models by using CPLEX 12.5 software. The results are compared to an existing model in literature and it is found that the new models are superior to the existing model. Besides, it has been detected that some of the optimal solutions of mathematical models are larger than the solutions of heuristics in the literature.