New Formulations for the Team Orienteering Problem

Routing problems have many practical applications in distribution and logistics management. The Traveling Salesman Problem (TSP) and its variants lie at the heart of routing problems. The Orienteering Problem (OP) is a subset selection version of well-known TSP which comes from an outdoor sport played on mountains. In the OP, the traveller must finish its journey within a predetermined time (cost, distance), and gets a gain (profit, reward) from the visited nodes. The objective is to maximize the total gain that the traveller collects during the predetermined time. The OP is also named as the selective TSP since not all cities have to be visited. The Team Orienteering Problem (TOP) is the extension of OP by multiple-traveller. As far as we know, there exist a few formulations for the TOP. In this paper we present two new integer linear programming formulations (ILPFs) for the TOP with O(n2) binary variables and O(n2) constraints, where n is the number of nodes on the underlying graph. The proposed formulations can be directly used for the OP when we take the number of traveller as one. We demonstrate that, additional restrictions and/or side conditions can be easily imported for both of the formulations. The performance of our formulations is tested on the benchmark instances from the literature. The benchmark instances are solved via CPLEX 12.6 by using the proposed and existing formulations. The computational experiments demonstrate that both of the new formulations outperform the existing one. The new formulations are capable of solving optimally most of the benchmark instances, which have solved by using special heuristics so far. As a result, the proposed formulations can be used to find the optimal solution of small- and moderate-size real life OP and TOP by using an optimizer. Keywords: Traveling salesman problem, orienteering problem, modeling;