The team orienteering problem with service times and mandatory & incompatible nodes

The Team Orienteering Problem with Service Times and Mandatory & Incompatible Nodes (TOP-ST-MIN) is a variant of the classic Team Orienteering Problem (TOP), which includes three features that stem from two real-world problems previously studied by the authors: service time at nodes, mandatory nodes and physical or logical incompatibilities between pairs of nodes. We gather all these ingredients for the first time in the same model and prove that even finding a feasible solution to this problem is NP-complete unlike the TOP where finding a feasible solution is straightforward. Two versions of this variant are considered in our study. For such versions, we proposed two alternative mathematical formulations, a route-based and a flow-based formulations. Based on the flow-based formulation, we developed a Cutting-Plane Algorithm (CPA) exploiting five families of valid inequalities, which are either new or have generally not been used as such in the TOP literature and separated by means of new algorithmic methods. Extensive computational experiments showed that the CPA outperforms CPLEX in solving the new benchmark instances, generated in such a way to evaluate the impact of the three novel features that characterise the problem. The CPA is also competitive for the TOP since it is able to solve almost the same number of instances as the state-of-art algorithms.