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The Steiner Team Orienteering Problem (STOP) is defined on a digraph in whicharcs are associated with traverse times, and whose vertices are labeled aseither mandatory or profitable, being the latter provided with rewards(profits). Given a homogeneous fleet of vehicles M, the goal is to find up to m= |M| disjoint routes (from an origin vertex to a destination one) thatmaximize the total sum of rewards collected while satisfying a given limit onthe route's duration. Naturally, all mandatory vertices must be visited. Inthis work, we show that solely finding a feasible solution for STOP is NP-hardand propose a Large Neighborhood Search (LNS) heuristic for the problem. Thealgorithm is provided with initial solutions obtained by means of thematheuristic framework known as Feasibility Pump (FP). In our implementation,FP uses as backbone a commodity-based formulation reinforced by three classesof valid inequalities. To our knowledge, two of them are also introduced inthis work. The LNS heuristic itself combines classical local searches from theliterature of routing problems with a long-term memory component based on PathRelinking. We use the primal bounds provided by a state-of-the-artcutting-plane algorithm from the literature to evaluate the quality of thesolutions obtained by the heuristic. Computational experiments show theefficiency and effectiveness of the proposed heuristic in solving a benchmarkof 387 instances. Overall, the heuristic solutions imply an average percentagegap of only 0.54% when compared to the bounds of the cutting-plane baseline. Inparticular, the heuristic reaches the best previously known bounds on 382 ofthe 387 instances. Additionally, in 21 of these cases, our heuristic is evenable to improve over the best known bounds.

Coupling Feasibility Pump and Large Neighborhood Search to solve the Steiner team orienteering problem