The min max multi-trip drone location arc routing problem

This paper studies the Min Max Multi-Trip drone Location Arc Routing Problem (MM-MT-dLARP), an arc routing problem that combines trucks and drones. We have a set of lines (usually curved) that have to be flown over by drones to perform a service (inspection, for example). There is a depot from which the trucks leave, each one carrying a drone, and a set of potential launching points where the truck can launch and pickup the drone. Drones have limited autonomy, but they can make several flights. We consider a min-max objective, in which the makespan, or time necessary to complete the service, must be minimized. Using aerial drones instead of ground vehicles allows to travel off the network: drones can enter a line through any of its points, service only a portion of that line and then exit through another of its points, without following the lines of the network. This allows for finding better solutions but also increases the difficulty of the problem. This issue can be addressed by digitizing the MM-MT-dLARP instances, approximating each line by a polygonal chain with a finite number of intermediate points, and requiring that drones can only enter and exit aline through those intermediate points. Thus, an instance of a discrete Min Max Multi-Trip Location Arc Routing Problem (MM-MT-LARP) is obtained. Here, an integer formulation for the MM-MT-LARP is proposed, some families of valid inequalities are proved to be facet-inducing of a relaxed polyhedron, and a branch-and-cut algorithm based on the strengthened formulation is developed. This algorithm has only been applied to small instances without intermediate points on the lines. In addition, we have developed a matheuristic algorithm for the MM-MT-dLARP that combines a construction phase, four local search procedures integrated into a Variable Neighborhood Descent (VND) algorithm, and a set of rules for selecting intermediate points to improve the solutions. We present the results obtained on a set of randomly generated instances involving up to 6 launching points and 88 original lines.