Routing for unmanned aerial vehicles: Touring dimensional sets

In this paper we deal with an extension of the crossing postman problem to design routes that have to visit different shapes of dimensional elements rather than edges. This problem models the design of routes of drones or other vehicles that must visit a number of geographical elements to deliver some good or service and then move directly to the next using straight line displacements. We present two families of mathematical programming formulations. The first one is time-dependent and captures a number of characteristics of real applications at the price of using three indexes variables. The second family of formulations is not time-dependent, instead it uses connectivity properties to ensure the proper definition of routes. We compare them on a testbed of instances with different shapes of elements: second order cone (SOC) representable and polyhedral neighborhoods and polygonal chains. The computational results reported in this paper show that our models are useful and our formulations can solve to optimality medium size instances of sizes similar to other combinatorial problems including neighborhoods that have already been studied in the literature. To address larger instances we also present a heuristic algorithm that runs in two phases: clustering and Variable Neighborhood Search. This algorithm performs very well since it provides promising feasible solutions and, in addition, it can be used to initialize the solvers with feasible solutions.