Polyhedral analysis and a new algorithm for the length constrained K –drones rural postman problem

The Length Constrained K –Drones Rural Postman Problem (LC K –DRPP) is a continuous optimization problem where a set of curved or straight lines of a network have to be traversed, in order to be serviced, by a fleet of homogeneous drones, with total minimum cost. Since the range and endurance of drones is limited, we consider here that the length of each route is constrained to a given limit L . Drones are not restricted to travel on the network, and they can enter and exit a line through any of its points, servicing only a portion of that line. Therefore, shorter solutions are obtained with “aerial” drones than with “ground” vehicles that are restricted to the network. If a LC K –DRPP instance is digitized by approximating each line with a polygonal chain, and it is assumed that drones can only enter and exit a line through the points of the chain, an instance of the Length Constrained K –vehicles Rural Postman Problem (LC K–RPP) is obtained. This is a discrete arc routing problem, and therefore can be solved with combinatorial optimization techniques. However, when the number of points in each polygonal chain is very large, the LC K –RPP instance can be so large that it is very difficult to solve, even for heuristic algorithms. Therefore, it is necessary to implement a procedure that generates smaller LC K –RPP instances by approximating each line by a few but “significant” points and segments. In this paper, we present a new formulation for the LC K –RPP with two binary variables for each edge and each drone representing the first and second traversals of the edge, respectively. We make a polyhedral study of the set of solutions of a relaxed formulation and prove that several families of inequalities induce facets of the polyhedron. We design and implement a branch–and–cut algorithm for the LC K –RPP that incorporates the separation of these inequalities. This B &C is the main routine of an iterative algorithm that, by solving a LC K –RPP instance at each step, finds good solutions for the original LC K –DRPP. The computational results show that the proposed method is effective in finding good solutions for LC K –DRPP, and that the branch–and–cut algorithm for the LC K –RPP outperforms the only published exact method for this problem.