Solving the length constrained [formula omitted]-drones rural postman problem

Abstract

In this paper we address the Length Constrained K-Drones Rural Postman Problem (LC K-DRPP). This is a continuous optimization problem where a fleet of homogeneous drones have to jointly service (traverse) a set of (curved or straight) lines of a network. Unlike the vehicles in classical arc routing problems, a drone can enter a line through any of its points, service a portion of that line, exit through another of its points, then travel directly to any point on another line, and so on. Moreover, since the range of the drones is restricted, the length of each route is limited by a maximum distance. Some applications for drone arc routing problems include inspection of pipelines, railway or power transmission lines and traffic monitoring.To deal with this problem, LC K-DRPP instances are digitized by approximating each line by a polygonal chain with a finite number of points and allowing drones to enter and exit each line only at these points. In this way we obtain an instance of the Length Constrained K-vehicles Rural Postman Problem (LC K-RPP). If the number of points used to discretize the lines is large, the LC K-RPP instance can be extremely large and, hence, very difficult to solve optimally. Even heuristic algorithms can fail in providing feasible solutions in reasonable computing times. An alternative is to generate smaller LC K-RPP instances by approximating each line with few but “significant” segments.We present a formulation and some valid inequalities for the LC K-RPP. Based on this, we have designed and implemented a branch-and-cut algorithm for its solution. Moreover, in order to be capable of providing good solutions for large LC K-RPP instances, we propose a matheuristic algorithm that begins by finding good solutions for the LC K-RPP instance obtained by approximating each line by a single segment. Then, to find better solutions, some promising intermediate points are sequentially incorporated. Extensive computational experiments to assess the performance of both algorithms are performed on several sets of instances from the literature.