Unmanned aerial vehicles, or drones, have the potential to significantly reduce the cost and time of making last-mile deliveries and responding to emergencies. Despite this potential, little work has gone into developing vehicle routing problems (VRPs) specifically for drone delivery scenarios. Existing VRPs are insufficient for planning drone deliveries: either multiple trips to the depot are not permitted, leading to solutions with excess drones, or the effect of battery and payload weight on energy consumption is not considered, leading to costly or infeasible routes. We propose two multitrip VRPs for drone delivery that address both issues. One minimizes costs subject to a delivery time limit, while the other minimizes the overall delivery time subject to a budget constraint. We mathematically derive and experimentally validate an energy consumption model for multirotor drones, demonstrating that energy consumption varies approximately linearly with payload and battery weight. We use this approximation to derive mixed integer linear programs for our VRPs. We propose a cost function that considers our energy consumption model and drone reuse, and apply it in a simulated annealing (SA) heuristic for finding suboptimal solutions to practical scenarios. To assist drone delivery practitioners with balancing cost and delivery time, the SA heuristic is used to show that the minimum cost has an inverse exponential relationship with the delivery time limit, and the minimum overall delivery time has an inverse exponential relationship with the budget. Numerical results confirm the importance of reusing drones and optimizing battery size in drone delivery VRPs.
Abstract The vehicle routing problem with multiple trips ( VRPMT ) is a variants of the standard ( VRP ), where each vehicle can be used more than once during the working period. For this NP-Hard problem, we propose a variable neighborhood search Algorithm in which four neighborhood structure are designed to find the planning of trips. The algorithm was tested over a set of benchmark problems and the obtained solutions were compared with five previously proposed algorithms. Encouraging results are obtained. We consider the Multi Trip Vehicle Routing Problem, in which a set of geographically scattered customers have to be served by a fleet of vehicles. Each vehicle can perform several trips during the working day. The objective is to minimize the total travel time while respecting temporal and capacity constraints. We investigate the exact solution of the vehicle routing problem with time windows, where multiple trips are allowed for the vehicles. In contrast to previous works in the literature, we specifically consider the case in which it is mandatory to visit all customers and there is no limitation on duration. We develop two branch-and-price frameworks based on two set covering formulations: a traditional one where columns (variables) represent routes, that is, a sequence of consecutive trips, and a second one in which columns are single trips. One important difficulty related to the latter is the way mutual temporal exclusion of trips can be handled. It raises the issue of time discretization when solving the pricing problem. Our dynamic programming algorithm is based on concept of groups of labels and representative labels. We provide computational results on modified small-sized instances (25 customers) from Solomon’s benchmarks in order to evaluate and compare the two methods. Results show that some difficult instances are out of reach for the first branch-and-price implementation, while they are consistently solved with the second. The vehicle routing problem with multiple routes consists in determining the routing of a fleet of vehicles when each vehicle can perform multiple routes during its operations day. This problem is relevant in applications where the duration of each route is limited, for example when perishable goods are transported. In this work, we assume that a fixed-size fleet of vehicles is available and that it might not be possible to serve all customer requests, due to time constraints. Accordingly, the objective is first to maximize the number of served customers and then, to minimize the total distance traveled by the vehicles. An adaptive large neighborhood search, exploiting the ruin-and-recreate principle, is proposed for solving this problem. The various destruction and reconstruction operators take advantage of the hierarchical nature of the problem by working either at the customer, route or workday level. Computational results on Euclidean instances, derived from well-known benchmark instances, demonstrate the benefits of this multi-level approach. 