The vehicle routing problem with drones and drone speed selection

Joint parcel delivery by trucks and drones has enjoyed significant attention for some time, as the advantages of one delivery method offset the disadvantages of the other. This paper focuses on the vehicle routing problem with drones and drone speed selection (VRPD-DSS), which considers speed-dependent energy consumption and drone-charging in detail. For this purpose, we formulate a comprehensive mixed-integer problem that aims to minimize the operational costs consisting of fuel consumption costs of the trucks, labor costs for the drivers, and energy costs of the drones. The speed at which a drone performs a flight must be selected from a discrete set. We introduce preprocessing steps to eliminate dominated speeds for a flight to reduce the problem size and use valid inequalities to accelerate the solution process. The consideration of speed-dependent energy consumption leads to the fact that it is advisable to perform different flights at different speeds and not to consistently operate a drone at maximum speed. Instead, drone speed should be selected to balance drone range and speed of delivery. Our extensive computational study of a rural real-world setting shows that, by modeling energy consumption realistically, the savings in operational costs compared to truck-only delivery are significant but smaller than those identified in previously published work. Our analysis further reveals that the greatest savings stem from the fact that overall delivery time decreases compared to truck-only delivery, allowing costly truck-driver time to be reduced. The additional energy costs of the drone, however, are largely negligible.