# Vehicle Routing Problem with Drones Considering Time Windows and Dynamic Demand
## 1 Introduction  
With the rapid development of drone technology, logistics companies have become increasingly interested in the field of drone delivery [1]. Drones have numerous advantages, such as not being restricted by ground obstacles, fast delivery, low cost, etc., ensuring that they can avoid traffic congestion and deliver goods to their destinations more quickly. However, the drone's load capacity is small and the battery technology cannot support extended deliveries. These limitations render it impossible for the drone to complete the delivery service independently. Therefore, most logistics and delivery companies are more inclined to adopt a collaborative delivery model of trucks and drones [2]. Under this cooperative model, trucks are not only used for large-scale delivery tasks but also serve as a \"mobile platform\" for drones, which undertake small-scale delivery tasks. This model can fully exploit the advantages of trucks and drones, effectively avoiding traffic congestion and road restrictions, significantly reducing labor and time costs, and comprehensively improving service efficiency. Different from traditional truck delivery, in this model, trucks and drones are synchronously combined for delivery. Trucks and drones travel at different speeds, hence impacting the delivery time window. Additionally, in the actual delivery process, the distribution center continues to receive new demands after the truck leaves, resulting in constant changes in the status of the delivery system, which requires readjustment of the existing delivery route. If the delivery route cannot be planned and adjusted reasonably and quickly, it will not only greatly increase the operating costs of the enterprise, but also lower the customer service level. In summary, it is of great practical significance to study the vehicle routing problem with dynamic demand and time window under a collaborative delivery model of trucks and drones. The vehicle routing problem (VRP) is a classic combinatorial optimization problem that has received extensive attention in the research community [3]. With the complexity of the real situation, many varieties of VRPs have been derived, including VRP with capacity constraints (CVRP) [4], VRP with time window constraints (VRPTW) [5], heterogeneous VRP (HVRP) [6], VRP with dynamic demand (DDVRP) [7], and VRP with dynamic demand and time window (DDVRPTW) [8]. Among these, DDVRPTW was introduced relatively late, with [PERSON] first combining time windows with dynamic vehicle routing problems in 2000 [9]. Since then, scholars have proposed different models and solutions [10; 11]. For example, [PERSON] et al. [12] proposed a method based on bender decomposition to apply electric vehicle delivery to the VRP problem with dynamic demand. Based on the intelligent network connection and queuing theory, [PERSON] et al. [13] proposed a predictive charging queuing probability and a path optimization model to reduce the waiting time of electric logistics vehicles, select appropriate charging locations and reduce delivery costs. However, electric vehicles (road-based ground vehicles) often lead to delivery inefficiencies due to factors such as traffic congestion and road network constraints, making it difficult to reduce delivery costs. In addition, dynamic customer orders and strict time constraints make it impossible for dispatchers to efficiently consolidate multiple orders of different sizes, while traditional vehicles can only deliver a small number of packages owing to their poor ability to quickly respond to the ever-changing dynamic demands, resulting in low flexibility and customer coverage in the delivery model. To solve this problem, [PERSON] et al. [14] proposed a novel delivery service in which the truck executes the demand of the customer who has placed an order in advance while emerging dynamic demand is replenished by the drone. However, this delivery mode limits the usage scenarios of drones, seriously affects the service scope of drones, and is not conducive to the reduction of total delivery costs.