# Vehicle Routing Problem with Drones Considering Time Windows and Dynamic Demand
## 1 Introduction
[PERSON] and [PERSON] [15] studied the problem of dynamic vehicle routing with drones under same-day delivery and proposed a geographic partition-based approximation of the policy function to decide whether an order is delivered by drone or vehicle. However, they ignored vehicle capacity constraints, which may lead to unfeasible scheduling situations such as overloaded trucks during the actual delivery. Although there has been some research on DDVRPTW, most of these studies only considered one vehicle type. However, in DDVRP, which considers the collaborative delivery model of trucks and drones, dynamic customers are often served independently by trucks or drones, in other words, only one kind of vehicle is used to serve dynamic customers. Additionally, time window constraints are not taken into account. With the development in drone technology and the increase in delivery demand, we can more effectively deal with the changing order quantity, capacity, and location and ensure that the delivery time is within the specified time window by comprehensively considering the two factors of dynamic demand and time window. For this reason, we chose to study the Vehicle Routing Problem with Drones considering Time Windows and Dynamic Demand (VRPTWTWD). The purpose was to solve the practical challenges of delivery logistics, improve delivery efficiency, and promote the application of drone delivery. Our research question includes some technically feasible manipulations to increase drone utilization, for example, allowing drones to carry multiple packages per trip, synchronizing with trucks to meet dynamic customer demands, and allowing direct launch and retrieve drones at distribution centers. Some important and practical factors should also be considered, especially the impact of dynamic degrees on vehicle scheduling and routing. The higher the dynamic degree of customers, the greater the challenge of optimizing delivery routes. The above factors make the VRPTWTWD problem more complicated; therefore, we established a mixed integer programming model for VRPTWTWD and proposed a Simulated Annealing Chimp Optimization Algorithm (SACHOA) with a sine-cosine operator to solve it. The algorithm uses the sine-cosine operator and the simulated annealing mechanism to avoid local optimal and improve the convergence accuracy of the algorithm. To adapt to the Hamiltonian loop routing with a branching structure under the two vehicle types, a variable-dimensional matrix encoding method, which makes the algorithm have strongeradaptability when dealing with diverse scenarios, is introduced. This study verified the effectiveness and stability of the proposed strategy and algorithm through examples and took a self-operated brand store in Shenyang as a case for verification and analysis. Finally, we conducted a sensitivity analysis on the degree of dynamics to ensure the practical feasibility and applicability of the research results. The results of this study have significant practical implications for enhancing logistics delivery efficiency and reducing logistics costs, particularly in meeting personalized customer demands and enhancing customer experience. The remainder of this paper is organized as follows: Section 2 presents a review of the relevant literature, followed by a mathematical formulation of the problem in Section 3. Sections 4 and 5 describe the proposed solution approach and computational results, respectively. Section 6 concludes this study and suggests potential extensions for future studies.