# Vehicle Routing Problem with Drones Considering Time Windows and Dynamic Demand
## 6 Conclusions  
This study aimed to solve the practical problems of untimely dynamic delivery network responses and high delivery costs caused by the dynamics of customer orders and the strictness of time constraints during actual delivery. This study establishes a two-stage optimization model based on different demand response strategies with the goal of minimizing the total delivery cost and proposes the simulated annealing chimp optimization algorithm (SACHOA) with a sine-cosine operator to solve this problem. Using test examples and case analysis, the following conclusions are drawn:  
* The number of customers served by trucks is positively correlated with the delivery cost, and the more concentrated their delivery locations are, the smaller the delivery cost is.
* Integrating sine-cosine operators and simulated annealing mechanisms can effectively improve the performance of the algorithm. In addition, SACHOA exhibits certain stability and can effectively solve the VRPDTW problem.
* Response strategies based on different demands have certain practical effectiveness.
* The delivery cost will increase with the increase in the dynamic degree under the condition that the total number of customers remains unchanged.  
This study expands on the theory of VRPD and aims to promote research in the field of the joint distribution of trucks and drones. Through the proposed model and algorithm, we provide logistics companies with an effective tool for optimizing dynamic distribution networks, which is expected to reduce distribution costs and improve distribution efficiency. Ultimately, the research conclusions can provide useful references for relevant decision-makers and support future distribution strategies and decision-making.  
Future research can extend our lines of investigation in several directions. First, this research can be extended to three-dimensional space, where drone obstacle avoidance and collision-free path problems are introduced to better represent the real delivery situation encountered by drones. In addition, there may be more complex factors in practical applications, such as traffic congestion, which will require further research. Finally, there is space for further improvement of the efficiency of the algorithm, and comparison and integration with other optimization algorithms can also be considered.  
Figure 12: Changes in delivery costs under different dynamics.  
**Author Contributions:** Conceptualization, [PERSON] ([PERSON]) and J.H.; methodology, [PERSON]; software, [PERSON]; validation, [PERSON] ([PERSON]) and J.H.; formal analysis, [PERSON] ([PERSON]); investigation, [PERSON] and Y.L. ([PERSON]); resources, [PERSON] ([PERSON]); data curation, [PERSON] and Y.L. ([PERSON]); writing--original draft preparation, [PERSON] and Y.L. ([PERSON]); writing--review and editing, [PERSON] ([PERSON]); visualization, [PERSON]; supervision, [PERSON] ([PERSON]); project administration, [PERSON] ([PERSON]); funding acquisition, [PERSON] ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 70431003.  
Not applicable.  
The data in this article can be obtained from the VRP Website (dorronsoro.es accessed on 7 November 2023).  
The authors would like to thank the anonymous reviewers for their constructive comments.  
The authors declare no conflict of interest.