Abstract
The logistics market stands to benefit from the accessibility and increased use of new technologies. As technology continues to advance, drones have emerged as a notable innovation. Within the logistics field, there is growing interest in leveraging drones, particularly for handling small and medium-sized orders such as mobile phones. The appeal of drones lies in their economic and environmental advantages, attributed to their reduced energy consumption. These unmanned aerial vehicles are considered a valuable component of the ongoing technological revolution in transportation, with the potential to enhance the efficiency of last-mile deliveries. To explore their applicability, mixed vehicle-drone distribution models have surfaced as a promising alternative to traditional delivery methods. These models enable companies to minimize transportation costs by leveraging the strengths of both vehicles and drones. In this study, we present a solution to the vehicle routing model with multiple drones (VRPm-D). Our objective is to efficiently transport a specified quantity of products from a central depot to customers by devising optimal routes for both trucks and drones. The study focuses on employing a VRPm-D model to facilitate the transportation process, involving a predetermined fleet of vehicles and drones. The vehicles start and end their routes at a central depot. The primary goal is to minimize the time taken by trucks utilizing drones to cater to the needs of every customer effectively while considering the payload capacity and energy endurance constraints of the drones. To address these objectives, we propose a hybrid vehicle-drone routing problem formulated using the genetic clustering algorithm (HVDRP-GCA). This approach aims to optimize the routes and schedules for both vehicles and drones, taking into account the aforementioned constraints. Our results demonstrate that the processing time exponentially increases as the number of customers grows, particularly noticeable in routes with five or more customers. Importantly, the HVDRP-GCA model outperforms existing methods in the literature, providing favorable outcomes. The results highlight the dominance of our proposed model compared to previous approaches found in the literature.
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References
Ahmadi Malakot, R., Sahraeian, R., & Hosseini, S. M. H. (2022). Optimizing the sales level of perishable goods in a two-echelon green supply chain under uncertainty in manufacturing cost and price. Journal of Industrial and Production Engineering, 39(8), 581–596.
Amorosi, L., Puerto, J., & Valverde, C. (2021). Coordinating drones with mothership vehicles: The mothership and drone routing problem with graphs. Computers & Operations Research, 136, 105445.
Bai, X., Cao, M., Yan, W., & Ge, S. S. (2019). Efficient routing for precedence-constrained package delivery for heterogeneous vehicles. IEEE Transactions on Automation Science and Engineering, 17(1), 248–260.
Bakir, I., & Tiniç, G. Ö. (2020). Optimizing drone-assisted last-mile deliveries: The vehicle routing problem with flexible drones. Optimization-Ouline. Org, 1–28.
Boysen, N., Briskorn, D., Fedtke, S., & Schwerdfeger, S. (2018). Drone delivery from trucks: Drone scheduling for given truck routes. Networks, 72(4), 506–527.
Cavani, S., Iori, M., & Roberti, R. (2021). Exact methods for the traveling salesman problem with multiple drones. Transportation Research Part c: Emerging Technologies, 130, 103280.
Chen, C., Demir, E., & Huang, Y. (2021). An adaptive large neighborhood search heuristic for the vehicle routing problem with time windows and delivery robots. European Journal of Operational Research, 294(3), 1164–1180.
Cheng, C., Adulyasak, Y., & Rousseau, L. M. (2020). Drone routing with energy function: Formulation and exact algorithm. Transportation Research Part b: Methodological, 139, 364–387.
Chípuli, G. P., & de la Mota, I. F. (2021). Analysis, design and reconstruction of a VRP model in a collapsed distribution network using simulation and optimization. Case Studies on Transport Policy, 9(4), 1440–1458.
Daknama, R., & Kraus, E. (2017). Vehicle routing with drones. ar**v:1705.06431.
Das, D. N., Sewani, R., Wang, J., & Tiwari, M. K. (2020). Synchronized truck and drone routing in package delivery logistics. IEEE Transactions on Intelligent Transportation Systems, 22(1), 5772–5782.
Dell’Amico, M., Montemanni, R., & Novellani, S. (2021). Modeling the flying sidekick traveling salesman problem with multiple drones. Networks, 78(3), 303–327.
Euchi, J., & Frifita, S. (2017). Hybrid metaheuristic to solve the “one-to-many-to-one” problem: Case of distribution of soft drink in Tunisia. Management Decision, 55(1), 136–155.
Dorling, K., Heinrichs, J., Messier, G. G., & Magierowski, S. (2016). Vehicle routing problems for drone delivery. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47(1), 70–85.
Euchi, J. (2021). Do drones have a realistic place in a pandemic fight for delivering medical supplies in healthcare systems problems? Chinese Journal of Aeronautics, 34(2), 182–190.
Euchi, J., & Kallel, A. (2021). Internalization of external congestion and CO2emissions costs related to road transport: The case of Tunisia. Renewable and Sustainable Energy Reviews, 142, 110858.
Euchi, J., & Sadok, A. (2021). Hybrid genetic-sweep algorithm to solve the vehicle routing problem with drones. Physical Communication, 44, 101236.
Euchi, J., & Yassine, A. (2022). A hybrid metaheuristic algorithm to solve the electric vehicle routing problem with battery recharging stations for sustainable environmental and energy optimization. Energy Systems, 14, 243–267.
Euchi, J., Zidi, S., & Laouamer, L. (2020). A hybrid approach to solve the vehicle routing problem with time windows and synchronized visits in-home health care. Arabian Journal for Science and Engineering, 45(12), 10637–10652.
Farajzadeh, F., Moadab, A., Valilai, O. F., & Houshmand, M. (2020). A novel mathematical model for a cloud-based drone enabled vehicle routing problem considering Multi-Echelon supply chain. IFAC-PapersOnLine, 53(2), 15035–15040.
Gonzalez-R, P. L., Canca, D., Andrade-Pineda, J. L., Calle, M., & Leon-Blanco, J. M. (2020). Truck-drone team logistics: A heuristic approach to multi-drop route planning. Transportation Research Part c: Emerging Technologies, 114, 657–680.
Gu, R., Poon, M., Luo, Z., Liu, Y., & Liu, Z. (2022). A hierarchical solution evaluation method and a hybrid algorithm for the vehicle routing problem with drones and multiple visits. Transportation Research Part c: Emerging Technologies, 141, 103733.
Hamdi, F., Messaoudi, L., & Euchi, J. (2023). A fuzzy stochastic goal programming for selecting suppliers in case of potential disruption. Journal of Industrial and Production Engineering, 40(8), 677–691.
Jeon, A., Kang, J., Choi, B., Kim, N., Eun, J., & Cheong, T. (2021). Unmanned aerial vehicle last-mile delivery considering backhauls. IEEE Access., 9, 85017–85033.
Kacem, A., & Dammak, A. (2021). Multi-objective scheduling on two dedicated processors. TOP, 29(3), 694–721.
Kang, M., & Lee, C. (2021). An exact algorithm for heterogeneous drone-truck routing problem. Transportation Science, 55(5), 1088–1112.
Karak, A., & Abdelghany, K. (2019). The hybrid vehicle-drone routing problem for pick-up and delivery services. Transportation Research Part C: Emerging Technologies, 102, 427–449.
Kitjacharoenchai, P., Min, B. C., & Lee, S. (2020). Two echelon vehicle routing problem with drones in last-mile delivery. International Journal of Production Economics, 225, 107598.
Kloster, K., Moeini, M., Vigo, D., & Wendt, O. (2022). The multiple traveling salesman problem in presence of drone-and robot-supported packet stations. European Journal of Operational Research, 305, 630–643.
Liu, Y., Liu, Z., Shi, J., Wu, G., & Pedrycz, W. (2020). Two-echelon routing problem for parcel delivery by cooperated truck and drone. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 51(12), 7450–7465.
Luo, Z., Poon, M., Zhang, Z., Liu, Z., & Lim, A. (2021). The multi-visit traveling salesman problem with multi-drones. Transportation Research Part c: Emerging Technologies, 128, 103172.
Masmoudi, M. A., Mancini, S., Baldacci, R., & Kuo, Y. H. (2022). Vehicle routing problems with drones equipped with multi-package payload compartments. Transportation Research Part e: Logistics and Transportation Review, 164, 102757.
Moshref-Javadi, M., & Winkenbach, M. (2021). Applications and Research avenues for drone-based models in logistics: A classification and review. Expert Systems with Applications, 177, 114854.
Moshref-Javadi, M., Hemmati, A., & Winkenbach, M. (2020). A truck and drones model for last-mile delivery: A mathematical model and heuristic approach. Applied Mathematical Modelling, 80, 290–318.
Murray, C. C., & Chu, A. G. (2015). The flying sidekick traveling salesman problem: Optimization of drone-assisted parcel delivery. Transportation Research Part c: Emerging Technologies, 54, 86–109.
Nizar, I., Jaafar, A., Hidila, Z., Barki, M., Illoussamen, E. H., & Mestari, M. (2021). Effective and safe trajectory planning for an autonomous UAV using a decomposition-coordination method. Journal of Intelligent & Robotic Systems, 103, 50.
Pachayappan, M., & Sudhakar, V. (2021). A solution to drone routing problems using docking stations for pickup and delivery services. Transportation Research Record, 2675(12), 1056–1074.
Perera, S., Dawande, M., Janakiraman, G., & Mookerjee, V. (2020). Retail deliveries by drones: How will logistics networks change? Production and Operations Management, 29(9), 2019–2034.
Poikonen, S., & Golden, B. (2020). Multi-visit drone routing problem. Computers & Operations Research, 113, 104802.
Poikonen, S., Wang, X., & Golden, B. (2017). The vehicle routing problem with drones: Extended models and connections. Networks, 70(1), 34–43.
Popović, D., Kovač, M., & Bjelić, N. (2019). A MIQP model for solving the vehicle routing problem with drones. In Proceedings of 4th Logistics International Conference–LOGIC (pp. 52–62).
Rabta, B., Wankmüller, C., & Reiner, G. (2018). A drone fleet model for last-mile distribution in disaster relief operations. International Journal of Disaster Risk Reduction, 28, 107–112.
Sacramento, D., Pisinger, D., & Ropke, S. (2019). An adaptive large neighborhood search metaheuristic for the vehicle routing problem with drones. Transportation Research Part c: Emerging Technologies, 102, 289–315.
Sadok, A. (2020). A genetic local search algorithm for the capacitated vehicle routing problem. International Journal of Advanced Computer Research, 10(48), 105.
Schermer, D., Moeini, M., & Wendt, O. (2018). Algorithms for solving the vehicle routing problem with drones. In Asian Conference on Intelligent Information and Database Systems (pp. 352–361). Springer, Cham.
Schermer, D., Moeini, M., & Wendt, O. (2019). A hybrid VNS/Tabu search algorithm for solving the vehicle routing problem with drones and en route operations. Computers & Operations Research, 109, 134–158.
Tamke, F., & Buscher, U. (2021). A branch-and-cut algorithm for the vehicle routing problem with drones. Transportation Research Part b: Methodological, 144, 174–203.
Wang, C., Lan, H., Saldanha-da-Gama, F., & Chen, Y. (2021). On optimizing a multi-mode last-mile parcel delivery system with vans, truck and dron, e. Electronics, 10(20), 2510.
Wang, X., Poikonen, S., & Golden, B. (2017). The vehicle routing problem with drones: Several worst-case results. Optimization Letters, 11(4), 679–697.
Wang, Z., & Sheu, J. B. (2019). Vehicle routing problem with drones. Transportation Research Part b: Methodological, 122, 350–364.
Yang, F., Dai, Y., & Ma, Z. J. (2020). There is a cooperative-rich vehicle routing problem in the last-mile logistics industry in rural areas. Transportation Research Part e: Logistics and Transportation Review, 141, 102024.
Zheng, Y., & Liu, Q. (2021). A review of distributed optimization: Problems, models, and algorithms. Neurocomputing, 483, 446–459.
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All authors contributed to the conception and design of this manuscript. AS: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing—original draft. JE: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Writing—review & editing, Visualization. PS: Project administration, review, Visualization.
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Sadok, A., Euchi, J. & Siarry, P. Vehicle routing with multiple UAVs for the last-mile logistics distribution problem: hybrid distributed optimization. Ann Oper Res (2024). https://doi.org/10.1007/s10479-024-06019-z
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DOI: https://doi.org/10.1007/s10479-024-06019-z