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Drone routing problem for shore-to-ship delivery services considering non-linear energy consumption

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  • Wang, Mengtong
  • Chen, Shukai
  • Meng, Qiang

Abstract

This study investigates the emerging application of unmanned aerial vehicles (UAVs), or drones, for shore-to-ship delivery services between onshore and offshore locations. However, deploying drones for shore-to-ship delivery can encounter unique operational challenges, including constantly moving target vessels and non-linear drone energy consumption. To address these issues, we propose a novel and practical drone routing problem for shore-to-ship delivery services (DRP-SSDS) considering the non-linear energy consumption related to payload, flight phase, and flight time. The proposed DRP-SSDS is formulated as a mixed-integer second-order cone programming (MISOCP) model that integrates continuous decisions on both time and location to realistically capture vessel movements within port waters. We then develop a tailored branch-and-price algorithm that can solve DRP-SSDS exactly and efficiently for medium-scale instances. Additionally, we design an effective heuristic method that can provide high-quality solutions in a reasonable time limit for large-scale instances. Extensive numerical experiments demonstrate the superiority of the proposed solution methods over the off-the-shelf optimization solver and a benchmark method across all tested instances.

Suggested Citation

  • Wang, Mengtong & Chen, Shukai & Meng, Qiang, 2026. "Drone routing problem for shore-to-ship delivery services considering non-linear energy consumption," Transportation Research Part B: Methodological, Elsevier, vol. 206(C).
    • Handle: RePEc:eee:transb:v:206:y:2026:i:c:s0191261526000226
    DOI: 10.1016/j.trb.2026.103410
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    1. G. Clarke & J. W. Wright, 1964. "Scheduling of Vehicles from a Central Depot to a Number of Delivery Points," Operations Research, INFORMS, vol. 12(4), pages 568-581, August.
    2. Amin Asadi & Sarah Nurre Pinkley, 2022. "A Monotone Approximate Dynamic Programming Approach for the Stochastic Scheduling, Allocation, and Inventory Replenishment Problem: Applications to Drone and Electric Vehicle Battery Swap Stations," Transportation Science, INFORMS, vol. 56(4), pages 1085-1110, July.
    3. Amine Masmoudi, M. & Mancini, Simona & Baldacci, Roberto & Kuo, Yong-Hong, 2022. "Vehicle routing problems with drones equipped with multi-package payload compartments," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 164(C).
    4. Cheng, Chun & Adulyasak, Yossiri & Rousseau, Louis-Martin, 2020. "Drone routing with energy function: Formulation and exact algorithm," Transportation Research Part B: Methodological, Elsevier, vol. 139(C), pages 364-387.
    5. Jean-François Cordeau & Gianpaolo Ghiani & Emanuela Guerriero, 2014. "Analysis and Branch-and-Cut Algorithm for the Time-Dependent Travelling Salesman Problem," Transportation Science, INFORMS, vol. 48(1), pages 46-58, February.
    6. Yan Xia & Rajan Batta & Rakesh Nagi, 2017. "Controlling a Fleet of Unmanned Aerial Vehicles to Collect Uncertain Information in a Threat Environment," Operations Research, INFORMS, vol. 65(3), pages 674-692, June.
    7. Gohram Baloch & Fatma Gzara, 2020. "Strategic Network Design for Parcel Delivery with Drones Under Competition," Transportation Science, INFORMS, vol. 54(1), pages 204-228, January.
    8. Yang, Ying & Hao, Xiaodeng & Wang, Shuaian, 2025. "The drone scheduling problem in shore-to-ship delivery: A time discretization-based model with an exact solving approach," Transportation Research Part B: Methodological, Elsevier, vol. 191(C).
    9. Katharina Glock & Anne Meyer, 2020. "Mission Planning for Emergency Rapid Mapping with Drones," Transportation Science, INFORMS, vol. 54(2), pages 534-560, March.
    10. Xia, Jun & Wang, Kai & Wang, Shuaian, 2019. "Drone scheduling to monitor vessels in emission control areas," Transportation Research Part B: Methodological, Elsevier, vol. 119(C), pages 174-196.
    11. Claudio Gambella & Joe Naoum-Sawaya & Bissan Ghaddar, 2018. "The Vehicle Routing Problem with Floating Targets: Formulation and Solution Approaches," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 554-569, August.
    12. Iman Dayarian & Martin Savelsbergh & John-Paul Clarke, 2020. "Same-Day Delivery with Drone Resupply," Transportation Science, INFORMS, vol. 54(1), pages 229-249, January.
    13. Minh Anh Nguyen & Minh Hoàng Hà, 2023. "The Parallel Drone Scheduling Traveling Salesman Problem with Collective Drones," Transportation Science, INFORMS, vol. 57(4), pages 866-888, July.
    14. Roberto Roberti & Mario Ruthmair, 2021. "Exact Methods for the Traveling Salesman Problem with Drone," Transportation Science, INFORMS, vol. 55(2), pages 315-335, March.
    15. Michael W. Levin & David Rey, 2023. "Branch-and-Price for Drone Delivery Service Planning in Urban Airspace," Transportation Science, INFORMS, vol. 57(4), pages 843-865, July.
    16. Munjeong Kang & Chungmok Lee, 2021. "An Exact Algorithm for Heterogeneous Drone-Truck Routing Problem," Transportation Science, INFORMS, vol. 55(5), pages 1088-1112, September.
    17. Shakiba Enayati & Haitao Li & James F. Campbell & Deng Pan, 2023. "Multimodal Vaccine Distribution Network Design with Drones," Transportation Science, INFORMS, vol. 57(4), pages 1069-1095, July.
    18. Wang, Mengtong & Miao, Lixin & Zhang, Canrong, 2021. "A branch-and-price algorithm for a green location routing problem with multi-type charging infrastructure," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 156(C).
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