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An Efficient Hybrid Graph Network Model for Traveling Salesman Problem with Drone

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Abstract

Traveling salesman problem with drone (TSPD) is an expansion of the conventional traveling salesman problem, in which parcels are delivered to consumers not only by vehicles but also by drones. The traditional algorithm is difficult to develop when dealing with the dynamic and complex route environment, and the solving time and accuracy cannot meet the needs of modern logistics. Deep reinforcement learning (DRL) has the advantages of large-scale decision making, self-adaptation, and online optimization, so it has been widely used. Instead of employing traditional algorithms to solve the TSPD, we propose a hybrid graph network model for the TSPD that is based on the DRL mechanism. By taking advantage of different network modules, the DRL model can improve its search performance in solution space. The experimental results show that our model outperforms a solely attention-based model in terms of solution quality and computing efficiency. In addition, our model has better generalization ability compared with the current DRL models.

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References

  1. Bamburry D (2015) Drones: designed for product delivery. Des Manag Rev 26(1):40–48

    Google Scholar 

  2. Amideo AE, Scaparra MP, Kotiadis K (2019) Optimising shelter location and evacuation routing operations: the critical issues. Eur J Oper Res 279(2):279–295

    Article  MathSciNet  MATH  Google Scholar 

  3. Otero Arenzana A, Escribano Macias JJ, Angeloudis P (2020) Design of hospital delivery networks using unmanned aerial vehicles. Transp Res Rec 2674(5):405–418

    Article  Google Scholar 

  4. Qiu H, Wang S, Yin Y, Wang D, Wang Y (2022) A deep reinforcement learning-based approach for the home delivery and installation routing problem. Int J Prod Econ 244:108362

    Article  Google Scholar 

  5. Murray CC, Raj R (2020) The multiple flying sidekicks traveling salesman problem: parcel delivery with multiple drones. Transp Res Part C Emerg Technol 110:368–398

    Article  Google Scholar 

  6. Agatz N, Bouman P, Schmidt M (2018) Optimization approaches for the traveling salesman problem with drone. Transp Sci 52(4):965–981

    Article  Google Scholar 

  7. Gendreau M, Laporte G, Solomon MM (1995) Single-vehicle routing and scheduling to minimize the number of delays. Transp Sci 29(1):56–62

    Article  MATH  Google Scholar 

  8. Bouman P, Agatz N, Schmidt M (2018) Dynamic programming approaches for the traveling salesman problem with drone. Networks 72(4):528–542

    Article  MathSciNet  Google Scholar 

  9. Karamyar F, Sadeghi J, Yazdi MM (2018) A Benders decomposition for the location-allocation and scheduling model in a healthcare system regarding robust optimization. Neural Comput Appl 29(10):873–886

    Article  Google Scholar 

  10. Freitas JC, Penna PHV, Toffolo TA (2021) Exact and heuristic approaches to drone delivery problems. http://arxiv.org/abs/2108.01996

  11. Google (2018) OR-Tools. Available: https://developers.google.com

  12. Vesselinova N, Steinert R, Perez-Ramirez DF, Boman M (2020) Learning combinatorial optimization on graphs: a survey with applications to networking. IEEE Access 8:120388–120416

    Article  Google Scholar 

  13. Scarselli F, Gori M, Tsoi AC, Hagenbuchner M, Monfardini G (2008) The graph neural network model. IEEE Trans Neural Netw 20(1):61–80

    Article  Google Scholar 

  14. Wang Q, Tang C (2021) Deep reinforcement learning for transportation network combinatorial optimization: a survey. Knowl-Based Syst 233:107526

    Article  Google Scholar 

  15. Poikonen S, Wang X, Golden B (2017) The vehicle routing problem with drones: extended models and connections. Networks 70(1):34–43

    Article  MathSciNet  MATH  Google Scholar 

  16. Macrina G, Pugliese LDP, Guerriero F, Laporte G (2020) Drone-aided routing: a literature review. Transp Res Part C Emerg Technol 120:102762

    Article  Google Scholar 

  17. Chung SH, Sah B, Lee J (2020) Optimization for drone and drone-vehicle combined operations: a review of the state of the art and future directions. Comput Oper Res 123:105004

    Article  MathSciNet  MATH  Google Scholar 

  18. Murray CC, Chu AG (2015) The flying sidekick traveling salesman problem: optimization of drone-assisted parcel delivery. Transp Res Part C Emerg Technol 54:86–109

    Article  Google Scholar 

  19. Carlsson JG, Song S (2018) Coordinated logistics with a vehicle and a drone. Manage Sci 64(9):4052–4069

    Article  Google Scholar 

  20. Ha QM, Deville Y, Pham QD, Hà MH (2018) On the min-cost traveling salesman problem with drone. Transp Res Part C Emerg Technol 86:597–621

    Article  Google Scholar 

  21. Chang YS, Lee HJ (2018) Optimal delivery routing with wider drone-delivery areas along a shorter vehicle-route. Expert Syst Appl 104:307–317

    Article  Google Scholar 

  22. Bello I, Pham H, Le QV, Norouzi M, Bengio S (2016) Neural combinatorial optimization with reinforcement learning. http://arxiv.org/abs/1611.09940

  23. Nazari M, Oroojlooy A, Snyder L, Takác M (2018) Reinforcement learning for solving the vehicle routing problem. Adv Neural Inf Process Syst 31

  24. Kool W, Van Hoof H, Welling M (2018) Attention, learn to solve routing problems!. http://arxiv.org/abs/1803.08475

  25. Bogyrbayeva A, Yoon T, Ko H, Lim S, Yun H, Kwon C (2023) A deep reinforcement learning approach for solving the traveling salesman problem with drone. Transp Res Part C Emerg Technol 148:103981

    Article  Google Scholar 

  26. Wu Y, Song W, Cao Z, Zhang J, Lim A (2021) Learning improvement heuristics for solving routing problems. IEEE Trans Neural Netw Learn Syst 33(9):5057–5069

    Article  MathSciNet  Google Scholar 

  27. Ferrandez SM, Harbison T, Weber T, Sturges R, Rich R (2016) Optimization of a vehicle-drone in tandem delivery network using k-means and genetic algorithm. J Ind Eng Manag (JIEM) 9(2):374–388

    Google Scholar 

  28. He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 770–778

  29. Ioffe S, Szegedy C (2015) Batch normalization: Accelerating deep network training by reducing internal covariate shift. In: International conference on machine learning, pp 448–456. PMLR

  30. Kipf TN, Welling M (2016) Semi-supervised classification with graph convolutional networks. http://arxiv.org/abs/1609.02907

  31. Qin Z, Sun W, Deng H, Li D, Wei Y, Lv B et al. (2022) cosFormer: Rethinking Softmax in Attention. http://arxiv.org/abs/2202.08791

  32. Williams RJ (1992) Simple statistical gradient-following algorithms for connectionist reinforcement learning. Mach Learn 8(3):229–256

    Article  MATH  Google Scholar 

  33. Euchi J, Sadok A (2021) Hybrid genetic-sweep algorithm to solve the vehicle routing problem with drones. Phys Commun 44:101236

    Article  Google Scholar 

  34. Huang SH, Huang YH, Blazquez CA, Chen CY (2022) Solving the vehicle routing problem with drone for delivery services using an ant colony optimization algorithm. Adv Eng Inform 51:101536

    Article  Google Scholar 

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Acknowledgements

This work is supported in part by the National Natural Science Foundation of China (11761042).

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Authors and Affiliations

  1. Department of Mathematics, Kunming University of Science and Technology, Kunming, 650093, People’s Republic of China

    Yang Wang, Xiaoxiao Yang & Zhibin Chen

Authors
  1. Yang Wang
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  2. Xiaoxiao Yang
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  3. Zhibin Chen
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Contributions

Conceptualization, Z.C.; Formal analysis, Y.W, Z.C. and X.Y.; Funding acquisition, Z.C.; Investigation, Y.W and Z.C.; Methodology, Y.W, Z.C. and X.Y.; Supervision, X.Y. and Z.C.; Writing—original draft, Y.W; Writing—review—editing, Y.W, Z.C. and X.Y. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Zhibin Chen.

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Wang, Y., Yang, X. & Chen, Z. An Efficient Hybrid Graph Network Model for Traveling Salesman Problem with Drone. Neural Process Lett 55, 10353–10370 (2023). https://doi.org/10.1007/s11063-023-11330-0

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  • DOI: https://doi.org/10.1007/s11063-023-11330-0

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