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The Last-Mile Delivery Process with Trucks and Drones Under Uncertain Energy Consumption

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Abstract

We address the problem of delivering parcels in an urban area, within a given time horizon, by conventional vehicles, i.e., trucks, equipped with drones. Both the trucks and the drones perform deliveries, and the drones are carried by the trucks. We focus on the energy consumption of the drones that we assume to be influenced by atmospheric events. Specifically, we manage the delivery process in a such a way as to avoid energy disruption against adverse weather conditions. We address the problem under the field of robust optimization, thus preventing energy disruption in the worst case. We consider several polytopes to model the uncertain energy consumption, and we propose a decomposition approach based on Benders’ combinatorial cuts. A computational study is carried out on benchmark instances. The aim is to assess the quality of the computed solutions in terms of solution reliability, and to analyze the trade-off between the risk-adverseness of the decision maker and the transportation cost.

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References

  1. Agatz, N., Bouman, P., Schmidt, M.: Optimization approaches for the traveling salesman problem with drone. Transp. Sci. 52(4), 739–1034 (2018)

    Article  Google Scholar 

  2. Agárdi, A., Kovács, L., Bányai, T.: Vehicle routing in drone-based package delivery services. In: Szita Tóthné, K., Jármai, K., Voith K. (eds.) Solutions for Sustainable Development - 1st International Conference on Engineering Solutions for Sustainable Development. London: CRC Press, pp. 151–1597 (2020)

  3. Ben-Tal, A., Nemirovski, A.: Robust solutions of uncertain linear programs. Oper. Res. Lett. 25, 1–13 (1999)

    Article  MathSciNet  Google Scholar 

  4. Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Math. Program. 98, 49–71 (2003)

    Article  MathSciNet  Google Scholar 

  5. Bertsimas, D., Sim, M.: The price of robustness. Oper. Res. 52, 35–53 (2004)

    Article  MathSciNet  Google Scholar 

  6. Chang, Y.S., Lee, H.J.: Optimal delivery routing with wider drone-delivery areas along a shorter truck-route. Expert Syst. Appl. 104, 307–317 (2018)

    Article  Google Scholar 

  7. Di Puglia Pugliese, L., Guerriero, F.: A survey of resource constrained shortest path problems: exact solution approaches. Networks 62(3), 183–200 (2013)

    Article  MathSciNet  Google Scholar 

  8. Di Puglia Pugliese, L., Guerriero, F.: Last-mile deliveries by using drones and classical vehicles. In: Sforza, A., Sterle, C. (eds.) International Conference on Optimization and Decision Science, ODS 2017. Springer Proceedings in Mathematics and Statistics, Springer New York LLC, pp. 557–565 (2017)

  9. Di Puglia Pugliese, L., Guerriero, F., Macrina, G.: Using drones for parcels delivery process. Procedia Manuf. 42, 488–497 (2020)

    Article  Google Scholar 

  10. Di Puglia Pugliese, L., Guerriero, F., Poss, M.: The resource constrained shortest path problem with uncertain data: a robust formulation and optimal solution approach. Comput. Oper. Res. 107, 140–155 (2019). https://doi.org/10.1016/j.cor.2019.03.010

    Article  MathSciNet  MATH  Google Scholar 

  11. Di Puglia Pugliese, L., Macrina, G., Guerriero, F.: Trucks and drones cooperation in the last-mile delivery process. Networks (2020). https://doi.org/10.1002/net.22015

    Article  Google Scholar 

  12. Dorling, K., Heinrichs, J., Messier, G.G., Magierwski, S.: Vehicle routing problem for drone delivery. IEEE Trans. Syst. Man Cybern. Syst. 47(1) (2017)

  13. Dumitrescu, I., Boland, N.: Improved preprocessing, labeling and scaling algorithms for the weight-constrained shortest path problem. Networks 42(3), 135–153 (2003)

    Article  MathSciNet  Google Scholar 

  14. Ferrandez, M., Harbison, T., Weber, T., Sturges, R., Rich, R.: Optimization of a truck-drone in tandem delivery network using k-means and genetic algorithm. J. Ind. Eng. Manag. 9(2) (2016)

  15. Ha, Q.M., Deville, Y., Pham, Q.D., Ha’, M.H.: On the min-cost traveling salesman problem with drone. Transp. Res. Part C Emerg. Technol. 86, 597–621 (2018)

  16. Ham, A.M.: Integrated scheduling of m-truck, m-drone, and m-depot constrained by time-window, drop-pickup, and m-visit using constraint programming. Transp. Res. Part C Emerg. Technol. 91, 1–14 (2018)

    Article  Google Scholar 

  17. Jeonga, H.Y., Leea, S., Songa, B.D.: Truck-drone hybrid delivery routing: payload-energy dependency and no-fly zones. Int. J. Prod. Econ. (2019). https://doi.org/10.1016/j.ijpe.2019.01.010

    Article  Google Scholar 

  18. Kima, S.J., Lima, G.J., Cho, J.: Drone flight scheduling under uncertainty on battery duration and air temperature. Comput. Ind. Eng. 117, 291–302 (2018)

    Article  Google Scholar 

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

  20. Moshref-Javadi, M., Lee, S., Winkenbach, M.: Design and evaluation of a multi-trip delivery model with truck and drones. Transp. Res. Part E Logist. Transp. Rev. 136, 101887 (2020)

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  23. Poikonen, S., Golden, B.L., Wasil, E.A.: A branch-and-bound approach to the traveling salesman problem with a drone. INFORMS J. Comput. 31(2), 335–346 (2019)

    Article  MathSciNet  Google Scholar 

  24. Poikonen, S., Wang, X., Golden, B.: Combinatorial benders’ cuts for mixed-integer linear programming. Oper. Res. 54(4), 756–766 (2006)

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

    Article  MathSciNet  Google Scholar 

  26. Roberti, R., Ruthmair, M.: Exact methods for the traveling salesman problem with drone. Transp. Sci. 55(2), 275–552 (2021)

    Article  Google Scholar 

  27. Rojas Viloria, D., Solano-Charris, E.L., Muñoz-Villamizar, A., Montoya-Torres, J.R.: Unmanned aerial vehicles/drones in vehicle routing problems: a literature review. Int. Trans. Oper. Res. 28(4), 1626–1657 (2021)

    Article  MathSciNet  Google Scholar 

  28. Schermer, D., Moeini, M., Wendt, O.: Algorithms for solving the vehicle routing problem with drones. In: Nguyen, N., Hoang, D., Hong, T., Pham, H., Trawiński, B. (eds.) Intelligent Information and Database Systems. ACIIDS 2018. Lecture Notes in Computer Science, Springer, Cham (2018)

  29. Solomon, M.M.: Algorithms for the vehicle routing and scheduling problems with time window constraints. Oper. Res. 35, 254–265 (1987)

    Article  MathSciNet  Google Scholar 

  30. Thibbotuwawa, A.: Unmanned aerial vehicle fleet mission planning subject to changing weather conditions. Ph.D. thesis, Faculty of Engineering and Science, Production, Operations Research (2019)

  31. Ulmer, M.W., Thomas, B.W.: Same-day delivery with a heterogeneous fleet of drones and vehicles. Networks 72(4), 475–505 (2018)

    Article  MathSciNet  Google Scholar 

  32. Wang, X., Poikonen, S., Golden, B.: The vehicle routing problem with drones: several worst-case results. Optim. Lett. 11(4), 679 (2017)

    Article  MathSciNet  Google Scholar 

  33. Yoo, W., Yu, E., Jung, J.: Drone delivery: factors affecting the public’s attitude and intention to adopt. Telematics Inform. 35, 1687–1700 (2018)

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Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and useful suggestions.

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Authors and Affiliations

  1. Istituto di Calcolo e Reti ad Alte Prestazioni, Consiglio Nazionale delle Ricerche, Rende, Italy

    Luigi Di Puglia Pugliese

  2. Department of Mechanical, Energy and Management Engineering, University of Calabria, Rende, Italy

    Francesca Guerriero

  3. Dipartimento di Informatica, University of Pisa, Pisa, Italy

    Maria Grazia Scutellá

Authors
  1. Luigi Di Puglia Pugliese
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  2. Francesca Guerriero
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  3. Maria Grazia Scutellá
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Correspondence to Luigi Di Puglia Pugliese.

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Communicated by Jörg Rambau.

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Di Puglia Pugliese, L., Guerriero, F. & Scutellá, M.G. The Last-Mile Delivery Process with Trucks and Drones Under Uncertain Energy Consumption. J Optim Theory Appl 191, 31–67 (2021). https://doi.org/10.1007/s10957-021-01918-8

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  • DOI: https://doi.org/10.1007/s10957-021-01918-8

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