{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T09:53:57Z","timestamp":1768384437372,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:00:00Z","timestamp":1767916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Future Internet"],"abstract":"<jats:p>The increasing demand for rapid and sustainable parcel delivery has motivated the exploration of innovative logistics systems that integrate drones with traditional ground vehicles. Among these, the single-truck, multiple-drone last-mile delivery configuration has attracted significant attention due to its potential to reduce both delivery time and environmental impact. However, optimizing such systems remains computationally challenging because of the nonlinear energy consumption behavior of drones, which depends on factors such as payload weight and travel time, among others. This study investigates the energy-aware optimization of truck\u2013drone collaborative delivery systems, with a particular focus on the mathematical formulation as mixed-integer nonlinear problem (MINLP) formulations and linearization of drone energy consumption constraints. Building upon prior models proposed in the literature in the field, we analyze the MINLP computational complexity and introduce alternative linearization strategies that preserve model accuracy while improving performance solvability. The resulting linearized mixed-integer linear problem (MILP) formulations are solved using the PuLP software, a Python library solver, to evaluate the efficacy of linearization on computation time and solution quality across diverse problem instance sizes from a benchmark of instances in the literature. Thus, extensive computational results drawn from a standard dataset benchmark from the literature by running the solver in a cluster infrastructure demonstrated that the designed linearization methods can reduce optimization time of nonlinear solvers to several orders of magnitude without compromising energy estimation accuracy, enabling the model to handle larger problem instances effectively. This performance improvement opens the door to a real-time or near-real-time solution of the problem, allowing the delivery system to dynamically react to operational changes and uncertainties during delivery.<\/jats:p>","DOI":"10.3390\/fi18010045","type":"journal-article","created":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T14:15:50Z","timestamp":1767968150000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linearization Strategies for Energy-Aware Optimization of Single-Truck, Multiple-Drone Last-Mile Delivery Systems"],"prefix":"10.3390","volume":"18","author":[{"given":"Ornela","family":"Gordani","sequence":"first","affiliation":[{"name":"Faculty of Natural Sciences, University of Tirana, 1016 Tirana, Albania"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-9039-1310","authenticated-orcid":false,"given":"Eglantina","family":"Kalluci","sequence":"additional","affiliation":[{"name":"Faculty of Natural Sciences, University of Tirana, 1016 Tirana, Albania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6569-5497","authenticated-orcid":false,"given":"Fatos","family":"Xhafa","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Universitat Polit\u00e8cnica de Catalunya, 08028 Barcelona, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1109\/TSMC.2016.2582745","article-title":"Vehicle Routing Problems for Drone Delivery","volume":"47","author":"Dorling","year":"2017","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1287\/trsc.2016.0675","article-title":"50th Anniversary Invited Article\u2014City Logistics: Challenges and Opportunities","volume":"50","author":"Savelsbergh","year":"2016","journal-title":"Transp. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3437","DOI":"10.3390\/smartcities7060134","article-title":"Drone-Assisted Last-Mile Delivery Under Windy Conditions: Zero Pollution Solutions","volume":"7","author":"Serdarasan","year":"2024","journal-title":"Smart Cities"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.trb.2020.06.011","article-title":"Drone Routing with Energy Function: Formulation and Exact Algorithm","volume":"139","author":"Cheng","year":"2020","journal-title":"Transp. Res. Part B"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Chen, S., Mo, Y., Wu, X., Xiao, J., and Liu, Q. (2024). Reinforcement Learning-Based Energy-Saving Path Planning for UAVs in Turbulent Wind. Electronics, 13.","DOI":"10.3390\/electronics13163190"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Bi, Z., Guo, X., Wang, J., Qin, S., and Liu, G. (2024). Truck\u2013Drone Delivery Optimization Based on Multi-Agent Reinforcement Learning. Drones, 8.","DOI":"10.3390\/drones8010027"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Chen, Y., Baek, D., Bocca, A., Macii, A., Macii, E., and Poncino, M. (2018, January 7\u201311). A Case for a Battery-Aware Model of Drone Energy Consumption. Proceedings of the 2018 IEEE International Telecommunications Energy Conference (INTELEC), Turino, Italy.","DOI":"10.1109\/INTLEC.2018.8612333"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Asghari, M., Fathollahi-Fard, A.M., Mirzapour Al-e hashem, S.M.J., and Dulebenets, M.A. (2022). Transformation and Linearization Techniques in Optimization: A State-of-the-Art Survey. Mathematics, 10.","DOI":"10.3390\/math10020283"},{"key":"ref_9","unstructured":"(2025, October 25). Gurobi Optimization, LLC. Gurobi Optimizer. Available online: https:\/\/www.gurobi.com."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.trc.2015.03.005","article-title":"The Flying Sidekick Traveling Salesman Problem: Optimization of Drone-Assisted Parcel Delivery","volume":"54","author":"Murray","year":"2015","journal-title":"Transp. Res. Part C"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1038\/s41467-017-02411-5","article-title":"Energy Use and Life Cycle Greenhouse Gas Emissions of Drones for Commercial Package Delivery","volume":"9","author":"Stolaroff","year":"2018","journal-title":"Nat. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.ijpe.2019.01.010","article-title":"Truck-drone hybrid delivery routing: Payload-energy dependency and No-Fly zones","volume":"214","author":"Jeong","year":"2019","journal-title":"Int. J. Prod. Econ."},{"key":"ref_13","unstructured":"Leishman, G.J. (2006). Principles of Helicopter Aerodynamics, Cambridge University Press. [2nd ed.]. Chapters 2 and 5."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1058","DOI":"10.1016\/j.ejor.2018.11.021","article-title":"Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programming-based solution methods","volume":"275","author":"Ngueveu","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Beal, L.D.R., Hill, D.C., Martin, R.A., and Hedengren, J.D. (2018). GEKKO Optimization Suite. Processes, 6.","DOI":"10.3390\/pr6080106"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/978-1-4614-1927-3_10","article-title":"Using Piecewise Linear Functions for Solving MINLPs","volume":"Volume 154","author":"Lee","year":"2012","journal-title":"Mixed Integer Nonlinear Programming"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3270","DOI":"10.4236\/am.2014.520305","article-title":"The Best Piecewise Linearization of Nonlinear Functions","volume":"5","author":"Mazarei","year":"2014","journal-title":"Appl. Math."},{"key":"ref_18","unstructured":"Huchette, J., and Vielma, J.P. (2019). Nonconvex Piecewise Linear Functions: Advanced Formulations and Simple Modeling Tools. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3225","DOI":"10.1109\/TCSI.2014.2327313","article-title":"Optimal Polygonal L1 Linearization and Fast Interpolation of Nonlinear Systems","volume":"61","author":"Gallego","year":"2014","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1287\/opre.35.2.254","article-title":"Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints","volume":"35","author":"Solomon","year":"1987","journal-title":"Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1287\/ijoc.11.1.63","article-title":"Analyzing Infeasible Mixed-Integer and Integer Linear Programs","volume":"11","author":"Guieu","year":"1999","journal-title":"INFORMS J. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1007\/s12532-021-00202-0","article-title":"Computational Aspects of Infeasibility Analysis in Mixed Integer Programming","volume":"13","author":"Witzig","year":"2021","journal-title":"Math. Program. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Li, Y., Peyman, M., Panadero, J., Juan, A.A., and Xhafa, F. (2022). IoT Analytics and Agile Optimization for Solving Dynamic Team Orienteering Problems with Mandatory Visits. Mathematics, 10.","DOI":"10.3390\/math10060982"}],"container-title":["Future Internet"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-5903\/18\/1\/45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T05:10:46Z","timestamp":1768367446000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-5903\/18\/1\/45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,9]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["fi18010045"],"URL":"https:\/\/doi.org\/10.3390\/fi18010045","relation":{},"ISSN":["1999-5903"],"issn-type":[{"value":"1999-5903","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,9]]}}}