{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T04:30:28Z","timestamp":1786681828119,"version":"build-2736575974"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T00:00:00Z","timestamp":1786492800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","award":["2026JBCG010"],"award-info":[{"award-number":["2026JBCG010"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Aiming at the multi-constraint three-dimensional bin packing problem for mixed vehicle types in urban logistics, where traditional exact algorithms are limited by NP-hard computational complexity and practical engineering constraints, this study proposes a progressive optimization framework for vehicle loading and fleet allocation optimization. First, a heuristic loading algorithm based on the extreme point method and greedy strategy is developed to maximize single-vehicle loading efficiency by balancing space and weight utilization. Second, an NSGA-II based evolutionary framework with sequential encoding is constructed to minimize fleet size while improving loading balance for single-vehicle-type optimization. Third, a three-stage hybrid algorithm integrating greedy packing, enumerative search, and tail vehicle replacement is designed to optimize mixed-vehicle fleet composition and minimize total transportation cost. Experimental results demonstrate that the proposed heuristic achieves high composite loading performance across vehicle types, and the evolutionary framework significantly reduces fleet size compared with theoretical lower bounds. Under mixed-fleet optimization, the model identifies cost-effective vehicle configurations that outperform single-type dispatching strategies. Sensitivity analysis reveals that cargo composition, particularly the number of fragile items, is the most critical factor affecting system performance, while validation on 16 vehicle types confirms the robustness and practical generalizability of the method. This study verifies the effectiveness and stability of heuristic-evolutionary hybrid optimization methods, providing a reliable decision-making reference for logistics enterprises in vehicle selection, cargo allocation, and transportation planning.<\/jats:p>","DOI":"10.3390\/e28080905","type":"journal-article","created":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T14:45:06Z","timestamp":1786545906000},"page":"905","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multi-Constraint Three-Dimensional Bin Packing Optimization for Mixed Vehicle Types: A Heuristic Approach"],"prefix":"10.3390","volume":"28","author":[{"given":"Yiting","family":"Hao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongqing","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4318-1780","authenticated-orcid":false,"given":"Wenhao","family":"Gui","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,12]]},"reference":[{"key":"ref_1","unstructured":"World Bank (2023). Connecting to Compete 2023: Trade Logistics in an Uncertain Global Economy, World Bank Group."},{"key":"ref_2","unstructured":"Kearney, and Council of Supply Chain Management Professionals (CSCMP) (2023). 34th Annual State of Logistics Report, CSCMP."},{"key":"ref_3","unstructured":"Eurostat (2022). Key Figures on European Transport\u20142022 Edition, Publications Office of the European Union."},{"key":"ref_4","unstructured":"China Federation of Logistics and Purchasing (2024). China Logistics Yearbook 2024, China Fortune Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2012.12.006","article-title":"Constraints in container loading: A state-of-the-art review","volume":"229","author":"Bortfeldt","year":"2013","journal-title":"Eur. J. Oper. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1287\/trsc.2016.0675","article-title":"City logistics: Challenges and opportunities","volume":"50","author":"Savelsbergh","year":"2016","journal-title":"Transp. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1109","DOI":"10.1016\/j.ejor.2005.12.047","article-title":"An improved typology of cutting and packing problems","volume":"183","author":"Schumann","year":"2007","journal-title":"Eur. J. Oper. Res."},{"key":"ref_8","first-page":"1","article-title":"Recent advances in packing problems","volume":"278","author":"Iori","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1111\/itor.12124","article-title":"A physical packing sequence algorithm for the container loading problem with static mechanical equilibrium conditions","volume":"23","author":"Ramos","year":"2016","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"109943","DOI":"10.1016\/j.cie.2024.109943","article-title":"Stability constraints in a 3D knapsack problem with non parallelepipedic items","volume":"189","author":"Brauner","year":"2024","journal-title":"Comput. Ind. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/0305-0483(95)00015-G","article-title":"Issues in the development of approaches to container loading","volume":"23","author":"Bischoff","year":"1995","journal-title":"Omega"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1111\/j.1475-3995.1997.tb00095.x","article-title":"A genetic algorithm for solving the container loading problem","volume":"4","author":"Gehring","year":"1997","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1016\/S0377-2217(00)00055-2","article-title":"A hybrid genetic algorithm for the container loading problem","volume":"131","author":"Bortfeldt","year":"2001","journal-title":"Eur. J. Oper. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0377-2217(90)90350-K","article-title":"A typology of cutting and packing problems","volume":"44","author":"Dyckhoff","year":"1990","journal-title":"Eur. J. Oper. Res."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1016\/S0377-2217(02)00132-7","article-title":"Heuristics for the container loading problem","volume":"141","author":"Pisinger","year":"2002","journal-title":"Eur. J. Oper. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"256","DOI":"10.1287\/opre.48.2.256.12386","article-title":"The three-dimensional bin packing problem","volume":"48","author":"Martello","year":"2000","journal-title":"Oper. Res."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Scheithauer, G. (2018). Introduction to Cutting and Packing Optimization, Springer.","DOI":"10.1007\/978-3-319-64403-5"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1287\/ijoc.1070.0250","article-title":"Extreme point-based heuristics for three-dimensional bin packing","volume":"20","author":"Crainic","year":"2008","journal-title":"INFORMS J. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1016\/S0377-2217(02)00134-0","article-title":"Heuristic algorithms for the three-dimensional bin packing problem","volume":"141","author":"Lodi","year":"2002","journal-title":"Eur. J. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Dorigo, M., and St\u00fctzle, T. (2004). Ant Colony Optimization, MIT Press.","DOI":"10.7551\/mitpress\/1290.001.0001"},{"key":"ref_21","unstructured":"Kennedy, J., and Eberhart, R. (December, January 27). Particle swarm optimization. Proceedings of the IEEE International Conference on Neural Networks, Perth, WA, Australia."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by simulated annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0305-0548(80)90001-5","article-title":"A heuristic for packing boxes into a container","volume":"7","author":"George","year":"1980","journal-title":"Comput. Oper. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4135","DOI":"10.1016\/j.asoc.2011.02.032","article-title":"Hybrid metaheuristics in combinatorial optimization: A survey","volume":"11","author":"Blum","year":"2011","journal-title":"Appl. Soft Comput."},{"key":"ref_25","first-page":"193","article-title":"Three-dimensional packing multi-objective optimization algorithm based on block merging strategy","volume":"46","author":"Yang","year":"2025","journal-title":"Packag. Eng."},{"key":"ref_26","first-page":"216","article-title":"Review of 3D packing optimization model and algorithm","volume":"45","author":"Liu","year":"2024","journal-title":"Packag. Eng."},{"key":"ref_27","first-page":"1275","article-title":"Deep study on the application of machine learning in bin packing problems","volume":"28","year":"2024","journal-title":"Comput. Sist."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multiobjective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"102554","DOI":"10.1016\/j.tre.2021.102554","article-title":"Stochastic fleet mix optimization: Evaluating electromobility in urban logistics","volume":"158","author":"Malladi","year":"2022","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Li, M., Shi, Y., and Zhu, B. (2022). Research on multi-center mixed fleet distribution path considering dynamic energy consumption integrated reverse logistics. Sustainability, 14.","DOI":"10.3390\/su14116613"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"(2025). Reinforcement learning based intelligent optimisation for bin packing problems: A review. Decis. Anal. J., 28, 100616.","DOI":"10.1016\/j.array.2025.100616"},{"key":"ref_32","unstructured":"Wang, L., and Geng, Y. (2023). Sustainable logistics: A systematic review of green vehicle routing and bin packing problems. Sustainability, 15."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1287\/trsc.1060.0165","article-title":"An exact approach for the vehicle routing problem with two-dimensional loading constraints","volume":"41","author":"Iori","year":"2007","journal-title":"Transp. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1002\/net.20192","article-title":"A tabu search heuristic for the vehicle routing problem with two-dimensional loading constraints","volume":"51","author":"Gendreau","year":"2006","journal-title":"Networks"},{"key":"ref_35","unstructured":"(2024). Technical Requirements for Lectotype of Urban Logistics Distribution Vehicles (Standard No. GB\/T 29912-2024). (In Chinese)."},{"key":"ref_36","unstructured":"(2016). Limits of Dimensions, Axle Load and Masses of Motor Vehicles (Standard No. GB 1589-2016). (In Chinese)."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1287\/mnsc.17.5.259","article-title":"The loading problem","volume":"17","author":"Eilon","year":"1971","journal-title":"Manag. Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/905\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,14]],"date-time":"2026-08-14T04:18:32Z","timestamp":1786681112000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/905"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,12]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080905"],"URL":"https:\/\/doi.org\/10.3390\/e28080905","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,12]]}}}