{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T13:19:56Z","timestamp":1763039996392,"version":"3.45.0"},"reference-count":29,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T00:00:00Z","timestamp":1762992000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program","doi-asserted-by":"publisher","award":["2024YFC3809403"],"award-info":[{"award-number":["2024YFC3809403"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a collaborative optimization method that combines the multi-objective grey wolf optimizer (MOGWO) and the dual-thread ant colony optimization (DTACO) algorithm to solve the vehicle routing problem with time windows (VRPTW). The aim is to simultaneously optimize two often conflicting objectives: minimizing the number of vehicles and travel costs. Traditional methods usually optimize the number of vehicles or travel costs separately, making it difficult to balance the two and leading to insufficient resource utilization or excessively high travel costs in practical applications. To address this issue, this paper presents a symmetrical collaborative framework that integrates multi-objective grey wolf optimization with the dual-thread ant colony system to achieve synchronous and balanced optimization of multiple objectives. Moreover, to solve the problem of premature convergence of the algorithm, a backtracking mechanism is proposed, and its effectiveness is verified through ablation experiments. Experimental results show that this method significantly outperforms single-objective optimization algorithms on the Solomon dataset. Compared with the best known solutions (BKSs), this paper finds better solutions in some datasets, and the performance on the remaining datasets is also close to BKSs. For example, in C204-50, the lowest cost is reduced by 0.48%, in R102-50, the lowest cost is reduced by 1.29% and the number of vehicles is reduced by 1, and in RC106-50, the lowest cost is reduced by 8.48% and the number of vehicles is also reduced by 1. Therefore, the proposed algorithm provides an efficient meta-heuristic framework for VRPTW, effectively balancing the dual objectives and highlighting the value of symmetrical collaboration in multi-objective optimization.<\/jats:p>","DOI":"10.3390\/sym17111949","type":"journal-article","created":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T12:57:13Z","timestamp":1763038633000},"page":"1949","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Research on Algorithm for Multi-Objective Symmetric Collaborative Optimization Vehicle Routing Problem with Time Windows"],"prefix":"10.3390","volume":"17","author":[{"given":"Yipeng","family":"Zhang","sequence":"first","affiliation":[{"name":"CCCC Second Harbor Engineering Company Ltd., Wuhan 430014, China"},{"name":"School of Automotive Engineering, Wuhan University of Technology, Wuhan 430070, China"}]},{"given":"Junya","family":"Yang","sequence":"additional","affiliation":[{"name":"CCCC Second Harbor Engineering Company Ltd., Wuhan 430014, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-6869-1170","authenticated-orcid":false,"given":"Pengyu","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Automotive Engineering, Wuhan University of Technology, Wuhan 430070, China"}]},{"given":"Guoning","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Automotive Engineering, Wuhan University of Technology, Wuhan 430070, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1287\/mnsc.6.1.80","article-title":"The truck dispatching problem","volume":"6","author":"Dantzig","year":"1959","journal-title":"Manag. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1002\/net.3230110211","article-title":"Complexity of vehicle routing and scheduling problems","volume":"11","author":"Lenstra","year":"1981","journal-title":"Networks"},{"key":"ref_3","unstructured":"Si, J. (2024). Research on Mobile Robot Path Planning Based on Ant Colony Algorithm and Dynamic Window Approach. [Master\u2019s Thesis, Shanghai Ocean University]."},{"key":"ref_4","first-page":"829","article-title":"Enhanced Gray Wolf Optimization Algorithm Integrating Multiple Improvement Methods","volume":"63","author":"Fei","year":"2025","journal-title":"J. Jilin Univ. (Sci. Ed.)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"109601","DOI":"10.1016\/j.cie.2023.109601","article-title":"A bi-objective time-dependent vehicle routing problem with delivery failure probabilities","volume":"185","author":"Menares","year":"2023","journal-title":"Comput. Ind. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"042034","DOI":"10.1088\/1742-6596\/2083\/4\/042034","article-title":"Path planning of intelligent mobile robot based on Dijkstra algorithm","volume":"2083","author":"Li","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"108123","DOI":"10.1016\/j.cie.2022.108123","article-title":"Global path planning based on a bidirectional alternating search A* algorithm for mobile robots","volume":"168","author":"Li","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_8","unstructured":"LaValle, S.M., and Kuffner, J.J. (2001). Rapidly-exploring random trees: Progress and prospects. Algorithmic and Computational Robotics, AK Peters\/CRC Press."},{"key":"ref_9","first-page":"104","article-title":"Solving Traveling Salesman Problem Using Branch and Bound Algorithm","volume":"28","author":"Guan","year":"2007","journal-title":"J. North Univ. China (Nat. Sci. Ed.)"},{"key":"ref_10","first-page":"109","article-title":"Vectorized Path Planning Algorithm in Dynamic Environment","volume":"44","author":"Xu","year":"2014","journal-title":"Period. Ocean Univ. China"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4378","DOI":"10.1016\/j.jfranklin.2023.01.033","article-title":"Dynamic path planning of mobile robot based on improved simulated annealing algorithm","volume":"360","author":"Shi","year":"2023","journal-title":"J. Frankl. Inst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"4158","DOI":"10.1007\/s11227-021-04031-9","article-title":"A new hybrid algorithm for path planning of mobile robot","volume":"78","author":"Zhang","year":"2022","journal-title":"J. Supercomput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"15568","DOI":"10.3934\/mbe.2023695","article-title":"Path planning for mobile robots in complex environments based on improved ant colony algorithm","volume":"20","author":"Shi","year":"2023","journal-title":"Math. Biosci. Eng."},{"key":"ref_14","unstructured":"Wang, Q. (2024). Collaborative Truck and UAV Delivery Route Optimization with Time Windows in Time-Dependent Networks. [Master\u2019s Thesis, Dalian Maritime University]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/s12532-016-0108-8","article-title":"Improved branch-cut-and-price for capacitated vehicle routing","volume":"9","author":"Pecin","year":"2017","journal-title":"Math. Program. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1016\/j.trb.2019.03.009","article-title":"A branch-and-price algorithm for the heterogeneous fleet green vehicle routing problem with time windows","volume":"122","author":"Yu","year":"2019","journal-title":"Transp. Res. Part B Methodol."},{"key":"ref_17","first-page":"186","article-title":"Application of Multi-Strategy Improved Particle Swarm Optimization Algorithm in VRPTW Problem","volume":"34","author":"Xie","year":"2024","journal-title":"Comput. Technol. Dev."},{"key":"ref_18","first-page":"36","article-title":"Optimization of Vehicle Distribution Path with Time Windows Based on Improved Genetic Algorithm","volume":"41","author":"Li","year":"2025","journal-title":"Bull. Sci. Technol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"105707","DOI":"10.1016\/j.asoc.2019.105707","article-title":"Quality of service objectives for vehicle routing problem with time windows","volume":"84","author":"Brito","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_20","first-page":"1","article-title":"Research on Solving VRPTW Using Genetic Algorithm Based on Adaptive Large Neighborhood Search","volume":"38","author":"Guo","year":"2023","journal-title":"J. Qingdao Univ. (Eng. Technol. Ed.)"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"106378","DOI":"10.1016\/j.asoc.2020.106378","article-title":"A new game-theoretical multi-objective evolutionary approach for cash-in-transit vehicle routing problem with time windows (A Real life Case)","volume":"93","author":"Ghannadpour","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_22","first-page":"58","article-title":"Research on Airport Terminal VRPTW Based on Multi-Objective Optimization Algorithm","volume":"25","author":"Huang","year":"2025","journal-title":"Sci. Technol. Ind."},{"key":"ref_23","first-page":"14287","article-title":"Global Path Planning for Robot Based on Improved Gray Wolf Optimization Algorithm","volume":"24","author":"Sun","year":"2024","journal-title":"Sci. Technol. Eng."},{"key":"ref_24","first-page":"309","article-title":"Hybrid Multi-Objective Gray Wolf Algorithm for Solving Multi-Objective VRPTW Problem","volume":"60","author":"Chen","year":"2024","journal-title":"Comput. Eng. Appl."},{"key":"ref_25","first-page":"66","article-title":"Multi-Objective Path Planning Method Based on Fused Ant Colony and A* Algorithm","volume":"43","author":"Li","year":"2024","journal-title":"Comput. Technol. Autom."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100675","DOI":"10.1016\/j.swevo.2020.100675","article-title":"An improved ant colony optimization algorithm to the periodic vehicle routing problem with time window and service choice","volume":"55","author":"Wang","year":"2020","journal-title":"Swarm Evol. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"93882","DOI":"10.1109\/ACCESS.2020.2984660","article-title":"A hybrid swarm intelligence algorithm for vehicle routing problem with time windows","volume":"8","author":"Shen","year":"2020","journal-title":"IEEE Access"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"42083","DOI":"10.1109\/ACCESS.2024.3378089","article-title":"A hybrid heuristic harmony search algorithm for the vehicle routing problem with time windows","volume":"12","author":"Zhang","year":"2024","journal-title":"IEEE Access"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"127470","DOI":"10.1109\/ACCESS.2024.3401487","article-title":"Self-competition particle swarm optimization algorithm for the vehicle routing problem with time window","volume":"12","author":"Wang","year":"2024","journal-title":"IEEE Access"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1949\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T13:17:32Z","timestamp":1763039852000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1949"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,13]]},"references-count":29,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111949"],"URL":"https:\/\/doi.org\/10.3390\/sym17111949","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,11,13]]}}}