{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T15:55:53Z","timestamp":1782834953833,"version":"3.54.5"},"reference-count":46,"publisher":"SAGE Publications","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IDA"],"published-print":{"date-parts":[[2024,4,1]]},"abstract":"<jats:p>In this paper, the mathematical model of Vehicle Routing Problem with Time Windows (VRPTW) is established based on the directed graph, and a 3-stage multi-modal multi-objective differential evolution algorithm (3S-MMDEA) is proposed. In the first stage, in order to expand the range of individuals to be selected, a generalized opposition-based learning (GOBL) strategy is used to generate a reverse population. In the second stage, a search strategy of reachable distribution area is proposed, which divides the population with the selected individual as the center point to improve the convergence of the solution set. In the third stage, an improved individual variation strategy is proposed to legalize the mutant individuals, so that the individual after variation still falls within the range of the population, further improving the diversity of individuals to ensure the diversity of the solution set. Based on the synergy of the above three stages of strategies, the diversity of individuals is ensured, so as to improve the diversity of solution sets, and multiple equivalent optimal paths are obtained to meet the planning needs of different decision-makers. Finally, the performance of the proposed method is evaluated on the standard benchmark datasets of the problem. The experimental results show that the proposed 3S-MMDEA can improve the efficiency of logistics distribution and obtain multiple equivalent optimal paths. The method achieves good performance, superior to the most advanced VRPTW solution methods, and has great potential in practical projects.<\/jats:p>","DOI":"10.3233\/ida-227410","type":"journal-article","created":{"date-parts":[[2024,2,23]],"date-time":"2024-02-23T11:29:16Z","timestamp":1708687756000},"page":"485-506","source":"Crossref","is-referenced-by-count":7,"title":["Three-stage multi-modal multi-objective differential evolution algorithm for vehicle routing problem with time windows"],"prefix":"10.1177","volume":"28","author":[{"given":"Hai-Fei","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, Jiangsu, China"},{"name":"Key Laboratory of Advanced Process Control for Light Industry (Jiangnan University), Ministry of Education, Wuxi, Jiangsu, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong-Wei","family":"Ge","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, Jiangsu, China"},{"name":"Key Laboratory of Advanced Process Control for Light Industry (Jiangnan University), Ministry of Education, Wuxi, Jiangsu, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ting","family":"Li","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, Jiangsu, China"},{"name":"Key Laboratory of Advanced Process Control for Light Industry (Jiangnan University), Ministry of Education, Wuxi, Jiangsu, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"ShuZhi","family":"Su","sequence":"additional","affiliation":[{"name":"School of Computer Science and Engineering, Anhui University of Science & Technology, Huainan, Anhui, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"YuBing","family":"Tong","sequence":"additional","affiliation":[{"name":"Department of Radiology, University of Pennsylvania, Philadelphia, PA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","reference":[{"key":"10.3233\/IDA-227410_ref1","doi-asserted-by":"crossref","unstructured":"Y. Deng et al., Multi-objective Path Optimization Method in Terminal Building Based on Improved Genetic Algorithm, in: 2020 Chinese Automation Congress (CAC), 2020, pp. 3181\u2013318.","DOI":"10.1109\/CAC51589.2020.9327639"},{"key":"10.3233\/IDA-227410_ref2","doi-asserted-by":"crossref","first-page":"107230","DOI":"10.1016\/j.cie.2021.107230","article-title":"Path planning optimization of indoor mobile robot based on adaptive ant colony algorithm","volume":"156","author":"Miao","year":"2021","journal-title":"Computers & Industrial Engineering"},{"key":"10.3233\/IDA-227410_ref3","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.swevo.2019.03.011","article-title":"A novel scalable test problem suite for multi-modal multi-objective optimization","volume":"48","author":"Yue","year":"2019","journal-title":"Swarm and Evolutionary Computation"},{"issue":"4","key":"10.3233\/IDA-227410_ref4","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1109\/TEVC.2018.2879406","article-title":"A multi-modal multi-objective evolutionary algorithm using two-archive and recombination strategies","volume":"23","author":"Liu","year":"2019","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"10.3233\/IDA-227410_ref6","doi-asserted-by":"crossref","unstructured":"Y. Wang et al., A novel multi-objective competitive swarm optimization algorithm for multi-modal multi objective problems, in: 2019 IEEE Congress on Evolutionary Computation (CEC), IEEE, 2019, pp. 271\u2013278.","DOI":"10.1109\/CEC.2019.8790218"},{"issue":"2","key":"10.3233\/IDA-227410_ref7","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1287\/opre.35.2.254","article-title":"Algorithms for the vehicle routing and scheduling problem with time window constraints","volume":"35","author":"Solomon","year":"1987","journal-title":"Operations Research"},{"issue":"9","key":"10.3233\/IDA-227410_ref8","doi-asserted-by":"crossref","first-page":"21033","DOI":"10.3390\/s150921033","article-title":"A combination of genetic algorithm and particle swarm optimization for vehicle routing problem with time windows","volume":"15","author":"Xu","year":"2015","journal-title":"Sensors"},{"issue":"6","key":"10.3233\/IDA-227410_ref9","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1007\/s00500-015-1642-4","article-title":"Adaptive memetic algorithm for minimizing distance in the vehicle routing problem with time windows","volume":"20","author":"Nalepa","year":"2016","journal-title":"Soft Computing"},{"key":"10.3233\/IDA-227410_ref10","doi-asserted-by":"crossref","unstructured":"C. Jose et al., An ACS-based memetic algorithm for the heterogeneous vehicle routing problem with time windows, Expert Systems With Applications 157 (2020).","DOI":"10.1016\/j.eswa.2020.113379"},{"key":"10.3233\/IDA-227410_ref12","first-page":"1","article-title":"Multi-depot multi-trip vehicle routing problem with time windows and release dates","volume":"135","author":"Lu","year":"2020","journal-title":"Transportation Research Part E"},{"issue":"4","key":"10.3233\/IDA-227410_ref13","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1002\/net.22028","article-title":"A solution approach for multi-trip vehicle routing problems with time windows, fleet sizing, and depot location","volume":"78","author":"Cueto","year":"2021","journal-title":"Networks"},{"key":"10.3233\/IDA-227410_ref14","doi-asserted-by":"crossref","unstructured":"M. Cheng, Y. Cai and S. Fu, A Preliminary Study of Evolutionary Multitasking for Multi-objective Vehicle Routing Problem With Time Windows, in: The 2021 6th International Conference on Computational Intelligence and Applications (ICCIA), IEEE, 2021.","DOI":"10.1109\/ICCIA52886.2021.00058"},{"key":"10.3233\/IDA-227410_ref15","unstructured":"H. Shu et al., Two-stage multi-objective evolutionary algorithm based on classified population for tri-objective VRPTW, International Journal of Unconventional Computing 16 (2021)."},{"key":"10.3233\/IDA-227410_ref16","doi-asserted-by":"crossref","unstructured":"Y. Hou, Y. Wu and H. Han, Solution Evaluation-Oriented Multi-objective Differential Evolution Algorithm for MOVRPTW, in: The 2021 8th International Conference on Information, Cybernetics, and Computational Social Systems (ICCSS), Piscataway: IEEE, 2021, pp. 50\u201355.","DOI":"10.1109\/ICCSS53909.2021.9721956"},{"issue":"1","key":"10.3233\/IDA-227410_ref19","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s11831-021-09574-x","article-title":"Review of vehicle routing problems: Models, classification and solving algorithms","volume":"29","author":"Zhang","year":"2022","journal-title":"Archives Computational Methods Engineering"},{"key":"10.3233\/IDA-227410_ref20","doi-asserted-by":"crossref","unstructured":"S. Akyo and B. Alatas, Plant intelligence based meta heuristic optimization algorithms, Artificial Intelligence Review 47(4) (2016).","DOI":"10.1007\/s10462-016-9486-6"},{"key":"10.3233\/IDA-227410_ref21","doi-asserted-by":"crossref","first-page":"51","DOI":"10.33383\/2019-029","article-title":"Comparative assessment of light-based intelligent search and optimization algorithms","volume":"3","author":"Alatas","year":"2020","journal-title":"Light & Engineering"},{"key":"10.3233\/IDA-227410_ref22","doi-asserted-by":"crossref","unstructured":"B. Alatas and H. Bingol, A physics based novel approach for travelling tournament problem: Optics inspired optimization, Information Technology and Control 3 (2019).","DOI":"10.5755\/j01.itc.48.3.20627"},{"key":"10.3233\/IDA-227410_ref23","doi-asserted-by":"crossref","unstructured":"H. Bingol and B. Alatas, Chaos based optics inspired optimization algorithms as global solution search approach, Chaos, Solitons & Fractals 141 (2020).","DOI":"10.1016\/j.chaos.2020.110434"},{"key":"10.3233\/IDA-227410_ref24","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.dam.2020.08.017","article-title":"Abranch-and-cut algorithm for the soft-clustered vehicle-routing problem","volume":"288","author":"He\u00dfler","year":"2021","journal-title":"Discrete Applied Mathematics"},{"issue":"1","key":"10.3233\/IDA-227410_ref25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2014.07.048","article-title":"Rich vehicle routing problems: From a taxonomy to a definition European","volume":"241","author":"Lahyani","year":"2015","journal-title":"Journal of Operational Research"},{"issue":"1","key":"10.3233\/IDA-227410_ref26","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.swevo.2011.03.001","article-title":"Multi-objective evolutionary algorithms: A survey of the state of the art","volume":"1","author":"Zhou","year":"2011","journal-title":"Swarm and Evolutionary Computation"},{"issue":"2","key":"10.3233\/IDA-227410_ref27","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A fast and elitist multi-objective genetic algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"10.3233\/IDA-227410_ref28","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.swevo.2019.03.011","article-title":"A novel scalable test problem suite for multi-modal multi-objective optimization","volume":"48","author":"Yue","year":"2019","journal-title":"Swarm and Evolutionary Computation"},{"issue":"4","key":"10.3233\/IDA-227410_ref29","doi-asserted-by":"crossref","first-page":"660","DOI":"10.1109\/TEVC.2018.2879406","article-title":"A multi-modal multi-objective evolutionary algorithm using two-archive and recombination strategies","volume":"23","author":"Liu","year":"2019","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"10.3233\/IDA-227410_ref30","doi-asserted-by":"crossref","unstructured":"J.J. Liang, C.T. Yue and B.Y. Qu, Multi-modal multi-objective optimization: A preliminary study, in: Proceedings of the IEEE Congress on Evolutionary Computation, Vancouver: IEEE, 2016, pp. 2454\u20132461.","DOI":"10.1109\/CEC.2016.7744093"},{"issue":"7","key":"10.3233\/IDA-227410_ref31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11432-018-9754-6","article-title":"A self-organizing multi-modal multi-objective pigeon-inspired optimization algorithm","volume":"62","author":"Hu","year":"2019","journal-title":"Science China Information Sciences"},{"key":"10.3233\/IDA-227410_ref32","doi-asserted-by":"crossref","unstructured":"K. Deb and S. Tiwari, Omni-optimizer: A procedure for single and multi-objective optimization, in: Proceedings of the International Conference on Evolutionary Multi-Criterion Optimization, Berlin: Springer, 2005, pp. 47\u201361.","DOI":"10.1007\/978-3-540-31880-4_4"},{"key":"10.3233\/IDA-227410_ref33","doi-asserted-by":"crossref","unstructured":"L. Yan et al., A performance enhanced niching multi-objective bat algorithm for multi-modal multi-objective problems, in: Proceedings of the IEEE Congress on Evolutionary Computation, Wellington: IEEE, 2019, pp. 1275\u20131282.","DOI":"10.1109\/CEC.2019.8790304"},{"issue":"5","key":"10.3233\/IDA-227410_ref34","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1109\/TEVC.2017.2754271","article-title":"A multi-objective particle swarm optimizer using ring topology for solving multi-modal multi-objective problems","volume":"22","author":"Yue","year":"2018","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"10.3233\/IDA-227410_ref35","doi-asserted-by":"crossref","unstructured":"J.J. Liang et al., A self-organizing multi-objective particle swarm optimization algorithm for multi-modal multi-objective problems, in: Proceedings of the International Conference on Swarm Intelligence, Cham: Springer, 2018, pp. 550\u2013560.","DOI":"10.1007\/978-3-319-93815-8_52"},{"issue":"4","key":"10.3233\/IDA-227410_ref36","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential evolution: A simple and efficient adaptive scheme for global optimization over continuous spaces","volume":"11","author":"Storn","year":"1997","journal-title":"Journal of Global Optimization"},{"key":"10.3233\/IDA-227410_ref38","doi-asserted-by":"crossref","first-page":"105683","DOI":"10.1016\/j.engappai.2022.105683","article-title":"Controlling highway toll stations using deep learning, queuing theory, and differential evolution","volume":"119","author":"Petrovi","year":"2023","journal-title":"Engineering Applications of Artificial Intelligence"},{"key":"10.3233\/IDA-227410_ref39","doi-asserted-by":"crossref","first-page":"99646","DOI":"10.1109\/ACCESS.2022.3206947","article-title":"Two-stage vehicle routing optimization for logistics distribution based on HSA-HGBS algorithm","volume":"10","author":"Sun","year":"2022","journal-title":"IEEE Access"},{"key":"10.3233\/IDA-227410_ref40","doi-asserted-by":"crossref","unstructured":"H.F. Zhang et al., Combining Affinity Propagation with Differential Evolution for Three-echelon logistics distribution optimization, Applied Soft Computing 131C(109878) (2022).","DOI":"10.1016\/j.asoc.2022.109787"},{"key":"10.3233\/IDA-227410_ref41","doi-asserted-by":"crossref","unstructured":"H. Wang et al., Space transformation search: a new evolutionary technique, in: Genetic & Evolutionary Computation Conference, DBLP, 2009, pp. 537\u2013544.","DOI":"10.1145\/1543834.1543907"},{"issue":"9","key":"10.3233\/IDA-227410_ref42","doi-asserted-by":"crossref","first-page":"100849","DOI":"10.1016\/j.swevo.2021.100849","article-title":"Differential evolution using improved crowding distance for multi-modal multi-objective optimization","volume":"62","author":"Yue","year":"2021","journal-title":"Swarm and Evolutionary Computation"},{"issue":"2","key":"10.3233\/IDA-227410_ref43","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/TEVC.2003.810758","article-title":"Performance assessment of multi-objective optimizers: An analysis and review","volume":"7","author":"Zitzler","year":"2003","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"10.3233\/IDA-227410_ref44","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1016\/j.swevo.2018.10.016","article-title":"Multi-modal multi-objective optimization with differential evolution","volume":"44","author":"Liang","year":"2019","journal-title":"Swarm and Evolutionary Computation"},{"issue":"1","key":"10.3233\/IDA-227410_ref45","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1109\/TEVC.2010.2087271","article-title":"Differential evolution with composite trial vector generation strategies and control parameters","volume":"15","author":"Yong","year":"2011","journal-title":"IEEE Transactions on Evolutionary Computation"},{"issue":"3","key":"10.3233\/IDA-227410_ref46","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1109\/JSYST.2014.2300201","article-title":"A local search-based multi-objective optimization algorithm for multi-objective vehicle routing problem with time windows","volume":"9","author":"Zhou","year":"2017","journal-title":"IEEE Systems Journal"},{"key":"10.3233\/IDA-227410_ref47","doi-asserted-by":"crossref","unstructured":"J.P.C. Guti\u00e9rrez, D. Landa-Silva and J.A. Moreno-P\u00e9rez, Nature of real-world multi-objective vehicle routing with evolutionary algorithms, in: IEEE International Conference on Systems, 2011, pp. 257\u2013264.","DOI":"10.1109\/ICSMC.2011.6083675"},{"issue":"1","key":"10.3233\/IDA-227410_ref49","doi-asserted-by":"crossref","first-page":"114779","DOI":"10.1016\/j.eswa.2021.114779","article-title":"NSGA-II with objective-specific variation operators for multi-objective vehicle routing problem with time windows","volume":"176","author":"Srivastava","year":"2021","journal-title":"Expert Systems with Applications"},{"issue":"6","key":"10.3233\/IDA-227410_ref50","doi-asserted-by":"crossref","first-page":"712","DOI":"10.1109\/TEVC.2007.892759","article-title":"MOEA\/D: A multi-objective evolutionary algorithm based on decomposition","volume":"11","author":"Zhang","year":"2008","journal-title":"IEEE Transactions on Evolutionary Computation"},{"issue":"2","key":"10.3233\/IDA-227410_ref51","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/TEVC.2003.810758","article-title":"Performance assessment of multi-objective optimizers: an analysis and review","volume":"7","author":"Zitzler","year":"2003","journal-title":"IEEE Transactions on Evolutionary Computation"},{"key":"10.3233\/IDA-227410_ref52","doi-asserted-by":"crossref","unstructured":"X. Yao et al., Multi-modal multi-objective evolutionary algorithm for multiple path planning, Computers & Industrial Engineering 169(Pt2) (2022).","DOI":"10.1016\/j.cie.2022.108145"}],"container-title":["Intelligent Data Analysis"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/IDA-227410","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:20:24Z","timestamp":1777454424000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/IDA-227410"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,1]]},"references-count":46,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.3233\/ida-227410","relation":{},"ISSN":["1088-467X","1571-4128"],"issn-type":[{"value":"1088-467X","type":"print"},{"value":"1571-4128","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,1]]}}}