{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T20:20:32Z","timestamp":1760646032630,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2013,4,9]],"date-time":"2013-04-09T00:00:00Z","timestamp":1365465600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>State-of-the-art heuristic algorithms to solve the vehicle routing problem with time windows (VRPTW) usually present slow speeds during the early iterations and easily fall into local optimal solutions. Focusing on solving the above problems, this paper analyzes the particle encoding and decoding strategy of the particle swarm optimization algorithm, the construction of the vehicle route and the judgment of the local optimal solution. Based on these, a hybrid chaos-particle swarm optimization algorithm (HPSO) is proposed to solve VRPTW. The chaos algorithm is employed to re-initialize the particle swarm. An efficient insertion heuristic algorithm is also proposed to build the valid vehicle route in the particle decoding process. A particle swarm premature convergence judgment mechanism is formulated and combined with the chaos algorithm and Gaussian mutation into HPSO when the particle swarm falls into the local convergence. Extensive experiments are carried out to test the parameter settings in the insertion heuristic algorithm and to evaluate that they are corresponding to the data\u2019s real-distribution in the concrete problem. It is also revealed that the HPSO achieves a better performance than the other state-of-the-art algorithms on solving VRPTW.<\/jats:p>","DOI":"10.3390\/e15041247","type":"journal-article","created":{"date-parts":[[2013,4,9]],"date-time":"2013-04-09T11:15:06Z","timestamp":1365506106000},"page":"1247-1270","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["A Hybrid Chaos-Particle Swarm Optimization Algorithm for the Vehicle Routing Problem with Time Window"],"prefix":"10.3390","volume":"15","author":[{"given":"Wenbin","family":"Hu","sequence":"first","affiliation":[{"name":"School of Computer, Wuhan University, Wuhan City, Hubei Province 430072, China"},{"name":"State Key Laboratory of Software Engineering, Wuhan University, Wuhan City, Hubei Province 430072, China"}]},{"given":"Huanle","family":"Liang","sequence":"additional","affiliation":[{"name":"School of Computer, Wuhan University, Wuhan City, Hubei Province 430072, China"}]},{"given":"Chao","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Computer, Wuhan University, Wuhan City, Hubei Province 430072, China"}]},{"given":"Bo","family":"Du","sequence":"additional","affiliation":[{"name":"School of Computer, Wuhan University, Wuhan City, Hubei Province 430072, China"}]},{"given":"Qi","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Computer, Wuhan University, Wuhan City, Hubei Province 430072, China"}]}],"member":"1968","published-online":{"date-parts":[[2013,4,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/0377-2217(88)90330-X","article-title":"Vechicle routing considerations in dsitribtion system design","volume":"37","author":"Bookbinder","year":"1988","journal-title":"Eur. J. Oper. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.ijpe.2009.10.004","article-title":"Stochastic optimization of medical supply location and distribution in disaster management","volume":"126","author":"Mete","year":"2010","journal-title":"Int. J. Prod. Econ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1080\/09511920701241624","article-title":"An agent-based dynamic routing strategy for automated material handling systems","volume":"21","author":"Lau","year":"2008","journal-title":"Int. J. Comput. Integrated Manuf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1080\/13675560600649982","article-title":"Redesigning distribution operations: A case study on integrating inventory management and vehicle routes design","volume":"9","author":"Oliveria","year":"2006","journal-title":"Int. J. Logist. Res. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1080\/03081060701390841","article-title":"The vehicle routing problem: The case of the hong kong postal service","volume":"30","author":"Ji","year":"2007","journal-title":"Transp. Plan. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1080\/07408179708966379","article-title":"A computerized approach to the New York Cityschool bus routing problem","volume":"29","author":"Braca","year":"1997","journal-title":"IIE Trans."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shaw, P. (1998). Using constraint programming and local search methods to vehicle routing problem. Princ. Pract. Constraint Program. (CP\u201998), 417\u2013431.","DOI":"10.1007\/3-540-49481-2_30"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"626","DOI":"10.1287\/opre.42.4.626","article-title":"Optimal solution of vehicle routing problems using minimum K-trees","volume":"42","author":"Fisher","year":"1994","journal-title":"Oper. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0305-0548(96)00065-2","article-title":"An adaptive memory heuristic for a class of vehicle routing problems with minmax objective","volume":"24","author":"Golden","year":"1999","journal-title":"Comput. Oper. Res."},{"key":"ref_10","first-page":"13","article-title":"Simulated annealing meta-heuristics for the vehicle routing problem with time windows","volume":"93","author":"Chiang","year":"1996","journal-title":"Ann. Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/BF02579017","article-title":"Tabu search heuristics for the vehicle routing problem with time windows","volume":"10","author":"Gendreau","year":"2002","journal-title":"TOP."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1153","DOI":"10.1016\/S0305-0548(98)00100-2","article-title":"A tabu search heuristic for the heterogeneous fleet vehicle routing problem","volume":"26","author":"Gendreau","year":"1999","journal-title":"Comput. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/S0925-5273(00)00174-2","article-title":"A hybrid genetic algorithm for the vehicle scheduling problem with due times and time deadlines","volume":"73","author":"Park","year":"2001","journal-title":"Int J Prod Econ"},{"key":"ref_14","unstructured":"Gambardella, L.M., Taillard, \u00c9., and Agazzi, G. (1999). New Ideals in Optimization, McGraw-Hill Ltd."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1007\/11890584_10","article-title":"A reactive greedy randomized variable neighborhood tabu search for the vehicle routing problem with time windows","volume":"4030","author":"Repoussis","year":"2006","journal-title":"Hybrid Metaheuristics Lect. Notes Comput. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s10732-007-9045-z","article-title":"A reactive variable neighborhood tabu search for the heterogeneous fleet vehicle routing problem with time windows","volume":"14","author":"Paraskevopoulos","year":"2008","journal-title":"J. Heuristics"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1007\/s10951-010-0213-x","article-title":"Tabu search techniques for the heterogeneous vehicle routing problem with time windows and carrier-dependent costs","volume":"14","author":"Ceschia","year":"2011","journal-title":"J. Sched."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1057\/palgrave.jors.2601935","article-title":"Metaheuristics applied to mixed and simultaneous extensions of vehicle routing problems with backhauls","volume":"56","author":"Crispim","year":"2005","journal-title":"J. Oper. Res. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1016\/j.cor.2005.03.014","article-title":"Heuristic algorithms for the vehicle routing problem with simulaneous pick-up and delivery","volume":"34","author":"Bianchessi","year":"2007","journal-title":"Comput. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.trc.2006.03.002","article-title":"Neighborhood search heuristics for a dynamic vehicle dispatching problem with pick-ups and deliveries","volume":"14","author":"Gendreau","year":"2006","journal-title":"Transp. Res. Part C."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2717","DOI":"10.1016\/j.eswa.2010.08.061","article-title":"A local search metaheuristic algorithm for the vehicle routing problem with simulanteous pick-ups and deliveries","volume":"38","author":"Zachariadis","year":"2011","journal-title":"Exp. Syst. Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.cor.2011.03.006","article-title":"An improved LNS algorithm for real-time vehicle routing problem with time windows","volume":"39","author":"Hong","year":"2012","journal-title":"Comput. Oper. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/j.amc.2005.09.040","article-title":"A hybird simulated annealing for capacitated vehicle routing problems with the independent route length","volume":"176","author":"Safaei","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/j.jmsy.2011.04.005","article-title":"A new mathematical model for a competitive vehicle routing problem with time windows solved by simulated annealing","volume":"30","author":"Gazanfari","year":"2011","journal-title":"J. Manuf. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"954","DOI":"10.1016\/j.cor.2010.10.011","article-title":"An ant colony algorithm hybridized with insertion heuristics for the time dependent vehicle routing problem with time windows","volume":"38","author":"Balseiro","year":"2011","journal-title":"Comput. Oper. Res."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"6809","DOI":"10.1016\/j.eswa.2010.03.045","article-title":"A new saving-based ant algorithm for the vehicle routing problem with simultaneous pickup and delivery","volume":"37","author":"Catay","year":"2010","journal-title":"Exp. Syst. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3215","DOI":"10.1016\/j.cor.2009.02.017","article-title":"An ant colony system (ACS) for vehicle routing problem with simulataneous delivery and pickup","volume":"36","author":"Gajpal","year":"2009","journal-title":"Comput. Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1096","DOI":"10.1016\/j.asoc.2010.04.001","article-title":"Multi-objective vehicle routing problem with time windows using goal programming and genetic algorithm","volume":"10","author":"Ghoseiri","year":"2010","journal-title":"Appl. Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"680","DOI":"10.1016\/j.cie.2007.06.031","article-title":"A vehicle routing problem solved by using a hybrid genetic algorithm","volume":"53","author":"Jeon","year":"2007","journal-title":"Computers & Industrial Engineering."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1631\/jzus.2006.A0607","article-title":"Hybrid discrete particle swarm optimization algorithm for capacitated vehicle routing problem","volume":"7","author":"Chen","year":"2006","journal-title":"J. Zhejiang Univ. Sci. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.cie.2008.06.012","article-title":"Particle swarm optimization and two solution representations for solving the capacitated vehicle routing problem","volume":"56","author":"Kachitvichyanukul","year":"2009","journal-title":"Comput. Ind. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"801","DOI":"10.4028\/www.scientific.net\/MSF.471-472.801","article-title":"Particle swarm optimization for vehicle routing problem with time windows","volume":"471","author":"Zhao","year":"2004","journal-title":"Materials Science Forum."},{"key":"ref_33","first-page":"1890","article-title":"A hybrid particle swarm optimization algorithm for vehicle routing problem with time windows","volume":"40","author":"Zhang","year":"2006","journal-title":"J. Shanghai Jiaotong Univ."},{"key":"ref_34","first-page":"230","article-title":"Modified Particle Swarm Optimization algorithm for vehicle routing problem with time windows","volume":"46","author":"Wu","year":"2010","journal-title":"Comput. Eng. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1109\/TSMCC.2011.2148712","article-title":"Optimizing the vehicle routing problem with time windows: A discrete particle swarm optimization approach","volume":"42","author":"Gong","year":"2012","journal-title":"Syst. Man Cybern. Part C Appl. Rev. IEEE Trans."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1051\/jphyscol:1978505","article-title":"Un attracteur \u00e9trange (?) du type attracteur de H\u00e9non","volume":"39","author":"Lozi","year":"1978","journal-title":"J. Phys. Colloq."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1103\/PhysRevLett.45.709","article-title":"Optical turbulence: Chaotic behavior of transmitted light from a ring cavity","volume":"45","author":"Ikeda","year":"1980","journal-title":"Phys. Rev. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"J\u00fcrgens, H., Peitgen, H.O., and Saupe, D. (1992). Chaos and Fractals: New Frontiers of Science, Springer-Verlag.","DOI":"10.1007\/978-1-4757-4740-9"},{"key":"ref_39","unstructured":"Solomon Benchmark Problems-VRPTW. Available online: http:\/\/www.idsia.ch\/luca\/macs-vrptw\/problmes\/welcome.htm."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/4\/1247\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:46:01Z","timestamp":1760219161000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/15\/4\/1247"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,9]]},"references-count":39,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,4]]}},"alternative-id":["e15041247"],"URL":"https:\/\/doi.org\/10.3390\/e15041247","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2013,4,9]]}}}