{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T05:14:21Z","timestamp":1787030061663,"version":"build-2736575974"},"reference-count":57,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"GraceChain Software Ltd-SDUST-GLOBAL OPTIMUM FRESH Cross-Border Fresh Supply Chain Platform joint research project"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In an era where people in the world are concerned about environmental issues, companies must reduce distribution costs while minimizing the pollution generated during the distribution process. For today\u2019s multi-depot problem, a mixed-integer programming model is proposed in this paper to minimize all costs incurred in the entire transportation process, considering the impact of time-varying speed, loading, and waiting time on costs. Time is directional; hence, the problems considered in this study are modeled based on asymmetry, making the problem-solving more complex. This paper proposes a genetic algorithm combined with simulated annealing to solve this issue, with the inner and outer layers solving for the optimal waiting time and path planning problem, respectively. The mutation operator is replaced in the outer layer by a neighbor search approach using a solution acceptance mechanism similar to simulated annealing to avoid a local optimum solution. This study extends the path distribution problem (vehicle-routing problem) and provides an alternative approach for solving time-varying networks.<\/jats:p>","DOI":"10.3390\/sym15010124","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T02:12:48Z","timestamp":1672625568000},"page":"124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["A Genetic Algorithm for the Waitable Time-Varying Multi-Depot Green Vehicle Routing Problem"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6502-472X","authenticated-orcid":false,"given":"Chien-Ming","family":"Chen","sequence":"first","affiliation":[{"name":"College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shi","family":"Lv","sequence":"additional","affiliation":[{"name":"College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jirsen","family":"Ning","sequence":"additional","affiliation":[{"name":"Qingdao GraceChain Software Ltd., Qingdao 266100, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3740-2102","authenticated-orcid":false,"given":"Jimmy Ming-Tai","family":"Wu","sequence":"additional","affiliation":[{"name":"College of Computer Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/j.cie.2016.04.002","article-title":"Composite particle algorithm for sustainable integrated dynamic ship routing and scheduling optimization","volume":"96","author":"De","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"102723","DOI":"10.1016\/j.tre.2022.102723","article-title":"Optimization model for sustainable food supply chains: An application to Norwegian salmon","volume":"161","author":"De","year":"2022","journal-title":"Transp. Res. Part Logist. Transp. Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"126518","DOI":"10.1016\/j.amc.2021.126518","article-title":"Solving the production transportation problem via a deterministic annealing neural network method","volume":"411","author":"Wu","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_4","unstructured":"Hardcastle, J. (2015). Walmart, General Mills, Anheuser-Busch improve freight efficiency, cut emissions. Environ. Lead., Available online: http:\/\/www.en-vironmentalleader.com\/2015\/05\/13\/walmart-general-mills-anheuser-busch-improve-freight-efficiency-cut-emissions\/#ixzz473YFXy9e."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4354","DOI":"10.1109\/TNNLS.2019.2955137","article-title":"A deterministic annealing neural network algorithm for the minimum concave cost transportation problem","volume":"31","author":"Wu","year":"2019","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_6","unstructured":"Namasudra, S., and Sharma, P. (2022). Achieving a decentralized and secure cab sharing system using blockchain technology. IEEE Trans. Intell. Transp. Syst., 1\u201310."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/j.matcom.2021.10.032","article-title":"Golden eagle optimizer with double learning strategies for 3D path planning of UAV in power inspection","volume":"193","author":"Pan","year":"2022","journal-title":"Math. Comput. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"15501329221104332","DOI":"10.1177\/15501329221104332","article-title":"A provably secure and lightweight mutual authentication protocol in fog-enabled social Internet of vehicles","volume":"18","author":"Li","year":"2022","journal-title":"Int. J. Distrib. Sens. Netw."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1504\/EJIE.2019.10022249","article-title":"A green vehicle routing problem with time windows considering the heterogeneous fleet of vehicles: Two metaheuristic algorithms","volume":"13","author":"Rezaei","year":"2019","journal-title":"Eur. J. Ind. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"104813","DOI":"10.1016\/j.knosys.2019.06.021","article-title":"Time-dependent vehicle routing problem with time windows of city logistics with a congestion avoidance approach","volume":"188","author":"Liu","year":"2020","journal-title":"Knowl.-Based Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.tre.2016.01.011","article-title":"The heterogeneous green vehicle routing and scheduling problem with time-varying traffic congestion","volume":"88","author":"Xiao","year":"2016","journal-title":"Transp. Res. Part E Logist. Transp. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1016\/j.trb.2011.02.004","article-title":"The Pollution-Routing Problem","volume":"45","author":"Laporte","year":"2011","journal-title":"Transp. Res. Part B Methodol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1385","DOI":"10.1016\/j.cie.2009.07.006","article-title":"Optimizing goods assignment and the vehicle routing problem with time-dependent travel speeds","volume":"57","author":"Kuo","year":"2009","journal-title":"Comput. Ind. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/S0377-2217(02)00147-9","article-title":"Vehicle dispatching with time-dependent travel times","volume":"144","author":"Ichoua","year":"2003","journal-title":"Eur. J. Oper. Res."},{"key":"ref_15","first-page":"1626","article-title":"Solving TDVRP based on a parallel-simulated annealing algorithm","volume":"21","author":"Mu","year":"2015","journal-title":"Comput. Integr. Manuf. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ma, X., Liao, L., Li, Z., Lai, R.X., and Zhang, M. (2022). Applying Federated Learning in Software-Defined Networks: A Survey. Symmetry, 14.","DOI":"10.3390\/sym14020195"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"986","DOI":"10.1109\/TITS.2019.2900490","article-title":"Hybridizing Basic Variable Neighborhood Search With Particle Swarm Optimization for Solving Sustainable Ship Routing and Bunker Management Problem","volume":"21","author":"De","year":"2020","journal-title":"IEEE Trans. Intell. Transp. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1016\/0305-0483(81)90055-4","article-title":"Adapting the savings algorithm for varying inter-customer travel times","volume":"9","author":"Beasley","year":"1981","journal-title":"Omega"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1287\/opre.12.4.568","article-title":"Scheduling of Vehicles from a Central Depot to a Number of Delivery Points","volume":"12","author":"Wright","year":"1964","journal-title":"Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1057\/jors.1992.49","article-title":"Modelling Intra-City Time-Dependent Travel Speeds for Vehicle Scheduling Problems","volume":"43","author":"Hill","year":"1992","journal-title":"J. Oper. Res. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1002\/1097-0037(200009)36:2<80::AID-NET2>3.0.CO;2-B","article-title":"Efficient modeling of travel in networks with time-varying link speeds","volume":"36","author":"Horn","year":"2015","journal-title":"Networks"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"104606","DOI":"10.1016\/j.engappai.2021.104606","article-title":"A hybrid algorithm for time-dependent vehicle routing problem with soft time windows and stochastic factors","volume":"109","author":"Jie","year":"2022","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1108\/IMDS-06-2020-0345","article-title":"Low-carbon VRP for cold chain logistics considering real-time traffic conditions in the road network","volume":"122","author":"Bai","year":"2022","journal-title":"Ind. Manag. Data Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1287\/trsc.3.3.192","article-title":"The Multiple Terminal Delivery Problem with Probabilistic Demands","volume":"3","author":"Tillman","year":"1969","journal-title":"Transp. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1057\/jors.1972.53","article-title":"Computer Scheduling of Vehicles from One or More Depots to a Number of Delivery Points","volume":"23","author":"Wren","year":"1972","journal-title":"J. Oper. Res. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/S0377-2217(82)80011-8","article-title":"A modular algorithm for an extended vehicle scheduling problem","volume":"11","author":"Raft","year":"1982","journal-title":"Eur. J. Oper. Res."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105269","DOI":"10.1016\/j.cor.2021.105269","article-title":"An efficient variable neighborhood search with tabu shaking for a class of multi-depot vehicle routing problems","volume":"133","author":"Aksen","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1109\/TASE.2005.853472","article-title":"Multi-depot vehicle routing problem: A one-stage approach","volume":"2","author":"Lim","year":"2005","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"794","DOI":"10.1016\/j.cie.2011.05.012","article-title":"The multi-depot split delivery vehicle routing problem: An integer programming-based heuristic, new test problems, and computational results","volume":"61","author":"Gulczynski","year":"2011","journal-title":"Comput. Ind. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.ejor.2012.02.007","article-title":"Heuristics for the multi-depot petrol station replenishment problem with time windows","volume":"220","author":"Cornillier","year":"2012","journal-title":"Eur. J. Oper. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"756","DOI":"10.1016\/j.ejor.2014.10.048","article-title":"A hybrid metaheuristic algorithm for the multi-depot covering tour vehicle routing problem","volume":"242","author":"Allahyari","year":"2015","journal-title":"Eur. J. Oper. Res."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.endm.2018.03.022","article-title":"A GVNS Algorithm for Solving the Multi-Depot Vehicle Routing Problem","volume":"66","author":"Bezerra","year":"2018","journal-title":"Electron. Notes Discret. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1016\/j.ejor.2020.01.008","article-title":"A memory-based iterated local search algorithm for the multi-depot open vehicle routing problem","volume":"284","year":"2020","journal-title":"Eur. J. Oper. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/BF01919767","article-title":"Shortest route with time dependent length of edges and limited delay possibilities in nodes","volume":"21","author":"Halpern","year":"1977","journal-title":"Math. Methods Oper. Res."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Orda, A., and Rom, R. (1997). Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length. J. ACM, 37.","DOI":"10.1145\/79147.214078"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Cai, X., Kloks, T., and Wong, C.K. (1996). Shortest Path Problems with Time Constraints, Springer.","DOI":"10.1007\/3-540-61550-4_153"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1002\/(SICI)1097-0037(199705)29:3<141::AID-NET2>3.0.CO;2-H","article-title":"Time-varying shortest path problems with constraints","volume":"29","author":"Cai","year":"1997","journal-title":"Networks"},{"key":"ref_38","first-page":"41","article-title":"Introduction Algorithms for Minimum-Cost Paths in Time-Dependent Networks with Waiting Policies","volume":"44","author":"Dean","year":"2004","journal-title":"DBLP"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s00778-018-0499-4","article-title":"Go slow to go fast: Minimal on-road time route scheduling with parking facilities using historical trajectory","volume":"27","author":"Li","year":"2018","journal-title":"VLDB J. Int. J. Very Large Data Bases"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1002\/net.21885","article-title":"Time-dependent shortest paths with discounted waits","volume":"74","author":"Omer","year":"2019","journal-title":"Networks"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1287\/ijoc.2020.0985","article-title":"Time-Dependent Shortest Path Problems with Penalties and Limits on Waiting","volume":"33","author":"He","year":"2021","journal-title":"INFORMS J. Comput."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"105211","DOI":"10.1016\/j.cor.2021.105211","article-title":"Time-dependent multi-depot green vehicle routing problem with time windows considering temporal-spatial distance","volume":"129","author":"Fan","year":"2021","journal-title":"Comput. Oper. Res."},{"key":"ref_43","unstructured":"Hickman, J., Hassel, D., Joumard, R., Samaras, Z., and Sorenson, S.C. (1999). Methodology for Calculating Transport Emissions and Energy Consumption, Transport Research Laboratory."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"2050","DOI":"10.1016\/j.dam.2007.04.022","article-title":"An iterated local search algorithm for the vehicle routing problem with convex time penalty functions","volume":"156","author":"Ibaraki","year":"2008","journal-title":"Discret. Appl. Math."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"e12264","DOI":"10.1111\/exsy.12264","article-title":"Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem","volume":"35","author":"Majumder","year":"2018","journal-title":"Expert Syst."},{"key":"ref_46","first-page":"918","article-title":"The Optimization Research of Vehicle Routing Problem with Heterogeneous Fleet, Simultaneous Pickup-Delivery Considering Temporal-Spatial Distance","volume":"15","author":"Wang","year":"2018","journal-title":"Chin. J. Manag."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1789","DOI":"10.3390\/sym13101789","article-title":"Research and Application of Improved Clustering Algorithm in Retail Customer Classification","volume":"13","author":"Liu","year":"2021","journal-title":"Symmetry"},{"key":"ref_48","first-page":"218","article-title":"Vehicle Routing Algorithm Based on Spatiotemporal Clustering","volume":"41","author":"Qi","year":"2014","journal-title":"Comput. Sci."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Liao, L., Leung, V.C.M., Li, Z., and Chao, H.C. (2021). Genetic Algorithms with Variant Particle Swarm Optimization Based Mutation for Generic Controller Placement in Software-Defined Networks. Symmetry, 13.","DOI":"10.3390\/sym13071133"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"24570","DOI":"10.1109\/JSEN.2021.3115471","article-title":"Matching sensor ontologies with multi-context similarity measure and parallel compact differential evolution algorithm","volume":"21","author":"Xue","year":"2021","journal-title":"IEEE Sens. J."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Namasudra, S., Dhamodharavadhani, S., and Rathipriya, R. (2021). Nonlinear neural network based forecasting model for predicting COVID-19 cases. Neural Process. Lett., 1\u201321.","DOI":"10.1007\/s11063-021-10495-w"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Pan, J.S., Hu, P., Sn\u00e1\u0161el, V., and Chu, S.C. (2022). A survey on binary metaheuristic algorithms and their engineering applications. Artif. Intell. Rev., 1\u201367.","DOI":"10.1007\/s10462-022-10328-9"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"59504","DOI":"10.1109\/ACCESS.2019.2913757","article-title":"Selecting hyper-parameters of Gaussian process regression based on non-inertial particle swarm optimization in Internet of Things","volume":"7","author":"Kang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Wu, T.Y., Lin, J.C.W., Zhang, Y., and Chen, C.H. (2019). A grid-based swarm intelligence algorithm for privacy-preserving data mining. Appl. Sci., 9.","DOI":"10.3390\/app9040774"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"120329","DOI":"10.1016\/j.energy.2021.120329","article-title":"Binary fish migration optimization for solving unit commitment","volume":"226","author":"Pan","year":"2021","journal-title":"Energy"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Kong, L., Chen, C.M., Shih, H.C., Lin, C.W., He, B.Z., and Pan, J.S. (2014). An energy-aware routing protocol using cat swarm optimization for wireless sensor networks. Advanced Technologies, Embedded and Multimedia for Human-Centric Computing, Springer.","DOI":"10.1007\/978-94-007-7262-5_36"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1002\/(SICI)1097-0037(199709)30:2<105::AID-NET5>3.0.CO;2-G","article-title":"A tabu search heuristic for periodic and multi-depot vehicle routing problems","volume":"30","author":"Cordeau","year":"1997","journal-title":"Networks"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:55:23Z","timestamp":1760118923000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":57,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010124"],"URL":"https:\/\/doi.org\/10.3390\/sym15010124","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,1]]}}}