{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:10:15Z","timestamp":1760145015302,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T00:00:00Z","timestamp":1717632000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Economic globalization and the rapid development of the Internet make logistics systems more and more diversified, people and enterprises have greatly increased their requirements for logistics systems, and fourth party logistics has received more and more attention from people and related enterprises. In order to further study the routing problem under uncertain stochastic environments, this paper considers the fourth party logistics routing problem from a single manufacturer to multiple distributors with uncertain times and random supplies under the complete information symmetry scenario and symmetric transportation volume decision space. Then, an uncertain stochastic programming model is established with the minimum total cost as its core objective, and the total transportation time, manufacturer\u2019s supply, and distributor\u2019s demand as constraints. In order to solve the optimal path of the above problems, this paper transforms the uncertain stochastic programming model into a classical mathematical programming model based on the distribution functions of uncertain time and random supply. Finally, two numerical examples are given to verify the effectiveness of the proposed model.<\/jats:p>","DOI":"10.3390\/sym16060701","type":"journal-article","created":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T03:44:35Z","timestamp":1717645475000},"page":"701","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Symmetric Fourth Party Logistics Routing Problem with Multiple Distributors in Uncertain Random Environments"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-4119-5695","authenticated-orcid":false,"given":"Xinyu","family":"Gao","sequence":"first","affiliation":[{"name":"School of Mathematical Science and Physics, North China Electric Power University, Beijing 102206, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0462-9813","authenticated-orcid":false,"given":"Xin","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematical Science and Physics, North China Electric Power University, Beijing 102206, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0780-8267","authenticated-orcid":false,"given":"Yang","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"key":"ref_1","first-page":"120","article-title":"Research on pricing decision and coordination mechanism of supply chain under the role of fourth party logistics","volume":"12","author":"Fan","year":"2022","journal-title":"China Storage Transp."},{"key":"ref_2","first-page":"659","article-title":"A two-sided logistics matching method considering trading psychology and matching effort under a 4PL","volume":"14","author":"Yuan","year":"2023","journal-title":"Int. J. Ind. Eng. Comput."},{"key":"ref_3","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":"Arch. Comput. Method E."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107484","DOI":"10.1016\/j.cie.2021.107484","article-title":"Supply chain network design considering customer psychological behavior-a 4PL perspective","volume":"159","author":"Huang","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"108002","DOI":"10.1016\/j.cie.2022.108002","article-title":"Fourth-party logistics network design under uncertainty environment","volume":"167","author":"Yin","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Jiang, S., Chen, J., Huang, M., Zhang, Y., and Yin, M. (2021, January 22\u201324). Fourth party logistics network design considering quantity discount. Proceedings of the 33rd Chinese Control and Decision Conference, Kunming, China.","DOI":"10.1109\/CCDC52312.2021.9602597"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1080\/13675567.2015.1107533","article-title":"Understanding the 4PL approach within an agricultural supply chain using matrix model and cross-case analysis","volume":"19","author":"Mehmann","year":"2016","journal-title":"Int. J. Log. Res. Appl."},{"key":"ref_8","first-page":"529","article-title":"Fourth party logistics firm assessment using a novel neutrosophic MCDM","volume":"42","author":"Kabak","year":"2022","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Li, J., Li, R., and Liu, Y. (2020, January 22\u201324). Multi-period reliable network design of fourth party logistics base on krill herd algorithm. Proceedings of the 32nd Chinese Control and Decision Conference, Hefei, China.","DOI":"10.1109\/CCDC49329.2020.9164808"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"34518","DOI":"10.1007\/s11356-022-24364-z","article-title":"Green winner determination method based on environmental performance and minimum adjustment consensus in 4PL transportation service procurement","volume":"30","author":"Yuan","year":"2023","journal-title":"Environ. Sci. Pollut. Res."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"106189","DOI":"10.1016\/j.cor.2023.106189","article-title":"Metaheuristics with variable diversity control and neighborhood search for the Heterogeneous Site-Dependent Multi-depot Multi-trip Periodic Vehicle Routing Problem","volume":"153","author":"Vieira","year":"2023","journal-title":"Comput. Oper. Res."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"105856","DOI":"10.1016\/j.cor.2022.105856","article-title":"Hybrid adaptive large neighborhood search for vehicle routing problems with depot location decisions","volume":"146","author":"Voigt","year":"2022","journal-title":"Comput. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s12293-021-00352-7","article-title":"Solving vehicle routing problem by memetic search with evolutionary multitasking","volume":"14","author":"Shang","year":"2022","journal-title":"Memetic Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4472","DOI":"10.1080\/00207543.2022.2154406","article-title":"Data-driven Game-theoretic Model Based on Blockchain for Managing Resource Allocation and Vehicle Routing in Modular Integrated Construction","volume":"61","author":"Eltoukhy","year":"2023","journal-title":"Int. J. Prod. Res."},{"key":"ref_15","first-page":"274","article-title":"Improved crow algorithm for soft time window multimodal 4PL routing problem","volume":"58","author":"Tang","year":"2022","journal-title":"Comput. Eng. Appl."},{"key":"ref_16","unstructured":"Li, X., Ying, W., Liu, W., Chen, J., and Huang, B. (2003, January 8). The decision optimization model of 4PL. Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Washington, DC, USA."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Yang, S., Zhu, J., Wang, Q., and Huang, M. (2020, January 22\u201324). Routing problem with stochastic delay time under fourth party logistics. Proceedings of the 2020 Chinese Control and Decision Conference, Hefei, China.","DOI":"10.1109\/CCDC49329.2020.9164104"},{"key":"ref_18","first-page":"583","article-title":"Fourth party logistics routing problem with time window in uncertain environments","volume":"47","author":"Ren","year":"2018","journal-title":"Inf. Control"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Liu, Y., and Liu, B. (2023). A modified uncertain maximum likelihood estimation with applications in uncertain statistics. Commun. Stat. Theor. Methods.","DOI":"10.1080\/03610926.2023.2248534"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, Y., and Liu, B. (2023). Estimation of uncertainty distribution function by the principle of least squares. Commun. Stat. Theor. Methods.","DOI":"10.1080\/03610926.2023.2269451"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Liu, Y. (2022). Moment estimation for uncertain regression model with application to factors analysis of grain yield. Commun. Stat. Simul. Comput.","DOI":"10.1080\/03610918.2022.2160461"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s10700-023-09415-5","article-title":"Solution method and parameter estimation of uncertain partial differential equation with application to China\u2019s population","volume":"23","author":"Yang","year":"2024","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Liu, B. (2009). Theory and Practice of Uncertain Programming, Springer. [2nd ed.].","DOI":"10.1007\/978-3-540-89484-1"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"107169","DOI":"10.1016\/j.cie.2021.107169","article-title":"A robust optimization model under uncertain environment: An application in production planning","volume":"155","author":"Singh","year":"2021","journal-title":"Comput. Ind. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1016\/j.orl.2022.02.001","article-title":"Constrained Markov decision processes with uncertain costs","volume":"50","author":"Varagapriya","year":"2022","journal-title":"Oper. Res. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2350105","DOI":"10.1142\/S0218126623501050","article-title":"Application of uncertain programming in hardware\/software partitioning: Model and algorithm","volume":"32","author":"Chen","year":"2023","journal-title":"J. Circuit. Syst. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"6375","DOI":"10.1007\/s00500-019-04132-x","article-title":"The risk path selection problem in uncertain network","volume":"24","author":"Li","year":"2020","journal-title":"Soft Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"110949","DOI":"10.1016\/j.asoc.2023.110949","article-title":"Neutrosophic hyperbolic programming strategy for uncertain multi-objective transportation problem","volume":"149","author":"Kumar","year":"2023","journal-title":"Appl. Soft Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2640","DOI":"10.3934\/jimo.2024019","article-title":"Uncertain multi-objective programming model of tourist route considering tourist preference","volume":"20","author":"Huang","year":"2024","journal-title":"J. Ind. Manag. Optim."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/701\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:54:30Z","timestamp":1760108070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/701"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,6]]},"references-count":29,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060701"],"URL":"https:\/\/doi.org\/10.3390\/sym16060701","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,6,6]]}}}