{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T04:29:39Z","timestamp":1773116979255,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,22]],"date-time":"2024-10-22T00:00:00Z","timestamp":1729555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Interval-valued linguistic neutrosophic sets (IVLNSs), Z-numbers, and the trapezium cloud model are powerful tools for expressing uncertainty and randomness. This paper aims to combine these methodologies. First, we review relevant concepts and operators, introducing a novel combination of IVLNSs and Z-numbers, which establishes a new form of expression. Subsequently, we propose the Z-interval-valued linguistic neutrosophic set-trapezium\u2013trapezium cloud (Z-IVLNS-TTC) model, designed to minimize information loss and distortion during quantification. A novel method for calculating the objective weight vector is then developed using multi-objective programming (MOP). Drawing inspiration from the TOPSIS method, we propose a new approach for calculating the distance between Z-IVLNS-TTCs based on the p-norm. Finally, a group decision-making problem is presented to demonstrate the practical application of the proposed method. To validate the effectiveness and feasibility of the method, sensitivity analysis and comparisons with existing approaches are conducted.<\/jats:p>","DOI":"10.3390\/e26110892","type":"journal-article","created":{"date-parts":[[2024,10,22]],"date-time":"2024-10-22T11:31:57Z","timestamp":1729596717000},"page":"892","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Decision-Making Model with Cloud Model, Z-Numbers, and Interval-Valued Linguistic Neutrosophic Sets"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9184-9493","authenticated-orcid":false,"given":"Huakun","family":"Chen","sequence":"first","affiliation":[{"name":"School of Automation, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Shaanxi Province Key Laboratory of Flight Control and Simulation Technology, Xi\u2019an 710072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5700-2638","authenticated-orcid":false,"given":"Jingping","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Automation, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Shaanxi Province Key Laboratory of Flight Control and Simulation Technology, Xi\u2019an 710072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8912-3418","authenticated-orcid":false,"given":"Yongxi","family":"Lyu","sequence":"additional","affiliation":[{"name":"School of Automation, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Shaanxi Province Key Laboratory of Flight Control and Simulation Technology, Xi\u2019an 710072, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3827-3832","authenticated-orcid":false,"given":"Qianlei","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Automation, Northwestern Polytechnical University, Xi\u2019an 710072, China"},{"name":"Shaanxi Province Key Laboratory of Flight Control and Simulation Technology, Xi\u2019an 710072, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"105703","DOI":"10.1016\/j.asoc.2019.105703","article-title":"Divergence-based cross entropy and uncertainty measures of Atanassov\u2019s intuitionistic fuzzy sets with their application in decision making\u2014ScienceDirect","volume":"84","author":"Song","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","article-title":"Interval valued intuitionistic fuzzy sets","volume":"31","author":"Atanassov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"106556","DOI":"10.1016\/j.asoc.2020.106556","article-title":"An Atanassov intuitionistic fuzzy programming method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations","volume":"95","author":"Wan","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2139","DOI":"10.1016\/j.microrel.2006.01.007","article-title":"Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly","volume":"46","author":"Mhs","year":"2006","journal-title":"Microelectron. Reliab."},{"key":"ref_7","first-page":"1571","article-title":"Multi-criteria decision-making method based on aggregation operators for intuitionistic linguistic fuzzy numbers","volume":"25","author":"Wang","year":"2010","journal-title":"Control Decis."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.knosys.2014.06.023","article-title":"A new outranking choice method for group decision making under Atanassov\u2019s interval-valued intuitionistic fuzzy environment","volume":"70","author":"Xu","year":"2014","journal-title":"Knowl.-Based Syst."},{"key":"ref_9","unstructured":"Smarandache, F. (2005). A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability, American Research Press. [5th ed.]."},{"key":"ref_10","unstructured":"Smarandache, F., Wang, H., Zhang, Y.Q., and Sunderraman, R. (2014). Interval Neutrosophic Sets and Logic: Theory and Applications in Computing, Hexis."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Yang, L., and Li, B. (2016). A Multi-Criteria Decision-Making Method Using Power Aggregation Operators for Single-Valued Neutrosophic Sets, Infinite Study.","DOI":"10.14257\/ijdta.2016.9.2.04"},{"key":"ref_12","first-page":"1139","article-title":"TODIM method with multi-valued neutrosophic sets","volume":"30","author":"Wang","year":"2015","journal-title":"Control Decis."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1016\/j.asoc.2018.07.020","article-title":"An aggregation method for solving group multi-criteria decision-making problems with single-valued neutrosophic sets","volume":"71","author":"Sodenkamp","year":"2018","journal-title":"Appl. Soft Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40815-016-0195-8","article-title":"The Aggregation Operators Based on Archimedean t-Conorm and t-Norm for Single-Valued Neutrosophic Numbers and their Application to Decision Making","volume":"18","author":"Liu","year":"2016","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1109\/JAS.2017.7510424","article-title":"Power Aggregation Operators of Simplified Neutrosophic Sets and Their Use in Multi-attribute Group Decision Making","volume":"6","author":"Liu","year":"2017","journal-title":"IEEE\/Caa J. Autom. Sin."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1016\/j.asoc.2014.08.070","article-title":"An outranking approach for multi-criteria decision-making problemswith simplified neutrosophic sets","volume":"25","author":"Chen","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1007\/s00521-016-2575-2","article-title":"Linear assignment method for interval neutrosophic sets","volume":"29","author":"Yang","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2923","DOI":"10.1016\/j.ins.2011.02.022","article-title":"A Note on Z-numbers","volume":"181","author":"Zadeh","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"890","DOI":"10.1109\/TFUZZ.2022.3191359","article-title":"Pythagorean Fuzzy Sets to Solve Z-Numbers in Decision-Making Model","volume":"31","author":"Jia","year":"2023","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4102","DOI":"10.1109\/TFUZZ.2021.3138649","article-title":"A Novel Solution for Z-Numbers Based on Complex Fuzzy Sets and Its Application in Decision-Making System","volume":"30","author":"Jia","year":"2022","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"10636","DOI":"10.1109\/TII.2024.3397392","article-title":"Multitarget Assignment Under Uncertain Information Through Decision Support Systems","volume":"20","author":"Jia","year":"2024","journal-title":"IEEE Trans. Ind. Inform."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.ins.2021.11.016","article-title":"A novel method to research linguistic uncertain Z-numbers","volume":"586","author":"Jia","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"104437","DOI":"10.1016\/j.engappai.2021.104437","article-title":"Polar coordinate system to solve an uncertain linguistic Z-number and its application in multicriteria group decision-making","volume":"105","author":"Jia","year":"2021","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.ins.2015.07.055","article-title":"The general theory of decisions","volume":"327","author":"Aliev","year":"2016","journal-title":"Inf. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1109\/TFUZZ.2016.2612303","article-title":"Approximate Reasoning on a Basis of Z-Number-Valued If\u2013Then Rules","volume":"25","author":"Aliev","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1016\/j.ins.2018.12.012","article-title":"Z-Advanced numbers processes","volume":"480","author":"Allahviranloo","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1109\/TFUZZ.2016.2555999","article-title":"FN-TOPSIS: Fuzzy Networks for Ranking Traded Equities","volume":"25","author":"Yaakob","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1109\/TFUZZ.2015.2453994","article-title":"Extended fuzzy logic: Sets and systems","volume":"24","author":"Sabahi","year":"2016","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.ins.2019.05.090","article-title":"An uncertain Z-number multicriteria group decision-making method with cloud models","volume":"501","author":"Peng","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"(2015). Atanassov\u2019s Interval-Valued Intuitionistic Linguistic Multicriteria Group Decision-Making Method Based on the Trapezium Cloud Model. IEEE Trans. Fuzzy Syst. Publ. IEEE Neural Netw. Counc., 23, 542\u2013554.","DOI":"10.1109\/TFUZZ.2014.2317500"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.ins.2021.02.042","article-title":"A Multicriteria Group Decision-making Method Based on AIVIFSs, Z-numbers, and Trapezium Clouds","volume":"566","author":"Jia","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_32","first-page":"780","article-title":"Improved two-tuple linguistic representation model based on new linguistic evaluation scale","volume":"25","author":"Bao","year":"2010","journal-title":"Control. Decis."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Li, D., Liu, C., and Gan, W. (2009). A New Cognitive Model: Cloud Model, Wiley Subscription Services, Inc.","DOI":"10.1002\/int.20340"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"105610","DOI":"10.1016\/j.engappai.2022.105610","article-title":"A new situation assessment method for aerial targets based on linguistic fuzzy sets and trapezium clouds","volume":"117","author":"Jia","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"103123","DOI":"10.1016\/j.oregeorev.2019.103123","article-title":"A minimization approach to depth and shape determination of mineralized zones from potential field data using the Nelder-Mead simplex algorithm","volume":"114","author":"Abdelrahman","year":"2019","journal-title":"Ore Geol. Rev."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10726-014-9385-7","article-title":"An Uncertain Linguistic Multi-criteria Group Decision-Making Method Based on a Cloud Model","volume":"24","author":"Wang","year":"2015","journal-title":"Group Decis. Negot."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1300","DOI":"10.1007\/s40815-016-0257-y","article-title":"Hesitant Uncertain Linguistic Z-Numbers and Their Application in Multi-criteria Group Decision-Making Problems","volume":"19","author":"Peng","year":"2017","journal-title":"Int. J. Fuzzy Syst."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/892\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:18:03Z","timestamp":1760113083000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/11\/892"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,22]]},"references-count":37,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["e26110892"],"URL":"https:\/\/doi.org\/10.3390\/e26110892","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,22]]}}}