{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T17:46:35Z","timestamp":1769190395667,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,1,29]],"date-time":"2019-01-29T00:00:00Z","timestamp":1548720000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key R&amp;D project of Shandong Province,China:","award":["2017XCG0605"],"award-info":[{"award-number":["2017XCG0605"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Schweizer\u2013Sklar (SS) operation can make information aggregation more flexible, and the Muirhead mean (MM) operator can take into account the correlation between inputs by a variable parameter. Because traditional MM is only available for real numbers and single-valued neutrosophic set (SVNS) can better express incomplete and uncertain information in decision systems, in this paper, we applied MM operators to single-valued neutrosophic sets (SVNSs) and presented two new MM aggregation operators with the SS operation, i.e., a single-valued neutrosophic SS Muirhead mean (SVNSSMM) operator and a weighted single-valued neutrosophic SS MM (WSVNSSMM) operator. We listed some properties of them and some particular cases about various parameter values. We also proposed the multi-criteria decision-making method based on the WSVNSSMM operator in SVNS. At last, we illustrated the feasibility of this method using a numerical example of company investment.<\/jats:p>","DOI":"10.3390\/sym11020152","type":"journal-article","created":{"date-parts":[[2019,1,29]],"date-time":"2019-01-29T11:27:52Z","timestamp":1548761272000},"page":"152","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Multi-Criteria Decision-Making Method Based on Single-Valued Neutrosophic Schweizer\u2013Sklar Muirhead Mean Aggregation Operators"],"prefix":"10.3390","volume":"11","author":[{"given":"Huanying","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Information, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}]},{"given":"Fei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Information, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}]},{"given":"Yushui","family":"Geng","sequence":"additional","affiliation":[{"name":"School of Information, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"},{"name":"Graduate School, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,29]]},"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":"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_4","doi-asserted-by":"crossref","unstructured":"Atanassov, K.T. (1989). More on Intuitionistic Fuzzy Sets, Elsevier.","DOI":"10.1016\/0165-0114(89)90215-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0165-0114(94)90331-X","article-title":"Operators over interval valued intuitionistic fuzzy sets","volume":"64","author":"Atanassov","year":"1994","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0165-0114(86)90077-1","article-title":"Interval valued fuzzy sets based on normal forms","volume":"20","author":"Turksen","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","unstructured":"Smarandache, F. (2019, January 02). Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis. Available online: https:\/\/www.google.com.tw\/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=2ahUKEwjF8vj495HgAhUJIIgKHTsxBFEQFjACegQIDxAB&url=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F266366834_Neutrosophy_Neutrosophic_probability_set_and_logic_Analytic_synthesis_and_synthetic_analysis&usg=AOvVaw38P0cCji8UDaORB1bDykIZ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2459","DOI":"10.3233\/IFS-130916","article-title":"A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets","volume":"26","author":"Ye","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","unstructured":"Wang, H., Smarandache, F., Sunderraman, R., and Zhang, Y.-Q. (2005). Interval Neutrosophic Sets and Logic: Theory and Applications in Computing: Theory and Applications in Computing, Infinite Study."},{"key":"ref_10","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 problems with simplified neutrosophic sets","volume":"25","author":"Peng","year":"2014","journal-title":"Appl. Soft Comput."},{"key":"ref_11","unstructured":"Haibin, W., Smarandache, F., Zhang, Y., and Sunderraman, R. (2010). Single Valued Neutrosophic Sets, Infinite Study."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1073\/pnas.28.12.535","article-title":"Statistical Metrics","volume":"28","author":"Menger","year":"1942","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_13","unstructured":"Schweizer, B., and Sklar, A. (2011). Probabilistic Metric Spaces, Courier Corporation."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Liu, P., and Wang, P. (2018). Some interval-valued intuitionistic fuzzy Schweizer\u2013Sklar power aggregation operators and their application to supplier selection. Int. J. Syst. Sci., 1\u201324.","DOI":"10.1080\/00207721.2018.1442510"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3080","DOI":"10.1016\/j.fss.2009.05.002","article-title":"Generalized arithmetic operators and their relationship to t-norms in interval-valued fuzzy set theory","volume":"160","author":"Deschrijver","year":"2009","journal-title":"Fuzzy Sets Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s11432-006-0175-y","article-title":"A fuzzy logic system based on Schweizer-Sklar t-norm","volume":"49","author":"Zhang","year":"2006","journal-title":"Sci. China Ser. F Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1016\/j.amc.2013.12.133","article-title":"Some intuitionistic uncertain linguistic Heronian mean operators and their application to group decision making","volume":"230","author":"Liu","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2001","DOI":"10.1007\/s00521-014-1688-8","article-title":"Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean","volume":"25","author":"Liu","year":"2014","journal-title":"Neural Comput. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.knosys.2013.12.009","article-title":"2-Dimension uncertain linguistic power generalized weighted aggregation operator and its application in multiple attribute group decision making","volume":"57","author":"Liu","year":"2014","journal-title":"Knowl. Based Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"849","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_21","doi-asserted-by":"crossref","first-page":"453","DOI":"10.3846\/1392-8619.2009.15.453-463","article-title":"Multi-attribute decision-making method research based on interval vague set and TOPSIS method","volume":"15","author":"Liu","year":"2009","journal-title":"Technol. Econ. Dev. Econ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1080\/00207540903490171","article-title":"Research on the supplier selection of a supply chain based on entropy weight and improved ELECTRE-III method","volume":"49","author":"Liu","year":"2011","journal-title":"Int. J. Prod. Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1109\/3468.983429","article-title":"The power average operator","volume":"31","author":"Yager","year":"2001","journal-title":"IEEE Trans. Syst. Man Cybern. Part A Syst. Hum."},{"key":"ref_24","first-page":"267","article-title":"Sulle medie multiple di potenze","volume":"5","author":"Bonferroni","year":"1950","journal-title":"Bollettino dell\u2019Unione Matematica Italiana"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1007\/s00521-015-2122-6","article-title":"Some neutrosophic uncertain linguistic number Heronian mean operators and their application to multi-attribute group decision making","volume":"28","author":"Liu","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1109\/TSMCB.2010.2072918","article-title":"Intuitionistic fuzzy Bonferroni means","volume":"41","author":"Xu","year":"2011","journal-title":"IEEE Trans. Syst. Man Cybern. B Cybern."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1007\/s12559-017-9453-9","article-title":"Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making","volume":"9","author":"Liu","year":"2017","journal-title":"Cogn. Comput."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1181","DOI":"10.1142\/S0219622016500346","article-title":"Multi-valued neutrosophic number Bonferroni mean operators with their applications in multiple attribute group decision making","volume":"15","author":"Liu","year":"2016","journal-title":"Int. J. Inf. Technol. Decis. Mak."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2514","DOI":"10.1109\/TCYB.2016.2634599","article-title":"Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers","volume":"47","author":"Liu","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.3233\/IFS-131105","article-title":"Multi-attribute decision-making approach based on intuitionistic trapezoidal fuzzy generalized heronian OWA operator","volume":"27","author":"Chen","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/j.cie.2017.04.033","article-title":"Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators","volume":"108","author":"Liu","year":"2017","journal-title":"Comput. Ind. Eng."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"4158","DOI":"10.5897\/AJBM11.2522","article-title":"Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making","volume":"6","author":"Yu","year":"2012","journal-title":"Afr. J. Bus. Manag."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2177","DOI":"10.3233\/IFS-141182","article-title":"An approach to intuitionistic fuzzy multiple attribute decision making based on Maclaurin symmetric mean operators","volume":"27","author":"Qin","year":"2014","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1017\/S001309150003460X","article-title":"Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters","volume":"21","author":"Muirhead","year":"1902","journal-title":"Proc. Edinburgh Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Liu, P., Rong, L., Chu, Y., and Li, Y. (2014). Intuitionistic linguistic weighted Bonferroni mean operator and its application to multiple attribute decision making. Sci. World J., 2014.","DOI":"10.1155\/2014\/545049"},{"key":"ref_36","first-page":"59","article-title":"A second letter to Martin Folkes, Esq.; concerning the roots of equations, with the demonstartion of other rules in algebra","volume":"36","author":"Maclaurin","year":"1730","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/s00521-015-2048-z","article-title":"Multiple attribute decision-making method based on some normal neutrosophic Bonferroni mean operators","volume":"28","author":"Liu","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1080\/03081079.2012.761609","article-title":"Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment","volume":"42","author":"Ye","year":"2013","journal-title":"Int. J. Gener. Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/152\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:29:34Z","timestamp":1760185774000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,29]]},"references-count":38,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020152"],"URL":"https:\/\/doi.org\/10.3390\/sym11020152","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1,29]]}}}