{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,28]],"date-time":"2025-10-28T15:03:27Z","timestamp":1761663807959,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2018,11,1]],"date-time":"2018-11-01T00:00:00Z","timestamp":1541030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51508319"],"award-info":[{"award-number":["51508319"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The subject of this study is to explore the role of cardinality of hesitant fuzzy element (HFE) in distance measures on hesitant fuzzy sets (HFSs). Firstly, three parameters, i.e., credibility factor, conservative factor, and a risk factor are introduced, thereafter, a series of novel distance measures on HFSs are proposed using these three parameters. These newly proposed distance measures handle the relationship between the cardinal number and the element values of hesitant fuzzy set well, and are suitable to combine subjective and objective decision-making information. When using these functions, decision makers with different risk preferences are allowed to give different values for these three parameters. In particular, this study transfers the hesitance degree index to a credibility of the values in HFEs, which is consistent with people\u2019s intuition. Finally, the practicability of the newly proposed distance measures is verified by two examples.<\/jats:p>","DOI":"10.3390\/sym10110557","type":"journal-article","created":{"date-parts":[[2018,11,1]],"date-time":"2018-11-01T11:31:47Z","timestamp":1541071907000},"page":"557","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Novel Parameterized Distance Measures on Hesitant Fuzzy Sets with Credibility Degree and Their Application in Decision-Making"],"prefix":"10.3390","volume":"10","author":[{"given":"Jiaru","family":"Li","sequence":"first","affiliation":[{"name":"College of Transport and Communications, Shanghai Maritime University, Shanghai 201306, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5798-5662","authenticated-orcid":false,"given":"Fangwei","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Transport and Communications, Shanghai Maritime University, Shanghai 201306, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiang","family":"Li","sequence":"additional","affiliation":[{"name":"Faculty of Public Security, Shanghai Police College, Shanghai 200137, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jing","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Macau, Macau 999078, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4467-9671","authenticated-orcid":false,"given":"Janney","family":"Yee","sequence":"additional","affiliation":[{"name":"College of Social Sciences, University of Birmingham, Birmingham 1610681, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuhong","family":"Wang","sequence":"additional","affiliation":[{"name":"Shool of Computing, National University of Singapore, Singapore 118414, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shujun","family":"Xiao","sequence":"additional","affiliation":[{"name":"Logistics Engineering College, Shanghai Maritime University, Shanghai 201306, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,11,1]]},"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":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning (I)","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/0020-0255(75)90046-8","article-title":"The concept of a linguistic variable and its application to approximate reasoning (II)","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/0020-0255(75)90017-1","article-title":"The concept of a linguistic variable and its application to approximate reasoning (III)","volume":"9","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"1","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/0165-0114(89)90205-4","article-title":"Interval valued intuitionistic fuzzy sets","volume":"3","author":"Atanassov","year":"1989","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Torra, V., and Narukawa, Y. (2009, January 20\u201324). On hesitant fuzzy sets and decision. Proceedings of the 2009 IEEE International Conference on Fuzzy Systems, Jeju Island, Korea.","DOI":"10.1109\/FUZZY.2009.5276884"},{"key":"ref_8","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"181","DOI":"10.5391\/IJFIS.2014.14.3.181","article-title":"Group decision making using intuitionistic hesitant fuzzy sets","volume":"3","author":"Beg","year":"2014","journal-title":"Int. J. Fuzzy Logic Intell. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1881","DOI":"10.3233\/JIFS-161811","article-title":"Interval-valued dual hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making","volume":"33","author":"Wei","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/0888-613X(87)90015-6","article-title":"Measures of similarity among fuzzy concepts: A comparative analysis","volume":"1","author":"Zwick","year":"1987","journal-title":"Int. J. Approx. Reason."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/0165-0114(93)90141-4","article-title":"A comparative assessment of measures of similarity of fuzzy values","volume":"56","author":"Pappis","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0165-0114(95)00365-7","article-title":"New similarity measures on fuzzy sets and on elements","volume":"85","author":"Wang","year":"1997","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2128","DOI":"10.1016\/j.ins.2011.01.028","article-title":"Distance and similarity measures for hesitant fuzzy sets","volume":"181","author":"Xu","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1002\/int.20474","article-title":"On distance and correlation measures of hesitant fuzzy information","volume":"26","author":"Xu","year":"2011","journal-title":"Int. J. Intell. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5837","DOI":"10.1016\/j.apm.2012.11.016","article-title":"Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making","volume":"37","author":"Peng","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1002\/int.21654","article-title":"Hesitant fuzzy sets: State of the art and future directions","volume":"29","author":"Torra","year":"2014","journal-title":"Int. J. Intell. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.engappai.2014.12.012","article-title":"New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making","volume":"40","author":"Li","year":"2015","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3903","DOI":"10.3233\/JIFS-169561","article-title":"Novel distance and similarity measures for hesitant fuzzy sets and their applications to multiple attribute decision making","volume":"34","author":"Tang","year":"2018","journal-title":"J. Intell. Fuzzy Sys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1016\/j.asoc.2015.09.035","article-title":"Hesitant fuzzy worth: An innovative ranking methodology for hesitant fuzzy subsets","volume":"38","author":"Alcantud","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.knosys.2012.03.011","article-title":"Hesitant fuzzy prioritized operators and their application to multiple attribute decision making","volume":"31","author":"Wei","year":"2012","journal-title":"Knowl.-Based Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"13051","DOI":"10.1016\/j.eswa.2012.05.056","article-title":"A state-of the-art survey of TOPSIS applications","volume":"17","author":"Behzadian","year":"2012","journal-title":"Expert Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Muscat, J. (2014). Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer International Publishing.","DOI":"10.1007\/978-3-319-06728-5"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/11\/557\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:27:15Z","timestamp":1760196435000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/11\/557"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,1]]},"references-count":23,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2018,11]]}},"alternative-id":["sym10110557"],"URL":"https:\/\/doi.org\/10.3390\/sym10110557","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,11,1]]}}}