{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T18:17:53Z","timestamp":1775585873196,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,3,5]],"date-time":"2025-03-05T00:00:00Z","timestamp":1741132800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2025,3,5]],"date-time":"2025-03-05T00:00:00Z","timestamp":1741132800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"DOI":"10.13039\/501100017596","name":"Natural Science Basic Research Program of Shaanxi Province","doi-asserted-by":"publisher","award":["Grant No. 2023-JC-QN-0046"],"award-info":[{"award-number":["Grant No. 2023-JC-QN-0046"]}],"id":[{"id":"10.13039\/501100017596","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013101","name":"Scientific research plan projects of Shaanxi Education Department","doi-asserted-by":"crossref","award":["Grant No. 22JK0299"],"award-info":[{"award-number":["Grant No. 22JK0299"]}],"id":[{"id":"10.13039\/501100013101","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"crossref","award":["Grant No. LY20A010012"],"award-info":[{"award-number":["Grant No. LY20A010012"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["Grant No.12271319"],"award-info":[{"award-number":["Grant No.12271319"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Comput Intell Syst"],"DOI":"10.1007\/s44196-025-00769-9","type":"journal-article","created":{"date-parts":[[2025,3,5]],"date-time":"2025-03-05T12:15:37Z","timestamp":1741176937000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Hesitant Fuzzy $$\\beta $$-Covering $$({\\mathcal {I}},$$ $${\\mathcal {O}})$$ Rough Set Models and Applications to Multi-attribute Decision-Making"],"prefix":"10.1007","volume":"18","author":[{"given":"Jingyi","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Songtao","family":"Shao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoyan","family":"Mao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaohong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,3,5]]},"reference":[{"issue":"6","key":"769_CR1","doi-asserted-by":"publisher","first-page":"3149","DOI":"10.1007\/s12559-024-10340-x","volume":"16","author":"W Bai","year":"2024","unstructured":"Bai, W., Zhang, C., Zhai, Y., Sangaiah, A.K., Wang, B., Li, W.: Cognitive analysis of medical decision-making: an extended MULTIMOORA-based multigranulation probabilistic model with evidential reasoning. Cogn. Comput. 16(6), 3149\u20133167 (2024)","journal-title":"Cogn. Comput."},{"issue":"1","key":"769_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-024-02962-6","volume":"44","author":"H Bu","year":"2025","unstructured":"Bu, H., Wang, J., Shao, S., Zhang, X.: A novel model of fuzzy rough sets based on grouping functions and its application. Comput. Appl. Math. 44(1), 1\u201331 (2025)","journal-title":"Comput. Appl. Math."},{"issue":"3\u20134","key":"769_CR3","doi-asserted-by":"publisher","first-page":"1488","DOI":"10.1016\/j.na.2009.08.033","volume":"72","author":"H Bustince","year":"2010","unstructured":"Bustince, H., Fernandez, J., Mesiar, R., Montero, J., Orduna, R.: Overlap functions. Nonlinear Anal.: Theory Methods Appl. 72(3\u20134), 1488\u20131499 (2010)","journal-title":"Nonlinear Anal.: Theory Methods Appl."},{"issue":"6","key":"769_CR4","doi-asserted-by":"publisher","first-page":"10005","DOI":"10.3233\/JIFS-223842","volume":"44","author":"C Fu","year":"2023","unstructured":"Fu, C., Qin, K., Yang, L., Hu, Q.: Hesitant fuzzy $$\\beta $$-covering (t, i) rough set models: an application to multi-attribute decision-making. J. Intell. Fuzzy Syst. 44(6), 10005\u201310025 (2023)","journal-title":"J. Intell. Fuzzy Syst."},{"issue":"3","key":"769_CR5","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1007\/s12559-019-09678-4","volume":"12","author":"H Garg","year":"2020","unstructured":"Garg, H., Rani, D.: Generalized geometric aggregation operators based on t-norm operations for complex intuitionistic fuzzy sets and their application to decision-making. Cogn. Comput. 12(3), 679\u2013698 (2020)","journal-title":"Cogn. Comput."},{"key":"769_CR6","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.fss.2014.11.023","volume":"287","author":"D G\u00f3mez","year":"2016","unstructured":"G\u00f3mez, D., Rodriguez, J.T., Montero, J., Bustince, H., Barrenechea, E.: n-Dimensional overlap functions. Fuzzy Sets Syst. 287, 57\u201375 (2016)","journal-title":"Fuzzy Sets Syst."},{"issue":"3","key":"769_CR7","doi-asserted-by":"publisher","first-page":"3173","DOI":"10.3233\/JIFS-213261","volume":"43","author":"N Han","year":"2022","unstructured":"Han, N., Qiao, J.: On (go, o)-fuzzy rough sets derived from overlap and grouping functions. J. Intell. Fuzzy Syst. 43(3), 3173\u20133187 (2022)","journal-title":"J. Intell. Fuzzy Syst."},{"key":"769_CR8","doi-asserted-by":"publisher","DOI":"10.1016\/j.fss.2024.108893","volume":"481","author":"N Han","year":"2024","unstructured":"Han, N., Qiao, J., Li, T., Ding, W.: Multigranulation fuzzy probabilistic rough sets induced by overlap functions and their applications. Fuzzy Sets Syst. 481, 108893 (2024)","journal-title":"Fuzzy Sets Syst."},{"key":"769_CR9","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/j.ins.2018.08.051","volume":"471","author":"F Jia","year":"2019","unstructured":"Jia, F., Liu, P.: A novel three-way decision model under multiple-criteria environment. Inf. Sci. 471, 29\u201351 (2019)","journal-title":"Inf. Sci."},{"key":"769_CR10","doi-asserted-by":"publisher","first-page":"18","DOI":"10.1016\/j.ijar.2022.01.012","volume":"144","author":"H Jiang","year":"2022","unstructured":"Jiang, H., Hu, B.Q.: On (o, g)-fuzzy rough sets based on overlap and grouping functions over complete lattices. Int. J. Approx. Reason. 144, 18\u201350 (2022)","journal-title":"Int. J. Approx. Reason."},{"issue":"2","key":"769_CR11","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1109\/TFUZZ.2014.2310495","volume":"23","author":"D Liang","year":"2014","unstructured":"Liang, D., Liu, D.: A novel risk decision making based on decision-theoretic rough sets under hesitant fuzzy information. IEEE Trans. Fuzzy Syst. 23(2), 237\u2013247 (2014)","journal-title":"IEEE Trans. Fuzzy Syst."},{"issue":"4","key":"769_CR12","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s10700-013-9162-0","volume":"12","author":"H Liao","year":"2013","unstructured":"Liao, H., Xu, Z.: A VIKOR-based method for hesitant fuzzy multi-criteria decision making. Fuzzy Optim. Decis. Mak. 12(4), 373\u2013392 (2013)","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"769_CR13","doi-asserted-by":"publisher","first-page":"2084","DOI":"10.1007\/s40815-017-0432-9","volume":"20","author":"H Liao","year":"2018","unstructured":"Liao, H., Xu, Z., Herrera-Viedma, E., Herrera, F.: Hesitant fuzzy linguistic term set and its application in decision making: a state-of-the-art survey. Int. J. Fuzzy Syst. 20, 2084\u20132110 (2018)","journal-title":"Int. J. Fuzzy Syst."},{"issue":"5","key":"769_CR14","doi-asserted-by":"publisher","first-page":"2144","DOI":"10.1002\/int.22374","volume":"36","author":"M Matzenauer","year":"2021","unstructured":"Matzenauer, M., Reiser, R., Santos, H., Bedregal, B., Bustince, H.: Strategies on admissible total orders over typical hesitant fuzzy implications applied to decision making problems. Int. J. Intell. Syst. 36(5), 2144\u20132182 (2021)","journal-title":"Int. J. Intell. Syst."},{"key":"769_CR15","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1016\/j.ins.2021.02.010","volume":"562","author":"R Paiva","year":"2021","unstructured":"Paiva, R., Santiago, R., Bedregal, B., Palmeira, E.: Lattice-valued overlap and quasi-overlap functions. Inf. Sci. 562, 180\u2013199 (2021)","journal-title":"Inf. Sci."},{"key":"769_CR16","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/BF01001956","volume":"11","author":"Z Pawlak","year":"1982","unstructured":"Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341\u2013356 (1982)","journal-title":"Int. J. Comput. Inf. Sci."},{"key":"769_CR17","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1016\/j.ijar.2021.02.001","volume":"132","author":"J Qiao","year":"2021","unstructured":"Qiao, J.: On (io, o)-fuzzy rough sets based on overlap functions. Int. J. Approx. Reason. 132, 26\u201348 (2021)","journal-title":"Int. J. Approx. Reason."},{"key":"769_CR18","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.fss.2021.03.004","volume":"446","author":"J Qiao","year":"2022","unstructured":"Qiao, J.: Constructions of quasi-overlap functions and their generalized forms on bounded partially ordered sets. Fuzzy Sets Syst. 446, 68\u201392 (2022)","journal-title":"Fuzzy Sets Syst."},{"key":"769_CR19","doi-asserted-by":"crossref","unstructured":"Seikh, M.R., Mandal, U.: Some picture fuzzy aggregation operators based on Frank t-norm and t-conorm: application to MADM process. Informatica 45(3), 447-461 (2021)","DOI":"10.31449\/inf.v45i3.3025"},{"key":"769_CR20","doi-asserted-by":"crossref","unstructured":"Su, Z., Xu, Z., Zhang, S.: Multi-attribute decision-making method based on probabilistic hesitant fuzzy entropy. In: Hesitant Fuzzy and Probabilistic Information Fusion: Theory and Applications, pp. 73\u201398. Springer Nature Singapore, Editor: Li, X (2024)","DOI":"10.1007\/978-981-97-3140-4_4"},{"issue":"6","key":"769_CR21","first-page":"529","volume":"25","author":"V Torra","year":"2010","unstructured":"Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529\u2013539 (2010)","journal-title":"Int. J. Intell. Syst."},{"key":"769_CR22","doi-asserted-by":"crossref","unstructured":"Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: 2009 IEEE International Conference on Fuzzy Systems, pp. 1378\u20131382. IEEE, Piscataway, NJ, USA (2009)","DOI":"10.1109\/FUZZY.2009.5276884"},{"issue":"10","key":"769_CR23","doi-asserted-by":"publisher","first-page":"2445","DOI":"10.1016\/j.camwa.2008.03.054","volume":"56","author":"D-G Wang","year":"2008","unstructured":"Wang, D.-G., Meng, Y.-P., Li, H.-X.: A fuzzy similarity inference method for fuzzy reasoning. Comput. Math. Appl. 56(10), 2445\u20132454 (2008)","journal-title":"Comput. Math. Appl."},{"key":"769_CR24","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/j.ijar.2020.07.006","volume":"125","author":"H Wang","year":"2020","unstructured":"Wang, H.: Constructions of overlap functions on bounded lattices. Int. J. Approx. Reason. 125, 203\u2013217 (2020)","journal-title":"Int. J. Approx. Reason."},{"issue":"6","key":"769_CR25","doi-asserted-by":"publisher","first-page":"411","DOI":"10.3390\/axioms13060411","volume":"13","author":"J Wang","year":"2024","unstructured":"Wang, J., Zhang, X.: Intuitionistic fuzzy granular matrix: novel calculation approaches for intuitionistic fuzzy covering-based rough sets. Axioms 13(6), 411 (2024)","journal-title":"Axioms"},{"issue":"1\u20134","key":"769_CR26","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/S0020-0255(02)00180-9","volume":"144","author":"W-Z Wu","year":"2002","unstructured":"Wu, W.-Z., Zhang, W.-X.: Neighborhood operator systems and approximations. Inf. Sci. 144(1\u20134), 201\u2013217 (2002)","journal-title":"Inf. Sci."},{"issue":"3","key":"769_CR27","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1016\/j.ijar.2010.09.002","volume":"52","author":"M Xia","year":"2011","unstructured":"Xia, M., Xu, Z.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52(3), 395\u2013407 (2011)","journal-title":"Int. J. Approx. Reason."},{"key":"769_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2023.121786","volume":"238","author":"S Xian","year":"2024","unstructured":"Xian, S., Ma, D., Feng, X.: Z hesitant fuzzy linguistic term set and their applications to multi-criteria decision making problems. Expert Syst. Appl. 238, 121786 (2024)","journal-title":"Expert Syst. Appl."},{"key":"769_CR29","doi-asserted-by":"publisher","DOI":"10.1016\/j.eswa.2023.121459","volume":"237","author":"G Xin","year":"2024","unstructured":"Xin, G., Ying, L.: Multi-attribute decision-making based on comprehensive hesitant fuzzy entropy. Expert Syst. Appl. 237, 121459 (2024)","journal-title":"Expert Syst. Appl."},{"key":"769_CR30","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/j.knosys.2013.05.011","volume":"52","author":"Z Xu","year":"2013","unstructured":"Xu, Z., Zhang, X.: Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl. Based Syst. 52, 53\u201364 (2013)","journal-title":"Knowl. Based Syst."},{"key":"769_CR31","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1007\/s00500-013-1127-2","volume":"18","author":"X Yang","year":"2014","unstructured":"Yang, X., Song, X., Qi, Y., Yang, J.: Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput. 18, 1067\u20131077 (2014)","journal-title":"Soft Comput."},{"issue":"1\u20134","key":"769_CR32","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/S0020-0255(98)00012-7","volume":"109","author":"Y Yao","year":"1998","unstructured":"Yao, Y.: Constructive and algebraic methods of the theory of rough sets. Inf. Sci. 109(1\u20134), 21\u201347 (1998)","journal-title":"Inf. Sci."},{"issue":"3","key":"769_CR33","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1515\/dema-1983-0319","volume":"16","author":"W Zakowski","year":"1983","unstructured":"Zakowski, W.: Approximations in the space (u, $$\\pi $$). Demonstratio mathematica 16(3), 761\u2013770 (1983)","journal-title":"Demonstratio mathematica"},{"key":"769_CR34","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1007\/s13042-016-0541-z","volume":"9","author":"C Zhang","year":"2018","unstructured":"Zhang, C., Li, D., Liang, J.: Hesitant fuzzy linguistic rough set over two universes model and its applications. Int. J. Mach. Learn. Cybern. 9, 577\u2013588 (2018)","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"769_CR35","doi-asserted-by":"publisher","first-page":"665","DOI":"10.1016\/j.ins.2019.01.033","volume":"507","author":"C Zhang","year":"2020","unstructured":"Zhang, C., Li, D., Liang, J.: Multi-granularity three-way decisions with adjustable hesitant fuzzy linguistic multigranulation decision-theoretic rough sets over two universes. Inf. Sci. 507, 665\u2013683 (2020)","journal-title":"Inf. Sci."},{"key":"769_CR36","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s13042-017-0753-x","volume":"10","author":"C Zhang","year":"2019","unstructured":"Zhang, C., Li, D., Zhai, Y., Yang, Y.: Multigranulation rough set model in hesitant fuzzy information systems and its application in person-job fit. Int. J. Mach. Learn. Cybern. 10, 717\u2013729 (2019)","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"769_CR37","doi-asserted-by":"publisher","first-page":"3287","DOI":"10.1007\/s00500-015-2008-7","volume":"21","author":"H Zhang","year":"2017","unstructured":"Zhang, H., Shu, L., Liao, S., Xiawu, C.: Dual hesitant fuzzy rough set and its application. Soft Comput. 21, 3287\u20133305 (2017)","journal-title":"Soft Comput."},{"key":"769_CR38","doi-asserted-by":"publisher","first-page":"11357","DOI":"10.1007\/s00500-019-04037-9","volume":"23","author":"H Zhang","year":"2019","unstructured":"Zhang, H., Shu, L., Xiong, L.: On novel hesitant fuzzy rough sets. Soft Comput. 23, 11357\u201311371 (2019)","journal-title":"Soft Comput."},{"issue":"11","key":"769_CR39","doi-asserted-by":"publisher","first-page":"593","DOI":"10.3390\/axioms11110593","volume":"11","author":"X Zhang","year":"2022","unstructured":"Zhang, X., Liang, R., Bustince, H., Bedregal, B., Fernandez, J., Li, M., Ou, Q.: Pseudo overlap functions, fuzzy implications and pseudo grouping functions with applications. Axioms 11(11), 593 (2022)","journal-title":"Axioms"},{"key":"769_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2024.120451","volume":"666","author":"X Zhang","year":"2024","unstructured":"Zhang, X., Ou, Q., Wang, J.: Variable precision fuzzy rough sets based on overlap functions with application to tumor classification. Inf. Sci. 666, 120451 (2024)","journal-title":"Inf. Sci."},{"issue":"1","key":"769_CR41","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1002\/int.21754","volume":"31","author":"N Zhao","year":"2016","unstructured":"Zhao, N., Xu, Z., Ren, Z.: On typical hesitant fuzzy prioritized \u201cor\u2019\u2019 operator in multi-attribute decision making. Int. J. Intell. Syst. 31(1), 73\u2013100 (2016)","journal-title":"Int. J. Intell. Syst."},{"issue":"1","key":"769_CR42","doi-asserted-by":"publisher","first-page":"2387","DOI":"10.3233\/JIFS-190959","volume":"41","author":"J-J Zhou","year":"2021","unstructured":"Zhou, J.-J., Li, X.-Y.: Hesitant fuzzy $$\\beta $$ covering rough sets and applications in multi-attribute decision making. J. Intell. Fuzzy Syst. 41(1), 2387\u20132402 (2021)","journal-title":"J. Intell. Fuzzy Syst."},{"issue":"1","key":"769_CR43","doi-asserted-by":"publisher","DOI":"10.1155\/2012\/879629","volume":"2012","author":"B Zhu","year":"2012","unstructured":"Zhu, B., Xu, Z., Xia, M.: Dual hesitant fuzzy sets. J. Appl. Math. 2012(1), 879629 (2012)","journal-title":"J. Appl. Math."}],"container-title":["International Journal of Computational Intelligence Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44196-025-00769-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s44196-025-00769-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44196-025-00769-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,5]],"date-time":"2025-03-05T12:15:42Z","timestamp":1741176942000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s44196-025-00769-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,5]]},"references-count":43,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["769"],"URL":"https:\/\/doi.org\/10.1007\/s44196-025-00769-9","relation":{},"ISSN":["1875-6883"],"issn-type":[{"value":"1875-6883","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,5]]},"assertion":[{"value":"11 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 March 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}],"article-number":"49"}}