{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,26]],"date-time":"2025-12-26T22:26:59Z","timestamp":1766788019362,"version":"build-2065373602"},"reference-count":39,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"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>In this paper, we present four categories of covering-based intuitionistic hesitant fuzzy rough set (CIHFRS) models using intuitionistic hesitant fuzzy \u03b2-neighborhoods (IHF \u03b2-neighborhoods) and intuitionistic hesitant fuzzy complementary \u03b2-neighborhoods (IHFC \u03b2-neighborhoods. Through theoretical analysis of covering-based IHFRS models, we propose the intuitionistic hesitant fuzzy TOPSIS (IHF-TOPSIS) technique for order of preference by similarity to an ideal solution, addressing multicriteria decision-making (MCDM) challenges concerning the assessment of IHF data. A compelling example aptly showcases the suggested approach. Furthermore, we address MCDM problems regarding the assessment of IHF information based on CIHFRS models. Through comparison and analysis, it is evident that addressing MCDM problems by assessing IHF data using CIHFRS models proves more effective than utilizing intuitionistic fuzzy data with CIFRS models or hesitant fuzzy information with CHFRS models. IHFS emerges as a unique and superior tool for addressing real-world challenges. Additionally, covering-based rough sets (CRSs) have been successfully applied to decision problems due to their robust capability in handling unclear data. In this study, by combining CRSs with IHFS, four classes of CIFRS versions are established using IHF \u03b2-neighborhoods and IHFC \u03b2-neighborhoods. A corresponding approximation axiomatic system is developed for each. The roughness and precision degrees of CBIHFRS models are specifically talked about. The relationship among these four types of IHFRS versions and existing related versions is presented based on theoretical investigations. A method for MCDM problems through IHF information, namely, IHF-TOPSIS, is introduced to further demonstrate its effectiveness and applicability. By conducting a comparative study, the effectiveness of the suggested approach is evaluated.<\/jats:p>","DOI":"10.3390\/sym16060693","type":"journal-article","created":{"date-parts":[[2024,6,5]],"date-time":"2024-06-05T03:49:28Z","timestamp":1717559368000},"page":"693","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Covering-Based Intuitionistic Hesitant Fuzzy Rough Set Models and Their Application to Decision-Making Problems"],"prefix":"10.3390","volume":"16","author":[{"given":"Muhammad Kamraz","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8217-3308","authenticated-orcid":false,"family":"Kamran","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Islamia College Peshawar, Peshawar 25120, Khyber Pakhtunkhwa, Pakistan"}]},{"given":"Muhammad","family":"Sajjad Ali Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khushal Khan Khattak University, Karak 27200, Khyber Pakhtunkhwa, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9578-2073","authenticated-orcid":false,"given":"Ahmad","family":"Aloqaily","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7986-886X","authenticated-orcid":false,"given":"Nabil","family":"Mlaiki","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,4]]},"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":"102678","DOI":"10.1016\/j.mex.2024.102678","article-title":"Some distance measures for pythagorean cubic fuzzy sets: Application selection in optimal treatment for depression and anxiety","volume":"12","author":"Rahim","year":"2024","journal-title":"MethodsX"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"60026","DOI":"10.1109\/ACCESS.2021.3073206","article-title":"A development of complex multi-fuzzy hypersoft set with application in MCDM based on entropy and similarity measure","volume":"9","author":"Saeed","year":"2021","journal-title":"IEEE Access"},{"key":"ref_4","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_5","unstructured":"Atanassov, K., and Gargov, G. (1990). Intuitionistic Fuzzy Logic, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0165-0114(93)90166-F","article-title":"Intuitionistic fuzzy prolog","volume":"53","author":"Atanassov","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","first-page":"15","article-title":"Intuitionistic fuzzy sets in group decision making","volume":"2","author":"Szmidt","year":"1996","journal-title":"Notes IFS"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0165-0114(98)00235-8","article-title":"An application of intuitionistic fuzzy sets in medical diagnosis","volume":"117","author":"De","year":"2001","journal-title":"Fuzzy Sets Syst."},{"key":"ref_9","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intel. Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s10726-011-9261-7","article-title":"Some hesitant fuzzy aggregation operators with their application in group decision making","volume":"22","author":"Xia","year":"2013","journal-title":"Group Decis. Negot."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.knosys.2013.05.011","article-title":"Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information","volume":"52","author":"Xu","year":"2013","journal-title":"Knowl.-Based Syst."},{"key":"ref_12","first-page":"4506","article-title":"Multi-criteria decision making based on Choquet integral under hesitant fuzzy environment","volume":"7","author":"Yu","year":"2011","journal-title":"J. Comput. Inf. Syst."},{"key":"ref_13","first-page":"3051","article-title":"Generalized power average operator of hesitant fuzzy numbers and its application in multiple attribute decision making","volume":"9","author":"Liu","year":"2013","journal-title":"J. Comput. Inf. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.ins.2013.01.002","article-title":"Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making","volume":"234","author":"Zhang","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1080\/00207721.2014.993744","article-title":"The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making","volume":"46","author":"Peng","year":"2015","journal-title":"Int. Syst. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF01001956","article-title":"Rough sets","volume":"11","author":"Pawlak","year":"1982","journal-title":"Int. J. Comput. Inf. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Pawlak, Z. (1991). Rough Sets: Theoretical Aspects Data Analysis, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-3534-4"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1080\/03081079008935107","article-title":"Rough fuzzy sets and fuzzy rough sets","volume":"17","author":"Dubois","year":"1990","journal-title":"Int. J. Gen. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Som, T., Castillo, O., Tiwari, A.K., and Shreevastava, S. (2023). Fuzzy, Rough and Intuitionistic Fuzzy Set Approaches for Data Handling, Springer.","DOI":"10.1007\/978-981-19-8566-9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"892","DOI":"10.1016\/j.ins.2007.09.006","article-title":"Rough fuzzy approximation on two 967 universes of discourse","volume":"178","author":"Li","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1016\/j.ijar.2012.05.004","article-title":"NMGRS: Neighborhood-based multigranulation rough sets","volume":"53","author":"Lin","year":"2012","journal-title":"Int. J. Approx. Reason."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/s10462-018-9649-8","article-title":"Two types of coverings based multigranulation rough fuzzy sets and applications to decision making","volume":"53","author":"Zhan","year":"2020","journal-title":"Artif. Intell. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1404","DOI":"10.1016\/j.ijar.2014.01.002","article-title":"On multi-granulation covering rough sets","volume":"55","author":"Liu","year":"2014","journal-title":"Int. J. Approx. Reason."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"10199","DOI":"10.1109\/ACCESS.2021.3049605","article-title":"EDAS method for multi-criteria group decision making based on intuitionistic fuzzy rough aggregation operators","volume":"9","author":"Chinram","year":"2021","journal-title":"IEEE Access"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Hwang, C.L., and Yoon, K.S. (1981). Multiple Attibute Decision Methods and Applications, Springer.","DOI":"10.1007\/978-3-642-48318-9_4"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.ins.2013.03.046","article-title":"Multigranulation rough sets: From partition to covering","volume":"241","author":"Lin","year":"2013","journal-title":"Inf. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.cie.2019.01.004","article-title":"Fuzzy \u03b2-covering based (I,T)-fuzzy rough set models and applications to multi-attribute decision-making","volume":"128","author":"Zhang","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1080\/00207720500382365","article-title":"IF interpretations of multi-criteria multi-person and multi-measurement tool decision-making","volume":"36","author":"Atanassov","year":"2015","journal-title":"Int. J. Syst. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1007\/s10726-009-9164-z","article-title":"A deviation-based approach to IF multiple attribute group decision-making","volume":"19","author":"Xu","year":"2010","journal-title":"Group Decis. Negot."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/j.knosys.2016.06.006","article-title":"An intuitionistic fuzzy graded covering rough set","volume":"107","author":"Huang","year":"2016","journal-title":"Knowl.-Based Syst."},{"key":"ref_31","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, Republic of Korea.","DOI":"10.1109\/FUZZY.2009.5276884"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1016\/j.ijar.2010.09.002","article-title":"Hesitant fuzzy information aggregation in decision making","volume":"52","author":"Xia","year":"2011","journal-title":"Int. J. Approx."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1859","DOI":"10.1007\/s12559-021-09956-0","article-title":"Three-way decision models based on multi-granulation rough intuitionistic hesitant fuzzy sets","volume":"14","author":"Xue","year":"2022","journal-title":"Cogn. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2063","DOI":"10.1016\/j.patrec.2005.03.018","article-title":"Distance measure between intuitionistic fuzzy sets","volume":"26","author":"Wang","year":"2005","journal-title":"Pattern Recognit. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.ijar.2007.08.008","article-title":"Dynamic intuitionistic fuzzy multi-atribute decision-making","volume":"48","author":"Xu","year":"2008","journal-title":"Int. J. Approx. Reason."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/S0165-0114(98)00244-9","article-title":"Distances between intuitionistic fuzzy sets","volume":"114","author":"Szmidt","year":"2000","journal-title":"Fuzzy Sets Syst."},{"key":"ref_37","first-page":"31","article-title":"A survey on multi criteria decision making methods and its applications","volume":"1","author":"Aruldoss","year":"2013","journal-title":"Am. J. Inf. Syst."},{"key":"ref_38","first-page":"56","article-title":"An analysis of multi-criteria decision making methods","volume":"10","author":"Velasquez","year":"2013","journal-title":"Int. J. Oper."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2387","DOI":"10.3233\/JIFS-190959","article-title":"Hesitant fuzzy \u03b2 covering rough sets and applications in multi-attribute decision making","volume":"41","author":"Zhou","year":"2021","journal-title":"J. Intell. Fuzzy Syst."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/693\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:53:47Z","timestamp":1760108027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/6\/693"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,4]]},"references-count":39,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["sym16060693"],"URL":"https:\/\/doi.org\/10.3390\/sym16060693","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,6,4]]}}}