{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,31]],"date-time":"2026-01-31T06:41:38Z","timestamp":1769841698984,"version":"3.49.0"},"reference-count":41,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T00:00:00Z","timestamp":1583107200000},"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>We lay the theoretical foundations of a novel model, termed picture hesitant fuzzy rough sets, based on picture hesitant fuzzy relations. We also combine this notion with the ideas of multi-granulation rough sets. As a consequence, a new multi-granulation rough set model on two universes, termed a multi-granulation picture hesitant fuzzy rough set, is developed. When the universes coincide or play a symmetric role, the concept assumes the standard format. In this context, we put forward two new classes of multi-granulation picture hesitant fuzzy rough sets, namely, the optimistic and pessimistic multi-granulation picture hesitant fuzzy rough sets. Further, we also investigate the relationships among these two concepts and picture hesitant fuzzy rough sets.<\/jats:p>","DOI":"10.3390\/sym12030362","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T07:50:53Z","timestamp":1583135453000},"page":"362","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Multi-Granulation Picture Hesitant Fuzzy Rough Sets"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8895-7568","authenticated-orcid":false,"given":"Bibin","family":"Mathew","sequence":"first","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Calicut, Calicut 673 601, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6333-2884","authenticated-orcid":false,"given":"Sunil Jacob","family":"John","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Institute of Technology Calicut, Calicut 673 601, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4533-9281","authenticated-orcid":false,"given":"Jos\u00e9 Carlos R.","family":"Alcantud","sequence":"additional","affiliation":[{"name":"BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, 37007 Salamanca, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,2]]},"reference":[{"key":"ref_1","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_2","first-page":"7","article-title":"Rough set theory and its application","volume":"3","author":"Pawlak","year":"2002","journal-title":"J. Telecommun. Inf. Technol."},{"key":"ref_3","first-page":"33","article-title":"Hesitant fuzzy rough sets through hesitant fuzzy relations","volume":"8","author":"Deepak","year":"2014","journal-title":"Ann. Fuzzy Math. Inform."},{"key":"ref_4","first-page":"1","article-title":"Intuitionistic fuzzy rough sets","volume":"8","author":"Jena","year":"2002","journal-title":"Notes Intuit. Fuzzy Sets"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1007\/s00500-013-1127-2","article-title":"Constructive and axiomatic approaches to hesitant fuzzy rough set","volume":"18","author":"Yang","year":"2014","journal-title":"Soft Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s00500-014-1490-7","article-title":"On interval-valued hesitant fuzzy rough approximation operators","volume":"20","author":"Zhang","year":"2016","journal-title":"Soft Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/S0020-0255(02)00379-1","article-title":"Generalized fuzzy rough sets","volume":"151","author":"Wu","year":"2003","journal-title":"Inf. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Alcantud, J.C.R., Feng, F., and Yager, R.R. (2019). An N-soft set approach to rough sets. IEEE Trans. Fuzzy Syst.","DOI":"10.1109\/TFUZZ.2019.2946526"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gal, T., Stewart, T.J., and Hanne, T. (1999). The use of rough sets and fuzzy sets in MCDM. Multicriteria Decision Making: Advances in MCDM Models, Algorithms, Theory, and Applications, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4615-5025-9"},{"key":"ref_10","first-page":"167","article-title":"On rough sets under generalized equivalence relations","volume":"5","author":"Inuiguchi","year":"2001","journal-title":"Bull. Int. Rough Set Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/S0020-0255(97)10046-9","article-title":"Extensions and intensions in the rough set theory","volume":"107","author":"Bonikowski","year":"1998","journal-title":"Inf. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Inuiguchi, M., and Tanino, T. (2002). Generalized rough sets and rule extraction. Rough Sets and Current Trends in Computing, Springer.","DOI":"10.1007\/3-540-45813-1_13"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0165-0114(97)00077-8","article-title":"Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic","volume":"19","author":"Zadeh","year":"1997","journal-title":"Fuzzy Sets Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1016\/j.ins.2014.12.010","article-title":"Concept learning via granular computing: A cognitive viewpoint","volume":"298","author":"Li","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_15","unstructured":"Pedrycz, W. (2016). Granular Computing, Analysis and Design of Intelligent Systems, CRC Press Taylor & Francis Group."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1109\/3477.990878","article-title":"Granular computing: A granular signature of data","volume":"32","author":"Pedrycz","year":"2002","journal-title":"IEEE Trans. Syst. Man Cybern. Part B (Cybern.)"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"582","DOI":"10.1109\/TSMCB.2011.2170067","article-title":"An optimization of allocation of information granularity in the interpretation of data structures: Toward granular fuzzy clustering","volume":"42","author":"Pedrycz","year":"2012","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"366","DOI":"10.1109\/TCYB.2014.2361772","article-title":"Granular computing approach to two way learning based on formal concept analysis in fuzzy datasets","volume":"46","author":"Xu","year":"2014","journal-title":"IEEE Trans. Cybern."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1803","DOI":"10.1007\/s00500-015-1882-3","article-title":"Hesitant fuzzy rough set over two universes and its application in decision making","volume":"21","author":"Zhang","year":"2017","journal-title":"Soft Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1080\/0308107042000193561","article-title":"Rough set models on two universes","volume":"33","author":"Pei","year":"2004","journal-title":"Int. J. Gen. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/s00500-010-0562-6","article-title":"Variable precision rough set model over two universes and its properties","volume":"15","author":"Shen","year":"2011","journal-title":"Soft Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/j.ins.2009.11.023","article-title":"MGRS: A multigranulation rough set","volume":"180","author":"Qian","year":"2010","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.knosys.2012.05.019","article-title":"On the structure of the multigranulation rough set model","volume":"36","author":"She","year":"2012","journal-title":"Knowl.-Based Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1007\/s11390-012-1294-0","article-title":"Hierarchical structures on multigranulation spaces","volume":"27","author":"Yang","year":"2012","journal-title":"J. Comput. Sci. Technol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s13042-011-0054-8","article-title":"On multigranulation rough sets in incomplete information system","volume":"3","author":"Yang","year":"2012","journal-title":"Int. J. Mach. Learn. Cybern."},{"key":"ref_26","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. Approxim. Reason."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s41066-016-0032-3","article-title":"Granular computing: From granularity optimization to multi-granularity joint problem solving","volume":"2","author":"Wang","year":"2017","journal-title":"Granul. Comput."},{"key":"ref_28","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_29","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.ins.2019.12.008","article-title":"Aggregation of infinite chains of intuitionistic fuzzy sets and their application to choices with temporal intuitionistic fuzzy information","volume":"514","author":"Alcantud","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_30","first-page":"4","article-title":"Picture fuzzy sets","volume":"30","author":"Cuong","year":"2014","journal-title":"J. Comput. Sci. Cybemetics"},{"key":"ref_31","unstructured":"Cuong, B. (2013, January 15\u201318). Picture fuzzy sets-first results Part 1. Proceedings of the Third World Congress on Information and Communication WICT\u20192013, Hanoi, Vietnam."},{"key":"ref_32","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"Int. J. Intell. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Wang, R., and Li, Y. (2018). Picture hesitant fuzzy set and its application to multiple criteria decision-making. Symmetry, 10.","DOI":"10.3390\/sym10070295"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.ins.2019.04.054","article-title":"Covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decision-making","volume":"494","author":"Zhang","year":"2019","journal-title":"Inf. Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1016\/j.ijar.2012.02.004","article-title":"An efficient rough feature selection algorithm with a multi-granulation view","volume":"53","author":"Liang","year":"2012","journal-title":"Int. J. Approx. Reason."},{"key":"ref_36","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. Reason."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/j.ins.2003.08.005","article-title":"Constructive and axiomatic approaches of fuzzy approximation operators","volume":"159","author":"Wu","year":"2004","journal-title":"Inf. Sci."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0165-0114(92)90114-J","article-title":"Fuzzy rough sets","volume":"45","author":"Nanda","year":"1992","journal-title":"Fuzzy Sets Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0165-0114(01)00032-X","article-title":"A comparative study of fuzzy rough sets","volume":"126","author":"Radzikowska","year":"2002","journal-title":"Fuzzy Sets Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.fss.2012.06.001","article-title":"Fuzzy rough sets, fuzzy preorders and fuzzy topologies","volume":"210","author":"Tiwari","year":"2013","journal-title":"Fuzzy Sets Syst."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"292710","DOI":"10.1155\/2015\/292710","article-title":"A dual hesitant fuzzy multigranulation rough set over two-universe model for medical diagnoses","volume":"2015","author":"Zhang","year":"2015","journal-title":"Comput. Math. Methods Med."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/362\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:03:13Z","timestamp":1760173393000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/3\/362"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,2]]},"references-count":41,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["sym12030362"],"URL":"https:\/\/doi.org\/10.3390\/sym12030362","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3,2]]}}}