{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T05:28:25Z","timestamp":1767677305788,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,3]],"date-time":"2019-04-03T00:00:00Z","timestamp":1554249600000},"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>An m-polar fuzzy model plays a vital role in modeling of real-world problems that involve multi-attribute, multi-polar information and uncertainty. The m-polar fuzzy models give increasing precision and flexibility to the system as compared to the fuzzy and bipolar fuzzy models. An m-polar fuzzy set assigns the membership degree to an object belonging to      [ 0 , 1 ]  m     describing the m distinct attributes of that element. Granular computing deals with representing and processing information in the form of information granules. These information granules are collections of elements combined together due to their similarity and functional\/physical adjacency. In this paper, we illustrate the formation of granular structures using m-polar fuzzy hypergraphs and level hypergraphs. Further, we define m-polar fuzzy hierarchical quotient space structures. The mappings between the m-polar fuzzy hypergraphs depict the relationships among granules occurring at different levels. The consequences reveal that the representation of the partition of a universal set is more efficient through m-polar fuzzy hypergraphs as compared to crisp hypergraphs. We also present some examples and a real-world problem to signify the validity of our proposed model.<\/jats:p>","DOI":"10.3390\/sym11040483","type":"journal-article","created":{"date-parts":[[2019,4,4]],"date-time":"2019-04-04T03:13:42Z","timestamp":1554347622000},"page":"483","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["An m-Polar Fuzzy Hypergraph Model of Granular Computing"],"prefix":"10.3390","volume":"11","author":[{"given":"Anam","family":"Luqman","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of the Punjab, New Campus, P.O. Box 54590, Lahore, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7217-7962","authenticated-orcid":false,"given":"Muhammad","family":"Akram","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the Punjab, New Campus, P.O. Box 54590, Lahore, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5047-9908","authenticated-orcid":false,"given":"Ali N.A.","family":"Koam","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Jazan University, New Campus, P.O. Box 2097, Jazan, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,3]]},"reference":[{"key":"ref_1","unstructured":"Lin, T.Y. (1997, January 8\u201312). Granular computing: From rough sets and neighborhood systems to information granulation and computing with words. Proceedings of the European Congress on Intelligent Techniques and Soft Computing, Aachen, Germany."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhang, L., and Zhang, B. (2014). Hierarchy and Multi-Granular Computing, Quotient Space Based Problem Solving, Tsinghua University Press.","DOI":"10.1016\/B978-0-12-410387-0.00002-0"},{"key":"ref_3","unstructured":"Berge, C. (1973). Graphs and Hypergraphs, North-Holland Publishing Company."},{"key":"ref_4","unstructured":"Liu, Q., Jin, W.B., Wu, S.Y., and Zhou, Y.H. (2005, January 25\u201327). Clustering research using dynamic modeling based on granular computing. Proceedings of the IEEE International Conference on Granular Computing, Beijing, China."},{"key":"ref_5","unstructured":"Wong, S.K.M., and Wu, D. (2002, January 25\u201328). Automated mining of granular database scheme. Proceedings of the IEEE International Conference on Fuzzy Systems, St. Louis, MO, USA."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Chen, G., Zhong, N., and Yao, Y. (2008, January 26\u201328). A hypergraph model of granular computing. Proceedings of the IEEE International Conference on Granular Computing, Hangzhou, China.","DOI":"10.1109\/GRC.2008.4664749"},{"key":"ref_7","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_8","unstructured":"Zhang, W.-R. (1994, January 18\u201321). Bipolar fuzzy sets and relations: A computational framework forcognitive modeling and multiagent decision analysis. Proceedings of the First International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference, the Industrial Fuzzy Control and Intellige, San Antonio, TX, USA."},{"key":"ref_9","first-page":"416530","article-title":"m-Polar fuzzy sets: An extension of bipolar fuzzy sets","volume":"2014","author":"Chen","year":"2014","journal-title":"Sci. World J."},{"key":"ref_10","unstructured":"Rosenfeld, A. (1975). Fuzzy graphs. Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/978-3-030-03751-2_1","article-title":"m-Polar fuzzy graphs","volume":"Volume 371","author":"Akram","year":"2019","journal-title":"Studies in Fuzziness and Soft Computing"},{"key":"ref_12","unstructured":"Kaufmann, A. (1977). Introduction a la Thiorie des Sous-Ensemble Flous, Masson."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1665","DOI":"10.3233\/JIFS-16468","article-title":"On the connection of fuzzy hypergraph with fuzzy information system","volume":"44","author":"Gong","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/j.ins.2017.11.024","article-title":"An application of fuzzy hypergraphs and hypergraphs in granular computing","volume":"429","author":"Wang","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2747","DOI":"10.3233\/JIFS-16859","article-title":"Novel applications of m-polar fuzzy hypergraphs","volume":"32","author":"Akram","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"351","DOI":"10.3233\/JIFS-161668","article-title":"Transversals of m-polar fuzzy hypergraphs with applications","volume":"33","author":"Akram","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Akram, M., and Shahzadi, G. (2018). Hypergraphs in m-polar fuzzy environment. Mathematics, 6.","DOI":"10.3390\/math6020028"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s12597-017-0306-9","article-title":"Intuitionistic single-valued neutrosophic hypergraphs","volume":"54","author":"Akram","year":"2017","journal-title":"Opsearch"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1699","DOI":"10.3233\/JIFS-17228","article-title":"Bipolar neutrosophic hypergraphs with applications","volume":"33","author":"Akram","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mordeson, J.N., and Nair, P.S. (2001). Fuzzy Graphs and Fuzzy Hypergraphs, Physica Verlag. [2nd ed.].","DOI":"10.1007\/978-3-7908-1854-3"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.ipl.2013.05.004","article-title":"Intuitionistic fuzzy shortest hyperpath in a network","volume":"113","author":"Parvathi","year":"2013","journal-title":"Inf. Process. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.ins.2018.03.026","article-title":"Knowledge distance measure in multigranulation spaces of fuzzy equivalence relation","volume":"448","author":"Yang","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhang, H., Zhang, R., Huang, H., and Wang, J. (2018). Some picture fuzzy Dombi Heronian mean operators with their application to multi-attribute decision-making. Symmetry, 10.","DOI":"10.3390\/sym10110593"},{"key":"ref_24","unstructured":"Zhang, L., and Zhang, B. (2007). The Theory and Applications of Problem Solving-Quotient Space Based Granular Computing, Tsinghua University Press."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.ijar.2004.11.003","article-title":"The structural analysis of fuzzy sets","volume":"40","author":"Zhang","year":"2005","journal-title":"J. Approx. Reason."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Peters, J.F., Skowron, A., Grzymala-Busse, J.W., Kostek, B., Swiniarski, R.W., and Szczuka, M.S. (2004). A partition model of granular computing. Transactions on Rough Sets I. Lecture Notes in Computer Science, Springer.","DOI":"10.1007\/b98175"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/483\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:42:48Z","timestamp":1760186568000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/483"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,3]]},"references-count":26,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040483"],"URL":"https:\/\/doi.org\/10.3390\/sym11040483","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,4,3]]}}}