{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T17:06:25Z","timestamp":1791911185509,"version":"4.3.4"},"reference-count":17,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2020,6,30]],"date-time":"2020-06-30T00:00:00Z","timestamp":1593475200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007224","name":"National Foundation for Science and Technology Development","doi-asserted-by":"publisher","award":["102.05-2018.02"],"award-info":[{"award-number":["102.05-2018.02"]}],"id":[{"id":"10.13039\/100007224","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Clustering is an unsupervised machine learning technique with many practical applications that has gathered extensive research interest. Aside from deterministic or probabilistic techniques, fuzzy C-means clustering (FCM) is also a common clustering technique. Since the advent of the FCM method, many improvements have been made to increase clustering efficiency. These improvements focus on adjusting the membership representation of elements in the clusters, or on fuzzifying and defuzzifying techniques, as well as the distance function between elements. This study proposes a novel fuzzy clustering algorithm using multiple different fuzzification coefficients depending on the characteristics of each data sample. The proposed fuzzy clustering method has similar calculation steps to FCM with some modifications. The formulas are derived to ensure convergence. The main contribution of this approach is the utilization of multiple fuzzification coefficients as opposed to only one coefficient in the original FCM algorithm. The new algorithm is then evaluated with experiments on several common datasets and the results show that the proposed algorithm is more efficient compared to the original FCM as well as other clustering methods.<\/jats:p>","DOI":"10.3390\/a13070158","type":"journal-article","created":{"date-parts":[[2020,6,30]],"date-time":"2020-06-30T09:04:59Z","timestamp":1593507899000},"page":"158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":45,"title":["Fuzzy C-Means Clustering Algorithm with Multiple Fuzzification Coefficients"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2371-858X","authenticated-orcid":false,"given":"Tran Dinh","family":"Khang","sequence":"first","affiliation":[{"name":"Department of Information Systems, Hanoi University of Science and Technology, Hanoi 10000, Vietnam"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nguyen Duc","family":"Vuong","sequence":"additional","affiliation":[{"name":"Department of Information Systems, Hanoi University of Science and Technology, Hanoi 10000, Vietnam"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9937-1749","authenticated-orcid":false,"given":"Manh-Kien","family":"Tran","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9761-0336","authenticated-orcid":false,"given":"Michael","family":"Fowler","sequence":"additional","affiliation":[{"name":"Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Everitt, B.S., Landau, S., Leese, M., and Stahl, D. (2011). Cluster Analysis, John Wiley & Sons, Ltd.. [5th ed.].","DOI":"10.1002\/9780470977811"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0098-3004(84)90020-7","article-title":"FCM: The fuzzy c-mean clustering algorithm","volume":"10","author":"Bezdek","year":"1984","journal-title":"Comput. Geosci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1109\/MCI.2018.2881643","article-title":"Fuzzy Clustering: A Historical Perspective","volume":"14","author":"Ruspini","year":"2019","journal-title":"IEEE Comput. Intell. Mag."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.procs.2016.03.014","article-title":"Performance Analysis of Various Fuzzy Clustering Algorithms: A Review","volume":"79","author":"Gosain","year":"2016","journal-title":"Procedia Comput. 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Proceedings of the Conference: FUZZ-IEEE 2010, IEEE International Conference on Fuzzy Systems, Barcelona, Spain.","DOI":"10.1109\/FUZZY.2010.5584108"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/7\/158\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:45:01Z","timestamp":1760175901000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/13\/7\/158"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,30]]},"references-count":17,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2020,7]]}},"alternative-id":["a13070158"],"URL":"https:\/\/doi.org\/10.3390\/a13070158","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,30]]}}}