{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T20:42:17Z","timestamp":1771360937729,"version":"3.50.1"},"reference-count":44,"publisher":"SAGE Publications","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IFS"],"published-print":{"date-parts":[[2016,3,10]]},"DOI":"10.3233\/ifs-151986","type":"journal-article","created":{"date-parts":[[2016,3,16]],"date-time":"2016-03-16T13:50:20Z","timestamp":1458136220000},"page":"2169-2180","source":"Crossref","is-referenced-by-count":12,"title":["Type-2 fuzzy cross-entropy and entropy measures and their applications"],"prefix":"10.1177","volume":"30","author":[{"given":"Dengbao","family":"Yao","sequence":"first","affiliation":[{"name":"School of Economics and Management, Southeast University, Nanjing, Jiangsu, China"}]},{"given":"Xiaoxing","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Southeast University, Nanjing, Jiangsu, China"}]},{"given":"Xu","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Economics and Management, Southeast University, Nanjing, Jiangsu, China"}]},{"given":"Cuicui","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Foundation, Anhui Sanlian University, Hefei, Anhui, China"}]}],"member":"179","reference":[{"key":"10.3233\/IFS-151986_ref1","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1016\/j.ins.2010.12.014","article-title":"Type-2 fuzzy nueral networks with fuzzy clustering and differential evolution optimization","volume":"181","author":"Aliev","year":"2011","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref2","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.cie.2015.03.005","article-title":"An interval type-2 fuzzy technique for order preference by similarity to ideal solutions using a likelihood-based comparison approach for multiple criteria decision analysis","volume":"85","author":"Chen","year":"2015","journal-title":"Computers and Industrial Engineering"},{"key":"10.3233\/IFS-151986_ref3","doi-asserted-by":"crossref","first-page":"2102","DOI":"10.1016\/j.ins.2008.04.009","article-title":"Interval type-2 fuzzy membership function generation methods for pattern recognition","volume":"179","author":"Choi","year":"2009","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref4","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.compind.2010.10.006","article-title":"Industrial applications of type-2 fuzzy sets and systems: A concise review","volume":"62","author":"Derel","year":"2011","journal-title":"Computers in Industry"},{"key":"10.3233\/IFS-151986_ref5","unstructured":"Ghorahaee M.K. , Developing an MCDM method for robot selection with interval type-2 fuzzy sets, Robotics and Computer-Integrated Manufacturing (2015). DOI: 10.1016\/j.rcim.2015.04.007i"},{"key":"10.3233\/IFS-151986_ref6","doi-asserted-by":"crossref","first-page":"860","DOI":"10.1016\/j.asoc.2011.10.004","article-title":"Computing with words for hierarchical competency based selection of personal in construction companies","volume":"12","author":"Gilan","year":"2012","journal-title":"Applied Soft Computing"},{"key":"10.3233\/IFS-151986_ref7","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.ins.2014.01.050","article-title":"Similarity measures for general type-2 fuzzy sets based on the a-plane representation","volume":"277","author":"Hao","year":"2014","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.fss.2013.07.011","article-title":"On type-2 fuzzy relations and interval-valued type-2 fuzzy sets","volume":"236","author":"Hu","year":"2014","journal-title":"Fuzzy Sets & Systems"},{"key":"10.3233\/IFS-151986_ref9","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1002\/int.20131","article-title":"Fuzzy entropy on intuitionistic fuzzy sets","volume":"21","author":"Hung","year":"2006","journal-title":"International Journal of Intelligent Systems"},{"key":"10.3233\/IFS-151986_ref10","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1142\/S0218488504003235","article-title":"Similarity measures between type-2 fuzzy sets","volume":"12","author":"Hung","year":"2004","journal-title":"International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems"},{"key":"10.3233\/IFS-151986_ref11","doi-asserted-by":"crossref","first-page":"1788","DOI":"10.1016\/j.mcm.2010.12.057","article-title":"Similarity, inclusion and entropy measures between type-2 fuzzy sets based on the Sugeno integral","volume":"53","author":"Hwang","year":"2011","journal-title":"Mathematical and Computer Modelling"},{"key":"10.3233\/IFS-151986_ref12","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1142\/S0218488512500225","article-title":"On similarity, inclusion measure and entropy between Type-2 fuzzy sets","volume":"20","author":"Hwang","year":"2012","journal-title":"International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems"},{"key":"10.3233\/IFS-151986_ref13","first-page":"89","article-title":"Applications of type-2 fuzzy logic systems to forecasting of time-series","volume":"120","author":"Karnik","year":"1999","journal-title":"InformationSciences"},{"key":"10.3233\/IFS-151986_ref14","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/S0165-0114(00)00079-8","article-title":"Operations on type-2 fuzzy sets","volume":"122","author":"Karnik","year":"2001","journal-title":"Fuzzy Sets & Systems"},{"key":"10.3233\/IFS-151986_ref15","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0165-0114(95)00032-G","article-title":"Principles of uncertainty: What are they? why do we need them?","volume":"74","author":"Klir","year":"1995","journal-title":"Fuzzy Sets & Systems"},{"key":"10.3233\/IFS-151986_ref16","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.asoc.2013.12.020","article-title":"Distinguishability of interval type-2 fuzzy sets data by analyzing upper and lower membership functions","volume":"17","author":"Livi","year":"2014","journal-title":"Applied Soft Computing"},{"key":"10.3233\/IFS-151986_ref17","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.knosys.2013.04.011","article-title":"A novel cross-entropy and entropy measures of IFSs and their applications","volume":"48","author":"Mao","year":"2013","journal-title":"Knowledge-Based Systems"},{"key":"10.3233\/IFS-151986_ref18","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1016\/j.asoc.2014.04.017","article-title":"A review on type-2 fuzzy logic applications in clustering, classification and pattern recognition","volume":"21","author":"Melin","year":"2014","journal-title":"Applied Soft Computing"},{"key":"10.3233\/IFS-151986_ref19","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.ins.2006.05.003","article-title":"Advances in type-2 fuzzy sets and systems","volume":"177","author":"Mendel","year":"2007","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref20","doi-asserted-by":"crossref","first-page":"988","DOI":"10.1016\/j.ins.2006.06.008","article-title":"Computing with words and its relationship with fuzziness","volume":"177","author":"Mendel","year":"2007","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref21","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/91.995115","article-title":"Type-2 fuzzy sets made simple","volume":"10","author":"Mendel","year":"2002","journal-title":"IEEE Transactions on Fuzzy Systems"},{"key":"10.3233\/IFS-151986_ref22","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1016\/S0019-9958(76)80011-3","article-title":"Some properties of fuzzy sets of type-2","volume":"31","author":"Mizumoto","year":"1976","journal-title":"Information and Control"},{"key":"10.3233\/IFS-151986_ref23","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.ins.2014.02.092","article-title":"Footprint of uncertainty for type-2 fuzzy sets","volume":"272","author":"Mo","year":"2014","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref24","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.ins.2011.07.036","article-title":"MiniMax e-star cluster validity index for type-2 fuzziness","volume":"184","author":"Ozkan","year":"2012","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref25","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.ins.2012.11.016","article-title":"Uncertainties with Atanassov\u2019s intuitionistic fuzzy sets: Fuzziness and lack of knowledge","volume":"228","author":"Pal","year":"2013","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref26","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.ins.2014.11.022","article-title":"Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment","volume":"297","author":"Qin","year":"2015","journal-title":"Information Sciences"},{"issue":"11","key":"10.3233\/IFS-151986_ref27","doi-asserted-by":"crossref","first-page":"3652","DOI":"10.1016\/j.patcog.2015.04.010","article-title":"Interval type-2 credibilistic clustering for pattern recognition","volume":"48","author":"Kalhori","journal-title":"Pattern Recognition"},{"key":"10.3233\/IFS-151986_ref28","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1016\/S0167-8655(97)00028-7","article-title":"A note on fuzzy information measures","volume":"18","author":"Shang","year":"1997","journal-title":"Pattern Recognition Letter"},{"key":"10.3233\/IFS-151986_ref29","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1016\/S0165-0114(98)00402-3","article-title":"Entropy for intuitionistic fuzzy sets","volume":"118","author":"Szmidt","year":"2001","journal-title":"Fuzzy Sets & Systems"},{"key":"10.3233\/IFS-151986_ref30","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.fss.2013.01.002","article-title":"Inclusion and subsethood measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets","volume":"224","author":"Tak\u00e1\u010d","year":"2013","journal-title":"Fuzzy Sets & Systems"},{"key":"10.3233\/IFS-151986_ref31","first-page":"529","article-title":"Hesitant fuzzy sets","volume":"25","author":"Torra","year":"2010","journal-title":"International Journal of Intelligent Systems"},{"key":"10.3233\/IFS-151986_ref32","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/j.patrec.2006.07.004","article-title":"Intuitionistic fuzzy information-applications to pattern recognition","volume":"28","author":"Vlachos","year":"2007","journal-title":"Pattern Recognition Letters"},{"key":"10.3233\/IFS-151986_ref33","doi-asserted-by":"crossref","first-page":"5378","DOI":"10.1016\/j.ins.2007.07.012","article-title":"Uncertainty measures for interval type-2 fuzzy sets","volume":"177","author":"Wu","year":"2007","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref34","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/S0019-0578(07)60228-6","article-title":"A simplified type-2 fuzzy logic controller for real-time control","volume":"45","author":"Wu","year":"2006","journal-title":"ISA Transactions"},{"key":"10.3233\/IFS-151986_ref35","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1002\/int.21548","article-title":"Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making","volume":"27","author":"Xu","year":"2012","journal-title":"International Journal of Intelligent Systems"},{"key":"10.3233\/IFS-151986_ref36","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1080\/01969728008927629","article-title":"Fuzzy subsets of type II in decisions","volume":"10","author":"Yager","year":"1980","journal-title":"Journal of Cybernetics"},{"key":"10.3233\/IFS-151986_ref37","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1016\/j.camwa.2008.10.028","article-title":"On similarity and inclusion measures between type-2 fuzzy sets with an application to clustering","volume":"57","author":"Yang","year":"2009","journal-title":"Computers & Mathematics with Applications"},{"key":"10.3233\/IFS-151986_ref38","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/S0165-0114(99)00146-3","article-title":"Cluster analysis based on fuzzy relations","volume":"120","author":"Yang","year":"2001","journal-title":"Fuzzy Sets & Systems"},{"key":"10.3233\/IFS-151986_ref39","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/0022-247X(68)90078-4","article-title":"Probability measures of fuzzy events","volume":"23","author":"Zadeh","year":"1968","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"10.3233\/IFS-151986_ref40","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0020-0255(75)90036-5","article-title":"The concept of a linguistic variable and its application to approximate reasoning","volume":"8","author":"Zadeh","year":"1975","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref41","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.ins.2014.02.066","article-title":"Zarandi and I.B. Turksen, Interval type-2 relative entropy fuzzy C-means clustering","volume":"272","author":"Zarinbal","year":"2014","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref42","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/j.ins.2010.09.020","article-title":"Uncertainty measures for general Type-2 fuzzy sets","volume":"181","author":"Zhai","year":"2011","journal-title":"Information Sciences"},{"key":"10.3233\/IFS-151986_ref43","first-page":"1","article-title":"Uncertainty measures for Hesitant fuzzy information","author":"Zhao","year":"2015","journal-title":"International Journal of Intelligent Systems"},{"key":"10.3233\/IFS-151986_ref44","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s00500-013-1101-z","article-title":"A new approach to similarity and inclusion measures between general type-2 fuzzy sets","volume":"18","author":"Zhao","year":"2014","journal-title":"Soft Computing"}],"container-title":["Journal of Intelligent & Fuzzy Systems"],"original-title":[],"link":[{"URL":"https:\/\/content.iospress.com\/download?id=10.3233\/IFS-151986","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T06:50:12Z","timestamp":1770619812000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/IFS-151986"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,10]]},"references-count":44,"journal-issue":{"issue":"4"},"URL":"https:\/\/doi.org\/10.3233\/ifs-151986","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,10]]}}}