{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:38:03Z","timestamp":1777703883627,"version":"3.51.4"},"reference-count":19,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2019,5,8]],"date-time":"2019-05-08T00:00:00Z","timestamp":1557273600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2019,10,25]]},"abstract":"<jats:p>\n                    In recent years, interval type-2 fuzzy set has played an increasingly important role in decision-making. The purpose of this paper is firstly to propose fuzzy factor and interval factor of interval type-2 fuzzy sets (IT2FSs) based on\n                    <jats:italic>\u03b1<\/jats:italic>\n                    -cuts. Then, entropy and cross-entropy formulas of IT2FSs are put forward to measure the fuzzy information and mutual information between IT2FSs based on the two factors. Furthermore, IT2FS comprehensive cross-entropy formula is brought forward to measure the relative uncertain relationships among IT2FSs based on IT2FS cross-entropy formula and total probability formula. In the next moment, a utility function of IT2FS is proposed to compare the ordering relationship among IT2FSs. On the other hands, an optimal linear programming model is constructed based on IT2FS entropy, comprehensive cross-entropy and utility function to solve the optimal alternative. Finally, a simple example is utilized to testify the practicability and effectiveness of the proposed approach.\n                  <\/jats:p>","DOI":"10.3233\/jifs-18678","type":"journal-article","created":{"date-parts":[[2019,5,17]],"date-time":"2019-05-17T11:46:31Z","timestamp":1558093591000},"page":"5143-5156","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":3,"title":["Comprehensive cross-entropy and entropy measures and their applications under interval type-2 fuzzy environment"],"prefix":"10.1177","volume":"37","author":[{"given":"Kongchun","family":"Mei","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Anhui University, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Peng","sequence":"additional","affiliation":[{"name":"Key Laboratory of Intelligent Computing and Signal processing, Ministry of Education, Anhui University, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junjun","family":"Mao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Intelligent Computing and Signal processing, Ministry of Education, Anhui University, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2019,5,8]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(75)90036-5"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1109\/91.995115"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(65)90241-X"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(68)90078-4"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1002\/int.21714"},{"key":"e_1_3_2_7_2","first-page":"183","article-title":"Uncertainty Measurement for the Interval Type-2 Fuzzy Set","author":"Greenfield S.","year":"2016","unstructured":"S.Greenfield, Uncertainty Measurement for the Interval Type-2 Fuzzy Set, International Conference on Artificial Intelligence and Soft Computing. 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