{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T11:09:48Z","timestamp":1778584188630,"version":"3.51.4"},"reference-count":33,"publisher":"World Scientific Pub Co Pte Lt","issue":"supp02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Unc. Fuzz. Knowl. Based Syst."],"published-print":{"date-parts":[[2007,4]]},"abstract":"<jats:p> This paper presents the independence of fuzzy variables as well as its applications in fuzzy random optimization. First, the independence of fuzzy variables is defined based on the concept of marginal possibility distribution function, and a discussion about the relationship between the independent fuzzy variables and the noninteractive (unrelated) fuzzy variables is included. Second, we discuss some properties of the independent fuzzy variables, and establish the necessary and sufficient conditions for the independent fuzzy variables. Third, we propose the independence of fuzzy events, and deal with its fundamental properties. Finally, we apply the properties of the independent fuzzy variables to a class of fuzzy random programming problems to study their convexity. <\/jats:p>","DOI":"10.1142\/s021848850700456x","type":"journal-article","created":{"date-parts":[[2007,5,21]],"date-time":"2007-05-21T07:43:34Z","timestamp":1179733414000},"page":"1-20","source":"Crossref","is-referenced-by-count":135,"title":["THE INDEPENDENCE OF FUZZY VARIABLES WITH APPLICATIONS TO FUZZY RANDOM OPTIMIZATION"],"prefix":"10.1142","volume":"15","author":[{"given":"YIAN-KUI","family":"LIU","sequence":"first","affiliation":[{"name":"College of Mathematics &amp; Computer Science, Hebei University, Baoding 071002, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JINWU","family":"GAO","sequence":"additional","affiliation":[{"name":"School of Information, Renmin University of China, Beijing 1000872, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(78)90029-5"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(78)90011-8"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(82)90032-X"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-5287-7"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/0165-0114(92)90226-T"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-0114(97)00371-0"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-48753-8"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-1633-4"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7908-1781-2"},{"key":"rf10","first-page":"509","volume":"36","author":"Liu Y.-K.","journal-title":"Mathematical and Computer Modelling"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488503002016"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488505003539"},{"key":"rf13","first-page":"445","volume":"10","author":"Liu B.","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-39987-2"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-006-0016-x"},{"key":"rf16","first-page":"295","volume":"14","author":"Liu Y.-K.","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"rf17","first-page":"1","volume":"1","author":"Liu B.","journal-title":"Journal of Uncertain Systems"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1013771608623"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1023\/A:1023447217758"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-0255(03)00079-3"},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1016\/S0305-0548(03)00235-1"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2004.03.010"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1007\/s10700-006-0013-0"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-017-3087-7"},{"key":"rf25","series-title":"Lecture Notes in Artificial Intelligence","first-page":"984","volume":"4114","author":"Liu Y.-K.","year":"2006"},{"key":"rf26","first-page":"195","volume":"1","author":"Liu Y.-K.","journal-title":"Journal of Information and Computing Science"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2004.07.027"},{"key":"rf28","series-title":"Lecture Notes in Artificial Intelligence","first-page":"377","volume":"3930","author":"Chen Y.","year":"2006"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2003.05.012"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cie.2005.03.003"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1142\/S0218488503002338"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2004.1280"},{"key":"rf33","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2005.04.015"}],"container-title":["International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S021848850700456X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T23:12:16Z","timestamp":1565133136000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S021848850700456X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,4]]},"references-count":33,"journal-issue":{"issue":"supp02","published-online":{"date-parts":[[2011,11,21]]},"published-print":{"date-parts":[[2007,4]]}},"alternative-id":["10.1142\/S021848850700456X"],"URL":"https:\/\/doi.org\/10.1142\/s021848850700456x","relation":{},"ISSN":["0218-4885","1793-6411"],"issn-type":[{"value":"0218-4885","type":"print"},{"value":"1793-6411","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,4]]}}}