{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T21:53:49Z","timestamp":1782856429134,"version":"3.54.5"},"reference-count":27,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2015,8,31]],"date-time":"2015-08-31T00:00:00Z","timestamp":1440979200000},"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":[[2015,8,31]]},"abstract":"<jats:p>\n                    The purpose of this paper is to introduce the concept of type 2 fuzzy multiset by integrating the type 2 fuzzy set and multiset models. Basic set operations of union, intersection and complement are extended to type 2 fuzzy multisets and made a comparative study on various set theoretic laws satisfied by these operations with set theoretic operations on type 1 and type 2 fuzzy sets. The notion of (\u03b1,\n                    <jats:italic>\u03bb<\/jats:italic>\n                    ) cut set is defined and some properties of cut sets are studied. Furthermore, we investigate the decision making based on type 2 fuzzy multisets. By means of (\u03b1,\n                    <jats:italic>\u03bb<\/jats:italic>\n                    ) cut sets, we propose an adjustable approach to type 2 fuzzy multiset based decision making. Moreover, we introduce the weighted type 2 fuzzy multiset and examine its application to decision making. Finally, by means of some illustrative examples we show the validity of type 2 fuzzy multiset, weighted type 2 fuzzy multiset in decision making problems.\n                  <\/jats:p>","DOI":"10.3233\/ifs-151761","type":"journal-article","created":{"date-parts":[[2016,1,15]],"date-time":"2016-01-15T12:24:04Z","timestamp":1452860644000},"page":"359-369","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Type 2 fuzzy multisets and its applications in decision making problems"],"prefix":"10.1177","volume":"30","author":[{"given":"Jaya","family":"Paul","sequence":"first","affiliation":[{"name":"Department of Mathematics, National Institute of Technology, Calicut, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sunil Jacob","family":"John","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Institute of Technology, Calicut, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2015,10,8]]},"reference":[{"issue":"1","key":"e_1_3_1_2_2","article-title":"Solution of the decision making problems using fuzzy soft relations","volume":"15","author":"Chaudhuri Arindam","year":"2009","unstructured":"ChaudhuriArindam, DeKajal and ChatterjeeDipak, Solution of the decision making problems using fuzzy soft relations, International Journal of Information Technology 15(1) (2009).","journal-title":"International Journal of Information Technology"},{"issue":"1","key":"e_1_3_1_3_2","first-page":"36","article-title":"Multiset theory","volume":"30","author":"Blizard W.D.","year":"1989","unstructured":"BlizardW.D., Multiset theory, Norte Dame Journal of Formal Logic 30(1) (1989), 36\u201366.","journal-title":"Norte Dame Journal of Formal Logic"},{"key":"e_1_3_1_4_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-141288"},{"key":"e_1_3_1_5_2","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.dss.2011.05.003","article-title":"A framework for dynamic multiple criteria decision making","volume":"52","author":"Campanella G.","year":"2011","unstructured":"CampanellaG. and RiberioR.A., A framework for dynamic multiple criteria decision making, Decision Support Systems 52 (2011), 52\u201360.","journal-title":"Decision Support Systems"},{"key":"e_1_3_1_6_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ins.2013.04.005","article-title":"Fuzzy decision making based on Interval type 2 fuzzy sets","volume":"242","author":"Chen S.","year":"2013","unstructured":"ChenS. and WangC., Fuzzy decision making based on Interval type 2 fuzzy sets, Information Sciences 242 (2013), 1\u201321.","journal-title":"Information Sciences"},{"key":"e_1_3_1_7_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-141246"},{"key":"e_1_3_1_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ins.2011.11.023"},{"key":"e_1_3_1_9_2","doi-asserted-by":"publisher","DOI":"10.3233\/IFS-141357"},{"issue":"1","key":"e_1_3_1_10_2","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/S0020-0255(99)00067-5","article-title":"Applications of type 2 fuzzy logic systems to forecasting of time series","volume":"120","author":"Karnik N.N.","year":"1999","unstructured":"KarnikN.N. and MendelJ.M., Applications of type 2 fuzzy logic systems to forecasting of time series, Information Sciences 120(1) (1999), 89\u2013111.","journal-title":"Information Sciences"},{"issue":"2","key":"e_1_3_1_11_2","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/S0165-0114(00)00079-8","article-title":"Operations of type 2 fuzzy sets","volume":"122","author":"Karnik N.N.","year":"2001","unstructured":"KarnikN.N. and MendelJ.M., Operations of type 2 fuzzy sets, Fuzzy Sets and Systems 122(2) (2001), 327\u2013348.","journal-title":"Fuzzy Sets and Systems"},{"key":"e_1_3_1_12_2","unstructured":"KlirG.J. and YuanB.o. Fuzzy sets and Fuzzy Logic: Theory and applications."},{"issue":"9","key":"e_1_3_1_13_2","doi-asserted-by":"crossref","first-page":"2224","DOI":"10.1016\/j.ins.2007.11.014","article-title":"An efficient centroid type-reduction strategy for general type 2 logic systems","volume":"178","author":"Liu F.","year":"2008","unstructured":"LiuF., An efficient centroid type-reduction strategy for general type 2 logic systems, Information Sciences 178(9) (2008), 2224\u20132236.","journal-title":"Information Sciences"},{"issue":"3","key":"e_1_3_1_14_2","first-page":"589","article-title":"Fuzzy soft sets","volume":"9","author":"Maji P.K.","year":"2001","unstructured":"MajiP.K., BiswasR. and RoyA.R., Fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 589\u2013602.","journal-title":"Journal of Fuzzy Mathematics"},{"issue":"2","key":"e_1_3_1_15_2","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1109\/91.995115","article-title":"Type 2 fuzzy sets made simple","volume":"10","author":"Mendel J.M.","year":"2002","unstructured":"MendelJ.M. and JohnR.I., Type 2 fuzzy sets made simple, IEEE Transactions on Fuzzy Systems 10(2) (2002), 117\u2013127.","journal-title":"IEEE Transactions on Fuzzy Systems"},{"key":"e_1_3_1_16_2","first-page":"327","article-title":"Interval type 2 fuzzy logic systems made simple","volume":"122","author":"Mendel J.M.","year":"2001","unstructured":"MendelJ.M., JohnR.I. and LiuF., Interval type 2 fuzzy logic systems made simple, IEEE Transactions on Fuzzy Systems 122 (2001), 327\u2013348.","journal-title":"IEEE Transactions on Fuzzy Systems"},{"issue":"5","key":"e_1_3_1_17_2","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1109\/TFUZZ.2009.2024411","article-title":"\u03b1-Plane representation for type 2 fuzzy sets: Theory and applications","volume":"17","author":"Mendel J.M.","year":"2009","unstructured":"MendelJ.M., LiuF. and ZhaiD., \u03b1-Plane representation for type 2 fuzzy sets: Theory and applications, IEEE Transactions on Fuzzy Systems 17(5) (2009), 1189\u20131207.","journal-title":"IEEE Transactions on Fuzzy Systems"},{"key":"e_1_3_1_18_2","unstructured":"MendelJ.M. Uncertain rules based fuzzy logic systems; Introduction and new directions Prentice-Hall USA 2001."},{"key":"e_1_3_1_19_2","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 M.","year":"1976","unstructured":"MizumotoM. and TanakaK., Some properties of Fuzzy sets of type 2, Informations and Control 31 (1976), 312\u2013340.","journal-title":"Informations and Control"},{"issue":"3","key":"e_1_3_1_20_2","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0165-0114(81)90056-7","article-title":"Fuzzy sets of type 2 under algebraic product and algebraic sum","volume":"5","author":"Mizumoto M.","year":"1981","unstructured":"MizumotoM. and TanakaK., Fuzzy sets of type 2 under algebraic product and algebraic sum, Fuzzy Sets and Systems 5(3) (1981), 227\u2013290.","journal-title":"Fuzzy Sets and Systems"},{"issue":"5","key":"e_1_3_1_21_2","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/BF01001956","article-title":"Rough sets","volume":"37","author":"Pawlak Z.","year":"1982","unstructured":"PawlakZ., Rough sets, International Journal of Computer and Information Sciences 37(5) (1982), 341\u2013356.","journal-title":"International Journal of Computer and Information Sciences"},{"key":"e_1_3_1_22_2","unstructured":"RobertR. Stoll Set theory and Logic W.H. Freeman and Company San Francisco and London 1963."},{"issue":"1","key":"e_1_3_1_23_2","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s12543-011-0064-y","article-title":"Multi Fuzzy sets: An extension of fuzzy sets","volume":"3","author":"Sebastian S.","year":"2011","unstructured":"SebastianS. and RamakrishnanT., Multi Fuzzy sets: An extension of fuzzy sets, Fuzzy Information and Engineering 3(1) (2011), 35\u201343.","journal-title":"Fuzzy Information and Engineering"},{"issue":"2","key":"e_1_3_1_24_2","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1109\/TFUZZ.2009.2013458","article-title":"An Interval type 2 rough set model for attribute reduction","volume":"17","author":"Wu H.","year":"2009","unstructured":"WuH., WuY. and LuoF., An Interval type 2 rough set model for attribute reduction, IEEE Transactions on Fuzzy Systems 17(2) (2009), 301\u2013315.","journal-title":"IEEE Transactions on Fuzzy Systems"},{"issue":"3","key":"e_1_3_1_25_2","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/j.camwa.2008.11.002","article-title":"Soft set theory applied to ideals in d-algebras","volume":"57","author":"Jun Young B.","year":"2009","unstructured":"JunYoung B., LeeK.J. and ParkC.H., Soft set theory applied to ideals in d-algebras, Computers and Mathematics with Applications 57(3) (2009), 367\u2013378.","journal-title":"Computers and Mathematics with Applications"},{"key":"e_1_3_1_26_2","first-page":"338","article-title":"Fuzzy sets","volume":"8","author":"Zadeh L.A.","year":"1965","unstructured":"ZadehL.A., Fuzzy sets, Information and Computation 8 (1965), 338\u2013353.","journal-title":"Information and Computation"},{"key":"e_1_3_1_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0255(75)90036-5"},{"key":"e_1_3_1_28_2","doi-asserted-by":"crossref","unstructured":"ZhangZhiming and ZhangShouhua Type 2 fuzzy soft sets and their applications in decision making Journal of Applied Mathematics (2012) Article ID 608681 35 pages 2012.","DOI":"10.1155\/2012\/608681"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/IFS-151761","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/IFS-151761","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/IFS-151761","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:39:08Z","timestamp":1777455548000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/IFS-151761"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,8,31]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,8,31]]}},"alternative-id":["10.3233\/IFS-151761"],"URL":"https:\/\/doi.org\/10.3233\/ifs-151761","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,8,31]]}}}