{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:49:13Z","timestamp":1760150953507,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,29]],"date-time":"2022-01-29T00:00:00Z","timestamp":1643414400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71872110"],"award-info":[{"award-number":["71872110"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we define the variance and semi-variances of regular interval type-2 fuzzy variables (RIT2-FVs) as well as derive a calculation formula of them based on the credibility distribution. Following the relationship between the variance and the semi-variances of the regular symmetric triangular interval type-2 fuzzy variables (RSTIT2-FVs), a special type of interval type-2 fuzzy variable is discovered and proved. Furthermore, for applying the two measures, we propose the operational law for the variance and semi-variances of the linear function of mutually independent RSTIT2-FVs. Some numerical examples are illustrated. The consequences of examples prove that the formulas we proposed can be effectively applied to the calculation of the variance of RSTIT2-FVs. The results indicate that they play a great role in the application of variance of type-2 fuzzy sets in various fields.<\/jats:p>","DOI":"10.3390\/sym14020278","type":"journal-article","created":{"date-parts":[[2022,1,30]],"date-time":"2022-01-30T00:12:56Z","timestamp":1643501576000},"page":"278","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Variance and Semi-Variances of Regular Interval Type-2 Fuzzy Variables"],"prefix":"10.3390","volume":"14","author":[{"given":"Wenjing","family":"Tang","sequence":"first","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]},{"given":"Yitao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Management, Shanghai University, Shanghai 200444, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","first-page":"53","article-title":"Twofold fuzzy sets: An approach to the representation of sets with fuzzy boundaries based on possibility and necessity measures","volume":"3","author":"Dubois","year":"1983","journal-title":"J. Fuzzy Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1023\/A:1013771608623","article-title":"Toward fuzzy optimization without mathematical ambiguity","volume":"1","author":"Liu","year":"2002","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_4","first-page":"363","article-title":"On weighted possibilistic mean and variance of fuzzy numbers","volume":"136","author":"Robert","year":"2002","journal-title":"Fuzzy Set Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1080\/00207720601157997","article-title":"Analysis of variance for fuzzy data","volume":"3","author":"Wu","year":"2007","journal-title":"Int. J. Syst. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s00500-009-0407-3","article-title":"The revised algorithms of fuzzy variance and an application to portfolio selection","volume":"14","author":"Tsao","year":"2010","journal-title":"Soft Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1007\/s40815-015-0072-x","article-title":"A new magnitude possibilistic mean value and variance of fuzzy numbers","volume":"18","author":"Gong","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"353","DOI":"10.3233\/JIFS-181467","article-title":"Calculation formulas and correlation inequalities for variance bounds and semi-variances of fuzzy intervals","volume":"36","author":"Gu","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3720","DOI":"10.3934\/math.2021221","article-title":"Some notes on possibilistic variances of generalized trapezoidal intuitionistic fuzzy numbers","volume":"6","author":"Zhang","year":"2021","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.mcm.2008.07.035","article-title":"Weighted possibilistic moments of fuzzy numbers with applications to GARCH modeling and option pricing","volume":"49","author":"Thavaneswaran","year":"2009","journal-title":"Math. Comput. Model."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1001509","DOI":"10.1016\/j.rinam.2021.100159","article-title":"Credibilistic variance and skewness of trapezoidal fuzzy variable and meanvariance-skewness model for portfolio selection","volume":"11","author":"Pahade","year":"2021","journal-title":"Results Appl. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1272266","DOI":"10.1155\/2021\/1272266","article-title":"(3, 2)-Fuzzy Sets and Their Applications to Topology and Optimal Choices","volume":"2021","author":"Ibrahim","year":"2021","journal-title":"Comput. Intel. Neurosci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s40314-021-01501-x","article-title":"Fuzzy soft covering-based multi-granulation fuzzy rough sets and their applications","volume":"40","author":"Atef","year":"2021","journal-title":"Comput. Appl. Math."},{"key":"ref_14","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\u2014I","volume":"8","author":"Zadeh","year":"1975","journal-title":"Inf. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"El Baf, F., Bouwmans, T., and Vachon, B. (2008, January 1\u20133). Type-2 fuzzy mixture of Gaussians model: Application to background modeling. Proceedings of the International Symposium on Visual Computing, Las Vegas, NV, USA.","DOI":"10.1007\/978-3-540-89639-5_74"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1109\/TFUZZ.2006.890668","article-title":"Wireless sensor network lifetime analysis using interval type-2 fuzzy logic systems","volume":"16","author":"Shu","year":"2008","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3869","DOI":"10.1007\/s00521-016-2262-3","article-title":"R&D project evaluation and project portfolio selection by a new interval type-2 fuzzy optimization approach","volume":"28","author":"Mohagheghi","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Guo, J., and Du, S. (2018). Modeling words for qualitative distance based on interval type-2 fuzzy sets. ISPRS. Int. J. Geo-Inf., 7.","DOI":"10.3390\/ijgi7080291"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101889","DOI":"10.1016\/j.scs.2019.101889","article-title":"The assessment of smart city projects using zslice type-2 fuzzy sets based interval agreement method","volume":"53","author":"Deveci","year":"2020","journal-title":"Sustain. Cites Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"108076","DOI":"10.1016\/j.asoc.2021.108076","article-title":"Interval type-2 hesitant fuzzy entropy-based WASPAS approach for aircraft type selection","volume":"114","author":"Deveci","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_21","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":"Inf. Sci."},{"key":"ref_22","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":"Inf. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1002\/tee.22224","article-title":"Design of a qualitative classification model through fuzzy support vector machine with type-2 fuzzy expected regression classifier preset","volume":"11","author":"Wei","year":"2016","journal-title":"IEEJ. Trans. Electr. Electr. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1891","DOI":"10.3233\/JIFS-161279","article-title":"A new approach for ranking of interval type-2 trapezoidal fuzzy numbers","volume":"32","author":"Gong","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1109\/TFUZZ.2018.2874018","article-title":"A constrained representation theorem for well-shaped interval type-2 fuzzy sets, and the corresponding constrained uncertainty measures","volume":"27","author":"Wu","year":"2018","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100693","DOI":"10.1016\/j.seps.2019.02.008","article-title":"Real options valuation of an IoT based healthcare device with interval type-2 fuzzy numbers","volume":"69","author":"Tolga","year":"2019","journal-title":"Soc.-Econ. Plan. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Li, H., and Cai, J. (2021). Arithmetic operations and expected Values of regular interval type-2 fuzzy variables. Symmetry, 13.","DOI":"10.3390\/sym13112196"},{"key":"ref_28","unstructured":"Liu, B. (2007). Uncertainty Theory, Springer. [2nd ed.]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1080\/00207727808941724","article-title":"Operations on fuzzy numbers","volume":"9","author":"Dubois","year":"1978","journal-title":"Int. J. Syst. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"71","DOI":"10.3233\/IFS-151712","article-title":"Fuzzy arithmetic on LR fuzzy numbers with applications to fuzzy programming","volume":"30","author":"Zhou","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_31","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 Trans. Fuzzy Syst."},{"key":"ref_32","unstructured":"Liu, B. (2004). Uncertainty Theory: An Introduction to Its Axiomatic Foundations, Springer."},{"key":"ref_33","unstructured":"Liu, B. (2010). Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2007.06.009","article-title":"Mean-semivariance models for fuzzy portfolio selection","volume":"217","author":"Huang","year":"2008","journal-title":"J. Comput. Appl. Mathem."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/278\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:11:09Z","timestamp":1760134269000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/278"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,29]]},"references-count":34,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020278"],"URL":"https:\/\/doi.org\/10.3390\/sym14020278","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,29]]}}}