{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:45:27Z","timestamp":1740149127561,"version":"3.37.3"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T00:00:00Z","timestamp":1671494400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T00:00:00Z","timestamp":1671494400000},"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":["62006155"],"award-info":[{"award-number":["62006155"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62102261"],"award-info":[{"award-number":["62102261"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shaoguan City Social Development Science and Technology Collaborative Innovation System Construction Project","award":["220606114533116"],"award-info":[{"award-number":["220606114533116"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Comput Intell Syst"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Intuitionistic fuzzy sets, characterized by membership degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bc<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, non-membership degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\upsilon $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c5<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and hesitation degree <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\pi $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c0<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, are a meaningful extension of fuzzy set. Inequalities on intuitionistic fuzzy sets\/values are very important in solving real problems. In this paper, some inequalities on intuitionistic fuzzy sets are derived from operations. Moreover, three unweighted intuitionistic fuzzy aggregation operators, including unweighted intuitionistic fuzzy Square, unweighted intuitionistic fuzzy Arithmetic and unweighted intuitionistic fuzzy Geometric, are developed. Later, some corresponding inequality relations on them are deeply explored. Finally, some inequalities on intuitionistic fuzzy value are constructed by equality <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu +\\upsilon +\\pi =1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03bc<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u03c5<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u03c0<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in critical definition and proved by some existing famous inequalities, which provide a novel basis for the intuitionistic fuzzy inequalities in operations and aggregation operators.<\/jats:p>","DOI":"10.1007\/s44196-022-00170-w","type":"journal-article","created":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T10:06:23Z","timestamp":1671530783000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Results for Intuitionistic Fuzzy Inequality"],"prefix":"10.1007","volume":"15","author":[{"given":"Xindong","family":"Peng","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Harish","family":"Garg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhigang","family":"Luo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,20]]},"reference":[{"key":"170_CR1","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","volume":"20","author":"K Atanassov","year":"1986","unstructured":"Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Set. Syst. 20, 87\u201396 (1986)","journal-title":"Fuzzy Set. Syst."},{"key":"170_CR2","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","volume":"8","author":"L Zadeh","year":"1965","unstructured":"Zadeh, L.: Fuzzy sets. Inf. Control 8, 338\u2013353 (1965)","journal-title":"Inf. Control"},{"key":"170_CR3","doi-asserted-by":"publisher","unstructured":"Singh, M., Rathi, R., Antony, J., Garza-Reyes, J.: Lean six sigma project selection in a manufacturing environment using hybrid methodology based on intuitionistic fuzzy MADM approach. IEEE Trans. Eng. Manag. https:\/\/doi.org\/10.1109\/TEM.2021.3049877","DOI":"10.1109\/TEM.2021.3049877"},{"key":"170_CR4","doi-asserted-by":"publisher","first-page":"115413","DOI":"10.1016\/j.eswa.2021.115413","volume":"183","author":"Y Yener","year":"2021","unstructured":"Yener, Y., Can, G.: A FWMA based novel intuitionistic fuzzy approach proposal: intuitionistic fuzzy advance MCDM and mathematical modeling integration. Expert Syst. Appl. 183, 115413 (2021)","journal-title":"Expert Syst. Appl."},{"key":"170_CR5","doi-asserted-by":"publisher","first-page":"121500","DOI":"10.1016\/j.energy.2021.121500","volume":"237","author":"A Mishra","year":"2021","unstructured":"Mishra, A., Mardani, A., Rani, P., Kamyab, H., Alrasheedi, M.: A new intuitionistic fuzzy combinative distance-based assessment framework to assess low-carbon sustainable suppliers in the maritime sector. Energy 237, 121500 (2021)","journal-title":"Energy"},{"key":"170_CR6","doi-asserted-by":"publisher","first-page":"1698","DOI":"10.1109\/TFUZZ.2021.3064695","volume":"30","author":"S Wan","year":"2022","unstructured":"Wan, S., Dong, J.: A novel extension of best-worst method with intuitionistic fuzzy reference comparisons. IEEE Trans. Fuzzy Syst. 30, 1698\u20131711 (2022)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"170_CR7","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1109\/TFUZZ.2020.3033062","volume":"30","author":"C Cheng","year":"2022","unstructured":"Cheng, C., Ding, W., Xiao, F., Pedrycz, W.: A majority rule-based measure for Atanassov type intuitionistic membership grades in MCDM. IEEE Trans. Fuzzy Syst. 30, 121\u2013132 (2022)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"170_CR8","doi-asserted-by":"publisher","first-page":"1557","DOI":"10.1109\/TFUZZ.2019.2917809","volume":"28","author":"D Jin","year":"2020","unstructured":"Jin, D., Bai, X.: Distribution information based intuitionistic fuzzy clustering for infrared ship segmentation. IEEE Trans. Fuzzy Syst. 28, 1557\u20131571 (2020)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"170_CR9","doi-asserted-by":"publisher","first-page":"2905","DOI":"10.1109\/TFUZZ.2019.2945251","volume":"28","author":"V Singh","year":"2020","unstructured":"Singh, V., Yadav, S., Mesiar, R.: Information measures in Atanassov\u2019s intuitionistic fuzzy environment and their application in decision making. IEEE Trans. Fuzzy Syst. 28, 2905\u20132917 (2020)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"170_CR10","doi-asserted-by":"publisher","first-page":"104512","DOI":"10.1016\/j.engappai.2021.104512","volume":"107","author":"L Pan","year":"2022","unstructured":"Pan, L., Deng, Y.: A novel similarity measure in intuitionistic fuzzy sets and its applications. Eng. Appl. Artif. Intell. 107, 104512 (2022)","journal-title":"Eng. Appl. Artif. Intell."},{"key":"170_CR11","doi-asserted-by":"publisher","first-page":"166","DOI":"10.1016\/j.ijar.2021.07.009","volume":"137","author":"J Duan","year":"2021","unstructured":"Duan, J., Li, X.: Similarity of intuitionistic fuzzy sets and its applications. Int. J. Approx. Reason. 137, 166\u2013180 (2021)","journal-title":"Int. J. Approx. Reason."},{"key":"170_CR12","doi-asserted-by":"publisher","first-page":"192","DOI":"10.1109\/TCYB.2020.2973379","volume":"52","author":"A Tan","year":"2022","unstructured":"Tan, A., Shi, S., Wu, W., Li, J., Pedrycz, W.: Granularity and entropy of intuitionistic fuzzy information and their applications. IEEE Trans. Cybern. 52, 192\u2013204 (2022)","journal-title":"IEEE Trans. Cybern."},{"key":"170_CR13","doi-asserted-by":"publisher","first-page":"1179","DOI":"10.1109\/TFUZZ.2006.890678","volume":"15","author":"Z Xu","year":"2007","unstructured":"Xu, Z.: Intuitionistic fuzzy aggregation operators. IEEE Trans. Fuzzy Syst. 15, 1179\u20131187 (2007)","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"170_CR14","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1016\/j.ins.2021.03.020","volume":"568","author":"K Kumar","year":"2021","unstructured":"Kumar, K., Chen, S.: Multiattribute decision making based on the improved intuitionistic fuzzy Einstein weighted averaging operator of intuitionistic fuzzy values. Inf. Sci. 568, 369\u2013383 (2021)","journal-title":"Inf. Sci."},{"key":"170_CR15","doi-asserted-by":"publisher","first-page":"1529","DOI":"10.1002\/int.22684","volume":"37","author":"T Senapati","year":"2022","unstructured":"Senapati, T., Chen, G., Yager, R.: Aczel-Alsina aggregation operators and their application to intuitionistic fuzzy multiple attribute decision making. Int. J. Intell. Syst. 37, 1529\u20131551 (2022)","journal-title":"Int. J. Intell. Syst."},{"key":"170_CR16","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1016\/j.ins.2021.04.068","volume":"571","author":"W Du","year":"2021","unstructured":"Du, W.: Subtraction and division operations on intuitionistic fuzzy sets derived from the Hamming distance. Inf. Sci. 571, 206\u2013224 (2021)","journal-title":"Inf. Sci."},{"key":"170_CR17","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1016\/S0165-0114(98)00191-2","volume":"114","author":"S De","year":"2000","unstructured":"De, S., Biswas, R., Roy, A.: Some operations on intuitionistic fuzzy sets. Fuzzy Set. Syst. 114, 477\u2013484 (2000)","journal-title":"Fuzzy Set. Syst."},{"key":"170_CR18","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/0165-0114(94)90229-1","volume":"61","author":"K Atanassov","year":"1994","unstructured":"Atanassov, K.: New operations defined over the intuitionistic fuzzy sets. Fuzzy Set. Syst. 61, 137\u2013142 (1994)","journal-title":"Fuzzy Set. Syst."},{"key":"170_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7908-1870-3","volume-title":"Intuitionistic Fuzzy Sets: Theory and Applications","author":"K Atanassov","year":"1999","unstructured":"Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Heidelberg (1999)"},{"key":"170_CR20","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-29127-2","volume-title":"On Intuitionistic Fuzzy Sets Theory","author":"K Atanassov","year":"2012","unstructured":"Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012)"},{"key":"170_CR21","doi-asserted-by":"publisher","first-page":"468","DOI":"10.1002\/int.21696","volume":"30","author":"Q Lei","year":"2015","unstructured":"Lei, Q., Xu, Z.: Derivative and differential operations of intuitionistic fuzzy numbers. Int. J. Intell. Syst. 30, 468\u2013498 (2015)","journal-title":"Int. J. Intell. Syst."},{"key":"170_CR22","doi-asserted-by":"publisher","first-page":"695","DOI":"10.3846\/tede.2020.11920","volume":"26","author":"X Peng","year":"2020","unstructured":"Peng, X., Huang, H.: Fuzzy decision making method based on CoCoSo with critic for financial risk evaluation. Technol. Econ. Dev. Econ. 26, 695\u2013724 (2020)","journal-title":"Technol. Econ. Dev. Econ."},{"key":"170_CR23","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s41066-019-00209-y","volume":"6","author":"M Seikh","year":"2021","unstructured":"Seikh, M., Mandal, U.: Intuitionistic fuzzy dombi aggregation operators and their application to multiple attribute decision-making. Granular Comput. 6, 473\u2013488 (2021)","journal-title":"Granular Comput."},{"key":"170_CR24","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1090\/S0002-9939-04-07729-9","volume":"133","author":"C Draghici","year":"2005","unstructured":"Draghici, C.: A general rearrangement inequality. Proc. Am. Math. Soc. 133, 735\u2013743 (2005)","journal-title":"Proc. Am. Math. Soc."},{"key":"170_CR25","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1080\/00029890.1987.12000678","volume":"94","author":"F Burk","year":"1987","unstructured":"Burk, F.: The geometric, logarithmic, and arithmetic mean inequality. Am. Math. Monthly. 94, 527\u2013528 (1987)","journal-title":"Am. Math. Monthly."},{"key":"170_CR26","doi-asserted-by":"publisher","first-page":"523","DOI":"10.7153\/jmi-07-47","volume":"7","author":"Q Wang","year":"2013","unstructured":"Wang, Q.: Some Nesbitt type inequalities with applications for the Zeta functions. J. Math. Inequal. 7, 523\u2013527 (2013)","journal-title":"J. Math. Inequal."},{"key":"170_CR27","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1080\/00029890.1989.11972154","volume":"96","author":"C Cooper","year":"1989","unstructured":"Cooper, C., Kennedy, R.: Chebyshev\u2019s inequality and natural density. Am. Math. Monthly. 96, 118\u2013124 (1989)","journal-title":"Am. Math. Monthly."},{"key":"170_CR28","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1080\/0025570X.1964.11975469","volume":"37","author":"E Tolsted","year":"1964","unstructured":"Tolsted, E.: An elementary derivation of the Cauchy, Holder, and Minkowski inequalities from Young\u2019s inequality. Math. Mag. 37, 2\u201312 (1964)","journal-title":"Math. Mag."},{"key":"170_CR29","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1016\/j.aml.2010.12.044","volume":"24","author":"M Wang","year":"2011","unstructured":"Wang, M., Chu, Y., Qiu, Y., Qiu, S.: An optimal power mean inequality for the complete elliptic integrals. Appl. Math. Lett. 24, 887\u2013890 (2011)","journal-title":"Appl. Math. Lett."},{"key":"170_CR30","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1016\/0022-247X(84)90081-7","volume":"100","author":"J Bertolo","year":"1984","unstructured":"Bertolo, J., Fernandez, D.: A multidimensional version of the Carlson inequality. J. Math. Anal. Appl. 100, 302\u2013306 (1984)","journal-title":"J. Math. Anal. Appl."},{"key":"170_CR31","first-page":"6566","volume":"218","author":"M Bakula","year":"2012","unstructured":"Bakula, M., Pe\u010dari\u0107, J., Peri\u0107, J.: On the converse Jensen inequality. Appl. Math. Comput. 218, 6566\u20136575 (2012)","journal-title":"Appl. Math. Comput."},{"key":"170_CR32","volume-title":"Wei\u2013Wei dual inequality and its applications","author":"Y Zhang","year":"2014","unstructured":"Zhang, Y.: Wei\u2013Wei dual inequality and its applications, 2nd edn. USTC Press, Hefei (2014)","edition":"2"},{"key":"170_CR33","doi-asserted-by":"publisher","first-page":"110","DOI":"10.4169\/002557010X496407","volume":"8","author":"R Pratt","year":"2010","unstructured":"Pratt, R.: Proof without words: a tangent inequality. Math. Magn. 8, 110 (2010)","journal-title":"Math. Magn."},{"key":"170_CR34","doi-asserted-by":"publisher","first-page":"181","DOI":"10.7153\/jmi-03-18","volume":"3","author":"J Paris","year":"2009","unstructured":"Paris, J., Vencovska, A.: A generalization of Muirhead\u2019s inequality. J. Math. Inequal. 3, 181\u2013187 (2009)","journal-title":"J. Math. Inequal."},{"key":"170_CR35","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1080\/03081087908817292","volume":"7","author":"JD da Silva","year":"1979","unstructured":"da Silva, J.D.: On the Schur inequality. Linear Multilinear A. 7, 343\u2013357 (1979)","journal-title":"Linear Multilinear A."},{"key":"170_CR36","first-page":"29","volume":"20","author":"S Bao","year":"2017","unstructured":"Bao, S.: Proof of Vasc inequalities based on characteristics of convexity and concavity of functions. Stud. Coll. Math. 20, 29\u201330 (2017)","journal-title":"Stud. Coll. Math."},{"key":"170_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00283-011-9266-8","volume":"34","author":"L Maligranda","year":"2012","unstructured":"Maligranda, L.: The AM-GM inequality is equivalent to the Bernoulli inequality. Math. Intell. 34, 1\u20132 (2012)","journal-title":"Math. Intell."},{"key":"170_CR38","volume-title":"Abel\u2019s Identities and Classical Inequalities and Their Applications","author":"Z Yang","year":"2018","unstructured":"Yang, Z.: Abel\u2019s Identities and Classical Inequalities and Their Applications, 6th edn. HIT Press, Harbin (2018)","edition":"6"},{"key":"170_CR39","first-page":"53","volume":"27","author":"M Vassilev-Missana","year":"2021","unstructured":"Vassilev-Missana, M.: Note on one inequality and its application in intuitionistic fuzzy sets theory. Part 1. Notes on Intuitionistic. Fuzzy Sets 27, 53\u201359 (2021)","journal-title":"Fuzzy Sets"},{"key":"170_CR40","first-page":"78","volume":"27","author":"M Vassilev-Missana","year":"2021","unstructured":"Vassilev-Missana, M.: Note on one inequality and its application in intuitionistic fuzzy sets theory. Part 2. Notes on Intuitionistic. Fuzzy Sets 27, 78\u201381 (2021)","journal-title":"Fuzzy Sets"},{"key":"170_CR41","unstructured":"Zhao, B.: Half concave and half convex theorem. http:\/\/mathlinks.ro\/community\/h64933 (2005)"}],"container-title":["International Journal of Computational Intelligence Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44196-022-00170-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s44196-022-00170-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s44196-022-00170-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,20]],"date-time":"2022-12-20T10:18:26Z","timestamp":1671531506000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s44196-022-00170-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,20]]},"references-count":41,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["170"],"URL":"https:\/\/doi.org\/10.1007\/s44196-022-00170-w","relation":{},"ISSN":["1875-6883"],"issn-type":[{"type":"electronic","value":"1875-6883"}],"subject":[],"published":{"date-parts":[[2022,12,20]]},"assertion":[{"value":"8 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 November 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 December 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interests regarding the publication for the paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"111"}}