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Intuitionistic fuzzy integrals based on Archimedean t-conorms and t-norms. (English) Zbl 1390.68664

Summary: Intuitionistic fuzzy set (IFS) is an extension of fuzzy sets. The basic element of an IFS is the ordered pair called intuitionistic fuzzy number (IFN). In order to solve decision making problems under intuitionistic fuzzy environments, many aggregation techniques for IFNs have been proposed, most of which can only deal with discrete intuitionistic fuzzy information. In this paper, we define two subtraction and division operations of IFNs, and develop a sequence of general integrals dealing with continuous intuitionistic fuzzy data based on Archimedean t-conorm and t-norm. Then we discuss some special cases, and investigate the basic properties of these intuitionistic fuzzy integrals.

MSC:

68T37 Reasoning under uncertainty in the context of artificial intelligence
28E10 Fuzzy measure theory
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[1] Atanassov, K., Intuitionistic fuzzy set, Fuzzy Sets Syst., 20, 87-96 (1986) · Zbl 0631.03040
[2] Atanassov, K., On Intuitionistic Fuzzy Sets Theory (2012), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1247.03112
[3] Beliakov, G.; Bustince, H.; Goswami, D. P.; Mukherjee, U. K.; Pal, N. R., On averaging operators for Atanassov’s intuitionistic fuzzy sets, Inf. Sci., 181, 1116-1124 (2011) · Zbl 1215.03064
[4] Beliakov, G.; Bustince, H.; James, S.; Calvo, T.; Fernandez, J., Aggregation for Atanassov’s intuitionistic and interval valued fuzzy sets: the median operator, IEEE Trans. Fuzzy Syst., 20, 487-498 (2012)
[5] Bustince, H.; Herrera, F.; Montero, J., Fuzzy Sets and Their Extensions: Representation, Aggregation and Models (2008), Springer-Verlag: Springer-Verlag Berlin
[6] Bustince, H.; Fernández, J.; Kolesárová, A.; Mesiar, R., Generation of linear orders for intervals by means of aggregation functions, Fuzzy Sets Syst., 220, 69-77 (2013) · Zbl 1284.03242
[7] Bustince, H.; Galar, M.; Bedregal, B.; Kolesárová, A.; Mesiar, R., A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications, IEEE Trans. Fuzzy Syst., 21, 1150-1162 (2013)
[8] Chen, N.; Xu, Z. S., Hesitant fuzzy ELECTRE II approach: a new way to handle multi-criteria decision making problems, Inf. Sci., 292, 175-197 (2015)
[9] Jiang, Y.; Xu, Z. S.; Yu, X. H., Group decision making based on incomplete intuitionistic multiplicative preference relations, Inf. Sci., 295, 33-52 (2015) · Zbl 1360.91069
[10] Khatibi, V.; Montazer, G. A., Intuitionistic fuzzy set vs. fuzzy set application in medical pattern recognition, Artif. Intell. Med., 47, 43-52 (2009)
[11] Klement, E. P.; Mesiar, R., Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms (2005), Elsevier: Elsevier New York
[12] Klir, G.; Yuan, B., Fuzzy Sets and Fuzzy Logic: Theory and Applications (1995), Prentice Hall: Prentice Hall Upper Saddle River · Zbl 0915.03001
[13] Lei, Q.; Xu, Z. S., Derivative and differential operations of intuitionistic fuzzy numbers, Int. J. Intell. Syst., 30, 468-498 (2015)
[14] Lei, Q.; Xu, Z. S.; Bustince, H.; Burusco, A., Definite integrals of Atanassov’s intuitionistic fuzzy information, IEEE Trans. Fuzzy Syst. (2014)
[15] Li, D. F., Decision and Game Theory in Management With Intuitionistic Fuzzy Sets (2014), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1291.90003
[16] Lopes, N. V.; Mogadouro do Couto, P. A.; Bustince, H.; Melo-Pinto, P., Automatic histogram threshold using fuzzy measures, IEEE Trans. Image Process., 19, 199-204 (2010) · Zbl 1371.94386
[17] Merigó, J. M.; Gil-Lafuente, A. M.; Yager, R. R., An overview of fuzzy research with bibliometric indicators, Appl. Soft Comput., 27, 420-433 (2015)
[18] Paternain, D.; Jurio, A.; Barrenechea, E.; Bustince, H.; Bedregal, B.; Szmidt, E., An alternative to fuzzy methods in decision-making problems, Expert Syst. Appl., 39, 7729-7735 (2012)
[19] Pedrycz, W., Granular Computing: Analysis and Design of Intelligent Systems (2013), CRC Press/Francis Taylor: CRC Press/Francis Taylor Boca Raton
[20] Szmidt, E., Distances and Similarities in Intuitionistic Fuzzy Sets (2013), Springer-Verlag: Springer-Verlag Berlin, Heidelberg
[21] Szmidt, E.; Kacprzyk, J., On an enhanced method for a more meaningful ranking of intuitionistic fuzzy alternatives, Artificial Intelligence and Soft Computing, Lecture Notes in Computer Science, 6113, 232-239 (2010)
[22] Szmidt, E.; Kacprzyk, J., Ranking of intuitionistic fuzzy alternatives in a multi-criteria decision making problem, (Proceedings of the 28th North American Fuzzy Information Processing Society Annual Conference. Proceedings of the 28th North American Fuzzy Information Processing Society Annual Conference, Cincinnati, OH (Jun. 14-17, 2009))
[23] Vlachos, K. I.; Sergiadis, G. D., Intuitionistic fuzzy information-applications to pattern recognition, Pattern Recognit. Lett., 28, 197-206 (2007)
[24] Xia, M. M.; Xu, Z. S.; Zhu, B., Generalized intuitionistic fuzzy Bonferroni means, Int. J. Intell. Syst., 27, 23-47 (2012)
[25] Xia, M. M.; Xu, Z. S.; Zhu, B., Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowl. Based Syst., 31, 78-88 (2012)
[26] Xu, Z. S., Intuitionistic Fuzzy Aggregation and Clustering (2013), Springer-Verlag: Springer-Verlag Berlin, Heidelberg
[27] Xu, Z. S., Intuitionistic fuzzy aggregation operators, IEEE Trans. Fuzzy Syst., 15, 1179-1187 (2007)
[28] Xu, Z. S.; Yager, R. R., Some geometric aggregation operators based on intuitionistic fuzzy sets, Int. J. Gen. Syst., 35, 417-433 (2006) · Zbl 1113.54003
[29] Xu, Z. S.; Cai, X. Q., Intuitionistic Fuzzy Information Aggregation: Theory and Applications (2012), Science Press: Science Press Beijing, Springer-Verlag, Berlin, Heidelberg
[30] Yager, R. R.; Kacprzyk, J., The Ordered Weighted Averaging Operators: Theory and Applications (1997), Kluwer: Kluwer Norwell, MA · Zbl 0948.68532
[31] Yu, D.; Shi, S., Researching the development of Atanassov intuitionistic fuzzy set: using a citation network analysis, Appl. Soft Comput., 32, 189-198 (2015)
[32] Zadeh, L. A., Fuzzy sets, Inf. Control, 8, 338-356 (1965) · Zbl 0139.24606
[33] Zhang, X. M.; Xu, Z. S., A new method for ranking intuitionistic fuzzy values and its application in multi-attribute decision making, Fuzzy Optim. Decis. Mak., 12, 135-146 (2012) · Zbl 1254.90320
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