×

zbMATH — the first resource for mathematics

On averaging operators for Atanassov’s intuitionistic fuzzy sets. (English) Zbl 1215.03064
Let \(A =\{\langle x, \mu_{A}(x), \nu_{A}(x)\rangle\mid x \in E\}\) be an intuitionistic fuzzy set (IFS), where \(E\) is a given universe, the functions \(\mu_{A}: E \rightarrow [0,1]\) and \(\nu_{A}: E \rightarrow [0,1]\) define the degrees of membership and of non-membership of the element \(x\in E\) to the set \(A\), respectively, and for every \(x\in E\), \(0 \leq \mu_{A}(x) + \nu_{A}(x) \leq 1.\) The authors introduce the following two new averaging operators over IFSs:
\[ \text{IWAM}_w(A_1,\dots, A_n)= w_1A_1+ w_2A_2+\cdots + w_nA_n = \bigg\langle 1- \prod_{i=1}^n (1-\mu_{A_i})^{w_i}, \prod_{i=1}^n (\nu_{A_i})^{w_i} \bigg\rangle \]
and
\[ \text{IOWA}_w(A_1,\dots,A_n)= w_1A_{\sigma(1)}+ w_2A_{\sigma(2)}+\cdots+ w_nA_{\sigma(n)}= \!\bigg\langle 1-\prod_{i=1}^n (1-\mu_{A_{\sigma(i)}})^{w_i}, \prod_{i=1}^n (\nu_{A_{\sigma(i)}})^{w_i} \!\bigg\rangle, \]
where \(w = (w_1, w_2,\dots, w_n)\), \(w_i\in [0,1]\), \(\sum_{i=1}^n w_i=1\), and \(A_{\sigma(i)}\) is the \(i\)-th largest value according to the total order \(A_{\sigma(1)} \geq\dots \geq A_{\sigma(n)}\). These operators are called, respectively, Intuitionistic Weighted Arithmetic Mean with respect to a weighting vector \(w\), and Intuitionistic Ordered Weighted Averaging with respect to a weighting vector \(w\). Some extensions of these operators are given and their basic proterties are studied.
Reviewer’s remark: In the names of the two operators the word “fuzzy” should be inserted after the word “intuitionistic”, because without it these names would refer to Brouwer’s concept of intuitionism, and not to IFS theory.

MSC:
03E72 Theory of fuzzy sets, etc.
Software:
AIFS
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aczél, J., Lectures on functional equations and their applications, (1966), Academic Press New York · Zbl 0139.09301
[2] Atanassov, K., Intuitionistic fuzzy sets, Fuzzy sets and systems, 80, 87-96, (1986) · Zbl 0631.03040
[3] Atanassov, K., More on intuitionistic fuzzy sets, Fuzzy sets and systems, 33, 37-46, (1989) · Zbl 0685.03037
[4] Atanassov, K., New operations defined over the intuitionistic fuzzy sets, Fuzzy sets and systems, 61, 137-142, (1994) · Zbl 0824.04004
[5] Atanassov, K.; Gargov, G., Interval valued intuitionistic fuzzy sets, Fuzzy sets and systems, 31, 343-349, (1989) · Zbl 0674.03017
[6] Beliakov, G.; Warren, J., Appropriate choice of aggregation operators in fuzzy decision support system, IEEE transactions on fuzzy systems, 9, 773-784, (2001)
[7] Beliakov, G.; Pradera, A.; Calvo, T., Aggregation functions: A guide for practitioners, (2007), Springer Heidelberg, Berlin, New York · Zbl 1123.68124
[8] Bustince, H.; Burillo, P., Structures on intuitionistic fuzzy relations, Fuzzy sets and systems, 78, 293-303, (1996) · Zbl 0875.04006
[9] Bustince, H.; Burillo, P., Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems, 79, 403-405, (1996) · Zbl 0871.04006
[10] Deschrijver, G.; Kerre, E.E., On the relationship between some extensions of fuzzy set theory, Fuzzy sets and systems, 133, 227-235, (2003) · Zbl 1013.03065
[11] Deschrijver, G.; Kerre, E.E., On the composition of intuitionistic fuzzy relations, Fuzzy sets and systems, 136, 333-361, (2003) · Zbl 1028.03047
[12] Deschrijver, G.; Kerre, E.E., Implicators based on binary aggregation operators in interval-valued fuzzy set theory, Fuzzy sets and systems, 153, 229-248, (2005) · Zbl 1090.03024
[13] Deschrijver, G., Arithmetic operators in interval-valued fuzzy set theory, Information sciences, 177, 2906-2924, (2007) · Zbl 1120.03033
[14] Deschrijver, G.; Kerre, E.E., On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision, Information sciences, 177, 1860-1866, (2007) · Zbl 1121.03074
[15] Deschrijver, G.; Kerre, E.E., Aggregation operators in interval-valued fuzzy and atanassov’s intuitionistic fuzzy set theory, (), 183-203 · Zbl 1201.68118
[16] Dey, S.K.; Biswas, R.; Roy, A.R., Some operations on intuitionistic fuzzy sets, Fuzzy sets and systems, 114, 477-484, (2000) · Zbl 0961.03049
[17] Gau, W.L.; Buehrer, D.J., Vague sets, IEEE transactions on systems, man and cybernetics, 23, 610-614, (1993) · Zbl 0782.04008
[18] Goguen, J., L-fuzzy sets, Journal of mathematical analysis and applications, 18, 143-174, (1967) · Zbl 0145.24404
[19] Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation functions, (2009), Cambridge University Press Cambridge
[20] Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation functions: means, Information sciences, 181, 1-22, (2011) · Zbl 1206.68298
[21] ()
[22] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic theory and application, (1997), Prentice- Hall of India Pvt. Ltd. New Delhi
[23] Li, D.-F., TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets, IEEE transactions on fuzzy systems, 18, 299-311, (2010)
[24] Montero, J.; Gómez, D.; Bustince, H., On the relevance of some families of fuzzy sets, Fuzzy sets and systems, 158, 2429-2442, (2007) · Zbl 1157.03319
[25] M. Nachtegael, P. Sussner, T. Melange, E.E. Kerre, On the role of complete lattices in mathematical morphology: From tool to uncertainty model, Information Sciences, in press, doi:10.1016/j.ins.2010.03.009. · Zbl 1216.68326
[26] Szmidt, E.; Kacprzyk, J., Remarks on some application of intuitionistic fuzzy sets in decision making, Notes on IFS, 2, 2-31, (1996) · Zbl 0865.90003
[27] E. Szmidt, J. Kacprzyk, Group decision making under intuitionistic fuzzy preference relation, in: Proc IPMU 98, 1998, pp. 172-178.
[28] E. Szmidt, J. Kacprzyk, On measures on consensus under intuitionistic fuzzy relations, in: Proc IPMU 2000, 2000, pp. 1454-1461.
[29] E. Szmidt, J. Kacprzyk, Analysis of agreement in a group of experts via distances between intuitionistic fuzzy preferences, in: Proc. IPMU 2002, 2002, pp. 1859-1865.
[30] Szmidt, E.; Kacprzyk, J., An intuitionistic fuzzy set based approach to intelligent data analysis: an application to medical diagnosis, (), 57-70
[31] Tan, C.; Chen, X., Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert systems with applications, 37, 149-157, (2010)
[32] Torra, Y.; Narukawa, V., Modeling decisions, Information fusion and aggregation operators, (2007), Springer Berlin, Heidelberg
[33] Wei, G., Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied soft computing, 10, 423-431, (2010)
[34] Xu, Z., Choquet integrals of weighted intuitionistic fuzzy information, Information sciences, 180, 726-736, (2010) · Zbl 1186.68469
[35] Xu, Z., Intuitionistic fuzzy aggregation operations, IEEE transactions on fuzzy systems, 15, 1179-1187, (2007)
[36] Xu, Z.; Yager, R., Some geometric aggregation operators based on intuitionistic fuzzy sets, International journal of general systems, 35, 417-433, (2006) · Zbl 1113.54003
[37] Yager, R., On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE transactions on systems, man and cybernetics, 18, 183-190, (1988) · Zbl 0637.90057
[38] Yager, R., OWA aggregation of intuitionistic fuzzy sets, International journal of general systems, 38, 617-641, (2009) · Zbl 1187.68615
[39] Zadeh, L.A., Fuzzy sets, Information control, 8, 338-353, (1965) · Zbl 0139.24606
[40] Zadeh, L.A., Outline of a new approach to the analysis of complex systems and decision processes interval-valued fuzzy sets, IEEE transactions on systems, man and cybernetics, SMC-3, 28-44, (1973) · Zbl 0273.93002
[41] Zadeh, L.A., Is there a need for fuzzy logic?, Information sciences, 178, 2751-2779, (2008) · Zbl 1148.68047
[42] Zadeh, L.A., Toward extended fuzzy logic - A first step, Fuzzy sets and systems, 160, 3175-3181, (2009) · Zbl 1185.03042
[43] Zhao, H.; Xu, Z.; Ni, M.; Liu, S., Generalized aggregation operators for intuitionistic fuzzy sets, International journal of intelligent systems, 25, 1-30, (2010) · Zbl 1185.68660
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.