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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:
 3e+72 Theory of fuzzy sets, etc.
AIFS
Full Text:
##### References:
  Aczél, J., Lectures on functional equations and their applications, (1966), Academic Press New York · Zbl 0139.09301  Atanassov, K., Intuitionistic fuzzy sets, Fuzzy sets and systems, 80, 87-96, (1986) · Zbl 0631.03040  Atanassov, K., More on intuitionistic fuzzy sets, Fuzzy sets and systems, 33, 37-46, (1989) · Zbl 0685.03037  Atanassov, K., New operations defined over the intuitionistic fuzzy sets, Fuzzy sets and systems, 61, 137-142, (1994) · Zbl 0824.04004  Atanassov, K.; Gargov, G., Interval valued intuitionistic fuzzy sets, Fuzzy sets and systems, 31, 343-349, (1989) · Zbl 0674.03017  Beliakov, G.; Warren, J., Appropriate choice of aggregation operators in fuzzy decision support system, IEEE transactions on fuzzy systems, 9, 773-784, (2001)  Beliakov, G.; Pradera, A.; Calvo, T., Aggregation functions: A guide for practitioners, (2007), Springer Heidelberg, Berlin, New York · Zbl 1123.68124  Bustince, H.; Burillo, P., Structures on intuitionistic fuzzy relations, Fuzzy sets and systems, 78, 293-303, (1996) · Zbl 0875.04006  Bustince, H.; Burillo, P., Vague sets are intuitionistic fuzzy sets, Fuzzy sets and systems, 79, 403-405, (1996) · Zbl 0871.04006  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  Deschrijver, G.; Kerre, E.E., On the composition of intuitionistic fuzzy relations, Fuzzy sets and systems, 136, 333-361, (2003) · Zbl 1028.03047  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  Deschrijver, G., Arithmetic operators in interval-valued fuzzy set theory, Information sciences, 177, 2906-2924, (2007) · Zbl 1120.03033  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  Deschrijver, G.; Kerre, E.E., Aggregation operators in interval-valued fuzzy and atanassov’s intuitionistic fuzzy set theory, (), 183-203 · Zbl 1201.68118  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  Gau, W.L.; Buehrer, D.J., Vague sets, IEEE transactions on systems, man and cybernetics, 23, 610-614, (1993) · Zbl 0782.04008  Goguen, J., L-fuzzy sets, Journal of mathematical analysis and applications, 18, 143-174, (1967) · Zbl 0145.24404  Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation functions, (2009), Cambridge University Press Cambridge  Grabisch, M.; Marichal, J.-L.; Mesiar, R.; Pap, E., Aggregation functions: means, Information sciences, 181, 1-22, (2011) · Zbl 1206.68298  ()  Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic theory and application, (1997), Prentice- Hall of India Pvt. Ltd. New Delhi  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)  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  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  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  E. Szmidt, J. Kacprzyk, Group decision making under intuitionistic fuzzy preference relation, in: Proc IPMU 98, 1998, pp. 172-178.  E. Szmidt, J. Kacprzyk, On measures on consensus under intuitionistic fuzzy relations, in: Proc IPMU 2000, 2000, pp. 1454-1461.  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.  Szmidt, E.; Kacprzyk, J., An intuitionistic fuzzy set based approach to intelligent data analysis: an application to medical diagnosis, (), 57-70  Tan, C.; Chen, X., Intuitionistic fuzzy Choquet integral operator for multi-criteria decision making, Expert systems with applications, 37, 149-157, (2010)  Torra, Y.; Narukawa, V., Modeling decisions, Information fusion and aggregation operators, (2007), Springer Berlin, Heidelberg  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)  Xu, Z., Choquet integrals of weighted intuitionistic fuzzy information, Information sciences, 180, 726-736, (2010) · Zbl 1186.68469  Xu, Z., Intuitionistic fuzzy aggregation operations, IEEE transactions on fuzzy systems, 15, 1179-1187, (2007)  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  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  Yager, R., OWA aggregation of intuitionistic fuzzy sets, International journal of general systems, 38, 617-641, (2009) · Zbl 1187.68615  Zadeh, L.A., Fuzzy sets, Information control, 8, 338-353, (1965) · Zbl 0139.24606  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  Zadeh, L.A., Is there a need for fuzzy logic?, Information sciences, 178, 2751-2779, (2008) · Zbl 1148.68047  Zadeh, L.A., Toward extended fuzzy logic - A first step, Fuzzy sets and systems, 160, 3175-3181, (2009) · Zbl 1185.03042  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
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