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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:
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