On averaging operators for Atanassov's intuitionistic fuzzy sets
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.
scientific article; zbMATH DE number 5870610
scientific article