Classification of artificial intelligence tools for civil engineering under the notion of complex fuzzy rough Frank aggregation operators
In this section, we have to discuss the notion of AOs called CFRF AOs. Now throughout this section,
\({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be a collection of CFRNs. Also, assume that \({\Omega }_{\mathbbm{w}}=\left({\Omega }_{{\mathbbm{w}}-1}, {\Omega }_{{\mathbbm{w}}-2}\dots , {\Omega }_{{\mathbbm{w}}-n}\right)\) denote the weight vector (WV) such that \(0\le {\Omega }_{{\mathbbm{w}}-j}\le 1\), and \(\sum_{j=1}^{n}{\Omega }_{{\mathbbm{w}}-j}=1.\)
Complex fuzzy rough Frank arithmetic AOs
This part of the article is devoted to defining the notion of CFRF arithmetic AOs. The overall discussion is given by:
Definition 10
For the family of CFRNs, the CFR Frank weighted average (CFRFWA) AO is the function \({\left({\hat{C} }_{om.}\right)}^{n}\to \left({\hat{C} }_{om.}\right)\) given by
$$CFRFWA \left({\hat{\text{C}} }_{om.-1}, {\hat{\text{C}} }_{om.-2}, \dots , {\hat{\text{C}} }_{om.-n}\right)=\begin{array}{c}n\\ \oplus \\ j=1\end{array}{\Omega }_{{\mathbbm{w}}-j}{\hat{\text{C}} }_{om.-j}$$
(1)
Theorem 2
Let \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be a collection of CFRNs, the aggregating result by using Eq. (1) is again CFRN given as
$$\begin{aligned} & CFRFWA\left({\hat{\text{C}} }_{om.-1}, {\hat{\text{C}} }_{om.-2}, \dots , {\hat{\text{C}} }_{om.-n}\right)\\ & \quad =\left(\begin{array}{c}1-\left({{\text{log}}}_{{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{{{\rotatebox{180}{{\rm h}}}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right)+\iota 1-\left({{\text{log}}}_{{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{{{\rotatebox{180}{{\rm h}}}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right),\\ 1-\left({{\text{log}}}_{{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{{{ { \bar{\d{\rm L}}} }}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right)+\iota 1-\left({{\text{log}}}_{{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{{{\sf (} \! \rotatebox{146}{\sf c}}}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{{{ { \bar{\d{\rm L}}} }}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right),\end{array}\right) \end{aligned}$$
(2)
Proof
Using the method of mathematical induction, for \(n=2\), we get
$$CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}\right)={\Omega }_{{\mathbbm{w}}-1}{\hat{C} }_{om.-1}\oplus {\Omega }_{{\mathbbm{w}}-2}{\hat{C} }_{om.-2}$$
$$=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{1}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-1}}\right)+\iota {1-{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{1- \frown\!\!\!\!\!\! {\smallint} }_{1}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{1}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-1}}\right)+\iota {1-{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{1- \frown\!\!\!\!\!\! {\smallint} }_{1}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-1}}\right)\end{array}\right)$$
$$\oplus \left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{2}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}-1}}\right)+\iota {1-{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{1- \frown\!\!\!\!\!\! {\smallint} }_{2}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}-1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{2}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}-1}}\right)+\iota {1-{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{1- \frown\!\!\!\!\!\! {\smallint} }_{2}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-2}-1}}\right)\end{array}\right)$$
$$=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{2}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{2}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{2}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{2}{\Omega }_{{\mathbbm{w}}-j}-1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{2}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{2}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{2}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{2}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)\end{array}\right)$$
Hence, Eq. (2) is true for \(n=2\).
Now suppose Eq. (2) holds for some \({\mathcalligra{k}}\) i.e.
$$\begin{aligned} & CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-{\mathcalligra{k}}}\right)\\ &\quad=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)\end{array}\right)\end{aligned}$$
Now we prove that Eq. (2) holds for \(n={\mathcalligra{k}}+1\).
$$CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-{\mathcalligra{k}}}, {\hat{C} }_{om.-{\mathcalligra{k}}+1}\right)=CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-{\mathcalligra{k}}}\right)\oplus {\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}{\hat{C} }_{om.-{\mathcalligra{k}}+1}$$
$$=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)\end{array}\right)$$
$$\oplus \left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{1}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-{\mathcalligra{k}}+1}}\right)+\iota {1-{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{1- \frown\!\!\!\!\!\! {\smallint} }_{1}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-{\mathcalligra{k}}+1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{1}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-{\mathcalligra{k}}+1}}\right)+\iota {1-{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{1- \frown\!\!\!\!\!\! {\smallint} }_{1}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-1}-{\mathcalligra{k}}+1}}\right)\end{array}\right)$$
where
$$A=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)}-1\right)\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}-1}}\right)}-1\right)}{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1}\right)\\ +\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)}-1\right)\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}-1}}\right)}-1\right)}{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1}\right)\end{array}\right)$$
And
$$B=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)}-1\right)\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}-1}}\right)}-1\right)}{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1}\right)\\ +\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)}-1\right)\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{{\Omega }_{{\mathbbm{w}}-{\mathcalligra{k}}+1}-1}}\right)}-1\right)}{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1}\right)\end{array}\right)$$
$$=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}+1}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}+1}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}+1}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}+1}{\Omega }_{{\mathbbm{w}}-j}-1}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}+1}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}+1}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\frac{\prod_{j=1}^{{\mathcalligra{k}}+1}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}}{{\left({{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }-1\right)}^{\sum_{j=1}^{{\mathcalligra{k}}+1}{\Omega }_{{\mathbbm{w}}-j}-1}}\right)\end{array}\right)$$
Hence, Eq. (2) is valid for \(n={\mathcalligra{k}}+1\). Therefore, Eq. (2) is valid for all \(n\).
Now we will discuss that CFRFWA AOs satisfy the following characteristics. Here \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right)\) and \({\hat{C} }_{om.-j}^{\divideontimes }=\left({{\mathfrak{L}}^{\divideontimes }}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {{\mathfrak{L}}^{\divideontimes }}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) represent two collections of CFRNs, then
1. (Idempotency): If all \({\hat{C} }_{om.-j}={\hat{C} }_{om.} \forall j\) then
$$CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)={\hat{C} }_{om.}$$
2. Boundedness: Let \({{\hat{C} }_{om.}}^{-}=\left(\underset{j}{{\text{min}}}\left\{{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}+\iota \underset{j}{{\text{min}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}, \underset{j}{{\text{min}}}\left\{{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}+\iota \underset{j}{{\text{min}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}\right),\) and \({{\hat{C} }_{om.}}^{+}=\left(\underset{j}{{\text{max}}}\left\{{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}+\iota \underset{j}{{\text{max}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}, \underset{j}{{\text{max}}}\left\{{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}+\iota \underset{j}{{\text{max}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}\right)\). Then
$${{\hat{C} }_{om.}}^{-}\le CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\le {{\hat{C} }_{om.}}^{+}$$
3. Monotonicity: If \({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\le {{\mathfrak{L}}^{\divideontimes }}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\), \({ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\le { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\), \({\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\le {{\mathfrak{L}}^{\divideontimes }}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\), \({ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\le { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\forall j\), then
$$CFRFWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\le CFRFWA\left({\hat{C} }_{om.-1}^{\divideontimes }, {\hat{C} }_{om.-2}^{\divideontimes }, \dots , {\hat{C} }_{om.-n}^{\divideontimes }\right)$$
Complex fuzzy rough Frank ordered weighted average (CFRFOWA) AOs
The notion of CFROWA AO is defined here. Moreover, we will analyze the characteristics of this delivered approach.
Definition 11
Let \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be a collection of CFRNs, then
$$CFRFOWA \left({\hat{\text{C}}}_{om.-1}, {\hat{\text{C}} }_{om.-2}, \ldots , {\hat{\text{C}} }_{om.-n}\right)=\begin{array}{c}n\\ \oplus \\ j=1\end{array}{\Omega }_{{\mathbbm{w}}-j}{\hat{\text{C}} }_{om.-\mathfrak{d}\left(j\right)}$$
(3)
Where, \(\left(\mathfrak{d}\left(1\right),\mathfrak{d}\left(2\right),\dots ,\mathfrak{d}\left(n\right)\right)\) represent the permutation of \(j=1, 2, .., n\) with \({\hat{C} }_{om.-\mathfrak{d}\left(j-1\right)}\ge {\hat{C} }_{om.-\mathfrak{d}\left(j\right)}\) for all \(j.\)
Theorem 3
Let \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be a collection of CFRNs, then the aggregated result by using Eq. (3) is again CFRN
$$\begin{aligned} & CFRFOWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\\ & \quad =\left(\begin{array}{c}1-\left({{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{\mathfrak{d}\left(j\right)}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right)+\iota 1-\left({{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{\mathfrak{d}\left(j\right)}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right),\\ 1-\left({{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{\mathfrak{d}\left(j\right)}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right)+\iota 1-\left({{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{\mathfrak{d}\left(j\right)}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\right),\end{array}\right)\end{aligned}$$
(4)
Proof
Proof is the same as Theorem 2.
Let \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right)\) and \({\hat{C} }_{om.-j}^{\divideontimes }=\left({{\mathfrak{L}}^{\divideontimes }}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {{\mathfrak{L}}^{\divideontimes }}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be two collections of CFRNs, then CFRFOWA AOs follow the characteristics given by
1. (Idempotency): If all \({\hat{C} }_{om.-j}={\hat{C} }_{om.} \forall j\) then
$$CFRFOWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)={\hat{C} }_{om.}$$
2. Boundedness: Let \({{\hat{C} }_{om.}}^{-}=\left(\underset{j}{{\text{min}}}\left\{{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}+\iota \underset{j}{{\text{min}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}, \underset{j}{{\text{min}}}\left\{{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}+\iota \underset{j}{{\text{min}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}\right),\) and \({{\hat{C} }_{om.}}^{+}=\left(\underset{j}{{\text{max}}}\left\{{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}+\iota \underset{j}{{\text{max}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}, \underset{j}{{\text{max}}}\left\{{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}+\iota \underset{j}{{\text{max}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}\right)\). Then
$${{\hat{C} }_{om.}}^{-}\le CFRFOWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\le {{\hat{C} }_{om.}}^{+}$$
3. Monotonicity: If \({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\le {{\mathfrak{L}}^{\divideontimes }}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\), \({ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\le { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\), \({\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\le {{\mathfrak{L}}^{\divideontimes }}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\), \({ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\le { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\forall j\), then
$$CFRFOWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\le CFRFOWA\left({\hat{C} }_{om.-1}^{\divideontimes }, {\hat{C} }_{om.-2}^{\divideontimes }, \dots , {\hat{C} }_{om.-n}^{\divideontimes }\right)$$
Complex fuzzy rough Frank hybrid weighted average (CFRHWA) AOs
Here in this section, by combining the characteristics of the CFRFWA and CFRFOWA, we define the notion of CFRFHWA AOs as:
Definition 12
Let \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be a collection of CFRNs, then CFRHWA AOs are given by
$$CFRFHWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)=\begin{array}{c}n\\ \oplus \\ j=1\end{array}{{\Omega }^{\between }}_{{\mathbbm{w}}-j}{{\hat{C} }^{\between }}_{om.-\mathfrak{d}\left(j\right)}$$
$$=\left(\begin{array}{c}1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{\mathfrak{d}\left(j\right)}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{\mathfrak{d}\left(j\right)}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right),\\ 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{\mathfrak{L}}_{\mathfrak{d}\left(j\right)}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)+\iota 1-{{\text{log}}}_{{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}\left(1+\prod_{j=1}^{n}{\left({{{{\sf (} \! \rotatebox{146}{\sf c}}}^{\divideontimes }}^{1-{ \frown\!\!\!\!\!\! {\smallint} }_{\mathfrak{d}\left(j\right)}^{{ { \bar{\d{\rm L}}} }^{\diamond }}}-1\right)}^{{\Omega }_{{\mathbbm{w}}-j}}\right)\end{array}\right)$$
where \({\Omega }_{\mathbbm{w}}^{*}=\left({\Omega }_{{\mathbbm{w}}-1}^{*}, {\Omega }_{{\mathbbm{w}}-2}^{*}\dots , {\Omega }_{{\mathbbm{w}}-n}^{*}\right)\) is the linked WV such that \(0\le {\Omega }_{{\mathbbm{w}}-j}^{*}\le 1\), and \(\sum_{j=1}^{n}{\Omega }_{{\mathbbm{w}}-j}^{*}=1, {\Omega }_{\mathbbm{w}}=\left({\Omega }_{{\mathbbm{w}}-1}, {\Omega }_{{\mathbbm{w}}-2}\dots , {\Omega }_{{\mathbbm{w}}-n}\right)\) is the WV of \({\hat{C} }_{om.-j}\left(j=1, 2, \dots , n\right)\) such that \(0\le {\Omega }_{{\mathbbm{w}}-j}\le 1\), and \(\sum_{j=1}^{n}{\Omega }_{{\mathbbm{w}}-j}=1\) is the \(jth\) major weighted CFR values of \({\hat{C} }_{om.-j}^{*}\left({\hat{C} }_{om.-j}^{*}=n{\Omega }_{{\mathbbm{w}}-j}{\hat{C} }_{om.-j}\right), n\) is the balancing coefficient.
Let \({\hat{C} }_{om.-j}=\left({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right)\) and \({\hat{C} }_{om.-j}^{\divideontimes }=\left({{\mathfrak{L}}^{\divideontimes }}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}, {{\mathfrak{L}}^{\divideontimes }}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}+\iota { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right) \left(j=1, 2, .., n\right)\) be two collections of CFRNs, then the following properties hold for the structure of CFRFHWA AOs.
1. (Idempotency): If all \({\hat{C} }_{om.-j}={\hat{C} }_{om.} \forall j\) then
$$CFRFHWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)={\hat{C} }_{om.}$$
2. Boundedness: Let \({{\hat{C} }_{om.}}^{-}=\left(\underset{j}{{\text{min}}}\left\{{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}+\iota \underset{j}{{\text{min}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}, \underset{j}{{\text{min}}}\left\{{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}+\iota \underset{j}{{\text{min}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}\right),\) and \({{\hat{C} }_{om.}}^{+}=\left(\underset{j}{{\text{max}}}\left\{{\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}+\iota \underset{j}{{\text{max}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\right\}, \underset{j}{{\text{max}}}\left\{{\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}+\iota \underset{j}{{\text{max}}}\left\{{ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\right\}\right)\). Then
$${{\hat{C} }_{om.}}^{-}\le CFRFHWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\le {{\hat{C} }_{om.}}^{+}$$
3. Monotonicity: If \({\mathfrak{L}}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\le {{\mathfrak{L}}^{\divideontimes }}_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\), \({ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\le { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{\rotatebox{180}{{\rm h}}}^{\diamond }}\), \({\mathfrak{L}}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\le {{\mathfrak{L}}^{\divideontimes }}_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\), \({ \frown\!\!\!\!\!\! {\smallint} }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\le { \frown\!\!\!\!\!\! {\smallint} \divideontimes }_{j}^{{ { \bar{\d{\rm L}}} }^{\diamond }}\forall j\), then
$$CFRFHWA\left({\hat{C} }_{om.-1}, {\hat{C} }_{om.-2}, \dots , {\hat{C} }_{om.-n}\right)\le CFRFHWA\left({\hat{C} }_{om.-1}^{\divideontimes }, {\hat{C} }_{om.-2}^{\divideontimes }, \dots , {\hat{C} }_{om.-n}^{\divideontimes }\right).$$
Remark 1
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1.
By using \({\Omega }_{\mathbbm{w}}={\left(\frac{1}{n}, \frac{1}{n}, \dots , \frac{1}{n}\right)}^{t}\) we have \({\hat{C} }_{om.-j}^{\divideontimes }=n\times \frac{1}{n}\times {\hat{C} }_{om.-j}={\hat{C} }_{om.-j}\) for \(j=1, 2, .., n\). Hence, CFRFHWA AOs degenerate into CFRFOWA AOs.
-
2.
By utilizing \({\Omega }_{{\mathbbm{w}}-j}^{*}={\left(\frac{1}{n}, \frac{1}{n}, \dots , \frac{1}{n}\right)}^{t}\), the CFRFWHA AOs degenerate into CFRFWA AOs.
