In this section, we construct the Schweizer-Sklar operational laws for CA-IF set and then propose the PAOs based on Schweizer-Sklar t-norm and t-conorm for CA-IF values, such as CA-IFSSPA, CA-IFSSPOA, CA-IFSSPG, and CA-IFSSPOG operators. Moreover, we explore some properties of them.
Definition 4
For any two CA-IFNs \(\check{\dddot{{\partial \partial }_{CF-1}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}\right)\right)\) and \(\check{\dddot{{\partial \partial }_{CF-2}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}\right)\right)\), the Schweizer-Sklar operational laws are defined as
$$\check{\dddot{{\partial \partial }_{CF-1}}}\oplus \check{\dddot{{\partial \partial }_{CF-2}}}=\left(\begin{array}{c}\left(1-{\left({\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}+{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}}\right)}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},1-{\left({\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}+{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right)}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right),\\ \left({\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}^{\dddot{{\varphi }_{sc}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},{\left({\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}^{\dddot{{\varphi }_{sc}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right)\end{array}\right),$$
(11)
$$\check{\dddot{{\partial \partial }_{CF-1}}}\otimes \check{\dddot{{\partial \partial }_{CF-2}}}=\left(\begin{array}{c}\left({\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}^{\dddot{{\varphi }_{sc}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},{\left({\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}^{\dddot{{\varphi }_{sc}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right),\\ \left(1-{\left({\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}+{\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}}\right)}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},1-{\left({\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}+{\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}\right)}^{\dddot{{\varphi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right)\end{array}\right),$$
(12)
$$\dddot{{^\circ{\rm F} }_{sc}}\check{\dddot{{\partial \partial }_{CF-1}}}=\left(\begin{array}{c}\left(1-{\left(\dddot{{^\circ{\rm F} }_{sc}}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},1-{\left(\dddot{{^\circ{\rm F} }_{sc}}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right),\\ \left({\left(\dddot{{^\circ{\rm F} }_{sc}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},{\left(\dddot{{^\circ{\rm F} }_{sc}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right)\end{array}\right),$$
(13)
$${\check{\dddot{{\partial \partial }_{CF-1}}}}^{\dddot{{^\circ{\rm F} }_{sc}}}=\left(\begin{array}{c}\left({\left(\dddot{{^\circ{\rm F} }_{sc}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},{\left(\dddot{{^\circ{\rm F} }_{sc}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right),\\ \left(1-{\left(\dddot{{^\circ{\rm F} }_{sc}}{\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},1-{\left(\dddot{{^\circ{\rm F} }_{sc}}{\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\dddot{{^\circ{\rm F} }_{sc}}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\right)\end{array}\right).$$
(14)
Definition 5
The CA-IFSSPA operator is defined by
$$\begin{aligned}&CA-IFSSPA\left(\check{\dddot{{\partial
\partial }_{CF-1}}},\check{\dddot{{\partial \partial
}_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm
E}}}}\right) \\ &\quad =\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-1}}}\\ & \qquad \oplus \frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-2}}}\oplus \dots \\& \qquad \oplus \frac{\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-{\rm
E}}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-{\rm E}}}}
,\end{aligned}$$
(15)
where \(\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF{-}\zeta }}}\right){=}\sum_{\begin{array}{c}\zeta {=}1,\\ \zeta \ne k\end{array}}^{\rm E}\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF{-}\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\) and
\(\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=1-dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\), with a
distance measure, which is defined as
$$dis\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}},\check{\dddot{{\partial \partial
}_{CF-k}}}\right)=\frac{1}{4}\left(\left|{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta
}}}-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-k}}}\right|+\left|{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta
}}}-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-k}}}\right|+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta
}}}-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-k}}}\right|+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta
}}}-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-k}}}\right|\right).$$
(16)
Theorem 1
The aggregated results
from (15) are also a
CA-IFN:
$$\begin{aligned}&CA-IFSSPA\left(\check{\dddot{{\partial
\partial }_{CF-1}}},\check{\dddot{{\partial \partial
}_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm
E}}}}\right)\\
&\quad=\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits_{\zeta
=1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta
}}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}},\\ 1-{\left(\sum\limits_{\zeta =1}^{\rm
E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta
}}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}}\end{array}\right),\\
\left(\begin{array}{c}{\left(\sum\limits_{\zeta =1}^{\rm
E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta
}}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}},\\ {\left(\sum\limits_{\zeta =1}^{\rm
E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta
}}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm
E}\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$
(17)
Proof
Based on the mathematical induction, we prove the Eq. (17).
For this, when \({\rm E}=2\), we have
$$\begin{aligned}&\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-1}}}\\ &\quad =\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\frac{\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}},\\ 1-{\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}}\end{array}\right),\\
\left(\begin{array}{c}{\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}},\\ {\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}}\end{array}\right)\end{array}\right),\end{aligned}$$
$$\begin{aligned}&\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-2}}}\\ &\quad =\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\frac{\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}}\right)}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}},\\ 1-{\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right)}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}}\end{array}\right),\\
\left(\begin{array}{c}{\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}},\\ {\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}^{\dddot{{\varphi
}_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta
}}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi
}_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$
Then,
$$\begin{aligned}& CA-IFSSPA\left(\check{\dddot{{\partial
\partial }_{CF-1}}},\check{\dddot{{\partial \partial
}_{CF-2}}}\right) =\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-1}}} \oplus \frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-2}}}\\ & \qquad=\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right)\\ &\quad\qquad \oplus \left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right),\end{aligned}$$
$$=\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\sum\limits_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum_{\zeta =1}^{2}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right).$$
Further, suppose Eq. (17) is also valid for \({ E}=k\)
$$\begin{aligned}& CA-IFSSPA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-k}}}\right)\\ &\quad =\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$
Then, when \({E}=k+1\), we have
$$\begin{aligned}&CA-IFSSPA\left(\check{\dddot{{\partial
\partial }_{CF-1}}},\check{\dddot{{\partial \partial
}_{CF-2}}},\dots ,\check{\dddot{{\partial \partial
}_{CF-k+1}}}\right)\\ &\quad =\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-1}}}\\ &\qquad \oplus \frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-2}}}\end{aligned}$$
$$\begin{aligned}&\oplus \dots \oplus
\frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-k}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-k}}}\\ &\quad \oplus \frac{\left(1+\mathop {}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop
{}\limits^{}
\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over
{{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial
}_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial
}_{CF-k+1}}},\end{aligned}$$
$$\begin{aligned}&=\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right)\\ &\quad \oplus \frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial }_{CF-k+1}}},\end{aligned}$$
$$\begin{aligned}&=\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits\limits\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right)\\ &\quad \oplus \left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-k+1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-k+1}}}\right)}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-k+1}}}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-k+1}}}^{\dddot{{\varphi }_{sc}}}-\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-k+1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}-1\right)\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right),\end{aligned}$$
$$=\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits\limits\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits\limits\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\sum\limits\limits\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits\limits\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits_{\zeta =1}^{k+1}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right),$$
i.e., when \(E=k+1,\) Eq. (17) is also kept. So, for any E, Eq. (17) is right.
Moreover, we give the properties of this aggregation operator, including idempotency, monotonicity, and boundedness:
Property (Idempotency) 1:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}=\check{\dddot{{\partial \partial }_{CF}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\right)\right),\zeta =\mathrm{1,2},\dots ,E\), then
$$CA-IFSSPA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)=\check{\dddot{{\partial \partial }_{CF}}}.$$
(18)
Property (Monotonicity) 2:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\le \check{\dddot{{^\circ{\rm C} }_{CF-\zeta }}},\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le CA-IFSSPA\left(\check{\dddot{{\mathrm{^\circ{\rm C} }}_{CF-1}}},\check{\dddot{{\mathrm{^\circ{\rm C} }}_{CF-2}}},\dots ,\check{\dddot{{\mathrm{^\circ{\rm C} }}_{CF-{\rm E}}}}\right).$$
(19)
Property (Boundedness) 3:
If \({\check{\dddot{{\partial \partial }_{CF}}}}^{-}=\left(\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)\right.,\)\(\left.\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right)\) and \({\check{\dddot{{\partial \partial }_{CF}}}}^{+}=\left(\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$${\check{\dddot{{\partial \partial }_{CF}}}}^{-}\le CA-IFSSPA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le {\check{\dddot{{\partial \partial }_{CF}}}}^{+}. $$
(20)
Definition 6
The CA-IFSSPOA operator is stated by
$$CA-IFSSPOA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)=\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial }_{CF-O\left(1\right)}}}\oplus \frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial }_{CF-O\left(2\right)}}}\oplus \dots \oplus \frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\check{\dddot{{\partial \partial }_{CF-O\left({\rm E}\right)}}},$$
(21)
where \(O\left(\zeta \right)\le O\left(\zeta -1\right)\) and \(\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)=\sum_{\begin{array}{c}\zeta =1,\\ \zeta \ne k\end{array}}^{\rm E}\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\), where \(\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=1-dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\), with a distance measure:
$$\begin{aligned}&dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=\frac{1}{4}\left(\left|{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-k}}}\right|\right.\\ &\quad +\left|{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-k}}}\right|+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-k}}}\right|\\ &\quad \left.+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-k}}}\right|\right).\end{aligned}$$
(22)
Theorem 2
The aggregated results from (21) are also a CA-IFN:
$$\begin{aligned} & CA-IFSSPOA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\\ &\quad =\left(\begin{array}{c}\left(\begin{array}{c}1-{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-O\left(\zeta \right)}}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-O\left(\zeta \right)}}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{RP-O\left(\zeta \right)}}}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-O\left(\zeta \right)}}}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$
(23)
Moreover, we give the properties of this aggregation operator, including idempotency, monotonicity, and boundedness:
Property (Idempotency) 4:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}=\check{\dddot{{\partial \partial }_{CF}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPOA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)=\check{\dddot{{\partial \partial }_{CF}}} .$$
(24)
Property (Monotonicity) 5:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\le \check{\dddot{{^\circ{\rm C} }_{CF-\zeta }}},\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPOA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le CA-IFSSPOA\left(\check{\dddot{{^\circ{\rm C} }_{CF-1}}},\check{\dddot{{^\circ{\rm C} }_{CF-2}}},\dots ,\check{\dddot{{^\circ{\rm C} }_{CF-{\rm E}}}}\right).$$
(25)
Property (Boundedness) 6:
If \({\check{\dddot{{\partial \partial }_{CF}}}}^{-}=\left(\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\right.\) \(\left.\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right)\) and \({\check{\dddot{{\partial \partial }_{CF}}}}^{+}=\left(\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$${\check{\dddot{{\partial \partial }_{CF}}}}^{-}\le CA-IFSSPOA\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le {\check{\dddot{{\partial \partial }_{CF}}}}^{+}.$$
(26)
Definition 7
The CA-IFSSPG operator is stated by
$$\begin{aligned}&CA-IFSSPG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\\ &\quad ={\check{\dddot{{\partial \partial }_{CF-1}}}}^{\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}}\otimes {\check{\dddot{{\partial \partial }_{CF-2}}}}^{\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}}\\ &\qquad\qquad \otimes \dots \otimes {\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}}^{\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}} ,\end{aligned}$$
(27)
where \(\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)=\sum_{\begin{array}{c}\zeta =1,\\ \zeta \ne k\end{array}}^{\rm E}\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\) and \(\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=1-dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\), with a distance measure:
$$\begin{aligned}& dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=\frac{1}{4}\left(\left|{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-k}}}\right|\right.\\ &\quad+\left|{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-k}}}\right|+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-k}}}\right|\\ &\quad\left.+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-k}}}\right|\right).\end{aligned}$$
(28)
Theorem 3
The aggregated results from (27) are also a CA-IFN:
$$\begin{aligned}& CA-IFSSPG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\\ & \quad =\left(\begin{array}{c}\left(\begin{array}{c}{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}1-{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$
(29)
Moreover, we give the properties of this aggregation operator, including idempotency, monotonicity, and boundedness:
Property (Idempotency) 7:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}=\check{\dddot{{\partial \partial }_{CF}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)=\check{\dddot{{\partial \partial }_{CF}}}.$$
(30)
Property (Monotonicity) 8:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\le \check{\dddot{{^\circ{\rm C} }_{CF-\zeta }}},\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le CA-IFSSPG\left(\check{\dddot{{^\circ{\rm C} }_{CF-1}}},\check{\dddot{{^\circ{\rm C} }_{CF-2}}},\dots ,\check{\dddot{{^\circ{\rm C} }_{CF-{\rm E}}}}\right).$$
(31)
Property (Boundedness) 9:
If \({\check{\dddot{{\partial \partial }_{CF}}}}^{-}=\left(\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right)\right.,\) \(\left.\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right)\) and \({\check{\dddot{{\partial \partial }_{CF}}}}^{+}=\left(\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$${\check{\dddot{{\partial \partial }_{CF}}}}^{-}\le CA-IFSSPG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le {\check{\dddot{{\partial \partial }_{CF}}}}^{+}.$$
(32)
Definition 8
The CA-IFSSPOG operator is stated by
$$CA-IFSSPOG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)={\check{\dddot{{\partial \partial }_{CF-O\left(1\right)}}}}^{\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}}\otimes {\check{\dddot{{\partial \partial }_{CF-O\left(2\right)}}}}^{\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}}\otimes \dots \otimes {\check{\dddot{{\partial \partial }_{CF-O\left({\rm E}\right)}}}}^{\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}} ,$$
(33)
where \(O\left(\zeta \right)\le O\left(\zeta -1\right),\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)=\sum_{\begin{array}{c}\zeta =1,\\ \zeta \ne k\end{array}}^{\rm E}\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\) and \(\mathcal{S}up\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=1-dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)\), with a distance measure:
$$\begin{aligned}& dis\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}},\check{\dddot{{\partial \partial }_{CF-k}}}\right)=\frac{1}{4}\left(\left|{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}}-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-k}}}\right|\right.\\ &\quad +\left|{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-k}}}\right|\\ &\quad \left.+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}}-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-k}}}\right|+\left|{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-k}}}\right|\right).\end{aligned}$$
(34)
Theorem 4
The aggregated results from (33) are also a CA-IFN:
$$\begin{aligned}& CA-IFSSPOG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\\ &\quad =\left(\begin{array}{c}\left(\begin{array}{c}{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{M}}}_{\check{\dddot{RP-O\left(\zeta \right)}}}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ {\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\dddot{\mathbb{M}}}_{\check{\dddot{IP-O\left(\zeta \right)}}}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right),\\ \left(\begin{array}{c}1-{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-O\left(\zeta \right)}}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}},\\ 1-{\left(\sum\limits\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-O\left(\zeta \right)}}}\right)}^{\dddot{{\varphi }_{sc}}}-\sum\limits\limits_{\zeta =1}^{\rm E}\left(\frac{\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$
(35)
Moreover, we give the properties of this aggregation operator, including idempotency, monotonicity, and boundedness:
Property (Idempotency) 10:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}=\check{\dddot{{\partial \partial }_{CF}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPOG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)=\check{\dddot{{\partial \partial }_{CF}}}.$$
(36)
Property (Monotonicity) 11:
If \(\check{\dddot{{\partial \partial }_{CF-\zeta }}}\le \check{\dddot{{^\circ{\rm C} }_{CF-\zeta }}},\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$$CA-IFSSPOG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le CA-IFSSPOG\left(\check{\dddot{{^\circ{\rm C} }_{CF-1}}},\check{\dddot{{^\circ{\rm C} }_{CF-2}}},\dots ,\check{\dddot{{^\circ{\rm C} }_{CF-{\rm E}}}}\right).$$
(37)
Property (Boundedness) 12:
If \({\check{\dddot{{\partial \partial }_{CF}}}}^{-}=\left(\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\right.\) \(\left.\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right)\) and \({\check{\dddot{{\partial \partial }_{CF}}}}^{+}=\left(\left(\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{max}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\left(\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},\underset{\zeta }{{\text{min}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}\), then
$${\check{\dddot{{\partial \partial }_{CF}}}}^{-}\le CA-IFSSPOG\left(\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}\right)\le {\check{\dddot{{\partial \partial }_{CF}}}}^{+}.$$
(38)