Schweizer-Sklar power aggregation operators based on complex intuitionistic fuzzy information and their application in decision-making

Abstract

In 1960, Schweizer and Sklar introduced the novel Schweizer-Sklar t-norm and t-conorm which is used in the construction of aggregation operators. Schweizer-Sklar norms are more general than algebraic norms and Einstein norms. Additionally, computing the power operators based on the Schweizer-Sklar norms for complex Atanassov intuitionistic fuzzy (CA-IF) set is very awkward and complicated. In this manuscript, firstly, we propose the Schweizer-Sklar operational laws for CA-IF values, and secondly, we develop the CA-IF Schweizer-Sklar power averaging (CA-IFSSPA) operator, CA-IF Schweizer-Sklar power ordered averaging (CA-IFSSPOA) operator, CA-IF Schweizer-Sklar power geometric (CA-IFSSPG) operator, and CA-IF Schweizer-Sklar power ordered geometric (CA-IFSSPOG) operator. Some suitable and dominant properties for the above operators are also discussed. Furthermore, to simplify the above operators, we develop the procedure of decision-making technique, called multi-attribute decision-making (MADM) methods based on the proposed operators based on CA-IF values. Finally, we compare the proposed methods with some existing methods to describe the efficiency and capability of the discovered approaches by some examples.

Introduction

Clustering analysis, pattern recognition, and decision-making techniques are a few valuable and dominant research scopes. Especially in the decision-making field, the MADM is an important branch. In order to express the attribute value in uncertain conditions, Zadeh [1] proposed fuzzy set (FS) to describe the fuzzy information by membership function which is any value [0, 1], and it is the modified version of the classical or crisp set, and then FS is extended to fuzzy superior Mandelbrot set [2] which is one of the best extensions of FS. However, there is membership function in FS, and it cannot express the non-membership function. For example, for a voting question, there are 10 people in a group, 5 people give affirmative votes and 2 people give negative votes. Obviously, FS cannot describe this decision problem. Further, Atanassov [3] proposed intuitionistic FS (IFS) which included the membership function ${\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)$ and the non-membership function ${\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right) ,$ and meets a prominent characteristic, such as$0\le {\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)\le 1$. When ${\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)=0,$ IFS reduces to FS. Additionally, Mahmood et al. [4] developed the TOPSIS and Hamacher Choquet integral aggregation operators for Atanassov IFS. Jia and Wang [5] developed the simple Choquet integral under the IFS.

IFS and FS can only express one-dimensional information. Another dimensional information, called the phase term, also played a valuable and important role. For example, to buy a car, the owner of a car company gives two kinds of data such as the name and production time of the car, where the name of the car stated the real part which can be easily handled by FS, but the production time of the car can be stated by the phase term, and now it is ignored. Therefore, the concept of complex FS (CFS) [6] is very famous for depicting the above real problem with complex-valued truth information, where the real and phase parts are contained in the unit interval, which is the extension of the FS. Moreover, Liu et al. [7] proposed the distance measures and cross-entropy measures for CFS, Mahmood et al. [8] proposed the interdependency and neighborhood operator for CFS, Alkouri and Salleh [9] proposed the complex Atanassov IFS (CA-IFS), where ${\dddot{\mathbb{M}}}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right)=\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right),{\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\right)$ is called a supporting degree and ${\dddot{\mathbb{N}}}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right)=\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right),{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\right)$ is called supporting-against degree with $0\le {\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)\le 1$ and $0\le {\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\le 1$. Additionally, Garg and Rani [10] proposed the information measures for CA-IFS, Ali et al. [11] developed the prioritized operators for complex intuitionistic fuzzy soft set.

In 1960, Schweizer and Sklar [12] proposed Schweizer-Sklar t-norm and t-conorm which can make the data aggregation procedure more reliable than others, and the PAO [13] can eliminate the effects of unreliable and unreasonable information from biased experts. Furthermore, Jiang et al. [14] explored the PAOs in the availability of the IFS, Rani and Garg [15] developed the PAOs for CA-IFS, Garg et al. [16] developed Schweizer-Sklar prioritized operators for IFS, and Zhang [17] developed the averaging Schweizer-Sklar operators for IFS. After a long assessment, we observed that the following queries are the fundamental parts of every decision-makers:

• (1) How to combine new t-norms and t-conorms based on CA-IFSs?

• (2) How to construct new operators based on CA-IFSs?

• (3) How to find the best optimal based on the new proposed operators?

To solve the above problems, which is very difficult, we find the by developing the Schweizer-Sklar power operators for CA-IFSs. Moreover, we also noticed that the CA-IFS is more general than FS, IFS, and CFS, because of its structure, where the Schweizer-Sklar norms are also very general, for example, their special cases, called algebraic norms and Einstein norms. Further, the PAO can eliminate the effects of unreliable and unreasonable information from biased experts. How to propose the Schweizer-Sklar power aggregation operator based on the CA-IFS is also very famous and valuable research works, and some advantages of the proposed operators are listed below:

• (1) Power averaging/geometric operators for FSs, IFSs, and CFSs are the special cases of the proposed ideas.

• (2) Schweizer-Sklar averaging/geometric operators for FSs, IFSs, and CFSs are the special cases of the proposed ideas.

• (3) Averaging/geometric operators for FSs, IFSs, and CFSs are the special cases of the proposed ideas.

• (4) Einstein averaging/geometric operators for FSs, IFSs, and CFSs are the special cases of the proposed ideas.

• (5) Einstein power averaging/geometric operators for FSs, IFSs, and CFSs are the special cases of the proposed ideas.

• (6) Schweizer-Sklar power averaging/geometric operators for FSs, IFSs, and CFSs are the special cases of the proposed ideas.

Keeping the advantages of the Schweizer-Sklar norms and PAOs, the main contributions of this manuscript can be summarized as follows:

• 1. To propose the novel Schweizer-Sklar operational laws for CA-IFS.

• 2. To construct the CA-IFSSPA, CA-IFSSPOA, CA-IFSSPG, and CA-IFSSPOG operators.

• 3. To discuss the main properties of the described operators.

• 4. To develop the MADM techniques for CA-IFS based on the developed operators.

• 5. To compare the proposed methods with some existing ones to describe the efficiency and capability of the discovered approaches by some examples. The geometrical abstract of this manuscript is listed in Fig. 1.

Fig. 1

Geometrical representation of the proposed works

This article is reviewed in the following structure. In “Preliminaries”, we revised the CA-IFSs, Schweizer-Sklar norms, and PAOs. In “Schweizer-Sklar power aggregation operators for CA-IFNs”, we constructed the Schweizer-Sklar operational laws for CA-IF set, and then we proposed the CA-IFSSPA, CA-IFSSPOA, CA-IFSSPG, and CA-IFSSPOG operators. Moreover, we explored the main properties of the described operators. In “Decision-making procedures”, the MADM methods are developed based on the proposed operators for the CA-IF set. In “Comparative analysis”, the comparison between derived works and some existing methods is discussed to describe the efficiency and capability of the discovered approaches by some examples. Concluding remarks are given in “Conclusion”.

Preliminaries

In this part, we briefly reviewed the old idea of CA-IFSs, Schweizer-Sklar norms, and PAOs. For convenience, the universal is stated by $X$.

Definition 1

[9] A CA-IFS $\check{\dddot{{\partial \partial }_{CF}}}$ is described by

$$\check{\dddot{{\partial \partial }_{CF}}}=\left\{\left({\dddot{\mathbb{M}}}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right),{\dddot{\mathbb{N}}}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right)\right):{\mathbb{u}}\in X\right\} ,$$

(1)

where ${\dddot{\mathbb{M}}}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right)=\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right),{\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\right)$ is called a supporting degree and ${\dddot{\mathbb{N}}}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right)=\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right),{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\right)$ is called supporting-against degree with $0\le {\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)\le 1$ and $0\le {\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\le 1$. Moreover, ${\dddot{\eta }}_{\check{\dddot{{\partial \partial }_{CF}}}}\left({\mathbb{u}}\right)=\left({\dddot{\eta }}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right),{\dddot{\eta }}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\right)=\left(1-\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{RP}}}\left({\mathbb{u}}\right)\right),1-\left({\dddot{\mathbb{M}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)+{\dddot{\mathbb{N}}}_{\check{\dddot{IP}}}\left({\mathbb{u}}\right)\right)\right)$ is represented by the refusal degree with a CA-IF number (CA-IFN) $\check{\dddot{{\partial \partial }_{CF-\zeta }}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-\zeta }}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-\zeta }}}\right)\right),\zeta =\mathrm{1,2},\dots ,{\rm E}$.

Furthermore, the algebraic operational laws 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 $\check{\dddot{{\partial \partial }_{CF-2}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}\right)\right)$ are defined as

$$\check{\dddot{{\partial \partial }_{CF-1}}}\oplus \check{\dddot{{\partial \partial }_{CF-2}}}=\left(\begin{array}{c}\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}}-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right),\\ \left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}\right)\end{array}\right),$$

(2)

$$\check{\dddot{{\partial \partial }_{CF-1}}}\otimes \check{\dddot{{\partial \partial }_{CF-2}}}=\left(\begin{array}{c}\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}{\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right),\\ \left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}}-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}{\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}\right)\end{array}\right),$$

(3)

$$\dddot{{\pi }_{sc}}\check{\dddot{{\partial \partial }_{CF-1}}}=\left(\left(1-{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\pi }_{sc}}},1-{\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\pi }_{sc}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}^{\dddot{{\pi }_{sc}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}^{\dddot{{\pi }_{sc}}}\right)\right),$$

(4)

$$\begin{aligned}{\check{\dddot{{\partial \partial }_{CF-1}}}}^{\dddot{{\pi }_{sc}}}&=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}^{\dddot{{\pi }_{sc}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}^{\dddot{{\pi }_{sc}}}\right),\left(1-{\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}\right)}^{\dddot{{\pi }_{sc}}},\right.\right.\\ &\left.\left.\quad 1-{\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}\right)}^{\dddot{{\pi }_{sc}}}\right)\right).\end{aligned}$$

(5)

Additionally, we get the score and accuracy functions, shown as

$$\begin{aligned}{\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)&=\frac{1}{2}\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)\right.\\ &\quad \left.-\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}\right)\right)\in \left[-\mathrm{1,1}\right], \end{aligned}$$

(6)

$$\begin{aligned}{\widehat{\mathcal{S}}}_{AV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)&=\frac{1}{2}\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-1}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{IP-1}}}\right)\right.\\ &\quad\left.+\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-1}}} +{\dddot{\mathbb{N}}}_{\check{\dddot{IP-1}}}\right)\right)\in \left[\mathrm{0,1}\right].\end{aligned}$$

(7)

Based on Eq. (6) and Eq. (7), 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 $\check{\dddot{{\partial \partial }_{CF-2}}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-2}}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-2}}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-2}}}\right)\right)$ can be compared by.

If ${\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)>{\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\Rightarrow \check{\dddot{{\partial \partial }_{CF-1}}}>\check{\dddot{{\partial \partial }_{CF-2}}}$;

If ${\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)<{\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\Rightarrow \check{\dddot{{\partial \partial }_{CF-1}}}<\check{\dddot{{\partial \partial }_{CF-2}}}$;

If ${\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)={\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)\Rightarrow$ If ${\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)>{\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\Rightarrow \check{\dddot{{\partial \partial }_{CF-1}}}>\check{\dddot{{\partial \partial }_{CF-2}}}$;

If ${\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-1}}}\right)<{\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-2}}}\right)\Rightarrow \check{\dddot{{\partial \partial }_{CF-1}}}<\check{\dddot{{\partial \partial }_{CF-2}}}$.

Definition 2

[12] The Schweizer-Sklar t-norm and t-conorm are defined by

$$\check{\dddot{{\partial }_{1}}}\oplus \check{\dddot{{\partial }_{2}}}=1-{\left({\left(1-\check{\dddot{{\partial }_{1}}}\right)}^{\dddot{{\pi }_{sc}}}+{\left(1-\check{\dddot{{\partial }_{2}}}\right)}^{\dddot{{\pi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\pi }_{sc}}}} ,$$

(8)

$$\check{\dddot{{\partial }_{1}}}\otimes \check{\dddot{{\partial }_{2}}}={\left({\check{\dddot{{\partial }_{1}}}}^{\dddot{{\pi }_{sc}}}+{\check{\dddot{{\partial }_{2}}}}^{\dddot{{\pi }_{sc}}}-1\right)}^{\frac{1}{\dddot{{\pi }_{sc}}}},$$

(9)

where $\check{\dddot{{\partial }_{1}}}$ and $\check{\dddot{{\partial }_{2}}}$ are the non-negative real numbers.

Definition 3

[13] $\left(\check{\dddot{{\partial }_{1}}},\check{\dddot{{\partial }_{2}}},\dots ,\check{\dddot{{\partial }_{\rm E}}}\right)$ is a group of non-negative real numbers, the PAO is stated by.

$$\begin{aligned}&PA\left(\check{\dddot{{\partial }_{1}}},\check{\dddot{{\partial }_{2}}},\dots ,\check{\dddot{{\partial }_{\rm E}}}\right)\\ &\qquad=\frac{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial }_{\zeta }}}\right)\right)\check{\dddot{{\partial }_{\zeta }}}}{\sum_{\zeta =1}^{\rm E}\left(1+\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial }_{\zeta }}}\right)\right)},\end{aligned}$$

(10)

where $\mathop {}\limits^{} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over {{\mathcal U}\Xi} }\left(\check{\dddot{{\partial }_{\zeta }}}\right)=\sum_{\begin{array}{c}\zeta =1,\\ \zeta \ne k\end{array}}^{\rm E}\mathcal{S}up\left(\check{\dddot{{\partial }_{\zeta }}},\check{\dddot{{\partial }_{k}}}\right)$ and $\mathcal{S}up\left(\check{\dddot{{\partial }_{\zeta }}},\check{\dddot{{\partial }_{k}}}\right)=1-dis\left(\check{\dddot{{\partial }_{\zeta }}},\check{\dddot{{\partial }_{k}}}\right)$, with the following axioms:

• 1. $\mathcal{S}up\left(\check{\dddot{{\partial }_{\zeta }}},\check{\dddot{{\partial }_{k}}}\right)\in \left[\mathrm{0,1}\right]$.

• 2. $\mathcal{S}up\left(\check{\dddot{{\partial }_{\zeta }}},\check{\dddot{{\partial }_{k}}}\right)=\mathcal{S}\mathcal{U}\Xi \mathcal{P}\left(\check{\dddot{{\partial }_{k}}},\check{\dddot{{\partial }_{\zeta }}}\right)$.

• 3. $\mathcal{S}up\left(\check{\dddot{{\partial }_{\zeta }}},\check{\dddot{{\partial }_{k}}}\right)\ge \mathcal{S}up\left(\check{\dddot{{\partial }_{l}}},\check{\dddot{{\partial }_{m}}}\right)\mathrm{If }\left|\check{\dddot{{\partial }_{\zeta }}}-\check{\dddot{{\partial }_{k}}}\right|\le \left|\check{\dddot{{\partial }_{l}}}-\check{\dddot{{\partial }_{m}}}\right|$.

Schweizer-Sklar power aggregation operators for CA-IFNs

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)

Decision-making procedures

In this section, we develop two MADM methods with the CA-IFS information based on the proposed aggregation operators and give the detailed decision steps and some examples.

For this, suppose $(\check{\dddot{{\partial \partial }_{Al-1}}},\check{\dddot{{\partial \partial }_{Al-2}}},\dots ,\check{\dddot{{\partial \partial }_{Al-m}}})$ is the set of evaluation alternatives, and they are evaluated by the finite attributes $\check{\dddot{{\partial \partial }_{CF-1}}},\check{\dddot{{\partial \partial }_{CF-2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-{\rm E}}}}$. $\check{\dddot{{\partial \partial }_{CF-i\zeta }}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right)\right),(i=\mathrm{1,2},\dots ,m,\zeta =\mathrm{1,2},\dots ,{\rm E})$ is the evaluation value of alternative $\check{\dddot{{\partial \partial }_{Al-i}}}$ under the attribute $\check{\dddot{{\partial \partial }_{CF-\zeta }}}$, which is expressed by CA-IF value, and $\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right) and \left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right)$ are supporting degree and supporting-against degree. The geometrical interpretation of the proposed algorithms are stated in Fig. 2. Based on this evaluation information, we can give the decision steps as follows:

Fig. 2

Geometrical representation of the proposed algorithm

Step 1: Standardize decision matrix. For the above decision problem, a matrix will be covering the information with benefit or cost types, if one evaluation value is the cost type, it should be converted to benefit type by

$$Z=\left\{\begin{array}{cc}\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right)\right)& \text{for benefit}\\ \left(\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right),\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right)\right)& \text{for cost}.\end{array}\right.$$

Step 2: Aggregate all attribute values for each alternative based on the CA-IFSSPA operator and CA-IFSSPG operator, and get the singleton CA-IF value:

$$\begin{aligned}&\check{\dddot{{\partial \partial }_{CF-i}}}=CA-IFSSPA\left(\check{\dddot{{\partial \partial }_{CF-i1}}},\check{\dddot{{\partial \partial }_{CF-i2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-i{\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-i\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-i\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\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-i\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-i\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-i\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-i\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\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-i\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-i\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-i\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-i\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\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-i\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-i\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-i\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-i\zeta }}}\right)\right)}\right){\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\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-i\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-i\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right)\end{aligned}$$

and

$$\begin{aligned}&\check{\dddot{{\partial \partial }_{CF-i}}}=CA-IFSSPG\left(\check{\dddot{{\partial \partial }_{CF-i1}}},\check{\dddot{{\partial \partial }_{CF-i2}}},\dots ,\check{\dddot{{\partial \partial }_{CF-i{\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-i\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-i\zeta }}}\right)\right)}\right){\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\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-i\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-i\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-i\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-i\zeta }}}\right)\right)}\right){\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\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-i\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-i\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-i\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-i\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\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-i\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-i\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-i\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-i\zeta }}}\right)\right)}\right){\left(1-{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\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-i\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-i\zeta }}}\right)\right)}\right)+1\right)}^{\frac{1}{\dddot{{\varphi }_{sc}}}}\end{array}\right)\end{array}\right).\end{aligned}$$

Step 3: Calculate the score values (if fail the score values then use accuracy values) of the aggregated information:

$${\widehat{\mathcal{S}}}_{SV}\left(\check{\dddot{{\partial \partial }_{CF-i}}}\right)=\frac{1}{2}\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i}}}\right)-\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i}}}\right)\right)\in \left[-\mathrm{1,1}\right],$$

$${\widehat{\mathcal{S}}}_{AV}\left(\check{\dddot{{\partial \partial }_{CF-i}}}\right)=\frac{1}{2}\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i}}}+{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i}}}\right)+\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i}}}+{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i}}}\right)\right)\in \left[\text{0,1}\right].$$

Step 4: Rank all the alternatives under the score values of each alternative to find the finest one.

Then some examples are given to describe the efficiency and capability of the discovered approaches.

An electronic commerce distributor expects to choose a valuable, suitable, and dominant third-party logistics facilitator. Initially, five facilitators (alternatives) $\check{\dddot{{\partial \partial }_{Al-i}}},i=1,2,3,4,5$ are considered and are evaluated by decision-makers concerning the following attributes, such as $\check{\dddot{{\partial \partial }_{CF-1}}}$: Customer satisfaction, $\check{\dddot{{\partial \partial }_{CF-2}}}$: Growth analysis, $\check{\dddot{{\partial \partial }_{CF-3}}}$: Market reputations, and $\check{\dddot{{\partial \partial }_{CF-4}}}$: operational experience in the industry. $\check{\dddot{{\partial \partial }_{CF-i\zeta }}}=\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right)\right),(i=1,2,3,4,5,\zeta =1,2,\text{3,4})$ is the evaluation value of alternative $\check{\dddot{{\partial \partial }_{Al-i}}}$ under the attribute $\check{\dddot{{\partial \partial }_{CF-\zeta }}}$, which is expressed by CA-IF value. Based on this information, the main steps of the decision-making procedure are stated below:

Step 1: Construct a matrix (see Table 1) and standardize it. Based on the different information type, standardize it:

Table 1 The decision matrix with CA-IF information

$$Z=\left\{\begin{array}{cc}\left(\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right),\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right)\right)& \text{for benefit}\\ \left(\left({\dddot{\mathbb{N}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{N}}}_{\check{\dddot{IP-i\zeta }}}\right),\left({\dddot{\mathbb{M}}}_{\check{\dddot{RP-i\zeta }}},{\dddot{\mathbb{M}}}_{\check{\dddot{IP-i\zeta }}}\right)\right)& \text{for cost}.\end{array}\right.$$

Because all are benefiting type, data in Table 1 are not required to be normalized.

Step 2: Aggregate all attribute values for each alternative based on the CA-IFSSPA operator and CA-IFSSPG operator and get the singleton CA-IF value shown in Table 2.

Table 2 Aggregated values by the CA-IFSSPA and CA-IFSSPG operators

Step 3: Calculate the score values of the aggregated information shown in Table 3.

Table 3 The score values

Step 4: Rank all the alternatives shown in Table 4.

Table 4 The ranking results by the deferent operators

From Table 4, we get the same ranking results, and then get the same best choice $\check{\dddot{{\partial \partial }_{Al-3}}}$ for the proposed CA-IFSSPA and CA-IFSSPG operators.

Furthermore, in order to show the supremacy of the proposed methods, we discuss more possibilities for the efficiency and capability of them. For this, we remove the phase term from the data in Table 1, and then the aggregated data are given in Table 5.

Table 5 Aggregated values with no phase term by the CA-IFSSPA and CA-IFSSPG operators

Based on the aggregated values, we get the score values shown in Table 6.

Table 6 The score values from Table 5

Rank all the alternatives under consideration of the score values of each alternative to find the finest optimal shown in Table 7.

Table 7 The ranking results from Table 6

From Table 7, we get the same ranking results, and then get the same best choice $\check{\dddot{{\partial \partial }_{Al-3}}}$ for the proposed CA-IFSSPA and CA-IFSSPG operators.

From these two examples, we can see that IFS is a special case of CIFS, and the proposed methods are more general than some existing methods.

Comparative analysis

To demonstrate the effectiveness and superiority of the derived operators, the comparison between derived works with some existing methods is necessary and important. Therefore, based on the illustrated examples, we compare the presented operators with some valuable operators to show the efficiency and capability of the discovered approaches. For this, we considered the following operators, such as PAOs based on the IFS proposed by Jiang et al. [14], PAOs for CA-IFS by Rani and Garg [15], Schweizer-Sklar prioritized operators for IFS by Garg et al. [16], and averaging Schweizer-Sklar operators for IFS by Zhang [17]. After the above consideration, the comparative analysis is listed in Table 8 (for the data in Table 1).

Table 8 The comparative analysis from the data in Table 1 (with phase term)

From Table 8, we can get the $\check{\dddot{{\partial \partial }_{Al-3}}}$ is the finest optimal under the proposed CA-IFSSPA operator, proposed CA-IFSSPG operator, and PAOs for CA-IFS by Rani and Garg [15], where, because of some limitations, the proposed methods from Jiang et al. [14], Garg et al. [16], and Zhang [17] have failed. Further, we do the comparative analysis by removing the phase term from the data in Table 1, and the results are listed in Table 9.

Table 9 The comparative analysis from the data in Table 1 (without phase term)

From Table 9, all methods including the proposed CA-IFSSPA operator, proposed CA-IFSSPG operator, Jiang et al. [14], method, Rani and Garg [15], method, Garg et al. [16], method, and Zhang [17], method get the same ranking results, and then get the finest optimal $\check{\dddot{{\partial \partial }_{Al-3}}}$ based on the data in Table 1 (without phase information).

From the above analysis, the proposed methods are more general than some existing methods.

Conclusion

The PAOs based on the Schweizer-Sklar norms for CA-IFSs are very important task for scholars, and to evaluate these problems, the main contributions of this manuscript can be summarized as follows:

• 1. Proposed the novel Schweizer-Sklar operational laws for the CA-IFS.

• 2. Developed the CA-IFSSPA, CA-IFSSPOA, CA-IFSSPG, and CA-IFSSPOG operators.

• 3. Explored the main properties of the described approaches.

• 4. Developed a procedure of the MADM technique based on the proposed operators.

• 5. Compared the derived methods with some existing ones, and get the proposed methods are more general than some existing ones.

Information discussed in this article is very reliable, but if we provides information in the shape of yes, abstinence, no, and refusal, then the CA-IFSs have been failed; for this, we aim to develop the complex spherical fuzzy hesitant sets and their modifications.

In the future, we will extend the proposed works to Aczel-Alsina information [18], correlation measures [19], neutrality operators [20], taxonomy methods [21], and medical diagnosis [22], assessment of smart systems in hydroponic vertical farming [23], the evaluation of metaverse integration [24], finite-interval-valued type-2 Gaussian fuzzy number [25], and portfolio allocation with the TODIM technique [26] and then try to utilize it in the field of artificial intelligence, neural networks, and machine learning.