Schweizer–Sklar Power Aggregation Operators Based on Complex Interval-Valued Intuitionistic Fuzzy Information for Multi-attribute Decision-Making

Abstract

This manuscript proposes the concept of Schweizer–Sklar operational laws under the consideration of the complex interval-valued intuitionistic fuzzy (CIVIF) set theory, where the Schweizer–Sklar norms are the essential and valuable modification of many norms, such as algebraic, Hamacher, and Lukasiewicz norms. Moreover, keeping the dominancy of the presented laws, we derive the concept of CIVIF Schweizer–Sklar power averaging (CIVIFSSPA), CIVIF Schweizer–Sklar power ordered averaging (CIVIFSSPOA), CIVIF Schweizer–Sklar power geometric (CIVIFSSPG), and CIVIF Schweizer–Sklar power ordered geometric (CIVIFSSPOG) operators, which are the combination of the three different structures for evaluating three different problems. Further, some reliable and feasible properties and results for derived work are also invented. Additionally, we also illustrate an application, called multi-attribute decision-making (MADM) scenario for evaluating some real-world problems with the help of discovered operators for showing the reliability and stability of the evaluated operators. Finally, we compare our mentioned operators with various prevailing operators for enhancing the worth and stability of the evaluated approaches.

1 Introduction

Evaluating some real-life engineering complex problems such as urban mobility governance are very awkward and complicated under the consideration of classical information, certain individuals have used different types of technique for managing ambiguous and unreliable information such as pattern recognition, medical diagnosis, clustering analysis, artificial intelligence, and pattern classification. But they have lost a lot of data during decision-making procedures under the presence of classical information. Furthermore, the procedure of the MADM technique is also a sub-part of the decision-making process which is used for evaluating or selecting the finest decision from the collection of decisions. But, because of classical information, we faced a bundle of problems; therefore, for depicting the discussed problem, the theory of fuzzy set (FS) is much more valuable and reliable which was derived by Zadeh [1] in 1965, because the FS has more possibilities as compared to classical set. Further, Atanassov [2] extended the theory of FS, and evaluated or derived the novel theory of intuitionistic FS (IFS) in 1986. IFS has two different types of grades such as truth “$T$” and falsity “$F$” with a characteristic: $0\le T+F\le 1$. We noticed that the theory of IFS has only two possibilities and the value of each possibility is contained in a unit interval. But in any situation, we noticed that experts need these two pieces of information in the shape of the interval because in the presence of interval-valued information experts can easily get has required results. Therefore, for depicting such sort of complicated and unreliable information the theory of interval-valued IFS (IVIFS) was computed by Atanassov [3] in 1999. Various applications are stated to employ different types of ideas in the field of IVIFS such as distance measures [4], aggregation operators [5], correlation measures [6], analysis of accuracy function [7], assessments o incomplete weighted [8], a not on correlation measures [9], and information measures [10].

Utilization of phase term in the field of truth grade is a very challenging task for scholars because, in the sense of FS theory, we have managed information with the help of truth grade whose value belongs to the unit interval. Utilizing the phase term in the field of truth is very important because in many situations, we faced a lot of problems that contained two-dimension information and the fuzzy truth grade is not enough to evaluate it. Therefore, the theory of complex FS (CFS) was derived by Ramot et al. [11] in 2002 by adjusting the phase term in the term of truth grade. Moreover, Alkouri and Salleh [12] extended the theory of CFS, and evaluated or derived the novel theory of complex IFS (CIFS) in 2012. CIFS has two different types of grades such as truth “$\left({T}_{R},{T}_{I}\right)$” and falsity “$\left({F}_{R},{F}_{I}\right)$” with a characteristic: $0\le {T}_{R}+{F}_{R}\le 1$ and $0\le {T}_{I}+{F}_{I}\le 1$. We noticed that the theory of CIFS has only two possibilities and the real value (imaginary part) of each possibility is contained in a unit interval. But in any situation, we noticed that experts need these two pieces of information in the shape of the interval because in the presence of interval-valued information, experts can easily get the required results. Therefore, for depicting such sort of complicated and unreliable information, the theory of complex IVIFS (CIVIFS) was computed by Garg and Rani [13] in 2019. Various applications are stated to employ different types of ideas in the field of CIFS and CIVIFS such as information measures [14], correlation measures [15], analysis of relation for CIFSs [16], and prioritized aggregation operators [17].

The key and basic idea of power aggregation information was evaluated by Xu and Yager [18] in 2010 for the first time. After the successful utilization of power aggregation operators, certain people have utilized them in the environment of different fields. Furthermore, various well-known and valuable scholars have derived different types of norms, and under the consideration of these norms, certain types of operators have been developed. Similarly, Deschrijver and Kerre [19] derived the novel type of norms called Schweizer–Sklar t-norm and t-conorm in 2002. Some valuable and dominant applications computed based on the above analysis have been described in the form, for instance, power aggregation operators for IVIFSs [20], Schweizer–Sklar power aggregation operators [21], and finally, the theory of TODIM and Schweizer–Sklar power aggregation operators were also derived by Zindani et al. [22].

After a long discussion about existing information, we noticed that to propose the theory of power aggregation operators for CIVIF set theory under the presence of the Schweizer–Sklar operational laws are a very challenging task for all researchers. Because many scholars have proposed the power aggregation, Schweizer–Sklar aggregation operators, Hamacher aggregation operators, and averaging/geometric aggregation operators for all existing ideas separately, but no one can combine them, because of the following queries, we aim to derive this information (proposed work), such as:

• 1. How we utilize the theory of power aggregation operators based on Schweizer–Sklar operational laws because for aggregating the collection of information, we have needed a strong structure.

• 2. How we combine two or more different types of aggregation operators.

• 3. How we aggregate the collection of information for evaluating the finest decision from the collection of preferences.

According to the above knowledge, we get these queries, because before no one can develop the theory of Schweizer–Sklar power aggregation operators based on CIVIFSs which are very awkward and unreliable tasks. Because the many operators are the special cases of the invented information due to its structure. Therefore, inspired by the above theories, we aim to derive the following ideas:

• 4. To evaluate a novel concept of Schweizer–Sklar operational laws for CIVIF information.

• 5. To expose four different ideas, such as CIVIFSSPA, CIVIFSSPOA, CIVIFSSPG, and CIVIFSSPOG operators. Some reliable and feasible properties and results for derived work are also invented.

• 6. To compute a MADM scenario for discovered operators under the presence of CIVIF information for showing the reliability and stability of the evaluated operators.

• 7. To compare our mentioned operators with various prevailing operators for enhancing the worth and stability of the evaluated approaches.

The main shape of this analysis is computed in the form: In Sect. 2, we revised the theory of CIVIFS, Schweizer–Sklar t-norm and t-conorm, power aggregation operator, and its operational laws. In Sect. 3, we computed the Schweizer–Sklar operational laws for CIVIF information. Furthermore, under the presence of Schweizer–Sklar operational laws, we derived the theory of CIVIFSSPA, CIVIFSSPOA, CIVIFSSPG, and CIVIFSSPOG operators. Some reliable and feasible properties and results for derived work are also invented. In Sect. 4, we computed a MADM “multi-attribute decision-making” scenario for discovered operators under the presence of CIVIF information for showing the reliability and stability of the evaluated operators. In Sect. 5, we compared our mentioned operators with various prevailing operators to discuss the worth and stability of the evaluated approaches. Some concluding remarks are given in Sect. 6.

2 Preliminaries

In this section, we aim to recall the theory of CIFS, CIVIFS, Schweizer–Sklar t-norm and t-conorm, power aggregation operator, and its operational laws.

Definition 1

[12] A CIFS under the consideration of universal set $U$ is presented by:

$$A=\left\{\left( x,{r}_{A}\left(x\right){e}^{2\pi i\left({w}_{{r}_{A}}\left(x\right)\right)},{k}_{A}\left(x\right){e}^{2\pi i\left({w}_{{k}_{A}}\left(x\right)\right)}\right):x\in U\right\}.$$

(1)

We noticed that the symbol ${r}_{A}\left(x\right){e}^{2\pi i\left({w}_{{r}_{A}}\left(x\right)\right)}$ and ${k}_{A}\left(x\right){e}^{2\pi i\left({w}_{{k}_{A}}\left(x\right)\right)}$ expressed the information of truth and falsity function in the shape of complex numbers with ${r}_{A}\left(x\right)+{k}_{A}\left(x\right)\le 1$ and ${w}_{{r}_{A}}\left(x\right)+{w}_{{k}_{A}}\left(x\right)\le 1.\forall x\in U$. The CIF number (CIFSN) is represented by: $A=\left( {r}_{A}\left(x\right){e}^{2\pi i\left({w}_{{r}_{A}}\left(x\right)\right)},{k}_{A}\left(x\right){e}^{2\pi i\left({w}_{{k}_{A}}\left(x\right)\right)}\right)$.

Definition 2

[13] A CIVIFS under the consideration of universal set $U$ is presented by:

$$A=\left\{\left( x,\left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]},[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}\right):x\in U\right\}.$$

(2)

We noticed that the symbol $\left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)]}$ and $[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)]}$ expressed the information of truth and falsity function in the shape of complex numbers, where ${r}_{A}^{-}\le {r}_{A}^{+},{k}_{A}^{-}\le {k}_{A}^{+}$ and ${r}_{A}^{+}\left(x\right)+{k}_{A}^{+}\left(x\right)\le 1$ and ${w}_{{r}_{A}}^{-}\le {w}_{{r}_{A}}^{+}, {w}_{{k}_{A}}^{-}\le {w}_{{k}_{A}}^{+}$ and ${w}_{{r}_{A}}^{+}\left(x\right)+{w}_{{k}_{A}}^{+}\left(x\right)\le 2\pi .\forall x\in U$. The CIVIF number (CIVIFSN) is represented by: $A=\left( \left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]},[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}\right).$

Definition 3

[13] For two CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),(\rho=\mathrm{1,2})$, $\lambda\ge 1,$ then we have

$${\beta }_{1}^{c}=\left(\left[{k}_{1}^{-}, {k}_{1}^{+}\right]{e}^{2\pi i\left[{w}_{{k}_{1}}^{-} ,{w}_{{k}_{1}}^{+}\right]}, \left[{r}_{1}^{-},{r}_{1}^{+}\right]{e}^{2\pi i\left[{w}_{{r}_{1}}^{-} , {w}_{{r}_{1}}^{+}\right]}\right),$$

(3)

$${\beta }_{1}\cup {\beta }_{2}=\left(\begin{array}{c} \left[{\mathrm{max}\{r}_{1}^{-},{r}_{2}^{-}\},{\mathrm{max}\{r}_{1}^{+},{r}_{2}^{+}\}\right]{e}^{2\pi i\left[{\mathrm{max}\{w}_{{r}_{1}}^{-},{w}_{{r}_{2}}^{-}\},{\mathrm{ max}\{w}_{{r}_{1}}^{+},{w}_{{r}_{2}}^{+}\}\right]},\\ \left[{\mathrm{min}\{k}_{1}^{-},{k}_{2}^{-}\right\},{\mathrm{min}\{k}_{1}^{+},{k}_{2}^{+}]{e}^{2\pi i\left[{\mathrm{min}\{w}_{{k}_{1}}^{-},{w}_{{k}_{2}}^{-}\right\},{\mathrm{min}\{w}_{{k}_{1}}^{+},{w}_{{k}_{2}}^{+}]}\end{array}\right),$$

(4)

$${\beta }_{1}\cap {\beta }_{2}=\left(\begin{array}{l} \left[{\mathrm{min}\{r}_{1}^{-},{r}_{2}^{-}\},{\mathrm{min}\{r}_{1}^{+},{r}_{2}^{+}\}\right]{e}^{2\pi i\left[{\mathrm{min}\{w}_{{r}_{1}}^{-},{w}_{{r}_{2}}^{-}\},{\mathrm{ min}\{w}_{{r}_{1}}^{+},{w}_{{r}_{2}}^{+}\}\right]},\\ \left[{\mathrm{max}\{k}_{1}^{-},{k}_{2}^{-}\right\},{\mathrm{max}\{k}_{1}^{+},{k}_{2}^{+}]{e}^{2\pi i\left[{\mathrm{max}\{w}_{{k}_{1}}^{-},{w}_{{k}_{2}}^{-}\right\},{\mathrm{max}\{w}_{{k}_{1}}^{+},{w}_{{k}_{2}}^{+}]}\end{array}\right),$$

(5)

$${\beta }_{1}\oplus {\beta }_{2}=\left(\begin{array}{l}\left[1-\prod_{\rho=1}^{2}\left(1-{r}_{\rho}^{-}\right),1-\prod_{\rho=1}^{2}\left(1-{r}_{\rho}^{+}\right)\right]{e}^{2\pi i\left[\left(1-\prod_{\rho=1}^{2}(1-\frac{{w}_{{r}_{\rho}}^{-}}{2\pi })\right),\left(1-\prod_{\rho=1}^{2}(1-\frac{{w}_{{r}_{\rho}}^{+}}{2\pi })\right)\right]},\\ \left[\prod_{\rho=1}^{2}{k}_{\rho}^{-},\prod_{\rho=1}^{2}{k}_{\rho}^{+}\right]{e}^{2\pi i\left[\left(\prod_{\rho=1}^{2}\frac{{w}_{{k}_{\rho}}^{-}}{2\pi }\right),\left(\prod_{\rho=1}^{2}\frac{{w}_{{k}_{\rho}}^{+}}{2\pi }\right)\right]}\end{array}\right),$$

(6)

$${\beta }_{1}\otimes {\beta }_{2}=\left(\begin{array}{l}\left[\prod_{\rho=1}^{2}{r}_{\rho}^{-},\prod_{\rho=1}^{2}{r}_{\rho}^{+}\right]{e}^{2\pi i\left[\left(\prod_{\rho=1}^{2}\frac{{w}_{{r}_{\rho}}^{-}}{2\pi }\right),\left(\prod_{\rho=1}^{2}\frac{{w}_{{r}_{\rho}}^{+}}{2\pi }\right)\right]},\\ \left[1-\prod_{\rho=1}^{2}\left(1-{k}_{\rho}^{-}\right),1-\prod_{\rho=1}^{2}\left(1-{k}_{\rho}^{+}\right)\right]{e}^{2\pi i\left[\left(1-\prod_{\rho=1}^{2}(1-\frac{{w}_{{k}_{\rho}}^{-}}{2\pi })\right),\left(1-\prod_{\rho=1}^{2}(1-\frac{{w}_{{k}_{\rho}}^{+}}{2\pi })\right)\right]}\end{array}\right),$$

(7)

$$\lambda{\beta }_{1}=\left(\begin{array}{c}\left[1-{\left(1-{r}_{1}^{-}\right)}^{\lambda},1-{\left(1-{r}_{1}^{+}\right)}^{\lambda}\right]{e}^{2\pi i\left[\left(1-{\left(1-\frac{{w}_{{r}_{1}}^{-}}{2\pi }\right)}^{ \lambda}\right),\left(1-{\left(1-\frac{{w}_{{r}_{1}}^{+}}{2\pi }\right)}^{ \lambda}\right)\right]},\\ \left[{\left({k}_{1}^{-}\right)}^{\lambda},{\left({k}_{1}^{+}\right)}^{\lambda}\right]{e}^{2\pi i\left[{\left(\frac{{w}_{{k}_{\rho}}^{-}}{2\pi }\right)}^{\lambda},{\left(\frac{{w}_{{k}_{\rho}}^{+}}{2\pi }\right)}^{\lambda}\right]}\end{array}\right),$$

(8)

$${\beta }_{1}^{\lambda}=\left(\begin{array}{l}\left[{\left({r}_{1}^{-}\right)}^{\lambda},{\left({r}_{1}^{+}\right)}^{\lambda}\right]{e}^{2\pi i\left[{\left(\frac{{w}_{{r}_{\rho}}^{-}}{2\pi }\right)}^{\lambda},{\left(\frac{{w}_{{r}_{\rho}}^{+}}{2\pi }\right)}^{\lambda}\right]}.\\ \left[1-{(1-{k}_{1}^{-})}^{\lambda},1-{(1-{k}_{1}^{+})}^{\lambda}\right]{e}^{2\pi i\left[\left(1-{\left(1-\frac{{w}_{{k}_{1}}^{-}}{2\pi }\right)}^{ \lambda}\right),\left(1-{\left(1-\frac{{w}_{{k}_{1}}^{+}}{2\pi }\right)}^{ \lambda}\right)\right]}\end{array}\right).$$

(9)

Definition 4

[13] For two CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),(\rho=\mathrm{1,2})$, then we have

$$T\left({\beta }_{\rho}\right)=\frac{1}{4}\left({r}_{\rho}^{-}+{r}_{\rho}^{+}+{w}_{{r}_{\rho}}^{-}+{w}_{{r}_{\rho}}^{+}-{k}_{\rho}^{-}-{k}_{\rho}^{+}-{w}_{{k}_{\rho}}^{-}-{w}_{{k}_{\rho}}^{+}\right),$$

(10)

$$H\left({\beta }_{\rho}\right)=\frac{1}{4}\left({r}_{\rho}^{-}+{r}_{\rho}^{+}+{w}_{{r}_{\rho}}^{-}+{w}_{{r}_{\rho}}^{+}+{k}_{\rho}^{-}+{k}_{\rho}^{+}+{w}_{{k}_{\rho}}^{-}+{w}_{{k}_{\rho}}^{+}\right),$$

(11)

which are described as a score and accuracy values and we know that $S\left({\beta }_{\rho}\right)\in \left[-\mathrm{2,2}\right]$ and $H\left({\beta }_{\rho}\right)\in \left[\mathrm{0,2}\right]$. Further, some rules are stated below:

• 1. If $S\left({\beta }_{1}\right)>S\left({\beta }_{2}\right)$, then ${\beta }_{1}>{\beta }_{2}$.

• 2. If $S\left({\beta }_{1}\right)<S\left({\beta }_{2}\right)$, then ${\beta }_{1}<{\beta }_{2}$.

• 3. If $S\left({\beta }_{1}\right)=S\left({\beta }_{2}\right)$, then

• 4. If $H\left({\beta }_{1}\right)>H\left({\beta }_{2}\right)$, then ${\beta }_{1}>{\beta }_{2}$.

• 5. If $H\left({\beta }_{1}\right)<H\left({\beta }_{2}\right)$, then ${\beta }_{1}<{\beta }_{2}$.

To compute some new theory, we need to review the idea of Schweizer–Sklar t-norm and Schweizer–Sklar t-conorm [19], such as:

$${T}_{ss,\pi}\left(x,y\right)={{(x}^{\pi}+{y}^{\pi}-1)}^{\frac{1}{\pi}},$$

(12)

$${{{T}_{ss,\pi}^{*}\left(x,y\right)=1-((1-x)}^{\pi}+(1-{y}^{\pi})-1)}^{\frac{1}{\pi}}.$$

(13)

We noticed that $\pi>0,x,y\in \left[\mathrm{0,1}\right]$, and where for $\pi=0$, we can easily obtain the algebraic t-norm and algebraic t-conorm such as: ${T}_{\pi}\left(x,y\right)=xy$ and ${T}_{\pi}^{*}\left(x,y\right)=x+y-xy$.

Definition 5

[18] For any collection of positive numbers ${\beta }_{\rho},\rho=\mathrm{1,2},\dots , \mathfrak{H}$, the final form of PA operator and PG operator are presented by:

$$PA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\frac{\sum_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right){\beta }_{\rho}}{\sum_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)},$$

(14)

$$PG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\prod_{\rho=1}^{\mathfrak{H}}{{\beta }_{\rho}}^{\frac{1+T\left({\beta }_{\rho}\right)}{\sum_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}.}$$

(15)

The value of $T\left({\beta }_{\rho}\right)=\sum_{\begin{array}{c}\rho=1\\ \rho\ne k\end{array}}^{\mathfrak{H}}Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ with $Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ represented the support of the ${\beta }_{\rho}$ and ${\beta }_{k}$ under the consideration of some valuable properties:

• 1. $Sup\left({\beta }_{\rho},{\beta }_{k}\right)\in \left[\mathrm{0,1}\right]$.

• 2. $Sup\left({\beta }_{\rho},{\beta }_{k}\right)=Sup\left({\beta }_{k},{\beta }_{\rho}\right)$.

• 3. $Sup\left({\beta }_{\rho},{\beta }_{k}\right)\ge Sup\left({\beta }_{m},{\beta }_{\mathfrak{H}}\right)$ if $|{\beta }_{\rho}-{\beta }_{k}|<|{\beta }_{m}-{\beta }_{\mathfrak{H}}|$.

3 Schweizer–Sklar Power Aggregation Operators Based on CIVIF Setting

The key influence of this section is to compute the Schweizer–Sklar operational laws for CIVIF information. Furthermore, under the presence of Schweizer–Sklar operational laws, we aim to derive the theory of power aggregation operators such as CIVIFSSPA, CIVIFSSPOA, CIVIFSSPG, and CIVIFSSPOG operators. Some reliable and feasible properties and results for derived work are also invented.

Definition 6

For two CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),(\rho=\mathrm{1,2})$, $\lambda\ge 1,$ then we have

$${\beta }_{1}\oplus {\beta }_{2}=\left(\begin{array}{l} \left[1-{\left({\left(1-{r}_{1}^{-}\right)}^{\pi}+{\left(1-{r}_{2}^{-}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{r}_{1}^{+}\right)}^{\pi}+{\left(1-{r}_{2}^{+}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\left(1-{w}_{{r}_{1}}^{-}\right)}^{\pi}+{\left(1-{w}_{{r}_{2}}^{-}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{w}_{{r}_{1}}^{+}\right)}^{\pi}+{\left(1-{w}_{{r}_{2}}^{+}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({{k}_{1}^{-}}^{\pi}+{{k}_{2}^{-}}^{\pi}-1\right)}^{\frac{1}{\pi}},{\left({{k}_{1}^{+}}^{\pi}+{{k}_{2}^{+}}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({{w}_{{k}_{1}}^{-}}^{\pi}+{{w}_{{k}_{2}}^{-}}^{\pi}-1\right)}^{\frac{1}{\pi}},{\left({{w}_{{k}_{1}}^{+}}^{\pi}+{{w}_{{k}_{2}}^{+}}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right) ,$$

(16)

$${\beta }_{1}\otimes {\beta }_{2}=\left(\begin{array}{l}\left[{\left({{r}_{1}^{-}}^{\pi}+{{r}_{2}^{-}}^{\pi}-1\right)}^{\frac{1}{\pi}},{\left({{r}_{1}^{+}}^{\pi}+{{r}_{2}^{+}}^{\pi}-1\right)}^{\frac{1}{\pi}}\right] \\ {e}^{2\pi i\left[{\left({{w}_{{r}_{1}}^{-}}^{\pi}+{{w}_{{r}_{2}}^{-}}^{\pi}-1\right)}^{\frac{1}{\pi}},{\left({{w}_{{r}_{1}}^{+}}^{\pi}+{{w}_{{r}_{2}}^{+}}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]},\\ \left[1-{\left({\left(1-{k}_{1}^{-}\right)}^{\pi}+{\left(1-{k}_{2}^{-}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{k}_{1}^{+}\right)}^{\pi}+{\left(1-{k}_{2}^{+}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\left(1-{w}_{{k}_{1}}^{-}\right)}^{\pi}+{\left(1-{w}_{{k}_{2}}^{-}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{w}_{{k}_{1}}^{+}\right)}^{\pi}+{\left(1-{w}_{{k}_{2}}^{+}\right)}^{\pi}-1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right) ,$$

(17)

$$\beth {\beta }_{\rho}=\left(\begin{array}{l} \left[1-{\left(\beth {\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}},1-{\left(\beth {\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\beth {\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}},1-{\left(\beth {\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\beth {\left({k}_{\rho}^{-}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}},{\left(\beth {\left({k}_{\rho}^{+}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\beth {\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}},{\left(\beth {\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\left(\beth -1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right),$$

(18)

$${\beta }_{\rho}^{\mathrm{\beth }}=\left(\begin{array}{l}\left[{\left(\mathrm{\beth }{\left({r}_{\rho}^{-}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}},{\left(\mathrm{\beth }{\left({r}_{\rho}^{+}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}}\right],{e}^{2\pi i\left[{\left(\mathrm{\beth }{\left({w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}},{\left(\mathrm{\beth }{\left({w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}}\right]}\\ \left[1-{\left(\mathrm{\beth }{\left(1-{k}_{\rho}^{-}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}},1-{\left(\mathrm{\beth }{\left(1-{k}_{\rho}^{+}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\mathrm{\beth }{\left(1-{w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}},1-{\left(\mathrm{\beth }{\left(1-{w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\left(\mathrm{\beth }-1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right).$$

(19)

Definition 7

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, the final form of the CIVIFSSPA operator is presented by:

$$CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\frac{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right){\beta }_{\rho}}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)} .$$

(20)

The value of $T\left({\beta }_{\rho}\right)=\sum_{\begin{subarray}{c}\rho=1\\ \rho\ne k\end{subarray}}^{\mathfrak{H}}Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ with $Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ represented the support of the ${\beta }_{\rho}$ and ${\beta }_{k}$.

Theorem 1

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, then using the information in Eq. (20), we prove that they again give the CIVIFNs, such as

$$CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}+1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right),$$

(21)

where ${\beth }_{\rho}=\frac{\left(1+T\left({\beta }_{\rho}\right)\right)}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}$.

Proof

Under the consideration of mathematical induction, we prove that the information in Eq. (21) is corrected. For $\mathfrak{H}=2$, we prove that Eq. (21) is correct, such as

$$\begin{aligned}{\beth }_{1}{\beta }_{1}&=\left(\begin{array}{c} \left[1-{\left({\beth }_{1}{\left(1-{r}_{1}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{1}{\left(1-{r}_{1}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\beth }_{1}{\left(1-{w}_{{r}_{1}}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{1}{\left(1-{w}_{{r}_{1}}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\beth }_{1}{\left({k}_{1}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{1}{\left({k}_{1}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\beth }_{1}{\left({w}_{{k}_{1}}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{1}{\left({w}_{{k}_{1}}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right),\\ {\beth }_{2}{\beta }_{2}&=\left(\begin{array}{c} \left[1-{\left({\beth }_{2}{\left(1-{r}_{2}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{2}{\left(1-{r}_{2}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\beth }_{2}{\left(1-{w}_{{r}_{2}}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{2}{\left(1-{w}_{{r}_{2}}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\beth }_{2}{\left({k}_{2}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{2}{\left({k}_{2}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\beth }_{2}{\left({w}_{{k}_{2}}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{2}{\left({w}_{{k}_{2}}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right).\end{aligned}$$

Then,

$$\begin{aligned}CIVIFSSPA \left({\beta }_{1},{\beta }_{2}\right)& =\frac{\left(1+T\left({\beta }_{1}\right)\right){\beta }_{1}}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}\oplus \frac{\left(1+T\left({\beta }_{2}\right)\right){\beta }_{2}}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)} \\ &=\left(\begin{array}{l} \left[1-{\left({\beth }_{1}{\left(1-{r}_{1}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{1}{\left(1-{r}_{1}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\beth }_{1}{\left(1-{w}_{{r}_{1}}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{1}{\left(1-{w}_{{r}_{1}}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\beth }_{1}{\left({k}_{1}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{1}{\left({k}_{1}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\beth }_{1}{\left({w}_{{k}_{1}}^{-}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{1}{\left({w}_{{k}_{1}}^{+}\right)}^{\pi}-\left({\beth }_{1}-1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right)\oplus \left(\begin{array}{c} \left[1-{\left({\beth }_{2}{\left(1-{r}_{2}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{2}{\left(1-{r}_{2}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\beth }_{2}{\left(1-{w}_{{r}_{2}}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{2}{\left(1-{w}_{{r}_{2}}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\beth }_{2}{\left({k}_{2}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{2}{\left({k}_{2}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\beth }_{2}{\left({w}_{{k}_{2}}^{-}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{2}{\left({w}_{{k}_{2}}^{+}\right)}^{\pi}-\left({\beth }_{2}-1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right)\\ =\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left({k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{2}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{2}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right). \end{aligned}$$

Further, we consider that the information in Eq. (21) is held for $\mathfrak{H}=k$, then

$$CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{k}\right)=\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right).$$

Further, we prove that the information in Eq. (21) also holds for $\mathfrak{H}=k+1$, then

$$\begin{aligned}CIVIFSSPA \left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{k+1}\right)& =\frac{{\oplus }_{\rho=1}^{k}\left(1+T\left({\beta }_{\rho}\right)\right){\beta }_{\rho}}{{\oplus }_{\rho=1}^{k}\left(1+T\left({\beta }_{\rho}\right)\right)}\oplus \frac{\left(1+T\left({\beta }_{k+1}\right)\right){\beta }_{k+1}}{{\oplus }_{\rho=1}^{k+1}\left(1+T\left({\beta }_{\rho}\right)\right)} \\ & =\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right)\oplus \frac{\left(1+T\left({\beta }_{k+1}\right)\right){\beta }_{k+1}}{{\oplus }_{\rho=1}^{k+1}\left(1+T\left({\beta }_{\rho}\right)\right)} \\ & =\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right)\oplus \left(\begin{array}{c} \left[1-{\left({\beth }_{k+1}{\left(1-{r}_{k+1}^{-}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{k+1}{\left(1-{r}_{k+1}^{+}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\beth }_{k+1}{\left(1-{w}_{{r}_{k+1}}^{-}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}},1-{\left({\beth }_{k+1}{\left(1-{w}_{{r}_{k+1}}^{+}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\beth }_{k+1}{\left({k}_{k+1}^{-}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{k+1}{\left({k}_{k+1}^{+}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\beth }_{k+1}{\left({w}_{{k}_{k+1}}^{-}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}},{\left({\beth }_{k+1}{\left({w}_{{k}_{k+1}}^{+}\right)}^{\pi}-\left({\beth }_{k+1}-1\right)\right)}^{\frac{1}{\pi}}\right]}\end{array}\right)\\ & =\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left({k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{k+1}{\beth }_{\rho}{\left({w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{k+1}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right).\end{aligned}$$

Hence, we derive that the information in Eq. (21) is satisfied for all possible values of $\mathfrak{H}$.

Further, we will discover the theory of idempotency, monotonicity, and boundedness under the consideration of the information in Eq. (21).

Property 1

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if ${\beta }_{\rho}=\beta =\left( \left[{r}^{-},{r}^{+}\right]{e}^{2\pi i[{w}_{r}^{-}, {w}_{r}^{+}]},[{k}^{-},{k}^{+}]{e}^{2\pi i[{w}_{k}^{-},{w}_{k}^{+}]}\right)$, then

$$CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\beta .$$

(22)

Proof

Let ${\beta }_{\rho}=\beta =\left( \left[{r}^{-},{r}^{+}\right]{e}^{2\pi i[{w}_{r}^{-}, {w}_{r}^{+}]},[{k}^{-},{k}^{+}]{e}^{2\pi i[{w}_{k}^{-},{w}_{k}^{+}]}\right)$, then

$$\begin{aligned}CIVIFSSPA \left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)&=\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{r}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{r}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{k}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{k}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right)\\ & =\left(\begin{array}{c} \left[1-{\left({\left(1-{r}^{-}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{r}^{+}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\left(1-{w}_{r}^{-}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{w}_{r}^{+}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\left({k}^{-}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}},{\left({\left({k}^{+}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\left({w}_{k}^{-}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}},{\left({\left({w}_{k}^{+}\right)}^{\pi}-1+1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right) \\ & =\left(\begin{array}{c} \left[1-{\left({\left(1-{r}^{-}\right)}^{\pi}\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{r}^{+}\right)}^{\pi}\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left({\left(1-{w}_{r}^{-}\right)}^{\pi}\right)}^{\frac{1}{\pi}},1-{\left({\left(1-{w}_{r}^{+}\right)}^{\pi}\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left({\left({k}^{-}\right)}^{\pi}\right)}^{\frac{1}{\pi}},{\left({\left({k}^{+}\right)}^{\pi}\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left({\left({w}_{k}^{-}\right)}^{\pi}\right)}^{\frac{1}{\pi}},{\left({\left({w}_{k}^{+}\right)}^{\pi}\right)}^{\frac{1}{\pi}}\right]}\end{array}\right) \\ & =\left(\begin{array}{c} \left[1-\left(1-{r}^{-}\right),1-\left(1-{r}^{+}\right)\right]\\ {e}^{2\pi i\left[1-\left(1-{w}_{r}^{-}\right),1-\left(1-{w}_{r}^{+}\right)\right]},\\ \left[{k}^{-},{k}^{+}\right]\\ {e}^{2\pi i\left[{w}_{k}^{-},{w}_{k}^{+}\right]}\end{array}\right)=\left(\begin{array}{c} \left[{r}^{-},{r}^{+}\right]\\ {e}^{2\pi i\left[{w}_{r}^{-},{w}_{r}^{+}\right]},\\ \left[{k}^{-},{k}^{+}\right]\\ {e}^{2\pi i\left[{w}_{k}^{-},{w}_{k}^{+}\right]}\end{array}\right) \\ & =\left( \left[{r}^{-},{r}^{+}\right]{e}^{2\pi i[{w}_{r}^{-}, {w}_{r}^{+}]},[{k}^{-},{k}^{+}]{e}^{2\pi i[{w}_{k}^{-},{w}_{k}^{+}]}\right)=\beta . \end{aligned}$$

Property 2

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if ${\beta }_{\rho}\le {\beta }_{\rho}{\prime}$, then

$$CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le CIVIFSSPA\left({\beta }_{1}{\prime},{\beta }_{2}{\prime},\dots ,{\beta }_{\mathfrak{H}}{\prime}\right).$$

(23)

Proof

Let if ${\beta }_{\rho}\le {\beta }_{\rho}{\prime}$, then ${r}_{\rho}^{-}\le {{r}{\prime}}_{\rho}^{-},{r}_{\rho}^{+}\le {{r}{\prime}}_{\rho}^{+},{w}_{{r}_{\rho}}^{-}\le {{w}{\prime}}_{{r}_{\rho}}^{-},{w}_{{r}_{\rho}}^{+}\le {{w}{\prime}}_{{r}_{\rho}}^{+}$ and ${k}_{\rho}^{-}\ge {{k}{\prime}}_{\rho}^{-},{k}_{\rho}^{+}\ge {{k}{\prime}}_{\rho}^{+},{w}_{{k}_{\rho}}^{-}\ge {{w}{\prime}}_{{k}_{\rho}}^{-},{w}_{{k}_{\rho}}^{+}\ge {{w}{\prime}}_{{k}_{\rho}}^{+}$, thus

$$\begin{aligned}{r}_{\rho}^{-}\le {{r}{\prime}}_{\rho}^{-} & \Rightarrow 1-{r}_{\rho}^{-}\ge 1-{{r}{\prime}}_{\rho}^{-}\Rightarrow {\left(1-{r}_{\rho}^{-}\right)}^{\pi}\ge {\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi}\\ & \Rightarrow \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}\ge \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi} \\ & \Rightarrow \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth \ge \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth \\ & \Rightarrow \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\ge \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1 \\ & \Rightarrow {\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\ge {\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}} \\ & \Rightarrow -{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\le -{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}} \\ & \Rightarrow 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\le 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}} \\ & \Rightarrow 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\le 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{r}{\prime}}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}} \\ & \Rightarrow 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\le 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{w}{\prime}}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}} \\ & \Rightarrow 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\le 1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{{w}{\prime}}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}. \end{aligned}$$

Further, we derive that

$$\begin{aligned}{k}_{\rho}^{-}\ge {{k}{\prime}}_{\rho}^{-} & \Rightarrow {\left({k}_{\rho}^{-}\right)}^{\pi}\ge {\left({{k}{\prime}}_{\rho}^{-}\right)}^{\pi}\Rightarrow {\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}\ge {\beth }_{\rho}{\left({{k}{\prime}}_{\rho}^{-}\right)}^{\pi} \\ & \Rightarrow \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}\ge \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({{k}{\prime}}_{\rho}^{-}\right)}^{\pi} \\ & \Rightarrow \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth \ge \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({{k}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth \\ & \Rightarrow \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\ge \sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({{k}{\prime}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1 \\ & \Rightarrow {\left(\sum_{\rho=1}^{\mathfrak{H}}{\mathrm{\beth }}_{\rho}{\left({k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\mathrm{\beth }+1\right)}^{\frac{1}{\pi}}\ge {\left(\sum_{\rho=1}^{\mathfrak{H}}{\mathrm{\beth }}_{\rho}{\left({{k}^{\mathrm{^{\prime}}}}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\mathrm{\beth }+1\right)}^{\frac{1}{\pi}}.\end{aligned}$$

Then, using the information in Eq. (10) and Eq. (11), we can easily determine our final results, such as:

$$CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le CIVIFSSPA\left({\beta }_{1}^{\prime},{\beta }_{2}^{\prime},\dots ,{\beta }_{\mathfrak{H}}^{\prime}\right).$$

Property 3

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if ${\beta }_{\rho}^{-}=\left( \left[\underset{\rho}{\mathrm{min}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{max}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{+}]}\right)$ and ${\beta }_{\rho}^{+}=\left( \left[\underset{\rho}{\mathrm{max}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{min}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{+}]}\right)$, then

$${\beta }_{\rho}^{-}\le CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le {\beta }_{\rho}^{+}.$$

(24)

Proof

Using property 1 and property 2, we have

$$\begin{aligned}&CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le CIVIFSSPA\left({\beta }_{1}^{+},{\beta }_{2}^{+},\dots ,{\beta }_{\mathfrak{H}}^{+}\right)={\beta }_{\rho,}^{+}\\ & \quad CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\ge CIVIFSSPA\left({\beta }_{1}^{-},{\beta }_{2}^{-},\dots ,{\beta }_{\mathfrak{H}}^{-}\right)={\beta }_{\rho}^{-}.\end{aligned}$$

Then,

$${\beta }_{\rho}^{-}\le CIVIFSSPA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le {\beta }_{\rho}^{+}.$$

Definition 8

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, the final form of the CIVIFSSPOA operator is presented by:

$$CIVIFSSPOA\left({\beta }_{0\left(1\right)},{\beta }_{0\left(2\right)},\dots ,{\beta }_{0\left(\mathfrak{H}\right)}\right)=\frac{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right){\beta }_{0\left(\rho\right)}}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)} .$$

(25)

The value of $T\left({\beta }_{\rho}\right)=\sum_{\begin{subarray}{c}\rho=1\\ \rho\ne k\end{subarray}}^{\mathfrak{H}}Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ with $Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ represented the support of the ${\beta }_{\rho}$ and ${\beta }_{k}$ with permutation $0\left(\rho\right)\le 0\left(\rho-1\right)$.

Theorem 2

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, then using the information in Eq. (25), we prove that they again give the CIVIFNs, such as

$$CIVIFSSPOA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\left(\begin{array}{c} \left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{0\left(\rho\right)}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{r}_{0\left(\rho\right)}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{r}_{0\left(\rho\right)}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{r}_{0\left(\rho\right)}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{0\left(\rho\right)}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({k}_{0\left(\rho\right)}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{k}_{0\left(\rho\right)}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{k}_{0\left(\rho\right)}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right),$$

(26)

where ${\beth }_{\rho}=\frac{\left(1+T\left({\beta }_{\rho}\right)\right)}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}$.

Proof

Straightforward.

Further, we will discover the theory of idempotency, monotonicity, and boundedness under the consideration of the information in Eq. (26).

Property 4

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if${\beta }_{\rho}=\beta =\left( \left[{r}^{-},{r}^{+}\right]{e}^{2\pi i[{w}_{r}^{-}, {w}_{r}^{+}]},[{k}^{-},{k}^{+}]{e}^{2\pi i[{w}_{k}^{-},{w}_{k}^{+}]}\right)$, then

$$CIVIFSSPOA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\beta .$$

(27)

Property 5

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if${\beta }_{\rho}\le {\beta }_{\rho}{\prime}$, then

$$CIVIFSSPOA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le CIVIFSSPOA\left({\beta }_{1}{\prime},{\beta }_{2}{\prime},\dots ,{\beta }_{\mathfrak{H}}{\prime}\right).$$

(28)

Property 6

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if ${\beta }_{\rho}^{-}=\left( \left[\underset{\rho}{\mathrm{min}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{max}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{+}]}\right)$ and ${\beta }_{\rho}^{+}=\left( \left[\underset{\rho}{\mathrm{max}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{min}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{+}]}\right)$, then

$${\beta }_{\rho}^{-}\le CIVIFSSPOA\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le {\beta }_{\rho}^{+} .$$

(29)

Definition 9

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, the final form of the CIVIFSSPG operator is presented by:

$$CIVIFSSPG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)={\otimes }_{\rho=1}^{\mathfrak{H}}{\beta }_{\rho}^{\left(\frac{\left(1+T\left({\beta }_{\rho}\right)\right)}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}\right)}.$$

(30)

The value of $T\left({\beta }_{\rho}\right)=\sum_{\begin{subarray}{c}\rho=1\\ \rho\ne k\end{subarray}}^{\mathfrak{H}}Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ with $Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ represented the support of the ${\beta }_{\rho}$ and ${\beta }_{k}$.

Theorem 3

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, then using the information in Eq. (30), we prove that they again give the CIVIFNs, such as

$$CIVIFSSPG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\left(\begin{array}{c}\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({r}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({r}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right] \\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{r}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{r}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{k}_{\rho}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{k}_{\rho}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{k}_{\rho}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{k}_{\rho}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right),$$

(31)

where ${\beth }_{\rho}=\frac{\left(1+T\left({\beta }_{\rho}\right)\right)}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}$.

Property 7

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if${\beta }_{\rho}=\beta =\left( \left[{r}^{-},{r}^{+}\right]{e}^{2\pi i[{w}_{r}^{-}, {w}_{r}^{+}]},[{k}^{-},{k}^{+}]{e}^{2\pi i[{w}_{k}^{-},{w}_{k}^{+}]}\right)$, then

$$CIVIFSSPG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\beta .$$

(32)

Property 8

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if${\beta }_{\rho}\le {\beta }_{\rho}{\prime}$, then

$$CIVIFSSPG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le CIVIFSSPG\left({\beta }_{1}^{\prime},{\beta }_{2}^{\prime},\dots ,{\beta }_{\mathfrak{H}}{\prime}\right) .$$

(33)

Property 9

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if ${\beta }_{\rho}^{-}=\left( \left[\underset{\rho}{\mathrm{min}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{max}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{+}]}\right)$ and ${\beta }_{\rho}^{+}=\left( \left[\underset{\rho}{\mathrm{max}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{min}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{+}]}\right)$, then

$${\beta }_{\rho}^{-}\le CIVIFSSPG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le {\beta }_{\rho}^{+} .$$

(34)

Definition 10

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, the final form of the CIVIFSSPOG operator is presented by:

$$CIVIFSSPOG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)={\otimes }_{\rho=1}^{\mathfrak{H}}{\beta }_{0\left(\rho\right)}^{\left(\frac{\left(1+T\left({\beta }_{\rho}\right)\right)}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}\right)}.$$

(35)

The value of $T\left({\beta }_{\rho}\right)=\sum_{\begin{subarray}{c}\rho=1\\ \rho\ne k\end{subarray}}^{\mathfrak{H}}Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ with $Sup\left({\beta }_{\rho},{\beta }_{k}\right)$ represented the support of the ${\beta }_{\rho}$ and ${\beta }_{k}$ with permutations $0\left(\rho\right)\le 0\left(\rho-1\right).$

Theorem 4

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, then using the information in Eq. (35), we prove that they again give the CIVIFNs, such as

$$CIVIFSSPOG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\left(\begin{array}{l}\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({r}_{0\left(\rho\right)}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({r}_{0\left(\rho\right)}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right] \\ {e}^{2\pi i\left[{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{r}_{0\left(\rho\right)}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left({w}_{{r}_{0\left(\rho\right)}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]},\\ \left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{k}_{0\left(\rho\right)}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{k}_{0\left(\rho\right)}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]\\ {e}^{2\pi i\left[1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{k}_{0\left(\rho\right)}}^{-}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}},1-{\left(\sum_{\rho=1}^{\mathfrak{H}}{\beth }_{\rho}{\left(1-{w}_{{k}_{0\left(\rho\right)}}^{+}\right)}^{\pi}-\sum_{\rho=1}^{\mathfrak{H}}\beth +1\right)}^{\frac{1}{\pi}}\right]}\end{array}\right),$$

(36)

where ${\mathrm{\beth }}_{\rho}=\frac{\left(1+T\left({\beta }_{\rho}\right)\right)}{{\oplus }_{\rho=1}^{\mathfrak{H}}\left(1+T\left({\beta }_{\rho}\right)\right)}$.

Property 10

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if${\beta }_{\rho}=\beta =\left( \left[{r}^{-},{r}^{+}\right]{e}^{2\pi i[{w}_{r}^{-}, {w}_{r}^{+}]},[{k}^{-},{k}^{+}]{e}^{2\pi i[{w}_{k}^{-},{w}_{k}^{+}]}\right)$, then

$$CIVIFSSPOG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)=\beta .$$

(37)

Property 11

For any collection of CIVIFNs${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if${\beta }_{\rho}\le {\beta }_{\rho}{\prime}$, then

$$CIVIFSSPOG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le CIVIFSSPOG\left({\beta }_{1}{\prime},{\beta }_{2}{\prime},\dots ,{\beta }_{\mathfrak{H}}{\prime}\right).$$

(38)

Property 12

For any collection of CIVIFNs ${\beta }_{\rho}=\left( \left[{r}_{\rho}^{-},{r}_{\rho}^{+}\right]{e}^{2\pi i[{w}_{{r}_{\rho}}^{-} , {w}_{{r}_{\rho}}^{+}]},[{k}_{\rho}^{-},{k}_{\rho}^{+}]{e}^{2\pi i[{w}_{{k}_{\rho}}^{-} ,{w}_{{k}_{\rho}}^{+}]}\right),\rho=\mathrm{1,2},\dots ,\mathfrak{H}$, if ${\beta }_{\rho}^{-}=\left( \left[\underset{\rho}{\mathrm{min}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{max}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{k}_{\rho}}^{+}]}\right)$ and ${\beta }_{\rho}^{+}=\left( \left[\underset{\rho}{\mathrm{max}}{r}_{\rho}^{-},\underset{\rho}{\mathrm{max}}{r}_{\rho}^{+}\right]{e}^{2\pi i[\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{-},\underset{\rho}{\mathrm{max}}{w}_{{r}_{\rho}}^{+}]},[\underset{\rho}{\mathrm{min}}{k}_{\rho}^{-},\underset{\rho}{\mathrm{min}}{k}_{\rho}^{+}]{e}^{2\pi i[\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{-},\underset{\rho}{\mathrm{min}}{w}_{{k}_{\rho}}^{+}]}\right)$, then

$${\beta }_{\rho}^{-}\le CIVIFSSPOG\left({\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}\right)\le {\beta }_{\rho}^{+} .$$

(39)

4 Multi-attribute Decision-Making

The decision-making technique is a very valuable and effective procedure for evaluating the finest preference from the collection of preferences. Here, we aim to evaluate the idea of the MADM scenario under the availability of discovered operators for CIVIF information for showing the reliability and stability of the evaluated operators.

For this, we have a family of alternatives ${\beta }_{1},{\beta }_{2},\dots ,{\beta }_{\mathfrak{H}}$ with a group of finite values of attributes for each alternative such as: ${\beta }_{1}^{AT},{\beta }_{2}^{AT},\dots ,{\beta }_{m}^{AT}$. For performing the decision-making procedure, we require to construct a decision matrix by including the CIVIFNs such as $A=\left( \left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]},[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}\right)$. We noticed that the symbols $\left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)]}$ and $[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)]}$ expressed the information of truth and falsity function in the shape of complex numbers, where ${r}_{A}^{-}\le {r}_{A}^{+},{k}_{A}^{-}\le {k}_{A}^{+}$ and ${r}_{A}^{+}\left(x\right)+{k}_{A}^{+}\left(x\right)\le 1$ and ${w}_{{r}_{A}}^{-}\le {w}_{{r}_{A}}^{+}, {w}_{{k}_{A}}^{-}\le {w}_{{k}_{A}}^{+}$ and ${w}_{{r}_{A}}^{+}\left(x\right)+{w}_{{k}_{A}}^{+}\left(x\right)\le 2\pi .\forall x\in U$. For this, we aim to rearrange a procedure for evaluating the finest decision from the collection of preferences under the consideration of below procedure such as:

Step 1: Considering the cost and benefit types of information, we arrange the CIVIFNs in the shape of a matrix, if all data are cost types, then using the below information normalize it, such as:

$${Z}{\prime}=\left\{\begin{array}{cc}\left( \left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]},[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}\right)& \text{for be }\mathfrak{H}\; \text{efit}\\ \left(\left[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}, \left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]}\right)& \text{for\; cost}.\end{array}\right.$$

In an alternative scenario, provided all data categories yield advantages.

Step 2: Considering the theory of the CIVIFSSPA operator and CIVIFSSPG operator, we theme to aggregate our mentioned or constructed matrix of information.

Step 3: Compute the score values (Eq. 10) (in the case of failure, we use accuracy values (Eq. 11)) of all obtain aggregated information.

Step 4: Perform the ranking results based on each score value and examine the finest values from the collection of preferences.

Further, we proceed our work by considering some artificial information and trying to justify the above procedure.

4.1 Illustrative Example

In this example, we evaluate the problem of green suppliers, often referred to as sustainable or environmentally friendly suppliers, offering goods and services that are less harmful to the environment than those offered by conventional suppliers. Here are a few distinct categories of eco-friendly vendors:

Renewable Energy Providers (${\beta }_{1}$): These companies provide electricity from clean, renewable sources including solar, wind, hydro-, and geothermal energy. They assist businesses in lowering their carbon footprint and reliance on fossil fuels.

Suppliers of Organic and Sustainable Agriculture (${\beta }_{2}$): These suppliers supply dairy products, cereals, fruits, and vegetables, among other organic and sustainable agricultural products. They use organic agricultural methods, steer clear of chemical pesticides, and place a high priority on soil health and biodiversity.

Eco-Friendly Packaging Suppliers (${\beta }_{3}$): These companies concentrate on offering environmentally friendly packaging options, such as packaging produced from recycled or recyclable materials, biodegradable or compostable materials, or packaging created from.

Suppliers of Green Building Products (${\beta }_{4}$): These companies provide eco-friendly products for construction and remodeling projects. They could provide ecologically friendly paints and coatings, resilient flooring, water-saving fixtures, or energy-efficient insulation.

Vendors Who Provide Trash Management and Recycling Services (${\beta }_{5}$): These vendors are experts in these fields. Through recycling initiatives, composting services, or waste-to-energy technologies, they assist businesses in managing and reducing their trash.

This all information represents the five forms of alternatives. Based on the following attributes or criteria, we will try to select the best one for us, such as ${\beta }_{1}^{AT}$: growth factor, ${\beta }_{2}^{AT}$: Political impact, ${\beta }_{3}^{AT}$: suitable for our work, and ${\beta }_{4}^{AT}$: Environmental impact. For evaluating this problem, we aim to rearrange a procedure for evaluating the finest decision from the collection of preferences under the consideration of the below procedure such as:

Step 1: Considering the cost and benefit types of information, we arrange the CIVIFNs in the shape of a matrix see Table 1, if all data are cost types, then using the below information normalize it, such as:

$${Z}{\prime}=\left\{\begin{array}{cc}\left( \left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]},[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}\right)& \mathrm{for\, be \,}\mathfrak{H}\mathrm{efit}\\ \left(\left[{k}_{A}^{-}\left(x\right),{k}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{k}_{A}}^{-}\left(x\right),{w}_{{k}_{A}}^{+}\left(x\right)\right]}, \left[{r}_{A}^{-}\left(x\right),{r}_{A}^{+}\left(x\right)\right]{e}^{2\pi i\left[{w}_{{r}_{A}}^{-}\left(x\right),{w}_{{r}_{A}}^{+}\left(x\right)\right]}\right)& \mathrm{for\, cost}.\end{array}\right.$$

Table 1 CIVIF information matrix

In an alternative scenario, provided all data categories yield advantages. All information in Table 1 is arranged in the shape of benefit types, so we do not require to normalize it.

Step 2: Considering the theory of the CIVIFSSPA operator and CIVIFSSPG operator, we theme to aggregate our mentioned or constructed matrix of information, see Table 2.

Table 2 Aggerated information matrix

Step 3: Compute the score values (Eq. 10) (in the case of failure, we use accuracy values (Eq. 11)) of all obtain aggregated information, see Table 3.

Table 3 Score values matrix

Step 4: Perform the ranking results based on each score value and examine the finest values from the collection of preferences, see Table 4.

Table 4 Ranking values matrix

From Table 4, we get two different types of finest decisions for two different types of operators such as ${\beta }_{5}$ for CIVIFSSPA operator and ${\beta }_{1}$ for CIVIFSSPG operator. Further, we compare our presented information with some existing information to show the validity and supremacy of the derived work.

5 Comparative Analysis

Comparing the proposed work with various prevailing operators are an essential part of every valuable manuscript. Here, we compared our mentioned operators with various prevailing operators for enhancing the worth and stability of the evaluated approaches. For this, we consider some prevailing operators which were computed under the consideration of different ideas, such as aggregation operators for CIVIFS evaluated by Garg and Rani [13], power aggregation operators for IVIFS derived by He et al. [20], Schweizer–Sklar power aggregation operators for IVIFS were derived by Liu and Wang [21], and finally, the theory of TODIM and Schweizer–Sklar power aggregation operators for IVIFSs were also derived by Zindani et al. [22]. In the presence of information in Table 1, the comparative analysis is stated in Table 5.

Table 5 Comparative analysis (using information in Table 1)

Information in Table 5 stated the comparative analysis between proposed and existing information which were stated by different scholars, according to the theory of Garg and Rani [13] and according to the theory of proposed work, we get two types of best decisions such as ${\beta }_{5}$ for CIVIFSSPA operator and information in Ref. [13], but we get the best decision as a ${\beta }_{1}$ according to the theory of the CIVIFSSPG operator. The main problem, we have discussed below, why some existing information have been failed, such as

• 1. First, we try to briefly discuss the main problem with the theory of He et al. [20]. He et al. [20] evaluated the power aggregation operators for IVIFS, where the power operators are very skillful and reliable, but due to IVIFS, they have less worth, because the theory of IVIFS is the special cases of the CIVIFS, therefore, it is clear that the theory derived in Ref. [20] has been failed to evaluate our required or considered problem.

• 2. Second, we try to briefly discuss the main problem with the theory of Liu and Wang [21]. Liu and Wang [21] evaluated the Schweizer–Sklar power aggregation operators for IVIFS, where the Schweizer–Sklar operators are very skillful and reliable, but due to IVIFS, they have less worth, because the theory of IVIFS is the special cases of the CIVIFS; therefore, it is clear that the theory derived in Ref. [21] has been failed to evaluate our required or considered problem.

• 3. Third, we try to briefly discuss the main problem with the theory of Zindani et al. [22]. Zindani et al. [22] evaluated the TODIM and Schweizer–Sklar power aggregation operators for IVIFSs, where the TODIM and Schweizer–Sklar power aggregation operators are very skillful and reliable, but due to IVIFS, they have less worth, because the theory of IVIFS is the special cases of the CIVIFS, therefore, it is clear that the theory derived in Ref. [22] has been failed to evaluate our required or considered problem.

Therefore, we clear that the theory of Liu and Wang [21], and Zindani et al. [22] have been failed for depicting or evaluating the information in Table 1. Because the operators based on IVIFS which were evaluated or derived by He et al. [20], Liu and Wang [21], and Zindani et al. [22] are a special case of the deceived work. Hence, the proposed operators based on CIVIFS are novel and very well for depicting awkward and unreliable information in decision-making problems.

6 Conclusion

In the presence of CIVIFS, Schweizer–Sklar norms, and power aggregation operators, we derived the following ideas:

• 1. We computed the Schweizer–Sklar operational laws for CIVIF information.

• 2. We derived the theory of CIVIFSSPA, CIVIFSSPOA, CIVIFSSPG, and CIVIFSSPOG operators. Some reliable and feasible properties and results for derived work are also invented.

• 3. We computed a MADM scenario for discovered operators under the presence of CIVIF information for showing the reliability and stability of the evaluated operators.

• 4. We compared our mentioned operators with various prevailing operators for enhancing the worth and stability of the evaluated approaches.

In the future, we concentrate to derive some new aggregation operators, similarity measures, and different types of techniques for different types of fuzzy sets and their extensions. Fuzzy logic has been employed widely in solving different engineering problems [38, 39]. These future developments would allow to contribute to the solution of complex real-life problems such as decision-making with multiple stakeholders related to urban mobility governance. Further, we aim to employ it in the field of decision-making [23, 24], pattern recognition [25, 26], q-Rung orthopair fuzzy information [27], preference decision-making [28] or group decision-making [29,30,31], and decision-making [32,33,34,35,36,37] to improve the worth of derived theory.