Aczel–Alsina Power Aggregation Operators for Complex Picture Fuzzy (CPF) Sets with Application in CPF Multi-Attribute Decision Making

Abstract

Complex picture fuzzy sets are the updated version of the complex intuitionistic fuzzy sets. A complex picture fuzzy set covers three major grades such as membership, abstinence, and falsity with a prominent characteristic in which the sum of the triplet will be contained in the unit interval. In this scenario, we derive the power aggregation operators based on the Aczel–Alsina operational laws for managing the complex picture of fuzzy values. These complex picture fuzzy power aggregation operators are complex picture fuzzy Aczel–Alsina power averaging, complex picture fuzzy Aczel–Alsina weighted power averaging, complex picture fuzzy Aczel–Alsina power geometric, and complex picture fuzzy Aczel–Alsina weighted power geometric operators. We also investigate their theoretical properties. To justify these complex picture fuzzy power aggregation operators, we illustrate a procedure of a decision-making technique in the presence of complex picture fuzzy values and derive an algorithm to evaluate some multi-attribute decision-making problems. Finally, a practical example is examined to illustrate the decision-making procedure under the consideration of derived operators, and their performance is compared with that of various operators to show the supremacy and validity of the proposed approaches.

1. Introduction

For illustrating the valuable preference from the collection of preferences, the major idea of the multi-attribute decision-making (MADM) procedure is a great and dominant technique, which is the modified version of the simple decision-making scenario. In our daily-life problems, we evaluate or face different types of decision-making problems where our major or initial or first step is to learn how to make a valuable, excellent, or great decision. Usually, the expert brings classical information without evaluating the uncertainty and ambiguity in it. However, in sequence to treat the ambiguity in the information, a hypothesis such as fuzzy sets (FSs) [1] is one of the most effective and worthy ideas for describing the value of truth information whose range is in the unit interval. The FSs were widely applied in a diversity of ways, such as [2,3]. But here, a lot of individuals have raised the question of where the falsity of the information is. To accommodate the above problem, Atanassov [4] invented the theory of intuitionistic FSs (IFSs) as an effective idea that describes the value of truth and falsity information whose range is in a unit interval with the prominent characteristic $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$. The theory of IFSs is a generalized form of FSs, but IFSs also have their restrictions; for instance, during an election, experts have faced four types of problems such as yes, abstinence, no, and neutral or refusal, where the theory of IFSs has been neglected to cope with some problems [5]. Therefore, to accommodate the above problems, Cuong [6] invented the theory of picture FSs (PFSs) as a worthy idea that described the value of truth, abstinence, and falsity of information whose range is in a unit interval with a prominent characteristic $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$, where the dynamic distance measures were derived in [7].

The amplitude and phase terms are generally famous and valuable because in many situations we have used them. For instance, if we are taking any kind of software, then we need to give two types of information regarding each software such as the name and version of the software, where both are represented by the amplitude and phase terms. In the presence of FSs theory, we fail to cope with it; therefore, Ramot et al. [8] derived a new structure of complex FSs (CFSs), where the value of the truth grade in CFSs is computed in the form of polar coordinates. A systematic review of CFSs with various applications was given by Yazdanbakhsh and Dick [9]. To utilize the grade of falsity information in the structure of CFSs is a very complicated and awkward task for individuals. Therefore, Alkouri and Salleh [10] derived a new shape, called complex IFSs (CIFSs). In CIFSs, there are truth grades such as $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$ and falsity grades such as $\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$ with a prominent characteristic, such as $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$ and $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$. The FSs, IFSs, and CFSs are special cases of the CIFSs. The theory of CIFSs is an extension of CFSs, but CIFSs also have their restrictions and limitations. Therefore, to accommodate the above problems, Akram et al. [11] invented the theory of complex picture FSs (CPFSs), which is an effective and worthy idea that describes the value of truth, abstinence, and falsity as stated by: $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)},\stackrel{=}{{℃}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$ and $\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$ with a dominant and well-known condition, such as $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$ and $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$.

The main structure of Aczel–Alsina t-norm and t-conorm was derived by Aczel and Alsina [12] in 1982. Further, the power aggregation operators (AOs) were evaluated by Yager [13] in 2001. Additionally, Aczel–Alsina AOs for IFSs were invented by Senapati et al. [14]. Aczel–Alsina geometric operators for IFSs were derived by Senapati et al. [15]. Aczel–Alsina prioritized AOs for IFS were considered by Sarfraz et al. [16]. Aczel–Alsina AOs for Pythagorean FSs were proposed by Hussain et al. [17], and Aczel–Alsina AOs for complex Pythagorean FSs were considered by Jin et al. [18]. Aczel–Alsina weighted AOs of neutrosophic Z-numbers were given by Ye et al. [19]. Aczel–Alsina averaging operators for picture FSs (PFSs) were derived by Senapati [20], and Aczel–Alsina geometric operators for PFSs were considered by Naeem et al. [21]. Power AOs for IFSs were derived by Xu [22]. Jiang et al. [23] demonstrated the power AOs based on entropy measures for IFSs, Rani and Garg [24] examined the power AOs for CIFSs, and Liu et al. [25] proposed the power AOs for CPFSs. Mahmood et al. [26] considered Aczel–Alsina AOs for bipolar CFSs. We mention that there is no research on power AOs for CPFSs based on Aczel–Alsina operational laws, and these should be our main focus in this paper.

It is known that the theory of FSs, IFSs, PFSs, CFSs, and CIFSs has a lot of applications in decision-making, pattern recognition, medical diagnosis, and cluster analysis. But in many situations, it may fail because of its limitations. Since the theory of CPFSs is more modified and valuable than that of FSs, IFSs, CFSs, and CIFSs, in which CPFSs cover three major grades of membership, abstinence, and falsity with a prominent characteristic such that the sum of the triplet will be contained in the unit interval. Additionally, the theory of Aczel–Alsina information and power AOs has a lot of advantages. This is because the theory of averaging and geometric AOs are the particular cases of the Aczel–Alsina information. Moreover, the algebraic t-norm and t-conorm are also special cases of the Aczel–Alsina operators. Furthermore, it is an important task to show how to develop the theory of power AOs based on Aczel–Alsina operational laws for CPFSs. Even though combining these ideas into one structure is a complicated task, if we can combine these ideas, then it will be beneficial for researchers. After all, with the help of this information, we can easily derive those required results of averaging, geometric, and power based on Aczel–Alsina AOs. In fact, these should be the special cases of our derived theory in Aczel–Alsina AOs for CPFSs. Thus, the contributions in this paper can be described as follows:

• To explore the basic Aczel–Alsina operational law for complex picture fuzzy (CPF) values.
• To derive the CPF Aczel–Alsina power averaging (CPFAAP-A), CPF Aczel–Alsina weighted power averaging (CPFAAWP-A), CPF Aczel–Alsina power geometric (CPFAAP-G), and CPF Aczel–Alsina weighted power geometric (CPFAAWP-G) operators.
• To examine the three basic properties of the above operators, such as idempotency, monotonicity, and boundedness.
• To justify the above problem, we illustrate a procedure of decision making in the presence of the CPF values (CPFVs) and derive an algorithm to evaluate the MADM problems. Furthermore, these can be extended to CPF Maclaurin symmetric mean and power generalized Maclaurin symmetric mean operators.
• To illustrate a practical example of a decision-making procedure under the consideration of derived operators and compare their performance with various operators to show the supremacy and validity of the derived approaches.

The remainder of this paper is as follows. In Section 2, we review some prevailing or existing information such as CPFSs, power aggregation (P-A) operators, and some Aczel–Alsina operational laws. In Section 3, we derive the idea of CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators and discover their properties. To justify the above problem, in Section 4, we illustrate a procedure of a decision-making technique in the presence of CPFVs and derive an algorithm to evaluate the MADM problems. Finally, in Section 5, a numerical or practical example is examined to illustrate the decision-making procedure under the consideration of derived operators, and their performance is compared with various operators to show the supremacy and validity of the derived approaches. Some concluding analysis is stated in Section 6.

2. Preliminaries

In this section, we review some prevailing and existing information such as CPFS, P-A operator, and some Aczel–Alsina operational laws based on a universal set $\widetilde{{\mathbb{X}}_U}$.

Definition 1 ([11).

Using a fixed set $\widetilde{{\mathbb{X}}_U}$, we examine the idea of CPFS as

$$\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}=\left\{\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right),\stackrel{=}{{℃}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right),\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right):\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\in \widetilde{{\mathbb{X}}_U}\right\}$$

(1)

Here, we talk about the triplet such as truth, abstinence, and falsity which are stated by:  $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)},\stackrel{=}{{℃}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$  and  $\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$  with  $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$  and  $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$. Furthermore, the complicated structure  $\stackrel{=}{{ℝ}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{ℝ}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{ℝ}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}=\left(1-\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)\right)e^{i2\pi \left(1-\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)\right)}$is used as a neutral grade and  $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)},\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)},\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)}\right),ℐ=1,2,\dots ,\ell$ states the CPF values (CPFVs).

Definition 2 ([13]).

By considering $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},ℐ=1,2,\dots ,\ell$, used as non-negative information, the P-A operator is defined as

$$A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\frac{{{\displaystyle \sum }}_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}{{{\displaystyle \sum }}_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}$$

(2)

Notice that  $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne ℐ\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)$states the relationship between  $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}$  and  $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}$  with the following characteristics:

(1)

$\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)\in \left[0,1\right]$.

(2)

$\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)$.

(3)

$\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)\ge \mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_k}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_l}}\right)$ when $\left|\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}-\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right|\le \left|\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_k}}-\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_l}}\right|$.

Definition 3.

The existing form of Aczel–Alsina operational laws for any two CPFVs are defined as:

$$\stackrel{{\displaystyle =}}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\left(\begin{array}{c}(1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}) e^{i2\pi \left(1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}, \\ \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right) e^{i2\pi \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)} \end{array}\right)$$

(3)

$$\stackrel{{\displaystyle =}}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\otimes \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\left(\begin{array}{c}\left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right) e^{i2\pi \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}, \\ (1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{℃}_{\stackrel{=}{R_1}}})\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{℃}_{\stackrel{=}{R_2}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}})e^{i2\pi (1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{℃}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}})},\\ (1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}})\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}) e^{i2\pi (1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}})\right)}^{\mathbb{U}}+{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_2}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}})} \end{array}\right)$$

(4)

$$\stackrel{=}{{\mathcal{O}}_S}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\left(\begin{array}{c}(1-e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}) e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right) e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)$$

(5)

$${\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{O}}_S}}\left(\begin{array}{c}\left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right) e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ 1-\left(e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(1-e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right) e^{i2\pi (1-e^{-{\left(\stackrel{=}{{\mathcal{O}}_S}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}})} \end{array}\right)$$

(6)

Definition 4 ([11]).

The existing form of score and accuracy function for any two CPFVs are stated below:

$$\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)=\frac{1}{3}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}+\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}-\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}-\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right),\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\in \left[-1,1\right]$$

(7)

$$\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)=\frac{1}{3}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}+\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}\_+\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}+\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right),\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\in \left[0,1\right]$$

(8)

Here, we describe some characteristics of the above information such as

• If $\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)>\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)\Rightarrow \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$;
• If $\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)<\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)\Rightarrow \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}<\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$;
• If $\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)=\stackrel{=}{{\mathcal{Y}}_{SV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)\Rightarrow$;

  (i)

  If $\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)>\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)\Rightarrow \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$;

  (ii)

  If $\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)<\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)\Rightarrow \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}<\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$;

  (iii)

  If $\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)=\stackrel{=}{{\mathcal{Y}}_{AV}}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)\Rightarrow \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$.

Theorem 1.

The existing form of the well-known results/properties for any two CPFVs are stated as follows:

• $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\oplus \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}$;
• $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\otimes \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\otimes \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}$;
• $\stackrel{=}{{\mathcal{O}}_S}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=\stackrel{=}{{\mathcal{O}}_S}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathcal{O}}_S}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$;
• $\left(\stackrel{=}{{\mathcal{O}}_{S_1}}+\stackrel{=}{{\mathcal{O}}_{S_1}}\right)\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\stackrel{=}{{\mathcal{O}}_{S_1}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathcal{O}}_{S_2}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}$;
• ${\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\otimes \stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)}^{\stackrel{=}{{\mathcal{O}}_S}}={\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{O}}_S}}\otimes {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\stackrel{=}{{\mathcal{O}}_S}}$;
• ${\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{O}}_{S_1}}}\otimes {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{O}}_{S_1}}}={\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{O}}_{S_1}}+\stackrel{=}{{\mathcal{O}}_{S_2}}}$.

3. The Proposed Aczel-Alsina Power AOs for CPFVs

In this section, we derive some power AOs based on Aczel–Alsina operational laws for managing the CPF values (CPFVs). These are CPF Aczel–Alsina power averaging (CPFAAP-A), CPF Aczel–Alsina weighted power averaging (CPFAAWP-A), CPF Aczel–Alsina power geometric (CPFAAP-G), and CPF Aczel–Alsina weighted power geometric (CPFAAWP-G) operators. We then give some properties of them.

Definition 5.

The computational form of the CPFAAP-A operator is defined as:

$$CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\stackrel{=}{{\mathcal{K}}_1}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathcal{K}}_2}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\oplus \dots \oplus \stackrel{=}{{\mathcal{K}}_{\ell }}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}={\oplus }_{ℐ=1}^{\ell }\left(\stackrel{=}{{\mathcal{K}}_{ℐ}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)$$

(9)

$$\stackrel{=}{{\mathcal{K}}_{ℐ}}=PA\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\frac{\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}$$

(10)

where $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne ℐ\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)$, $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=1-d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)$. Further, $d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\frac{1}{6}\left(\left|\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_s}}}\right|+\left|\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_s}}}\right|+\left|\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}-\stackrel{=}{{℃}_{\stackrel{=}{R_s}}}\right|+\left|\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}-\stackrel{=}{{℃}_{\stackrel{=}{I_s}}}\right|+\left|\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_s}}}\right|+\left|\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_s}}}\right|\right)$ illustrates the distance measure. The information in Equations (9) and (10) can be seen as a generalization of Aczel–Alsina power averaging operators of FSs, IFSs, PFSs, CFSs, and CIFSs.

Theorem 2.

In the consideration of Equations (9) and (10), we prove that the finalized result of the above theory is again in the form of CPFVs with

$$\begin{array}{l}CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\\ =\left(\begin{array}{c}\left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{=}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

(11)

Proof. 

(Using Mathematical induction). For $\ell =2$, we have

$$\stackrel{=}{{\mathcal{K}}_1}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\left(\begin{array}{c}\left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)$$

$$\stackrel{=}{{\mathcal{K}}_2}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\left(\begin{array}{c}\left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)$$

Thus, $CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=\stackrel{=}{{\mathcal{K}}_1}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathcal{K}}_2}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$

$$\begin{array}{l}=(\begin{array}{c}\left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_1}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_1}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\oplus \left(\begin{array}{c}\left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_2}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_2}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^2\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^2\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ e^{-{\left({\displaystyle \sum }_{ℐ=1}^2\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}{e}^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^2\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^2\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^2\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

Our assumption holds. Further, we try to assume that for $\ell =k$ is also hold. Then we have

$$\begin{array}{l}CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_k}}\right)\\ =\left(\begin{array}{c}\left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{=}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

For $\ell =k+1$, we have that

$$CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{k+1}}}\right)={\oplus }_{ℐ=1}^{k+1}\left(\stackrel{=}{{\mathcal{K}}_{ℐ}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)={\oplus }_{ℐ=1}^k\left(\stackrel{=}{{\mathcal{K}}_{ℐ}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\oplus \stackrel{=}{{\mathcal{K}}_{k+1}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{k+1}}}$$

$$\begin{array}{l}=(\begin{array}{c}\left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi (e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^k\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}})\end{array})\\ \oplus \left(\begin{array}{c}\left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_{k+1}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{k+1}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left(\stackrel{=}{{\mathcal{K}}_{k+1}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{k+1}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_{k+1}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{k+1}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_{k+1}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{k+1}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_{k+1}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{k+1}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left(\stackrel{=}{{\mathcal{K}}_{k+1}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{k+1}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{k+1}\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{k+1}\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ e^{-{\left({\displaystyle \sum }_{ℐ=1}^{k+1}\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}{e}^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{k+1}\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{k+1}\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{k+1}\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

Our assumption holds for all positive information. □

Example 1.

To clarify the above problem with the help of practical information, we use some information in the form of CPFVs, such as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\left(0.4e^{i2\pi \left(0.3\right)},0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\left(0.5e^{i2\pi \left(0.4\right)},0.3e^{i2\pi \left(0.2\right)},0.1e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}=\left(0.7e^{i2\pi \left(0.8\right)},0.1e^{i2\pi \left(0.1\right)},0.01e^{i2\pi \left(0.01\right)}\right)$ and $\mathbb{U}=3$. Then, $d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=0.06667;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.21333;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.18;$ $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-0.06667=0.93333;$ $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.78667;\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.82.$ Thus, we have $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne 1\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)+\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.72;$ $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1.7533; \mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.60667;$ ${\displaystyle \sum }_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)=8.08.$ We the have $\stackrel{=}{{\mathcal{K}}_1}=\frac{\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^3\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}=\frac{1+1.72}{8.08}=0.33663$; $\stackrel{=}{{\mathcal{K}}_2}=0.34076;\stackrel{=}{{\mathcal{K}}_3}=0.32261.$ Thus, we get $CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=\left(0.3226e^{i2\pi \left(0.38522\right)},0.45549e^{i2\pi \left(0.43987\right)},0.23502e^{i2\pi \left(0.22707\right)}\right).$

Proposition 1 (Idempotency).

If $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\stackrel{=}{\mathbb{G}}$, then

$$CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\stackrel{=}{\mathbb{G}}$$

(12)

Proof. 

Notice that we have $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\stackrel{=}{\mathbb{G}}=\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\right)},\stackrel{=}{{℃}_{\stackrel{=}{R}}}{e}^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\right)},\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\right)}\right)$. Then

$$\begin{array}{l}CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\\ =\left(\begin{array}{c}\left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left(\begin{array}{c}\left(1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\left(1-e^{\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\right)}\right)e^{i2\pi \left(1-e^{\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\right)}\right)},\\ \left(e^{\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R}}}\right)}\right)e^{i2\pi \left(e^{\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\right)}\right)},\left(e^{\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\right)}\right)e^{i2\pi \left(e^{\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\right)}\right)}\end{array}\right)=\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\right)},\stackrel{=}{{℃}_{\stackrel{=}{R}}}{e}^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\right)},\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\right)}\right).\hspace{1em}\square \end{array}$$

Our next concern is if the CPFAAP-A operator $CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)$ satisfies the property of monotonicity. That is, the statement “If $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\le {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}^{\prime }=\left({\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}}^{\prime }{e}^{i2\pi \left({\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}}^{\prime }\right)},{\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}}^{\prime }{e}^{i2\pi \left({\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}}^{\prime }\right)},{\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}}^{\prime }{e}^{i2\pi \left({\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}}^{\prime }\right)}\right)$, then $CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\le CPFAAP-A\left({\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\prime },{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\prime },\dots ,{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}}^{\prime }\right)$” is true or not. Our answer is that the statement is not true, i.e., If $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\le {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}^{\prime },$ then

$$CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\nleqq CPFAAP-A\left({\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\prime },{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\prime },\dots ,{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}}^{\prime }\right)$$

(13)

Our counter-example is as follows: Consider the three major CPFVs with $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\left(0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.2\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\left(0.4e^{i2\pi \left(0.4\right)},0.4e^{i2\pi \left(0.4\right)},0.2e^{i2\pi \left(0.2\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}=\left(0.1e^{i2\pi \left(0.1\right)},0.89e^{i2\pi \left(0.89\right)},0.01e^{i2\pi \left(0.01\right)}\right)$ and ${\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\prime }=\left(0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.2\right)}\right),{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\prime }=\left(0.4e^{i2\pi \left(0.4\right)},0.4e^{i2\pi \left(0.4\right)},0.2e^{i2\pi \left(0.2\right)}\right),{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}}^{\prime }=\left(0.11e^{i2\pi \left(0.11\right)},0.11e^{i2\pi \left(0.11\right)},0.01e^{i2\pi \left(0.01\right)}\right)$ with the help of $\mathbb{U}=3$, we have $d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=0.13333$; $d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.32667;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.32667$; $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=0.86667$; $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.67333;\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.67333$. Thus, we have that $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne 1\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)+\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.54$; $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1.54; \mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.34667$; ${\displaystyle \sum }_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)=7.42667$. Therefore, $\stackrel{=}{{\mathcal{K}}_1}=\frac{\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^3\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}=\frac{1+1.54}{7.42667}=0.34201$; $\stackrel{=}{{\mathcal{K}}_2}=0.34201; \stackrel{=}{{\mathcal{K}}_3}=0.31598.$ Then, we get $CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=\left(0.14763e^{i2\pi \left(0.14763\right)},0.59616e^{i2\pi \left(0.59616\right)},0.24578e^{i2\pi \left(0.24587\right)}\right)$. Further, in the same way, we derive the followings, such as $CPFAAP-A\left({\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\prime },{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\prime },{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}}^{\prime }\right)=\left(0.14599e^{i2\pi \left(0.14559\right)},0.47078e^{i2\pi \left(0.47078\right)},0.24192e^{i2\pi \left(0.24192\right)}\right)$. Then, it is clear that our required result is on hold. Hence, $CPFAAP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\nleqq CPFAAP-A\left({\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\prime },{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\prime },\dots ,{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}}^{\prime }\right)$. From this, we notice that the property of monotonicity cannot be satisfied under the consideration of the CPFAAP-A operator.

We next propose the CPFAAWP-A operator.

Definition 6.

The computational form of the CPFAAWP-A operator is given as follows:

$$CPFAAWP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\stackrel{=}{{\mathcal{K}}_1}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\oplus \stackrel{=}{{\mathcal{K}}_2}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\oplus \dots \oplus \stackrel{=}{{\mathcal{K}}_{\ell }}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}={\oplus }_{ℐ=1}^{\ell }\left(\stackrel{=}{{\mathcal{K}}_{ℐ}}\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)$$

(14)

$$\stackrel{=}{{\mathcal{K}}_{ℐ}}=WPA\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\frac{{\mathsf{\Phi }}_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^{\ell }{\mathsf{\Phi }}_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}$$

(15)

Notice that ${\mathsf{\Phi }}_{ℐ}\in \left[0, 1\right]$ with ${\displaystyle \sum }_{ℐ=1}^{\ell }{\mathsf{\Phi }}_{ℐ}=1$ names weight vectors.

Theorem 3. 

In the consideration of Equations (14) and (15), we prove that the finalized result of the above theory is again in the form of CPFV, such that

$$\begin{array}{l}CPFAAWP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\\ =\left(\begin{array}{c}\left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

(16)

Proof. 

The proof is similar as Theorem 2 by using the mathematical induction. □

Example 2.

To clarify Theorem 3 with the help of practical information To verify the above problem with the help of practical information, we decided to use some practical information in the form of CPFVs such as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\left(0.4e^{i2\pi \left(0.3\right)},0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\left(0.5e^{i2\pi \left(0.4\right)},0.3e^{i2\pi \left(0.2\right)},0.1e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}=\left(0.7e^{i2\pi \left(0.8\right)},0.1e^{i2\pi \left(0.1\right)},0.01e^{i2\pi \left(0.01\right)}\right)$ and $\mathbb{U}=3$. Then, $\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=0.06667;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.21333;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.18;$ $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-0.06667=0.93333$; $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.78667;\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.82$. Thus, $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne 1\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)+\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.72$. Using the value of weight vectors 0.4, 0.4, and 0.2, we have $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1.7533; \mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.60667$; ${\displaystyle \sum }_{ℐ=1}^{\ell }{\Phi }_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)=2.710667.$ Thus, $\stackrel{=}{{\mathcal{K}}_1}=\frac{{\Phi }_1\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^3{\Phi }_{=}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}=0.401377;$ $\stackrel{=}{{\mathcal{K}}_2}=0.406296;\stackrel{=}{{\mathcal{K}}_3}=0.192327.$ Then, we get $CPFAAWP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=\left(0.281354e^{i2\pi \left(0.296406\right)},0.493241e^{i2\pi \left(0.464818\right)},0.309996e^{i2\pi \left(0.284854\right)}\right).$

Proposition 2 (Idempotency).

If $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\stackrel{=}{\mathbb{G}}$, then

$$CPFAAWP-A\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\stackrel{=}{\mathbb{G}}$$

(17)

Proof. 

It is similar as the proof of Proposition 1.  □

Similarly, the proposed CPFAAWP-A operator cannot satisfy the property of monotonicity. We next propose the CPFAAP-G operator.

Definition 7.

The computational form of the CPFAAP-G operator is defined as follows:

$$CPFAAP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)={\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{K}}_1}}\otimes {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\stackrel{=}{{\mathcal{K}}_2}}\otimes \dots \otimes {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}}^{\stackrel{=}{{\mathcal{K}}_{\ell }}}={\otimes }_{ℐ=1}^{\ell }\left({\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}^{\stackrel{=}{{\mathcal{K}}_{ℐ}}}\right)$$

(18)

$$\stackrel{=}{{\mathcal{K}}_{ℐ}}=PA\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\frac{\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}{{{\displaystyle \sum }}_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}$$

(19)

Theorem 4.

In the consideration of Equations (18) and (19), we prove that the finalized result of the above theory is again in the form of CPFV with

$$\begin{array}{l}CPFAAP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\\ =\left(\begin{array}{c}\left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(1-\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

(20)

Proof. 

The proof is similar as Theorem 2 by using the mathematical induction.  □

Example 3.

To clarify Theorem 4 with the help of practical information, we decide to use some practical information in the form of CPFVs such as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\left(0.4e^{i2\pi \left(0.3\right)},0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\left(0.5e^{i2\pi \left(0.4\right)},0.3e^{i2\pi \left(0.2\right)},0.1e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}=\left(0.7e^{i2\pi \left(0.8\right)},0.1e^{i2\pi \left(0.1\right)},0.01e^{i2\pi \left(0.01\right)}\right)$ and $\mathbb{U}=3$. Then, $d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=0.06667;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.21333;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.18;$ $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-0.06667=0.93333;$ $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.78667;\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.82$. Thus, $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne 1\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)+\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.72$; $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1.7533; \mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.60667;$ ${\displaystyle \sum }_{ℐ=1}^{\ell }\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)=8.08$. We have $\stackrel{=}{{\mathcal{K}}_1}=\frac{\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^3\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}=\frac{1+1.72}{8.08}=0.33663$, and $\stackrel{=}{{\mathcal{K}}_2}=0.34076;\stackrel{=}{{\mathcal{K}}_3}=0.32261.$ Then, we get $CPFAAP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=\left(0.72869e^{i2\pi \left(0.66238\right)},0.11046e^{i2\pi \left(0.08287\right)},0.06735e^{i2\pi \left(0.03939\right)}\right).$

Proposition 3 (Idempotency).

If $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\stackrel{=}{\mathbb{G}}$, then

$$CPFAAP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\stackrel{=}{\mathbb{G}}$$

(21)

Proof. 

It is similar as the proof of Proposition 1.  □

Similarly, the proposed CPFAAP-G operator cannot satisfy the property of monotonicity. We next propose the CPFAAWP-G operator.

Definition 8.

The computational form of the CPFAAWP-G operator is illustrated as:

$$C-IFAAWPG\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)={\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}}^{\stackrel{=}{{\mathcal{K}}_1}}\otimes {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}}^{\stackrel{=}{{\mathcal{K}}_2}}\otimes \dots \otimes {\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}}^{\stackrel{=}{{\mathcal{K}}_{\ell }}}={\otimes }_{ℐ=1}^{\ell }\left({\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}^{\stackrel{=}{{\mathcal{K}}_{ℐ}}}\right)$$

(22)

$$\stackrel{=}{{\mathcal{K}}_{ℐ}}=WPA\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\frac{{\mathsf{\Phi }}_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}}{{{\displaystyle \sum }}_{ℐ=1}^{\ell }{\mathsf{\Phi }}_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}$$

(23)

Theorem 5.

In the consideration of Equations (22) and (23), we prove that the finalized result of the above theory is again in the form of CPFV, such that

$$\begin{array}{l}CPFAAWP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)\\ =\left(\begin{array}{c}\left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(1-e^{-{\left({{\displaystyle \sum }}_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)},\\ \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)e^{i2\pi \left(1-e^{-{\left({\displaystyle \sum }_{ℐ=1}^{\ell }\stackrel{=}{{\mathcal{K}}_{ℐ}}{\left(-\mathbb{I}\mathbb{O}\mathbb{G}(1-\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}})\right)}^{\mathbb{U}}\right)}^{\frac{1}{\mathbb{U}}}}\right)}\end{array}\right)\end{array}$$

(24)

Proof. 

The proof is similar as Theorem 2 by using the mathematical induction.  □

Example 4.

To clarify Theorem 5 with the help of practical information, we use some practical information in the form of CPFVs such as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}=\left(0.4e^{i2\pi \left(0.3\right)},0.2e^{i2\pi \left(0.2\right)},0.2e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}=\left(0.5e^{i2\pi \left(0.4\right)},0.3e^{i2\pi \left(0.2\right)},0.1e^{i2\pi \left(0.1\right)}\right),\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}=\left(0.7e^{i2\pi \left(0.8\right)},0.1e^{i2\pi \left(0.1\right)},0.01e^{i2\pi \left(0.01\right)}\right)$ and $\mathbb{U}=3$. Then, $\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=0.06667;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.21333;d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.18;$ $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-d\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1-0.06667=0.93333$; $\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.78667;\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=0.82$. Thus, $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)={\displaystyle \sum }_{\begin{array}{c}s=1,\\ s\ne 1\end{array}}^{\ell }\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_s}}\right)=\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)+\mathbb{S}up\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.72$. Using the value of weight vectors 0.4, 0.4, and 0.2, we have that $\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}\right)=1.7533; \mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=1.60667$; ${\displaystyle \sum }_{ℐ=1}^{\ell }{\Phi }_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)=2.710667$. We have $\stackrel{=}{{\mathcal{K}}_1}=\frac{{\Phi }_1\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}\right)\right)}{{{\displaystyle \sum }}_{ℐ=1}^3{\Phi }_{ℐ}\left(1+\mathbb{V}\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}\right)\right)}=0.401377$ and $\stackrel{=}{{\mathcal{K}}_2}=0.406296;\stackrel{=}{{\mathcal{K}}_3}=0.192327.$ Then, we get $CPFAAWP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}\right)=\left(0.723231e^{i2\pi \left(0.657451\right)},0.111293e^{i2\pi \left(0.085498\right)},0.065739e^{i2\pi \left(0.040331\right)}\right).$

Proposition 4 (Idempotency):

If $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\stackrel{=}{\mathbb{G}}$, then

$$CPFAAWP-G\left(\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}}\right)=\stackrel{=}{\mathbb{G}}$$

(25)

Proof. 

It is similar as the proof of Proposition 1.  □

Similarly, the proposed CPFAAWP-G operator cannot satisfy the property of monotonicity. Totally, the proposed CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators have the property of idempotency, but there is no property of monotonicity. In general, the properties of monotonicity and boundedness may be neglected for most AOs under the consideration of Aczel–Alsina operational laws. Overall, the advantages of the derived works can be listed as follows:

• By setting the value of $\stackrel{=}{{℃}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=0$ in the invented theory, the invented theory will be reduced for CIFSs.
• By setting the value of $\stackrel{=}{{℃}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=0$ in the invented theory, the invented theory will be reduced for CFSs.
• By setting the value of $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=0$ in the invented theory, the invented theory will be reduced for PFSs.
• By setting the value of $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=0$ in the invented theory, the invented theory will be reduced for IFSs.
• By setting the value of $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=0$ in the invented theory, the invented theory will be reduced for FSs.
• Furthermore, the Aczel–Alsina aggregation operators for FSs, IFSs, PFSs, CFSs, CIFSs, and CPFSs are the special cases of the proposed theory, when we remove the power aggregation operators.
• The power aggregation operators for FSs, IFSs, PFSs, CFSs, CIFSs, and CPFSs are the special cases of the proposed theory, when we remove the Aczel–Alsina information.
• The simple averaging and geometric aggregation operators for FSs, IFSs, PFSs, CFSs, CIFSs, and CPFSs are the special cases of the proposed theory, when we use the algebraic information instead of Aczel–Alsina and power aggregation operators.

4. Strategic CPF MADM Methods

In this section, we organize a MADM procedure based on our proposed AOs, such as CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators, to state the effectiveness and efficacy of the described theory. To resolve the above problem, we aim to compute a decision matrix whose values are in the form of CPF information. For this, we suggest the finite values of alternatives such as $\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}=\left\{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_m}}\right\}$ concerning finite values of attributes with $\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}^{\prime }}=\left\{\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}^{\prime }},\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}^{\prime }},\dots ,\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\ell }}^{\prime }}\right\}$ under the weighted information such as $\stackrel{=}{\mathcal{K}}={\left({\stackrel{=}{\mathcal{K}}}_1,{\stackrel{=}{\mathcal{K}}}_2,\dots ,{\stackrel{=}{\mathcal{K}}}_{\ell }\right)}^T,{\displaystyle \sum }_{ℐ=1}^{\ell }{\stackrel{=}{\mathcal{K}}}_{ℐ}=1$, where the order of the attributes and weight vectors will be the same. Furthermore, here, we describe the values of the decision matrix that we talk about in the triplet, such as truth, abstinence, and falsity, which are stated by: $\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)},\stackrel{=}{{℃}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$ and $\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}$ with $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$ and $0\le \stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\le 1$. Furthermore, the complicated structure $\stackrel{=}{{ℝ}_{\stackrel{=}{{\mathbb{G}}_{\mathfrak{w}}}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)=\stackrel{=}{{ℝ}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)e^{i2\pi \left(\stackrel{=}{{ℝ}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)}=\left(1-\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)\right)e^{i2\pi \left(1-\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{℃}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)+\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I}}}\left(\widetilde{{\mathfrak{u}}_{\mathfrak{g}}}\right)\right)\right)}$ is used as a neutral grade and $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}}=\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐ}}}}\right)},\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)},\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐ}}}}\right)}\right),ℐ=1,2,\dots ,\ell$ states the CPF values (CPFVs). Finally, with the help of the below procedure, we aim to evaluate the above problems, as shown in

Table 1. The procedure of decision-making application.

Organized Algorithm for Evaluating the Decision-Making Problem

Notice that here we construct a decision-making procedure based on the proposed theory, whose main steps are stated below: Step 1: During the construction of every decision matrix, experts have faced two types of information like cost and benefit. In this problem, if we have cost type of data, then normalize it, such as Equation (26) But, if we have a benefit type of data, then proceed with the procedure of decision-making application. Step 2: To use the idea of CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators, we concentrate to aggregate the information matrix into a single set theory. Step 3: To use the idea of score and accuracy values, we focus to more evaluate the single aggregated values into real-valued information. Step 4: To use the score values, we examine the ranking values for evaluating the finest preferences from the collection of finite preferences.

.

We next demonstrate a real-life problem and try to solve it with the help of our derived theory.

Example 5.

Here, we expose a company that wants to buy different types of cars out of five different and valuable models with $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{ℐ}}},ℐ=1,2,3,4,5$, which are stated or used as alternatives, such as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}$: Motor cars, $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}$: Bus cars, $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$: Ambulances cars, $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}}$: VIP cars, $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_5}}$: Small cars (other cars). To choose the best one, we take our decision under the presence of the following criteria with $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}^{\prime }}$: elasticity, $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}^{\prime }}$: consistency, $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}^{\prime }}$: rates, and $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}^{\prime }}$: protection. Therefore, with the help of the investigated operators, we aim to decide which one is the best and which one is not under the consideration of the following weight vectors such as $0.3,0.3,0.1,0.3$ for four attributes in each alternative. Finally, with the help of the below procedure, we aim to evaluate the above problem according to Table 1.

Step 1: During the construction of every decision matrix (see

Table 2. Complex picture fuzzy information matrix.

Alternatives/Attributes	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}^{\prime }}$	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}^{\prime }}$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	$\left(\left(0.5,0.4\right),\left(0.3,0.2\right),\left(0.1,0.1\right)\right)$	$\left(\left(0.51,0.41\right),\left(0.31,0.21\right),\left(0.11,0.11\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	$\left(\left(0.4,0.3\right),\left(0.4,0.2\right),\left(0.1,0.3\right)\right)$	$\left(\left(0.41,0.31\right),\left(0.41,0.21\right),\left(0.11,0.31\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	$\left(\left(0.6,0.7\right),\left(0.2,0.1\right),\left(0.1,0.1\right)\right)$	$\left(\left(0.61,0.71\right),\left(0.21,0.11\right),\left(0.11,0.11\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	$\left(\left(0.3,0.2\right),\left(0.2,0.1\right),\left(0.2,0.4\right)\right)$	$\left(\left(0.31,0.21\right),\left(0.21,0.11\right),\left(0.21,0.41\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$	$\left(\left(0.1,0.2\right),\left(0.1,0.2\right),\left(0.1,0.1\right)\right)$	$\left(\left(0.11,0.21\right),\left(0.11,0.21\right),\left(0.11,0.11\right)\right)$
Alternatives/Attributes	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}^{\prime }}$	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}^{\prime }}$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	$\left(\left(0.52,0.42\right),\left(0.32,0.22\right),\left(0.12,0.12\right)\right)$	$\left(\left(0.53,0.43\right),\left(0.33,0.23\right),\left(0.13,0.13\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	$\left(\left(0.42,0.32\right),\left(0.42,0.22\right),\left(0.12,0.32\right)\right)$	$\left(\left(0.43,0.33\right),\left(0.43,0.23\right),\left(0.13,0.33\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	$\left(\left(0.62,0.72\right),\left(0.22,0.12\right),\left(0.12,0.12\right)\right)$	$\left(\left(0.63,0.73\right),\left(0.23,0.13\right),\left(0.13,0.13\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	$\left(\left(0.32,0.22\right),\left(0.22,0.12\right),\left(0.22,0.42\right)\right)$	$\left(\left(0.33,0.23\right),\left(0.23,0.13\right),\left(0.23,0.43\right)\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$	$\left(\left(0.12,0.22\right),\left(0.12,0.22\right),\left(0.12,0.12\right)\right)$	$\left(\left(0.13,0.23\right),\left(0.13,0.23\right),\left(0.13,0.13\right)\right)$

), experts have faced two types of information such as cost and benefit. In this problem, if we have cost-type data, then normalize it, such as

$$Z^{DM}=\{\begin{array}{cc}\left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐk}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐk}}}}\right)},\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)},\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐk}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐk}}}}\right)}\right)& Benefit\\ \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{R_{ℐk}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{N}}_{\stackrel{=}{I_{ℐk}}}}\right)},\stackrel{=}{{℃}_{\stackrel{=}{R_{ℐ}}}}{e}^{i2\pi \left(\stackrel{=}{{℃}_{\stackrel{=}{I_{ℐ}}}}\right)},\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{R_{ℐk}}}}{e}^{i2\pi \left(\stackrel{=}{{\mathbb{M}}_{\stackrel{=}{I_{ℐk}}}}\right)}\right)& Cost\end{array}$$

(26)

However, if we have benefit-type data, then we should proceed with the procedure of decision-making application. But unfortunately, the data in Table 2 are in the form of benefits, so they did not require evaluation.

Step 2: To use the idea of CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators, we concentrate on aggregating the information matrix into a single set theory (see

Table 3. Covered data of power weight vectors.

$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\stackrel{=}{{\mathcal{K}}_{\mathbf{1}}}=\mathbf{0.2494}$	$\stackrel{=}{{\mathcal{K}}_1}=0.2494$	$\stackrel{=}{{\mathcal{K}}_1}=0.2494$	$\stackrel{=}{{\mathcal{K}}_1}=0.2494$	$\stackrel{=}{{\mathcal{K}}_1}=0.2494$
$\stackrel{=}{{\mathcal{K}}_{\mathbf{2}}}=\mathbf{0.2506}$	$\stackrel{=}{{\mathcal{K}}_2}=0.2506$	$\stackrel{=}{{\mathcal{K}}_2}=0.2506$	$\stackrel{=}{{\mathcal{K}}_2}=0.2506$	$\stackrel{=}{{\mathcal{K}}_2}=0.2506$
$\stackrel{=}{{\mathcal{K}}_{\mathbf{3}}}=\mathbf{0.2506}$	$\stackrel{=}{{\mathcal{K}}_3}=0.2506$	$\stackrel{=}{{\mathcal{K}}_3}=0.2506$	$\stackrel{=}{{\mathcal{K}}_3}=0.2506$	$\stackrel{=}{{\mathcal{K}}_3}=0.2506$
$\stackrel{=}{{\mathcal{K}}_{\mathbf{4}}}=\mathbf{0.2494}$	$\stackrel{=}{{\mathcal{K}}_4}=0.2494$	$\stackrel{=}{{\mathcal{K}}_4}=0.2494$	$\stackrel{=}{{\mathcal{K}}_4}=0.2494$	$\stackrel{=}{{\mathcal{K}}_4}=0.2494$

and

Table 4. Covered aggregated data.

Alternatives	CPFAAP-A Operator	CPFAAP-G Operator
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	$\left(\begin{array}{c}\left(0.2699,0.2079\right),\left(0.6052,0.5125\right),\\ \left(0.3897,0.3897\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.7494,0.6823\right),\left(0.1517,0.1\right),\\ \left(0.052,0.052\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	$\left(\begin{array}{c}\left(0.2079,0.1517\right),\left(0.6823,0.5125\right),\\ \left(0.3897,0.6052\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.6823,0.6052\right),\left(0.2079,0.1\right),\\ \left(0.052,0.1517\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	$\left(\begin{array}{c}\left(0.3396,0.4206\right),\left(0.5125,0.3897\right),\\ \left(0.3897,0.3897\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.8095,0.8642\right),\left(0.1,0.052\right),\\ \left(0.052,0.052\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	$\left(\begin{array}{c}\left(0.1517,0.1\right),\left(0.5125,0.3897\right),\\ \left(0.5125,0.6823\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.6052,0.5125\right),\left(0.1,0.052\right),\\ \left(0.1,0.2079\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$	$\left(\begin{array}{c}\left(0.052,0.1\right),\left(0.3897,0.5125\right),\\ \left(0.3897,0.3897\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.3897,0.5125\right),\left(0.052,0.1\right),\\ \left(0.052,0.052\right)\end{array}\right)$
Alternatives	CPFAAWP-A Operator	CPFAAWP-G Operator
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	$\left(\begin{array}{c}\left(0.2692,0.2073\right),\left(0.6043,0.5113\right),\\ \left(0.3881,0.3881\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.7488,0.6815\right),\left(0.1512,0.0995\right),\\ \left(0.0516,0.0516\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	$\left(\begin{array}{c}\left(0.2073,0.1512\right),\left(0.6815,0.5113\right),\\ \left(0.3881,0.6043\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.6815,0.6043\right),\left(0.2073,0.0995\right),\\ \left(0.0516,0.1512\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	$\left(\begin{array}{c}\left(0.3389,0.4198\right),\left(0.5113,0.3881\right),\\ \left(0.3881,0.3881\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.8089,0.8637\right),\left(0.0995,0.0516\right),\\ \left(0.0516,0.0516\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	$\left(\begin{array}{c}\left(0.1512,0.0995\right),\left(0.5113,0.3881\right),\\ \left(0.5113,0.6815\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.6043,0.5113\right),\left(0.0995,0.0516\right),\\ \left(0.0995,0.2073\right)\end{array}\right)$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$	$\left(\begin{array}{c}\left(0.0516,0.0995\right),\left(0.3881,0.5113\right),\\ \left(0.3881,0.3881\right)\end{array}\right)$	$\left(\begin{array}{c}\left(0.3881,0.5113\right),\left(0.0516,0.0995\right),\\ \left(0.0516,0.0516\right)\end{array}\right)$

).

Using the power-weighted vectors in Table 3, we derive the data in Table 4.

Step 3: To use the idea of score and accuracy values, we focus on evaluating the single aggregated values into real-valued information, as shown in

Table 5. Covered score information.

Alternatives	CPFAAP-A Operator	CPFAAP-G Operator	CPFAAWP-A Operator	CPFAAWP-G Operator
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	$-0.4731$	$0.3587$	$-0.4717$	$0.3588$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	$-0.61$	$0.2586$	$-0.6089$	$0.2587$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	$-0.3071$	$0.4726$	$-0.3056$	$0.4728$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	$-0.6151$	$0.2193$	$-0.6139$	$0.2192$
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$	$-0.5099$	$0.2154$	$-0.5082$	$0.2151$

.

Step 4: To use the score values, we examine the ranking values for evaluating the finest preferences from the collection of finite preferences, as shown in

Table 6. Covered ranking values.

CPFAAP-A Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_5}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}}$
CPFAAP-G Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
CPFAAWP-A Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAWP-G Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$

.

Here, we get the best optimal as the $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$. Furthermore, to check the supremacy and worth of the derived operators, we remove the phase information from Table 2, and then we find the score values of the aggregated values, as shown in

Table 7. Covered score values (without phase terms).

Alternatives	CPFAAP-A Operator	CPFAAP-G Operator	CPFAAWP-A Operator	CPFAAWP-G Operator
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}$	−0.3625	0.2729	−0.3616	0.273
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}$	−0.4321	0.2112	−0.4311	0.2113
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}$	−0.2813	0.3288	−0.2803	0.3289
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$	−0.4366	0.2026	−0.4358	0.2026
$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$	−0.3637	0.1429	−0.3623	0.1425

.

To use the score values, we examine the ranking values for evaluating the finest preferences from the collection of finite preferences, as shown in

Table 8. Covered ranking values.

CPFAAP-A Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAP-G Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
CPFAAWP-A Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAWP-G Operator	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$

.

The finest optimal is $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$. Further, we also discuss the stability or influence of the proposed work with the help of parameters by using the CPF information matrix in Table 2 with phase information and without phase information, as shown in

Table 9. Evaluation of parameters for different values (with phase data).

Paramete	Operato	Score Value	Ranking Value
$\mathbb{U}=\mathbf{1}$	CPFAAP-A	−0.4735, −0.6104, −0.3077, −0.6155, −0.5105	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.359, 0.259, 0.473, 0.2196, 0.2159	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.4722, −0.6093, −0.3062, −0.6143, −0.5088	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.3592,0.2591,0.4733,0.2196,0.2157	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{3}$	CPFAAP-A	−0.4727, −0.6097, −0.3066, −0.6147, −0.5093	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.3583, 0.2583, 0.4722, 0.2189, 0.2149	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.4712, −0.6085, −0.305, −0.6134, −0.5075	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.3584, 0.2584, 0.4723, 0.2188, 0.2145	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{5}$	CPFAAP-A	−0.4718, −0.609, −0.3056, −0.6139, −0.5082	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.3577, 0.2577, 0.4714, 0.2182, 0.2139	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.4703, −0.6077, −0.3038, −0.6125, −0.5062	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.3576, 0.2577, 0.4714, 0.218, 0.2133	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{7}$	CPFAAP-A	−0.4709, −0.6082, −0.3045, −0.6131, −0.507	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.357, 0.2571, 0.4707, 0.2176, 0.213	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.4693, −0.6069, −0.3027, −0.6117, −0.5049	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.3568, 0.257, 0.4705, 0.2172, 0.2122	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{10}$	CPFAAP-A	−0.4697, −0.6072, −0.303, −0.612, −0.5054	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.3561, 0.2563, 0.4696, 0.2166, 0.2116	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.4679, −0.6057, −0.301, −0.6104, −0.5031	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.3557, 0.2559, 0.4693, 0.2161, 0.2107	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{\mathbf{4}}}$

.

The finest optimal is $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$ according to the theory of all derived information. Similarly, we find the influence for values shown in Table 2 without phase information, as seen in

Table 10. Evaluation of parameters for different values (without phase data).

Paramete	Operato	Score Value	Ranking Value
$\mathbb{U}=\mathbf{1}$	CPFAAP-A	−0.3628, −0.4324, −0.2816, −0.4369, −0.3642	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.2731, 0.2115, 0.329, 0.2028, 0.1433	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.3619, −0.4315, −0.2807, −0.4361, −0.3629	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.2733, 0.2116, 0.3292, 0.2029, 0.143	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{3}$	CPFAAP-A	−0.3622, −0.4318, −0.2809, −0.4364, −0.3632	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.2726, 0.211, 0.3285, 0.2024, 0.1424	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.3612, −0.4308, −0.2799, −0.4355, −0.3617	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.2727, 0.211, 0.3286, 0.2024, 0.142	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{5}$	CPFAAP-A	−0.3616, −0.4311, −0.2803, −0.4358, −0.3622	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.2721, 0.2105, 0.328, 0.2019, 0.1416	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.3605, −0.4301, −0.2791, −0.4349, −0.3606	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.2721, 0.2105, 0.328, 0.2018, 0.1409	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{7}$	CPFAAP-A	−0.361, −0.4305, −0.2796, −0.4353, −0.3613	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.2716, 0.21, 0.3274, 0.2014, 0.1407	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.3598, −0.4294, −0.2783, −0.4343, −0.3595	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.2716, 0.2099, 0.3274, 0.2013, 0.14	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
$\mathbb{U}=\mathbf{10}$	CPFAAP-A	−0.36, −0.4297, −0.2786, −0.4346, −0.3599	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAP-G	0.271, 0.2093, 0.3267, 0.2008, 0.1395	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
	CPFAAWP-A	−0.3588, −0.4285, −0.2772, −0.4334, −0.358	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
	CPFAAWP-G	0.2708, 0.2091, 0.3265, 0.2005, 0.1386	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$

.

The finest optimal is $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$ according to the theory of all derived information. Additionally, we compare the derived theory with various existing operators to show the worth and effectiveness of the invented information in the next section.

5. Comparative Analysis

In this section, under the consideration of the numerical or practical example that is examined to illustrate in the above section under the consideration of the derived operators, we compare their performance with various operators to show the supremacy and validity of the derived approaches. For comparing the derived theory, we select the following important operators for which Aczel–Alsina AOs for IFSs were invented by Senapati et al. [14]; Aczel–Alsina geometric operators for IFS were derived by Senapati et al. [15]; Aczel–Alsina prioritized AOs for IFS was discovered by Sarfraz et al. [16]; Aczel–Alsina averaging operators for PFS was derived by Senapati [19]; Aczel–Alsina geometric operators for PFS was derived by Naeem et al. [20]; Aczel–Alsina averaging/geometric operators for CIFS was invented by Mahmood et al. [21]; power AOs for IFS was derived by Xu [22]; Jiang et al. [23] demonstrated the power AOs based on entropy measures for IFSs; Rani and Garg [24] examined the power AOs for CIFS; and Liu et al. [25] proposed the power AOs for CPFSs. Then, in

Table 11. Comparison information for data is in Table 2 (with phase terms).

Method	Score Value	Ranking Value
Senapati et al. [14]	$XXXXX$	$XXXXX$
Senapati et al. [15]	$XXXXX$	$XXXXX$
Sarfraz et al. [16]	$XXXXX$	$XXXXX$
Senapati [19]	$XXXXX$	$XXXXX$
Naeem et al. [20]	$XXXXX$	$XXXXX$
Mahmood et al. [21]	$XXXXX$	$XXXXX$
Xu [22]	$XXXXX$	$XXXXX$
Jiang et al. [23]	$XXXXX$	$XXXXX$
Rani and Garg [24]	$XXXXX$	$XXXXX$
Liu et al. [25]	0.0573, −0.1095, 0.2574, −0.1429, −0.076	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAP-A	−0.4731, −0.61, −0.3071, −0.6151, −0.5099	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAP-G	0.3587, 0.2586, 0.4726, 0.2193, 0.2154	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
CPFAAWP-A	−0.4717, −0.6089, −0.3056, −0.6139, −0.5082	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAWP-G	0.3588, 0.2587, 0.4728, 0.2192, 0.2151	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$

, we illustrate the comparison information for the CPF information matrix in Table 2 (with phase terms).

Again, we get the best optimal as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$ according to the theory of Liu et al. [25] and our derived operators (four main operators). Furthermore, by excluding the phase information from Table 2, the comparison values are illustrated in

Table 12. Comparison information for data in Table 2 (without phase terms).

Methods	Score Values	Ranking Values
Senapati et al. [14]	$XXXXX$	$XXXXX$
Senapati et al. [15]	$XXXXX$	$XXXXX$
Sarfraz et al. [16]	$XXXXX$	$XXXXX$
Senapati [19]	−0.3588,−0.4285,−0.2772,−0.4334,−0.358	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}}$
Naeem et al. [20]	−0.3588,−0.4285,−0.2772,−0.4334,−0.358	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_1}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}}$
Mahmood et al. [21]	$XXXXX$	$XXXXX$
Xu [22]	$XXXXX$	$XXXXX$
Jiang et al. [23]	$XXXXX$	$XXXXX$
Rani and Garg [24]	$XXXXX$	$XXXXX$
Liu et al. [25]	0.0429, −0.0571, 0.143, −0.0572, −0.0569	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_5}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}}$
CPFAAP-A	−0.3625, −0.4321, −0.2813, −0.4366, −0.3637	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_5}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_2}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_4}}$
CPFAAP-G	0.2729, 0.2112, 0.3288, 0.2026, 0.1429	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$
CPFAAWP-A	−0.3616, −0.4311, −0.2803, −0.4358, −0.3623	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}$
CPFAAWP-G	0.273, 0.2113, 0.3289, 0.2026, 0.1425	$\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{3}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{1}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{2}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{4}}}}>\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_{\mathbf{5}}}}$

.

Again, we get the best optimal as $\stackrel{=}{{\mathbb{G}}_{{\mathfrak{w}}_3}}$ according to the theory of Senapati [19], Naeem et al. [20], Liu et al. [25], and our derived theory (four main operators). But the theory proposed by Senapati et al. [14], Senapati et al. [15], Sarfraz et al. [16], Mahmood et al. [21], Xu [22], Jiang et al. [23], and Rani and Garg [24] has failed. Totally, our proposed theory (four main operators) is actually more feasible for depicting awkward and unreliable information in real-life problems.

6. Conclusions

The theory of complex picture fuzzy (CPF) information is very valuable and dominant, because it is the extended version of fuzzy sets in which the extension covers the three major grades of membership, abstinence, and falsity with a prominent characteristic such that the sum of the triplet will be contained in the unit interval. On the other hand, the idea of averaging, geometric, Aczel–Alsina, and power aggregation operators is very helpful and beneficial for depicting unreliable and vague information in real-life problems. Thus, we proposed these novel works inspired by the above ideas in this paper. The major influences of the derived theory and methods are stated as follows:

• We exposed the theory of Aczel–Alsina operational laws for the invented theory such as CPF information.
• We derived the CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators and discover their properties, and then illustrated the procedure of the decision-making technique in the presence of the CPF values and derived an algorithm to evaluate the MADM problems.
• We give an example to illustrate the decision-making procedure based on the derived CPFAAP-A, CPFAAWP-A, CPFAAP-G, and CPFAAWP-G operators.
• We compared the derived theory with the existing operators, which showed the supremacy and validity of our proposed approaches.

In the future, we will concentrate on developing some new ideas based on fuzzy sets, lattice-ordered fuzzy sets, and their generalizations. We should try to apply these in the areas of pattern recognition, green supplier selection, clustering, and multi-attribute decision-making problems to improve the quality of the proposed theory.