An approach to decision making with interval-valued complex Pythagorean fuzzy model for evaluating personal risk of mental patients

Abstract

It is prominent important for managers to assess the personal risk of mental patients. The evaluation process refers to numerous indexes, and the evaluation values are general portrayed by uncertainty information. In order to conveniently model the complicated uncertainty information in realistic decision making, interval-valued complex Pythagorean fuzzy set is proposed. Firstly, with the aid of Einstein t-norm and t-conorm, four fundamental operations for interval-valued complex Pythagorean fuzzy number (IVCPFN) are constructed along with some operational properties. Subsequently, according to these proposed operations, the weighted average and weighted geometric forms of aggregation operators are initiated for fusing IVCPFNs, and their anticipated properties are also examined. In addition, a multiple attribute decision making issue is examined under the framework of IVCPFNs when employing the novel suggested operators. Ultimately, an example regarding the assessment on personal risk of mental patients is provided to reveal the practicability of the designed approach, and the attractiveness of our results is further found through comparing with other extant approaches.The main novelty of the coined approach is that it not only can preserve the original assessment information adequately by utilizing the IVCPFNs, but also can aggregate IVCPFNs effectively.

Keywords

Multiple attribute decision making Einstein operation interval-valued complex Pythagorean fuzzy number aggregation operators personal risk

1 Introduction

As a kind of social defense procedure, compulsory medical procedure plays a very significant role in safeguarding social public security and human rights. However, the traditional medical care, security compulsory hospitalization and rescue compulsory medical care possess certain limitations, and even because of the expansion of the application, leading to the case of “called mental illness”. For this, the personal danger of mental patients, that is, the possibility of continuing to harm society is regarded as one of the applicable conditions of compulsory medical treatment [1]. Since personal danger is a very abstract notion with useful connotation and extension, which is difficult to grasp in judicial practice. Therefore, the construction of a comprehensive evaluation framework of mental patients’ personal risk can provide a feasible path for the scientific quantification of mental patients’ personal risk, and offer a reasonable basis for the judicial application of compulsory medical procedures [1, 2]. The assessment on personal risk of mental patients includes diverse aspects, such as the physical condition of the respondent, the extent of the disease, the performance before, during and after the crime, and so on. In this respect, it is not easy to evaluate the mental patients with the least harmfulness that is dominant in all perspectives. The compound solution of multiple attribute decision making (MADM) is an effective method to cope with this issue.

Multiple attribute decision making (MADM) is a research hotspot in decision science, which refers to the decision makers provide their assessment values for the given alternatives under multiple attributes, and then determine the finest alternative. Considering the complexity of the realistic decision making circumstances and fuzziness of human cognition, it is difficult to employ the exact number that depicts the assessment values. To handle this issue, at present, intuitionistic fuzzy set (IFS) [3] and its extension forms are widely employed. Pythagorean fuzzy set (PFS) [4] and interval-valued PFS (IVPFS) [5] are available extension of intuitionistic fuzzy set (IFS), by relaxing the constraint condition that the sum of membership degree (MD) and nonmembership degree (NMD) is exceeded to the upper limit of interval [0,1], but square sum of MD and NMD is bounded to interval [0,1]. Therefore, PFS have a stronger capacity than IFS in signifying the fuzzy indeterminacy in MADM issues [6 –9]. Ever since PFS was initiated by Yager [4], which has obtained great concentrations both in the theoretical basis and in applications, especially in the aspects of decision making [10 –15]. Liang et al. [10] combined with TODIM and VIKOR approaches to Pythagorean fuzzy numbers (PFNs), and researched the assessment problem of internet banking website quality of Ghanaian banking industry. Peng and Yang [11] created novel subtraction and division operations between PFNs and enriched some existing operations [3], then they established the Pythagorean fuzzy superiority and inferiority ranking technique to address MAGDM issue concerning the selection of internet stocks. Within Pythagorean fuzzy setting, on the basis of cosine and cotangent trigonometric functions, Verma and Merigó [12] put forward several trigonometric similarity measures and employed them to manage realistic decision issues. In order to fuse the Pythagorean fuzzy evaluation values during the process in decision making, Ma and Xu [13] constructed Pythagorean fuzzy weighted averaging and geometric operators, on this basis, they pointed out that the information of MD and NMD should be treated equally in some situations and exploited neutrality operations for PFNs. Considering the possible association among attributes and the interactional influence between ND and NMD of the assembled PFNs, Wang and Li [14] set up interactional power Bonferroni mean AOs under the framework of PFNs. Based on Archimedean norm, some new Pythagorean fuzzy interactive AOs were set up by Wang and Garg [15] Enlightened by the work [7], Zhang [5] extended PFN and created mathematical expression structure of interval PFN (IVPFN) including basic operations, operations rules, score and accuracy function, order relations and distance measures for IVPFNs. Additionally, he established a Pythagorean fuzzy QUALIFLEX model to address hierarchical decision making issues within IVPFNs, and also deal with heterogeneous information. As a useful extended form of PFN and interval intuitionistic fuzzy number, IVPFNs have been widely applied in complex decision making environments. In the light of addition and multiplication between real numbers, Peng and Yang [16] introduced the interval-valued Pythagorean fuzzy weighted average (IVPFWA) operator and weighted geometric (IVPFWG) operator, discussed their basic features, and by combing them and ELECTRE method to address uncertainty decision analysis. Liang et al. [17] introduced interval-valued Pythagorean fuzzy extended Bonferroni mean (IVPFEBM) and weighted IVPFEBM (WIVPFEBM) operators, their fundamental features are that they can capture the partition structure of relationship among the attributes. Another kind of interval-valued Pythagorean fuzzy weighted geometric (IPFWG) operator was explored by Garg [18] based on the operational laws of IVPFNs [5]. In the aggregation process, in order to consider all the elements in the interval [0, 1], Wang and Li [19] defined continuous interval-valued Pythagorean fuzzy ordered weighted quadratic averaging (C-IVPFOWQA) operator and weighted C-IVPFOWQA (WC-IVPFOWQA) operator and used them in MADM. By integrating IVPFSs with game theory, Han et al. [20] constructed a novel decision model, which can manage group decision-making issues under the interactive and dynamic situations. Xian et al. [21] extended the IVPFS to linguistic decision environment and proposed the interval-valued Pythagorean uncertain linguistic (IVPUL) Euclidean distance and IVPUL operator, and applied them to settle the assessment problem on material supplier.

From the literature survey, we can perceive that IFSs, PFSs or IVPFSs are limited as to enough to deal with periodic fuzzy and inconsistent information in MADM. Under this situation, Ramot et al [22] constructed the conception of complex fuzzy set (CFS) in which domain of membership degree was expanded from [0, 1] to unit disc. Moreover, membership degree contains two parts named amplitude term and phase term, where the phase portrays the periodicity of information. Alkouri and Salleh [23] extended CFS to propose the notion of complex intuitionistic fuzzy set (CIFS) by adding nonmembership degree to CFS. Rani and Garg [24] established several power AOs in CIFS circumstances and their application in the MADM process. Base on the CFS and CIFS, Rani and Garg [25] defined the conception of complex interval valued intuitionistic fuzzy (CIVIF) set and proposed CIVIF weighted averaging (CIVIFWA) and CIVIF weighted geometric (CIVIFWG) AOs, a group decision making issue is provided to indicate the effectiveness of the proposed AOs. By relaxing the restrictions of CIFS, Ullah et al. [26] explored the idea of complex Pythagorean fuzzy set (CPFS) and developed several distance measures for CPFSs. In the complex Pythagorean fuzzy setting, some AOs are studied to fuse the CPFSs during the process of decision making, such as complex Pythagorean fuzzy prioritized AOs [27], complex Pythagorean Dombi fuzzy AOs [28], complex Pythagorean fuzzy Yager AOs [29]. Ma et al. [30] extended VIKOR to CPFS and created the complex Pythagorean fuzzy VIKOR technique and its application in realistic decision making issues. Akram et al. [31] explored two approaches called as complex Pythagorean fuzzy ELECTRE I approach and complex Pythagorean fuzzy TOPSIS approach respectively, and researched the performance of the two methods in solving MADM problem.

As the realistic decision making contexts are extremely complex and uncertain, as well as DMs’ cognitive structure is limited, sometimes DMs are not completely justifiable to quantify their MDs and NMDs by means of single valued. Hence, to make decision more freely and conveniently to provide their preferences, DMs can employ the forms of interval value to depict their MDs and NMDs [32]. For this aim, we propose interval-valued CPFS (IVCPFS) which is parallel to the complex interval-valued intuitionistic fuzzy set (CIVIFS) [25]. As an effective extended form of CIVIFS in the perspectives of constrained condition, IVCPFS also plays a prominent role in dealing with decision making issues. To our knowledge, there is little research on IVCPFS, especially in aspects of information fusion in MADM. Therefore, it is necessary to put forward some feasible and effective MADM methods under the framework of interval valued complex Pythagorean fuzzy information. Einstein operation was firstly defined by Klement et al. [33], the original Einstein operation is performed mainly between real numbers. In the recent era, it has been expanded to other fuzzy environments, such as IFSs [34], PFSs [35], Hesitant fuzzy sets [36], and so on. Consequently, it is a meaningful topic to extend Einstein operation to IVCPFSs.

From the above discussion, we can conclude that the IVCPFSs can more efficiently portrait imprecise and fuzzy information by combining the IVPFSs and CFSs, and Einstein operations are also meaningful alternatives for establishing AOs. Take these ideas into consideration, the motivation and goal of this work are displayed as bellow:

To propose some new operations for IVCPFSs by incorporating Einstein operation into IVCPFSs.

To explore weighted average and geometric AOs under the framework of IVCPFSs, and also check their valuable properties and special cases.

To build a new technique for coping with the MADM issues with the aid of suggested operators.

To reveal the efficiency and superiority of the methodology and solve an actual issue focusing on personal risk of mental patients, and compare the results with relevant methodologies [16–19 , 32].

For it, this manuscript is structured as below: Several conceptions about IVPFS and IVCPFS are displayed and proposed in Section 2. A novel operation of IVCPFS and its fundamental operational rules are exploited in Section 3. IVCPFEWA operator and IVCPFEWG operator as well as their useful properties are established in Section 4. A novel MADM technique based on suggested AOs is designed in Section 5. In Section 6, we employ a realistic example to check the validity of the founded approach and compare with the related ranking techniques. Ultimately, a conclusion is shown in Section 7.

2 Preliminaries

In this part, we recall some preconditions for interval-valued Pythagorean fuzzy set and Einstein operations, and develop interval-valued complex Pythagorean fuzzy set.

Definition 1. [5]. Suppose Z is a given set. An IVPFS B is an object owning the expression: $B = {〈 z, (μ_{B} (z), ν_{B} (z)) 〉 | z \in Z} .$ (1) where $μ_{B} (z) = [μ_{B}^{-} (z), μ_{B}^{+} (z)] \subseteq [0, 1]$ , $ν_{B} (z) = [ν_{B}^{-} (z), ν_{B}^{+} (z)] \subseteq [0, 1]$ respectively indicate MD and NMD of z to B, and satisfy ${(μ_{B}^{+} (z))}^{2} + {(ν_{B}^{+} (z))}^{2} ⩽ 1 .$ $π_{B} (z) = [π_{B}^{-} (z), π_{B}^{+} (z)]$ represents the degree of indeterminacy, where $π_{B}^{-} (z) = \sqrt{1 - {(μ_{B}^{+} (z))}^{2} - {(ν_{B}^{+} (z))}^{2}}$ and $π_{B}^{+} (z) = \sqrt{1 - {(μ_{B}^{-} (z))}^{2} - {(ν_{B}^{-} (z))}^{2}}$ .

Definition 2. [26]. Suppose Z is a given set. A CPFS C is an object owning the expression: $C = {〈 z, (μ_{C} (z), θ_{C} (z)) 〉 | z \in Z} .$ (2) where μ_C (z) = η_C (z) · e^{i·2πΨ_{η_C(z)}}, θ_C (z) = λ_C (z) · e^{i·2πΨ_{λ_C(z)}} and $i = \sqrt{- 1}$ with the conditions η_C (z), λ_C (z), Ψ_{η_C(z)}, Ψ_{λ_C(z)} ∈ [0, 1] and η_C (z) ² + λ_C (z) ² ⩽ 1, (Ψ_{η_C(z)}) ² + (Ψ_{λ_C(z)}) ² ⩽ 1.η_C (z), λ_C (z) respectively indicate amplitude terms for MD and NMD of z to C, Ψ_{η_C(z)}, Ψ_{λ_C(z)} respectively mean phase terms for MD and NMD of z to C.

Combining the advantages of CPFS and IVPFS, we will introduce a broader set, namely interval-valued complex Pythagorean fuzzy set (IVCPFS).

Definition 3. Let Z be a given set. An IVCPFS C is an object owning the expression:

$C = {〈 z, ([μ_{C}^{-} (z), μ_{C}^{+} (z)], [θ_{C}^{s} - (z), θ_{C}^{+} (z)]) 〉 | z \in Z} .$ (3) where $μ_{C}^{-} (z) = η_{C}^{-} (z) \cdot e^{i \cdot 2 π Ψ_{η_{C}^{-} (z)}}$ , $μ_{C}^{+} (z) = η_{C}^{+} (z) \cdot e^{i \cdot 2 π Ψ_{η_{C}^{+} (z)}}$ and $0 ⩽ η_{C}^{-} ⩽ η_{C}^{+} ⩽ 1$ , $0 ⩽ Ψ_{η_{C}^{-} (z)} ⩽ Ψ_{η_{C}^{+} (z)} ⩽ 1$ . $θ_{C}^{-} (z) = λ_{C}^{-} (z) \cdot e^{i \cdot 2 π Ψ_{λ_{C}^{-} (z)}}$ , $θ_{C}^{+} (z) = λ_{C}^{+} (z) \cdot e^{i \cdot 2 π Ψ_{λ_{C}^{+} (z)}}$ and $0 ⩽ λ_{C}^{-} ⩽ λ_{C}^{+} ⩽ 1$ , $0 ⩽ Ψ_{λ_{C}^{-} (z)} ⩽ Ψ_{λ_{C}^{+} (z)} ⩽ 1$ , obeying the conditions $(η_{C}^{+} (z))^{2} + (λ_{C}^{+} (z))^{2} ⩽ 1$ , . Additional, $C = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}$ , $[λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]})$ is named as interval-valued complex Pythagorean fuzzy number (IVCPFN).

Definition 4. Assume $ς = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}, [λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]})$ is an IVCPFN, $\begin{matrix} CS (γ) = (((η^{-})^{2} + (η^{+})^{2}) - ((λ^{-})^{2} + (λ^{+})^{2}) \\ + ((Ψ_{η}^{-})^{2} + (Ψ_{η}^{+})^{2}) - ((Ψ_{λ}^{-})^{2} + (Ψ_{λ}^{+})^{2}))) / 4 \end{matrix}$

and $\begin{matrix} CH (γ) = (((η^{-})^{2} + (η^{+})^{2}) + ((λ^{-})^{2} + (λ^{+})^{2}) \\ + ((Ψ_{η}^{-})^{2} + (Ψ_{η}^{+})^{2}) + ((Ψ_{λ}^{-})^{2} + (Ψ_{λ}^{+})^{2}))) / 4 \end{matrix}$

respectively mean its score value and accuracy value. For any two IVCPFNs ς₁, ς₂, we have

If CS (ς₁) > CS (ς₂), then ς₁ > ς₂;

If CS (ς₁) = CS (ς₂) and CH (ς₁) > CH (ς₂), then ς₁ > ς₂.

3 Einstein operation laws for interval-valued complex Pythagorean fuzzy set

Einstein t-norm and t-conorm [4] is an essential mean to define the operation rules. Einstein product ⊗ and Einstein sum ⊕ are two fundamental Einstein operations, which are provided as below [4]: $Ξ (y, z) = y \oplus z = \frac{y + z}{1 + yz}$ (4)

$Θ (y, z) = y \otimes z = \frac{yz}{1 + (1 - y) (1 - z)}$ (5)

In which, y, z are two real numbers and y, z ∈ [0, 1].

With the aid of the Einstein operations provided in Equations (4) and (5), we can present the basic operations with IVCPFNs as below:

Definition 5. Let $ς_{1} = ([η_{1}^{-}, η_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{1}}^{-}, Ψ_{η_{1}}^{+}]}, [λ_{1}^{-}, λ_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{1}}^{-}, Ψ_{λ_{1}}^{+}]})$ , $ς_{2} = ([η_{2}^{-}, η_{2}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{2}}^{-}, Ψ_{η_{2}}^{+}]}$ , $[λ_{2}^{-}, λ_{2}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{2}}^{-}, Ψ_{λ_{2}}^{+}]})$ be two IVCPFNs and τ > 0, then the Einstein operations for IVCPFNs are stated as

(1) $(ς_{1})^{c} = ([λ_{1}^{-}, λ_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{1}}^{-}, Ψ_{λ_{1}}^{+}]}, [η_{1}^{-}, η_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{1}}^{-}, Ψ_{η_{1}}^{+}]});$

(2) $\begin{matrix} ς_{1} \land ς_{2} = ([min (η_{1}^{-}, η_{2}^{-}), min (η_{1}^{+}, η_{2}^{+})] \cdot e^{i \cdot 2 π [min (Ψ_{η_{1}}^{-}, Ψ_{η_{2}}^{-}), min (Ψ_{η_{1}}^{+}, Ψ_{η_{2}}^{+})]}, \\ [max (λ_{1}^{-}, λ_{2}^{-}), max (λ_{1}^{+}, λ_{2}^{+})] \cdot e^{i \cdot 2 π [max (Ψ_{λ_{1}}^{-}, Ψ_{λ_{2}}^{-}), max (Ψ_{λ_{1}}^{+}, Ψ_{λ_{2}}^{+})]}); \end{matrix}$

(3) $\begin{matrix} ς_{1} \lor ς_{2} = ([max (η_{1}^{-}, η_{2}^{-}), max (η_{1}^{+}, η_{2}^{+})] \cdot e^{i \cdot 2 π [max (Ψ_{η_{1}}^{-}, Ψ_{η_{2}}^{-}), m xa (Ψ_{η_{1}}^{+}, Ψ_{η_{2}}^{+})]}, \\ [min (λ_{1}^{-}, λ_{2}^{-}), min (λ_{1}^{+}, λ_{2}^{+})] \cdot e^{i \cdot 2 π [min (Ψ_{λ_{1}}^{-}, Ψ_{λ_{2}}^{-}), min (Ψ_{λ_{1}}^{+}, Ψ_{λ_{2}}^{+})]}); \end{matrix}$

(4) $ς_{1} \oplus ς_{2} = (\begin{matrix} [\sqrt{\frac{(η_{1}^{-})^{2} + (η_{2}^{-})^{2}}{1 + (η_{1}^{-})^{2} (η_{2}^{-})^{2}}}, \sqrt{\frac{(η_{1}^{+})^{2} + (η_{2}^{+})^{2}}{1 + (η_{1}^{+})^{2} (η_{2}^{+})^{2}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(Ψ_{η_{1}}^{-})^{2} + (Ψ_{η_{2}}^{-})^{2}}{1 + (Ψ_{η_{1}}^{-})^{2} (Ψ_{η_{2}}^{-})^{2}}}, \sqrt{\frac{(Ψ_{η_{1}}^{+})^{2} + (Ψ_{η_{2}}^{+})^{2}}{1 + (Ψ_{η_{1}}^{+})^{2} (Ψ_{η_{2}}^{+})^{2}}}]}, \\ [\frac{(λ_{1}^{-}) (λ_{2}^{-})}{\sqrt{1 + (1 - (λ_{1}^{-})^{2}) (1 - (λ_{2}^{-})^{2})}}, \frac{(λ_{1}^{+}) (λ_{2}^{+})}{\sqrt{1 + (1 - (λ_{1}^{+})^{2}) (1 - (λ_{2}^{+})^{2})}}] \cdot e^{i \cdot 2 π [\frac{(Ψ_{λ_{1}}^{-}) (Ψ_{λ_{2}}^{-})}{\sqrt{1 + (1 - (Ψ_{λ_{1}}^{-})^{2}) (1 - (Ψ_{λ_{2}}^{-})^{2})}}, \frac{(Ψ_{λ_{1}}^{+}) (Ψ_{λ_{2}}^{+})}{\sqrt{1 + (1 - (Ψ_{λ_{1}}^{+})^{2}) (1 - (Ψ_{λ_{2}}^{+})^{2})}}]} \end{matrix});$

(5) $ς_{1} \otimes ς_{2} = (\begin{matrix} [\frac{(η_{1}^{-}) (η_{2}^{-})}{\sqrt{1 + (1 - (η_{1}^{-})^{2}) (1 - (η_{2}^{-})^{2})}}, \frac{(η_{1}^{+}) (η_{2}^{+})}{\sqrt{1 + (1 - (η_{1}^{+})^{2}) (1 - (η_{2}^{+})^{2})}}] \cdot e^{i \cdot 2 π [\frac{(Ψ_{η_{1}}^{-}) (Ψ_{η_{2}}^{-})}{\sqrt{1 + (1 - (Ψ_{η_{1}}^{-})^{2}) (1 - (Ψ_{η_{2}}^{-})^{2})}}, \frac{(Ψ_{η_{1}}^{+}) (Ψ_{η_{2}}^{+})}{\sqrt{1 + (1 - (Ψ_{η_{1}}^{+})^{2}) (1 - (Ψ_{η_{2}}^{+})^{2})}}]}, \\ [\sqrt{\frac{(λ_{1}^{-})^{2} + (λ_{2}^{-})^{2}}{1 + (λ_{1}^{-})^{2} (λ_{2}^{-})^{2}}}, \sqrt{\frac{(λ_{1}^{+})^{2} + (λ_{2}^{+})^{2}}{1 + (λ_{1}^{+})^{2} (λ_{2}^{+})^{2}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(Ψ_{λ_{1}}^{-})^{2} + (Ψ_{λ_{2}}^{-})^{2}}{1 + (Ψ_{λ_{1}}^{-})^{2} (Ψ_{λ_{2}}^{-})^{2}}}, \sqrt{\frac{(Ψ_{λ_{1}}^{+})^{2} + (Ψ_{λ_{2}}^{+})^{2}}{1 + (Ψ_{λ_{1}}^{+})^{2} (Ψ_{λ_{2}}^{+})^{2}}}]} \end{matrix});$

(6) $τ ς_{1} = (\begin{matrix} [\sqrt{\frac{(1 + (η_{1}^{-})^{2})^{τ} - (1 - (η_{1}^{-})^{2})^{τ}}{(1 + (η_{1}^{-})^{2})^{τ} + (1 - (η_{1}^{-})^{2})^{τ}}}, \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{τ} - (1 - (η_{1}^{+})^{2})^{τ}}{(1 + (η_{1}^{+})^{2})^{τ} + (1 - (η_{1}^{+})^{2})^{τ}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{1}}^{-})^{2})^{τ} - (1 - (Ψ_{η_{1}}^{-})^{2})^{τ}}{(1 + (Ψ_{η_{1}}^{-})^{2})^{τ} + (1 - (Ψ_{η_{1}}^{-})^{2})^{τ}}}, \sqrt{\frac{(1 + (Ψ_{η_{1}}^{+})^{2})^{τ} - (1 - (Ψ_{η_{1}}^{+})^{2})^{τ}}{(1 + (Ψ_{η_{1}}^{+})^{2})^{τ} + (1 - (Ψ_{η_{1}}^{+})^{2})^{τ}}}]}, \\ [\frac{\sqrt{2} (λ_{1}^{-})^{τ}}{\sqrt{(2 - (λ_{1}^{-})^{2})^{τ} + ((λ_{1}^{-})^{2})^{τ}}}, \frac{\sqrt{2} (λ_{1}^{+})^{τ}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{τ} + ((λ_{1}^{+})^{2})^{τ}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{1}}^{-})^{τ}}{\sqrt{(2 - (Ψ_{λ_{1}}^{-})^{2})^{τ} + ((Ψ_{λ_{1}}^{-})^{2})^{τ}}}, \frac{\sqrt{2} (Ψ_{λ_{1}}^{+})^{τ}}{\sqrt{(2 - (Ψ_{λ_{1}}^{+})^{2})^{τ} + ((Ψ_{λ_{1}}^{+})^{2})^{τ}}}]} \end{matrix});$

(7) $(ς_{1})^{τ} = (\begin{matrix} [\frac{\sqrt{2} (η_{1}^{-})^{τ}}{\sqrt{(2 - (η_{1}^{-})^{2})^{τ} + ((η_{1}^{-})^{2})^{τ}}}, \frac{\sqrt{2} (η_{1}^{+})^{τ}}{\sqrt{(2 - (η_{1}^{+})^{2})^{τ} + ((η_{1}^{+})^{2})^{τ}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{η_{1}}^{-})^{τ}}{\sqrt{(2 - (Ψ_{η_{1}}^{-})^{2})^{τ} + ((Ψ_{η_{1}}^{-})^{2})^{τ}}}, \frac{\sqrt{2} (Ψ_{η_{1}}^{+})^{τ}}{\sqrt{(2 - (Ψ_{η_{1}}^{+})^{2})^{τ} + ((Ψ_{η_{1}}^{+})^{2})^{τ}}}]}, \\ [\sqrt{\frac{(1 + (λ_{1}^{-})^{2})^{τ} - (1 - (λ_{1}^{-})^{2})^{τ}}{(1 + (λ_{1}^{-})^{2})^{τ} + (1 - (λ_{1}^{-})^{2})^{τ}}}, \sqrt{\frac{(1 + (λ_{1}^{+})^{2})^{τ} - (1 - (λ_{1}^{+})^{2})^{τ}}{(1 + (λ_{1}^{+})^{2})^{τ} + (1 - (λ_{1}^{+})^{2})^{τ}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{λ_{1}}^{-})^{2})^{τ} - (1 - (Ψ_{λ_{1}}^{-})^{2})^{τ}}{(1 + (Ψ_{λ_{1}}^{-})^{2})^{τ} + (1 - (Ψ_{λ_{1}}^{-})^{2})^{τ}}}, \sqrt{\frac{(1 + (Ψ_{λ_{1}}^{+})^{2})^{τ} - (1 - (Ψ_{η_{1}}^{+})^{2})^{τ}}{(1 + (Ψ_{λ_{1}}^{+})^{2})^{τ} + (1 - (Ψ_{η_{1}}^{+})^{2})^{τ}}}]} \end{matrix}) .$

Theorem 1. The calculation results of (ς₁) ^c, ς₁ ∧ ς₂, ς₁ ∨ ς₂, ς₁ ⊕ ς₂, ς₁ ⊗ ς₂, τς₁ and (ς₁) ^τ by employing the Definition 5 are again IVCPFN.

Proof. We only verify the part (4), the remainder can be verified using similar mean.

For the part (4). Firstly, we verify the upper of the real term of the result ς₁ ⊕ ς₂. Since

$ς_{1} = ([η_{1}^{-}, η_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{1}}^{-}, Ψ_{η_{1}}^{+}]}, [λ_{1}^{-}, λ_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{1}}^{-}, Ψ_{λ_{1}}^{+}]})$ , $ς_{2} = ([η_{2}^{-}, η_{2}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{2}}^{-}, Ψ_{η_{2}}^{+}]}, [λ_{2}^{-}, λ_{2}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{2}}^{-}, Ψ_{λ_{2}}^{+}]})$ are two IVCPFNs, then $0 ⩽ η_{1}^{+}, λ_{1}^{+}, η_{2}^{+}, λ_{2}^{+} ⩽ 1$ , and $0 ⩽ (η_{1}^{+})^{2}, (λ_{1}^{+})^{2}, (η_{2}^{+})^{2}, (λ_{2}^{+})^{2} ⩽ 1$ . Based on the Equations (4) and (5), we get $0 ⩽ Ξ ((η_{1}^{+})^{2}, (η_{2}^{+})^{2}) ⩽ 1$ and $0 ⩽ Θ ((λ_{1}^{+})^{2}, (λ_{2}^{+})^{2}) ⩽ 1$ , further we have $0 ⩽ \sqrt{Ξ ((η_{1}^{+})^{2}, (η_{2}^{+})^{2})} ⩽ 1$ and $0 ⩽ \sqrt{Θ ((λ_{1}^{+})^{2}, (λ_{2}^{+})^{2})} ⩽ 1$ . It follows that $0 ⩽ \sqrt{\frac{(η_{1}^{+})^{2} + (η_{2}^{+})^{2}}{1 + (η_{1}^{+})^{2} (η_{2}^{+})^{2}}} ⩽ 1$ and $0 ⩽ \frac{(λ_{1}^{+}) (λ_{2}^{+})}{\sqrt{1 + (1 - (λ_{1}^{+})^{2}) (1 - (λ_{2}^{+})^{2})}} ⩽ 1$ .

Furthermore, $(η_{1}^{+})^{2} ⩽ 1 - (λ_{1}^{+})^{2}$ and $(η_{2}^{+})^{2} ⩽ 1 - (λ_{2}^{+})^{2}$ , then we can deduce that $0 ⩽ {(\sqrt{\frac{(η_{1}^{+})^{2} + (η_{2}^{+})^{2}}{1 + (η_{1}^{+})^{2} (η_{2}^{+})^{2}}})}^{2}$ $+ {(\frac{(λ_{1}^{+}) (λ_{2}^{+})}{\sqrt{1 + (1 - (λ_{1}^{+})^{2}) (1 - (λ_{2}^{+})^{2})}})}^{2} = \frac{(η_{1}^{+})^{2} + (η_{2}^{+})^{2}}{1 + (η_{1}^{+})^{2} (η_{2}^{+})^{2}} + \frac{(λ_{1}^{+})^{2} (λ_{2}^{+})^{2}}{1 + (1 - (λ_{1}^{+})^{2}) (1 - (λ_{2}^{+})^{2})} ⩽ \frac{(η_{1}^{+})^{2} + (η_{2}^{+})^{2} + (λ_{1}^{+})^{2} (λ_{2}^{+})^{2}}{1 + (η_{1}^{+})^{2} (η_{2}^{+})^{2}} ⩽ \frac{(η_{1}^{+})^{2} + (η_{2}^{+})^{2} + (1 - (η_{1}^{+})^{2}) (1 - (η_{2}^{+})^{2})}{1 + (η_{1}^{+})^{2} (η_{2}^{+})^{2}} = 1$ .

So the upper of the real term of the result ς₁ ⊕ ς₂ is right. Obviously, the lower of the real term of the result ς₁ ⊕ ς₂ is also kept. Meanwhile, the imaginary part of the result can be checked analogously. This finishes the proof of Theorem 1.

Example 3.1. Consider two IVCPFNs ς₁ = ([0.32, 0.75] · e^{i·2π[0.22,0.46]}, [0.21, 0.45] · e^{i·2π[0.34,0.81]}), ς₂ = ([0.14, 0.44] · e^{i·2π[0.34,0.55]}, [0.37, 0.80] · e^{i·2π[0.29,0.70]}) and real number τ = 4. Then in the light of Definition 5, we get

(1) $\begin{matrix} ς_{1} \oplus ς_{2} = (\begin{matrix} [\sqrt{\frac{0 . 32^{2} + 0 . 14^{2}}{1 + 0 . 32^{2} 0 . 14^{2}}}, \sqrt{\frac{0 . 75^{2} + 0 . 44^{2}}{1 + 0 . 75^{2} 0 . 44^{2}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{0 . 22^{2} + 0 . 34^{2}}{1 + 0 . 22^{2} 0 . 34^{2}}}, \sqrt{\frac{0 . 46^{2} + 0 . 55^{2}}{1 + 0 . 46^{2} 0 . 55^{2}}}]}, \\ [\frac{0.21 \times 0.37}{\sqrt{1 + (1 - 0 . 21^{2}) (1 - 0 . 37^{2})}}, \frac{0.45 \times 0.80}{\sqrt{1 + (1 - 0 . 45^{2}) (1 - 0 . 80^{2})}}] \cdot e^{i \cdot 2 π [\frac{0.34 \times 0.29}{\sqrt{1 + (1 - 0 . 34^{2}) (1 - 0 . 29^{2})}}, \frac{0.81 \times 0.70}{\sqrt{1 + (1 - 0 . 81^{2}) (1 - 0 . 70^{2})}}]} \end{matrix}) \\ = ([0.3489, 0.8257] \cdot e^{i \cdot 2 π [0.4038, 0.6951]}, [0.0575, 0.3173] \cdot e^{i \cdot 2 π [0.0733, 0.5230]}) \end{matrix}$

(2) $\begin{matrix} ς_{1} \otimes ς_{2} = (\begin{matrix} [\frac{0.32 \times 0.14}{\sqrt{1 + (1 - 0 . 32^{2}) (1 - 0 . 14^{2})}}, \frac{0.75 \times 0.44}{\sqrt{1 + (1 - 0 . 75^{2}) (1 - 0 . 44^{2})}}] \cdot e^{i \cdot 2 π [\frac{0.22 \times 0.34}{\sqrt{1 + (1 - 0 . 22^{2}) (1 - 0 . 34^{2})}}, \frac{0.46 \times 0.55}{\sqrt{1 + (1 - 0 . 46^{2}) (1 - 0 . 55^{2})}}]}, \\ [\sqrt{\frac{0 . 21^{2} + 0 . 37^{2}}{1 + 0 . 21^{2} 0 . 37^{2}}}, \sqrt{\frac{0 . 45^{2} + 0 . 80^{2}}{1 + 0 . 45^{2} 0 . 80^{2}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{0 . 34^{2} + 0 . 29^{2}}{1 + 0 . 34^{2} 0 . 29^{2}}}, \sqrt{\frac{0 . 81^{2} + 0 . 70^{2}}{1 + 0 . 81^{2} 0 . 70^{2}}}]} \end{matrix}) \\ = ([0.0327, 0.2837] \cdot e^{i \cdot 2 π [0.0551, 0.2032]}, [0.4242, 0.8636] \cdot e^{i \cdot 2 π [0.4447, 0.9313]}) \end{matrix}$

(3) $\begin{matrix} 4 ς_{1} = (\begin{matrix} [\sqrt{\frac{(1 + 0 . 32^{2})^{4} - (1 - 0 . 32^{2})^{4}}{(1 + 0 . 32^{2})^{4} + (1 - 0 . 32^{2})^{4}}}, \sqrt{\frac{(1 + 0 . 75^{2})^{4} - (1 - 0 . 75^{2})^{4}}{(1 + 0 . 75^{2})^{4} + (1 - 0 . 75^{2})^{4}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + 0 . 22^{2})^{4} - (1 - 0 . 22^{2})^{4}}{(1 + 0 . 22^{2})^{4} + (1 - 0 . 22^{2})^{4}}}, \sqrt{\frac{(1 + 0 . 46^{2})^{4} - (1 - 0 . 46^{2})^{4}}{(1 + 0 . 46^{2})^{4} + (1 - 0 . 46^{2})^{4}}}]}, \\ [\frac{\sqrt{2} \times 0 . 21^{4}}{\sqrt{(2 - 0 . 21^{2})^{4} + (0 . 21^{2})^{4}}}, \frac{\sqrt{2} \times 0 . 45^{4}}{\sqrt{(2 - 0 . 45^{2})^{4} + (0 . 45^{2})^{4}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \times 0 . 34^{4}}{\sqrt{(2 - 0 . 34^{2})^{4} + (0 . 34^{2})^{4}}}, \frac{\sqrt{2} \times 0 . 81^{4}}{\sqrt{(2 - 0 . 81^{2})^{4} + (0 . 81^{2})^{4}}}]} \end{matrix}) \\ = ([0.6240, 0.9939] \cdot e^{i \cdot 2 π [0.4375, 0.8342]}, [0.0007, 0.0179] \cdot e^{i \cdot 2 π [0.0053, 0.3279]}) \end{matrix}$

(4) $\begin{matrix} (ς_{1})^{4} = (\begin{matrix} [\frac{\sqrt{2} \times 0 . 32^{4}}{\sqrt{(2 - 0 . 32^{2})^{4} + (0 . 32^{2})^{4}}}, \frac{\sqrt{2} \times 0 . 75^{4}}{\sqrt{(2 - 0 . 75^{2})^{4} + (0 . 75^{2})^{4}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \times 0 . 22^{4}}{\sqrt{(2 - 0 . 22^{2})^{4} + (0 . 22^{2})^{4}}}, \frac{\sqrt{2} \times 0 . 46^{4}}{\sqrt{(2 - 0 . 46^{2})^{4} + (0 . 46^{2})^{4}}}]}, \\ [\sqrt{\frac{(1 + 0 . 21^{2})^{4} - (1 - 0 . 21^{2})^{4}}{(1 + 0 . 21^{2})^{4} + (1 - 0 . 21^{2})^{4}}}, \sqrt{\frac{(1 + 0 . 45^{2})^{4} - (1 - 0 . 45^{2})^{4}}{(1 + 0 . 45^{2})^{4} + (1 - 0 . 45^{2})^{4}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + 0 . 34^{2})^{4} - (1 - 0 . 34^{2})^{4}}{(1 + 0 . 34^{2})^{4} + (1 - 0 . 34^{2})^{4}}}, \sqrt{\frac{(1 + 0 . 81^{2})^{4} - (1 - 0 . 81^{2})^{4}}{(1 + 0 . 81^{2})^{4} + (1 - 0 . 81^{2})^{4}}}]} \end{matrix}) \\ = ([0.0041, 0.2140] \cdot e^{i \cdot 2 π [0.0009, 0.0198]}, [0.4180, 0.8221] \cdot e^{i \cdot 2 π [0.6586, 0.9981]}) \end{matrix}$ In the following, we are able to provide several operational properties for IVCPFNs:

Theorem 2. Suppose $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2) are two IVCPFNs, and τ > 0, τ₁ > 0, τ₂ > 0, then we have

$\begin{matrix} (1) ς_{1} \oplus ς_{2} = ς_{2} \oplus ς_{1}; \\ (2) τ_{1} ς_{1} \oplus τ_{2} ς_{1} = (τ_{1} + τ_{2}) ς_{1}; \\ (3) τ (ς_{1} \oplus ς_{2}) = τ ς_{1} \oplus τ ς_{2}; \\ (4) ς_{1} \otimes ς_{2} = ς_{2} \otimes ς_{1}; \\ (5) (ς_{1} \otimes ς_{2})^{τ} = (ς_{1})^{τ} \otimes (ς_{2})^{τ}; \\ (6) (ς_{1}^{τ_{1}})^{τ_{2}} = ς_{1}^{τ_{1} τ_{2}}; \\ (7) ς_{1}^{τ_{1}} \otimes ς_{1}^{τ_{2}} = ς_{1}^{(τ_{1} + τ_{2})} . \end{matrix}$

Proof. We only verify the formula (2), the rest can be proved by similar manner. As $ς_{1} = ([η_{1}^{-}, η_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{1}}^{-}, Ψ_{η_{1}}^{+}]}, [λ_{1}^{-}, λ_{1}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{1}}^{-}, Ψ_{λ_{1}}^{+}]})$ is an IVCPFN, in light of Definition 5 we have $τ_{1} ς_{1} = (\begin{matrix} [\sqrt{\frac{(1 + (η_{1}^{-})^{2})^{τ_{1}} - (1 - (η_{1}^{-})^{2})^{τ_{1}}}{(1 + (η_{1}^{-})^{2})^{τ_{1}} + (1 - (η_{1}^{-})^{2})^{τ_{1}}}}, \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{τ_{1}} - (1 - (η_{1}^{+})^{2})^{τ_{1}}}{(1 + (η_{1}^{+})^{2})^{τ_{1}} + (1 - (η_{1}^{+})^{2})^{τ_{1}}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{1}}^{-})^{2})^{τ_{1}} - (1 - (Ψ_{η_{1}}^{-})^{2})^{τ_{1}}}{(1 + (Ψ_{η_{1}}^{-})^{2})^{τ_{1}} + (1 - (Ψ_{η_{1}}^{-})^{2})^{τ_{1}}}}, \sqrt{\frac{(1 + (Ψ_{η_{1}}^{+})^{2})^{τ_{1}} - (1 - (Ψ_{η_{1}}^{+})^{2})^{τ_{1}}}{(1 + (Ψ_{η_{1}}^{+})^{2})^{τ_{1}} + (1 - (Ψ_{η_{1}}^{+})^{2})^{τ_{1}}}}]}, \\ [\frac{\sqrt{2} (λ_{1}^{-})^{τ_{1}}}{\sqrt{(2 - (λ_{1}^{-})^{2})^{τ_{1}} + ((λ_{1}^{-})^{2})^{τ_{1}}}}, \frac{\sqrt{2} (λ_{1}^{+})^{τ_{1}}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{τ_{1}} + ((λ_{1}^{+})^{2})^{τ_{1}}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{1}}^{-})^{τ_{1}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{-})^{2})^{τ_{1}} + ((Ψ_{λ_{1}}^{-})^{2})^{τ_{1}}}}, \frac{\sqrt{2} (Ψ_{λ_{1}}^{+})^{τ_{1}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{+})^{2})^{τ_{1}} + ((Ψ_{λ_{1}}^{+})^{2})^{τ_{1}}}}]} \end{matrix})$ $τ_{2} ς_{1} = (\begin{matrix} [\sqrt{\frac{(1 + (η_{1}^{-})^{2})^{τ_{2}} - (1 - (η_{1}^{-})^{2})^{τ_{2}}}{(1 + (η_{1}^{-})^{2})^{τ_{2}} + (1 - (η_{1}^{-})^{2})^{τ_{2}}}}, \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{τ_{2}} - (1 - (η_{1}^{+})^{2})^{τ_{2}}}{(1 + (η_{1}^{+})^{2})^{τ_{2}} + (1 - (η_{1}^{+})^{2})^{τ_{2}}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{1}}^{-})^{2})^{τ_{2}} - (1 - (Ψ_{η_{1}}^{-})^{2})^{τ_{2}}}{(1 + (Ψ_{η_{1}}^{-})^{2})^{τ_{2}} + (1 - (Ψ_{η_{1}}^{-})^{2})^{τ_{2}}}}, \sqrt{\frac{(1 + (Ψ_{η_{1}}^{+})^{2})^{τ_{2}} - (1 - (Ψ_{η_{1}}^{+})^{2})^{τ_{2}}}{(1 + (Ψ_{η_{1}}^{+})^{2})^{τ_{2}} + (1 - (Ψ_{η_{1}}^{+})^{2})^{τ_{2}}}}]}, \\ [\frac{\sqrt{2} (λ_{1}^{-})^{τ_{2}}}{\sqrt{(2 - (λ_{1}^{-})^{2})^{τ_{2}} + ((λ_{1}^{-})^{2})^{τ_{2}}}}, \frac{\sqrt{2} (λ_{1}^{+})^{τ_{2}}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{τ_{2}} + ((λ_{1}^{+})^{2})^{τ_{2}}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{1}}^{-})^{τ_{2}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{-})^{2})^{τ_{2}} + ((Ψ_{λ_{1}}^{-})^{2})^{τ_{2}}}}, \frac{\sqrt{2} (Ψ_{λ_{1}}^{+})^{τ_{2}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{+})^{2})^{τ_{2}} + ((Ψ_{λ_{1}}^{+})^{2})^{τ_{2}}}}]} \end{matrix})$

and $(τ_{1} + τ_{2}) ς_{1} = (\begin{matrix} [\sqrt{\frac{(1 + (η_{1}^{-})^{2})^{(τ_{1} + τ_{2})} - (1 - (η_{1}^{-})^{2})^{(τ_{1} + τ_{2})}}{(1 + (η_{1}^{-})^{2})^{(τ_{1} + τ_{2})} + (1 - (η_{1}^{-})^{2})^{(τ_{1} + τ_{2})}}}, \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})} - (1 - (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})}}{(1 + (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})} + (1 - (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{1}}^{-})^{2})^{(τ_{1} + τ_{2})} - (1 - (Ψ_{η_{1}}^{-})^{2})^{(τ_{1} + τ_{2})}}{(1 + (Ψ_{η_{1}}^{-})^{2})^{(τ_{1} + τ_{2})} + (1 - (Ψ_{η_{1}}^{-})^{2})^{(τ_{1} + τ_{2})}}}, \sqrt{\frac{(1 + (Ψ_{η_{1}}^{+})^{2})^{(τ_{1} + τ_{2})} - (1 - (Ψ_{η_{1}}^{+})^{2})^{(τ_{1} + τ_{2})}}{(1 + (Ψ_{η_{1}}^{+})^{2})^{(τ_{1} + τ_{2})} + (1 - (Ψ_{η_{1}}^{+})^{2})^{(τ_{1} + τ_{2})}}}]}, \\ [\frac{\sqrt{2} (λ_{1}^{-})^{(τ_{1} + τ_{2})}}{\sqrt{(2 - (λ_{1}^{-})^{2})^{(τ_{1} + τ_{2})} + ((λ_{1}^{-})^{2})^{(τ_{1} + τ_{2})}}}, \frac{\sqrt{2} (λ_{1}^{+})^{(τ_{1} + τ_{2})}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{τ} + ((λ_{1}^{+})^{2})^{(τ_{1} + τ_{2})}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{1}}^{-})^{(τ_{1} + τ_{2})}}{\sqrt{(2 - (Ψ_{λ_{1}}^{-})^{2})^{(τ_{1} + τ_{2})} + ((Ψ_{λ_{1}}^{-})^{2})^{(τ_{1} + τ_{2})}}}, \frac{\sqrt{2} (Ψ_{λ_{1}}^{+})^{(τ_{1} + τ_{2})}}{\sqrt{(2 - (Ψ_{λ_{1}}^{+})^{2})^{(τ_{1} + τ_{2})} + ((Ψ_{λ_{1}}^{+})^{2})^{(τ_{1} + τ_{2})}}}]} \end{matrix})$ Let $\begin{matrix} τ_{1} ς_{1} \oplus τ_{2} ς_{1} = \\ ς = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}, [λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]}) \end{matrix}$ , we first compute the upper of the real term of the membership η⁺ for ς. For IVCPFNs λ₁ς₁ and λ₂ς₁, we respectively set $\sqrt{\frac{(1 + (η_{1}^{+})^{2})^{τ_{1}} - (1 - (η_{1}^{+})^{2})^{τ_{1}}}{(1 + (η_{1}^{+})^{2})^{τ_{1}} + (1 - (η_{1}^{+})^{2})^{τ_{1}}}} = ς_{τ_{1}}^{⌢} +$ and $\sqrt{\frac{(1 + (η_{1}^{+})^{2})^{τ_{2}} - (1 - (η_{1}^{+})^{2})^{τ_{2}}}{(1 + (η_{1}^{+})^{2})^{τ_{2}} + (1 - (η_{1}^{+})^{2})^{τ_{2}}}} = ς_{τ_{2}}^{⌢} +$ . According to Definition 5 we get $η^{+} = \sqrt{\frac{(ς_{τ_{1}}^{⌢} +)^{2} + (ς_{τ_{2}}^{⌢} +)^{2}}{1 + (ς_{τ_{1}}^{⌢} +)^{2} (ς_{τ_{2}}^{⌢} +)^{2}}}$ , further calculation we have $η^{+} = \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})} - (1 - (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})}}{(1 + (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})} + (1 - (η_{1}^{+})^{2})^{(τ_{1} + τ_{2})}}}$ , which indicates that the upper of the real term of the membership for IVCPFNs τ₁ς₁ ⊕ τ₂ς₁ and (τ₁ + τ₂) ς₁ are equal. By the same manner we can obtain the lower of the real term of the membership and imaginary term of the membership for IVCPFNs τ₁ς₁ ⊕ τ₂ς₁ and (τ₁ + τ₂) ς₁ are also equal respectively. Also, for the upper of the real term of the nonmembership value λ⁺ of ς, we set $\frac{\sqrt{2} (λ_{1}^{+})^{τ_{1}}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{τ_{1}} + ((λ_{1}^{+})^{2})^{τ_{1}}}} = ς_{τ_{1}}^{⌣} +$ , and $\frac{\sqrt{2} (λ_{1}^{+})^{τ_{2}}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{τ_{2}} + ((λ_{1}^{+})^{2})^{τ_{2}}}} = ς_{τ_{2}}^{⌣} +$ . Applying Definition 5 we obtain $λ^{+} = \frac{(ς_{τ_{1}}^{⌣} +) (ς_{τ_{2}}^{⌣} +)}{\sqrt{1 + (1 - (ς_{τ_{1}}^{⌣} +)^{2}) (1 - (ς_{τ_{2}}^{⌣} +)^{2})}}$ , which implies that the upper of the real term of the nonmembership for IVCPFNs τ₁ς₁ ⊕ τ₂ς₁ and (τ₁ + τ₂) ς₁ are equal. Similarity, we can obtain the lower of the real term of the nonmembership and imaginary term of the nonmembership for IVCPFNs τ₁ς₁ ⊕ τ₂ς₁ and (τ₁ + τ₂) ς₁ are also equal respectively. Therefore, we can get τ₁ς₁ ⊕ τ₂ς₁ = (τ₁ + τ₂) ς₁, the result holds.

4 Interval-valued complex Pythagorean fuzzy Einstein aggregation operators

In this par, with the aid of operations of IVCPFNs provided in Section 3, we design certain aggregation operators for fusing diverse IVCPFNs.

4.1 Algebraic aggregation operators

Definition 6. Suppose $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n) is a group of IVCPFNs. Then, the structure of interval-valued complex Pythagorean fuzzy Einstein weighted average (IVCPFEWA) operator is described as $IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) = \oplus_{l = 1}^{n} ɛ_{l} ς_{l}$ (6) where, ɛ = (ɛ₁, ɛ₂, …, ɛ_n) ^T is the weight vector of ς_l (l = 1, 2, 3, …, n), obeying ɛ_l ∈ [0, 1], $\sum_{l = 1}^{n} ɛ_{l} = 1$ .

Theorem 3. Suppose $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n) is a group of IVCPFNs, then the result employing Equation (6) is again a IVCPFN and is provided by $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) = \oplus_{l = 1}^{n} ɛ_{l} ς_{l} \\ (\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix}); \end{matrix}$ (7)

Proof. To verify Equation (7), we employ mathematical induction on n.

(1) For n = 2, Because $ɛ_{1} ς_{1} = (\begin{matrix} [\sqrt{\frac{(1 + (η_{1}^{-})^{2})^{ɛ_{1}} - (1 - (η_{1}^{-})^{2})^{ɛ_{1}}}{(1 + (η_{1}^{-})^{2})^{ɛ_{1}} + (1 - (η_{1}^{-})^{2})^{ɛ_{1}}}}, \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{ɛ_{1}} - (1 - (η_{1}^{+})^{2})^{ɛ_{1}}}{(1 + (η_{1}^{+})^{2})^{ɛ_{1}} + (1 - (η_{1}^{+})^{2})^{ɛ_{1}}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}} - (1 - (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}}}{(1 + (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}} + (1 - (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}}}}, \sqrt{\frac{(1 + (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}} - (1 - (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}}}{(1 + (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}} + (1 - (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}}}}]}, \\ [\frac{\sqrt{2} (λ_{1}^{-})^{ɛ_{1}}}{\sqrt{(2 - (λ_{1}^{-})^{2})^{ɛ_{1}} + ((λ_{1}^{-})^{2})^{ɛ_{1}}}}, \frac{\sqrt{2} (λ_{1}^{+})^{ɛ_{1}}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{ɛ_{1}} + ((λ_{1}^{+})^{2})^{ɛ_{1}}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{1}}^{-})^{ɛ_{1}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{-})^{2})^{ɛ_{1}} + ((Ψ_{λ_{1}}^{-})^{2})^{ɛ_{1}}}}, \frac{\sqrt{2} (Ψ_{λ_{1}}^{+})^{ɛ_{1}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{+})^{2})^{ɛ_{1}} + ((Ψ_{λ_{1}}^{+})^{2})^{ɛ_{1}}}}]} \end{matrix})$ $ɛ_{2} ς_{2} = (\begin{matrix} [\sqrt{\frac{(1 + (η_{2}^{-})^{2})^{ɛ_{2}} - (1 - (η_{2}^{-})^{2})^{ɛ_{2}}}{(1 + (η_{2}^{-})^{2})^{ɛ_{2}} + (1 - (η_{2}^{-})^{2})^{ɛ_{2}}}}, \sqrt{\frac{(1 + (η_{2}^{+})^{2})^{ɛ_{2}} - (1 - (η_{2}^{+})^{2})^{ɛ_{2}}}{(1 + (η_{2}^{+})^{2})^{ɛ_{2}} + (1 - (η_{2}^{+})^{2})^{ɛ_{2}}}}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}} - (1 - (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}}}{(1 + (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}} + (1 - (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}}}}, \sqrt{\frac{(1 + (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}} - (1 - (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}}}{(1 + (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}} + (1 - (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}}}}]}, \\ [\frac{\sqrt{2} (λ_{2}^{-})^{ɛ_{2}}}{\sqrt{(2 - (λ_{2}^{-})^{2})^{ɛ_{2}} + ((λ_{2}^{-})^{2})^{ɛ_{2}}}}, \frac{\sqrt{2} (λ_{2}^{+})^{ɛ_{2}}}{\sqrt{(2 - (λ_{2}^{+})^{2})^{ɛ_{2}} + ((λ_{2}^{+})^{2})^{ɛ_{2}}}}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{2}}^{-})^{ɛ_{2}}}{\sqrt{(2 - (Ψ_{λ_{2}}^{-})^{2})^{ɛ_{2}} + ((Ψ_{λ_{2}}^{-})^{2})^{ɛ_{2}}}}, \frac{\sqrt{2} (Ψ_{λ_{2}}^{+})^{ɛ_{2}}}{{\sqrt{(2 - (Ψ_{λ_{2}}^{+})^{2})^{τ_{2}} + ((Ψ_{λ_{2}}^{+})^{2})^{τ_{2}}}}^{ɛ_{2}}}]} \end{matrix})$ , then according to Definition 5, we obtain $τ_{1} ς_{1} \oplus τ_{2} ς_{2} = (\begin{matrix} [\begin{matrix} \sqrt{\frac{(1 + (η_{1}^{-})^{2})^{ɛ_{1}} (1 + (η_{2}^{-})^{2})^{ɛ_{2}} - (1 + (η_{1}^{-})^{2})^{ɛ_{1}} (1 + (η_{2}^{-})^{2})^{ɛ_{2}}}{(1 + (η_{1}^{-})^{2})^{ɛ_{1}} (1 + (η_{2}^{-})^{2})^{ɛ_{2}} + (1 + (η_{1}^{-})^{2})^{ɛ_{1}} (1 + (η_{2}^{-})^{2})^{ɛ_{2}}}}, \\ \sqrt{\frac{(1 + (η_{1}^{+})^{2})^{ɛ_{1}} (1 + (η_{2}^{+})^{2})^{ɛ_{2}} - (1 + (η_{1}^{+})^{2})^{ɛ_{1}} (1 + (η_{2}^{+})^{2})^{ɛ_{2}}}{(1 + (η_{1}^{+})^{2})^{ɛ_{1}} (1 + (η_{2}^{+})^{2})^{ɛ_{2}} + (1 + (η_{1}^{+})^{2})^{ɛ_{1}} (1 + (η_{2}^{+})^{2})^{ɛ_{2}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}} - (1 + (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}}}{(1 + (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}} + (1 + (Ψ_{η_{1}}^{-})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{-})^{2})^{ɛ_{2}}}}, \sqrt{\frac{(1 + (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}} - (1 + (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}}}{(1 + (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}} + (1 + (Ψ_{η_{1}}^{+})^{2})^{ɛ_{1}} (1 + (Ψ_{η_{2}}^{+})^{2})^{ɛ_{2}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} (λ_{1}^{-})^{ɛ_{1}} (λ_{2}^{-})^{ɛ_{2}}}{\sqrt{(2 - (λ_{1}^{-})^{2})^{ɛ_{1}} (2 - (λ_{2}^{-})^{2})^{ɛ_{2}} + ((λ_{1}^{-})^{2})^{ɛ_{1}} ((λ_{2}^{-})^{2})^{ɛ_{2}}}}, \\ \frac{\sqrt{2} (λ_{1}^{+})^{ɛ_{1}} (λ_{2}^{+})^{ɛ_{2}}}{\sqrt{(2 - (λ_{1}^{+})^{2})^{ɛ_{1}} (2 - (λ_{2}^{+})^{2})^{ɛ_{2}} + ((λ_{1}^{+})^{2})^{ɛ_{1}} ((λ_{2}^{+})^{2})^{ɛ_{2}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{1}}^{-})^{ɛ_{1}} (Ψ_{λ_{2}}^{-})^{ɛ_{2}}}{\sqrt{(2 - (Ψ_{λ_{1}}^{-})^{2})^{ɛ_{1}} (2 - (Ψ_{λ_{2}}^{-})^{2})^{ɛ_{2}} + ((Ψ_{λ_{1}}^{-})^{2})^{ɛ_{1}} ((Ψ_{λ 2}^{-})^{2})^{ɛ_{2}}}}, \frac{\sqrt{2} (Ψ_{λ_{1}}^{+})^{ɛ_{1}} (Ψ_{λ_{2}}^{+})^{ɛ_{2}}}{(2 - (Ψ_{λ_{1}}^{+})^{2})^{ɛ_{1}} (2 - (Ψ_{λ_{2}}^{+})^{2})^{ɛ_{2}} + ((Ψ_{λ_{1}}^{+})^{2})^{ɛ_{1}} ((Ψ_{λ 2}^{+})^{2})^{ɛ_{2}}}]} \end{matrix})$ $= (\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{2} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{2} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{2} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{2} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{2} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{2} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{2} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{2} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{2} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{2} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{2} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{2} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{2} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{2} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{2} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{2} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{2} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{2} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{2} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{2} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{2} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix}) = IVCPFEWA (ς_{1}, ς_{2});$

(2) Assume Equation (7) is true for n = n₀, i.e., $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n_{0}}) = \oplus_{l = 1}^{n_{0}} ɛ_{l} ς_{l} \\ = (\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix}); \end{matrix}$

Then by Definition 5, for n = n₀ + 1, we acquire $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n_{0}}, ς_{n_{0} + 1}) = \oplus_{l = 1}^{n_{0}} ɛ_{l} ς_{l} \oplus ɛ_{n_{0} + 1} ς_{n_{0} + 1} \\ = (\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0}} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix}) \oplus \end{matrix}$ $(\begin{matrix} [\begin{matrix} \sqrt{\frac{(1 + (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} - (1 - (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}}}{(1 + (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} + (1 - (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}}}}, \\ \sqrt{\frac{(1 + (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} - (1 - (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}}}{(1 + (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} + (1 - (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{(1 + (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}} - (1 - (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}}}{(1 + (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}} + (1 - (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}}}}, \sqrt{\frac{(1 + (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}} - (1 - (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}}}{(1 + (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}} + (1 - (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} (λ_{n_{0} + 1}^{-})^{ɛ_{n_{0} + 1}}}{\sqrt{(2 - (λ_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} + ((λ_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}}}}, \\ \frac{\sqrt{2} (λ_{n_{0} + 1}^{+})^{ɛ_{n_{0} + 1}}}{\sqrt{(2 - (λ_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} + ((λ_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} (Ψ_{λ_{n_{0} + 1}}^{-})^{ɛ_{n_{0} + 1}}}{\sqrt{(2 - (Ψ_{λ_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}} + ((Ψ_{λ_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}}}}, \frac{\sqrt{2} (Ψ_{λ_{n_{0} + 1}}^{+})^{ɛ_{n_{0} + 1}}}{\sqrt{(2 - (Ψ_{λ_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}} + ((Ψ_{λ_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}}}}]} \end{matrix})$ $= (\begin{matrix} [\begin{matrix} \sqrt{\frac{\begin{matrix} \prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} (1 + (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} - \\ \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}} (1 - (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} \end{matrix}}{\begin{matrix} \prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} (1 + (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} + \\ \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}} (1 - (η_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} \end{matrix}}}, \\ \sqrt{\frac{\begin{matrix} \prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} (1 + (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} - \\ \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}} (1 - (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} \end{matrix}}{\begin{matrix} \prod_{l = 1}^{n_{0}} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} (1 + (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} + \\ \prod_{l = 1}^{n_{0}} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}} (1 - (η_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} \end{matrix}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} (1 + (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}} - \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} (1 - (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}}}{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} (1 + (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}} + \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} (1 - (Ψ_{η_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}}}}, \sqrt{\frac{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} (1 + (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}} - \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} (1 - (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}}}{\prod_{l = 1}^{n_{0}} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} (1 + (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}} + \prod_{l = 1}^{n_{0}} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} (1 - (Ψ_{η_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(λ_{l}^{-})}^{ɛ_{l}} (λ_{n_{0} + 1}^{-})^{ɛ_{n_{0} + 1}}}{\sqrt{\begin{matrix} \prod_{l = 1}^{n_{0}} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} (2 - (λ_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} + \\ \prod_{l = 1}^{n_{0}} {((λ_{l}^{-})^{2})}^{ɛ_{l}} ((λ_{n_{0} + 1}^{-})^{2})^{ɛ_{n_{0} + 1}} \end{matrix}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(λ_{l}^{+})}^{ɛ_{l}} (λ_{n_{0} + 1}^{+})^{ɛ_{n_{0} + 1}}}{\sqrt{\begin{matrix} \prod_{l = 1}^{n_{0}} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} (2 - (λ_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} + \\ \prod_{l = 1}^{n_{0}} {((λ_{l}^{+})^{2})}^{ɛ_{l}} ((λ_{n_{0} + 1}^{+})^{2})^{ɛ_{n_{0} + 1}} \end{matrix}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}} (Ψ_{λ_{n_{0} + 1}}^{-})^{ɛ_{n_{0} + 1}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} (2 - (Ψ_{λ_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}} + \prod_{l = 1}^{n_{0}} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} ((Ψ_{λ_{n_{0} + 1}}^{-})^{2})^{ɛ_{n_{0} + 1}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n_{0}} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}} (Ψ_{λ_{n_{0} + 1}}^{+})^{ɛ_{n_{0} + 1}}}{\sqrt{\prod_{l = 1}^{n_{0}} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} (2 - (Ψ_{λ_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}} + \prod_{l = 1}^{n_{0}} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} ((Ψ_{λ_{n_{0} + 1}}^{+})^{2})^{ɛ_{n_{0} + 1}}}}]} \end{matrix}$ $= (\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n_{0} + 1} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0} + 1} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0} + 1} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{n_{0} + 1} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0} + 1} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0} + 1} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n_{0} + 1} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0} + 1} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0} + 1} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{n_{0} + 1} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n_{0} + 1} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n_{0} + 1} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n_{0} + 1} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0} + 1} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n_{0} + 1} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0} + 1} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n_{0} + 1} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0} + 1} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n_{0} + 1} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n_{0} + 1} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n_{0} + 1} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix});$

So the result is true for n = n₀ + 1. Thus, we verified the result of Equation (7) is right for all n by employing mathematical induction principle.

To show the use of IVCPFEWA operator, an example is provided as below:

Example 4.1. There are four IVCPFNs ς₁ = ([0.22, 0.65] · e^{i·2π[0.12,0.56]}, [0.31, 0.44] · e^{i·2π[0.24,0.66]}), ς₂ = ([0.16, 0.35] · e^{i·2π[0.32,0.48]}, [0.24, 0.54] · e^{i·2π[0.34,0.50]}), ς₃ = ([0.56, 0.66] · e^{i·2π[0.72,0.80]}, [0.40, 0.55] · e^{i·2π[0.24,0.56]}), ς₄ = ([0.11, 0.32] · e^{i·2π[0.18,0.45]}, [0.43, 0.68] · e^{i·2π[0.62,0.71]}), that is, $η_{1}^{-} = 0.22$ , $η_{2}^{-} = 0.16$ , $η_{3}^{-} = 0.56$ , $η_{4}^{-} = 0.11$ , $η_{1}^{+} = 0.65, η_{2}^{+} = 0.35$ , $η_{3}^{+} = 0.66, η_{4}^{+} = 0.32$ , $Ψ_{η_{1}}^{-} = 0.12, Ψ_{η_{2}}^{-} = 0.32, Ψ_{η_{3}}^{-} = 0.72, Ψ_{η_{4}}^{-} = 0.18, Ψ_{η_{1}}^{+} = 0.56, Ψ_{η_{2}}^{+} = 0.48, Ψ_{η_{3}}^{+} = 0.80, Ψ_{η_{4}}^{+} = 0.45$ , $λ_{1}^{-} = 0.31$ , $λ_{2}^{-} = 0.24$ , $λ_{3}^{-} = 0.40$ , $λ_{4}^{-} = 0.43$ , $λ_{1}^{+} = 0.44$ , $λ_{2}^{+} = 0.54$ , $λ_{3}^{+} = 0.55$ , $η_{4}^{+} = 0.68$ , $Ψ_{λ_{1}}^{-} = 0.24, Ψ_{λ_{2}}^{-} = 0.34, Ψ_{λ_{3}}^{-} = 0.24, Ψ_{λ_{4}}^{-} = 0.62$ , $Ψ_{λ_{1}}^{+} = 0.66, Ψ_{λ_{2}}^{+} = 0.50, Ψ_{λ_{3}}^{+} = 0.56, Ψ_{λ_{4}}^{+} = 0.71$ . Let ɛ = (0.3, 0.1, 0.2, 0.4) ^T be their weight. With the aid of Theorem 3, we obtain $\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{4} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{4} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{4} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{4} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{4} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{4} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \\ = [\sqrt{\frac{1.0790 - 0.9070}{1.0790 + 0.9070}}, \sqrt{\frac{1.2568 - 0.7151}{1.2568 + 0.7151}}] \\ = [0.2943, 0.5242] \end{matrix}$

By the same manner, we can derive $[\sqrt{\frac{\prod_{l = 1}^{4} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{4} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{4} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{4} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{4} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{4} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}] = [0.3761, 0.5835] .$

Analogously, we derive $[\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{4} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{4} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{4} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{4} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] = [0.3633, 0.5623]$ $[\frac{\sqrt{2} \prod_{l = 1}^{4} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{4} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{4} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{4} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{4} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}] = [0.3697, 0.6418]$

Then according to Equation (7), we get $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, ς_{3}, ς_{4}) = \\ ([0.2943, 0.5242] \cdot e^{i \cdot 2 π [0.3761, 0.5835]}, \\ [0.3633, 0.5623] \cdot e^{i \cdot 2 π [0.3697, 0.6418]}) \end{matrix} .$

In what follows, we shall discuss several properties for the AO of IVCPFEWA.

Theorem 4. (Idempotency). If any group of IVCPFNs $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n) are equal to $ς = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}, [λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]})$ , then $IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) = ς$ (8)

Proof. Since ς_l = ς (l = 1, 2, 3, . . . , n), based on Theorem 3, then we have $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) = \\ (\begin{matrix} [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ Ψ_{η}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix}); \end{matrix}$ $\begin{matrix} (\begin{matrix} [\begin{matrix} \sqrt{\frac{{(1 + (η^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} - {(1 - (η^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}{{(1 + (η^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {(1 - (η^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}}, \\ \sqrt{\frac{{(1 + (η^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} - {(1 - (η^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}{{(1 + (η^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {(1 - (η^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{{(1 + (Ψ_{η}^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} - {(1 - (Ψ_{η}^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}{{(1 + (Ψ_{η}^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {(1 - (Ψ_{η}^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}}, \sqrt{\frac{{(1 + (Ψ_{η}^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} - {(1 - (Ψ_{η}^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}{{(1 + (Ψ_{η}^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {(1 - (Ψ_{η}^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}}]}, \\ [\begin{matrix} \frac{\sqrt{2} {(λ^{-})}^{\sum_{l = 1}^{n} ɛ_{l}}}{\sqrt{{(2 - (λ^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {((λ^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}}, \\ \frac{\sqrt{2} {(λ^{+})}^{\sum_{l = 1}^{n} ɛ_{l}}}{\sqrt{{(2 - (λ^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {((λ^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} {(Ψ_{λ}^{-})}^{\sum_{l = 1}^{n} ɛ_{l}}}{\sqrt{{(2 - (Ψ_{λ}^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {((Ψ_{λ}^{-})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}}, \frac{\sqrt{2} {(Ψ_{λ}^{+})}^{\sum_{l = 1}^{n} ɛ_{l}}}{\sqrt{{(2 - (Ψ_{λ}^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}} + {((Ψ_{λ}^{+})^{2})}^{\sum_{l = 1}^{n} ɛ_{l}}}}]} \end{matrix}) \\ = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}, [λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]}) \\ = ς \end{matrix}$

Therefore, Equation (8) is kept.

Theorem 5. (Monotonicity). For any two groups IVCPFNs $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ and $ς_{l}^{'} = ([η_{l}^{' -}, η_{l}^{' +}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{' -}, Ψ_{η_{l}}^{' +}]}, [λ_{l}^{' -}, λ_{l}^{' +}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{' -}, Ψ_{λ_{l}}^{' +}]})$ , (l = 1, 2, 3, . . . , n). If η^- ⩽ η′-, dη⁺ ⩽ η^′+, $Ψ_{η}^{-} ⩽ Ψ_{η}^{' -}$ , $Ψ_{η}^{+} ⩽ Ψ_{η}^{' +}$ , λ^- ⩾ λ^′-, λ⁺ ⩾ λ^′+, $Ψ_{λ}^{-} ⩾ Ψ_{λ}^{' -}$ and $Ψ_{λ}^{+} ⩾ Ψ_{λ}^{' +}$ , then $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) \\ ⩽ IVCPFEWA (ς_{1}^{'}, ς_{2}^{'}, \dots, ς_{n}^{'}) \end{matrix}$ (9)

Proof. Let function $h (z) = \frac{1 - z}{1 + z}, z \in [0, 1]$ , because $h^{'} (z) = \frac{- 2}{(1 + z)^{2}} < 0$ , so h (z) is an decreasing function about z. Since $(η_{l}^{+})^{2} ⩽ (η_{l}^{' +})^{2}$ (l = 1, 2, 3, . . . , n), then $h ((η_{l}^{+})^{2}) ⩾ h ((η_{l}^{' +})^{2})$ , that is $\frac{1 - (η_{l}^{+})^{2}}{1 + (η_{l}^{+})^{2}} ⩾ \frac{1 - (η_{l}^{' +})^{2}}{1 + (η_{l}^{' +})^{2}}$ , further ${(\frac{1 - (η_{l}^{+})^{2}}{1 + (η_{l}^{+})^{2}})}^{ɛ_{l}} ⩾ {(\frac{1 - (η_{l}^{' +})^{2}}{1 + (η_{l}^{' +})^{2}})}^{ɛ_{l}}$ . Thus, $\prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{+})^{2}}{1 + (η_{l}^{+})^{2}})}^{ɛ_{l}} ⩾ \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{' +})^{2}}{1 + (η_{l}^{' +})^{2}})}^{ɛ_{l}}$ $\Leftrightarrow 1 + \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{+})^{2}}{1 + (η_{l}^{+})^{2}})}^{ɛ_{l}} ⩾ 1 + \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{' +})^{2}}{1 + (η_{l}^{' +})^{2}})}^{ɛ_{l}}$ $\Leftrightarrow \frac{2}{1 + \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{+})^{2}}{1 + (η_{l}^{+})^{2}})}^{ɛ_{l}}} ⩽ \frac{2}{1 + \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{' +})^{2}}{1 + (η_{l}^{' +})^{2}})}^{ɛ_{l}}}$ $\Leftrightarrow \frac{2}{1 + \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{+})^{2}}{1 + (η_{l}^{+})^{2}})}^{ɛ_{l}}} - 1 ⩽ \frac{2}{1 + \prod_{l = 1}^{n} {(\frac{1 - (η_{l}^{' +})^{2}}{1 + (η_{l}^{' +})^{2}})}^{ɛ_{l}}} - 1$ $\begin{matrix} \Leftrightarrow \frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}} \\ ⩽ \frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{' +})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{' +})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}} \end{matrix}$ $\begin{matrix} \Leftrightarrow \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \\ ⩽ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{' +})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{' +})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}$

By the same manner we obtain $\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}} \\ ⩽ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (η_{l}^{' -})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (η_{l}^{' -})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (η_{l}^{-})^{2})}^{ɛ_{l}}}}, \end{matrix}$ $\sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}} ⩽ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{' +})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{' +})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{' +})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{' +})^{2})}^{ɛ_{l}}}}$ and $\sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}} ⩽ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{' -})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{' -})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{η_{l}}^{' -})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{η_{l}}^{' -})^{2})}^{ɛ_{l}}}}$

Also, Let function $f (z) = \frac{2 - z}{z}, z \in [0, 1]$ , because $f^{'} (z) = \frac{- 2}{z^{2}} < 0$ , so f (z) is an decreasing function about z. Since $λ_{l}^{+} ⩾ λ_{l}^{' +}$ (l = 1, 2, 3, . . . , n), then $f ((λ_{l}^{+})^{2}) ⩽ f ((λ_{l}^{' +})^{2})$ , that is $\frac{2 - (λ_{l}^{+})^{2}}{(λ_{l}^{+})^{2}} ⩽ \frac{2 - (λ_{l}^{' +})^{2}}{(λ_{l}^{' +})^{2}}$ , further ${(\frac{2 - (λ_{l}^{+})^{2}}{(λ_{l}^{+})^{2}})}^{ɛ_{l}} ⩽ {(\frac{2 - (λ_{l}^{' +})^{2}}{(λ_{l}^{' +})^{2}})}^{ɛ_{l}}$ . Thus, $\prod_{l = 1}^{n} {(\frac{2 - (λ_{l}^{+})^{2}}{(λ_{l}^{+})^{2}})}^{ɛ_{l}} ⩽ \prod_{l = 1}^{n} {(\frac{2 - (λ_{l}^{' +})^{2}}{(λ_{l}^{' +})^{2}})}^{ɛ_{l}}$ $\Leftrightarrow 1 + \prod_{l = 1}^{n} {(\frac{2 - (λ_{l}^{+})^{2}}{(λ_{l}^{+})^{2}})}^{ɛ_{l}} ⩽ 1 + \prod_{l = 1}^{n} {(\frac{2 - (λ_{l}^{' +})^{2}}{(λ_{l}^{' +})^{2}})}^{ɛ_{l}}$ $\Leftrightarrow \frac{2}{1 + \prod_{l = 1}^{n} {(\frac{2 - (λ_{l}^{+})^{2}}{(λ_{l}^{+})^{2}})}^{ɛ_{l}}} ⩾ \frac{2}{1 + \prod_{l = 1}^{n} {(\frac{2 - (λ_{l}^{' +})^{2}}{(λ_{l}^{' +})^{2}})}^{ɛ_{l}}}$ $\begin{matrix} \Leftrightarrow \frac{2 \prod_{l = 1}^{n} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{+})^{2})}^{ɛ_{l}}} \\ ⩾ \frac{2 \prod_{l = 1}^{n} {((λ_{l}^{' +})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(2 - (λ_{l}^{' +})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{' +})^{2})}^{ɛ_{l}}} \end{matrix}$ $\begin{matrix} \Leftrightarrow \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{+})^{2})}^{ɛ_{l}}}} \\ ⩾ \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ_{l}^{' +})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ_{l}^{' +})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{' +})^{2})}^{ɛ_{l}}}} \end{matrix}$

By the same means we derive $\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{-})^{2})}^{ɛ_{l}}}} \\ ⩾ \frac{\sqrt{2} \prod_{l = 1}^{n} {(λ_{l}^{' -})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (λ_{l}^{' -})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((λ_{l}^{' -} s)^{2})}^{ɛ_{l}}}}, \end{matrix}$ $\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}} \\ ⩾ \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ_{l}}^{' +})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ_{l}}^{' +})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ_{l}}^{' +})^{2})}^{ɛ_{l}}}} \end{matrix}$

and $\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}} \\ ⩾ \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{λ_{l}}^{' -})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{λ_{l}}^{' -})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((Ψ_{λ_{l}}^{' -})^{2})}^{ɛ_{l}}}} . \end{matrix}$

Assume that $\begin{matrix} IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) \\ = ς = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}, [λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]}) \end{matrix}$

and $\begin{matrix} IVCPFEWA (ς_{1}^{'}, ς_{2}^{'}, \dots, ς_{n}^{'}) \\ = ς^{'} = ([η^{' -}, η^{' +}] \cdot e^{i \cdot 2 π [Ψ_{η}^{' -}, Ψ_{η}^{' +}]}, [λ^{' -}, λ^{' +}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{' -}, Ψ_{λ}^{' +}]}) . \end{matrix}$

From above discussion, we have η^- ⩽ η^′-, η⁺ ⩽ η^′+, $Ψ_{η}^{-} ⩽ Ψ_{η}^{' -}$ , $Ψ_{η}^{+} ⩽ Ψ_{η}^{' +}$ , λ^- ⩾ λ^′-, λ⁺ ⩾ λ^′+, $Ψ_{λ}^{-} ⩾ Ψ_{λ}^{' -}$ , $Ψ_{λ}^{+} ⩾ Ψ_{λ}^{' +}$ , employing Definition 4, we get CS (ς) ⩽ CS (ς′).

If CS (ς) < CS (ς′), then ς < ς′, the result is kept.

If CS (ς) = CS (ς′), then $\begin{matrix} (((η^{-})^{2} + (η^{+})^{2}) - ((λ^{-})^{2} + (λ^{+})^{2}) \\ + ((Ψ_{η}^{-})^{2} + (Ψ_{η}^{+})^{2}) - ((Ψ_{λ}^{-})^{2} + (Ψ_{λ}^{+})^{2}))) / 4 \\ = (((η^{' -})^{2} + (η^{' +})^{2}) - ((λ^{' -})^{2} + (λ^{' +})^{2}) \\ + ((Ψ_{η}^{' -})^{2} + (Ψ_{η}^{' +})^{2}) - ((Ψ_{λ}^{' -})^{2} + (Ψ_{λ}^{' +})^{2}))) / 4, \end{matrix}$

Because η^- ⩽ η^′-, η⁺ ⩽ η^′+, $Ψ_{η}^{-} ⩽ Ψ_{η}^{' -}$ , $Ψ_{η}^{+} ⩽ Ψ_{η}^{' +}$ , λ^- ⩾ λ^′-, λ⁺ ⩾ λ^′+, $Ψ_{λ}^{-} ⩾ Ψ_{λ}^{' -}$ , $Ψ_{λ}^{+} ⩾ Ψ_{λ}^{' +}$ , then we have $\begin{matrix} ((η^{-})^{2} + (η^{+})^{2}) + ((Ψ_{η}^{-})^{2} + (Ψ_{η}^{+})^{2}) = \\ ((η^{' -})^{2} + (η^{' +})^{2}) + ((Ψ_{η}^{' -})^{2} + (Ψ_{η}^{' +})^{2}) \end{matrix}$

and $\begin{matrix} ((λ^{-})^{2} + (λ^{+})^{2}) + ((Ψ_{λ}^{-})^{2} + (Ψ_{λ}^{+})^{2}) = \\ ((λ^{' -})^{2} + (λ^{' +})^{2}) + ((Ψ_{λ}^{' -})^{2} + (Ψ_{λ}^{' +})^{2}) \end{matrix}$

According to Definition 4, we get CH (ς) = CH (ς′). Which implies that ς = ς′, thus the result is kept.

Theorem 6. (Boundedness). For any group of IVCPFNs $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n), then $ς_{-} ⩽ IVCPFEWA (ς_{1}, ς_{2}, \dots, ς_{n}) ⩽ ς_{+}$ (10)

in which $\begin{matrix} ς_{-} = ([\underset{l}{min {} η_{l}^{-}}, \underset{l}{min {} η_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{min {} Ψ_{η_{l}}^{-}}, \underset{l}{min {} Ψ_{η_{l}}^{+}}]}, \\ [\underset{l}{max {} λ_{l}^{-}}, \underset{l}{max {} λ_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{max {} Ψ_{λ_{l}}^{-}}, \underset{l}{max {} Ψ_{λ_{l}}^{+}}]}) \end{matrix}$ , and $\begin{matrix} ς_{+} = ([\underset{l}{max {} η_{l}^{-}}, \underset{l}{max {} η_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{max {} Ψ_{η_{l}}^{-}}, \underset{l}{m xa {} Ψ_{η_{l}}^{+}}]}, \\ [\underset{l}{min {} λ_{l}^{-}}, \underset{l}{min {} λ_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{min {} Ψ_{λ_{l}}^{-}}, \underset{l}{min {} Ψ_{λ_{l}}^{+}}]}) . \end{matrix}$

Proof. By employing analogous method as utilized in Theorem 5, we can obtain this theorem.

4.2 Geometric aggregation operators

Definition 7. Suppose $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n) is a group of IVCPFNs. Then, the structure of interval-valued complex Pythagorean fuzzy Einstein weighted geometric (IVCPFEWG) operator is described as $IVCPFEWG (ς_{1}, ς_{2}, \dots, ς_{n}) = \otimes_{l = 1}^{n} ς_{l}^{ɛ_{l}}$ (11) where, ɛ = (ɛ₁, ɛ₂, …, ɛ_n) ^T is the weight vector of ς_l (l = 1, 2, 3, …, n), obeying ɛ_l ∈ [0, 1], $\sum_{l = 1}^{n} ɛ_{l} = 1$ .

Theorem 7. Suppose $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n) is a group of IVCPFNs, then the result employing Equation (11) is also a IVCPFN, and $\begin{matrix} IVCPFEWG (ς_{1}, ς_{2}, \dots, ς_{n}) = \\ (\begin{matrix} [\begin{matrix} \frac{\sqrt{2} \prod_{l = 1}^{n} {(η_{l}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (η_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{k = 1}^{n} {((η_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \frac{\sqrt{2} \prod_{l = 1}^{n} {(η_{l}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (η_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {((η_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{η_{l}}^{-})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{k = 1}^{n} {((Ψ_{η_{l}}^{-})^{2})}^{ɛ_{l}}}}, \frac{\sqrt{2} \prod_{l = 1}^{n} {(Ψ_{η_{l}}^{+})}^{ɛ_{l}}}{\sqrt{\prod_{l = 1}^{n} {(2 - (Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{k = 1}^{n} {((Ψ_{η_{l}}^{+})^{2})}^{ɛ_{l}}}}]}, \\ [\begin{matrix} \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (λ_{l}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (λ_{l}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (λ_{l}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (λ_{l}^{-})^{2})}^{ɛ_{l}}}}, \\ \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (λ_{l}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (λ_{l}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (λ_{l}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (λ_{l}^{+})^{2})}^{ɛ_{l}}}} \end{matrix}] \cdot e^{i \cdot 2 π [\sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{λ_{l}}^{-})^{2})}^{ɛ_{l}}}}, \sqrt{\frac{\prod_{l = 1}^{n} {(1 + (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} - \prod_{l = 1}^{n} {(1 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}{\prod_{l = 1}^{n} {(1 + (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}} + \prod_{l = 1}^{n} {(1 - (Ψ_{λ_{l}}^{+})^{2})}^{ɛ_{l}}}}]} \end{matrix}); \end{matrix}$ (12)

Proof. This can be verified by following Theorem 3.

Similar to the features of IVCPFEWA operator, certain properties for IVCPFEWG operator can be determined as below.

Theorem 8. (Idempotency). If any group of IVCPFNs $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n) are equal to $ς = ([η^{-}, η^{+}] \cdot e^{i \cdot 2 π [Ψ_{η}^{-}, Ψ_{η}^{+}]}, [λ^{-}, λ^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ}^{-}, Ψ_{λ}^{+}]})$ , then $IVCPFEWG (ς_{1}, ς_{2}, \dots, ς_{n}) = ς$ (13)

Theorem 9. (Monotonicity). For any two groups IVCPFNs $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ and $ς_{l}^{'} = ([η_{l}^{' -}, η_{l}^{' +}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{' -}, Ψ_{η_{l}}^{' +}]}, [λ_{l}^{' -}, λ_{l}^{' +}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{' -}, Ψ_{λ_{l}}^{' +}]})$ , (l = 1, 2, 3, . . . , n). If η^- ⩽ η^′-, η⁺ ⩽ η^′+, $Ψ_{η}^{-} ⩽ Ψ_{η}^{' -}$ , $Ψ_{η}^{+} ⩽ Ψ_{η}^{' +}$ , λ^- ⩾ λ^′-, λ⁺ ⩾ λ^′+, $Ψ_{λ}^{-} ⩾ Ψ_{λ}^{' -}$ and $Ψ_{λ}^{+} ⩾ Ψ_{λ}^{' +}$ s then $\begin{matrix} IVCPFEWG (ς_{1}, ς_{2}, \dots, ς_{n}) ⩽ \\ IVCPFEWG (ς_{1}^{'}, ς_{2}^{'}, \dots, ς_{n}^{'}) \end{matrix}$ (14)

Theorem 10. (Boundedness). For any group of IVCPFNs $ς_{l} = ([η_{l}^{-}, η_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{l}}^{-}, Ψ_{η_{l}}^{+}]}, [λ_{l}^{-}, λ_{l}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{l}}^{-}, Ψ_{λ_{l}}^{+}]})$ (l = 1, 2, 3, . . . , n), then $ς_{-} ⩽ IVCPFEWG (ς_{1}, ς_{2}, \dots, ς_{n}) ⩽ ς_{+}$ (15) which $\begin{matrix} ς_{-} = ([\underset{l}{min {} η_{l}^{-}}, \underset{l}{min {} η_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{min {} Ψ_{η_{l}}^{-}}, \underset{l}{min {} Ψ_{η_{l}}^{+}}]}, \\ [\underset{l}{max {} λ_{l}^{-}}, \underset{l}{max {} λ_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{max {} Ψ_{λ_{l}}^{-}}, \underset{l}{max {} Ψ_{λ_{l}}^{+}}]}), \end{matrix}$

and $\begin{matrix} ς_{+} = ([\underset{l}{max {} η_{l}^{-}}, \underset{l}{max {} η_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{max {} Ψ_{η_{l}}^{-}}, \underset{l}{m xa {} Ψ_{η_{l}}^{+}}]}, \\ [\underset{l}{min {} λ_{l}^{-}}, \underset{l}{min {} λ_{l}^{+}}] \cdot e^{i \cdot 2 π [\underset{l}{min {} Ψ_{λ_{l}}^{-}}, \underset{l}{min {} Ψ_{λ_{l}}^{+}}]}) . \end{matrix}$

Example 4.2. Consider the information shown in Example 4.1, we can use the IVCPFEWG operator to resolve Example 4.1, and the outcome is listed below $\begin{matrix} IVCPFEWG (ς_{1}, ς_{2}, ς_{3}, ς_{4}) \\ = ([0.1972, 0.4688] \cdot e^{i \cdot 2 π [0.2281, 0.5482]}, \\ [0.3753, 0.5811] \cdot e^{i \cdot 2 π [0.4470, 0.6511]}) . \end{matrix}$

5 Decision making approach applying the instituted operators

In this part, a novel technique was designed to settle MADM issues by employing the created AOs. Let the group of alternatives be denoted by S = {S₁, S₂, …, S_m}, and the weight vector of attribute A = {A₁, A₂, …, A_n} is ɛ = (ɛ₁, ɛ₂, …, ɛ_n) ^T, and obeying ɛ_l ∈ [0, 1], $\sum_{l = 1}^{n} ɛ_{l} = 1$ . When the alternative S_t ∈ S with respect to the attribute A_t ∈ A is appraised, the decision maker provides his (her) preference information in IVCPFNs form $r_{tl} = ([η_{tl}^{-}, η_{tl}^{+}] \cdot e^{i \cdot 2 π [Ψ_{η_{tl}}^{-}, Ψ_{η_{tl}}^{+}]}, [λ_{tl}^{-}, λ_{tl}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{tl}}^{-}, Ψ_{λ_{tl}}^{+}]})$ . Thus, the interval-valued complex Pythagorean fuzzy judgment matrix R = (r_tl) _m×n (t = 1, 2, 3, . . . , m ; l = 1, 2, 3, . . . , n) is constructed. Then our aim is to sort the alternatives, the steps of the established approach are carried out:

Step 1. Normalize judgment matrix R = (r_tl) _m×n to keep the equilibrium for the natural measurements of different preference values, the normalized judgment matrix denoted as D = (d_tl) _m×n is acquired via the following way: $d_{tl} = {\begin{matrix} (r_{tl}), if A_{t} is benefit attribute, \\ (r_{tl})^{c}, if A_{t} is cost attribute . \end{matrix}$

in which $\begin{matrix} (r_{tl})^{c} = ([λ_{tl}^{-}, λ_{tl}^{+}] \cdot e^{i \cdot 2 π [Ψ_{λ_{tl}}^{-}, Ψ_{λ_{tl}}^{+}]}, [η_{tl}^{-}, η_{tl}^{+}] \\ \cdot e^{i \cdot 2 π [Ψ_{η_{tl}}^{-}, Ψ_{η_{tl}}^{+}]}), (t = 1, 2, 3, . . ., m; l = 1, 2, 3, . . ., n) . \end{matrix}$

Step 2. With the help of judgment matrix D = (d_tl) _m×n and the IVCPFEWA operator $ς_{t} = IVCPFEWA (d_{t 1}, d_{t 2}, \dots, d_{tn}) .$ (16)

or IVCPFEWG operator $ς_{t} = IVCPFEWG (d_{t 1}, d_{t 2}, \dots, d_{tn}) .$ (17) to gain the overall performance value ς_t for every option S_t (t = 1, 2, 3, …, m).

Step 3. Employing Definition 4 to determine the score index CS (ς_t) of every performance value ς_t, and rank option S_t (t = 1, 2, 3, …, m).

Step 4. End.

The procedure of decision making can be seen in Fig. 1.

Fig. 1

The procedure of decision making.

6 Illustrative example

Psychiatry can’t control the mental patients’ condition effectively, which makes some serious mental patients have the possibility of harming themselves or others’ personal safety, which leads to the need to implement compulsory medical treatment for these mental patients in practice [1]. Different from the general crime, the main cause of mental illness is the actor’s own condition, so the evaluation of the possibility of mental illness “continuing to harm the society” should be different from the evaluation of the possibility of general recidivism from the index and the weight of the attribute [2].

However, in the assessment of the personal risk of mental patients, due to abstract and complex concept of personal danger, and the influence of uncertain factors in decision maker’s cognitive, the preference value provided by decision makers usually can not be accurately determined, it is perhaps to be an interval, and it is described by the form of two dimensions. Therefore, the interval-valued complex Pythagorean fuzzy set can successful represent this kind of uncertain information. In order to evaluate the personal risk of mental patients involving many indicators, then the following four important indicators [1] are considered.

(1) Physical and mental aspects: it involves physical condition, degree of mental illness, types of mental illness, personality status, and so on.

(2) Guardians aspects: it consists of whether there is a guardian, guardian wishes, guardian health, time, economic status, and so on.

(3) Performance before, during and after the crime: it consists of consistent performance, criminal nature, criminal means, post crime attitude, post crime behavior, and so on.

(4) Social aspects: it involves living environment, help and education conditions, community evaluation, activity places, and so on.

Example 6.1. After initial analyzing, there are five possible alternatives {S₁, S₂, S₃, S₄, S₅} are obtained. The four attributes mentioned above are adopted by the DMs to examine the five alternatives: (1) A₁ indicates physical and mental aspects; (2) A₂ indicates guardians aspects; (3) A₃ indicates performance before, during and after the crime and (4) A₄ indicates social aspects. According to Ref [1], their respective importance is given as ɛ = (0 . 4, 0 . 3, 0 . 2, 0 . 1) ^T. The experts provide the judge matrix R = (r_tl) _5×4 in form of IVCPFNs, is summarized in Table 1.

Table 1
Judge matrix R = (r_tl) _5×4

A ₁ A ₂ A ₃ A ₄

S ₁ $\begin{matrix} ([0.52, 0.74] \cdot e^{i \cdot 2 π [0.51, 0.72]}, \\ \cdot e^{i \cdot 2 π [0.20, 0.41]}) \end{matrix}$ $\begin{matrix} ([0.82, 0.90] \cdot e^{i \cdot 2 π [0.31, 0.62]}, \\ [0.23, 0.31] \cdot e^{i \cdot 2 π [0.43, 0.55]}) \end{matrix}$ $\begin{matrix} ([0.65, 0.73] \cdot e^{i \cdot 2 π [0.22, 0.30]}, \\ [0.44, 0.62] \cdot e^{i \cdot 2 π [0.76, 0.92]}) \end{matrix}$ $\begin{matrix} ([0.41, 0.73] \cdot e^{i \cdot 2 π [0.71, 0.91]}, \\ [0.46, 0.58] \cdot e^{i \cdot 2 π [0.18, 0.32]}) \end{matrix}$

S ₂ $\begin{matrix} ([0.55, 0.73] \cdot e^{i \cdot 2 π [0.81, 0.91]}, \\ [0.22, 0.46] \cdot e^{i \cdot 2 π [0.24, 0.31]}) \end{matrix}$ $\begin{matrix} ([0.55, 0.64] \cdot e^{i \cdot 2 π [0.55, 0.80]}, \\ [0.11, 0.34] \cdot e^{i \cdot 2 π [0.45, 0.62]}) \end{matrix}$ $\begin{matrix} ([0.61, 0.75] \cdot e^{i \cdot 2 π [0.54, 0.61]}, \\ [0.44, 0.68] \cdot e^{i \cdot 2 π [0.45, 0.57]}) \end{matrix}$ $\begin{matrix} ([0.45, 0.63] \cdot e^{i \cdot 2 π [0.66, 0.81]}, \\ [0.26, 0.39] \cdot e^{i \cdot 2 π [0.41, 0.54]}) \end{matrix}$

S ₃ $\begin{matrix} ([0.60, 0.90] \cdot e^{i \cdot 2 π [0.67, 0.72]}, \\ [0.14, 0.31] \cdot e^{i \cdot 2 π [0.26, 0.34]}) \end{matrix}$ $\begin{matrix} ([0.53, 0.66] \cdot e^{i \cdot 2 π [0.58, 0.73]}, \\ [0.33, 0.57] \cdot e^{i \cdot 2 π [0.41, 0.54]}) \end{matrix}$ $\begin{matrix} ([0.82, 0.90] \cdot e^{i \cdot 2 π [0.57, 0.63]}, \\ [0.25, 0.31] \cdot e^{i \cdot 2 π [0.42, 0.51]}) \end{matrix}$ $\begin{matrix} ([0.76, 0.83] \cdot e^{i \cdot 2 π [0.77, 0.90]}, \\ [0.42, 0.56] \cdot e^{i \cdot 2 π [0.13, 0.25]}) \end{matrix}$

S ₄ $\begin{matrix} ([0.81, 0.91] \cdot e^{i \cdot 2 π [0.55, 0.71]}, \\ [0.20, 0.30] \cdot e^{i \cdot 2 π [0.46, 0.58]}) \end{matrix}$ $\begin{matrix} ([0.66, 0.71] \cdot e^{i \cdot 2 π [0.62, 0.82]}, \\ [0.32, 0.53] \cdot e^{i \cdot 2 π [0.43, 0.51]}) \end{matrix}$ $\begin{matrix} ([0.56, 0.68] \cdot e^{i \cdot 2 π [0.40, 0.53]}, \\ [0.44, 0.55] \cdot e^{i \cdot 2 π [0.71, 0.82]}) \end{matrix}$ $\begin{matrix} ([0.44, 0.60] \cdot e^{i \cdot 2 π [0.80, 0.91]}, \\ [0.32, 0.57] \cdot e^{i \cdot 2 π [0.22, 0.30]}) \end{matrix}$

S ₅ $\begin{matrix} ([0.72, 0.91] \cdot e^{i \cdot 2 π [0.56, 0.76]}, \\ [0.16, 0.31] \cdot e^{i \cdot 2 π [0.22, 0.51]}) \end{matrix}$ $\begin{matrix} ([0.42, 0.65] \cdot e^{i \cdot 2 π [0.83, 0.92]}, \\ [0.14, 0.35] \cdot e^{i \cdot 2 π [0.22, 0.31]}) \end{matrix}$ $\begin{matrix} ([0.22, 0.34] \cdot e^{i \cdot 2 π [0.11, 0.34]}, \\ [0.71, 0.83] \cdot e^{i \cdot 2 π [0.63, 0.83]}) \end{matrix}$ $\begin{matrix} ([0.75, 0.80] \cdot e^{i \cdot 2 π [0.67, 0.82]}, \\ [0.43, 0.52] \cdot e^{i \cdot 2 π [0.41, 0.50]}) \end{matrix}$

	A ₁	A ₂	A ₃	A ₄
S ₁	$\begin{matrix} ([0.52, 0.74] \cdot e^{i \cdot 2 π [0.51, 0.72]}, \\ \cdot e^{i \cdot 2 π [0.20, 0.41]}) \end{matrix}$	$\begin{matrix} ([0.82, 0.90] \cdot e^{i \cdot 2 π [0.31, 0.62]}, \\ [0.23, 0.31] \cdot e^{i \cdot 2 π [0.43, 0.55]}) \end{matrix}$	$\begin{matrix} ([0.65, 0.73] \cdot e^{i \cdot 2 π [0.22, 0.30]}, \\ [0.44, 0.62] \cdot e^{i \cdot 2 π [0.76, 0.92]}) \end{matrix}$	$\begin{matrix} ([0.41, 0.73] \cdot e^{i \cdot 2 π [0.71, 0.91]}, \\ [0.46, 0.58] \cdot e^{i \cdot 2 π [0.18, 0.32]}) \end{matrix}$
S ₂	$\begin{matrix} ([0.55, 0.73] \cdot e^{i \cdot 2 π [0.81, 0.91]}, \\ [0.22, 0.46] \cdot e^{i \cdot 2 π [0.24, 0.31]}) \end{matrix}$	$\begin{matrix} ([0.55, 0.64] \cdot e^{i \cdot 2 π [0.55, 0.80]}, \\ [0.11, 0.34] \cdot e^{i \cdot 2 π [0.45, 0.62]}) \end{matrix}$	$\begin{matrix} ([0.61, 0.75] \cdot e^{i \cdot 2 π [0.54, 0.61]}, \\ [0.44, 0.68] \cdot e^{i \cdot 2 π [0.45, 0.57]}) \end{matrix}$	$\begin{matrix} ([0.45, 0.63] \cdot e^{i \cdot 2 π [0.66, 0.81]}, \\ [0.26, 0.39] \cdot e^{i \cdot 2 π [0.41, 0.54]}) \end{matrix}$
S ₃	$\begin{matrix} ([0.60, 0.90] \cdot e^{i \cdot 2 π [0.67, 0.72]}, \\ [0.14, 0.31] \cdot e^{i \cdot 2 π [0.26, 0.34]}) \end{matrix}$	$\begin{matrix} ([0.53, 0.66] \cdot e^{i \cdot 2 π [0.58, 0.73]}, \\ [0.33, 0.57] \cdot e^{i \cdot 2 π [0.41, 0.54]}) \end{matrix}$	$\begin{matrix} ([0.82, 0.90] \cdot e^{i \cdot 2 π [0.57, 0.63]}, \\ [0.25, 0.31] \cdot e^{i \cdot 2 π [0.42, 0.51]}) \end{matrix}$	$\begin{matrix} ([0.76, 0.83] \cdot e^{i \cdot 2 π [0.77, 0.90]}, \\ [0.42, 0.56] \cdot e^{i \cdot 2 π [0.13, 0.25]}) \end{matrix}$
S ₄	$\begin{matrix} ([0.81, 0.91] \cdot e^{i \cdot 2 π [0.55, 0.71]}, \\ [0.20, 0.30] \cdot e^{i \cdot 2 π [0.46, 0.58]}) \end{matrix}$	$\begin{matrix} ([0.66, 0.71] \cdot e^{i \cdot 2 π [0.62, 0.82]}, \\ [0.32, 0.53] \cdot e^{i \cdot 2 π [0.43, 0.51]}) \end{matrix}$	$\begin{matrix} ([0.56, 0.68] \cdot e^{i \cdot 2 π [0.40, 0.53]}, \\ [0.44, 0.55] \cdot e^{i \cdot 2 π [0.71, 0.82]}) \end{matrix}$	$\begin{matrix} ([0.44, 0.60] \cdot e^{i \cdot 2 π [0.80, 0.91]}, \\ [0.32, 0.57] \cdot e^{i \cdot 2 π [0.22, 0.30]}) \end{matrix}$
S ₅	$\begin{matrix} ([0.72, 0.91] \cdot e^{i \cdot 2 π [0.56, 0.76]}, \\ [0.16, 0.31] \cdot e^{i \cdot 2 π [0.22, 0.51]}) \end{matrix}$	$\begin{matrix} ([0.42, 0.65] \cdot e^{i \cdot 2 π [0.83, 0.92]}, \\ [0.14, 0.35] \cdot e^{i \cdot 2 π [0.22, 0.31]}) \end{matrix}$	$\begin{matrix} ([0.22, 0.34] \cdot e^{i \cdot 2 π [0.11, 0.34]}, \\ [0.71, 0.83] \cdot e^{i \cdot 2 π [0.63, 0.83]}) \end{matrix}$	$\begin{matrix} ([0.75, 0.80] \cdot e^{i \cdot 2 π [0.67, 0.82]}, \\ [0.43, 0.52] \cdot e^{i \cdot 2 π [0.41, 0.50]}) \end{matrix}$

6.1 Assessment steps adopting constructed AOs

Step 1. As all the attributes are the same benefit types, so the normalization is not conducted. Hence, the judgment matrix is D = R = (r_tl) _5×4.

Step 2. In light of judgment matrix D = (d_tl) _5×4 shown in Table 1 and the IVCPFEWA operator, the overall performance value ς_t can be acquired as below $\begin{matrix} ς_{1} = ([0.6574, 0.8015] \cdot e^{i \cdot 2 π [0.4449, 0.6725]}, \\ [0.3624, 0.4665] \cdot e^{i \cdot 2 π [0.3334, 0.5295]}), \end{matrix}$ $\begin{matrix} ς_{2} = ([0.5541, 0.7008] \cdot e^{i \cdot 2 π [0.6882, 0.8335]}, \\ [0.2101, 0.4505] \cdot e^{i \cdot 2 π [0.3486, 0.4609]}), \end{matrix}$ $\begin{matrix} ς_{3} = ([0.6597, 0.8466] \cdot e^{i \cdot 2 π [0.6392, 0.7351]}, \\ [0.2281, 0.3982] \cdot e^{i \cdot 2 π [0.3078, 0.4134]}), \end{matrix}$ $\begin{matrix} ς_{4} = ([0.7016, 0.8068] \cdot e^{i \cdot 2 π [0.5835, 0.7527]}, \\ [0.2838, 0.4319] \cdot e^{i \cdot 2 π [0.4617, 0.5674]}), \end{matrix}$ $\begin{matrix} ς_{5} = ([0.5849, 0.7803] \cdot e^{i \cdot 2 π [0.6395, 0.7937]}, \\ [0.2342, 0.4216] \cdot e^{i \cdot 2 π [0.2928, 0.4927]}) . \end{matrix}$

Step 3. Adopting Definition 4 to determine the score index, the result is provided as below $\begin{matrix} CS (ς_{1}) = 0.2469, CS (ς_{2}) = 0.3464, CS (ς_{3}) = \\ 0.4061, CS (ς_{4}) = 0.3120, CS (ς_{5}) = 0.3572 \end{matrix}$

then we can conclude $CS (ς_{3}) > CS (ς_{2}) > CS (ς_{5}) > CS (ς_{4}) > CS (ς_{1}),$

it follows that $S_{3} ≻ S_{2} ≻ S_{5} ≻ S_{4} ≻ S_{1},$

Obviously, S₃ is the best option.

Additionally, if the IVCPFEWA operator in Step 2 is replaced with the IVCPFEWG operator, then the following calculation procedures can be listed.

Step 1 is the same.

Step 2. With the aid of the IVCPFEWG operator, the overall performance value ς_t can be derived as below $\begin{matrix} ς_{1} = ([0.6168, 0.7849] \cdot e^{i \cdot 2 π [0.3887, 0.6049]}, \\ [0.3876, 0.4963] \cdot e^{i \cdot 2 π [0.4519, 0.6304]}), \end{matrix}$ $\begin{matrix} ς_{2} = ([0.5509, 0.6963] \cdot e^{i \cdot 2 π [0.6582, 0.8509]}, \\ [0.2625, 0.4815] \cdot e^{i \cdot 2 π [0.3765, 0.5020]}), \end{matrix}$ $\begin{matrix} ς_{3} = ([0.6350, 0.8204] \cdot e^{i \cdot 2 π [0.6315, 0.7223]}, \\ [0.2662, 0.4349] \cdot e^{i \cdot 2 π [0.3389, 0.4400]}), \end{matrix}$ $\begin{matrix} ς_{4} = ([0.6725, 0.7719] \cdot e^{i \cdot 2 π [0.5595, 0.7237]}, \\ [0.3099, 0.4641] \cdot e^{i \cdot 2 π [0.5023, 0.6108]}), \end{matrix}$ $\begin{matrix} ς_{5} = ([0.4982, 0.6889] \cdot e^{i \cdot 2 π [0.4861, 0.7121]}, \\ [0.3805, 0.5156] \cdot e^{i \cdot 2 π [0.3661, 0.5645]}) . \end{matrix}$

Step 3. Using Definition 4 to get the score index, the result is listed as below $\begin{matrix} CS (ς_{1}) = 0.1288, CS (ς_{2}) = 0.2941, CS (ς_{3}) = \\ 0.3571, CS (ς_{4}) = 0.2370, CS (ς_{5}) = 0.1507 \end{matrix}$

then we get $CS (ς_{3}) > CS (ς_{2}) > CS (ς_{4}) > CS (ς_{5}) > CS (ς_{1}),$

it indicates that $S_{3} ≻ S_{2} ≻ S_{4} ≻ S_{5} ≻ S_{1} .$

Apparently, S₃ is the finest option also, the result is agreed with IVCPFEWA operator.

6.2 Comparison with extant approaches

As IVPFNs are the particular IVCPFNs, in order to further reveal the feasibility and validity of the established approach, we compare our assessment approach with several extant aggregated approaches [16–19 , 32], respectively.

Example 6.2. [17] Assume that a committee intents to choose the suitable investment project with the least number of associated risk, where there are four options S₁, S₂, S₃ and S₄, i.e., online shopping mall S₁, online auction S₂, online bookstore S₃ and online crowd sourcing platform S₄, suppose that there are three attributes, i.e., shortage of skills and trained experts A₁, insufficient funds A₂ and poor maintenance procedure A₃. The weight vector of the attributes A₁, A₂ and A₃ is denoted as τ = (0 . 30, 0 . 45, 0 . 25) ^T. The assessment matrix in the manner of IVPFNs listed in Table 2

Table 2
The interval-valued Pythagorean fuzzy assessment matrix

A ₁ A ₂ A ₃

S ₁ ([0.1, 0.3],[0.4, 0.6]) ([0.6, 0.8],[0.2, 0.3]) ([0.2, 0.3],[0.6, 0.9])

S ₂ ([0.6, 0.7],[0.4, 0.5]) ([0.4, 0.5],[0.5, 0.8]) ([0.4, 0.5],[0.6, 0.7])

S ₃ ([0.3, 0.5],[0.6, 0.7]) ([0.4, 0.8],[0.4, 0.5]) ([0.7, 0.8],[0.4, 0.5])

S ₄ ([0.8, 0.9],[0.2, 0.3]) ([0.1, 0.3],[0.8, 0.9]) ([0.5, 0.8],[0.4, 0.5])

	A ₁	A ₂	A ₃
S ₁	([0.1, 0.3],[0.4, 0.6])	([0.6, 0.8],[0.2, 0.3])	([0.2, 0.3],[0.6, 0.9])
S ₂	([0.6, 0.7],[0.4, 0.5])	([0.4, 0.5],[0.5, 0.8])	([0.4, 0.5],[0.6, 0.7])
S ₃	([0.3, 0.5],[0.6, 0.7])	([0.4, 0.8],[0.4, 0.5])	([0.7, 0.8],[0.4, 0.5])
S ₄	([0.8, 0.9],[0.2, 0.3])	([0.1, 0.3],[0.8, 0.9])	([0.5, 0.8],[0.4, 0.5])

The following calculations are based on the assessment matrix shown in Table 2. A comparison of the ranking orders of the alternatives obtained by different methods [16 –19] is shown in Table 3, from Table 3, we can see that we employ distinct methods to obtain the distinct ranking results within same assessment values. The optimal option are consistent by using IVPFWA [16], IVPFWG [16], WIVPFEBM [17], IPFWG [18] and the proposed IVCPFEWA operator, i.e., S₂, which means that the effectiveness of our created methods. Moreover, the optimal option is S₄ by using WC-IVPFOWQA [19], which is different from our proposed method, the reason is that the decision result obtained by [19] is effected on the parameter value λ.

Table 3

A comparison of the ranking orders by different methods for Example 6.2

Methods	Overall score values of the alternatives				Ranking orders
	S ₁	S ₂	S ₃	S ₄
IVPFWA [16]	0.0115	0.0945	–0.0885	0.0603	S₂ ≻ S₄ ≻ S₁ ≻ S₃
IVPFWG [16]	0.0264	0.0906	–0.0754	0.0737	S₂ ≻ S₄ ≻ S₁ ≻ S₃
WIVPFEBM [17] (p = q = 1)	–0.5373	–0.4672	–0.5736	–0.5361	S₂ ≻ S₄ ≻ S₁ ≻ S₃
IVPFWG [18]	–0.1259	0.0667	–0.1409	–0.1789	S₂ ≻ S₁ ≻ S₃ ≻ S₄
WC-IVPFOWQA [19] (λ = 0.3)	0.0594	0.0510	–0.1390	0.2193	S₄ ≻ S₁ ≻ S₂ ≻ S₃
Proposed IVCPFEWA operator	–0.0951	0.0765	–0.1284	–0.1220	S₂ ≻ S₁ ≻ S₄ ≻ S₃

To further display that our created methods are practicable, we employ the following example.

Example 6 .3. [25] Assume a MADM issue consisting four alternatives S₁, S₂, S₃ and S₄ and four attributes A₁, A₂, A₃ and A₄. Weighting vector of attributes is assigned as τ = (0 . 3, 0 . 2, 0 . 1, 0 . 4) ^T, the assessment matrix is formed by CIVIFNs and seen in [25] (Table 4).

Table 4

A comparison of the ranking outcomes by different methods for Example 6.3

Methods	Overall score values of the alternatives				Ranking orders
	S ₁	S ₂	S ₃	S ₄
CIVIFWA [25]	0.2682	0.3222	0.3024	0.2374	S₂ ≻ S₃ ≻ S₁ ≻ S₄
CIVIFWG [25]	0.2432	0.3024	0.2736	0.2232	S₂ ≻ S₃ ≻ S₁ ≻ S₄
IVCq-ROFWA [32]	0.0935	0.1107	0.1111	0.0685	S₃ ≻ S₂ ≻ S₁ ≻ S₄
IVCq-ROFWG [32]	0.0802	0.1016	0.0987	0.0621	S₂ ≻ S₃ ≻ S₁ ≻ S₄
Proposed IVCPFEWG operator	0.0814	0.1022	0.0995	0.0625	S₂ ≻ S₃ ≻ S₁ ≻ S₄

A comparison of the ranking outcomes of the alternatives acquired by CIVIFWA [25], CIVIFWG [25], CIVq-ROFWA [32], CIVq-ROFWG [32] and our constructed IVCPFEWG operator is provided in Table 4.

From Table 4, we can realize that although the ranking order is S₃ ≻ S₂ ≻ S₁ ≻ S₄ through IVCq-ROFWA [32], the ranking results are the same by using CIVIFWA [25], CIVIFWG [25], CIVq-ROFWG [32] and our created methods. Which manifests that the presented methods are available and our established methods can deal with the MADM issues with CIVIFNs. Further visualization of the comparison results can be seen in Fig. 2.

Fig. 2

Spearman correlation: presented methods vs others.

As shown in Fig. 2, the Spearman’s coefficient of CIVq-ROFWA [32] is 0.8, while others are all 1, which further show that our created methods are effective and well.

6.3 Further comparison with extant approaches

In Examples 6.1–6.3, we have shown the effectiveness and feasibility of the proposed two MADM methods. In this part, we further show the merits of the proposed MADM methods by comparing with extant methods [16–19 , 32].

Example 6.4. We continue to visit the Example 6.1 to explain. The ranking results are described in Table 5 by using our created methods and extant methods [16–19 , 32].

Table 5
Decision outcomes with different operators

Methods Overall score values of the alternatives Ranking orders

S ₁ S ₂ S ₃ S ₄ S ₅

IVPFWA [16] Cannot be computed No

IPFWG [16] Cannot be computed No

WIVPFEBM [17] (p = q = 1) Cannot be computed No

IVPFWG [18] Cannot be computed No

WC-IVPFOWQA [19] (λ = 0.3) Cannot be computed No

CIVIFWA [25] Cannot be computed No

CIVIFWG [25] Cannot be computed No

CIVq-ROFWA [32] 0.2637 0.3538 0.4135 0.3251 0.3818 S₃ ≻ S₅ ≻ S₂ ≻ S₄ ≻ S₁

CIVq-ROFWG [32] 0.1047 0.2843 0.3485 0.2216 0.1091 S₃ ≻ S₂ ≻ S₄ ≻ S₅ ≻ S₁

Proposed IVCPFEWA operator 0.2469 0.3464 0.4061 0.3120 0.3572 S₃ ≻ S₅ ≻ S₂ ≻ S₄ ≻ S₁

Proposed IVCPFEWG operator 0.1288 0.2941 0.3571 0.2370 0.1507 S₃ ≻ S₂ ≻ S₄ ≻ S₅ ≻ S₁

Methods	Overall score values of the alternatives	Ranking orders
IVPFWA [16]	Cannot be computed	No
IPFWG [16]	Cannot be computed	No
WIVPFEBM [17] (p = q = 1)	Cannot be computed	No
IVPFWG [18]	Cannot be computed	No
WC-IVPFOWQA [19] (λ = 0.3)	Cannot be computed	No
CIVIFWA [25]	Cannot be computed	No
CIVIFWG [25]	Cannot be computed	No
CIVq-ROFWA [32]	0.2637	0.3538	0.4135	0.3251	0.3818	S₃ ≻ S₅ ≻ S₂ ≻ S₄ ≻ S₁
CIVq-ROFWG [32]	0.1047	0.2843	0.3485	0.2216	0.1091	S₃ ≻ S₂ ≻ S₄ ≻ S₅ ≻ S₁
Proposed IVCPFEWA operator	0.2469	0.3464	0.4061	0.3120	0.3572	S₃ ≻ S₅ ≻ S₂ ≻ S₄ ≻ S₁
Proposed IVCPFEWG operator	0.1288	0.2941	0.3571	0.2370	0.1507	S₃ ≻ S₂ ≻ S₄ ≻ S₅ ≻ S₁

From Table 5, we can perceive that IVPFWA [16], IVPFWG [16], WIVPFEBM [17], IPFWG [18], WC-IVPFOWQA [19], CIVIFWA [25] and CIVIFWG [25] are all failed to settle Example 6.4. The reason is that the assessment values are portrayed by IVPFNs in [16 –19] and portrayed by CIVIFNs in [25] respectively. These approaches [16–19 , 25] can not tackle interval-valued complex Pythagorean fuzzy information. Thus, the application range of our proposed methods is wider than the current methods [16–19 , 25]. Additional, we have noticed that the best and worst options are same respectively by employing methods [32] and our established techniques, but the following Fig. 3 shows that our methods have high discrimination than methods [32] from the point of view of score value. Therefore, the proposed method is more scientific and reasonable in solving MADM problems than some existing methods.

Fig. 3

Score values with different operators.

7 Conclusions

The main contributions of this work are provided as below:

We expanded the Einstein operations to IVCPFNs and coined four basic operations and corresponding operation properties for IVCPFNs.

We constructed the aggregation operators that is IVCPFEWA and IVCPFEWG, and also proved their basic properties.

We designed a new approach to model MADM issue by employing the found aggregation operators.

We provided a real word issue concerning on the evaluation of the personal risk of mental patients to explain the application and efficiency of the propounded approach under IVCPFNs, and the comparative discussion is performed with extant decision techniques [16–19 , 32]. Therefore, we can conclude that the constructed method exhibits a valid and successful manner to manage the complex uncertainty under realistic decision circumstances.

For coming research, we shall extend the propounded approach to tackle the MADM issues under diverse indeterminacy circumstances in which the assessment information are portrayed by interval-valued q-rung orthopair fuzzy sets [37], linguistic terms with weakened hedges [38, 39], neutrosophic sets [40, 41], and so on [42 –51]. Besides, we also extend it to other diverse application domains, such as risk evaluation [52], smart phone selection [53] and brain hemorrhage patients evaluation [54].

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China [grant number 62006155], the Humanities and Social Sciences of Ministry of Education in China [grant number 18YJCZH054] and the Scientific Research Funds Project of Liaoning Province Education Department (grant numbers. LJ2019QL014; LJ2020JCL018).

References

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