Enhancing Interval-Valued Pythagorean Fuzzy Decision-Making through Dombi-Based Aggregation Operators

Abstract

The success of any endeavor or process is heavily contingent on the ability to reconcile and satisfy balance requirements, which are often characterized by symmetry considerations. In practical applications, the primary goal of decision-making processes is to efficiently manage the symmetry or asymmetry that exists within different sources of information. In order to address this challenge, the primary aim of this study is to introduce novel Dombi operation concepts that are formulated within the framework of interval-valued Pythagorean fuzzy aggregation operators. In this study, an updated score function is presented to resolve the deficiency of the current score function in an interval-valued Pythagorean fuzzy environment. The concept of Dombi operations is used to introduce some interval-valued Pythagorean fuzzy aggregation operators, including the interval-valued Pythagorean fuzzy Dombi weighted arithmetic (IVPFDWA) operator, the interval-valued Pythagorean fuzzy Dombi ordered weighted arithmetic (IVPFDOWA) operator, the interval-valued Pythagorean fuzzy Dombi weighted geometric (IVPFDWG) operator, and the interval-valued Pythagorean fuzzy Dombi ordered weighted geometric (IVPFDOWG) operator. Moreover, the study investigates many important properties of these operators that provide new semantic meaning to the evaluation. In addition, the suggested score function and newly derived interval-valued Pythagorean fuzzy Dombi aggregation (IVPFDA) operators are successfully employed to select a subject expert in a certain institution. The proposed approach is demonstrated to be successful through empirical validation. Lastly, a comparative study is conducted to demonstrate the validity and applicability of the suggested approaches in comparison with current techniques. This research contributes to the ongoing efforts to advance the field of evaluation and decision-making by providing novel and effective tools and techniques.

1. Introduction

Selecting the best possible option from a set of candidates based on a number of criteria or qualities is the goal of multi-criteria decision-making. Generally, it is assumed that all data used to evaluate an option in terms of its qualities and the relative weights of those features are presented in the form of crisp numbers. However, in many real-life situations, there are problems taken into account where the objectives and constraints are conventionally vague and imprecise in nature. To cope with this problem, Zadeh [1] introduced the concept of fuzzy sets (FSs) as a generalization of crisp sets.

In fuzzy sets, information about the degree to which certain criteria are met is obtained in the form of membership functions, whose complements are taken as insufficient degrees. In 1970, Bellman and Zadeh [2] discussed the multiple attribute decision-making process in the FS theory. In addition, the theory of solving decision-making problems under fuzzy environments was developed in [3,4]. In 1982, Dombi [5] discussed a general class of fuzzy operators. The theory of intuitionistic fuzzy sets has been widely adopted and has a significant impact on various disciplines, including decision-making, engineering, information technology, pattern recognition, and medical diagnostics. The concept of intuitionistic fuzzy sets, which represents a generalization of fuzzy sets, was first proposed by Atanassov in 1986 [6]. It meets the condition that the sum of the membership and non-membership degrees of a given element must not exceed unity.

In 1986, Turksen [7] put forward the idea of the interval-valued fuzzy set (IVFS) as a modification to fuzzy sets. In real-life situations, where information is often inadequate and imprecise, decision-makers may find it challenging to express their opinions with precise numerical values. To address this challenge, Atanassov [8] introduced the IV IFS concept in 1989. In this approach, the membership and non-membership degrees are represented by an interval within [0, 1] rather than a precise numerical value. In 2000, De et al. [9] introduced three basic operations for an IFS, namely concentration, dilation, and normalization. The intuitionistic fuzzy set theory has attracted more attention from many researchers since its inception. However, later on, many deficiencies were found in this technique, which led the mathematician, Atanassov, to explore some higher order fuzzy sets.

The significance of the concepts of IFSs and IVIFSs is quite evident as many mathematicians have introduced different types of aggregation operators and information measures based on these sets, which have been effectively utilized in addressing MCDM problems in various circumstances [10,11,12,13,14,15]. Despite the numerous advantages of these theories, there still exist real-world situations that cannot be addressed by the above-mentioned strategies. For instance, if a decision-maker assigns a membership degree of 0.7 and a non-membership degree of 0.5 to an element, their sum would exceed 1. To broaden the scope of membership and non-membership degrees, Yager [16] introduced the concept of Pythagorean Fuzzy Sets (PFSs), where the sum of the squares of the membership and non-membership is less than or equal to 1.

In 2014, Zhang and Xu [17] discussed decision-making using PFSs. In addition, more develop methods in the framework of the interval-valued intuitionistic fuzzy (IVIF) environment can be found in [18,19]. In 2017, Chen and Huang [20] discussed decision-making based on particle swarm optimization techniques. In 2016, Zhang [21] presented the concept of the IV Pythagorean fuzzy set. Moreover, one can view the recent developments of IVPFSs in [22,23,24,25,26,27,28,29,30,31,32,33]. In 2016, Peng and Yang [34] discussed the fundamental properties of aggregation operators of IVPFSs. Furthermore, different Dombi Operators in various sets were discussed in [35,36,37,38,39,40]. In addition, many useful strategies were invented to address the issue of energy crises in [41,42,43]

Dombi operators can take multi-purpose aggregation, decision-making abilities, and operational characteristics and turn them into a highly adaptable tool for gathering imprecise information. The information is converted into a single value with the help of the averaging operators. Dombi operators are incredibly adaptable when it comes to operational conditions, and they are also highly effective at solving decision-making issues. The main benefit of our approach is that it can handle the interrelationship between the arguments, which the other approaches cannot do. Our approach is therefore more versatile. Furthermore, it is important to note that the existing techniques lack the ability to dynamically adjust the parameter in accordance with the decision-makers’ risk aversion, which makes the MADM solution difficult to implement in real life. However, the techniques presented in this article are very capable of addressing this weakness in this situation.

Our research work aims to achieve several primary objectives, which are as follows:

• Develop an updated score function that overcomes the deficiencies of existing score functions in IVPF environment. This will involve the integration of advanced mathematical and statistical techniques to create a more robust and accurate scoring system.
• Formulate fundamental Dombi operations for IVPFSs. This will involve the development of mathematical models that describe the relationships between different elements of IVPFSs, allowing for more accurate analysis and prediction of outcomes.
• Initiate the study of IVPFD aggregation operators. This will involve exploring the ways in which different IVPFD operators can be combined to create more effective aggregation methods for IVPFSs.
• Prove many key properties of the newly defined operators. This will involve rigorous mathematical analysis and formal proofs aimed at demonstrating the validity and effectiveness of the proposed operators.
• Present an algorithm to solve Multiple Attribute Decision-Making (MADM) problems using IVPFD aggregation operators. This will involve developing a step-by-step process for using the new operators to analyze and evaluate complex decision-making problems.
• Select the best subject expert in a certain institution using the newly suggested technique. This will involve applying the proposed algorithm to real-world scenarios, such as selecting the most qualified candidate for a job or identifying the most suitable expert for a specific project.
• Present a comparative analysis to show the validity of the proposed technique compared with existing strategies. This will involve testing the effectiveness of the proposed algorithm against existing methods, using real-world data sets and scenarios.

After a brief discussion of IVPFSs, the rest of the work is organized as: in Section 2, some basic definitions are defined to understand the novelty of the work presented in this article. In Section 3, the deficiency of existing score functions to solve MADM problems is discussed and an improved score function is determined to counter this problem under the IVPF environment. In Section 4, some Dombi aggregation operators for these sets are developed. In Section 5, a numerical example is given to show the validity of this new approach. In addition, a comparative analysis is presented to depict the validity and feasibility of this new strategy with existing methods. Finally, some concrete conclusions about the paper are summarized in Section 6.

2. Preliminaries

In this section, we briefly review some basic concepts, properties, operations, and magnitude comparison methods of IVPFSs over the non-empty universal set X that are important to the achievement of this study.

Definition 1

[7]. An IVFS A defined on X is given by A = {〈x, [µ_A^L(x), µ_A^U(x)]〉: x ∈ X}, where 0 ≤ µ_A^L(x) ≤ µ_A^U(x) ≤ 1.

Definition 2

[14]. Let X be a classical set. A PFS A on X is represented as: A = {〈x,µ_A(x),ν_A(x)〉: x ∈ X}, where µ_A: X→[0,1] and ν_A: X→[0,1] denote the membership and non-membership functions, respectively, that satisfy the condition $0\le {\mu }_A^2\left(x\right)+{\nu }_A^2\left(x\right)\le 1$. The hesitancy margin of the PFS A is defined as: π_A(x) = $\sqrt{1-{\mu }_A^2\left(x\right)-{\nu }_A^2\left(x\right)}$.

Definition 3

[18]. An IVPFS A on X is defined as: A = {〈x, [µ_A^L(x), µ_A^U(x)], [ν_A^L(x), ν_A^U(x)]〉: x ∈ X}, where [µ_A^L(x), µ_A^U(x)] and [ν_A^L(x), ν_A^U(x)], respectively, represent the membership and non-membership degree of A that are able to satisfy the condition 1 ≥ µ_A^U(x) ≥ µ_A^L(x) ≥ 0 and 0 ≤ ν_A^L(x) ≤ ν_A^U(x) ≤ 1. Moreover, 0 ≤ (µ_A^L(x))² + (ν_A^L(x))² ≤ 1 and 0 ≤ (µ_A^U(x))² + (ν_A^U(x))² ≤ 1. The hesitancy degree of the IVPFS A is defined as:

$$\begin{array}{c}{\pi }_A\left(x\right)=\left[{{\pi }_A}^L\right(x),{{\pi }_A}^U(x\left)\right]\\ =\left[\sqrt{1-{\left({\mu }_A^L\left(x\right)\right)}^2-{\left({\nu }_A^L\left(x\right)\right)}^2}, \sqrt{1-{\left({\mu }_A^U\left(x\right)\right)}^2-{\left({\nu }_A^U\left(x\right)\right)}^2}\right]\end{array}$$

Definition 4

[35]. Consider the three IVPF g = 〈[${\mathsf{\mu }}_g^L$, ${\mathsf{\mu }}_g^U$],[${ \nu }_g^L$, ${\nu }_g^U$]〉, g₁ = 〈[${\mathsf{\mu }}_{g1}^L$, ${\mathsf{\mu }}_{g1}^U$],[${ \nu }_{g1}^L$, ${\nu }_{g1}^U$]〉 and g₂ = 〈[${\mathsf{\mu }}_{g2}^L$, ${\mathsf{\mu }}_{g2}^U$],[${ \nu }_{g2}^L$, ${\nu }_{g2}^U$]〉. The basic operations on these numbers are defined in the subsequent way:

i. 

g₁∪g₂ = 〈[max{${\mu }_{g1}^L$, ${\mu }_{g2}^L$,}, max{${\mu }_{g1}^U$, ${\mu }_{g2}^U$,}], [min{${\nu }_{g1}^L$, ${\nu }_{g2}^L$}, min{${\nu }_{g1}^U$,${\nu }_{g2}^U$,}]〉

ii. 

g₁∩ g₂ = 〈[min{${\mu }_{g1}^L$, ${\mu }_{g2}^L$,}, min{${\mu }_{g1}^U$, ${\mu }_{g2}^U$,}], [max{${\nu }_{g1}^L$, ${\nu }_{g2}^L$}, max{${\nu }_{g1}^U$,${\nu }_{g2}^U$,}] 〉

iii. 

g₁⨁g₂ $=\langle \begin{array}{c}\left[\sqrt{{\left({\mu }_{g_1}^L\right)}^2+{\left({\mu }_{g_2}^L\right)}^2-{\left({\mu }_{g_1}^L\right)}^2{\left({\mu }_{g_2}^L\right)}^2},\sqrt{{\left({\mu }_{g_1}^U\right)}^2+{\left({\mu }_{g_2}^U\right)}^2-{\left({\mu }_{g_1}^U\right)}^2{\left({\mu }_{g_2}^U\right)}^2}\hspace{0.17em}\right],\\ \left[{\nu }_{g_1}^L{\nu }_{g_2}^L,{\nu }_{g_1}^U{\nu }_{g_2}^U\right]\end{array}\rangle$

iv.  

g₁⨂g₂ = $\langle \begin{array}{c}\left[{\mu }_{g_1}^L{\mu }_{g_2}^L,{\mu }_{g_1}^U{\mu }_{g_2}^U\right],\\ \left[\sqrt{{\left({\nu }_{g_1}^L\right)}^2+{\left({\nu }_{g_2}^L\right)}^2-{\left({\nu }_{g_1}^L\right)}^2{\left({\nu }_{g_2}^L\right)}^2},\sqrt{{\left({\nu }_{g_1}^U\right)}^2+{\left({\nu }_{g_2}^U\right)}^2-{\left({\nu }_{g_1}^U\right)}^2{\left({\nu }_{g_2}^U\right)}^2}\hspace{0.17em}\right]\end{array}\rangle$

v.  

χg =  $\langle \left[\sqrt{1-{\left(1-{\left({\mu }_g^L\right)}^2\right)}^{\mathcal{X}}} ,\sqrt{1-{\left(1-{\left({\mu }_g^U\right)}^2\right)}^{\mathcal{X}}} \right],\left[{\left({\nu }_g^L\right)}^{\mathcal{X}},{\left({\nu }_g^U\right)}^{\mathcal{X}}\right]\rangle$, for all χ > 0.

vi. 

g^χ = $\langle \left[{\left({\mu }_g^L\right)}^{\mathcal{X}},{\left({\mu }_g^U\right)}^{\mathcal{X}}\right],\left[\sqrt{1-{\left(1-{\left({\nu }_g^L\right)}^2\right)}^{\mathcal{X}}} ,\sqrt{1-{\left(1-{\left({\nu }_g^U\right)}^2\right)}^{\mathcal{X}}} \right]\rangle$, for all χ > 0.

vii. 

g^C = $\langle \left[{\nu }_g^L,{\nu }_g^U\right],\left[{\mu }_g^L,{\mu }_g^U\right]\rangle .$

Some certain types of triangular norms and conorms are discussed in the following definition.

Definition 5

[5]. Let ã and $\tilde{b}$ be any two real numbers. The Dombi t-norms and Dombi t-conorms are described in the following way:

i. 

Dom(ã, $\tilde{b}$) = $\frac{1}{1+{\left\{\left\{{\left(\frac{1-ã}{ã}\right)}^{\mathrm{Ќ}}+{\left(\frac{1-\tilde{b}}{\tilde{b}}\right)}^{\mathrm{Ќ}}\right\}\right\}}^{\frac{1}{\mathrm{Ќ}}}}$

ii. 

Dom^C(ã, $\tilde{b}$) = $\frac{1}{1+{\left\{\left\{{\left(\frac{ã}{1-ã}\right)}^{\mathrm{Ќ}}+{\left(\frac{\tilde{b}}{1-\tilde{b}}\right)}^{\mathrm{Ќ}}\right\}\right\}}^{\frac{1}{\mathrm{Ќ}}}}$

where $\mathsf{Ќ}\ge 1$ and $\left(\mathsf{ã}, \tilde{b}\right)\in \left[0,1\left]\times \right[0,1\right].$ In the above discussion, (i) represents the Dombi product and (ii) represents the Dombi sum.

Definition 6

[35]. Let g_ċ = (µ_ċ, ν_ċ) (ċ = 1, 2, 3, …, j) be j number of PFEs. A Pythagorean fuzzy Dombi weighted averaging (PFDWA) operator is characterized by a function PFDWA: PFE^j → PFE such that

$$\begin{array}{c}{\mathit{PFDWA}}_{ꝕ}(g_1,g_2,\dots ,g_j)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{g}_{\mathsf{ċ}}\right)\\ =\left(\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\mu }_{\mathsf{ċ}}^2}{1-{\mu }_{\mathsf{ċ}}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} , \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\nu }_{\mathsf{ċ}}^2}{{\nu }_{\mathsf{ċ}}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right)\end{array}$$

where $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ is the weight vector of $g_{\mathsf{ċ}}\left(\mathsf{ċ}=1,2,3,\dots ,j\right)$ with ${ꝕ}_{\mathsf{ċ}}>0 ,$ $\sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1$.

Definition 7

[35]. Let g_ċ = (µ_ċ, ν_ċ) (ċ = 1, 2, 3, …, j) be j number of PFEs. A PFD ordered weighted averaging operator of dimension j is characterized by a function PFDOWA: PFE^j → PFE with the associated weighted vector ꝕ = ${\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$such that ${ꝕ}_{\mathsf{ċ}}>0, \sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1$. Then,

$$\begin{array}{c}{\mathit{PFDOWA}}_{ꝕ}(g_1,g_2,\dots ,g_j)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{g}_{\rho \left(\mathsf{ċ}\right)}\right)\\ =\left(\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\mu }_{\rho \left(\mathsf{ċ}\right)}^2}{1-{\mu }_{\rho \left(\mathsf{ċ}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\nu }_{\rho \left(\mathsf{ċ}\right)}^2}{{\nu }_{\rho \left(\mathsf{ċ}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right)\end{array}$$

where ($\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(j\right)$) represents the permutations of $\mathsf{ċ}=1,2,3,\dots ,j$, and $g_{\rho \left(\mathsf{ċ}-1\right)}\ge$ $g_{\rho \left(\mathsf{ċ}\right)}\forall$ $\mathsf{ċ}=1,2,\dots ,j$.

Definition 8

[35]. Let g_ċ = (µ_ċ, ν_ċ) (ċ = 1, 2, 3, …, j) be j number of PFEs. A PFD weighted geometric operator is characterized by a function PFDWG: PFE^j→PFE such that

$$\begin{array}{c}{\mathit{PFDWG}}_{ꝕ}(g_1,g_2,\dots ,g_j)={\otimes }_{\mathsf{ċ}=1}^j{\left(g_{\mathsf{ċ}}\right)}^{{ꝕ}_{\mathsf{ċ}}}\\ =\left(\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\mu }_{\mathsf{ċ}}^2}{{\mu }_{\mathsf{ċ}}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} , \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\nu }_{\mathsf{ċ}}^2}{1-{\nu }_{\mathsf{ċ}}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right)\end{array}$$

where $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ is the weight vector of $g_{\mathsf{ċ}}\left(\mathsf{ċ}=1,2,3,\dots ,j\right)$ with ${ꝕ}_{\mathsf{ċ}}>0, \sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1$.

Definition 9

[34]. Let g_ċ = (µ_ċ, ν_ċ) (ċ = 1, 2, 3, …, j) be j number of PFEs. A PFD ordered weighted geometric operator of dimension j is characterized by a function PFDOWG: PFE^j →PFE with the associated weighted vector $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ such that ${ꝕ}_{\mathsf{ċ}}>0 ,$ $\sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1$. Then,

$$\begin{array}{c}{\mathit{PFDOWG}}_{ꝕ}(g_1,g_2,\dots ,g_{\mathrm{j}})={\otimes }_{\mathsf{ċ}=1}^j\left(g_{\rho \left(\mathsf{ċ}\right)}{}^{{ꝕ}_{\mathsf{ċ}}}\right)\\ =\left(\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\mu }_{\rho \left(\mathsf{ċ}\right)}^2}{{\mu }_{\rho \left(\mathsf{ċ}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\nu }_{\rho \left(\mathsf{ċ}\right)}^2}{1-{\nu }_{\rho \left(\mathsf{ċ}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right)\end{array}$$

where $\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(j\right)\right)$ represents the permutations of $\mathsf{ċ}=1,2,3,\dots ,j$, and $g_{\rho \left(\mathsf{ċ}-1\right)}\ge$ $g_{\rho \left(\mathsf{ċ}\right)} \forall$ $\mathsf{ċ}=1,2,\dots ,j$.

Definition 10

[27]. The score function for IVPF number $\alpha =\langle \left[a,b\right], \left[c,d\right]\rangle$ is defined in the following way:

$$S\left(\alpha \right)=\frac{\left(a^2-c^2\right)\left(1+\sqrt{1-a^2-c^2}\right)+\left(b^2-d^2\right)\left(1+\sqrt{1-b^2-d^2}\right)}{4}$$

where $S\left(\alpha \right)\in \left[-1,1\right].$ Particulary, if $S\left(\alpha \right)=1$, then $\alpha$ is the largest IVPFN, i.e., $\langle \left[1,1\right],\left[0,0\right]\rangle$; if $S\left(\alpha \right)$ = −1, then $\alpha$ is the least IVPFN, i.e., $\langle \left[0,0\right],\left[1,1\right]\rangle$. Moreover, the score function satisfies the following properties for any two IVPFNs $\alpha$ and $\beta$

• If $S\left(\alpha \right)<S\left(\beta \right)$, then $\alpha \prec \beta$
• If $S\left(\alpha \right)>S\left(\beta \right),$ then $\alpha \succ \beta$
• If $S\left(\alpha \right)=S\left(\beta \right)$, then $\alpha \sim \beta$

3. Deficiency of the Existing Score Function of IVPFS and Its Improvement

In this section, we provide an example that indicates the deficiencies in the score function of the IVPFS developed in [28] and improve it in the subsequent discussion.

Example 1.

Let ${\varsigma }_1=\langle \left[0.4,0.7\right],\left[0.4,0.7\right]\rangle$ and ${\varsigma }_{2 }$ = $\langle \left[0.3,0.4\right],\left[0.3,0.4\right]\rangle$ be any two IVPFNs. Then, by applying Definition 10 on ${\varsigma }_1$ and ${\varsigma }_2$, we obtain

$$S\left({\varsigma }_1\right)=S\left({\varsigma }_2\right)=0$$

Then, in view of the property (iii) of Definition 10, we have ${\varsigma }_1\sim {\varsigma }_2$, but ${\varsigma }_1\ne {\varsigma }_2$.

This indicates the deficiency in the score function under consideration. The above discussion leads us to improve this score function in the following definition.

Definition 11.

Consider the IVPFN $\alpha =\langle \left[a,b\right],\left[c,d\right]\rangle$. The improved score function $S\left(\alpha \right)$is defined for $\alpha$ as:

$$\begin{gathered} S\left(\alpha \right)=\frac{1}{4} (\left(a^2-c^2+\frac{1}{2}\right)\left(1+\sqrt{1-a^2-c^2}\right) \\ +\left(b^2-d^2+\frac{1}{3}\right)\left(1+\sqrt{1-b^2-d^2}\right)) \end{gathered}$$

(1)

where $h(\alpha$) is the range of the score function and −0.29 $\le h\left(\alpha \right)\le 0.58.$

Moreover, the above score function satisfies the following comparison law for any two IVPFNs $\alpha$ and $\beta$

• If $S\left(\alpha \right)<S\left(\beta \right)$, then $\alpha \prec \beta$
• If $S\left(\alpha \right)>S\left(\beta \right)$, then $\alpha \succ \beta$
• If $S\left(\alpha \right)=S\left(\beta \right)$, then $\alpha \sim \beta$

Consider the following example to demonstrate the effectiveness of the score function proposed for IVPFN.

Example 2.

The application of the proposed score function $S\left(\alpha \right)$ in Equation (1) of Example 1 gives that $S\left({\varsigma }_1\right)=0.392$ and $S\left({\varsigma }_2\right)=0.386$. Thus, in view of property ii of Definition 11, we have ${\varsigma }_1\succ$ ${\varsigma }_2$. This fact suggests that the alternative ${\varsigma }_1$ is better than the alternative ${\varsigma }_2.$

So, we conclude that the suggested score function offers a more appropriate and efficient approach for decision analysis compared with the current score function, which has proven to be inadequate.

4. Dombi Operations on Interval-Valued Pythagorean Fuzzy Numbers

In this section, we present the Dombi operations in the framework of the IVPF environment.

Definition 12.

Let 1 $\le \vartheta$, 0 $\le \mathcal{X}$ and $g_1=\langle \left[{\mathsf{\mu }}_{g_1}^L,{\mathsf{\mu }}_{g_1}^U\right],\left[{\nu }_{g_1}^L,{\nu }_{g_1}^U\right]\rangle$ and $g_2=\langle \left[{\mathsf{\mu }}_{g_2}^L,{\mathsf{\mu }}_{g_2}^U\right],\left[{\nu }_{g_2}^L,{\nu }_{g_2}^U\right]\rangle$ be any two IVPFEs. The Dombi operations of t-norms and t-conorms of IVPFEs are explained in the following way:

i. 

$g_1\oplus g_{2 }=\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}{1-{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}\right)}^{\vartheta }+{\left(\frac{{\left({\mathsf{\mu }}_{g_2}^L\right)}^2}{1-{\left({\mathsf{\mu }}_{g_2}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}{1-{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}\right)}^{\vartheta }+{\left(\frac{{\left({\mathsf{\mu }}_{g_2}^U\right)}^2}{1-{\left({\mathsf{\mu }}_{g_2}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right],\\ \left[\sqrt{\frac{1}{1+{\left\{{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_1}^L\right)}^2}\right)}^{\vartheta }+{\left(\frac{1-{\left({\nu }_{g_2}^L\right)}^2}{{\left({\nu }_{g_2}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_1}^U\right)}^2}\right)}^{\vartheta }+{\left(\frac{1-{\left({\nu }_{g_2}^U\right)}^2}{{\left({\nu }_{g_2}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right)$

ii. 

$g_1\otimes g_{2 }=\left(\begin{array}{c}\left[\sqrt{\frac{1}{1+{\left\{{\left(\frac{1-{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}{{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}\right)}^{\vartheta }+{\left(\frac{1-{\left({\mathsf{\mu }}_{g_2}^L\right)}^2}{{\left({\mathsf{\mu }}_{g_2}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\left(\frac{1-{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}{{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}\right)}^{\vartheta }+{\left(\frac{1-{\left({\mathsf{\mu }}_{g_2}^U\right)}^2}{{\left({\mathsf{\mu }}_{g_2}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right],\\ \left[\sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{\left({\nu }_{g_1}^L\right)}^2}{1-{\left({\nu }_{g_1}^L\right)}^2}\right)}^{\vartheta }+{\left(\frac{1-{\left({\nu }_{g_2}^L\right)}^2}{{\left({\nu }_{g_2}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\left(\frac{{\left({\nu }_{g_1}^U\right)}^2}{1-{\left({\nu }_{g_1}^U\right)}^2}\right)}^{\vartheta }+{\left(\frac{{\left({\nu }_{g_2}^U\right)}^2}{1-{\left({\nu }_{g_2}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right)$

iii. 

$\mathcal{X}. g_1=\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}{1-{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}{1-{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right],\\ \left[\sqrt{\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_1}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_1}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right)$

iv. 

$g_1{}^{\mathcal{X}}=\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{1-{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}{{\left({\mathsf{\mu }}_{g_1}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{1-{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}{{\left({\mathsf{\mu }}_{g_1}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right],\\ \left[\sqrt{\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{{\left({\nu }_{g_1}^L\right)}^2}{1-{\left({\nu }_{g_1}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{\mathcal{X}{\left(\frac{{\left({\nu }_{g_1}^U\right)}^2}{1-{\left({\nu }_{g_1}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right)$

In the following definition, we propose a Dombi arithmetic aggregation operator with IVPFEs, namely, the IVPFDWA operator.

Definition 13.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$) be $j$ number of IVPFEs. The IVPFDWA operator is characterized by a function $IVPFDWA: IVPFE^j$\to$IVPFE such that

$$IVPFDW{A}_{ꝕ}\left(g_1,g_2,g_3,\dots ,g_j\right)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{g}_{\mathsf{ċ}}\right)$$

where $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ is the weight vector of $g_{\mathsf{ċ}}\left(\mathsf{ċ}=1,2,3,\dots ,j\right)$ with ${ꝕ}_{\mathsf{ċ}}>0$ and $\sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$.

Theorem 1.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$ be $j$ number of IVPFEs. The aggregated value of these IVPFEs in the framework of the IVPFDWA operator is an IVPFE and is determined in the following way:

$$\begin{array}{c}IVPFDW{A}_{ꝕ}\left(g_1,g_2,g_3,\dots ,g_j\right)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{g}_{\mathsf{ċ}}\right)\\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right],\\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right)\end{array}$$

(2)

where $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ is the weight vector of $g_{\mathsf{ċ}}$ and 0$\le {ꝕ}_{\mathsf{ċ}}, \sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$.

Proof. 

The proof of this result is established through the use of mathematical induction. □

The application of Definition 12 for $\mathsf{ċ}$ = 2 gives the following outcome:

$$\begin{array}{c}IVPFDW{A}_{\mathsf{ꝕ}}\left(g_1,g_2\right)={ꝕ}_1{g}_1\oplus {ꝕ}_2{g}_2\\ =\left(\begin{array}{c}\left[\begin{array}{c}\sqrt{1-\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{{\left({\mu }_{g_1}^L\right)}^2}{1-{\left({\mu }_{g_1}^L\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{{\left({\mu }_{g_2}^L\right)}^2}{1-{\left({\mu }_{g_2}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \\ \sqrt{1-\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{{\left({\mu }_{g_1}^U\right)}^2}{1-{\left({\mu }_{g_1}^U\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{{\left({\mu }_{g_2}^U\right)}^2}{1-{\left({\mu }_{g_2}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \end{array}\right],\\ \left[\begin{array}{c}\sqrt{\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_1}^L\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{1-{\left({\nu }_{g_2}^L\right)}^2}{{\left({\nu }_{g_2}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\\ \sqrt{\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_1}^U\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{1-{\left({\nu }_{g_2}^U\right)}^2}{{\left({\nu }_{g_2}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\end{array}\right]\\ \end{array}\right)\end{array}$$

This means that

$$\begin{array}{l}IVPFDW{A}_{\mathsf{ꝕ}}\left(g_1,g_2\right)\\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right],\\ \\ \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right)\end{array}$$

Thus, Equation (2) holds for $\mathsf{ċ}$ = 2.

Suppose that Equation (2) holds for $\mathsf{ċ}$ = p. Therefore, we have

$$\begin{array}{c}IVPFDW{A}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_p\right)={\oplus }_{\mathsf{ċ}=1}^p\left({ꝕ}_{\mathsf{ċ}}{g}_{\mathsf{ċ}}\right)\\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right)\end{array}$$

Moreover, for ċ = p + 1, we have

$$\begin{array}{c}IVPFDW{A}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_{p, }{g}_{p+1}\right)={\oplus }_{\mathsf{ċ}=1}^p\left({ꝕ}_{\mathsf{ċ}}{g}_{\mathsf{ċ}}\right)\oplus \left({ꝕ}_{p+1}{g}_{p+1}\right)\\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right)\\ \oplus \left(\begin{array}{c}\left[\begin{array}{c}\sqrt{1-\frac{1}{1+{\left\{{ꝕ}_{p+1}{\left(\frac{{\left({\mu }_{g_{p+1}}^L\right)}^2}{1-{\left({\mu }_{g_{p+1}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\sqrt{1-\frac{1}{1+{\left\{{ꝕ}_{p+1}{\left(\frac{{\left({\mu }_{g_{p+1}}^U\right)}^2}{1-{\left({\mu }_{g_{p+1}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \\ \end{array}\right], \\ \left[\sqrt{\frac{1}{1+{\left\{{ꝕ}_{p+1}{\left(\frac{1-{\left({\nu }_{p+1}^L\right)}^2}{{\left({\nu }_{g_{p+1}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{ꝕ}_{p+1}{\left(\frac{1-{\left({\nu }_{g_{p+1}}^U\right)}^2}{{\left({\nu }_{g_{p+1}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right)\end{array}$$

This means that

$$\begin{array}{l}IVPFDW{A}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_{p, }{g}_{p+1}\right)\\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{p+1}{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{p+1}{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{p+1}{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{p+1}{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right)\end{array}$$

Thus, Equation (2) is true for $\mathsf{ċ}=p+1.$ Hence, we conclude that Equation (2) is true for any $\mathsf{ċ}\in N.$

The following example describes the above-stated fact.

Example 3.

Consider the IVPFEs ${\varsigma }_1$ = $\langle \left[0.1,0.5\right],\left[0.4,0.6\right]\rangle ,{ \varsigma }_2=\langle \left[0.3,0.4],[0.5,0.7\right]\rangle$, and ${\varsigma }_3=\langle \left[0.0,0.3],[0.1,0.6\right]\rangle$. Let $ꝕ=$(0.2,0.5,0.3)^T be the weighted vector of ${\varsigma }_{\mathsf{ċ} }\left(\mathsf{ċ}=1,2,3\right)$ and $\vartheta =4$. Then,

$$\begin{gathered} IVPFDW{A}_{\mathsf{ꝕ}}\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)={\oplus }_{\mathsf{ċ}=1}^3\left({ꝕ}_{\mathsf{ċ}}{\varsigma }_{\mathsf{ċ}}\right) \\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^L\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_1}^U\right)}^2}{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right) \end{gathered}$$

Consequently,

$$IVPFDW{A}_{ꝕ}\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)=\langle \left[0.28,0.39],[0.12,0.62\right]\rangle$$

In the following definition, we propose a Dombi arithmetic aggregation operator with IVPFEs, namely, the IVPFDOWA operator.

Definition 14.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$ be$j$ number of IVPFEs. A IVPFDOWA operator of dimension j is characterized by a function.

IVPFDOWA: IVPFE^j $\to$ IVPFE with the associated weighted vector $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ such that with ${ꝕ}_{\mathsf{ċ}}>0$ and $\sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$. Therefore, $IVPFDOW{A}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_j\right)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{g}_{\rho \left(\mathsf{ċ}\right)}\right)$, where ($\rho \left(1\right),\rho \left(2\right),\rho \left(3\right),\dots ,\rho \left(j\right)$) represents the permutations of $1,2,3,\dots ,j$, respectively, and $g_{\rho \left(\mathsf{ċ}-1\right)}\ge g_{\rho \left(\mathsf{ċ}\right)}$ $\forall$ $\mathsf{ċ}=1,2,\dots ,j$.

Theorem 2.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$ be the j number of IVPFEs. Then, the aggregated value of these IVPFEs in the framework of IVPFDOWA operator is an IVPFE and is determined in the following way:

$$\begin{gathered} IVPFDOW{A}_{ꝕ}\left(g_1,g_2,g_3,\dots ,g_j\right)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{g}_{\rho \left(\mathsf{ċ}\right)}\right) \\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right), \end{gathered}$$

where $\mathsf{ꝕ}={\left({\mathsf{ꝕ}}_1,{\mathsf{ꝕ}}_2,{\mathsf{ꝕ}}_3,\dots ,{\mathsf{ꝕ}}_j\right)}^T$ is the weighted vector of $g_{\mathsf{ċ}}$ with 0 $\le {\mathsf{ꝕ}}_{\mathsf{ċ}}, \sum _{\mathsf{ċ}=1}^j{\mathsf{ꝕ}}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1.$ Moreover, ($\rho \left(1\right),\rho \left(2\right),\rho \left(3\right),\dots ,\rho \left(j\right)$) represent the permutations of 1, 2, 3, …, j, respectively, and $g_{\rho \left(\mathsf{ċ}-1\right)}\ge g_{\rho \left(\mathsf{ċ}\right)}$ $\forall$ $\mathsf{ċ}=1,2,\dots ,j.$

Proof. 

The proof of this theorem is analogous to Theorem 1. □

The following example describes the above-stated fact.

Example 4.

Three researchers ${\varsigma }_1,{\varsigma }_2,{\varsigma }_3$ of mathematics want to estimate the performance of a student. The estimated values from researchers with respect to research work for a student P specified by IVPF information such as:

${\varsigma }_1=\langle \left[0.1,0.3],[0.2,0.5\right]\rangle ,{\varsigma }_2=\langle \left[0.3,0.4],[0.4,0.7\right]\rangle ,\hspace{0.17em}\hspace{0.17em}{\varsigma }_3=\langle \left[0.4,0.4],[0.3,0.7\right]\rangle$ where the corresponding weighted vector is $ꝕ=$(0.3,0.3,0.4)^T and the operational parameter $\vartheta =3.$ To aggregate these values by the IVPDOWA operator, we firstly permute these numbers by using Equation (1) and obtain the following information

$${\varsigma }_1=0.31, {\varsigma }_2=0.2, \mathrm{and} {\varsigma }_3=0.27.$$

By applying Definition 14, we then permuted values of IVPEs as follows:

${\varsigma }_{\rho \left(1\right)}=\langle \left[0.1,0.3\right],\left[0.2,0.5\right]\rangle$, ${\varsigma }_{\rho \left(2\right)}=\langle \left[0.4,0.4\right],\left[0.3,0.7\right]\rangle$ and ${\varsigma }_{\rho \left(3\right)}=\langle \left[0.3,0.4\right],\left[0.4,0.7\right]\rangle$.

Now,

$$\begin{array}{l}IVPFDOW{A}_{\mathsf{ꝕ}}\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)={\oplus }_{\mathsf{ċ}=1}^j\left({ꝕ}_{\mathsf{ċ}}{\varsigma }_{\rho \left(\mathsf{ċ}\right)}\right)\\ =\left(\begin{array}{c}\left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^3\right\}}^{\frac{1}{3}}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^3\right\}}^{\frac{1}{3}}}} \right], \\ \left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^3\right\}}^{\frac{1}{3}}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^3\right\}}^{\frac{1}{3}}}}\right]\\ \end{array}\right)\end{array}$$

Consequently, $IVPFDOW{A}_{\mathsf{ꝕ}}\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)=\left(\left[0.34,0.39],[0.24,0.57\right]\right)$.

In the following definition, we propose a Dombi geometric aggregation operator with IVPFEs, namely, the IVPFDWG operator.

Definition 15.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$ be the j number of IVPFEs. The IVPFDWG operator is characterized by a function IVPFDWG: IVPFE^j $\to$ IVPFE such that

$$IVPFDW{G}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_j\right)={\otimes }_{\mathsf{ċ}=1}^j{\left(g_{\mathsf{ċ}}\right)}^{{ꝕ}_{\mathsf{ċ}}}$$

where $\mathsf{ꝕ}={\left({\mathsf{ꝕ}}_1,{\mathsf{ꝕ}}_2,{\mathsf{ꝕ}}_3,\dots ,{\mathsf{ꝕ}}_j\right)}^T$ is the weight vector of $g_{\mathsf{ċ}}$ such that ${\mathsf{ꝕ}}_{\mathsf{ċ}}>0$ and $\sum _{\mathsf{ċ}=1}^j{\mathsf{ꝕ}}_{\mathsf{ċ}}=1.$.

Theorem 3.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$ be the j number of IVPFEs. The aggregated value of these IVPFEs in the framework of the IVPFDWG operator is an IVPFE and is determined in the following way:

$$\begin{gathered} IVPFDW{G}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_j\right)={\otimes }_{\mathsf{ċ}=1}^j{\left(g_{\mathsf{ċ}}\right)}^{{ꝕ}_{\mathsf{ċ}}} \\ =\left(\begin{array}{c}\left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}{{\left(1-{\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right), \end{gathered}$$

where $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ is the weight vector of $g_{\mathsf{ċ}}$, 0 $\le {ꝕ}_{\mathsf{ċ}}, \sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1.$

Proof. 

Proof of this result is analogous to Theorem 1. □

The following example describes the above-stated fact.

Example 5.

The three analyzers ${\varsigma }_1,{\varsigma }_2,{\varsigma }_3$ want to check the working ability of a certain machine. The estimated values of the working ability of a certain machine are specified by IVPF information such as:

${\varsigma }_1=\left(\left[0.4,0.6\right],\left[0.5,0.6\right]\right),{\varsigma }_2=\left(\left[0.5,0.7\right],\left[0.2,0.5\right]\right),{\varsigma }_3=\left(\left[0.3,0.4\right],\left[0.6,0.77\right]\right)$ with the corresponding weighted vector $ꝕ=$(0.3,0.5,0.2)^T and the operational parameter $\vartheta =5.$ Then,

$$\begin{array}{l}IVPFDW{G}_{\mathsf{ꝕ}}\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)={\otimes }_{\mathsf{ċ}=1}^3{\left({\varsigma }_{\mathsf{ċ}}\right)}^{{ꝕ}_{\mathsf{ċ}}}\\ =\left(\begin{array}{c}\left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}{{\left({\mu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^5\right\}}^{\frac{1}{5}}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}{{\left({\mu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^5\right\}}^{\frac{1}{5}}}} \right], \\ \left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}{{\left(1-{\nu }_{g_{\mathsf{ċ}}}^L\right)}^2}\right)}^5\right\}}^{\frac{1}{5}}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^3{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}{1-{\left({\nu }_{g_{\mathsf{ċ}}}^U\right)}^2}\right)}^5\right\}}^{\frac{1}{5}}}}\right]\\ \end{array}\right)\end{array}$$

Consequently, $IVPFDW{G}_{\mathsf{ꝕ}}\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)=\left(\left[0.34,0.46],[0.54,0.72\right]\right)$.

In the following definition, we propose a Dombi geometric aggregation operator with IVPFEs, namely, the IVPFDOWG operator.

Definition 16.

Let $\mathsf{ċ}=1,2,3,\dots ,j$ and $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^L,{\mathsf{\mu }}_{g_{\mathsf{ċ}}}^U\right],\left[{\nu }_{g_{\mathsf{ċ}}}^L,{\nu }_{g_{\mathsf{ċ}}}^U\right]\rangle$ be the j number of IVPFEs. An IVPFDOWG operator of dimension j is characterized by a function $IVPFDOWG: IVPFE^j $\to$ IVPFE with the associated weighted vector $ꝕ={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ such that ${ꝕ}_{\mathsf{ċ}}>0$ and $\sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$. Therefore,

where $\rho \left(1\right),\rho \left(2\right),\rho \left(3\right),\dots ,\rho \left(j\right)$ represent the permutations of $1,2,3,\dots ,j$, respectively, and $g_{\rho \left(\mathsf{ċ}-1\right)}\ge g_{\rho \left(\mathsf{ċ}\right)}$ $\forall$ $\mathsf{ċ}=1,2,\dots ,j.$

Theorem 4.

Let $g_{\mathsf{ċ}}=\langle \left[{\mathsf{\mu }}_{g_{\rho \left(\mathsf{ċ}\right)}}^L,{\mathsf{\mu }}_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right],\left[{\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L,{\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right]\rangle$ (1, 2, 3, …, j) be j number of IVPFEs. Then. The amassed value of these IVPFEs in the framework of the IVPFDOWA operator is an IVPFE and is determined in the following way:

$$\begin{gathered} IVPFDOW{G}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_j\right)={\otimes }_{\mathsf{ċ}=1}^j{\left(g_{\rho \left(\mathsf{ċ}\right)}\right)}^{{ꝕ}_{\mathsf{ċ}}} \\ =\left(\begin{array}{c}\left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{\mathrm{j}}{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mathsf{\mu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{L}}\right)}^2}{{\left({\mathsf{\mu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{L}}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{\mathrm{j}}{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mathsf{\mu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{U}}\right)}^2}{{\left({\mathsf{\mu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{U}}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right],\\ \left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{\mathrm{j}}{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mathsf{\nu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{L}}\right)}^2}{1-{\left({\mathsf{\nu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{L}}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^{\mathrm{j}}{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\mathsf{\nu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{U}}\right)}^2}{1-{\left({\mathsf{\nu }}_{g_{\mathsf{\rho }\left(\mathsf{ċ}\right)}}^{\mathrm{U}}\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right) \end{gathered}$$

(3)

where ꝕ $={\left({ꝕ}_1,{ꝕ}_2,{ꝕ}_3,\dots ,{ꝕ}_j\right)}^T$ is the weighted vector of $g_{\mathsf{ċ}}$ such that 0$\le {ꝕ}_{\mathsf{ċ}}, \sum _{\mathsf{ċ}=1}^j{ꝕ}_{\mathsf{ċ}}=1$ and $\vartheta \ge 1.$ Moreover, $\rho \left(1\right),\rho \left(2\right),\rho \left(3\right),\dots ,\rho \left(j\right)$ represent the permutations of $1,2,3,\dots ,j$, respectively, where $g_{\rho \left(\mathsf{ċ}-1\right)}\ge g_{\rho \left(\mathsf{ċ}\right)}$ $\forall$ $\mathsf{ċ}=1,2,\dots ,j.$

Proof. 

The proof of this result is established through the use of mathematical induction. □

The application of definition 12 for $\mathsf{ċ}$ = 2 gives the following outcome:

$$\begin{gathered} IVPFDOW{G}_{\mathsf{ꝕ}}\left(g_1,g_2\right)=g_{\rho \left(1\right)}{}^{{ꝕ}_1}\otimes g_{\rho \left(2\right)}{}^{{ꝕ}_2} \\ =\left(\begin{array}{c}\left[\begin{array}{c}\sqrt{\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{1-{\left({\mu }_{g_{\rho \left(1\right)}}^L\right)}^2}{{\left({\mu }_{g_{\rho \left(1\right)}}^L\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{1-{\left({\mu }_{g_{\rho \left(2\right)}}^L\right)}^2}{{\left({\mu }_{g_{\rho \left(2\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\\ \sqrt{\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{1-{\left({\mu }_{g_{\rho \left(1\right)}}^U\right)}^2}{{\left({\mu }_{g_{\rho \left(1\right)}}^U\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{1-{\left({\mu }_{g_{\rho \left(2\right)}}^U\right)}^2}{{\left({\mu }_{g_{\rho \left(2\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\end{array}\right], \\ \left[\begin{array}{c}\sqrt{1-\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{{\left({\nu }_{g_{\rho \left(1\right)}}^L\right)}^2}{1-{\left({\nu }_{g_{\rho \left(1\right)}}^L\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{{\left({\nu }_{g_{\rho \left(2\right)}}^L\right)}^2}{1-{\left({\nu }_{g_{\rho \left(2\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\\ \sqrt{1-\frac{1}{1+{\left\{{ꝕ}_1{\left(\frac{{\left({\nu }_{g_{\rho \left(1\right)}}^U\right)}^2}{1-{\left({\nu }_{g_{\rho \left(1\right)}}^U\right)}^2}\right)}^{\vartheta }+{ꝕ}_2{\left(\frac{{\left({\nu }_{g_{\rho \left(2\right)}}^U\right)}^2}{1-{\left({\nu }_{g_{\rho \left(2\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\end{array}\right]\end{array}\right) \end{gathered}$$

This means that

$$\begin{gathered} IVPFDOW{G}_{\mathsf{ꝕ}}\left(g_1,g_2\right) \\ =\left(\begin{array}{c}\left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^2{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right) \end{gathered}$$

Thus, Equation (3) holds for $\mathsf{ċ}$ = 2.

Let us assume that Equation (3) holds for $\mathsf{ċ}$ $=p$. Therefore, we have

$$\begin{array}{l}IVPFDOW{G}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_p\right)={\oplus }_{\mathsf{ċ}=1}^p{\left(g_{\rho \left(\mathsf{ċ}\right)}\right)}^{{ꝕ}_{\mathsf{ċ}}}\\ =\left(\begin{array}{c}[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}]\\ \left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\end{array}\right)\end{array}$$

$\mathrm{Moreover}, \mathrm{for}$ $\mathsf{ċ}=p+1,$ we have

$$\begin{gathered} IVPFDOW{G}_{\mathsf{ꝕ}}\left(g_1,g_2,g_3,\dots ,g_{p ,}{g}_{p+1}\right)={\oplus }_{\mathsf{ċ}=1}^p{\left(g_{\rho \left(\mathsf{ċ}\right)}\right)}^{{ꝕ}_{\mathsf{ċ}}}\oplus {(g_{\rho \left(p+1\right)})}^{{ꝕ}_{p+1}} \\ =\left(\begin{array}{c}\left[\sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{{\left({\mu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \right], \\ \left[\sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}, \sqrt{1-\frac{1}{1+{\left\{{\sum }_{\mathsf{ċ}=1}^p{ꝕ}_{\mathsf{ċ}}{\left(\frac{{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}{1-{\left({\nu }_{g_{\rho \left(\mathsf{ċ}\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}}\right]\\ \end{array}\right) \\ \otimes (\begin{array}{c}\\ \\ \begin{array}{c}\left[\begin{array}{c}\sqrt{\frac{1}{1+{\left\{{ꝕ}_{p+1}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(p+1\right)}}^L\right)}^2}{{\left({\mu }_{g_{\rho \left(p+1\right)}}^L\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}},\sqrt{\frac{1}{1+{\left\{{ꝕ}_{p+1}{\left(\frac{1-{\left({\mu }_{g_{\rho \left(p+1\right)}}^U\right)}^2}{{\left({\mu }_{g_{\rho \left(p+1\right)}}^U\right)}^2}\right)}^{\vartheta }\right\}}^{\frac{1}{\vartheta }}}} \\ \end{array}\right], \\ \\ [\sqrt{1-\frac{1}{1+{\{{ꝕ}_{p+1}{(\frac{{({\nu }_{\rho (p+1}^{}}^{}}{}}^{}}^{}}}\end{array}\end{array} \end{gathered}$$