A Novel Intuitionistic Fuzzy Rough Sets-Based Clustering Model Based on Aczel–Alsina Aggregation Operators

Abstract

Based on the approximation spaces, the interval-valued intuitionistic fuzzy rough set (IVIFRS) plays an essential role in coping with the uncertainty and ambiguity of the information obtained whenever human opinion is modeled. Moreover, a family of flexible t-norm (TNrM) and t-conorm (TCNrM) known as the Aczel–Alsina t-norm (AATNrM) and t-conorm (AATCNrM) plays a significant role in handling information, especially from the unit interval. This article introduces a novel clustering model based on IFRS using the AATNrM and AATCNrM. The developed clustering model is based on the aggregation operators (AOs) defined for the IFRS using AATNrM and AATCNrM. The developed model improves the level of accuracy by addressing the uncertain and ambiguous information. Furthermore, the developed model is applied to the segmentation problem, considering the information about the income and spending scores of the customers. Using the developed AOs, suitable customers are targeted for marketing based on the provided information. Consequently, the proposed model is the most appropriate technique for the segmentation problems. Furthermore, the results obtained at different values of the involved parameters are studied.

1. Introduction

An essential tool to handle the complexities of the uncertainty and imprecision of data is known as IVIFRS. IVIFRS offers an advanced mathematical model to deal with uncertainty and ambiguity in information. IVIFRS is obtained by linking interval-valued intuitionistic fuzzy sets (IVIFS) and rough sets (RS). The IVIFRS allows a more nuanced representation of uncertainty by characterizing an element in the set by a range rather than a single value. Thus, IVIFRS can capture a vast spectrum of uncertainty and ambiguity. IVIFRS can model the uncertainty involved using the membership value (MV), non-MV (NMV), and hesitation value (HV) of an object. Consequently, the IVIFRS is very beneficial for specific real-life applications. The IVIFRS is mainly used to deal with decision-making problems, aggregate data, and analyze robust decisions in economics, artificial intelligence, engineering, etc. Using IVIFRS in clustering can be very beneficial in obtaining more accuracy. Moreover, incomplete information can be handled using the IVIFRS. Consequently, using the IVIFRS to develop the clustering model enhances the accuracy of clustering problems.

Moreover, a flexible pair of operational laws known as AATNrM and AATCNrM can deal with uncertain information with high accuracy and flexibility. Accuracy is enhanced using the AATNrM and AATCNrM with the help of parameter $\mathcal{h}$. The involvement of parameter $\mathcal{h}$ makes the AATNrM and AATCNrM adaptable for uncertainty handling, especially in various fuzzy frameworks. Consequently, various fields have benefited from the application of the AATNrM and AATCNrM, e.g., decision-making, pattern recognition, data mining, and so on, especially in the economics and business sectors. Due to the desirable flexibility, the AATNrM and AATCNrM can be integrated with IVIFRS to deal with uncertainties more accurately and nuancedly. Thus, in this article, the AOs are developed to deal with the clustering problems using AATNrM and AATCNrM integration with IVIFRS.

Clustering is an important technique used to identify similar data points and to group them into clusters. Clustering is often used in social sciences, marketing, biology, etc. However, traditional clustering techniques rely on data from real-life scenarios, which presents uncertainty and ambiguity. The inclusion of uncertainty and ambiguity may cause ambiguous results. To address these challenges, developing the clustering model to deal with uncertainty and ambiguity is crucial. This article develops the clustering model by integrating IVIFRS, AATNrM, and AATCNrM. IVIFRS helps to reduce uncertainty at the initial stages with the approximations of the MV, NMV, and HV in the form of intervals. Furthermore, the developed AOs using AATNrM and AATCNrM aggregate the information more accurately and nuancedly. The main contributions of this article are (1) new AOs developed for IVIFRS, (2) a new clustering model introduced based on IVIFRS and AATNrM and AATCNrM, which is flexible and more accurate, and (3) the problem of segmentation is handled using the developed clustering model.

The remainder of the article is as follows: Section 2 summarizes the related literature, Section 3 deals with the basic terminologies, Section 4 consists of the development of new AOs for IVIFRS, Section 5 consists of the application of the developed clustering model, and the last section concludes this study.

2. Literature Review

The uncertainty inherent in information handling, mainly when human opinion is involved, is a significant issue in real-life scenarios. Several models address the uncertainty. Numerous methods have already been researched to deal with ambiguity and uncertainty. When any object is described as a set with condition $\mathcal{r}\in \left[\mathrm{0,1}\right]$, the crisp set can be generalized to the fuzzy set (FS) by Zadeh [1] in 1965. This remarkable tool is called the fuzzy set (FS). Due to its ability to grade objects and indicate their membership in the set, FS is an essential tool for dealing with uncertainty. To demonstrate the degree to which an object is not part of the scenario (set), Atanassov [2] pointed out that there should be an extra degree called a non-membership grade (NMrD). As a result, Atanassov [2] introduced IFS, a generalization of the FS with the condition $sum\left(\mathrm{M}\mathrm{V},\mathrm{N}\mathrm{M}\mathrm{V}\right)\in \left[\mathrm{0,1}\right]$. Following the release of the IFS, it was much simpler and more specific to describe a real-world scenario using the set’s terms. Many fields, including business [3], economics [4], medicine [5], engineering [6], and so forth, have found extensive use for the IFS.

Several AOs built on IFS have been released and are crucial to information aggregation. Using IFS, Chen [7] created prioritized AOs and utilized them in multi-attribute group decision-making (MAGDM). To address the multi-attribute decision-making (MADM) problem, Liu et al. [8] developed AOs for IFS based on the Dombi operational laws. Akram et al. presented IF AOs based on the Hamacher operational laws and used them to address the MADM issue [9]. Büyüközkan and Güleryüz presented AOs for IFS and used them in the MAGDM for phone selection [10]. Ahmed et al. [11] developed AOs using IFS to deal with complex information. Al-Qubati et al. [12] used IFS to develop the AOs for MADM based on the Choquet integral. Atanassov [13] introduced IVIFS to include more information than the IFS. Because of its appealing characteristics, IVIFS can be used in various fields. Alcantud and Santos-García [14] developed transformation techniques based on IVIFS. Ali et al. [15] developed decision models using IVIFS.

When describing an object as a component of a real-world scenario, IVIFS is a valuable tool for minimizing vagueness. However, Pawlak [16] introduced the rough set (RS), another generalization of the crisp set based on guesswork to account for the imprecise information. Furthermore, by utilizing fuzzy rather than crisp relations to introduce the IFRS, Pawlak [16] created a bridge between the IFS and RS and covered more information with greater accuracy. IFRS has gained popularity among researchers as an intriguing tool. To aggregate data in IFRS and apply them to the MADM problem, Ref. [17] introduced AOs based on the Frank operational laws. Das [18] solved the MADM problem using IFRS, viewing it as a valuable tool. In Mishra et al. [19], the IFRS addresses the MADM problem, which is an intriguing application of the IFRS. A more generalized version of the IFRS, the interval-valued IFRS, is presented and used to address the MADM issue [20]. The problem of MADM in the medical field is addressed by applying IFRS [21]. Tiwari et al. [22] used attribute reduction and the IFRS to solve the MADM problem.

Several researches have been conducted to solve the clustering problems and their various applications. Liu et al. [23] discussed the clustering problem of developing new fuzzy AOs. Shehzadi et al. [24] studied the fuzzy C-means and its applications in clustering problems. Chatziliadis et al. [25] studied the role of the AOs in the clustering problems. Ding et al. [26] studied the two-step aggregation functions in clustering problems. Höger et al. [27] studied the applications of clustering techniques in control distributions. Wang et al. [28] discussed two stages of aggregation functions and their application in clustering. Jabbarova [29] studied forecasting techniques using fuzzy clustering. Gupta and Kumar [30] studied the clustering technique using similarity measures and pattern recognition. Sakumoto et al. [31] studied the clustering technique for metadata.

In the aggregation of the data, the t-norm (TNrM) and t-conorm (TCNrM) [32] play a crucial role. Multiple varieties of TNrM and TCNrM have been introduced based on various parameters. Some examples of TNrMs and TCNrMs are Dombi [33], Frank [34], Einstein [35], and Hamacher [36]. As a result, these TNrMs and TCNrMs have been the basis for developing multiple AOs. The Dombi TNrM and TCNrM are the foundation for the AOs [37]. The algebraic TNrM and TCNrM-based AOs are studied by [38]. The Einstein TNrM and TCNrM are the foundation for the AOs [39]. The Einstein TNrM and TCNrM are the foundation for the AOs [40,41]. Einstein’s TNrM and TCNrM are the foundation for the AOs [42]. Frank TNrM and TCNrM are the foundation for AOs in [43,44]. Frank TNrM and TCNrM are the foundation for the AOs in [45]. AATCNrM and AATNrM, two important operational rules established by Aczél and Alsina [46], help fuse information depending on parameters. All fuzzy frameworks can use the AATNrM and AATCNrM because of their adaptability. To solve the MADM problem, Senapati et al. [47] created AOs based on the AATNrM and the AATCNrM for IFS. Siab et al. [48] introduced the AOs for the linguistic IFS to solve MADM problems. Owing to AATNrM and AATCNrM’s importance, Refs. [49,50,51,52] also discuss their use.

3. Basic Notions

In this section, we have defined some basic concepts. IVIFS and its operations are defined first. Some basics of IVIFRS, AATNrM, and AATCNrM are defined.

Definition 1

 ([13]). An IVIFS $\mathrm{{\rm Z}}$ over the universe $\mathrm{P}$ is defined as follows:

$\mathrm{Z}=\left\{\mathrm{z},\left(\left[{\mathcal{r}}_{\mathrm{z}}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}}^{\mathrm{u}}\right],\left[{\mathcal{s}}_{\mathrm{z}}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}}^{\mathrm{u}}\right]\right):\mathrm{z}\in \mathrm{P}\right\}$

where ${\mathcal{s}}_{\mathrm{z}},{\mathcal{r}}_{\mathrm{z}}:\mathrm{P}\to \left[\mathrm{0,1}\right]$ are the MV and NMV with condition $0\le {{\mathcal{r}}_{\mathrm{z}}^{\mathrm{u}}+\mathcal{s}}_{\mathrm{z}}^{\mathrm{u}}\le 1$.

Consider $\mathrm{Z}=\left(\left[{\mathcal{r}}_z^l,{\mathcal{r}}_z^u\right],\left[{\mathcal{s}}_z^l,{\mathcal{s}}_z^u\right]\right),{\mathrm{Z}}_p=\left(\left[{\mathcal{r}}_{zp}^l,{\mathcal{r}}_{zp}^l\right],\left[{\mathcal{s}}_{zp}^l,{\mathcal{s}}_{zp}^u\right]\right)$ for $p=1,2$ be three interval-valued intuitionistic fuzzy values (IVIFVs), and $\mathcal{g}>0$ be any real number. Then, the basic operations for these IVIFVs are stated below.

• ${\mathrm{Z}}_1\cup {\mathrm{Z}}_2=\left(\begin{array}{c}\max \left(\left[{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{u}}\right],\left[{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{u}}\right]\right)\\ \min \left(\left[{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{u}}\right],\left[{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{u}}\right]\right)\end{array}\right)$

• ${\mathrm{Z}}_1\cap {\mathrm{Z}}_2=\left(\begin{array}{c}\min \left(\left[{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{u}}\right],\left[{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{u}}\right]\right)\\ \max \left(\left[{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{u}}\right],\left[{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{u}}\right]\right)\end{array}\right)$

• ${\mathrm{Z}}_1\oplus {\mathrm{Z}}_2=\left(\begin{array}{c}\left[\begin{array}{c}{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{l}}+{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{l}}-{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{l}}{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{l}},\\ {\mathcal{r}}_{\mathrm{z}1}^{\mathrm{u}}+{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{u}}-{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{u}}{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{u}}\end{array}\right],\\ \left[{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{l}}{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{u}}{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{u}}\right]\end{array}\right)$

• ${\mathrm{Z}}_1\otimes {\mathrm{Z}}_2=\left(\begin{array}{c}\left[{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{l}}{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}1}^{\mathrm{u}}{\mathcal{r}}_{\mathrm{z}2}^{\mathrm{u}}\right],\\ \left[\begin{array}{c}{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{l}}+{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{l}}-{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{l}}{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{l}},\\ {\mathcal{s}}_{\mathrm{z}1}^{\mathrm{u}}+{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{u}}-{\mathcal{s}}_{\mathrm{z}1}^{\mathrm{u}}{\mathcal{s}}_{\mathrm{z}2}^{\mathrm{u}}\end{array}\right]\end{array}\right)$

• ${\mathrm{Z}}^{\mathrm{c}}=\left(\left[{\mathcal{s}}_{\mathrm{z}}^{\mathrm{l}},{\mathcal{s}}_{\mathrm{z}}^{\mathrm{u}}\right],\left[{\mathcal{r}}_{\mathrm{z}}^{\mathrm{l}},{\mathcal{r}}_{\mathrm{z}}^{\mathrm{u}}\right]\right)$ for an IVIFV $\mathrm{Z}$

• $\mathcal{g}\mathrm{Z}=\left(\left[1-{\left(1-{\mathcal{r}}_{\mathrm{Z}}^{\mathrm{l}}\right)}^{\mathcal{g}},1-{\left(1-{\mathcal{r}}_{\mathrm{Z}}^{\mathrm{u}}\right)}^{\mathcal{g}}\right],\left[{\mathcal{s}}_{\mathrm{Z}}^{\mathrm{l}\mathcal{g}},{\mathcal{s}}_{\mathrm{Z}}^{\mathrm{u}\mathcal{g}}\right]\right)$

• ${\mathrm{Z}}^{\mathsf{\sigma }}=\left(\left[{\mathcal{r}}_{\mathrm{Z}}^{\mathrm{l}\mathcal{g}},{\mathcal{r}}_{\mathrm{Z}}^{\mathrm{u}\mathcal{g}}\right],\left[1-{\left(1-{\mathcal{s}}_{\mathrm{Z}}^{\mathrm{l}}\right)}^{\mathcal{g}},1-{\left(1-{\mathcal{s}}_{\mathrm{Z}}^{\mathrm{u}}\right)}^{\mathcal{g}}\right]\right)$

Definition 2

 ([53]). Consider a relation over the universe $P$ from $IVIFS\left(P\times P\right)$. Then

• $\mathfrak{R}$ is known as reflexive if ${\mathcal{r}}_{\mathfrak{R}}\left(\mathcal{o},\mathcal{o}\right)=1$ and ${\mathcal{s}}_{\mathfrak{R}}\left(\mathcal{o},\mathcal{o}\right)=0$ for all the objects of the universe.

• $\mathfrak{R}$ is known as symmetric if $\left(\mathcal{o},\mathcal{p}\right)\in \mathrm{P}\times \mathrm{P}$ then ${\mathcal{r}}_{\mathfrak{R}}\left(\mathcal{p},\mathcal{o}\right)={\mathcal{r}}_{\mathfrak{R}}\left(\mathcal{o},\mathcal{p}\right)$ for all $\mathcal{o},\mathcal{p}\in \mathrm{P}$. Further, ${\mathcal{s}}_{\mathfrak{R}}\left(\mathcal{p},\mathcal{o}\right)={\mathcal{s}}_{\mathfrak{R}}\left(\mathcal{o},\mathcal{p}\right)$

• $\mathfrak{R}$ is known as transitive if for all $\mathcal{o},\mathcal{p},\mathcal{q}\in \mathrm{P}$ if $\left(\mathcal{p},\mathcal{q}\right)\in \mathfrak{R}$ and $\left(\mathcal{q},\mathcal{o}\right)\in \mathfrak{R}$ then ${\mathcal{r}}_{\mathfrak{R}}\left(\mathcal{p},\mathcal{o}\right)\ge \bigvee \left[{\mathcal{r}}_{\mathfrak{R}}\left(\mathcal{p},\mathcal{q}\right)\wedge {\mathcal{r}}_{\mathfrak{R}}\left(\mathcal{q},\mathcal{o}\right)\right]$ and ${\mathcal{s}}_{\mathfrak{R}}\left(\mathcal{p},\mathcal{o}\right)\ge \bigwedge \left[{\mathcal{s}}_{\mathfrak{R}}\left(\mathcal{p},\mathcal{q}\right)\wedge {\mathcal{s}}_{\mathfrak{R}}\left(\mathcal{q},\mathcal{o}\right)\right]$.

Definition 3.

 Consider a relation $\mathfrak{R}\in IVIFS\left(P\times P\right)$ is defined over the universe $P,$ then the approximation space for IVIFVs is denoted by $\left(P,\mathfrak{R}\right)$. The interval-valued intuitionistic fuzzy lower approximation (IVIFLA) and interval-valued intuitionistic fuzzy upper approximation (IVIFUA) for any set $\mathcal{D}\subseteq IVIFS\left(P\right)$ are defined as

$$\begin{gathered} {\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)=\left\{\begin{array}{c}\mathcal{o},\left[{\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right),{\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)\right],\\ \left[{\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right),{\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)\right]|\mathcal{o}\in \mathrm{P}\end{array}\right\} \\ {\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)=\left\{\begin{array}{c}\mathcal{o},\left[{\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right),{\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)\right],\\ \left[{\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right),{\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)\right]|\mathcal{o}\in \mathrm{P}\end{array}\right\} \end{gathered}$$

where

$$\begin{gathered} {\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigvee }\left({\mathcal{r}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{l}}\left(\mathcal{o},\mathrm{l}\right)\vee {\mathcal{r}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{l}}\right) \\ {\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigvee }\left({\mathcal{r}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{u}}\left(\mathcal{o},\mathrm{l}\right)\vee {\mathcal{r}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{u}}\right) \\ {\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigwedge }\left({\mathcal{s}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{l}}\left(\mathcal{o},\mathrm{l}\right)\wedge {\mathcal{s}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{l}}\right) \\ {\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigwedge }\left({\mathcal{s}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{u}}\left(\mathcal{o},\mathrm{l}\right)\wedge {\mathcal{s}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{u}}\right) \\ {\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigwedge }\left({\mathcal{r}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{l}}\left(\mathcal{o},\mathrm{l}\right)\vee {\mathcal{r}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{l}}\right) \\ {\mathcal{r}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigwedge }\left({\mathcal{r}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{u}}\left(\mathcal{o},\mathrm{l}\right)\vee {\mathcal{r}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{u}}\right) \\ {\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{l}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigvee }\left({\mathcal{s}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{l}}\left(\mathcal{o},\mathrm{l}\right)\wedge {\mathcal{s}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{l}}\right) \\ {\mathcal{s}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}^{\mathrm{u}}\left(\mathcal{o}\right)=\underset{\mathrm{l}\in \mathrm{P}}{\bigvee }\left({\mathcal{s}}_{\mathfrak{R}\left(\mathcal{o}\right)}^{\mathrm{u}}\left(\mathcal{o},\mathrm{l}\right)\wedge {\mathcal{s}}_{\mathcal{D}\left(\mathcal{o}\right)}^{\mathrm{u}}\right) \end{gathered}$$

With condition ${0\le {\mathcal{r}}^{\mathrm{u}}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}\left(\mathcal{o}\right)+{{\mathcal{s}}^{\mathrm{u}}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)}\left(\mathcal{o}\right)\le 1$ and ${0\le {\mathcal{r}}^{\mathrm{u}}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right)}\left(\mathcal{o}\right)+{{\mathcal{s}}^{\mathrm{u}}}_{{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{l}\mathrm{A}}\left(\mathcal{D}\right)}\left(\mathcal{o}\right)\le 1$. The set $\left\{\left({\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{L}\mathrm{A}}\left(\mathcal{D}\right),{\mathfrak{R}}^{\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{U}\mathrm{A}}\left(\mathcal{D}\right)\right)\right\}$ is said to be an IFRS based on IVIFLA and IVIFUA.

Definition 4

 ([48]). The definition of the AATNrM and

${\mathbb{T}}_{\mathbb{X}}^{\mathcal{h}}\left(\mathcal{u},\mathrm{b}\right)=\left\{\begin{array}{c}{\mathbb{T}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right) \mathrm{if} \mathcal{h}=0\\ \min \left(\mathcal{u},\mathrm{b}\right) \mathrm{if} \mathcal{h}\to \infty \\ {\mathrm{e}}^{-{\left({\left(-\mathrm{l}\mathrm{n}\mathcal{u}\right)}^{\mathcal{h}}+{\left(-\mathrm{l}\mathrm{n}\mathrm{b}\right)}^{\mathcal{h}}\right)}^{1/\mathcal{h}}} \mathrm{otherwise}.\end{array}\right.$

and AATCNrM is defined as

${\mathrm{S}}_{\mathbb{X}}^{\mathcal{h}}\left(\mathcal{u},\mathrm{b}\right)=\left\{\begin{array}{c}{\mathrm{S}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right) \mathrm{if} \mathcal{h}=0\\ \max \left(\mathcal{u},\mathrm{b}\right) \mathrm{if} \mathcal{h}\to \infty \\ {1-\mathrm{e}}^{-{\left({\left(-\ln \left(1-\mathcal{u}\right)\right)}^{\mathcal{h}}+{\left(-\ln \left(1-\mathrm{b}\right)\right)}^{\mathcal{h}}\right)}^{1/\mathcal{h}}} \mathrm{otherwise}.\end{array}\right.$

where $\mathcal{h}\in \left[0,\mathrm{\infty }\right]$. Moreover, ${\mathbb{T}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right)$ and ${\mathrm{S}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right)$ shows the Drastic t-norm and Drastic t-conorm, respectively. The equations of the ${\mathbb{T}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right)$ and ${\mathrm{S}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right)$ are given below.

$$\begin{gathered} {\mathbb{T}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right)=\left\{\begin{array}{c}u \mathrm{if} \mathrm{b}=1\\ \mathrm{b} \mathrm{if} \mathcal{u}=1\\ 0 \mathrm{otherwise}\end{array}\right. \\ {\mathrm{S}}_{\mathrm{D}}\left(\mathcal{u},\mathrm{b}\right)=\left\{\begin{array}{c}\mathcal{u} \mathrm{if} \mathrm{b}=0\\ \mathrm{b} \mathrm{if} \mathcal{u}=0\\ 1 \mathrm{otherwise}\end{array}\right. \end{gathered}$$

The role of parameter $\mathcal{h}$ is very crucial as a controller. It controls the strictness of the logical operational laws. The decision-makers can vary the value of $\mathcal{h}$ according to the results of their choice.

4. New Aggregation Operators

In this section, we have defined new AOs for IVIFRS using the AATNrM and AATCNrM. Some basic properties of the defined AOs are also investigated in this section.

First, the operations for IVIFRVs are introduced based on the AATNrM and AATCNrM, as follows. (Note that in $\left[{\mathcal{r}}_p^{lla},{\mathcal{r}}_p^{ula}\right]{\mathcal{r}}_p^{lla}$ and ${\mathcal{r}}_p^{ula}$ shows the lower and upper limits of the MV, respectively, and is similar to the NMV.)

Definition 5.

 Let  ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2$  be the collection of IVIFRVs. Then

$$\begin{gathered} {\mathcal{R}}_1\oplus {\mathcal{R}}_2=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left({\left(-\ln \left(1-{\mathcal{r}}_1^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{r}}_2^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left({\left(-\ln \left(1-{\mathcal{r}}_1^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{r}}_2^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left({\left(-\ln {\mathcal{s}}_1^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{s}}_2^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left({\left(-\ln {\mathcal{s}}_1^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{s}}_2^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left({\left(-\ln \left(1-{\mathcal{r}}_1^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{r}}_2^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left({\left(-\ln \left(1-{\mathcal{r}}_1^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{r}}_2^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left({\left(-\ln {\mathcal{s}}_1^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{s}}_2^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left({\left(-\ln {\mathcal{s}}_1^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)}^{\mathcal{h}}+{\left(-\ln {\mathcal{s}}_2^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right) \\ {\mathcal{R}}_1\otimes {\mathcal{R}}_2=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-e^{-{\left({\left(-\mathit{ln}\left(1-{\mathcal{r}}_1^{lla}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{r}}_2^{lla}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}},},1-e^{-{\left({\left(-\mathit{ln}\left(1-{\mathcal{r}}_1^{ula}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{r}}_2^{ula}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}},}\right],\\ \left[e^{-{\left({\left(-\mathit{ln}{\mathcal{s}}_1^{lla}\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{s}}_2^{lla}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left({\left(-\mathit{ln}{\mathcal{s}}_1^{ula}\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{s}}_2^{ula}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-e^{-{\left({\left(-\mathit{ln}\left(1-{\mathcal{r}}_1^{lua}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{r}}_2^{lua}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}},},1-e^{-{\left({\left(-\mathit{ln}\left(1-{\mathcal{r}}_1^{uua}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{r}}_2^{uua}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}},}\right],\\ \left[e^{-{\left({\left(-\mathit{ln}{\mathcal{s}}_1^{lua}\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{s}}_2^{lua}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left({\left(-\mathit{ln}{\mathcal{s}}_1^{uua}\right)}^{\mathcal{h}}+{\left(-\mathit{ln}{\mathcal{s}}_2^{uua}\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right) \end{gathered}$$

Considering the weights of the information included, the IVIFRAAWA operator is defined as follows.

Definition 6.

 Let  ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$  be the collection of the IVIFRVs and  ${\mathcal{w}}_{\mathrm{p}}$  is the weight of the $pth$  IVIFRV such that  $\sum _{\mathrm{p}=1}^{\mathrm{k}}{\mathcal{w}}_{\mathrm{p}}=1$  (Note that  ${\mathcal{w}}_p$  for weight with this condition unless otherwise stated). Then

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\stackrel{\mathrm{k}}{\underset{\mathrm{p}=1}{\oplus }}{{\mathcal{w}}_{\mathrm{p}}\mathcal{R}}_{\mathrm{p}}$

Some fundamental properties of the IVIFRAAWA operator are investigated as follows.

In Theorem 1, we prove that the aggregated value obtained from the IVIFRAAWA operator is also IVIFRV.

Theorem 1.

 Let  ${\mathcal{R}}_p=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_p^{lla},{\mathcal{r}}_p^{ula}\right],\left[{\mathcal{s}}_p^{lla},{\mathcal{s}}_p^{ula}\right]\right),\\ \left(\left[{\mathcal{r}}_p^{lua},{\mathcal{r}}_p^{uua}\right],\left[{\mathcal{s}}_p^{lua},{\mathcal{s}}_p^{uua}\right]\right)\end{array}\right),p=1,2,\cdots ,k$  be the collection of the IVIFRVs and  ${\mathcal{w}}_p$  is the weight of the $pth$  IVIFRV. Then the value obtained after the aggregation is IVIFRV and

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

Proof. 

We prove Theorem 1 with the help of the mathematical induction method.

First, we check for $k=2$

$IVIFRAAWA\left({\mathcal{R}}_1,{\mathcal{R}}_2\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left({\mathcal{s}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^2{\left(-\mathit{ln}\left({\mathcal{s}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

Which is an IVIFRV.

Now, consider Equation (2) true for $k=n$

$IVIFRAAWA\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_n\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

We must prove Equation (2) true for $k=n+1$, which is as follows.

$IVIFRAAWA\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_n,{\mathcal{R}}_{n+1}\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[\begin{array}{c}1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lla}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left(1-{\mathcal{r}}_{n+1}^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},\\ 1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{ula}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left(1-{\mathcal{r}}_{n+1}^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\end{array}\right],\\ \left[\begin{array}{c}{e}^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lla}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left({\mathcal{s}}_{n+1}^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},\\ e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{ula}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left({\mathcal{s}}_{n+1}^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\end{array}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[\begin{array}{c}1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lua}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left(1-{\mathcal{r}}_{n+1}^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},\\ 1-e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{uua}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left(1-{\mathcal{r}}_{n+1}^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\end{array}\right],\\ \left[\begin{array}{c}{e}^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lua}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left({\mathcal{s}}_{n+1}^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},\\ e^{-{\left(\sum _{p=1}^n{\left(-\mathit{ln}\left({\mathcal{s}}_p^{uua}\right)\right)}^{\mathcal{h}}+{\left(-\mathit{ln}\left({\mathcal{s}}_{n+1}^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\end{array}\right]\end{array}\right)\end{array}\right)$

Next, we have,

$IVIFRAAWA\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{n+1}\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left({\mathcal{s}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^{n+1}{\left(-\mathit{ln}\left({\mathcal{s}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

Which is IVIFRV. Hence, proof is completed. □

In Theorem 2, we prove the idempotency of the IFRAAWA operator.

Theorem 2

 (Idempotency). Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$ be the collection of the IVIFRVs and ${\mathcal{w}}_{\mathrm{p}}$ is the weight of the $\mathrm{p}\mathrm{t}\mathrm{h}$ IVIFRV. Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\left(\begin{array}{c}\left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\mathcal{R},\forall \mathrm{p}=1,2,\cdots ,\mathrm{k}$. Then

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\left(\begin{array}{c}\left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\mathcal{R}$

Proof. 

As ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\left(\begin{array}{c}\left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)$, so we have

$$\begin{gathered} \mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\left(\mathcal{R},\mathcal{R},\cdots ,\mathcal{R}\right) \\ =\left(\begin{array}{c}\left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right) \\ =\left(\begin{array}{c}\left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left(1-{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right) \\ =\left(\begin{array}{c}\left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\mathcal{R} \end{gathered}$$

Hence, proof is completed. □

Theorem 3

 (Boundedness). Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$ be the collection of the IVIFRVs and ${\mathcal{w}}_{\mathrm{p}}$ is the weight of the$pth$ IVIFRV. Let ${\mathcal{R}}_{\mathrm{p}}^{\mathrm{s}}$ is the smallest and ${\mathcal{R}}_{\mathrm{p}}^{\mathrm{g}}$ is the greatest IVIFRV. Then

${\mathcal{R}}_{\mathrm{p}}^{\mathrm{s}}\le \mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)\le {\mathcal{R}}_{\mathrm{p}}^{\mathrm{g}}$

Proof. 

We can prove this theorem using Theorems 1 and 2. □

Theorem 4

 (Monotonicity). Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$ and ${\mathcal{R}}_{\mathrm{p}}^{\mathrm{v}}=\left(\begin{array}{c}\left(\left[{\mathrm{v}\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathrm{v}\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[\mathrm{v}{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},\mathrm{v}{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathrm{v}\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathrm{v}\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathrm{v}\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathrm{v}\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)$ be the collections of the IVIFRVs such that ${\mathcal{R}}_{\mathrm{p}}\le {\mathcal{R}}_{\mathrm{p}}^{\mathrm{v}}$. Then

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)\le \mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{A}\left({\mathcal{R}}_1^{\mathrm{v}},{\mathcal{R}}_2^{\mathrm{v}},\cdots ,{\mathcal{R}}_{\mathrm{k}}^{\mathrm{v}}\right)$

Proof. 

It is very easy to prove using Theorems 1 and 2. □

The IVIFRAAWA operator is based on weighted averaging mean. In a similar way, we can define the IVIFRAAWG operator which is based on the weighted geometric mean operator. Now, weighted geometric AOs based on the operational laws defined above in Equation (1) are defined as below.

Definition 7.

 Let  ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$  be the collection of the IVIFRVs and  ${\mathcal{w}}_{\mathrm{p}}$  is the weight of the $pth$  IVIFRV. Then

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{G}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\stackrel{\mathrm{k}}{\underset{\mathrm{p}=1}{\otimes }}{\mathcal{R}}_{\mathrm{p}}^{{\mathcal{w}}_{\mathrm{p}}}$

Some fundamental properties of the IVIFRAAWG operator are investigated as follows.

In Theorem 5, we state that the aggregated value obtained from the IVIFRAAWG operator is also IVIFRV.

Theorem 5.

 Let  ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$  be the collection of the IVIFRVs and  ${\mathcal{w}}_{\mathrm{p}}$  is the weight of the $pth$  IVIFRV. Then, the value obtained after the aggregation is IVIFRV and

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{G}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{1-\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({1-\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{1-\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({1-\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{1-\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({1-\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{1-\mathrm{e}}^{-{\left(\sum _{\mathrm{p}=1}^{\mathrm{k}}{\left(-\ln \left({1-\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

Proof. 

Its proof is same as proof of Theorem 1. □

Theorem 6

 (Idempotency). Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$ be the collection of the IVIFRVs and ${\mathcal{w}}_{\mathrm{p}}$ is the weight of the$pth$ IVIFRV. Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\left(\begin{array}{c}\left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\mathcal{R},\forall \mathrm{p}=1,2,\cdots ,\mathrm{k}$. Then

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{G}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)=\left(\begin{array}{c}\left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)=\mathcal{R}$

Proof. 

Its proof is same as proof of Theorem 2. □

Theorem 7

 (Boundedness). Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$ be the collection of the IVIFRVs and ${\mathcal{w}}_{\mathrm{p}}$ is the weight of the$pth$ IVIFRV. Let ${\mathcal{R}}_{\mathrm{p}}^{\mathrm{s}}$ is the smallest and ${\mathcal{R}}_{\mathrm{p}}^{\mathrm{g}}$ is the greatest IVIFRV. Then

${\mathcal{R}}_{\mathrm{p}}^{\mathrm{s}}\le \mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{G}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)\le {\mathcal{R}}_{\mathrm{p}}^{\mathrm{g}}$

Proof. 

It is very easy to prove using Theorems 1 and 2. □

Theorem 8

 (Monotonicity). Let ${\mathcal{R}}_{\mathrm{p}}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right),\mathrm{p}=1,2,\cdots ,\mathrm{k}$ and ${\mathcal{R}}_{\mathrm{p}}^{\mathrm{v}}=\left(\begin{array}{c}\left(\left[\mathrm{v}{\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},\mathrm{v}{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right],\left[\mathrm{v}{\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{l}\mathrm{a}},\mathrm{v}{\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{l}\mathrm{a}}\right]\right),\\ \left(\left[{\mathrm{v}\mathcal{r}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},\mathrm{v}{\mathcal{r}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right],\left[{\mathrm{v}\mathcal{s}}_{\mathrm{p}}^{\mathrm{l}\mathrm{u}\mathrm{a}},{\mathrm{v}\mathcal{s}}_{\mathrm{p}}^{\mathrm{u}\mathrm{u}\mathrm{a}}\right]\right)\end{array}\right)$ be the collections of the IVIFRVs such that ${\mathcal{R}}_{\mathrm{p}}\le {\mathcal{R}}_{\mathrm{p}}^{\mathrm{v}}$. Then

$\mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{G}\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_{\mathrm{k}}\right)\le \mathrm{I}\mathrm{V}\mathrm{I}\mathrm{F}\mathrm{R}\mathrm{A}\mathrm{A}\mathrm{W}\mathrm{G}\left({\mathcal{R}}_1^{\mathrm{v}},{\mathcal{R}}_2^{\mathrm{v}},\cdots ,{\mathcal{R}}_{\mathrm{k}}^{\mathrm{v}}\right)$

Proof. 

It is straightforward to prove. □

5. Application of Developed Aggregation Operators to the Segmentation Problem

In data analysis, clustering is essential for grouping similar data points based on their behaviors. Recognizing the different objects’ patterns, trends, and behaviors becomes easy by converting the data into clusters. Clustering has vast applications in various fields due to its desirable characteristics. Classical set theory is primarily used in market segmentation to understand the behaviors of different customers based on other attributes. Different methods are helpful in clustering, but none can deal with the information obtained from multiple experts. In this article, the developed model can deal with the information obtained from various experts. The workings of the developed model are given in detail.

Consider $\left\{{\mathcal{Q}}_1,{\mathcal{Q}}_2,\cdots ,{\mathcal{Q}}_q\right\}$ to be the set of $q$ alternatives from which one alternative is to be analyzed based on the set of attributes, and $\left\{{\mathcal{G}}_1,{\mathcal{G}}_2,\cdots ,{\mathcal{G}}_r\right\}$ is the set of $r$ attributes. Consider $\left\{{\mathcal{X}}_1,{\mathcal{X}}_2,\cdots ,{\mathcal{X}}_h\right\}$ to be the set of $h$ experts having weights ${\mathcal{W}}_n\in \left[\mathrm{0,1}\right] n=1,2,\cdots ,h$ such that $\sum _{n=1}^h{\mathcal{W}}_n=1$. The steps involved in segmenting the alternatives are as follows.

Step 1: Each expert provides the information for each object based on attributes. The experts give their opinions about objects based on their attributes. IVIFS allows them to assign the MV, NMV, and HV to objects of their own from unit intervals. Furthermore, the obtained values are converted into IVIFRVs using any desired interval-valued intuitionistic fuzzy rough (VIFR) relation. The decision matrix obtained is of the following shape.

$D{M}_h=\left(\begin{array}{cccc}{\mathcal{R}}_{11}& {\mathcal{R}}_{12}& \cdots & {\mathcal{R}}_{1r}\\ {\mathcal{R}}_{21}& {\mathcal{R}}_{22}& \cdots & {\mathcal{R}}_{2r}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathcal{R}}_{q1}& {\mathcal{R}}_{q2}& \cdots & {\mathcal{R}}_{qr}\end{array}\right)$

where ${\mathcal{R}}_{qr}=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_{qr}^{lla},{\mathcal{r}}_{qr}^{ula}\right],\left[{\mathcal{s}}_{qr}^{lla},{\mathcal{s}}_{qr}^{ula}\right]\right),\\ \left(\left[{\mathcal{r}}_{qr}^{lua},{\mathcal{r}}_{qr}^{uua}\right],\left[{\mathcal{s}}_{qr}^{lua},{\mathcal{s}}_{qr}^{uua}\right]\right)\end{array}\right).$

Step 2: The information obtained from the experts is in the form of the IVIFRVs such that $\left(\begin{array}{c}\left(\left[{\mathcal{r}}_p^{lla},{\mathcal{r}}_p^{ula}\right],\left[{\mathcal{s}}_p^{lla},{\mathcal{s}}_p^{ula}\right]\right),\\ \left(\left[{\mathcal{r}}_p^{lua},{\mathcal{r}}_p^{uua}\right],\left[{\mathcal{s}}_p^{lua},{\mathcal{s}}_p^{uua}\right]\right)\end{array}\right)$. If there exists any cost type of attribute, then we must change its type by taking the complement by interchanging the MV and NMV such that $\left(\begin{array}{c}\left(\left[{\mathcal{s}}_p^{lla},{\mathcal{s}}_p^{ula}\right],\left[{\mathcal{r}}_p^{lla},{\mathcal{r}}_p^{ula}\right]\right),\\ \left(\left[{\mathcal{s}}_p^{lua},{\mathcal{s}}_p^{uua}\right],\left[{\mathcal{r}}_p^{lua},{\mathcal{r}}_p^{uua}\right]\right)\end{array}\right)$.

Step 3: After the normalization of the information, we need the aggregated values of the objects based on the provided attributes. We can aggregate the information obtained in the form of the IVIFRVs using the following equations.

(1)$IVIFRAAWA\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_k\right)=\stackrel{k}{\underset{p=1}{\oplus }}{{\mathcal{w}}_p\mathcal{R}}_p=\left(\begin{array}{c}\left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{s}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[1-e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},1-e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left(1-{\mathcal{r}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{s}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{s}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

and

(2)$IVIFRAAWG\left({\mathcal{R}}_1,{\mathcal{R}}_2,\cdots ,{\mathcal{R}}_k\right)=\left(\begin{array}{c}\left(\begin{array}{c}\left[e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{r}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{r}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{1-e}^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({1-\mathcal{s}}_p^{lla}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{1-e}^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({1-\mathcal{s}}_p^{ula}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\\ \left(\begin{array}{c}\left[e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{r}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},e^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({\mathcal{r}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right],\\ \left[{1-e}^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({1-\mathcal{s}}_p^{lua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}},{1-e}^{-{\left(\sum _{p=1}^k{\left(-\mathit{ln}\left({1-\mathcal{s}}_p^{uua}\right)\right)}^{\mathcal{h}}\right)}^{{\displaystyle \frac{1}{\mathcal{h}}}}}\right]\end{array}\right)\end{array}\right)$

Note that we can use any of these developed AOs.

Step 4: The obtained aggregated values using the developed AOs are in the form of the IVIFRVs, as stated in Theorem 1. We need the defuzzification of the IVIFRVs to cluster the information based on the attributes. We use the following equation for the defuzzification of an IVIFRV ${\mathcal{R}}_k=\left(\begin{array}{c}\left(\left[{\mathcal{r}}_p^{lla},{\mathcal{r}}_p^{ula}\right],\left[{\mathcal{s}}_p^{lla},{\mathcal{s}}_p^{ula}\right]\right),\\ \left(\left[{\mathcal{r}}_p^{lua},{\mathcal{r}}_p^{uua}\right],\left[{\mathcal{s}}_p^{lua},{\mathcal{s}}_p^{uua}\right]\right)\end{array}\right)$.

(3)$Scr\left({\mathcal{R}}_k\right)=g_k={\displaystyle \frac{{\mathcal{r}}_p^{lla}+{\mathcal{r}}_p^{ula}+{\mathcal{r}}_p^{lua}+{\mathcal{r}}_p^{uua}+{\mathcal{s}}_p^{lla}+{\mathcal{s}}_p^{ula}+{\mathcal{s}}_p^{lua}+{\mathcal{s}}_p^{uua}}{4}}$

We obtain the following matrix as a result.

$ScrD=\left(\begin{array}{cccc}{g}_{11}& g_{12}& \cdots & g_{1r}\\ g_{21}& g_{22}& \cdots & g_{2r}\\ ⋮& ⋮& \ddots & ⋮\\ g_{q1}& g_{q2}& \cdots & g_{qr}\end{array}\right)$

Step 5: We standardize the information obtained from Step 4 using the following equation.

(4)$Scr{D}_{qr}^N={\displaystyle \frac{g_{qr}-{mean(\mathcal{X}}_k) }{{SD(\mathcal{X}}_k)}}$

where ${mean(\mathcal{X}}_k)$ and ${SD(\mathcal{X}}_k) for k=1,2,\cdots ,r$ show the mean value of the attributes and standard deviation, respectively.

Step 6: We defined a range of the desired number of clusters.

Step 7: We find clusters using any of the clustering algorithms, i.e., K-means clustering.

Step 8: Finally, we visualize the clusters obtained.

Figure 1 shows the stepwise working of the developed clustering model. In the following text, we have solved an example in support of our study.

Example 1.

 Consider a company that desires the segmentation of its customers based on some attributes. Consider there are five consumers and the following four attributes.

1. Age: the customers are analyzed based on their age. This would be better for generational preferences. The customers will be targeted according to their age.

2. Annual income: the customers are analyzed based on annual income. This would be better for purchasing power. The customers will be targeted according to their purchasing powers.

3. Purchasing frequency: the customers are analyzed based on their purchasing frequency. This would be better for offering the loyalty programs.

4. Proffered category: the customers are analyzed based on their preferred category. This would be better for understanding the customers’ preferences about products. The customers will be targeted according to their preferred products.

Three experts analyze each customer based on given attributes. Experts provide their own opinions based on the provided attributes. We must cluster the customers based on the information provided by experts. Consider $\left(\mathrm{0.28,0.31,0.41}\right)$ are the weights assigned to the experts concerning their significance. The matrices obtained from each expert are given below.

$$\begin{gathered} D{M}_1=\left(\begin{array}{c}{\mathcal{Q}}_1\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.21,0.21}\right],\\ \left[\mathrm{0.19,0.23}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.18,0.25}\right],\\ \left[\mathrm{0.23,0.31}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.25,0.26}\right],\\ \left[\mathrm{0.24,0.28}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.23,0.30}\right],\\ \left[\mathrm{0.28,0.36}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.30,0.31}\right],\\ \left[\mathrm{0.29,0.33}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.28,0.35}\right],\\ \left[\mathrm{0.33,0.41}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_2\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.22,0.23}\right],\\ \left[\mathrm{0.21,0.25}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.20,0.27}\right],\\ \left[\mathrm{0.25,0.33}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.23,0.24}\right],\\ \left[\mathrm{0.22,0.26}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.21,0.28}\right],\\ \left[\mathrm{0.26,0.34}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.26,0.27}\right],\\ \left[\mathrm{0.25,0.29}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.24,0.31}\right],\\ \left[\mathrm{0.29,0.37}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.29,0.30}\right],\\ \left[\mathrm{0.28,0.32}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.27,0.34}\right],\\ \left[\mathrm{0.32,0.40}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_3\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.26,0.27}\right],\\ \left[\mathrm{0.25,0.29}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.24,0.31}\right],\\ \left[\mathrm{0.29,0.37}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.25,0.26}\right],\\ \left[\mathrm{0.24,0.28}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.23,0.30}\right],\\ \left[\mathrm{0.28,0.36}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.27,0.28}\right],\\ \left[\mathrm{0.26,0.30}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.25,0.32}\right],\\ \left[\mathrm{0.30,0.38}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.31,0.32}\right],\\ \left[\mathrm{0.30,0.34}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.29,0.36}\right],\\ \left[\mathrm{0.35,0.42}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_4\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.28,0.29}\right],\\ \left[\mathrm{0.27,0.31}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.26,0.33}\right],\\ \left[\mathrm{0.31,0.39}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.28,0.29}\right],\\ \left[\mathrm{0.27,0.31}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.26,0.33}\right],\\ \left[\mathrm{0.31,0.39}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_5\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.27,0.28}\right],\\ \left[\mathrm{0.26,0.30}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.25,0.32}\right],\\ \left[\mathrm{0.30,0.38}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.29,0.30}\right],\\ \left[\mathrm{0.28,0.32}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.27,0.34}\right],\\ \left[\mathrm{0.32,0.40}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.32,0.33}\right],\\ \left[\mathrm{0.31,0.35}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.30,0.37}\right],\\ \left[\mathrm{0.35,0.43}\right]\end{array}\right)\end{array}\right)\end{array}\right) \\ D{M}_2=\left(\begin{array}{c}{\mathcal{Q}}_1\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.209,0.219}\right],\\ \left[\mathrm{0.189,0.229}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.179,0.249}\right],\\ \left[\mathrm{0.229,0.309}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.249,0.259}\right],\\ \left[\mathrm{0.239,0.279}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.229,0.299}\right],\\ \left[\mathrm{0.279,0.359}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.229,0.309}\right],\\ \left[\mathrm{0.289,0.329}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.279,0.349}\right],\\ \left[\mathrm{0.329,0.409}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_2\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.219,0.229}\right],\\ \left[\mathrm{0.209,0.249}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.199,0.269}\right],\\ \left[\mathrm{0.249,0.329}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.229,0.239}\right],\\ \left[\mathrm{0.219,0.259}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.209,0.279}\right],\\ \left[\mathrm{0.259,0.339}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.259,0.269}\right],\\ \left[\mathrm{0.249,0.289}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.239,0.309}\right],\\ \left[\mathrm{0.289,0.369}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.289,0.299}\right],\\ \left[\mathrm{0.279,0.319}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.269,0.339}\right],\\ \left[\mathrm{0.319,0.399}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_3\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.259,0.269}\right],\\ \left[\mathrm{0.25,0.289}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.239,0.309}\right],\\ \left[\mathrm{0.289,0.369}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.249,0.259}\right],\\ \left[\mathrm{0.239,0.279}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.229,0.299}\right],\\ \left[\mathrm{0.279,0.359}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.269,0.279}\right],\\ \left[\mathrm{0.259,0.299}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.249,0.319}\right],\\ \left[\mathrm{0.299,0.379}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.309,0.319}\right],\\ \left[\mathrm{0.299,0.339}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.289,0.359}\right],\\ \left[\mathrm{0.349,0.419}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_4\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.279,0.289}\right],\\ \left[\mathrm{0.289,0.309}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.259,0.329}\right],\\ \left[\mathrm{0.309,0.389}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.279,0.289}\right],\\ \left[\mathrm{0.269,0.309}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.259,0.329}\right],\\ \left[\mathrm{0.309,0.389}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_5\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.269,0.279}\right],\\ \left[\mathrm{0.259,0.299}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.249,0.319}\right],\\ \left[\mathrm{0.299,0.379}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.289,0.299}\right],\\ \left[\mathrm{0.279,0.319}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.269,0.339}\right],\\ \left[\mathrm{0.319,0.399}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.319,0.329}\right],\\ \left[\mathrm{0.309,0.349}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.299,0.369}\right],\\ \left[\mathrm{0.349,0.429}\right]\end{array}\right)\end{array}\right)\end{array}\right) \\ D{M}_3=\left(\begin{array}{c}{\mathcal{Q}}_1\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.209,0.219}\right],\\ \left[\mathrm{0.189,0.229}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.179,0.249}\right],\\ \left[\mathrm{0.229,0.309}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.249,0.259}\right],\\ \left[\mathrm{0.239,0.279}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.229,0.299}\right],\\ \left[\mathrm{0.279,0.359}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.229,0.309}\right],\\ \left[\mathrm{0.289,0.329}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.279,0.349}\right],\\ \left[\mathrm{0.329,0.409}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_2\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.219,0.229}\right],\\ \left[\mathrm{0.209,0.249}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.199,0.269}\right],\\ \left[\mathrm{0.249,0.329}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.229,0.239}\right],\\ \left[\mathrm{0.219,0.259}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.209,0.279}\right],\\ \left[\mathrm{0.259,0.339}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.259,0.269}\right],\\ \left[\mathrm{0.249,0.289}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.239,0.309}\right],\\ \left[\mathrm{0.289,0.369}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.289,0.299}\right],\\ \left[\mathrm{0.279,0.319}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.269,0.339}\right],\\ \left[\mathrm{0.319,0.399}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_3\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.259,0.269}\right],\\ \left[\mathrm{0.25,0.289}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.239,0.309}\right],\\ \left[\mathrm{0.289,0.369}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.249,0.259}\right],\\ \left[\mathrm{0.239,0.279}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.229,0.299}\right],\\ \left[\mathrm{0.279,0.359}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.269,0.279}\right],\\ \left[\mathrm{0.259,0.299}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.249,0.319}\right],\\ \left[\mathrm{0.299,0.379}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.309,0.319}\right],\\ \left[\mathrm{0.299,0.339}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.289,0.359}\right],\\ \left[\mathrm{0.349,0.419}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_4\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.279,0.289}\right],\\ \left[\mathrm{0.289,0.309}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.259,0.329}\right],\\ \left[\mathrm{0.309,0.389}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.279,0.289}\right],\\ \left[\mathrm{0.269,0.309}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.259,0.329}\right],\\ \left[\mathrm{0.309,0.389}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_5\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.269,0.279}\right],\\ \left[\mathrm{0.259,0.299}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.249,0.319}\right],\\ \left[\mathrm{0.299,0.379}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.289,0.299}\right],\\ \left[\mathrm{0.279,0.319}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.269,0.339}\right],\\ \left[\mathrm{0.319,0.399}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.239,0.249}\right],\\ \left[\mathrm{0.229,0.269}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.219,0.289}\right],\\ \left[\mathrm{0.269,0.349}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.319,0.329}\right],\\ \left[\mathrm{0.309,0.349}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.299,0.369}\right],\\ \left[\mathrm{0.349,0.429}\right]\end{array}\right)\end{array}\right)\end{array}\right) \end{gathered}$$

In case of the presence of the cost type attributes, we will normalize the above decision matrices. From the above decision matrices, we need the aggregated value for each customer. Using the developed Aos (Equations (1) and (2)), we find the aggregated values in the following.

$D{M}_A=\left(\begin{array}{c}{\mathcal{Q}}_1\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.21,0.21}\right],\\ \left[\mathrm{0.19,0.23}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.18,0.25}\right],\\ \left[\mathrm{0.23,0.31}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.25,0.26}\right],\\ \left[\mathrm{0.24,0.28}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.23,0.30}\right],\\ \left[\mathrm{0.28,0.36}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.30,0.31}\right],\\ \left[\mathrm{0.29,0.33}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.28,0.35}\right],\\ \left[\mathrm{0.33,0.41}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_2\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.22,0.23}\right],\\ \left[\mathrm{0.21,0.25}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.20,0.27}\right],\\ \left[\mathrm{0.25,0.33}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.23,0.24}\right],\\ \left[\mathrm{0.22,0.26}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.21,0.28}\right],\\ \left[\mathrm{0.26,0.34}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.26,0.27}\right],\\ \left[\mathrm{0.25,0.29}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.24,0.31}\right],\\ \left[\mathrm{0.29,0.37}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.29,0.30}\right],\\ \left[\mathrm{0.28,0.32}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.27,0.34}\right],\\ \left[\mathrm{0.32,0.40}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_3\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.26,0.27}\right],\\ \left[\mathrm{0.25,0.29}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.24,0.31}\right],\\ \left[\mathrm{0.29,0.37}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.25,0.26}\right],\\ \left[\mathrm{0.24,0.28}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.23,0.30}\right],\\ \left[\mathrm{0.28,0.36}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.27,0.28}\right],\\ \left[\mathrm{0.26,0.30}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.25,0.32}\right],\\ \left[\mathrm{0.30,0.38}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.31,0.32}\right],\\ \left[\mathrm{0.30,0.34}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.29,0.36}\right],\\ \left[\mathrm{0.35,0.42}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_4\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.28,0.29}\right],\\ \left[\mathrm{0.27,0.31}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.26,0.33}\right],\\ \left[\mathrm{0.31,0.39}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.28,0.29}\right],\\ \left[\mathrm{0.27,0.31}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.26,0.33}\right],\\ \left[\mathrm{0.31,0.39}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right)\\ {\mathcal{Q}}_5\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.27,0.28}\right],\\ \left[\mathrm{0.26,0.30}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.25,0.32}\right],\\ \left[\mathrm{0.30,0.38}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.29,0.30}\right],\\ \left[\mathrm{0.28,0.32}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.27,0.34}\right],\\ \left[\mathrm{0.32,0.40}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.24,0.25}\right],\\ \left[\mathrm{0.23,0.27}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.22,0.29}\right],\\ \left[\mathrm{0.27,0.35}\right]\end{array}\right)\end{array}\right),\left(\begin{array}{c}\left(\begin{array}{c}\left[\mathrm{0.32,0.33}\right],\\ \left[\mathrm{0.31,0.35}\right]\end{array}\right),\\ \left(\begin{array}{c}\left[\mathrm{0.30,0.37}\right],\\ \left[\mathrm{0.35,0.43}\right]\end{array}\right)\end{array}\right)\end{array}\right)$

Now, we do the defuzzification (Equation (3)) of the aggregated values as follows. Table 1 shows the score values obtained after defuzzification.

The standard normalized values are given in the following, Table 2.

Let us divide the customers into three clusters, i.e.,$\left\{\mathrm{0,1},2\right\}$. Using the K-means algorithm, we obtained the clusters of our dataset. Table 3 shows the obtained clusters in the following.

The visualization of the clusters can be noted in the following, Figure 2.

Sensitivity Analysis

A pair of flexible operational laws known as the AATNrM and AATCNrM can deal with uncertain information with high accuracy and a degree of flexibility. Accuracy is enhanced using the AATNrM and AATCNrM with the help of the involvement of a parameter $\mathcal{h}$. The involvement of parameter $\mathcal{h}$ makes the AATNrM and AATCNrM adaptable for uncertainty handling, especially in various fuzzy frameworks. Consequently, various fields have benefited from the applications of the AATNrM and AATCNrM, e.g., decision-making, pattern recognition, data mining, and so on, especially in economics and business sectors. Due to the desirable flexibility, the AATNrM and AATCNrM can be integrated with IVIFRS to deal with uncertainty in a more accurate and nuanced way. We have changed the values of parameter $\mathcal{h}$ and observed the results. Note that the obtained clusters remained the same for all the values of the parameter. We obtain the same clusters as provided in Table 3. The following Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 show the sensitivity analysis of the developed model.

6. Conclusions

We have introduced a clustering model based on developed AOs integrating IVIFRS AATNrM and AATCNrM. The introduced AOs enhance accuracy and robustness by allowing the clustering based on the information obtained from multiple experts. The integration of the IVIFRS AATNrM and AATCNrM enhances the ability to deal with information with more accuracy and uncertainty, especially in real-life problems. Some key points of the developed model are given as follows.

1. The developed model is based on the AATNrM and AATCNrM. The AATNrM and AATCNrM are very flexible operational laws due to the inclusion of the parameter. The variation of the parameter enhances flexibility, adaptability, and accuracy. In the case of the considered example, the same clusters are obtained at various values of the included parameter.

2. The developed model is based on the IVIFRS. The IVIFRS is a framework that represents the information obtained from real-life scenarios in the form of MV, NMV, and HV intervals. Moreover, the IVIFRS can deal with missing information with the help of approximations based on the relation of our own choice. Thus, the developed model enhances accuracy of the clustering technique.

3. The improved accuracy and the robustness of the model make it suitable for applications in various fields. Particularly, the developed model is ideal for segmentation problems. For example, they target customers for marketing based on their shopping behaviors. An example is considered in this study.

4. Customers are clustered using the developed clustering model with more accuracy. Different values of the parameter are used to cluster the customer for flexibility. Interestingly, the same clusters have been obtained at various values.

This study shows potential for dealing with uncertainty and ambiguity in clustering problems. However, this study is limited to dealing with the information obtained by the IVIFRS. Several frameworks deal with uncertainty by extracting information from real-life scenarios, such as Pythagorean FS [54,55]. Thus, the introduced model cannot deal with information obtained from these frameworks and is limited to the IVIFRS. Consequently, this model applies to various fields for accuracy improvement. In the future, we aim to apply the developed model to improve accuracy using the frameworks defined in [53,56]. This study can be used for any clustering problem. However, we aim to study its application in sustainable fuzzy environments [57] by clustering the areas/cities with the same weather situations, etc. In the future, we also strive to study various applications of the developed model in the energy sector.

Data Availability Statement

In this study, we did not use human/animal objects. Moreover, there is no data associated.

Conflicts of Interest

The author declares no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Stepwise working of the developed clustering model based on AATNrM and AATCNrM.

Figure 2. Visualization of the obtained clusters. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Figure 3. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in same cluster. So, they should be treated similarly.

Figure 4. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Figure 5. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Figure 6. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Figure 7. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Figure 8. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Figure 9. Visualization of the obtained clusters at [Forumla omitted. See PDF.]. We can develop strategies to entertain the customers according to the obtained clusters. Customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in cluster 1. Therefore, the strategy for these customers should be the same. Similarly, customers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] are in the same cluster. So, they should be treated similarly.

Table 1

Defuzzification of the IVIFRVs using Equation (3).

Customers/Attributes	Attribute 1	Attribute 2	Attribute 3	Attribute 4
${\mathcal{Q}}_1$	0.5302	0.4502	0.5502	0.6502
${\mathcal{Q}}_2$	0.4902	0.5102	0.5702	0.6302
${\mathcal{Q}}_3$	0.5702	0.5502	0.5902	0.6702
${\mathcal{Q}}_4$	0.5302	0.6102	0.6102	0.5302
${\mathcal{Q}}_5$	0.5902	0.6302	0.5302	0.6902

Table 2

Standardized normalized information.

Customers/Attributes	Attribute 1	Attribute 2	Attribute 3	Attribute 4
${\mathcal{Q}}_1$	−0.3078	−1.3608	−0.6325	0.2569
${\mathcal{Q}}_2$	−1.3338	−0.5443	0.0000	−0.0642
${\mathcal{Q}}_3$	0.7182	0.0000	0.6325	0.5779
${\mathcal{Q}}_4$	−0.3078	0.8165	1.2649	−1.6696
	1.2312	1.0887	−1.2649	0.8990

Table 3

Clusters obtained using K-means considering the clusters $\left\{\mathrm{0,1},2\right\}.$.

Customers/Attributes	Attribute 1	Attribute 2	Attribute 3	Attribute 4	Clusters
${\mathcal{Q}}_1$	−0.3078	−1.3608	−0.6325	0.2569	1
${\mathcal{Q}}_2$	−1.3338	−0.5443	0.0000	−0.0642	1
${\mathcal{Q}}_3$	0.7182	0.0000	0.6325	0.5779	2
${\mathcal{Q}}_4$	−0.3078	0.8165	1.2649	−1.6696	0
${\mathcal{Q}}_5$	1.2312	1.0887	−1.2649	0.8990	2

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