A Modern Strategy for the Selection of Efficacious Solar Panels at Industrial Level under an Interval-Valued Picture Fuzzy Set Hybrid Environment

Abstract

Solar panels hold a significant amount of importance at an industrial level because they contribute to environmental sustainability by reducing carbon emissions, contribute to cost savings through reduced electricity bills, provide energy independence and reliability, and serve as a long-term investment with attractive returns. One of the objectives of this research article is to address the limitations of the current score function utilized in multi-criteria decision-making (MCDM) problems within an interval-valued picture fuzzy (IVPF) environment and to present an enhanced score function. Moreover, IVPF hybrid arithmetic operators and IVPF hybrid geometric operators are introduced in this article. These operators are further thoroughly examined to identify their key characteristics. By conducting a case study, an algorithm is formulated to select the most suitable solar panel to maximize energy availability at the industrial level in the framework of the newly proposed IVPF hybrid operators. In addition, a comprehensive comparative study is conducted to demonstrate the significance and validity of recently proposed novel techniques compared with existing methods.

1. Introduction

Decision making enables individuals and organizations to select the best course of action among choices, ensuring efficient allocation and utilization of resources while managing potential risks. Multi-criteria decision-making (MCDM) stands out as a prominent category of decision-making problems that seek to identify the optimal alternative by taking into account multiple criteria during the selection process. The use of aggregation operators makes it simple to find solutions to problems that arise in the real world and may be applied in a wide variety of disciplines, including science, engineering, environmental and social studies, and many more. The primary objective of aggregation operators is to reduce the number of distinct values to a single value by combining them all together. This ensures that each value will be taken into consideration when arriving at the final result of the aggregation. Crisp sets were frequently utilized as a means of assisting individuals in decision making prior to the development of aggregation operators.

In real-world situations, being a part of a set is not always entirely evident, and one can only apply conventional mathematical procedures in a limited number of contexts. This is notably the case in the fields of languages, psychology, economics, and the social sciences, such as biology and sociology.

Partial set membership was first described by Zadeh [1] in 1965, who coined the term “fuzzy set” (FS) to describe it. In later developments, several operators were added to the fuzzy environment to help with fuzzy decision-making situations where there was more doubt and ambiguity in the information than in a crisp set. Atanassov [2] first addressed the concept of an intuitionistic fuzzy set (IFS) in 1986. In this phenomenon, each element is given a membership degree and a non-membership degree to find out more about how fuzzy decision information works. In 1988, fuzzy aggregation operators were first proposed by Yager [3]. Intuitionistic fuzzy aggregation operators were first described by Xu [4] in 2007. Using Einstein’s procedures, Wang and Liu [5] have developed novel intuitionistic fuzzy geometric aggregation operators. However, decision makers faced a lot of difficulties in fairly evaluating their own ideas when employing intuitionistic fuzzy sets (IFS) in practice, due to the availability of limited and ambiguous data. Atanassov and Gargov [6] came up with the idea of IV intuitionistic fuzzy sets (IVIFS) in order to solve the problems brought up by these circumstances. Following that, in 2007, Wei and Wang [7] developed this concept further by presenting geometric aggregation operations that were based on IVIFS. Additional advancements in the realm of IVIFS involved the creation of IV intuitionistic fuzzy aggregation operators by Wang et al. [8] in 2012. A brief look at recent developments in the IFS and IVIFS fields is covered in [9,10], respectively.

In certain circumstances, IFS may not be sufficient because human perspectives require more than the typical “yes” and “no” answers for membership degrees and non-membership degrees. Moreover, Cuong and Kreinovich emerged with the ideas of a picture fuzzy set (PFS) [11] and an IVPF set (IVPFS) [12] to deal with these kinds of circumstances. These sets give each part of the universe a certain amount of positive membership, neutral membership, and negative membership. This facilitates decision makers in a more flexible way to make decisions. It’s important to note that the IVPF set is a strong generalization of the PF set and gives even more freedom when making decisions.

The PFS correlation values suggested by Singh [13] were utilized in a cluster analysis. Thong [14] designed a novel hybrid model by combining PF clustering with fuzzy recommender systems. This model has applicability in medical analysis. Thong and Son [15] pioneered advanced fuzzy clustering algorithms based on PFSs, aimed at augmenting the precision of time series and weather forecasting. In [16], the authors provided a method for classifying representable picture t-norms and t-conorms that are compatible with PFSs. It is also important to discuss recent advancements in PFSs, such as those listed in [17,18,19,20,21,22,23,24,25,26].

In 2017, Garg [27] conversed about some PF aggregation operators. Wang et al. [28] explored some geometric aggregation operators based on PFS in 2017. Wei [29] presented the concept of PF aggregation operators to MADM in 2017. In addition, other helpful averaging operators were discussed in [30,31,32,33,34,35,36,37,38,39,40,41] for a variety of sets.

Europe and Asia have both felt the repercussions of the world’s energy crisis. It is not just Pakistan. The energy industry in Pakistan is in a dilemma since it has not been able to meet the country’s growing demand for power over the past few decades. Pakistan is dependent on imports of foreign oil and gas supplies. The recent rise in energy prices has shown us what might happen to the market in the future if the switch to low-carbon energy sources is not controlled or emphasized enough. In the modern arena, solar energy has emerged as a great source to fulfill the needs for energy globally. Solar power can be used by power plants to create electricity with low ecological impact. Solar power generation does not contribute to environmental degradation in many aspects.

Pakistan is a great place to harness solar power because of its abundance of sunny, hot cities. Solar power has been discussed as a possible economic game changer in recent years. Pakistan has a lot of potential in the field of solar power generation.

Currently, we rely almost exclusively on nonrenewable resources like oil, gas, coal, and uranium for our energy needs. Oil will be the first fossil fuel to deplete at the current consumption rate. One of the reasons there is not enough electricity to go around is a lack of investment in power-generation infrastructure. Companies that generate electricity often use outdated machinery, which reduces output. In addition, there have been a lot of useful techniques developed to deal with the issue of energy crises in [42,43,44,45].

Renewable and nonpolluting sources of energy include the sun, water, wind, geothermal, and biomass. We must also reduce our energy consumption by implementing new energy infrastructure, such as smart grid systems and smart cities. It is also crucial to upgrade to energy-efficient appliances as necessary. LEDs with solar panels, for instance, might be used to replace incandescent light bulbs.

Dealing with uncertainty is a crucial part of solving the energy crisis. Uncertainty can come from a number of places, such as fluctuating energy prices, unplanned interruptions in supply, technological developments, changes in rules and regulations, and environmental factors.

The ability to deal with indeterminacy, obscurity, and ambivalence is essential for effective solar panel selection, and IVPFSs are well suited for this task. They have the potential to provide a more accurate solution to the problem of perplexing data regarding the various factors that go into selecting a solar panel.

The IVPFS provides a more thorough and precise reading of the uncertain information when the positive membership function, neutral membership function, and negative membership function in a PFS are not exact real numbers but instead have uncertainty associated with them. In the present study, we develop a set of IVPF hybrid aggregation operators using classical arithmetic and geometric operations. We also designed a technique for selecting efficient solar panels at an industrial level in an IVPFS environment.

The following are the most substantial contributions we hope to make with this article:

• Use IVPF aggregation operators to come up with an algorithm that can solve MADM problems. This will provide an organized way to use the new operators to analyze and judge complex decision-making problems.
• Utilize the recently proposed technique for determining the best solar panel type within a certain industry. This will involve the implementation of the proposed method to real-world scenarios, such as determining the best solar panel type for reducing energy crises. Moreover, examine the advantages of the new approach by contrasting it with conventional ways.

The rest of the details in this article are as follows: The originality of the work in question can be evaluated with the help of a few concepts reviewed in Section 2. In Section 3, a new score function, the IVPFHA operator, and the IVPFHG operator are introduced. In addition, proofs of some of their fundamental properties are presented in Appendix A. In Section 4, IVPFS data is used to economically choose an industrial solar panel, and a comprehensive comparative study is conducted to prove the efficacy of new methods compared to previous methods.

2. Preliminaries

This section of the article looks at several foundational ideas that are related to understanding the novelty of this work. We also go through the fundamental set-theoretical operations of IVPFS that underpin our latest investigation.

Definition 1

[46]. The interval-valued fuzzy set $K$ of a universe $X$ is defined as $K=\left\{\right(\mathrm{𝔵},H_K\left(\mathrm{𝔵}\right)\left)\right|\mathrm{𝔵}\in X\},$ where $H_K$ is defined as the set of all subintervals of a closed unit interval.

Definition 2

[11]. Let $X$ be the universe. A PFS $B$ is defined as $B=\left\{\right(\mathrm{𝔵},{\mathrm{Ґ}}_B\left(\mathrm{𝔵}\right) , {\mathrm{Ғ}}_B\left(\mathrm{𝔵}\right), {\mathrm{ը}}_B\left(\mathrm{𝔵}\right)\left)\right|\mathrm{𝔵}\in X\},$ where ${\mathrm{Ґ}}_B:X\to \left[0,1\right]$, ${\mathrm{Ғ}}_B:X\to \left[0,1\right]$, and ${\mathrm{ը}}_B:X\to \left[0,1\right]$ represent positive membership, neutral membership, and negative membership functions, respectively, which satisfies the condition: $0\le {\mathrm{Ґ}}_B\left(\mathrm{𝔵}\right)+{\mathrm{Ғ}}_B\left(\mathrm{𝔵}\right)+{\mathrm{ը}}_B\left(\mathrm{𝔵}\right)\le 1,\forall \mathrm{𝔵}\in X.$ Moreover, the refusal membership of the PFS $B$ is defined by the following: ${\mathrm{џ}}_B\left(\mathrm{𝔵}\right)=1-{\mathrm{Ґ}}_B\left(\mathrm{𝔵}\right)-{\mathrm{Ғ}}_B\left(\mathrm{𝔵}\right)-{\mathrm{ը}}_B\left(\mathrm{𝔵}\right)$. It is noteworthy that the positive, neutral, and negative membership degrees of $\mathrm{𝔵}\in X$ are illustrated as $\mathrm{𝔵}=\left({\mathrm{Ґ}}_B\left(\mathrm{𝔵}\right),{\mathrm{Ғ}}_B\left(\mathrm{𝔵}\right),{\mathrm{ը}}_B\left(\mathrm{𝔵}\right)\right).$ This illustration is called a PF number (PFN).

Definition 3

[12]. Let $X$ be the universe. An IVPFS $B$ is defined as $B=\left\{\left(\mathrm{𝔵},\left[{\mathrm{Ґ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{Ґ}}_B^U\left(\mathrm{𝔵}\right)\right],\left[{\mathrm{Ғ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{Ғ}}_B^U\left(\mathrm{𝔵}\right)\right],\left[{\mathrm{ը}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{ը}}_B^U\left(\mathrm{𝔵}\right)\right]\right)\right|\mathrm{𝔵}\in X\},$ where $\left[{\mathrm{Ґ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{Ґ}}_B^U\left(\mathrm{𝔵}\right)\right]$, $\left[{\mathrm{Ғ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{Ғ}}_B^U\left(\mathrm{𝔵}\right)\right]$, and $\left[{\mathrm{ը}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{ը}}_B^U\left(\mathrm{𝔵}\right)\right]$, respectively, represent positive membership, neutral membership, and negative membership degrees of $\mathrm{𝔵}$ that admit the condition: $0\le {\mathrm{Ґ}}_B^L\left(\mathrm{𝔵}\right)\le {\mathrm{Ґ}}_B^U\left(\mathrm{𝔵}\right)\le 1$,$0\le {\mathrm{Ғ}}_B^L\left(\mathrm{𝔵}\right)\le {\mathrm{Ғ}}_B^U\left(\mathrm{𝔵}\right)\le 1$ and $0\le {\mathrm{ը}}_B^L\left(\mathrm{𝔵}\right)\le {\mathrm{ը}}_B^U\left(\mathrm{𝔵}\right)\le 1.$ Moreover, $0\le {\mathrm{Ґ}}_B^U\left(\mathrm{𝔵}\right)+{\mathrm{Ғ}}_B^U\left(\mathrm{𝔵}\right)+{\mathrm{ը}}_B^U\left(\mathrm{𝔵}\right)\le 1.$ The refusal degree of the IVPFS $B$ is described as follows: ${\mathrm{џ}}_B\left(\mathrm{𝔵}\right)=\left[{\mathrm{џ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{џ}}_B^U\left(\mathrm{𝔵}\right)\right],=\left[1-{\mathrm{Ґ}}_B^L\left(\mathrm{𝔵}\right)-{\mathrm{Ғ}}_B^L\left(\mathrm{𝔵}\right)-{\mathrm{ը}}_B^L\left(\mathrm{𝔵}\right),1-{\mathrm{Ґ}}_B^U\left(\mathrm{𝔵}\right)-{\mathrm{Ғ}}_B^U\left(\mathrm{𝔵}\right)-{\mathrm{ը}}_B^U\left(\mathrm{𝔵}\right)\right].$ For convenience, we write the intervals $\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right]$ and $\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]$instead of $\left[{\mathrm{Ґ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{Ґ}}_B^U\left(\mathrm{𝔵}\right)\right],\left[{\mathrm{Ғ}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{Ғ}}_B^U\left(\mathrm{𝔵}\right)\right]$ and $\left[{\mathrm{ը}}_B^L\left(\mathrm{𝔵}\right),{\mathrm{ը}}_B^U\left(\mathrm{𝔵}\right)\right]$, respectively, in this work. It is noteworthy that the positive membership, neutral membership, and negative membership degrees of $\mathrm{𝔵}\in X$ are illustrated as $\mathrm{𝔵}=\left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)$. This illustration is called an IV picture fuzzy number (IVPFN).

Definition 4

[47]. For a non-negative real number $\gamma$ and any two IVPFNs, ${\varsigma }_1=\left(\left[{\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{w}}}_1\right],\left[{\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{h}}}_1\right],\left[{\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{z}}}_1\right]\right)$ and ${\varsigma }_2=\left(\left[{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_2\right],\left[{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_2\right],\left[{\widehat{\mathfrak{y}}}_2,{\widehat{\mathfrak{z}}}_2\right]\right).$ The fundamental operations on these numbers are defined as follows:

• $\hspace{0.17em}\overline{{\varsigma }_1}=\left(\left[{\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{z}}}_1\right],\left[{\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{h}}}_1\right],\left[{\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{w}}}_1\right]\right)$;
• ${\varsigma }_1\bigcap {\varsigma }_2=\left(\left[min\left({\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{u}}}_2\right),min\left({\widehat{\mathfrak{w}}}_1,{\widehat{\mathfrak{w}}}_2\right)\right],\left[max\left({\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{f}}}_2\right),max\left({\widehat{\mathfrak{h}}}_1,{\widehat{\mathfrak{h}}}_2\right)\right],\left[max\left({\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{y}}}_2\right),max\left({\widehat{\mathfrak{z}}}_1,{\widehat{\mathfrak{z}}}_2\right)\right]\right)$;
• $\hspace{0.17em}{\varsigma }_1\bigcup {\varsigma }_2=\left(\left[max\left({\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{u}}}_2\right),max\left({\widehat{\mathfrak{w}}}_1,{\widehat{\mathfrak{w}}}_2\right)\right],\left[min\left({\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{f}}}_2\right),min\left({\widehat{\mathfrak{h}}}_1,{\widehat{\mathfrak{h}}}_2\right)\right],\left[min\left({\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{y}}}_2\right),min\left({\widehat{\mathfrak{z}}}_1,{\widehat{\mathfrak{z}}}_2\right)\right]\right)$;
• ${\varsigma }_1\oplus {\varsigma }_2=\left(\left[{\widehat{\mathfrak{u}}}_1+{\widehat{\mathfrak{u}}}_2-{\widehat{\mathfrak{u}}}_1{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_1+{\widehat{\mathfrak{w}}}_2-{\widehat{\mathfrak{w}}}_1{\widehat{\mathfrak{w}}}_2\right],\left[{\widehat{\mathfrak{f}}}_1{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_1{\widehat{\mathfrak{h}}}_2\right],\left[{\widehat{\mathfrak{y}}}_1{\widehat{\mathfrak{y}}}_2,{\widehat{\mathfrak{z}}}_1{\widehat{\mathfrak{z}}}_2\right]\right)$;
• ${\varsigma }_1\otimes {\varsigma }_2=\left(\left[{\widehat{\mathfrak{u}}}_1{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_1{\widehat{\mathfrak{w}}}_2\right],\left[{\widehat{\mathfrak{f}}}_1+{\widehat{\mathfrak{f}}}_2-{\widehat{\mathfrak{f}}}_1{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_1+{\widehat{\mathfrak{h}}}_2-{\widehat{\mathfrak{h}}}_1{\widehat{\mathfrak{h}}}_2\right],\left[{\widehat{\mathfrak{y}}}_1+{\widehat{\mathfrak{y}}}_2-{\widehat{\mathfrak{y}}}_1{\widehat{\mathfrak{y}}}_2,{\widehat{\mathfrak{z}}}_1+{\widehat{\mathfrak{z}}}_2-{\widehat{\mathfrak{z}}}_1{\widehat{\mathfrak{z}}}_2\right]\right)$;
• $\gamma {\varsigma }_1=\left(\left[1-{\left(1-{\widehat{\mathfrak{u}}}_1\right)}^{\gamma },1-{\left(1-{\widehat{\mathfrak{w}}}_1\right)}^{\gamma }\right],\left[{\widehat{\mathfrak{f}}}_1^{\gamma },{\widehat{\mathfrak{h}}}_1^{\gamma }\right],\left[{\widehat{\mathfrak{y}}}_1^{\gamma },{\widehat{\mathfrak{z}}}_1^{\gamma }\right]\right)$;
• ${\varsigma }_1^{\mathsf{\gamma }}=\left(\left[{\widehat{\mathfrak{u}}}_1^{\mathsf{\gamma }},{\widehat{\mathfrak{w}}}_1^{\mathsf{\gamma }}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_1\right)}^{\mathsf{\gamma }},1-{\left(1-{\widehat{\mathfrak{h}}}_1\right)}^{\mathsf{\gamma }}\right],\left[1-{\left(1-{\widehat{\mathfrak{y}}}_1\right)}^{\mathsf{\gamma }},1-{\left(1-{\widehat{\mathfrak{z}}}_1\right)}^{\mathsf{\gamma }}\right]\right).$

Definition 5

[40]. For any two IVPFNs, ${\varsigma }_1=\left(\left[{\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{w}}}_1\right],\left[{\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{h}}}_1\right],\left[{\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{z}}}_1\right]\right)$ and ${\varsigma }_2=\left(\left[{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_2\right],\left[{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_2\right],\left[{\widehat{\mathfrak{y}}}_2,{\widehat{\mathfrak{z}}}_2\right]\right)$admit the following properties:

• ${\varsigma }_1={\varsigma }_2\iff {\widehat{\mathfrak{u}}}_1={\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_1={\widehat{\mathfrak{w}}}_2,{\widehat{\mathfrak{f}}}_1={\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_1={\widehat{\mathfrak{h}}}_2,{\widehat{\mathfrak{y}}}_1={\widehat{\mathfrak{y}}}_2$ and ${\widehat{\mathfrak{z}}}_1={\widehat{\mathfrak{z}}}_2$;
• ${\varsigma }_1<{\varsigma }_2\iff {\widehat{\mathfrak{u}}}_1<{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_1<{\widehat{\mathfrak{w}}}_2,{\widehat{\mathfrak{f}}}_1>{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_1>{\widehat{\mathfrak{h}}}_2,{\widehat{\mathfrak{y}}}_1>{\widehat{\mathfrak{y}}}_2$ and ${\widehat{\mathfrak{z}}}_1>{\widehat{\mathfrak{z}}}_2.$

Definition 6

[40]. Consider the family of IVPFNs ${\varsigma }_{ϵ}=$ {$\left(\left[{\widehat{\mathfrak{u}}}_{ϵ},{\widehat{\mathfrak{w}}}_{ϵ}\right],\left[{\widehat{\mathfrak{f}}}_{ϵ},{\widehat{\mathfrak{h}}}_{ϵ}\right],\left[{\widehat{\mathfrak{y}}}_{ϵ},{\widehat{\mathfrak{z}}}_{ϵ}\right]\right);ϵ=1,2,3,\dots ,n\}$. The IVPFWA operator, IVPFOWA operator, IVPFWG operator, and IVPFOWG operator are described in the following way:

• The IV picture fuzzy weighted arithmetic (IVPFWA) operator is a function of  $J^n\to J$ such that

$$\begin{gathered} IVPFW{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨁}}{\varrho }_{ϵ}{\varsigma }_{ϵ}= \\ \left(\begin{array}{c}\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{ϵ}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{ϵ}\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{f}}}_{ϵ}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{h}}}_{ϵ}^{{\varrho }_{ϵ}}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{y}}}_{ϵ}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{z}}}_{ϵ}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

where the associated weight vector $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T,{\varrho }_{ϵ}\in \left[0,1\right]$ such that ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1.$

2.

The IV picture fuzzy ordered weighted arithmetic operator of dimension n is a function of $J^n\to J$, which has an associated weight vector $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T$ such that ${\varrho }_{ϵ}\in \left[0,1\right]$and ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$ are defined by the following rule:

$$\begin{gathered} IVPFOW{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,···,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨁}}{\varrho }_{ϵ}{\varsigma }_{\rho \left(ϵ\right)}= \\ \left(\begin{array}{c}\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{f}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{h}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{y}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{z}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

where the corresponding permutation $of\left(1,2,3,\dots ,n\right)$ is $\left(\mathsf{\rho }\left(1\right),\mathsf{\rho }\left(2\right),\dots ,\mathsf{\rho }\left(n\right)\right)$ such that ${\varsigma }_{{\mathsf{\rho }}_{\left(ϵ-1\right)}}\ge {\varsigma }_{{\mathsf{\rho }}_{ϵ}}$ $\forall ϵ=1,2,3,\dots ,n.$

3.

The IV picture fuzzy weighted geometric (IVPFWG) operator is a function of $J^n\to J$ such that

$$\begin{gathered} IVPFW{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨂}}{\left({\varsigma }_{ϵ}\right)}^{{\varrho }_{ϵ}}= \\ \left(\begin{array}{c}\left[{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{u}}}_{ϵ}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{w}}}_{ϵ}^{{\varrho }_{ϵ}}\right],\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{ϵ}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{ϵ}\right)}^{{\varrho }_{ϵ}}\right],\\ \left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{ϵ}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{ϵ}\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

where the associated weight vector $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_n\right)}^T,{\varrho }_{ϵ}\in \left[0,1\right]$ such that ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1.$ $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{{\varrho }_{ϵ},\dots ,\varrho }_n\right)}^T,{\varrho }_{ϵ}\in \left[0,1\right]suchthat{\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1.$

4.

The IV picture fuzzy ordered weighted geometric (IVPFOWG) operator is a function of $J^n\to J$, which has an associated weight vector $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots {,\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T$ such that ${\varrho }_{ϵ}\in \left[0,1\right]$and ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$ are defined by the following rule:

$$\begin{gathered} IVPFOW{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨂}}{\left({\varsigma }_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}= \\ \left(\begin{array}{c}\left[{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{u}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\widehat{\mathfrak{w}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\\ \left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

where the corresponding permutation $of\left(1,2,3,\dots ,n\right)$ is $\left(\mathsf{\rho }\left(1\right),\mathsf{\rho }\left(2\right),\dots ,\mathsf{\rho }\left(n\right)\right)$ such that ${\varsigma }_{{\mathsf{\rho }}_{\left(ϵ-1\right)}}\ge {\varsigma }_{{\mathsf{\rho }}_{ϵ}}$ $\forall ϵ=1,2,3,\dots ,n.$

Definition 7

[47]. The score function for the IV picture fuzzy number, which is defined as $\varsigma =\left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)$, is described as follows:

$$Q\left(\varsigma \right)=\frac{\widehat{\mathfrak{u}}-\widehat{\mathfrak{f}}-\widehat{\mathfrak{y}}+\widehat{\mathfrak{w}}-\widehat{\mathfrak{h}}-\widehat{\mathfrak{z}}}{3},Q\left(\varsigma \right)\in \left[-1,1\right]$$

(1)

Moreover, the above formula satisfies the following relations for any two IVPFNs ${\varsigma }_1$  and  ${\varsigma }_2$:

• If $Q\left({\varsigma }_1\right)<Q\left({\varsigma }_2\right)$, then ${\varsigma }_1\prec {\varsigma }_2;$
• If $Q\left({\varsigma }_1\right)>Q\left({\varsigma }_2\right)$, then ${\varsigma }_1\succ {\varsigma }_2$;
• If $Q\left({\varsigma }_1\right)=Q\left({\varsigma }_2\right)$, then ${\varsigma }_1\sim {\varsigma }_2.$

3. Characteristics of IVPF Hybrid Aggregation Operators

In this section, we introduce score function, IVPF weighted hybrid arithmetic operator, IVPF weighted hybrid geometric operator, and present an algorithm to solve the MADM problem under an IVPFS environment.

3.1. Modification of Existing Score Function for IVPFS

In this section, we demonstrate the shortcomings of the score function developed in [47] in the framework of the IVPFS environment. In view of this drawback, we present a potential solution to this issue in the subsequent study.

Example 1.

Let ${\varsigma }_1=\left(\left[0.05,0.13\right],\left[0.04,0.07\right],\left[0.01,0.06\right]\right)$ and ${\varsigma }_2=\left(\left[0.13,0.16\right],\left[0.09,0.10\right],\left[0.02,0.08\right]\right)$ be two IVPFNs. Then, the application of Definition $7$ on ${\varsigma }_1$ and ${\varsigma }_2$ yields that

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

Therefore, based on property 3 of Definition 7, we have  ${\varsigma }_1\sim {\varsigma }_2.$  This shows that the existing score function is invalid. In light of our discussion, we modify this function in the following way.

Definition 8.

Consider$\varsigma$ $=\left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)$ be an IVPFN, a refined score function $\theta$is described as follows:

$$\theta \left(\varsigma \right)=\frac{\widehat{\mathfrak{u}}\widehat{\mathfrak{w}}-\widehat{\mathfrak{y}}\widehat{\mathfrak{z}}}{2}+\frac{\widehat{\mathfrak{f}}\widehat{\mathfrak{h}}}{3}$$

(2)

where $\theta \left(\varsigma \right)\in \left[-0.5,0.5\right]$. Some key features of the recently defined score function to resolve the MCDM issues within the context of IVPFS expertise are as follows. Let ${\varsigma }_1$ and ${\varsigma }_2$ be any two IVPFNs, then:

• If $\theta \left({\varsigma }_1\right)<\theta \left({\varsigma }_2\right)$, then ${\varsigma }_1\prec {\varsigma }_2$;
• If $\theta \left({\varsigma }_1\right)>\theta \left({\varsigma }_2\right)$, then ${\varsigma }_1\succ {\varsigma }_2;$
• If $\theta \left({\varsigma }_1\right)=\theta \left({\varsigma }_2\right),$ then ${\varsigma }_1\sim {\varsigma }_2.$

The subsequent illustration demonstrates the efficacy of the modified score function for the IV picture fuzzy number.

Example 2.

The application of the modified score function in Example 1 gives that  $\theta \left({\varsigma }_1\right)$ $=0.004$and $\theta \left({\varsigma }_2\right)$ $=0.013.$ This shows that ${\varsigma }_2$ is preferable to ${\varsigma }_1$ in the framework of property 1 of Definition 8.

Consequently, the above discussion describes the accuracy of the modified score function compared to the existing score function.

3.2. IVPF Hybrid Operators

In this section, we define the IVPF hybrid arithmetic operator, IVPF hybrid geometric operator, and establish a key feature of these concepts.

Definition 9.

Let  ${\varsigma }_{ϵ}=${$\left(\left[{\widehat{\mathfrak{u}}}_{ϵ},{\widehat{\mathfrak{w}}}_{ϵ}\right],\left[{\widehat{\mathfrak{f}}}_{ϵ},{\widehat{\mathfrak{h}}}_{ϵ}\right],\left[{\widehat{\mathfrak{y}}}_{ϵ},{\widehat{\mathfrak{z}}}_{ϵ}\right]\right);ϵ=1,2,3,\dots ,n\}$be the family of IVPFNs. The interval-valued picture fuzzy hybrid arithmetic (IVPFHA) operator of dimension n is a function of $J^n\to J$ and has an associated weight vector of $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T$ with 0 $\le$ ${\varrho }_{ϵ}\le 1$; ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$ defined by the rule:

$$IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,···,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨁}}{\varrho }_{ϵ}{\dot{\varsigma }}_{\rho \left(ϵ\right)}$$

where ${\dot{\varsigma }}_{\rho \left(ϵ\right)}$is the $ϵthlargestof$ ${\dot{\varsigma }}_{ϵ}=n{\mathsf{\omega }}_{ϵ}{\varsigma }_{ϵ}\mathrm{and}\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\mathsf{\omega }}_{ϵ},\dots ,{\omega }_n\right)}^T$ is the weighted vector of ${\varsigma }_{ϵ}$, $0\le {\mathsf{\omega }}_{ϵ}\le 1,\sum _{\mathsf{ϵ}=1}^n{\mathsf{\omega }}_{ϵ}=1$.

Theorem 1.

Let ${\varsigma }_{ϵ}=${$\left(\left[{\widehat{\mathfrak{u}}}_{ϵ},{\widehat{\mathfrak{w}}}_{ϵ}\right],\left[{\widehat{\mathfrak{f}}}_{ϵ},{\widehat{\mathfrak{h}}}_{ϵ}\right],\left[{\widehat{\mathfrak{y}}}_{ϵ},{\widehat{\mathfrak{z}}}_{ϵ}\right]\right);ϵ=1,2,3,\dots ,n\}$be the family of IVPFNs. The aggregated value of these IVPFNs in the context of the IVPF hybrid arithmetic operator is also an IVPFN, which is figured out as follows:

$$\begin{gathered} IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\cdots ,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨁}}{\varrho }_{ϵ}{\dot{\varsigma }}_{\rho \left(ϵ\right)}= \\ \left(\begin{array}{c}\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

(3)

where ${\varsigma }_{\rho \left(ϵ\right)}$ is the $ϵth$ largest of ${\dot{\varsigma }}_{ϵ}=n{\mathsf{\omega }}_{ϵ}{\varsigma }_{ϵ},\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\mathsf{\omega }}_{ϵ},\dots ,{\omega }_n\right)}^T$ is the weighted vector of ${\varsigma }_{ϵ}with$ $0\le {\mathsf{\omega }}_{ϵ}\le 1,\sum _{\mathsf{ϵ}=1}^n{\mathsf{\omega }}_{ϵ}=1$, and $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{{\varrho }_{ϵ},\dots ,\varrho }_n\right)}^T$ is the aggregation-associated weight vector with 0 $\le$ ${\varrho }_{ϵ}\le 1$; ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$.

Proof. 

We prove the result by applying induction to $n$ in Equation (3). The implementation of $\mathrm{Definition}9$ for $n=2$ in Equation (3) provides that:

$$IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2\right)={\varrho }_1{\dot{\varsigma }}_{\rho \left(1\right)}\oplus {\varrho }_2{\dot{\varsigma }}_{\rho \left(2\right)}$$

By applying Definition $4$ in the above relation, we have

$$\begin{gathered} {\varrho }_1{\dot{\varsigma }}_{\rho \left(1\right)}=\left(\left[1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}\right)}^{\varrho },1-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(1\right)}\right)}^{\varrho }\right],\left[{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}^{\varrho },{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}^{\varrho }\right],\left[{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}^{\varrho },{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}^{\varrho }\right]\right) \\ {\varrho }_2{\dot{\varsigma }}_{\rho \left(2\right)}=\left(\left[1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(2\right)}\right)}^{\varrho },1-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}\right)}^{\varrho }\right],\left[{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}^{\varrho },{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}^{\varrho }\right],\left[{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}^{\varrho },{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}^{\varrho }\right]\right) \\ IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2\right)= \\ \left(\begin{array}{c}\left[\begin{array}{c}\left(2-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)-\left(1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right),\\ \left(2-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)-\left(1-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)\end{array}\right],\\ \left[{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}^{\varrho 1_1},{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}^{{\varrho }_2}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right],\left[{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}^{{\varrho }_1},{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}^{{\varrho }_2}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right]\end{array}\right) \end{gathered}$$

It follows that

$$\begin{gathered} IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2\right)= \\ \left(\begin{array}{c}\left[\right(1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2},\left(1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right],\\ \left[{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right],\left[{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}^{{\varrho }_2}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right],\left[{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}^{{\varrho }_1}\right],\left[{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}^{{\varrho }_2}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right]\end{array}\right) \end{gathered}$$

Thus, the result is true for $\mathrm{n}=2$.

Assume that the statement is true for $n=q,$ i.e.,

$$\begin{gathered} IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_q\right)= \\ \left(\begin{array}{c}\left[1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^q}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^q}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^q}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^q}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right), \end{gathered}$$

Moreover, consider Equation (3) for $n=q+1$, we have

$$\begin{gathered} IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_q,{\varsigma }_{q+1}\right)= \\ {{\varrho }_1\dot{\varsigma }}_{\rho \left(1\right)}⨁{\varrho }_2{\dot{\varsigma }}_{\rho \left(2\right)},\dots ,⨁{\varrho }_q{\dot{\varsigma }}_{\rho \left(q\right)},⨁{\varrho }_{q+1}{\dot{\varsigma }}_{\rho \left(q+1\right)} \end{gathered}$$

By applying Definition 4 in the above equation, we obtain

$$\begin{gathered} IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_q,{\varsigma }_{q+1}\right)= \\ \left(\begin{array}{c}\left[\begin{array}{c}\left(1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}})+(1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)-\left(1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\left)\right(1-{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right),\\ \left(1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}})+(1-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)-\left(1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\left)\right(1-{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)\end{array}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

It follows that

$$\begin{gathered} IVPFH{A}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_q,{\varsigma }_{q+1}\right) \\ =\left(\begin{array}{c}\left[1-{\displaystyle \prod _{ϵ=1}^{q+1}}{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^{q+1}}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

Thus, the statement holds by means of the induction method. □

Example 3.

Let ${\varsigma }_1=\left(\left[0.15,0.25\right],\left[0.21,0.27\right],\left[0.29,0.32\right]\right)$, ${\varsigma }_2=\left(\left[0.19,0.23\right],\left[0.02,0.07\right],\left[0.31,0.45\right]\right)$, and ${\varsigma }_3=\left(\left[0.41,0.46\right],\left[0.19,0.22\right],\left[0.30,0.37\right]\right)$ be three IVPFNs with the weighted vector of $\omega ={\left(0.35,0.25,0.40\right)}^T$ and the aggregation-associated vector of $\varrho ={\left(0.4,0.3,0.3\right)}^T.$

Firstly, we compute the values of ${\dot{\varsigma }}_{ϵ}$, $ϵ=1,2,3$ by means of Definition 9 in the following way:

$$\begin{array}{l}{\dot{\varsigma }}_1=3\left(0.35\right)\left(\left[0.15,0.25\right],\left[0.21,0.27\right],\left[0.29,0.32\right]\right)\\ =1.05\left(\left[0.15,0.25\right],\left[0.21,0.27\right],\left[0.29,0.32\right]\right)\\ =\left(\right[1-{\left(1-0.15\right)}^{1.05},1-{\left(1-0.25\right)}^{1.05}],\\ \left[{\left(0.21\right)}^{1.05},{\left(0.27\right)}^{1.05}\right],\left[{\left(0.29\right)}^{1.05},{\left(0.32\right)}^{1.05}\right])\end{array}$$

It follows that:

$${\dot{\varsigma }}_1=\left(\left[0.16,0.26\right],\left[0.19,0.25\right],\left[0.27,0.30\right]\right)$$

Likewise, we obtain ${\dot{\varsigma }}_2=\left(\left[0.15,0.18\right],\left[0.05,0.14\right],\left[0.41,0.55\right]\right)$ and ${\dot{\varsigma }}_3=\left(\left[0.47,0.52\right],\left[0.14,0.16\right],\left[0.24,0.30\right]\right)$.

In order to obtain the values of  $\theta \left({\dot{\varsigma }}_1\right),\theta \left({\dot{\varsigma }}_2\right)$, and  $\theta \left({\dot{\varsigma }}_3\right)$, permute the above obtained values of  ${\dot{\varsigma }}_1,{\dot{\varsigma }}_2$, and  ${\dot{\varsigma }}_3$  in the framework of Equation (2), we have

$$\theta \left({\dot{\varsigma }}_1\right)=-0.004,\theta \left({\dot{\varsigma }}_2\right)=-0.099and\theta \left({\dot{\varsigma }}_3\right)=0.094.$$

The above study suggests that $\theta \left({\dot{\varsigma }}_3\right)>\theta \left({\dot{\varsigma }}_1\right)>\theta \left({\dot{\varsigma }}_2\right)$and, hence, the permuted IVPFNs are ${\dot{\varsigma }}_{\rho \left(1\right)}=\left(\left[0.47,0.52\right],\left[0.14,0.16\right],\left[0.24,0.30\right]\right),$ ${\dot{\varsigma }}_{\rho \left(2\right)}=\left(\left[0.16,0.26\right],\left[0.19,0.25\right],\left[0.27,0.30\right]\right),$ and ${\dot{\varsigma }}_{\rho \left(3\right)}=\left(\left[0.15,0.18\right],\left[0.05,0.14\right],\left[0.41,0.55\right]\right).$ Consider the following IVPFN:

$$\left(\begin{array}{c}\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\left[{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\\ \left[{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right)$$

Furthermore, by applying Definition $9$ to the above attained IVPFNs, we obtain ${\prod }_{ϵ=1}^3{\left(1-{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}={\left(1-0.47\right)}^{0.40}\times {\left(1-0.157\right)}^{0.30}\times {\left(1-0.146\right)}^{0.30}=0.709$ and ${\prod }_{ϵ=1}^3{\left(1-{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}={\left(1-0.52\right)}^{0.40}\times {\left(1-0.26\right)}^{0.30}\times {\left(1-0.18\right)}^{0.30}=0.64$.

Similarly, we can obtain

$${\prod }_{ϵ=1}^3{\left({\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.11,{\prod }_{ϵ=1}^3{\left({\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.18,{\prod }_{ϵ=1}^3{\left({\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.29and{\prod }_{ϵ=1}^3{\left({\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.36.$$

Consequently,

$$IVPFHA\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)=\left(\left[0.297,0.36\right],\left[0.11,0.17\right],\left[0.29,0.36\right]\right)$$

Remark 1.

The IVPF weighted arithmetic and IVPFO weighted arithmetic operators can be deduced from the IVPF hybrid arithmetic operator by substituting  $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_n\right)}^T={\left(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n}\right)}^T$ and  $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)}^T={\left(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n}\right)}^T$, respectively, in Theorem 1.

Definition 10. 

Let ${\varsigma }_{ϵ}=$ {$\left(\left[{\widehat{\mathfrak{u}}}_{ϵ},{\widehat{\mathfrak{w}}}_{ϵ}\right],\left[{\widehat{\mathfrak{f}}}_{ϵ},{\widehat{\mathfrak{h}}}_{ϵ}\right],\left[{\widehat{\mathfrak{y}}}_{ϵ},{\widehat{\mathfrak{z}}}_{ϵ}\right]\right);ϵ=1,2,3,\dots ,n\}$be the family of IVPFNs. The IVPFHG operator of dimension n is a function of $J^n\to J$ and has an associated weight vector $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T$ with 0 $\le$ ${\varrho }_{ϵ}\le 1$; ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$ defined by the rule:

$$IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right)=\underset{ϵ=1}{\overset{n}{⨂}}{\left({\dot{\varsigma }}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}$$

where ${\dot{\varsigma }}_{\rho \left(ϵ\right)}$ is the $ϵth\mathrm{largest}\mathrm{of}$ ${\dot{\varsigma }}_{ϵ}={\left({\varsigma }_{ϵ}\right)}^{n{\mathsf{\omega }}_{ϵ}}and\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\mathsf{\omega }}_{ϵ},\dots ,{\omega }_n\right)}^T$ is the weighted vector of ${\varsigma }_{ϵ}$, $0\le {\mathsf{\omega }}_{ϵ}\le 1,\sum _{\mathsf{ϵ}=1}^n{\mathsf{\omega }}_{ϵ}=1$.

Theorem 2.

Let  ${\varsigma }_{ϵ}=$ {$\left(\left[{\widehat{\mathfrak{u}}}_{ϵ},{\widehat{\mathfrak{w}}}_{ϵ}\right],\left[{\widehat{\mathfrak{f}}}_{ϵ},{\widehat{\mathfrak{h}}}_{ϵ}\right],\left[{\widehat{\mathfrak{y}}}_{ϵ},{\widehat{\mathfrak{z}}}_{ϵ}\right]\right);ϵ=1,2,3,\dots ,n\}$be the family of IVPFNs. Then, the aggregated value of these IVPFNs in the context of the IVPF hybrid geometric operator is also an IVPFN, which is figured out as follows:

$$\begin{gathered} IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_n\right)={⨂}_{ϵ=1}^n{\left({\dot{\varsigma }}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}= \\ \left(\begin{array}{c}\left[{\prod }_{ϵ=1}^n{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\prod }_{ϵ=1}^n{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\left[1-{\prod }_{ϵ=1}^n{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\prod }_{ϵ=1}^n{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\\ \left[1-{\prod }_{ϵ=1}^n{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\prod }_{ϵ=1}^n{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right) \end{gathered}$$

(4)

where ${\varsigma }_{\rho \left(ϵ\right)}$ is the $ϵth$ largest of ${\dot{\varsigma }}_{ϵ}={\left({\varsigma }_{ϵ}\right)}^{n{\mathsf{\omega }}_{ϵ}},\omega ={\left({\omega }_1,{\omega }_2,\dots {,\mathsf{\omega }}_{ϵ},\dots ,{\omega }_n\right)}^T$ is the weighted vector of ${\varsigma }_{ϵ}with$ $0\le {\mathsf{\omega }}_{ϵ}\le 1,\sum _{\mathsf{ϵ}=1}^n{\mathsf{\omega }}_{ϵ}=1$, and $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T$ is the aggregation-associated weight vector with 0 $\le$ ${\varrho }_{ϵ}\le 1$; ${\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$.

Proof. 

We prove the result by applying induction on $n$ in Equation $\left(4\right)$. The implementation of $\mathrm{Definition}10$ for $n=2$ in Equation $\left(4\right)$ gives that

$$IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2\right)={\left({\dot{\varsigma }}_{\rho \left(1\right)}\right)}^{{\varrho }_1}⨂{\left({\dot{\varsigma }}_{\rho \left(2\right)}\right)}^{{\varrho }_2}$$

By applying Definition $4$ in the above relation, we have

$$\begin{gathered} {\dot{\varsigma }}_1^{{\varrho }_1}=\left(\begin{array}{c}\left[{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}^{{\varrho }_1},{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(1\right)}^{{\varrho }_1}\right],\left[1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1},1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right],\\ \left[1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1},1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right]\end{array}\right) \\ {\dot{\varsigma }}_2^{{\varrho }_2}=\left(\begin{array}{c}\left[{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(2\right)}^{{\varrho }_2},{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right],\left[1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2},1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right],\\ \left[1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2},1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right]\end{array}\right) \\ IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2\right)= \\ \left(\begin{array}{c}\left[{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(2\right)}^{{\varrho }_2},{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right],\\ \left[\begin{array}{c}\left(2-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)-\left(1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right),\\ \left(2-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)-\left(1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)\end{array}\right],\\ \left[\begin{array}{c}\left(2-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)-\left(1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right),\\ \left(2-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)-\left(1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right)\end{array}\right]\end{array}\right) \end{gathered}$$

It follows that

$$\begin{gathered} IVPFH{G}_{\varrho }= \\ \left(\begin{array}{c}\left[{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(1\right)}^{{\varrho }_1}{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(2\right)}^{\varrho 1_1},{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(1\right)}^{{\varrho }_2}{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(2\right)}^{{\varrho }_2}\right],\left[(1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2},1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}\right],\\ \left[\right(1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2},1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(1\right)}\right)}^{{\varrho }_1}{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(2\right)}\right)}^{{\varrho }_2}]\end{array}\right) \end{gathered}$$

Thus, the result is true for $n=2$.

Assume that the statement is true for $n=q,$ i.e.,

$$\begin{gathered} IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_q\right)= \\ \left(\begin{array}{c}\left[{\displaystyle \prod _{ϵ=1}^q}{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^q}{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\left[1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\\ \left[1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^q}{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right), \end{gathered}$$

Moreover, consider Equation (4) for $n=q+1$, we have

$$IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_q,{\varsigma }_{q+1}\right)={\dot{\varsigma }}_{\rho \left(1\right)}^{{\varrho }_1}⨂{\dot{\varsigma }}_{\rho \left(2\right)}^{{\varrho }_2},\dots ,⨂{\dot{\varsigma }}_{\rho \left(q\right)}^{{\varrho }_q}⨂{\dot{\varsigma }}_{\rho \left(q+1\right)}^{{\varrho }_{q+1}}$$

By applying Definition 4 in the above equation, we obtain

$$\begin{gathered} IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_{q+1}\right) \\ = \\ \left(\begin{array}{c}\left[{\prod }_{ϵ=1}^{q+1}{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\prod }_{ϵ=1}^{q+1}{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\\ \left[\begin{array}{c}\left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)+\left(1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)-\left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right),\\ \left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)+\left(1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)-\left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)\end{array}\right],\\ \left[\begin{array}{c}\left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)+\left(1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)-\left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right),\\ \left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)+\left(1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)-\left(1-{\prod }_{ϵ=1}^q{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right)\left(1-{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(q+1\right)}\right)}^{{\varrho }_{q+1}}\right)\end{array}\right]\end{array}\right) \end{gathered}$$

It follows that

$$\begin{gathered} IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,\dots ,{\varsigma }_{q+1}\right) \\ =\left(\begin{array}{c}\left[{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^{q+1}}{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\left[1-{\displaystyle \prod _{ϵ=1}^{q+1}}{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^{q+1}}{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\\ \left[1-{\displaystyle \prod _{ϵ=1}^{q+1}}{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^{q+1}}{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right), \end{gathered}$$

Thus, the statement holds by means of the induction method. □

Example 4.

Let  ${\varsigma }_1=\left(\left[0.23,0.36\right],\left[0.41,0.49\right],\left[0.01,0.07\right]\right)$, ${\varsigma }_2=\left(\left[0.30,0.39\right],\left[0.11,0.17\right],\left[0.04,0.22\right]\right)$, and ${\varsigma }_3=\left(\left[0.04,0.26\right],\left[0.20,0.25\right],\left[0.13,0.21\right]\right)$ be three IVPFNs with the weighted vector $\omega ={\left(0.24,0.26,0.5\right)}^T$and the aggregation-associated vector is $\varrho ={\left(0.32,0.41,0.27\right)}^T.$

Firstly, we compute the values of ${\dot{\varsigma }}_{ϵ}$, $ϵ=1,2,3$ by means of Definition 10 in the following way:

$$\begin{array}{l}{\dot{\varsigma }}_1={\left(\left[0.15,0.25\right],\left[0.21,0.27\right],\left[0.29,0.32\right]\right)}^{3\left(0.24\right)}\\ ={\left(\left[0.15,0.25\right],\left[0.21,0.27\right],\left[0.29,0.32\right]\right)}^{0.72}\\ =(\left[{\left(0.23\right)}^{0.72},{\left(0.36\right)}^{0.72}\right],\left[1-{\left(1-0.41\right)}^{0.72},1-{\left(1-0.49\right)}^{0.72}\right],\left[1-{\left(1-0.01\right)}^{0.72},1-{\left(1-0.07\right)}^{0.72}\right]\end{array}$$

It follows that

$${\dot{\varsigma }}_1=\left(\left[0.35,0.48\right],\left[0.32,0.38\right],\left[0.01,0.05\right]\right)$$

Likewise, we obtain

$${\dot{\varsigma }}_2=\left(\left[0.39,0.48\right],\left[0.09,0.13\right],\left[0.03,0.17\right]\right)\mathrm{and}{\dot{\varsigma }}_3=\left(\left[0.01,0.13\right],\left[0.28,0.35\right],\left[0.19,0.30\right]\right)$$

In order to obtain the values of  $\theta \left({\dot{\varsigma }}_1\right),\theta \left({\dot{\varsigma }}_2\right)$, and  $\theta \left({\dot{\varsigma }}_3\right)$, permute the above obtained values of  ${\dot{\varsigma }}_1,{\dot{\varsigma }}_2$, and  ${\dot{\varsigma }}_3$  in the framework of Equation (2). We have

$$\theta \left({\dot{\varsigma }}_1\right)=0.123,\theta \left({\dot{\varsigma }}_2\right)=0.095\mathrm{and}\theta \left({\dot{\varsigma }}_3\right)=0.006.$$

The above study suggests that $\theta \left({\dot{\varsigma }}_1\right)>\theta \left({\dot{\varsigma }}_2\right)>\theta \left({\dot{\varsigma }}_3\right)$ and, hence, the permuted IVPFNs are ${\dot{\varsigma }}_{\mathsf{\rho }\left(1\right)}=\left(\left[0.35,0.48\right],\left[0.32,0.38\right],\left[0.01,0.05\right]\right)$, ${\dot{\varsigma }}_{\mathsf{\rho }\left(2\right)}=\left(\left[0.39,0.48\right],\left[0.09,0.13\right],\left[0.03,0.17\right]\right)$, and ${\dot{\varsigma }}_{\mathsf{\rho }\left(3\right)}=\left(\left[0.01,0.13\right],\left[0.28,0.35\right],\left[0.19,0.30\right]\right)$. Consider the following IVPFN:

$$\left(\begin{array}{c}\left[{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{u}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}},{\displaystyle \prod _{ϵ=1}^n}{\dot{\widehat{\mathfrak{w}}}}_{\rho \left(ϵ\right)}^{{\varrho }_{ϵ}}\right],\left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\rho \left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right],\\ \left[1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}},1-{\displaystyle \prod _{ϵ=1}^n}{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}\right]\end{array}\right)$$

Furthermore, by applying Definition 10 to the above attained IVPFNs, we obtaine ${\prod }_{ϵ=1}^3{\left({\dot{\widehat{\mathfrak{u}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}={\left(0.35\right)}^{0.32}\times {\left(0.39\right)}^{0.41}\times {\left(0.01\right)}^{0.27}=0.14.$ and ${\prod }_{ϵ=1}^3{\left({\dot{\widehat{\mathfrak{w}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}={\left(0.48\right)}^{0.32}\times {\left(0.48\right)}^{0.41}\times {\left(0.13\right)}^{0.27}=0.34.$Similarly, we can obtain, ${\prod }_{ϵ=1}^3{\left(1-{\dot{\widehat{\mathfrak{f}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.78,{\prod }_{ϵ=1}^3{\left(1-{\dot{\widehat{\mathfrak{h}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.72,{\prod }_{ϵ=1}^3{\left(1-{\dot{\widehat{\mathfrak{y}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.93\mathfrak{u}n\mathfrak{h}{\prod }_{ϵ=1}^3{\left(1-{\dot{\widehat{\mathfrak{z}}}}_{\mathsf{\rho }\left(ϵ\right)}\right)}^{{\varrho }_{ϵ}}=0.83$.

Consequently,

$$IVPFH{G}_{\varrho }\left({\varsigma }_1,{\varsigma }_2,{\varsigma }_3\right)=\left(\left[0.14,0.34\right],\left[0.22,0.28\right],\left[0.07,0.17\right]\right).$$

Remark 2.

The IVPF weighted geometric and IVPFO weighted geometric operators can be deduced from the IVPF hybrid geometric operator by substituting $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_n\right)}^T={\left(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n}\right)}^T$ and  $\omega ={\left({\omega }_1,{\omega }_2,\dots ,{\omega }_n\right)}^T={\left(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n}\right)}^T$, respectively, in Theorem 2.

3.3. Method to Solve MADM Problem with IVPF Information

In this section, we develop a mechanism to solve the MADM problem using the suggested IV picture fuzzy hybrid operator, considering that the weights of the criteria are shown as real values. IVPFNs are used to illustrate the numbers of the attributes. Let $I$ = $I_1,I_2,\dots ,I_n$ be a discrete arrangement of alternatives, $V=V_1,V_2,\dots ,V_n$ be the arrangement of attributes, and $\varrho ={\left({\varrho }_1,{\varrho }_2,\dots ,{\varrho }_{ϵ},\dots ,{\varrho }_n\right)}^T$ be the aggregation weighted vector of the attribute $V_{ϵ}\left(ϵ=1,2,3,\dots ,n\right)$, where ${\varrho }_{ϵ}\in \left[0,1\right],{\sum }_{ϵ=1}^n{\varrho }_{ϵ}=1$ and $\mathsf{\omega }={\left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2,{\mathsf{\omega }}_3,\dots ,{\mathsf{\omega }}_{ϵ},\dots {,\mathsf{\omega }}_n\right)}^T$ are the weighted vector of the attribute, $V_{ϵ}\left(ϵ=1,2,3,\dots ,n\right)$ such that $0\le {\mathsf{\omega }}_{ϵ}\le 1,ϵ=1,2,3,\dots ,n,\sum _{\mathsf{ϵ}=1}^n{\mathsf{\omega }}_{ϵ}=1$. Suppose that $\tilde{E}={\left({\varsigma }_{iϵ}\right)}_{m\times n}$ be a decision matrix with entries as IVPFNs $\left({\mathrm{Ґ}}_{iϵ},{\mathrm{Ғ}}_{iϵ},{\mathrm{ը}}_{iϵ}\right)$. This matrix captures various aspects of decision making, including ${\mathrm{Ґ}}_{iϵ}$ which represents that the alternative $I_i$ satisfies the criteria, $V_{ϵ}$ and ${\mathrm{Ғ}}_{iϵ}$ represents the neutrality of the criteria, and $V_{ϵ}$ and ${\mathrm{ը}}_{iϵ}$ represents that the alternative $I_i$ does not satisfy the criteria $V_{ϵ}$ provided by the decision maker. The following steps are included in the proposed IVPFHA-based method for designing the MADM problem-solving techniques.

Step 1. The following procedures are followed to create a decision matrix based on the information provided by the decision maker in the framework of IVPFNs as its entries:

$$\tilde{E}={\left({\varsigma }_{iϵ}\right)}_{m\times n}$$

This implies that,

$$\left(\begin{array}{ccc}{\varsigma }_{11}& \cdots & {\varsigma }_{1n}\\ ⋮& \ddots & ⋮\\ {\varsigma }_{m1}& \cdots & {\varsigma }_{mn}\end{array}\right)$$

Step 2. Obtain the normalized decision matrix by normalizing the characteristics by changing the cost criterion values into benefit criterion values, if applicable, using the normalizing procedure described below:

$${\varsigma }_{iϵ}=\left\{\begin{array}{c}\left(\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right]\right)forcosttypecriterion\\ \left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)forbenefittypecriterion\end{array}\right.$$

(5)

Step 3.

a)

Compute ${\dot{\varsigma }}_{ϵ}=n{\mathsf{\omega }}_{ϵ}{\varsigma }_{ϵ}$ for the IVPFHA operator and the score values $\theta \left({\dot{\varsigma }}_{iϵ}\right)$ using Definition 2 to obtain the hybrid decision matrix $\tilde{H}$.

b)

Compute ${\dot{\varsigma }}_{ϵ}={\left({\varsigma }_{ϵ}\right)}^{n{\mathsf{\omega }}_{ϵ}}$ for the IVPFHG operator and the score values $\theta \left({\dot{\varsigma }}_{iϵ}\right)$ using Definition 2 to obtain the hybrid decision matrix $\tilde{H}$.

Step 4. Use the implementation of the IVPFHA operator on the hybrid decision matrix $\tilde{H}$ to obtain

$$y_i=IVPFH{A}_{\varrho }\left(I_1,I_2,\dots ,I_n\right)={\oplus }_{ϵ=1}^n\left({\varrho }_{ϵ}{\dot{I}}_{{\rho }_{ϵ}}\right),y_i\left(i=1,2,3,\dots ,n\right)$$

Likewise, by applying the IVPFHG operator on the hybrid decision matrix $\tilde{H}$ to obtain

$$y_i=IVPFH{G}_{\varrho }\left(I_1,I_2,\dots ,I_n\right)={\otimes }_{ϵ=1}^n{\left({\dot{I}}_{{\mathsf{\rho }}_{ϵ}}\right)}^{{\varrho }_{ϵ}},y_i\left(i=1,2,3,\dots ,n\right).$$

Step 5. Compute the score values of the alternatives $I_i$ by using $\theta \left(\varsigma \right)=\frac{\widehat{\mathfrak{u}}\widehat{\mathfrak{w}}-\widehat{\mathfrak{y}}\widehat{\mathfrak{z}}}{2}+\frac{\widehat{\mathfrak{f}}\widehat{\mathfrak{h}}}{3}$

Step 6. Rank all the alternatives and select the optimal one.

The pictorial view of the method is illustrated in Figure 1.

4. Results and Discussions

In this section, the recently developed strategy is applied to select an efficient solar panel, and a comprehensive comparative analysis of this study is presented.

4.1. Selection of Efficient Solar Panel at an Industrial Level under IVPFS Environment

A country needs a steady supply of reliable energy to power its manufacturing plants, farms, businesses, and residences. One of the most significant challenges facing developing nations is access to reliable power. Solar energy has become a popular energy source for companies because it is environmentally friendly, cost effective, and has a long lifespan. Solar panels are installed to harness the sun’s rays. Solar power is a renewable, eco-friendly energy solution that does not contribute to global warming by emitting greenhouse gases. A certain industry cannot function without its own backup power source in the event of a power outage. In this scenario, a certain industry is looking to purchase solar panels to improve its energy capacity. The best solar panels will be effective and will not damage your electronics. Many companies produce and install solar panels. A certain industry has four options $(I_1,I_2,I_3,I_4)$ to fulfill the needs of its solar plant:

(i)

$I_1$: Polycrystalline solar panel;

(ii)

$I_2$: Monocrystalline solar panel;

(iii)

$I_3$: Thin film solar panel;

(iv)

$I_4$: Concentrated solar panel.

The expert assesses the quality of solar panels on the basis of the following attributes:

• $(V_1$) Cost per unit of electricity: Cost optimization, such as lowering installation, maintenance, and operational costs, boosts the return on investment and helps transition to sustainable and cost-effective energy solutions.
• ${(V}_2)$ Climate and weather factors: Solar panel performance and durability are affected by environmental factors such as varying solar availability, high-temperature resilience, temperature changes, and extreme weather occurrences.
• $\left(V_3\right)$ Expected life of the solar panel: It determines the long-term durability, reliability, and financial returns of the solar energy system. A longer expected life translates to a higher return on investment and greater sustainability in electricity generation.
• ${(V}_4)$ Space efficiency: It allows for maximizing power generation within the limited available areas. With limited land or rooftop space, efficient utilization is crucial to achieve optimal energy production. High space efficiency enables the installation of more panels, increasing the overall capacity and output of the solar energy system.

In

Table 1. Corresponding weights of the attributes.

Attribute	Weights
Cost per unit electricity	$0.29$
Climate and weather factors	$0.21$
Expected life of solar panel	$0.27$
Space efficiency	$0.23$

, an expert assigns the corresponding weights to the attributes on the basis of the above discussion of their respective significances.

According to the corresponding aggregated weight vector $\varrho ={\left(0.29,0.21,0.27,0.23\right)}^T$ and associated weight vector $\mathsf{\omega }={\left(0.30,0.20,0.26,0.24\right)}^T$ of the alternative assumed hypothetically to aggregate information given by the expert, the following steps can be taken:

• Step 1: The decision matrix displays the information by means of the IVPFNs arranged by the expert in

  Table 2. Decision matrix displaying expert IVPF assessment of physical situation.

  	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$
  $I_1$	$\left(\begin{array}{c}\left[0.60,0.70\right],\\ \left[0.01,0.04\right],\\ \left[0.15,0.20\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.50,0.60\right],\\ \left[0.05,0.06\right],\\ \left[0.20,0.25\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.50,0.65\right],\\ \left[0.30,0.31\right],\\ \left[0.01,0.02\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.50,0.70\right],\\ \left[0.00,0.01\right],\\ \left[0.20,0.25\right]\end{array}\right)$
  $I_2$	$\left(\begin{array}{c}\left[0.60,0.65\right],\\ \left[0.01,0.10\right],\\ \left[0.20,0.25\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.40,0.50\right],\\ \left[0.01,0.05\right],\\ \left[0.10,0.15\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.55,0.60\right],\\ \left[0.20,0.25\right],\\ \left[0.10,0.15\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.60,0.75\right],\\ \left[0.05,0.07\right],\\ \left[0.10,0.15\right]\end{array}\right)$
  $I_3$	$\left(\begin{array}{c}\left[0.35,0.40\right],\\ \left[0.20,0.25\right],\\ \left[0.30,0.32\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.30,0.35\right],\\ \left[0.20,0.30\right],\\ \left[0.15,0.20\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.25,0.35\right],\\ \left[0.20,0.30\right],\\ \left[0.20,0.25\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.70,0.80\right],\\ \left[0.01,0.05\right],\\ \left[0.10,0.15\right]\end{array}\right)$
  $I_4$	$\left(\begin{array}{c}\left[0.70,0.80\right],\\ \left[0.01,0.05\right],\\ \left[0.01,0.10\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.40,0.45\right],\\ \left[0.30,0.35\right],\\ \left[0.10,0.12\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.40,0.45\right],\\ \left[0.30,0.37\right],\\ \left[0.01,0.10\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.80,0.82\right],\\ \left[0.02,0.05\right],\\ \left[0.01,0.10\right]\end{array}\right)$

  .
• Step 2:

  Table 3. Normalized decision matrix.

  	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$
  $I_1$	$\left(\begin{array}{c}\left[0.15,0.20\right],\\ \left[0.01,0.04\right],\\ \left[0.60,0.70\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.50,0.60\right],\\ \left[0.05,0.06\right],\\ \left[0.20,0.25\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.50,0.65\right],\\ \left[0.30,0.31\right],\\ \left[0.01,0.02\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.20,0.25\right],\\ \left[0.00,0.01\right],\\ \left[0.50,0.70\right]\end{array}\right)$
  $I_2$	$\left(\begin{array}{c}\left[0.20,0.25\right],\\ \left[0.01,0.10\right],\\ \left[0.60,0.65\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.40,0.50\right],\\ \left[0.01,0.05\right],\\ \left[0.10,0.15\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.55,0.60\right],\\ \left[0.20,0.25\right],\\ \left[0.10,0.15\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.10,0.15\right],\\ \left[0.05,0.07\right],\\ \left[0.60,0.75\right]\end{array}\right)$
  $I_3$	$\left(\begin{array}{c}\left[0.30,0.32\right],\\ \left[0.20,0.25\right],\\ \left[0.35,0.40\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.30,0.35\right],\\ \left[0.20,0.30\right],\\ \left[0.15,0.20\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.25,0.35\right],\\ \left[0.20,0.30\right],\\ \left[0.20,0.25\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.10,0.15\right],\\ \left[0.01,0.05\right],\\ \left[0.70,0.80\right]\end{array}\right)$
  $I_4$	$\left(\begin{array}{c}\left[0.01,0.10\right],\\ \left[0.01,0.05\right],\\ \left[0.70,0.80\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.40,0.45\right],\\ \left[0.30,0.35\right],\\ \left[0.10,0.12\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.40,0.45\right],\\ \left[0.30,0.37\right],\\ \left[0.01,0.10\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.01,0.10\right],\\ \left[0.02,0.05\right],\\ \left[0.80,0.82\right]\end{array}\right)$

  describes the normalized decision matrix obtained using the above information in Equation (5).
• Step 3:

  a)

  Obtain the decision matrix by computing the values of ${\dot{\varsigma }}_{ϵ}=n{\mathsf{\omega }}_{ϵ}{\varsigma }_{ϵ}$ and score values $\theta \left({\dot{\varsigma }}_{iϵ}\right)$ $i=(1,2,3,4),e=(1,2,3,4)$ for the IVPFHA operator.

  Table 4. Hybrid decision matrix under IVPFHA.

  	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$
  $I_1$	$\left(\begin{array}{c}\left[0.51,0.66\right],\\ \left[0.29,0.30\right],\\ \left[0.01,0.02\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.43,0.52\right],\\ \left[0.09,0.10\right],\\ \left[0.28,0.33\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.18,0.24\right],\\ \left[0.00,0.02\right],\\ \left[0.54,0.65\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.19,0.24\right],\\ \left[0.00,0.01\right],\\ \left[0.51,0.71\right]\end{array}\right)$
  $I_2$	$\left(\begin{array}{c}\left[0.56,0.61\right],\\ \left[0.19,0.24\right],\\ \left[0.09,0.13\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.34,0.43\right],\\ \left[0.02,0.09\right],\\ \left[0.16,0.22\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.24,0.29\right],\\ \left[0.00,0.06\right],\\ \left[0.54,0.56\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.10,0.14\right],\\ \left[0.06,0.08\right],\\ \left[0.61,0.76\right]\end{array}\right)$
  $I_3$	$\left(\begin{array}{c}\left[0.26,0.36\right],\\ \left[0.18,0.29\right],\\ \left[0.19,0.24\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.25,0.29\right],\\ \left[0.27,0.38\right],\\ \left[0.22,0.28\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.35,0.37\right],\\ \left[0.14,0.19\right],\\ \left[0.28,0.33\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.10,0.14\right],\\ \left[0.01,0.06\right],\\ \left[0.71,0.80\right]\end{array}\right)$
  $I_4$	$\left(\begin{array}{c}\left[0.41,0.46\right],\\ \left[0.29,0.36\right],\\ \left[0.01,0.09\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.34,0.38\right],\\ \left[0.38,0.43\right],\\ \left[0.15,0.18\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.01,0.12\right],\\ \left[0.00,0.03\right],\\ \left[0.65,0.77\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.01,0.10\right],\\ \left[0.02,0.06\right],\\ \left[0.80,0.83\right]\end{array}\right)$

  describes the information obtained from the above process.

  b)

  Obtain the decision matrix by computing the values of ${\dot{\varsigma }}_{ϵ}={\left({\varsigma }_{ϵ}\right)}^{n{\mathsf{\omega }}_{ϵ}}$ and score values $\theta \left({\dot{\varsigma }}_{iϵ}\right)$ $i=(1,2,3,4),e=(1,2,3,4)$ for the IVPFHG operator.

  Table 5. Hybrid decision matrix under IVPFHG.

  	${\mathit{V}}_1$	${\mathit{V}}_2$	${\mathit{V}}_3$	${\mathit{V}}_4$
  $I_1$	$\left(\begin{array}{c}\left[0.49,0.64\right],\\ \left[0.31,0.32\right],\\ \left[0.01,0.02\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.57,0.66\right],\\ \left[0.04,0.08\right],\\ \left[0.16,0.21\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.21,0.26\right],\\ \left[0.00,0.01\right],\\ \left[0.49,0.69\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.10,0.15\right],\\ \left[0.01,0.05\right],\\ \left[0.67,0.76\right]\end{array}\right)$
  $I_2$	$\left(\begin{array}{c}\left[0.54,0.58\right],\\ \left[0.21,0.26\right],\\ \left[0.10,0.16\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.48,0.57\right],\\ \left[0.01,0.04\right],\\ \left[0.08,0.12\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.11,0.16\right],\\ \left[0.05,0.07\right],\\ \left[0.59,0.74\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.15,0.19\right],\\ \left[0.01,0.12\right],\\ \left[0.67,0.69\right]\end{array}\right)$
  $I_3$	$\left(\begin{array}{c}\left[0.38,0.43\right],\\ \left[0.16,0.25\right],\\ \left[0.16,0.21\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.24,0.34\right],\\ \left[0.21,0.31\right],\\ \left[0.21,0.26\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.24,0.25\right],\\ \left[0.24,0.29\right],\\ \left[0.4,0.46\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.11,0.16\right],\\ \left[0.01,0.05\right],\\ \left[0.69,0.79\right]\end{array}\right)$
  $I_4$	$\left(\begin{array}{c}\left[0.48,0.52\right],\\ \left[0.25,0.29\right],\\ \left[0.08,0.10\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.39,0.44\right],\\ \left[0.31,0.38\right],\\ \left[0.01,0.10\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.01,0.11\right],\\ \left[0.02,0.05\right],\\ \left[0.79,0.81\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.00,0.06\right],\\ \left[0.01,0.06\right],\\ \left[0.76,0.85\right]\end{array}\right)$

  describes the information obtained from the above process.

• Step 4: The application of the IVPFHA operator and the corresponding weighted vector $\varrho ={\left(0.29,0.21,0.27,0.23\right)}^T$ on the hybrid decision matrix is presented in Table 4 and the application of the IVPFHG operator and the corresponding weighted vector $\varrho ={\left(0.29,0.21,0.27,0.23\right)}^T$ on the hybrid decision matrix is presented in Table 5. The aggregated values of the alternatives under IVPFHA (IVPFHG) operators are described in

  Table 6. Aggregated values of the alternatives under IVPFHA (IVPFHG) operators.

  	${\mathit{V}}_1$	${\mathit{V}}_2$
  $I_1$	$\left(\begin{array}{c}\left[0.35,0.45\right],\\ \left[0.00,0.05\right],\\ \left[0.15,0.21\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.28,0.36\right],\\ \left[0.11,0.13\right],\\ \left[0.38,0.50\right]\end{array}\right)$
  $I_2$	$\left(\begin{array}{c}\left[0.35,0.40\right],\\ \left[0.00,0.10\right],\\ \left[0.25,0.32\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.25,0.32\right],\\ \left[0.08,0.13\right],\\ \left[0.42,0.52\right]\end{array}\right)$
  $I_3$	$\left(\begin{array}{c}\left[0.25,0.30\right],\\ \left[0.09,0.19\right],\\ \left[0.30,0.35\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.23,0.28\right],\\ \left[0.16,0.23\right],\\ \left[0.40,0.48\right]\end{array}\right)$
  $I_4$	$\left(\begin{array}{c}\left[0.22,0.29\right],\\ \left[0.00,0.13\right],\\ \left[0.15,0.31\right]\end{array}\right)$	$\left(\begin{array}{c}\left[0.00,0.20\right],\\ \left[0.15,0.20\right],\\ \left[0.54,0.61\right]\end{array}\right)$

  .
• Step 5: Determine the score value of each alternative by means of the information obtained in Step 4 and Equation (2). These values are listed in

  Table 7. Score values of alternatives.

  	IVPFHA	IVPFHG
  $I_1$	0.063	−0.040
  $I_2$	0.030	−0.066
  $I_3$	−0.009	−0.052
  $I_4$	0.009	−0.155

  .

The graphical description of the ranking of the alternatives is depicted in Figure 2.

• Step 6:

  (a)

  In view of column 1 of Table 7, we note that $\theta \left(I_1\right)>\theta \left(I_2\right)>\theta \left(I_4\right)>\theta \left(I_3\right).$ This shows that $I_1\succ I_2\succ I_4\succ I_3$.

  (b)

  In view of column 2 of Table 7, we note that $\theta \left(I_1\right)>\theta \left(I_3\right)>\theta \left(I_2\right)>\theta \left(I_4\right).$ This shows that $I_1\succ I_2\succ I_4\succ I_3$.

From the above discussion, we conclude that polycrystalline solar panels are the best choice for a certain industry to fulfill the needs of its solar plant.

4.2. Comparative Analysis

A comparative study is conducted to demonstrate that the suggested method is feasible and beneficial for the subsequent discussion. The article’s newly specified strategies are compared and contrasted with currently used methods in

Table 8. Comparison table of proposed strategies with existing ones.

Methods	Order of Preference	Best Alternative
IVPFHA	$I_1\succ I_2\succ I_4\succ I_3$	$I_1$
IVPFHG	$I_1\succ I_3\succ I_2\succ I_4$	$I_1$
IVPFWA [40]	$I_1\succ I_2\succ I_4\succ I_3$	$I_1$
IVPFOWA [40]	$I_1\succ I_2\succ I_4\succ I_3$	$I_1$
IVPFWG [40]	$I_1\succ I_3\succ I_2\succ I_4$	$I_1$
IVPFWOG [40]	$I_1\succ I_3\succ I_2\succ I_4$	$I_1$

, namely, the IVPFWA operator, IVPFOWA operator, IVPFWG operator, and IVPFOWG operator.

It is quite evident that our approaches are more flexible and generic than some of the existing methods to tackle picture fuzzy MADM problems. It is also important to note that interval-valued picture fuzzy aggregation operators have a few restrictions that may cause data loss, whereas, the proposed aggregation operators are comprehensive in their consideration of all relevant factors, as the hybrid operators weigh the arguments and their position simultaneously. Thus, the IVPFS aggregation operators introduced by Masmali et al. [40] are in fact a specific case of IVPFH aggregation operators.

The graphical explanation of the comparative study is illustrated in Figure 3.

5. Conclusions

In this article, a refined score function has been formulated to address the limitations of the existing score function and resolve the MCDM problem within the framework of the IVPF environment. The concepts of IVPFHA and IVPFHG operators have been presented. Moreover, an algorithm has been developed to tackle the MCDM problem using these novel strategies. A case study has been conducted to show that solar energy is essential for solving environmental problems, maintaining power reliability, boosting the economy, and giving more people access to electricity. In addition, an appropriate mechanism has been designed to select the most suitable solar panel at the industrial level within the capacity of the recently proposed strategies. Lastly, a comparative analysis has been set up to demonstrate that the suggested techniques are valid and feasible.

It is important to note that the study presented in this article is, in fact, a theoretical one. Our main goal in future work will be to develop computer programming in order to create an advanced, practically effective decision analysis tool based on IVPF aggregation operators that is more useful and applicable in the real world.

In addition, the techniques described here can be applied in the selection of cultivating crops in the agricultural sector, as well as in the evaluation of software in the IT sector. These initiatives will allow the efficient and inexpensive resolution of a wide variety of critical MADM problems.