Stock Reordering Decision Making under Interval Valued Picture Fuzzy Knowledge

Abstract

Symmetrical and asymmetrical information plays a critical role in resolving many issues. The implications of symmetry and asymmetry in interval-valued picture fuzzy decision-making, lie in their ability to represent and manage complex data. Decision makers approach the problem of information asymmetry through various methods. Integrating symmetric and asymmetric data in the context of a specific physical phenomenon poses significant challenges. To address these challenges, interval-valued picture fuzzy (IVPF) sets have emerged as an effective tool for managing complex data. In decision-making processes, it is essential to consider the complementary and conflicting nature of the analyzed data. This article aims to refine the shortcomings of the existing score function for Multiple Criteria Decision-Making (MCDM) problems in an IVPF environment, and present an improved score function. The IVPF sets are leveraged to propose IVPF weighted arithmetic operators, IVPF ordered weighted arithmetic operators, IVPF weighted geometric operators, and IVPF ordered weighted geometric operators, which are analyzed in terms of their key features. To demonstrate the effectiveness of the proposed score function and newly defined operators, a case study involving the selection of the best food item for manufacturing, is conducted. Additionally, a comparative analysis is established to investigate the significance of the newly defined techniques in solving decision-making problems under IVPF knowledge.

1. Introduction

Multi-attribute decision making (MADM) problems are encountered in a variety of circumstances when it is necessary to select one option, action, or candidate among a number of possible alternatives based on a predetermined set of standards. It is easy to tackle real world problems in practically every discipline, including science, engineering, environmental and social sciences, and many others, by using aggregation operators. The foremost purpose of aggregation operators is to combine all of the individual values into a single value. This ensures that the final result of aggregation takes each value into consideration. Prior to the development of aggregation operators, crisp sets were commonly utilized as decision-making tools.

The fact of the matter is that in the real world, membership in a set is not always crystal clear, and traditional mathematical techniques have a limited amount of application. This is especially true in the biological and social sciences, as well as in the fields of linguistics, psychology, and economics.

In 1965, Zadeh [1] came up with the idea of partial belongingness of a set, and gave it the name fuzzy set (FS). Later on, a number of operators in the framework of fuzzy environment were designed to assist with fuzzy decision-making scenarios in which the available information contains a greater degree of uncertainty and ambiguity than a crisp set. In 1986, the notion of an intuitionistic fuzzy set (IFS) was presented by Atanassov [2]. In this phenomenon, a membership degree and a non-membership degree are assigned to each element in order to identify the nature of fuzzy decision information in more depth. Yager [3] introduced fuzzy aggregation operators in 1988. Xu [4] defined intuitionistic fuzzy aggregation operators in 2007. Wang and Liu [5] have devised novel intuitionistic fuzzy geometric aggregation operators that employ Einstein operations. However, in practical decision-making scenarios where intuitionistic fuzzy sets (IFS) are used, decision-makers may encounter difficulties in precisely assessing their opinions, due to the production of inadequate and unclear information. Additionally, the use of crisp numbers to characterize the degree of membership and non-membership can prove to be problematic. To address these challenges, Atanassov and Gargov [6] proposed the concept of IV intuitionistic fuzzy sets (IVIFS). Subsequently, Wei and Wang [7] extended this idea by introducing geometric aggregation operators based on IVIFS, in 2007. Further contributions to the field of IVIFS include the development of IV intuitionistic fuzzy aggregation operators by Wang et al. [8] in 2012. Recent advancements in the field of IFS and IVIFS are discussed in [9,10].

In certain scenarios, the application of IFS may be inadequate due to the involvement of human opinions that require additional responses beyond the typical membership degrees of “yes” and “no”. To address these situations, Cuong and Kreinovich introduced the concepts of Picture Fuzzy Sets (PFS) [11] and IVPF Sets (IVPFS) [12] in 2013 and 2014, respectively. These sets assign degrees of positive membership, neutral membership, and negative membership, to each element of the universe, thereby providing decision makers with a more flexible approach to decision making. It is worth noting that IVPFS is a powerful generalization of PFS, offering even greater flexibility for decision making.

Singh [13] presented a proposal for correlation coefficients concerning PFSs, which were subsequently applied in a clustering analysis. Thong [14] devised a new hybrid model that integrated PF clustering with intuitionistic fuzzy recommender systems for medical diagnosis. Thong and Son [15] introduced a variety of innovative fuzzy clustering algorithms grounded in PFSs, which were found to be useful in time series and weather forecasting. In [16], Cuong et al. proposed a classification scheme for representable picture t-norms and t-conorms applicable to PFSs. Additionally, one may refer to the latest advances in PFSs, documented in [17,18,19,20,21,22,23,24,25].

Garg [26] discussed some PF aggregation operators in 2017. Wang et al. [27] investigated some geometric aggregation operators based on PFSs in 2017. Wei [28] introduced PF aggregation operators and their applications to MADM in 2017. In addition, many useful averaging operators in various sets were discussed in [29,30,31,32,33,34,35,36,37,38].

Stock reordering refers to the process of managing inventory levels by reordering items when their supply runs low. This practice can contribute to a company’s profitability by reducing the cost of supplied items through economies of scale, and increasing sales by ensuring that popular items are always available for customers. Moreover, it can boost efficiency by streamlining inventory management processes and ensuring that supplies are always available when needed [39]. This enhances reliability and accountability, allowing businesses to operate more smoothly and respond more effectively to customer needs.

Stock reordering is relevant to businesses of all sizes, as it can improve productivity throughout the supply chain while minimizing waste and reducing costs. In addition, stock control systems can help businesses locate lost or stolen merchandise, reducing losses and ensuring that inventory levels remain accurate. Overall, stock reordering plays an important role in maintaining a competitive edge in today’s business environment by helping companies operate more efficiently and meet customer demands more effectively.

In addition, many useful strategies were invented to address the issue of stock reordering in [40,41,42].

Stock management pertains to the systematic coordination of the storage, procurement, and sale of goods and services. This discipline encompasses the effective oversight of inventory and associated processes. It is a crucial aspect of a company’s sustainability, as it ensures that the stock levels are optimally maintained to prevent stockouts and inaccurate accounting. Additionally, it facilitates the tracking of supplies and determination of precise pricing. By managing sudden changes in demand, it promotes customer satisfaction and preserves product quality, enhancing the overall customer experience and bolstering sales. Efficient stock control also minimizes the risk of dead stock, which can cause operational disruption, resulting in substantial financial loss for the organization, particularly in the event of raw material scarcity.

Dealing with uncertainty is one of the most important aspects of creating successful inventory procedures. The supply, demand, and information delays associated with manufacturing and distribution processes, as well as expenses related to inventory and backorders, are typically unknown. To address this, people have turned to fuzzy mathematics, which describes the various sources of uncertainty using fuzzy logic derived from the examination of different problems [43].

Compared to fuzzy and intuitionistic fuzzy sets, the significance of IVPFSs is quite evident. These sets provide enormous flexibility to effectively tackle decision-making problems in various disciplines, including decision analysis, engineering, project management, and management sciences. The prime relevance of IVPFSs makes them a valuable tool for decision making [44].

Efficient stock methods depend heavily on the ability to deal with ambiguity, uncertainty, and vagueness, which can be accurately and efficiently addressed by IVPFSs. Data regarding various parameters in the optimization of stock control are often ambiguous in nature, and IVPFSs can solve this uncertainty phenomenon more accurately.

Therefore, the study of stock control in an uncertainty phenomenon described by IVPFSs is highly motivating.

The IVPFS provides a more comprehensive and precise interpretation of uncertain information in situations where membership function values, neutral membership function, and non-membership function values in a PFS, are uncertain, rather than precise real numbers. Despite the scarcity of studies exploring the integration of IVF numbers with PF numbers to handle decision information presented in the form of an IVPF number, none of the above methodologies are appropriate for aggregating these IVPF numbers. Consequently, aggregating these IVPF numbers is an intriguing problem. To address this issue, we developed some IVPF aggregation operators based on classical arithmetic and geometric operations in this study. Additionally, we present an algorithm to solve the stock reordering decision-making problem within the framework of IVPFS environment. We establish a comparative analysis to reveal the efficacy of the proposed model.

The following are the most significant contributions that we aim to accomplish in this article:

• Propose the development of an improved score function that addresses the limitations of the current score function in the IVPF framework. This endeavor will involve the utilization of statistical and mathematical methodologies to construct a more reliable and precise scoring mechanism.
• Demonstrate the essential features of the newly introduced operators. This will require the use of rigorous and systematic mathematical reasoning to demonstrate several fundamental properties of the operators.
• Propose an algorithm that can handle MADM problems using IVPF aggregation operators. This will provide a structured framework for utilizing the novel operators in the analysis and assessment of intricate decision-making scenarios.
• Implement the recently suggested method for identifying the optimal food category within a food corporation. This will necessitate the application of the suggested technique in real-world situations, such as deciding the optimal food category for production.
• Conduct a comparative assessment to show the advantage of the developed strategy in comparison to established methods. This will entail a comprehensive evaluation of the suggested method against existing schemes, utilizing real-world datasets and scenarios to determine the efficacy of the proposed approach.

The remaining content of this article is structured as follows: in order to comprehend the originality of the work offered in this article, one must grasp a few concepts that are provided in Section 2. In Section 3, the inadequacy of the current score function to solve the MADM problem is explored, and an updated score function is proposed to tackle this problem in an IVPF environment. In Section 4, a few IVPF weighted aggregation operators for these sets are developed. In Section 5, an economic solution is given to select the best food item to manufacture in order to make a stock reordering decision under IVPFS knowledge. Finally, the validity and viability of these novel strategies in comparison to previous methods, are shown by a comparative analysis. Additionally, a summary of the specific outcomes of this study is presented.

2. Preliminaries

This section encompasses fundamental notions, characteristics, procedures, and techniques for comparing magnitudes of IVPFSs over the universal set X, which are vital for the attainment of the present investigation.

Definition 1

([45]). The IVFS $K$ of a universe $X$ is defined as $K=\left\{\right(℩,H_K\left(℩\right)\left)\right|℩\in X\},$ where $H_K$ is characterized by a membership function defined from $X$ to collection of all subintervals of a closed unit interval.

Definition 2 

([11]). A PFS A of a universe $X$ is an object of the form $A=\left\{\right(℩,{\mu }_A\left(℩\right)\hspace{1em},\hspace{1em}{\eta }_A\left(℩\right),\hspace{1em}{\nu }_A\left(℩\right)\left)\right|℩\in X\},$ where ${\mu }_A:X\to \left[\mathrm{0,1}\right]$, ${\eta }_A:X\to \left[\mathrm{0,1}\right]$ and ${\nu }_A:X\to \left[\mathrm{0,1}\right]$ represent positive, neutral, and negative membership functions, respectively, that satisfy the condition: $0\le {\mu }_A\left(℩\right)+{\eta }_A\left(℩\right)+{\nu }_A\left(℩\right)\le 1,\forall ℩\in X.$ Moreover, the refusal degree of the PFS $A$ is described as follows: ${\pi }_A\left(℩\right)=1-{\mu }_A\left(℩\right)-{\eta }_A\left(℩\right)-{\nu }_A\left(℩\right)$. It may be noted that the positive, neutral, and negative membership degrees of $℩\in X$ are symbolized as $℩=\left({\mu }_A\left(℩\right),{\eta }_A\left(℩\right),{\nu }_A\left(℩\right)\right).$ This representation is called a PF number (PFN).

Definition 3 

([12]). An IVPFS $A$ of a universe $X$ is an object of the form $A=\left\{\left(℩,\left[{\mu }_A^L\left(℩\right),{\mu }_A^U\left(℩\right)\right],\left[{\nu }_A^L\left(℩\right),{\nu }_A^U\left(℩\right)\right],\left[{\nu }_A^L\left(℩\right),{\nu }_A^U\left(℩\right)\right]\right)\right|℩\in X\},$ where $\left[{\mu }_A^L\left(℩\right),{\mu }_A^U\left(℩\right)\right]$, $\left[{\eta }_A^L\left(℩\right),{\eta }_A^U\left(℩\right)\right]$ and $\left[{\nu }_A^L\left(℩\right),{\nu }_A^U\left(℩\right)\right]$, respectively, represent the positive, neutral, and negative membership degrees of $℩$ that satisfy the condition:  $0\le {\mu }_A^L\left(℩\right)\le {\mu }_A^U\left(℩\right)\le 1$ ,$0\le {\eta }_A^L\left(℩\right)\le {\eta }_A^U\left(℩\right)\le 1$ and $0\le {\nu }_A^L\left(℩\right)\le {\nu }_A^U\left(℩\right)\le 1.$ Moreover, $0\le {\mu }_A^U\left(℩\right)+{\eta }_A^U\left(℩\right)+{\nu }_A^U\left(℩\right)\le 1.$ The hesitancy degree of the IVPFS $A$ is defined as ${\pi }_A\left(℩\right)=\left[{\pi }_A^L\left(℩\right),{\pi }_A^U\left(℩\right)\right],=\left[1-{\mu }_A^L\left(℩\right)-{\eta }_A^L\left(℩\right)-{\nu }_A^L\left(℩\right),1-{\mu }_A^U\left(℩\right)-{\eta }_A^U\left(℩\right)-{\nu }_A^U\left(℩\right)\right].$ For convenience, we write the intervals $\left[\widehat{\mu },\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right]$ and $\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]$ instead of $\left[{\mu }_A^L\left(℩\right),{\mu }_A^U\left(℩\right)\right],\left[{\eta }_A^L\left(℩\right),{\eta }_A^U\left(℩\right)\right]$ and $\left[{\nu }_A^L\left(℩\right),{\nu }_A^U\left(℩\right)\right]$, respectively, in this article. It may be noted that the positive, neutral, and negative membership degrees of $℩\in X$ are symbolized as $℩=\left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)$. This representation is called an IVPF number (IVPFN).

Definition 4 

([46]). For a non-negative real number $\gamma$ and any two IVPFNs, ${\sigma }_1=\left(\left[{\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{w}}}_1\right],\left[{\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{h}}}_1\right],\left[{\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{z}}}_1\right]\right)$ and ${\sigma }_2=\left(\left[{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_2\right],\left[{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_2\right],\left[{\widehat{\mathfrak{y}}}_2,{\widehat{\mathfrak{z}}}_2\right]\right).$ The basic operations on these numbers are defined in the subsequent way:

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

Definition 5 

([46]). The score function for IVPF number $\sigma =\left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\mathfrak{h}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)$ is defined in the following way:

$$S\left(\sigma \right)=\frac{\widehat{\mathfrak{u}}-\widehat{\mathfrak{f}}-\widehat{\mathfrak{y}}+\widehat{\mathfrak{w}}-\widehat{\mathfrak{h}}-\widehat{\mathfrak{z}}}{2},S\left(\sigma \right)\in \left[-\mathrm{1,1}\right]$$

(1)

Moreover, the above defined score function satisfies the following properties for any two IVPFNs ${\mathsf{\sigma }}_1$ and ${\sigma }_2$

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

3. Drawbacks of the Existing Score Function for IVPFS and Its Improvement

In this part, we offer an example that illustrates the deficiencies in the score function of IVPFS created in [46]. We then proceed to explore how these deficiencies might be improved upon in the following section.

Example 1.

Let ${\sigma }_1=\left(\left[\mathrm{0.07,0.09}\right],\left[\mathrm{0.06,0.07}\right],\left[\mathrm{0.01,0.02}\right]\right)$ and ${\sigma }_2=\left(\left[\mathrm{0.11,0.16}\right],\left[\mathrm{0.09,0.10}\right],\left[\mathrm{0.02,0.06}\right]\right)$ be 2 IVPFNs. Then, by applying $Definition$ $5$ on ${\sigma }_1$ and ${\sigma }_2,$ we get

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

Therefore, in light of the $prop\mathfrak{y}rty3$ of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 5$, we have ${\sigma }_1\sim {\sigma }_2.$ This demonstrates the inability of the current score function. Based on what we have discussed, we can refine the definition of this scoring function as follows.

Definition 6.

Let $\sigma =\left(\left[\widehat{\mathfrak{u}},\widehat{\mathfrak{w}}\right],\left[\widehat{\mathfrak{f}},\widehat{\widehat{\mathfrak{h}}}\right],\left[\widehat{\mathfrak{y}},\widehat{\mathfrak{z}}\right]\right)$ be an IVPFN, an improved score function $Z$ is defined as

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

(2)

where $Z\left(\sigma \right)\in \left[-\mathrm{1.33,1.33}\right]$

In particular, when $\widehat{\mathfrak{u}}=\widehat{\mathfrak{w}}$, $\widehat{\mathfrak{f}}=\widehat{\mathfrak{h}}$, and $\widehat{\mathfrak{y}}=\widehat{\mathfrak{z},}$ an IVPFN becomes a PFN, and the improved score function of the IVPFN becomes the score function of a PFN. That is, the improved score function of PFN becomes

$$\mathrm{Z}\left(\mathsf{\sigma }\right)=\frac{2\widehat{\mathfrak{u}}-2\widehat{\mathfrak{f}}-2\widehat{\mathfrak{y}}}{2}+\frac{\widehat{\mathfrak{u}}-\widehat{\mathfrak{y}}}{3}$$

The key features of newly defined score functions to solve MCDM problems in the framework of IVPFS knowledge, are described in the following way.

Let ${\mathsf{\sigma }}_1$ and ${\mathsf{\sigma }}_2$ be any two IVPFNs, then

• If $\mathrm{Z}\left({\mathsf{\sigma }}_1\right)<\mathrm{Z}\left({\mathsf{\sigma }}_2\right)$, then ${\mathsf{\sigma }}_1\prec {\mathsf{\sigma }}_2;$
• If $\mathrm{Z}\left({\mathsf{\sigma }}_1\right)>\mathrm{Z}\left({\mathsf{\sigma }}_2\right)$, then ${\mathsf{\sigma }}_1\succ {\mathsf{\sigma }}_2;$
• If $\mathrm{Z}\left({\mathsf{\sigma }}_1\right)=\mathrm{Z}\left({\mathsf{\sigma }}_2\right),$ then ${\mathsf{\sigma }}_1\sim {\mathsf{\sigma }}_2.$

The following example interprets the utility of the suggested score function for IVPFN.

Example 2.

If we apply the proposed function Z$\left(\sigma \right)$ as given in $Equation\left(1\right)$ to Example 1, then we get Z$\left({\sigma }_1\right)$ $=0.083$ and Z$\left({\sigma }_2\right)$ $=0.183.$ Hence, in light of $prop\mathfrak{y}rty1$ of $definition6$, we conclude that ${\sigma }_2$ is preferable to ${\sigma }_1.$

4. Aggregation Operators Based on Interval-Valued Picture Fuzzy Numbers

In this section, we introduce IVPF weighted arithmetic operators, IVPF ordered weighted arithmetic operators, IVPF weighted geometric operators, and IVPF ordered weighted geometric operators, and investigates some of their important properties in the framework of an IVPF environment.

Definition 7.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. The IVPF weighted arithmetic (IVPFWA) operator is a mapping $U^n\to U$ such that

$$IVPFW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)={\displaystyle \underset{\tau =1}{\overset{n}{⨁}}}{\phi }_{\tau }{\sigma }_{\tau }$$

where${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n,\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{and\phi }_{\tau }\in \left[\mathrm{0,1}\right],$ such that ${\sum }_{\tau =1}^n{\phi }_{\tau }=1.$

Theorem 1.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Then the aggregated value of these IVPFNs in the framework of an IVPFWA operator is also IVPFN, and is determined in the following way:

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

(3)

where ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$, $\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,$ ${\phi }_{\tau }\in \left[\mathrm{0,1}\right]$ such that ${\sum }_{\tau =1}^n{\phi }_{\tau }=1.$

Proof. 

We prove $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(3\right)$ by mathematical induction on n.

The application of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 7$ for $n=2$ gives the following outcome

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\right)={\mathsf{\phi }}_1{\mathsf{\sigma }}_1\oplus {\phi }_2{\sigma }_2$$

In light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $4$, we have

$${\mathsf{\phi }}_1{\mathsf{\sigma }}_1=\left(\right[1-{\left(1-{\widehat{\mathfrak{u}}}_1\right)}^{\mathsf{\phi }},1-{\left(1-{\widehat{\mathfrak{w}}}_1\right)}^{\phi }],\left[{\widehat{\mathfrak{f}}}_1^{\mathsf{\phi }},{\widehat{\mathfrak{h}}}_1^{\mathsf{\phi }}\right],\left[{\widehat{\mathfrak{y}}}_1^{\mathsf{\phi }},{\widehat{\mathfrak{z}}}_1^{\mathsf{\phi }}\right])$$

$${\mathsf{\phi }}_2{\mathsf{\sigma }}_2=\left(\right[1-{\left(1-{\widehat{\mathfrak{u}}}_2\right)}^{\mathsf{\phi }},1-{\left(1-{\widehat{\mathfrak{w}}}_2\right)}^{\phi }],\left[{\widehat{\mathfrak{f}}}_2^{\mathsf{\phi }},{\widehat{\mathfrak{h}}}_2^{\mathsf{\phi }}\right],\left[{\widehat{\mathfrak{y}}}_2^{\phi },{\widehat{\mathfrak{z}}}_2^{\phi }\right])$$

$$\begin{gathered} IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\right)=\left(\left[\left(2-{\left(1-{\widehat{\mathfrak{u}}}_1\right)}^{{\mathsf{\phi }}_1}-{\left(1-{\widehat{\mathfrak{u}}}_2\right)}^{{\mathsf{\phi }}_2}\right)-\left(1-{\left(1-{\widehat{\mathfrak{u}}}_1\right)}^{{\phi }_1}\right) \\ \left(1-{\left(1-{\widehat{\mathfrak{u}}}_2\right)}^{{\mathsf{\phi }}_2}\right),\left(2-{\left(1-{\widehat{\mathfrak{w}}}_1\right)}^{{\phi }_1}-{\left(1-{\widehat{\mathfrak{w}}}_2\right)}^{{\phi }_2}\right)-\left(1-{\left(1-{\widehat{\mathfrak{w}}}_1\right)}^{{\phi }_1}\right) \\ \left(1-{\left(1-{\widehat{\mathfrak{w}}}_2\right)}^{{\phi }_2}\right)\right],\left[{\widehat{\mathfrak{f}}}_1^{{\phi }_1}{\widehat{\mathfrak{f}}}_2^{\phi 1_1},{\widehat{\mathfrak{h}}}_1^{{\phi }_2}{\widehat{\mathfrak{h}}}_2^{{\phi }_2}\right],\left[{\widehat{\mathfrak{y}}}_1^{{\phi }_1}{\widehat{\mathfrak{y}}}_2^{{\phi }_1},{\widehat{\mathfrak{z}}}_1^{{\phi }_2}{\widehat{\mathfrak{z}}}_2^{{\phi }_2}\right]\right) \end{gathered}$$

It follows that

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\right)=\left(\begin{array}{c}\left[\right(1-{\left(1-{\widehat{\mathfrak{u}}}_1\right)}^{{\mathsf{\phi }}_1}{\left(1-{\widehat{\mathfrak{u}}}_2\right)}^{{\mathsf{\phi }}_2}),\left(1-{\left(1-{\widehat{\mathrm{b}}}_1\right)}^{{\mathsf{\phi }}_1}{\left(1-{\widehat{\mathrm{b}}}_2\right)}^{{\mathsf{\phi }}_2})\right],\\ \left[{\widehat{\mathrm{c}}}_1^{{\mathsf{\phi }}_1}{\widehat{\mathrm{c}}}_2^{{\mathsf{\phi }}_2}\right],\left[{\widehat{\mathfrak{h}}}_1^{{\mathsf{\phi }}_2}{\widehat{\mathfrak{h}}}_2^{{\mathsf{\phi }}_2}\right],\left[{\widehat{\mathfrak{y}}}_1^{{\mathsf{\phi }}_1}{\widehat{\mathfrak{y}}}_2^{{\mathsf{\phi }}_1}\right],\left[{\widehat{\mathfrak{z}}}_1^{{\mathsf{\phi }}_2}{\widehat{\mathfrak{z}}}_2^{{\mathsf{\phi }}_2}\right]\end{array}\right)$$

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ (3) is true for $n=2$.

Assume that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ (3) is true for $n=q.$ Then,

$$IVPFW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_q\right)=\left(\begin{array}{c}\left[1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{u}}}_{\tau }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{w}}}_{\tau }\right)}^{{\phi }_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^q}{\widehat{\mathfrak{f}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^q}{\widehat{\mathfrak{h}}}_{\tau }^{{\phi }_{\tau }}\right],\\ \left[{\displaystyle \prod _{\tau =1}^q}{\widehat{\mathfrak{y}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^q}{\widehat{\mathfrak{z}}}_{\tau }^{{\phi }_{\tau }}\right]\end{array}\right),$$

Moreover, for $n=q+1$, by the operational laws of IVPF number, we have

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_q,{\mathsf{\sigma }}_{q+1}\right)={{\phi }_1\sigma }_1⨁{\phi }_2{\sigma }_2,\dots ,⨁{\mathsf{\phi }}_q{\mathsf{\sigma }}_q,⨁{\mathsf{\phi }}_{q+1}{\mathsf{\sigma }}_{q+1}$$

It follows that

$$\begin{array}{l}IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_q,{\mathsf{\sigma }}_{q+1}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\left(\right[(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{u}}}_{\tau }\right)}^{{\phi }_{\tau }})+(1-{\left(1-{\widehat{\mathfrak{u}}}_{q+1}\right)}^{{\mathsf{\phi }}_{q+1}})-(1-{\displaystyle \prod _{\tau =1}^q}(1-{{\widehat{\mathfrak{u}}}_{\tau }(}^{{\phi }_{\tau }})(1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\left(1-{\widehat{\mathfrak{u}}}_{q+1}\right)}^{{\phi }_{q+1}}),(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{w}}}_{\tau }\right)}^{{\phi }_{\tau }})+(1-{\left(1-{\widehat{\mathfrak{w}}}_{q+1}\right)}^{{\phi }_{q+1}})-(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{w}}}_{\tau }\right)}^{{\phi }_{\tau }}\left)\right(1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{-\left(1-{\widehat{\mathfrak{w}}}_{q+1}\right)}^{{\phi }_{q+1}}\left)\right],\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{f}}}_{\tau }^{{\mathsf{\phi }}_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{h}}}_{\tau }^{{\mathsf{\phi }}_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{y}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{z}}}_{\tau }^{{\phi }_{\tau }}\right])\end{array}$$

It follows that

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_q,{\mathsf{\sigma }}_{q+1}\right)=\left(\begin{array}{c}\left[1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{u}}}_{\tau }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{w}}}_{\tau }\right)}^{{\phi }_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{f}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{h}}}_{\tau }^{{\phi }_{\tau }}\right],\\ \left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{y}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{z}}}_{\tau }^{{\phi }_{\tau }}\right]\end{array}\right)$$

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ (3) is true for $n=q+1$. Hence, we conclude that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ (3) is true for any k ∈ N. □

The following example describes the above-stated fact.

Example 3.

Consider 3 IVPFNs ${\sigma }_1=\left(\left[\mathrm{0.05,0.1}\right],\left[\mathrm{0.1,0.2}\right],\left[\mathrm{0.17,0.2}\right]\right)$, ${\sigma }_2=\left(\left[\mathrm{0.03,0.07}\right],\left[\mathrm{0.3,0.35}\right],\left[\mathrm{0.1,0.25}\right]\right)$, and ${\sigma }_3=\left(\left[\mathrm{0.07,0.19}\right],\left[\mathrm{0.1,0.2}\right],\left[\mathrm{0.1,0.2}\right]\right)$, associated with weight vector $\phi ={\left(\mathrm{0.4,0.3,0.3}\right)}^T$. Then, we have ${\displaystyle {\prod }_{\tau =1}^3}{\left(1-{\widehat{\mathfrak{u}}}_{\tau }\right)}^{{\phi }_{\tau }}={\left(1-0.05\right)}^{0.4}\times {\left(1-0.03\right)}^{0.3}\times {\left(1-0.07\right)}^{0.3}=0.950$. Similarly, we can obtain ${\displaystyle {\prod }_{\tau =1}^3}{\left(1-{\widehat{\mathfrak{w}}}_{\tau }\right)}^{{\phi }_{\tau }}$= 0.882, ${\displaystyle {\prod }_{\tau =1}^3}{\left({\widehat{\mathfrak{f}}}_{\tau }\right)}^{{\phi }_{\tau }}$= 0.139, ${\displaystyle {\prod }_{\tau =1}^3}{\left({\widehat{\mathfrak{h}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.236$, ${\displaystyle {\prod }_{\tau =1}^3}{\left({\widehat{\mathfrak{y}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.123$, and ${\displaystyle {\prod }_{\tau =1}^3}{\left({\widehat{\mathfrak{z}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.214.$

Consequently,

$$IVPFWA\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\right)=\left(\left[\mathrm{0.05,0.12}\right],\left[\mathrm{0.139,0.236}\right],\left[\mathrm{0.123,0.214}\right]\right)$$

Theorem 2.

(Idempotency) If all ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},...,n\right)$ are equal, i.e., ${\sigma }_{\tau }=\sigma$ for all $\tau$, then

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_n\right)=\sigma$$

Proof. 

Let $\mathsf{\sigma }=\left(\left[{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }},{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}\right],\left[{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }},{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}\right],\left[{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }},{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}\right]\right)$ yields that

$$\begin{gathered} IVPFW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)= \\ \left(\begin{array}{c}\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\\ \left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{f}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{h}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{y}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{z}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right]\end{array}\right)= \\ \left(\begin{array}{c}\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\sigma }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\sigma }\right)}^{{\phi }_{\tau }}\right],\\ \left[{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{f}}}_{\sigma }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{h}}}_{\sigma }^{{\phi }_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{y}}}_{\sigma }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{z}}}_{\sigma }^{{\phi }_{\tau }}\right]\end{array}\right)= \\ \left(\begin{array}{c}\left[1-{\left(1-{\widehat{\mathfrak{u}}}_{\sigma }\right)}^{{\sum }_{\tau =1}^n{\phi }_{\tau }},1-{\left(1-{\widehat{\mathfrak{w}}}_{\sigma }\right)}^{{\sum }_{\tau =1}^n{\phi }_{\tau }}\right],\\ \left[{\widehat{\mathfrak{f}}}_{\sigma }^{{\sum }_{\tau =1}^n{\phi }_{\tau }},{\widehat{\mathfrak{h}}}_{\sigma }^{{\sum }_{\tau =1}^n{\phi }_{\tau }}\right],\left[{\widehat{\mathfrak{y}}}_{\sigma }^{{\sum }_{\tau =1}^n{\phi }_{\tau }},{\widehat{\mathfrak{z}}}_{\sigma }^{{\sum }_{\tau =1}^n{\phi }_{\tau }}\right]\end{array}\right) \\ =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right)=\sigma \end{gathered}$$

□

Definition 8.

For any two IVPFNs, ${\sigma }_1=\left(\left[{\widehat{\mathfrak{u}}}_1,{\widehat{\mathfrak{w}}}_1\right],\left[{\widehat{\mathfrak{f}}}_1,{\widehat{\mathfrak{h}}}_1\right],\left[{\widehat{\mathfrak{y}}}_1,{\widehat{\mathfrak{z}}}_1\right]\right)$ and ${\sigma }_2=\left(\left[{\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_2\right],\left[{\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_2\right],\left[{\widehat{\mathfrak{y}}}_2,{\widehat{\mathfrak{z}}}_2\right]\right),$ the basic operations are defined as follows:

1.

${\sigma }_1={\sigma }_2\iff {\widehat{\mathfrak{u}}}_1={\widehat{\mathfrak{u}}}_2,{\widehat{\mathfrak{w}}}_1={\widehat{\mathfrak{w}}}_2,{\widehat{\mathfrak{f}}}_1={\widehat{\mathfrak{f}}}_2,{\widehat{\mathfrak{h}}}_1={\widehat{\mathfrak{h}}}_2,{\widehat{\mathfrak{y}}}_1={\widehat{\mathfrak{y}}}_2$ and${\widehat{\mathfrak{z}}}_1={\widehat{\mathfrak{z}}}_2;$

2.

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

Theorem 3.

(Boundedness) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Let ${\sigma }^{-}=\mathit{min}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{-},{\widehat{\mathfrak{w}}}^{-}\right],\left[{\widehat{\mathfrak{f}}}^{-},{\widehat{\mathfrak{h}}}^{-}\right],\left[{\widehat{\mathfrak{y}}}^{-},{\widehat{\mathfrak{z}}}^{-}\right]\right)$ and ${\sigma }^{+}=\mathit{max}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{+},{\widehat{\mathfrak{w}}}^{+}\right],\left[{\widehat{\mathfrak{f}}}^{+},{\widehat{\mathfrak{h}}}^{+}\right],\left[{\widehat{\mathfrak{y}}}^{+},{\widehat{\mathfrak{z}}}^{+}\right]\right)\hspace{1em}$ where ${\widehat{\mathfrak{u}}}^{-}=min{\widehat{\mathfrak{u}}}_{\tau }^{-},{\widehat{\mathfrak{w}}}^{-}=min{\widehat{\mathfrak{w}}}_{\tau }^{-},{\widehat{\mathfrak{f}}}^{-}=max{\widehat{\mathfrak{f}}}_{\tau }^{-},{\widehat{\mathfrak{h}}}^{-}=max{\widehat{\mathfrak{h}}}_{\tau }^{-},{\widehat{\mathfrak{y}}}^{-}=max{\widehat{\mathfrak{y}}}_{\tau }^{-},{\widehat{\mathfrak{z}}}^{-}max{\widehat{\mathfrak{z}}}_{\tau }^{-},{\widehat{\mathfrak{u}}}^{+}max{\widehat{\mathfrak{u}}}_{\tau }^{+},{\widehat{\mathfrak{w}}}^{+}=max{\widehat{\mathfrak{w}}}_{\tau }^{+},{\mathfrak{f}}^{+}=min{\widehat{\mathfrak{f}}}_{\tau }^{+},{\mathfrak{h}}^{+}=min{\widehat{\mathfrak{h}}}_{\tau }^{+},{\widehat{\mathfrak{y}}}^{+}=min{\widehat{\mathfrak{y}}}_{\tau }^{+},{\widehat{\mathfrak{z}}}^{+}=min{\widehat{\mathfrak{z}}}_{\tau }^{+}$, then

$${\sigma }^{-}\le IVPFW{A}_{\phi }\left({\sigma }_1,{\sigma }_2\dots ,{\sigma }_n\right)\le {\sigma }^{+}$$

Proof. 

In view of given conditions, we have

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{h}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{h}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{h}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}.$$

In light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 8$ and the above relations, we get

$${\mathsf{\sigma }}^{-}\le IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\dots ,{\mathsf{\sigma }}_n\right)\le {\mathsf{\sigma }}^{+}$$

□

Theorem 4.

(Monotonicity) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)$ and ${\sigma }_{\tau }^{’}=\left(\left[{\widehat{\mathfrak{u}}}_{\tau }^{’},{\widehat{\mathfrak{w}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{f}}}_{\tau }^{’},{\widehat{\mathfrak{h}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{y}}}_{\tau }^{’},{\widehat{\mathfrak{z}}}_{\tau }^{’}\right]\right)$ for $\tau =\mathrm{1,2},3,\dots ,n$ be two collections of IVPFNs. If ${\widehat{\mathfrak{u}}}_{\tau }\le {\widehat{\mathfrak{u}}}_{\tau }^{’},{\widehat{\mathfrak{w}}}_{\tau }\le {\widehat{\mathfrak{w}}}_{\tau }^{’},{\widehat{\mathfrak{f}}}_{\tau }\ge {\widehat{\mathfrak{f}}}_{\tau }^{’},{\widehat{\mathfrak{h}}}_{\tau }\ge {\widehat{\mathfrak{h}}}_{\tau }^{’},{\widehat{\mathfrak{y}}}_{\tau }\ge {\widehat{\mathfrak{y}}}_{\tau }^{’}$ and ${\widehat{\mathfrak{z}}}_{\tau }\ge {\widehat{\mathfrak{z}}}_{\tau }^{’}\forall \tau$, then $IVPFWA\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)\le IVPFWA\left({\sigma }_1^{’},{\sigma }_2^{’},{\sigma }_3^{’}\dots ,{\sigma }_n^{’}\right)$.

Proof. 

As we know that

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\dots ,{\mathsf{\sigma }}_n\right)={\mathsf{\phi }}_1{\mathsf{\sigma }}_1\oplus {\mathsf{\phi }}_2{\mathsf{\sigma }}_2,\dots ,\oplus {\mathsf{\phi }}_n{\mathsf{\sigma }}_n$$

and

$$IVPFW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1^{’},{\mathsf{\sigma }}_2^{’},{\mathsf{\sigma }}_3^{’}\dots ,{\mathsf{\sigma }}_n^{’}\right)={\mathsf{\phi }}_1{\mathsf{\sigma }}_1^{’}\oplus {\mathsf{\phi }}_2{\mathsf{\sigma }}_2^{’},\dots ,\oplus {\mathsf{\phi }}_n{\mathsf{\sigma }}_n^{’}$$

as ${\mathsf{\sigma }}_{\mathsf{\tau }}\le {\mathsf{\sigma }}_{\mathsf{\tau }}^{’}\forall \mathsf{\tau }$, thus,

$$IVPFWA\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\dots ,{\mathsf{\sigma }}_n\right)\le IVPFWA\left({\mathsf{\sigma }}_1^{’},{\mathsf{\sigma }}_2^{’},{\mathsf{\sigma }}_3^{’}\dots ,{\mathsf{\sigma }}_n^{’}\right)$$

□

In the following definition, we propose a weighted arithmetic aggregation operator with IVPFNs, namely, an IVPF ordered weighted arithmetic operator (IVPFOWA).

Definition 9.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Then IVPFOWA operator of dimension n is a mapping $U^n\to U$ that has an associated weight vector $\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T$ such that ${\phi }_{\tau }>0$ and ${\sum }_{\tau =1}^n{\phi }_{\tau }=1$. Therefore,

$$IVPFOW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,···,{\sigma }_n\right)={\displaystyle \underset{\tau =1}{\overset{n}{⨁}}}{\phi }_{\tau }{\sigma }_{\tau }$$

where ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$, $\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{\phi }_{\tau }\in \left[\mathrm{0,1}\right]$ such that ${\sum }_{i=1}^n{\phi }_{\tau }=1;$ $\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(n\right)\right)$ is a permutation of $\left(\mathrm{1,2},3,\dots ,n\right)$ such that ${\sigma }_{{\rho }_{\left(\tau -1\right)}}\ge {\sigma }_{{\rho }_{\tau }}$ $\forall \tau =\mathrm{1,2},3,\dots ,n.$

Theorem 5.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Then the aggregated value of these IVPFNs in the framework of an IVPFOWA operator is also an IVPFN, and is determined in the following way:

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

(4)

where ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$, $\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{and\phi }_{\tau }\in \left[\mathrm{0,1}\right]$ , such that${\sum }_{i=1}^n{\phi }_{\tau }=1;$ $\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(n\right)\right)$ is a permutation of $\left(\mathrm{1,2},3,\dots ,n\right)$ such that ${\sigma }_{{\rho }_{\left(\tau -1\right)}}\ge {\sigma }_{{\rho }_{\tau }}$ for all $\tau =\mathrm{1,2},3,\dots ,n.$

Proof. 

We prove $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(4\right)$ by mathematical induction on n. The application of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$9 for $n=2$ gives the following outcome:

$$IVPFOW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\right)={\mathsf{\phi }}_1{\mathsf{\sigma }}_{\rho \left(1\right)}\oplus {\phi }_2{\sigma }_{\rho \left(2\right)}$$

In light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $4$, we have

$${\mathsf{\phi }}_1{\mathsf{\sigma }}_{\rho \left(1\right)}=\left([1-{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(1\right)}\right)}^{\mathsf{\phi }},1-{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(1\right)}\right)}^{\phi }],\left[{\widehat{\mathfrak{f}}}_{\rho \left(1\right)}^{\mathsf{\phi }},{\widehat{\mathfrak{h}}}_{\rho \left(1\right)}^{\mathsf{\phi }}\right],\left[{\widehat{\mathfrak{y}}}_{\rho \left(1\right)}^{\mathsf{\phi }},{\widehat{\mathfrak{z}}}_{\rho \left(1\right)}^{\mathsf{\phi }}\right]\right)$$

$${\mathsf{\phi }}_2{\mathsf{\sigma }}_{\rho \left(2\right)}=\left([1-{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(2\right)}\right)}^{\mathsf{\phi }},1-{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(2\right)}\right)}^{\phi }],\left[{\widehat{\mathfrak{f}}}_{\rho \left(2\right)}^{\mathsf{\phi }},{\widehat{\mathfrak{h}}}_{\rho \left(2\right)}^{\mathsf{\phi }}\right],\left[{\widehat{\mathfrak{y}}}_{\rho \left(2\right)}^{\phi },{\widehat{\mathfrak{z}}}_{\rho \left(2\right)}^{\phi }\right]\right)$$

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

It follows that:

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

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(4\right)$ is true for $n=2$.

Assume that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(4\right)$ is true for $n=q.$ Then,

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

Moreover, for $n=q+1$, by the operational laws of IVPF numbers, we have

$$\begin{gathered} IVPFOW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_q,{\sigma }_{q+1}\right)= \\ {{\phi }_1\sigma }_{\rho \left(1\right)}⨁{\phi }_2{\sigma }_{\rho \left(2\right)},\dots ,⨁{\phi }_q{\sigma }_{\rho \left(q\right)},⨁{\phi }_{q+1}{\sigma }_{\rho \left(q+1\right)} \end{gathered}$$

It follows that:

$$\begin{gathered} IVPFOW{A}_{\phi }\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_q,{\mathsf{\sigma }}_{q+1}\right)= \\ \left(\begin{array}{c}\left[\begin{array}{c}\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }})+(1-{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(q+1\right)}\right)}^{{\mathsf{\phi }}_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\left)\right(1-{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right),\\ \left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }})+(1-{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\left)\right(1-{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right)\end{array}\right],\\ \left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}^{{\mathsf{\phi }}_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}^{{\mathsf{\phi }}_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }}\right]\end{array}\right) \end{gathered}$$

It follows that:

$$\begin{gathered} IVPFOW{A}_{\phi }\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,...,{\mathsf{\sigma }}_q,{\mathsf{\sigma }}_{q+1}\right) \\ =\left(\begin{array}{c}\left[1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }}\right],\\ \left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }}\right]\end{array}\right) \end{gathered}$$

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(4\right)$ is true for $n=q+1$. Hence, we conclude that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(4\right)$ is true for any k ∈ N. □

Example 4.

Consider three IVPFNs, ${\sigma }_1=\left(\left[\mathrm{0.05,0.15}\right],\left[\mathrm{0.15,0.25}\right],\left[\mathrm{0.25,0.35}\right]\right)$, ${\sigma }_2=\left(\left[\mathrm{0.03,0.13}\right],\left[\mathrm{0.13,0.23}\right],\left[\mathrm{0.21,0.31}\right]\right),$ and ${\sigma }_3=\left(\left[\mathrm{0.06,0.16}\right],\left[\mathrm{0.16,0.26}\right],\left[\mathrm{0.23,0.32}\right]\right),$ associated with weight vector $\phi ={\left(\mathrm{0.23,0.5,0.27}\right)}^T.$ To aggregate these values by IVPFOWA operator, we firstly permute these numbers in light of $Equation\left(2\right)$ to obtain the values of $Z\left({\sigma }_1\right),Z\left({\sigma }_2\right),andZ\left({\sigma }_3\right),$ as follows:

$$Z\left({\sigma }_1\right)=-0.467,Z\left({\sigma }_2\right)=-0.420,Z\left({\sigma }_3\right)=-0.432.$$

Thus, we have, $\mathrm{Z}\left({\mathsf{\sigma }}_2\right)>\mathrm{Z}\left({\mathsf{\sigma }}_3\right)>\mathrm{Z}\left({\mathsf{\sigma }}_1\right),$ and permuted IVPFNs are ${\mathsf{\sigma }}_{\mathsf{\rho }\left(1\right)}=\left(\left[\mathrm{0.03,0.13}\right],\left[\mathrm{0.13,0.23}\right],\left[\mathrm{0.21,0.31}\right]\right)$, ${\mathsf{\sigma }}_{\mathsf{\rho }\left(2\right)}=\left(\left[\mathrm{0.06,0.16}\right],\left[\mathrm{0.16,0.26}\right],\left[\mathrm{0.23,0.32}\right]\right)$, and ${\mathsf{\sigma }}_{\mathsf{\rho }\left(3\right)}=\left(\left[\mathrm{0.05,0.15}\right],\left[\mathrm{0.15,0.25}\right],\left[\mathrm{0.25,0.35}\right]\right)$. Now, by applying $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $9$ on the above IVPFNs obtained, we get ${\displaystyle \prod _{\tau =1}^3}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\rho }\left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }}={\left(1-0.06\right)}^{0.23}\times {\left(1-0.03\right)}^{0.5}\times {\left(1-0.05\right)}^{0.27}=0.950.$ Similarly, we can obtain ${\displaystyle \prod _{\tau =1}^3}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\rho }\left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }}=0.85,{\displaystyle \prod _{\tau =1}^3}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\rho }\left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }}=0.15,{\displaystyle \prod _{\tau =1}^3}{\left({\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}=0.25,{\displaystyle \prod _{\tau =1}^3}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\rho }\left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }}=0.23,$ and ${\displaystyle \prod _{\tau =1}^3}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\rho }\left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }}=0.32$.

Consequently,

$$IVPFOWA\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\right)=\left(\left[\mathrm{0.05,0.15}\right],\left[\mathrm{0.15,0.25}\right],\left[\mathrm{0.23,0.32}\right]\right)$$

Theorem 6.

(Idempotency) If all ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},...,n\right)$ are equal, i.e., ${\sigma }_{\rho \left(\tau \right)}=\sigma$ for all $\tau$, then

$$IVPFOW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)=\sigma$$

Proof. 

Let $\sigma =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right).$ Then ${\mathsf{\sigma }}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}=\mathsf{\sigma }\left(\mathsf{\tau }=\mathrm{1,2},3\dots ,n\right)$ yields that

$$\begin{array}{c}IVPFOW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)\\ =\left(\begin{array}{c}\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{f}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}^{{\mathsf{\phi }}_{\mathsf{\sigma }}\mathsf{\tau }},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{h}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\\ \left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{y}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{z}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right]\end{array}\right)\\ =\left(\begin{array}{c}\left[1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\tau }},1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\tau }}\right],\left[{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\tau }},{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\tau }}\right],\\ \left[{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\tau }},{\displaystyle \prod _{\tau =1}^n}{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\tau }}\right]\end{array}\right)\\ =\left(\begin{array}{c}\left[1-{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},1-{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right],\left[{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right],\\ \left[{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right]\end{array}\right)\\ =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right)=\sigma \end{array}$$

□

Theorem 7.

(Boundedness) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Let ${\sigma }^{-}=\mathit{min}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{-},{\widehat{\mathfrak{w}}}^{-}\right],\left[{\widehat{\mathfrak{f}}}^{-},{\widehat{\mathfrak{h}}}^{-}\right],\left[{\widehat{\mathfrak{y}}}^{-},{\widehat{\mathfrak{z}}}^{-}\right]\right)$ and ${\sigma }^{+}=\mathit{max}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{+},{\widehat{\mathfrak{w}}}^{+}\right],\left[{\widehat{\mathfrak{f}}}^{+},{\widehat{\mathfrak{h}}}^{+}\right],\left[{\widehat{\mathfrak{y}}}^{+},{\widehat{\mathfrak{z}}}^{+}\right]\right)\hspace{1em}$ where ${\widehat{\mathfrak{u}}}^{-}=min{\widehat{\mathfrak{u}}}_{\tau }^{-},{\widehat{\mathfrak{w}}}^{-}=min{\widehat{\mathfrak{w}}}_{\tau }^{-},{\widehat{\mathfrak{f}}}^{-}=max{\widehat{\mathfrak{f}}}_{\tau }^{-},{\widehat{\mathfrak{h}}}^{-}=max{\widehat{\mathfrak{h}}}_{\tau }^{-},{\widehat{\mathfrak{y}}}^{-}=max{\widehat{\mathfrak{y}}}_{\tau }^{-},{\widehat{\mathfrak{z}}}^{-}=max{\widehat{\mathfrak{z}}}_{\tau }^{-},{\widehat{\mathfrak{u}}}^{+}=max{\widehat{\mathfrak{u}}}_{\tau }^{+},{\widehat{\mathfrak{w}}}^{+}=max{\widehat{\mathfrak{w}}}_{\tau }^{+},{\mathfrak{f}}^{+}=min{\widehat{\mathfrak{f}}}_{\tau }^{+},{\mathfrak{h}}^{+}=min{\widehat{\mathfrak{h}}}_{\tau }^{+},{\widehat{\mathfrak{y}}}^{+}=min{\widehat{\mathfrak{y}}}_{\tau }^{+},{\widehat{\mathfrak{z}}}^{+}=min{\widehat{\mathfrak{z}}}_{\tau }^{+}$, then

$${\sigma }^{-}\le IVPFOW{A}_{\phi }\left({\sigma }_1,{\sigma }_2\dots ,{\sigma }_n\right)\le {\sigma }^{+}$$

Proof. 

In view of given conditions, we have

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{u}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{w}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{f}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{h}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{h}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{h}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{y}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{z}}}_{\mathsf{\rho }\left(\mathsf{\tau }\right)}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

In light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 8$ and the above relations, we get ${\mathsf{\sigma }}^{-}\le IVPFOW{A}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\dots ,{\mathsf{\sigma }}_n\right)\le {\mathsf{\sigma }}^{+}$. □

Theorem 8.

(Monotonicity) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)$ and ${\sigma }_{\tau }^{’}=\left(\left[{\widehat{\mathfrak{u}}}_{\tau }^{’},{\widehat{\mathfrak{w}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{f}}}_{\tau }^{’},{\widehat{\mathfrak{h}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{y}}}_{\tau }^{’},{\widehat{\mathfrak{z}}}_{\tau }^{’}\right]\right)$ for $\tau =\mathrm{1,2},3,\dots ,n$ be two collections of IVPFNs. If ${\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\le {\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\le {\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}^{’}$ and ${\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}^{’}\forall \tau$, then $IVPFOWA\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)\le IVPFOWA\left({\sigma }_1^{’},{\sigma }_2^{’},{\sigma }_3^{’}\dots ,{\sigma }_n^{’}\right)$.

Proof. 

As we know that

$$IVPFOW{A}_{\phi }\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)={\phi }_1{\sigma }_{\rho \left(1\right)}\oplus {\phi }_2{\sigma }_{\rho \left(2\right)},\dots ,\oplus {\phi }_n{\sigma }_{\rho \left(n\right)}$$

and

$$IVPFOW{A}_{\phi }\left({\sigma }_1^{’},{\sigma }_2^{’},{\sigma }_3^{’}\dots ,{\sigma }_n^{’}\right)={\phi }_1{\sigma }_{\rho \left(1\right)}^{’}\oplus {\phi }_2{\sigma }_{\rho \left(2\right)}^{’},\dots ,\oplus {\phi }_n{\sigma }_{\rho \left(n\right)}^{’}$$

as ${\sigma }_{\rho \left(\tau \right)}\le {\sigma }_{\rho \left(\tau \right)}^{\mathrm{’}}\forall \mathsf{\tau }$, thus,

$$IVPFOWA\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)\le IVPFOWA\left({\sigma }_1^{’},{\sigma }_2^{’},{\sigma }_3^{’}\dots ,{\sigma }_n^{’}\right)$$

□

In the following definition, we propose a geometric aggregation operator with IVPFNs, namely, an IVPF weighted geometric operator (IVPFWG).

Definition 10.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ represents number of IVPFNs. The IVPFWG operator is a mapping $U^n\to U$ such that

$$IVPFW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)={⨂}_{\tau =1}^n{\left({\sigma }_{\tau }\right)}^{{\phi }_{\tau }}$$

where,${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$,$\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{\phi }_{\tau }\in \left[\mathrm{0,1}\right]su\mathfrak{f}hth\mathfrak{u}t{\sum }_{\tau =1}^n{\phi }_{\tau }=1.$

Theorem 9.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Then the aggregated value of these IVPFNs in the framework of IVPFWG is also IVPFN and is determined in the following way:

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

(5)

where,${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$,$\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{\phi }_{\tau }\in \left[\mathrm{0,1}\right]$ such that ${\sum }_{\tau =1}^n{\phi }_{\tau }=1.$

Proof. 

We prove $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$   $\left(5\right)$ by mathematical induction on n.

The application of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $10$ for $n=2$ gives the following outcome

$$IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\right)={\mathsf{\sigma }}_1^{{\mathsf{\phi }}_1}⨂{\sigma }_2^{{\phi }_2}$$

In the light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $4$, we have

$${\mathsf{\sigma }}_1^{{\mathsf{\phi }}_1}=\left(\left[{\widehat{\mathfrak{u}}}_1^{{\mathsf{\phi }}_1},{\widehat{\mathfrak{w}}}_1^{{\mathsf{\phi }}_1}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_1\right)}^{{\mathsf{\phi }}_1},1-{\left(1-{\widehat{\mathfrak{h}}}_1\right)}^{{\mathsf{\phi }}_1}\right],\left[1-{\left(1-{\widehat{\mathfrak{y}}}_1\right)}^{{\mathsf{\phi }}_1},1-{\left(1-{\widehat{\mathfrak{z}}}_1\right)}^{{\mathsf{\phi }}_1}\right]\right)$$

$${\sigma }_2^{{\phi }_2}=\left(\left[{\widehat{\mathfrak{u}}}_2^{{\phi }_2},{\widehat{\mathfrak{w}}}_2^{{\phi }_2}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_2\right)}^{{\phi }_2},1-{\left(1-{\widehat{\mathfrak{h}}}_2\right)}^{{\phi }_2}\right],\left[1-{\left(1-{\widehat{\mathfrak{y}}}_2\right)}^{{\phi }_2},1-{\left(1-{\widehat{\mathfrak{z}}}_2\right)}^{{\phi }_2}\right]\right)$$

$$IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\right)=\left(\begin{array}{c}\left[{\widehat{\mathfrak{u}}}_1^{{\mathsf{\phi }}_1}{\widehat{\mathfrak{u}}}_2^{{\phi }_2},{\widehat{\mathfrak{w}}}_1^{{\mathsf{\phi }}_1}{\widehat{\mathfrak{w}}}_2^{{\phi }_2}\right],\\ \left[\begin{array}{c}\left(2-{\left(1-{\widehat{\mathfrak{f}}}_1\right)}^{{\mathsf{\phi }}_1}-{\left(1-{\widehat{\mathfrak{f}}}_2\right)}^{{\mathsf{\phi }}_2}\right)-\left(1-{\left(1-{\widehat{\mathfrak{f}}}_1\right)}^{{\mathsf{\phi }}_1}\right)\left(1-{\left(1-{\widehat{\mathfrak{f}}}_2\right)}^{{\mathsf{\phi }}_2}\right),\\ \left(2-{\left(1-{\widehat{\mathfrak{h}}}_1\right)}^{{\mathsf{\phi }}_1}-{\left(1-{\widehat{\mathfrak{h}}}_2\right)}^{{\mathsf{\phi }}_2}\right)-\left(1-{\left(1-{\widehat{\mathfrak{h}}}_1\right)}^{{\mathsf{\phi }}_1}\right)\left(1-{\left(1-{\widehat{\mathfrak{h}}}_2\right)}^{{\mathsf{\phi }}_2}\right)\end{array}\right],\\ \left[\begin{array}{c}\left(2-{\left(1-{\widehat{\mathfrak{y}}}_1\right)}^{{\mathsf{\phi }}_1}-{\left(1-{\widehat{\mathfrak{y}}}_2\right)}^{{\mathsf{\phi }}_2}\right)-\left(1-{\left(1-{\widehat{\mathfrak{y}}}_1\right)}^{{\mathsf{\phi }}_1}\right)\left(1-{\left(1-{\widehat{\mathfrak{y}}}_2\right)}^{{\mathsf{\phi }}_2}\right),\\ (2-{\left(1-{\widehat{\mathfrak{z}}}_1\right)}^{{\mathsf{\phi }}_1}-{\left(1-{\widehat{\mathfrak{z}}}_2\right)}^{{\mathsf{\phi }}_2})-(1-{\left(1-{\widehat{\mathfrak{z}}}_1\right)}^{{\mathsf{\phi }}_1})(1-{\left(1-{\widehat{\mathfrak{z}}}_2\right)}^{{\mathsf{\phi }}_2})\end{array}\right]\end{array}\right)$$

It follows that

$$IVPFW{G}_{\mathsf{\phi }}=\left(\begin{array}{c}\left[{\widehat{\mathfrak{u}}}_1^{{\mathsf{\phi }}_1}{\widehat{\mathfrak{u}}}_2^{\mathsf{\phi }{1}_1},{\widehat{\mathfrak{w}}}_1^{{\mathsf{\phi }}_2}{\widehat{\mathfrak{w}}}_2^{{\mathsf{\phi }}_2}\right],\\ \left[(1-{\left(1-{\widehat{\mathfrak{f}}}_1\right)}^{{\mathsf{\phi }}_1}{\left(1-{\widehat{\mathfrak{f}}}_2\right)}^{{\mathsf{\phi }}_2},1-{\left(1-{\widehat{\mathfrak{h}}}_1\right)}^{{\mathsf{\phi }}_1}{\left(1-{\widehat{\mathfrak{h}}}_2\right)}^{{\mathsf{\phi }}_2}\right],\\ \left[(1-{\left(1-{\widehat{\mathfrak{y}}}_1\right)}^{{\mathsf{\phi }}_1}{\left(1-{\widehat{\mathfrak{y}}}_2\right)}^{{\mathsf{\phi }}_2},1-{\left(1-{\widehat{\mathfrak{z}}}_1\right)}^{{\phi }_1}{\left(1-{\widehat{\mathfrak{z}}}_2\right)}^{{\mathsf{\phi }}_2}\right]\end{array}\right)$$

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(5\right)$ is true for $n=2$.

Assume that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(5\right)$ is true for $n=q$. Then,

$$\begin{gathered} IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_q\right)= \\ \left(\begin{array}{c}\left[{\displaystyle \prod _{\tau =1}^q}{\widehat{\mathfrak{u}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^q}{\widehat{\mathfrak{w}}}_{\tau }^{{\phi }_{\tau }}\right],\left[1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{f}}}_{\tau }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{h}}}_{\tau }\right)}^{{\phi }_{\tau }}\right],\\ \left[1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{y}}}_{\tau }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{z}}}_{\tau }\right)}^{{\phi }_{\tau }}\right]\end{array}\right), \end{gathered}$$

Moreover, for $n=q+1$, by the operational laws of IVPF number, we have

$$IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_q,{\mathsf{\sigma }}_{q+1}\right)={\mathsf{\sigma }}_1^{{\mathsf{\phi }}_1}⨂{\sigma }_2^{{\phi }_2},\dots ,⨂{\sigma }_q^{{\phi }_q}⨂{\mathsf{\sigma }}_{q+1}^{{\mathsf{\phi }}_{q+1}}$$

It follows that

$$\begin{gathered} IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_{q+1}\right) \\ \begin{array}{l}=(\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{u}}}_{\tau }^{{\mathsf{\phi }}_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{w}}}_{\tau }^{{\mathsf{\phi }}_{\tau }}\right],[\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{f}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)+\left(1-{\left(1-{\widehat{\mathfrak{f}}}_{q+1}\right)}^{{\phi }_{q+1}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{f}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{f}}}_{q+1}\right)}^{{\phi }_{q+1}}\right),\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{h}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)+\left(1-{\left(1-{\widehat{\mathfrak{h}}}_{q+1}\right)}^{{\phi }_{q+1}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{h}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{h}}}_{q+1}\right)}^{{\phi }_{q+1}}\right)],[\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{y}}}_{\tau }\right)}^{{\mathsf{\phi }}_{\tau }}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(1-{\left(1-{\widehat{\mathfrak{y}}}_{q+1}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{y}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{y}}}_{q+1}\right)}^{{\phi }_{q+1}}\right),\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{z}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(1-{\left(1-{\widehat{\mathfrak{z}}}_{q+1}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{z}}}_{\tau }\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{z}}}_{q+1}\right)}^{{\phi }_{q+1}}\right)\left]\right)\end{array} \end{gathered}$$

$$IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_{q+1}\right)=\left(\begin{array}{c}\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{u}}}_{\tau }^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{w}}}_{\tau }^{{\phi }_{\tau }}\right],\left[1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{f}}}_{\tau }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{h}}}_{\tau }\right)}^{{\phi }_{\tau }}\right],\\ \left[1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{y}}}_{\tau }\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{z}}}_{\tau }\right)}^{{\phi }_{\tau }}\right]\end{array}\right),$$

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(5\right)$ is true for $n=q+1.$ Hence, we conclude that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(5\right)$ is true for any $k\in N$. □

Example 5.

Consider three IVPFNs ${\sigma }_1=\left(\left[\mathrm{0.07,0.11}\right],\left[\mathrm{0.37,0.44}\right],\left[\mathrm{0.09,0.2}\right]\right),{\sigma }_2=\left(\left[\mathrm{0.20,0.25}\right],\left[\mathrm{0.1,0.17}\right],\left[\mathrm{0.3,0.33}\right]\right)$ and ${\sigma }_3=\left(\left[\mathrm{0.18,0.28}\right],\left[\mathrm{0.16,0.22}\right],\left[\mathrm{0.24,0.3}\right]\right)$ associated with weight vector $\phi ={\left(\mathrm{0.28,0.32,0.4}\right)}^T.$ Then, we have ${\displaystyle \prod _{\tau =1}^3}{\left({\widehat{\mathfrak{u}}}_{\tau }\right)}^{{\phi }_{\tau }}={\left(0.07\right)}^{0.28}\times {\left(0.20\right)}^{0.32}\times {\left(0.18\right)}^{0.4}=0.14.$ Similarly, we can obtain ${\displaystyle \prod _{\tau =1}^3}{\left({\widehat{\mathfrak{w}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.21,{\displaystyle \prod _{\tau =1}^3}{\left(1-{\widehat{\mathfrak{f}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.79,{\displaystyle \prod _{\tau =1}^3}{\left(1-{\widehat{\mathfrak{h}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.73,{\displaystyle \prod _{\tau =1}^3}{\left(1-{\widehat{\mathfrak{y}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.78$ and ${\displaystyle \prod _{\tau =1}^3}{\left(1-{\widehat{\mathfrak{z}}}_{\tau }\right)}^{{\phi }_{\tau }}=0.72.$

Consequently,

$$IVPFWG\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\right)=\left(\left[\mathrm{0.14,0.21}\right],\left[\mathrm{0.21,0.27}\right],\left[\mathrm{0.22,0.28}\right]\right)$$

Theorem 10.

(Idempotency) If all ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},...,n\right)$ are equal, i.e., ${\sigma }_{\tau }=\sigma$ for all $\tau$, then

$$IVPFW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)=\sigma$$

Proof. 

Let $\sigma =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right)$ yields that

$$\begin{gathered} IVPFW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)= \\ \left(\begin{array}{c}\left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{u}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{w}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\\ \left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{{\mathsf{\sigma }}_{\mathsf{\tau }}}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right]\end{array}\right) \\ =\left(\begin{array}{c}\left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\\ \left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right]\end{array}\right) \\ =\left(\begin{array}{c}\left[{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},1-{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right],\\ \left[1-{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},1-{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right]\end{array}\right) \\ =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right)=\sigma \end{gathered}$$

□

Theorem 11.

(Boundedness) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Let ${\sigma }^{-}=\mathit{min}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{-},{\widehat{\mathfrak{w}}}^{-}\right],\left[{\widehat{\mathfrak{f}}}^{-},{\widehat{\mathfrak{h}}}^{-}\right],\left[{\widehat{\mathfrak{y}}}^{-},{\widehat{\mathfrak{z}}}^{-}\right]\right)$ and ${\sigma }^{+}=\mathit{max}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{+},{\widehat{\mathfrak{w}}}^{+}\right],\left[{\widehat{\mathfrak{f}}}^{+},{\widehat{\mathfrak{h}}}^{+}\right],\left[{\widehat{\mathfrak{y}}}^{+},{\widehat{\mathfrak{z}}}^{+}\right]\right)\hspace{1em}$ where ${\widehat{\mathfrak{u}}}^{-}=min{\widehat{\mathfrak{u}}}_{\tau }^{-},{\widehat{\mathfrak{w}}}^{-}=min{\widehat{\mathfrak{w}}}_{\tau }^{-},{\widehat{\mathfrak{f}}}^{-}=max{\widehat{\mathfrak{f}}}_{\tau }^{-},{\widehat{\mathfrak{h}}}^{-}=max{\widehat{\mathfrak{h}}}_{\tau }^{-},{\widehat{\mathfrak{y}}}^{-}=max{\widehat{\mathfrak{y}}}_{\tau }^{-},{\widehat{\mathfrak{z}}}^{-}=max{\widehat{\mathfrak{z}}}_{\tau }^{-},{\widehat{\mathfrak{u}}}^{+}=max{\widehat{\mathfrak{u}}}_{\tau }^{+},{\widehat{\mathfrak{w}}}^{+}=max{\widehat{\mathfrak{w}}}_{\tau }^{+},{\mathfrak{f}}^{+}=min{\widehat{\mathfrak{f}}}_{\tau }^{+},{\mathfrak{h}}^{+}=min{\widehat{\mathfrak{h}}}_{\tau }^{+},{\widehat{\mathfrak{y}}}^{+}=min{\widehat{\mathfrak{y}}}_{\tau }^{+},{\widehat{\mathfrak{z}}}^{+}=min{\widehat{\mathfrak{z}}}_{\tau }^{+}$, then ${\sigma }^{-}\le IVPFW{G}_{\phi }\left({\sigma }_1,{\sigma }_2\dots ,{\sigma }_n\right)\le {\sigma }^{+}$.

Proof. 

In view of given conditions, we have

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{u}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\le {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{u}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{u}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{w}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\tau }}\le {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{w}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\le {\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left({\widehat{\mathfrak{w}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},$$

$$1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\tau }}^{-}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\tau }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\ge 1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\tau }}^{+}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}.$$

In the light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 8$ and the above relations, we get ${\mathsf{\sigma }}^{-}\le IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2\dots ,{\mathsf{\sigma }}_n\right)\le {\mathsf{\sigma }}^{+}.$ □

Theorem 12.

(Monotonicity) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)$ and ${\sigma }_{\tau }^{’}=\left(\left[{\widehat{\mathfrak{u}}}_{\tau }^{’},{\widehat{\mathfrak{w}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{f}}}_{\tau }^{’},{\widehat{\mathfrak{h}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{y}}}_{\tau }^{’},{\widehat{\mathfrak{z}}}_{\tau }^{’}\right]\right)$ for $\tau =\mathrm{1,2},3,\dots ,n$ be two collections of IVPFNs. If ${\widehat{\mathfrak{u}}}_{\tau }\le {\widehat{\mathfrak{u}}}_{\tau }^{’},{\widehat{\mathfrak{w}}}_{\tau }\le {\widehat{\mathfrak{w}}}_{\tau }^{’},{\widehat{\mathfrak{f}}}_{\tau }\ge {\widehat{\mathfrak{f}}}_{\tau }^{’},{\widehat{\mathfrak{h}}}_{\tau }\ge {\widehat{\mathfrak{h}}}_{\tau }^{’},{\widehat{\mathfrak{y}}}_{\tau }\ge {\widehat{\mathfrak{y}}}_{\tau }^{’}$ and ${\widehat{\mathfrak{z}}}_{\tau }\ge {\widehat{\mathfrak{z}}}_{\tau }^{’}\forall \tau$. Then $IVPFWG\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)\le IVPFWG\left({\sigma }_1^{’},{\sigma }_2^{’},{\sigma }_3^{’}\dots ,{\sigma }_n^{’}\right)$.

Proof. 

As we know that

$$IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\dots ,{\mathsf{\sigma }}_n\right)={\mathsf{\sigma }}_1^{{\mathsf{\phi }}_1}\otimes {\mathsf{\sigma }}_2^{{\mathsf{\phi }}_2},\dots ,\otimes {\mathsf{\sigma }}_n^{{\mathsf{\phi }}_n}$$

and

$$IVPFW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_n\right)={\left({\mathsf{\sigma }}_1^{’}\right)}^{{\mathsf{\phi }}_1}\otimes {\left({\mathsf{\sigma }}_2^{’}\right)}^{{\mathsf{\phi }}_2},\dots ,\otimes {\left({\mathsf{\sigma }}_n^{’}\right)}^{{\mathsf{\phi }}_n}$$

as ${\mathsf{\sigma }}_{\mathsf{\tau }}\le {\mathsf{\sigma }}_{\mathsf{\tau }}^{’}\forall \mathsf{\tau }$, thus,

$$IVPFWG\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\dots ,{\mathsf{\sigma }}_n\right)\le IVPFWG\left({\mathsf{\sigma }}_1^{’},{\mathsf{\sigma }}_2^{’},{\mathsf{\sigma }}_3^{’}\dots ,{\mathsf{\sigma }}_n^{’}\right)$$

□

In the following definition, we propose a geometric aggregation operator with IVPFNs, namely, an IVPF ordered weighted geometric (IVPFOWG) operator.

Definition 11.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a collection of IVPFNs. The IVPF ordered weighted geometric (IVPFOWG) operator is a mapping $U^n\to U$ such that

$$IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)={⨂}_{\tau =1}^n{\left({\sigma }_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}$$

where, ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$ , $\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{\phi }_{\tau }\in \left[\mathrm{0,1}\right]$ such that${\sum }_{\tau =1}^n{\phi }_{\tau }=1;\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(n\right)\right)$ is a permutation of $\left(\mathrm{1,2},3,\dots ,n\right)$ such that ${\sigma }_{{\rho }_{\left(\tau -1\right)}}\ge {\sigma }_{{\rho }_{\tau }}$for all $\tau =\mathrm{1,2},3,\dots ,n.$

Theorem 13.

Let ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a collection of IVPFNs. Then the aggregated value of these IVPFNs in the framework of an IVPFOWG operator is also an IVPFN, and is determined in the following way:

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

(6)

where, ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right),\tau =\mathrm{1,2},3,\dots ,n$, $and\phi ={\left({\phi }_1,{\phi }_2,\dots ,{\phi }_n\right)}^T,{\phi }_{\tau }\in \left[\mathrm{0,1}\right]$ such that ${\sum }_{\tau =1}^n{\phi }_{\tau }=1;\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(n\right)\right)$ is a permutation of $\left(\mathrm{1,2},3,\dots ,n\right)$ such that ${\sigma }_{{\rho }_{\left(\tau -1\right)}}\ge {\sigma }_{{\rho }_{\tau }}$ for all $\tau =\mathrm{1,2},3,\dots ,n.$

Proof. 

We prove $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(6\right)$ by mathematical induction on n.

The application of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $11$ for $n=2$ gives the following outcome

$$IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2\right)={\left({\sigma }_{\rho \left(1\right)}\right)}^{{\phi }_1}⨂{\left({\sigma }_{\rho \left(2\right)}\right)}^{{\phi }_2}$$

In light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $4$, we have

$${\sigma }_1^{{\phi }_1}=\left(\begin{array}{c}\left[{\widehat{\mathfrak{u}}}_{\rho \left(1\right)}^{{\phi }_1},{\widehat{\mathfrak{w}}}_{\rho \left(1\right)}^{{\phi }_1}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(1\right)}\right)}^{{\phi }_1},1-{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(1\right)}\right)}^{{\phi }_1}\right],\\ \left[1-{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(1\right)}\right)}^{{\phi }_1},1-{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(1\right)}\right)}^{{\phi }_1}\right]\end{array}\right)$$

$${\sigma }_2^{{\phi }_2}=\left(\begin{array}{c}\left[{\widehat{\mathfrak{u}}}_{\rho \left(2\right)}^{{\phi }_2},{\widehat{\mathfrak{w}}}_{\rho \left(2\right)}^{{\phi }_2}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(2\right)}\right)}^{{\phi }_2},1-{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(2\right)}\right)}^{{\phi }_2}\right],\\ \left[1-{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(2\right)}\right)}^{{\phi }_2},1-{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(2\right)}\right)}^{{\phi }_2}\right]\end{array}\right)$$

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

It follows that

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

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\left(6\right)$ is true for $n=2$.

Assume that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(6\right)$ is true for $n=q$. Then,

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

Moreover, for $n=q+1$, by the operational laws of IVPF number, we have

$$IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,...,{\sigma }_q,{\sigma }_{q+1}\right)={\mathsf{\sigma }}_{\rho \left(1\right)}^{{\mathsf{\phi }}_1}⨂{\sigma }_{\rho \left(2\right)}^{{\phi }_2},\dots ,⨂{\sigma }_{\rho \left(q\right)}^{{\phi }_q}⨂{\mathsf{\sigma }}_{\rho (q+1)}^{{\mathsf{\phi }}_{q+1}}$$

It follows

$$\begin{gathered} IVPFOW{G}_{\mathsf{\phi }}\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,\dots ,{\mathsf{\sigma }}_{q+1}\right)= \\ \left(\begin{array}{c}\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}^{{\mathsf{\phi }}_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}^{{\mathsf{\phi }}_{\tau }}\right],\\ \left[\begin{array}{c}\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)+\left(1-{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right),\\ \left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)+\left(1-{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right)\end{array}\right],\\ \left[\begin{array}{c}\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }}\right)+\left(1-{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(q+1\right)}\right)}^{{\phi }_{q+1}}\right),\\ \left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)+\left(1-{\left(1-{\widehat{\mathfrak{z}}}_{\rho (q+1)}\right)}^{{\phi }_{q+1}}\right)-\left(1-{\displaystyle \prod _{\tau =1}^q}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right)\left(1-{\left(1-{\widehat{\mathfrak{z}}}_{\rho (q+1)}\right)}^{{\phi }_{q+1}}\right)\end{array}\right]\end{array}\right) \end{gathered}$$

$$IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_{q+1}\right)=\left(\begin{array}{c}\left[{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }},{\displaystyle \prod _{\tau =1}^{q+1}}{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}^{{\phi }_{\tau }}\right],\left[1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right],\\ \left[1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }},1-{\displaystyle \prod _{\tau =1}^{q+1}}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\right]\end{array}\right),$$

Thus, $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(6\right)$ is true for $n=q+1.$ Hence, we conclude that $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ $\left(6\right)$ is true for any $k\in N$. □

Example 6.

Consider 3 IVPFNs ${\sigma }_1=\left(\left[\mathrm{0.04,0.09}\right],\left[\mathrm{0.2,0.27}\right],\left[\mathrm{0.01,0.03}\right]\right),$ ${\sigma }_2=\left(\left[\mathrm{0.3,0.36}\right],\left[\mathrm{0.17,0.24}\right],\left[\mathrm{0.10,0.14}\right]\right),$ and ${\sigma }_3=\left(\left[\mathrm{0.2,0.31}\right],\left[\mathrm{0.1,0.19}\right],\left[\mathrm{0.03,0.17}\right]\right),$ associated with weight vector $\phi ={\left(\mathrm{0.1,0.4,0.5}\right)}^T$. To aggregate these values by IVPFOWG operator, we firstly permute these numbers in the light of $Equation\left(2\right)$ to obtain the values of $Z\left({\sigma }_1\right),Z\left({\sigma }_2\right),andZ\left({\sigma }_3\right),$ as follows $Z\left({\sigma }_1\right)=-0.18,Z\left({\sigma }_2\right)=0.07,Z\left({\sigma }_3\right)=0.06$. Thus, $Z\left({\sigma }_2\right)>Z\left({\sigma }_3\right)>Z\left({\sigma }_1\right),$ and permuted IVPFNs are ${\sigma }_{\rho \left(1\right)}=\left(\left[\mathrm{0.3,0.36}\right],\left[\mathrm{0.17,0.24}\right],\left[\mathrm{0.10,0.14}\right]\right),{\sigma }_{\rho \left(2\right)}=\left(\left[\mathrm{0.2,0.31}\right],\left[\mathrm{0.1,0.19}\right],\left[\mathrm{0.03,0.17}\right]\right),$ and ${\sigma }_{\rho \left(3\right)}=\left(\left[\mathrm{0.04,0.09}\right],\left[\mathrm{0.2,0.27}\right],\left[\mathrm{0.01,0.03}\right]\right)$. Now, by applying $Definition$ $11$ to the above IVPFNs obtained, we get ${\displaystyle {\prod }_{\tau =1}^3}{\left({\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}={\left(0.04\right)}^{0.1}\times {\left(0.3\right)}^{0.4}\times {\left(0.2\right)}^{0.5}=0.09.$ Similarly, we can obtain ${\displaystyle {\prod }_{\tau =1}^3}{\left({\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}=0.17,{\displaystyle {\prod }_{\tau =1}^3}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}=0.84,{\displaystyle {\prod }_{\tau =1}^3}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}=0.76,{\displaystyle {\prod }_{\tau =1}^3}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}=\mathfrak{u}n\mathfrak{h}{\displaystyle {\prod }_{\tau =1}^3}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}=0.90$.

Consequently,

$$IVPFOWG\left({\mathsf{\sigma }}_1,{\mathsf{\sigma }}_2,{\mathsf{\sigma }}_3\right)=\left(\left[\mathrm{0.04,0.14}\right],\left[\mathrm{0.16,0.24}\right],\left[\mathrm{0.03,0.10}\right]\right)$$

Theorem 14.

(Idempotency) If all ${\sigma }_{\tau }\left(\tau =\mathrm{1,2},\dots ,n\right)$ are equal, i.e., ${\sigma }_{\rho \left(\tau \right)}=\sigma$ for all $\tau$, then $IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)=\sigma$.

Proof. 

Let $\sigma =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right).$ Then ${\sigma }_{\rho \left(\tau \right)}=\sigma \left(\tau =\mathrm{1,2},3\dots ,n\right)$ yields that

$$\begin{gathered} IVPFOWG\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right) \\ \begin{array}{l}=\left(\begin{array}{c}\left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{u}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{w}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\\ \left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{{\mathsf{\sigma }}_{\mathsf{\rho }}\left(\mathsf{\tau }\right)}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right]\end{array}\right)\\ =\left(\begin{array}{c}\left[{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\mathsf{\tau }}},{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right],\\ \left[1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}},1-{\displaystyle \prod _{\mathsf{\tau }=1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}\right)}^{{\mathsf{\phi }}_{\mathsf{\tau }}}\right]\end{array}\right)\\ =\left(\begin{array}{c}\left[{\widehat{\mathfrak{u}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},{\widehat{\mathfrak{w}}}_{\mathsf{\sigma }}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right],\left[1-{\left(1-{\widehat{\mathfrak{f}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},1-{\left(1-{\widehat{\mathfrak{h}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right],\\ \left[1-{\left(1-{\widehat{\mathfrak{y}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }},1-{\left(1-{\widehat{\mathfrak{z}}}_{\mathsf{\sigma }}\right)}^{{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }}\right]\end{array}\right)\\ =\left(\left[{\widehat{\mathfrak{u}}}_{\sigma },{\widehat{\mathfrak{w}}}_{\sigma }\right],\left[{\widehat{\mathfrak{f}}}_{\sigma },{\widehat{\mathfrak{h}}}_{\sigma }\right],\left[{\widehat{\mathfrak{y}}}_{\sigma },{\widehat{\mathfrak{z}}}_{\sigma }\right]\right)=\sigma \end{array} \end{gathered}$$

□

Theorem 15.

(Boundedness) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)\left(\tau =\mathrm{1,2},\dots ,n\right)$ be a number of IVPFNs. Let ${\sigma }^{-}=\mathit{min}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{-},{\widehat{\mathfrak{w}}}^{-}\right],\left[{\widehat{\mathfrak{f}}}^{-},{\widehat{\mathfrak{h}}}^{-}\right],\left[{\widehat{\mathfrak{y}}}^{-},{\widehat{\mathfrak{z}}}^{-}\right]\right)$ and ${\sigma }^{+}=\mathit{max}{\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}^{+},{\widehat{\mathfrak{w}}}^{+}\right],\left[{\widehat{\mathfrak{f}}}^{+},{\widehat{\mathfrak{h}}}^{+}\right],\left[{\widehat{\mathfrak{y}}}^{+},{\widehat{\mathfrak{z}}}^{+}\right]\right),\hspace{1em}$ where ${\widehat{\mathfrak{u}}}^{-}=min{\widehat{\mathfrak{u}}}_{\tau }^{-},{\widehat{\mathfrak{w}}}^{-}=min{\widehat{\mathfrak{w}}}_{\tau }^{-},{\widehat{\mathfrak{f}}}^{-}=max{\widehat{\mathfrak{f}}}_{\tau }^{-},{\widehat{\mathfrak{h}}}^{-}=max{\widehat{\mathfrak{h}}}_{\tau }^{-},{\widehat{\mathfrak{y}}}^{-}=max{\widehat{\mathfrak{y}}}_{\tau }^{-},{\widehat{\mathfrak{z}}}^{-}=max{\widehat{\mathfrak{z}}}_{\tau }^{-},{\widehat{\mathfrak{u}}}^{+}=max{\widehat{\mathfrak{u}}}_{\tau }^{+},{\widehat{\mathfrak{w}}}^{+}=max{\widehat{\mathfrak{w}}}_{\tau }^{+},{\mathfrak{f}}^{+}=min{\widehat{\mathfrak{f}}}_{\tau }^{+},{\mathfrak{h}}^{+}=min{\widehat{\mathfrak{h}}}_{\tau }^{+},{\widehat{\mathfrak{y}}}^{+}=min{\widehat{\mathfrak{y}}}_{\tau }^{+},and{\widehat{\mathfrak{z}}}^{+}=min{\widehat{\mathfrak{z}}}_{\tau }^{+}$, then ${\sigma }^{-}\le IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2\dots ,{\sigma }_n\right)\le {\sigma }^{+}$.

Proof. 

In view of given conditions, we have

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}^{-}\right)}^{{\phi }_{\tau }}\le {\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\le {\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}^{+}\right)}^{{\phi }_{\tau }},$$

$${\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}^{-}\right)}^{{\phi }_{\tau }}\le {\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\le {\displaystyle \prod _{\tau =1}^n}{\left({\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}^{+}\right)}^{{\phi }_{\tau }},$$

$$1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}^{-}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}^{+}\right)}^{{\phi }_{\tau }},$$

$$1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}^{-}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}^{+}\right)}^{{\phi }_{\tau }},$$

$$1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}^{-}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}^{+}\right)}^{{\phi }_{\tau }},$$

$$1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}^{-}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\right)}^{{\phi }_{\tau }}\ge 1-{\displaystyle \prod _{\tau =1}^n}{\left(1-{\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}^{+}\right)}^{{\phi }_{\tau }}.$$

In light of $\mathrm{D}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} 8$ and the above relations, we get ${\sigma }^{-}\le IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2\dots ,{\sigma }_n\right)\le {\sigma }^{+}$ □

Theorem 16.

(Monotonicity) Let ${\sigma }_{\tau }=\left(\left[{\widehat{\mathfrak{u}}}_{\tau },{\widehat{\mathfrak{w}}}_{\tau }\right],\left[{\widehat{\mathfrak{f}}}_{\tau },{\widehat{\mathfrak{h}}}_{\tau }\right],\left[{\widehat{\mathfrak{y}}}_{\tau },{\widehat{\mathfrak{z}}}_{\tau }\right]\right)$ and ${\sigma }_{\tau }^{’}=\left(\left[{\widehat{\mathfrak{u}}}_{\tau }^{’},{\widehat{\mathfrak{w}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{f}}}_{\tau }^{’},{\widehat{\mathfrak{h}}}_{\tau }^{’}\right],\left[{\widehat{\mathfrak{y}}}_{\tau }^{’},{\widehat{\mathfrak{z}}}_{\tau }^{’}\right]\right)$ for $\tau =\mathrm{1,2},3,\dots ,n$ be two collections of IVPFNs. If ${\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}\le {\widehat{\mathfrak{u}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}\le {\widehat{\mathfrak{w}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{f}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{h}}}_{\rho \left(\tau \right)}^{’},{\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{y}}}_{\rho \left(\tau \right)}^{’}$ and ${\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}\ge {\widehat{\mathfrak{z}}}_{\rho \left(\tau \right)}^{’}\forall \tau$, then $IVPFOWG\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)\le IVPFOWG\left({\sigma }_1^{’},{\sigma }_2^{’},{\sigma }_3^{’}\dots ,{\sigma }_n^{’}\right)$.

Proof. 

As we know that

$$IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)={\sigma }_{\rho \left(1\right)}^{{\phi }_1}\otimes {\sigma }_{\rho \left(2\right)}^{{\phi }_2},\dots ,\otimes {\sigma }_{\rho \left(n\right)}^{{\phi }_n}$$

and

$$IVPFOW{G}_{\phi }\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n\right)={\left({\sigma }_{\rho \left(1\right)}^{’}\right)}^{{\phi }_1}\otimes {\left({\sigma }_{\rho \left(2\right)}^{’}\right)}^{{\phi }_2},\dots ,\otimes {\left({\sigma }_{\rho \left(n\right)}^{’}\right)}^{{\phi }_n}$$

as ${\sigma }_{\rho \left(\tau \right)}\le {\sigma }_{\rho \left(\tau \right)}^{’}\forall \mathsf{\tau }$, thus,

$$IVPFOWG\left({\sigma }_1,{\sigma }_2,{\sigma }_3\dots ,{\sigma }_n\right)\le IVPFOWG\left({\mathsf{\sigma }}_1^{’},{\mathsf{\sigma }}_2^{’},{\mathsf{\sigma }}_3^{’}\dots ,{\mathsf{\sigma }}_n^{’}\right)$$

□

5. Algorithm to Solve MADM Problems with Interval Valued Picture Fuzzy Information

In this section, we address the solution to MADM problems using the proposed IVPF weighted aggregation operators, where the weights of the attributes are represented by real numbers, and the values of the attributes are represented by IVPFNs. Let $L$ = $L_1,L_2,\dots ,L_n$ be a discrete arrangement of alternatives, and $Q=Q_1,Q_2,\dots ,Q_n$ be the arrangement of attributes, $\mathsf{\phi }={\left({\mathsf{\phi }}_1,{\mathsf{\phi }}_2,\dots ,{\mathsf{\phi }}_n\right)}^T$ be the associated weighted vector of the attribute $Q_{\tau }\left(\tau =\mathrm{1,2},3,\dots ,n\right)$ where $\mathsf{\phi }\in \left[\mathrm{0,1}\right],{\sum }_{\tau =1}^n{\mathsf{\phi }}_{\tau }=1.$ Suppose that $\tilde{N}={\left({\mathsf{\sigma }}_{i\tau }\right)}_{m\times n}=\left({\mathsf{\mu }}_{i\tau },{\mathsf{\eta }}_{i\tau },{\mathsf{\nu }}_{i\tau }\right)$ is a decision matrix representing the information in the form of IVPFNs, where ${\mathsf{\mu }}_{i\tau }$ represents the degree of positive membership that the alternative $L_i$ satisfies the attribute $Q_{\tau }$ given by the decision maker, and ${\mathsf{\eta }}_{i\tau }$ represents the neutrality of the attribute $Q_{\tau },{\mathsf{\nu }}_{i\tau }$ represents the degree that the alternative $L_i$ doesn’t satisfy the attribute $Q_{\tau }$ given by the decision maker. The proposed approach, based on the IVPFWA operator to design the solution to MADM problems, comprises the following steps:

$\mathit{S}\mathit{t}\mathit{e}\mathit{p} 1.$ A decision matrix $\tilde{N}$ is established on the basis of the information obtained from a decision maker in the following way:

$$\tilde{N}={\left({\mathsf{\sigma }}_{i\tau }\right)}_{m\times n}$$

It follows that,

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

$\mathit{S}\mathit{t}\mathit{e}\mathit{p} 2.$ Normalize the data presented in the decision matrix above by converting the cost-type (C) rating values into benefit-type (B) rating values, if applicable, utilizing the normalization formula provided below:

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

(7)

$\mathit{S}\mathit{t}\mathit{e}\mathit{p} 3.$ The application of the IVPFWA operator on the information obtained in $step 2$ yields that,

$$y_i=IVPFW{A}_{\mathsf{\phi }}\left(L_1,L_2,\dots ,L_n\right)={\oplus }_{\tau =1}^n\left({\mathsf{\phi }}_{\tau }{L}_{\tau }\right),y_i\left(i=\mathrm{1,2},3,\dots ,n\right)$$

Similarly, by applying the IVPFOWA operator on the information obtained in $step 2$, we get,

$$y_i=IVPFOW{A}_{\mathsf{\phi }}\left(L_1,L_2,\dots ,L_n\right)={\oplus }_{\tau =1}^n\left({\mathsf{\phi }}_{\tau }{L}_{{\mathsf{\rho }}_{\tau }}\right),y_i\left(i=\mathrm{1,2},3,\dots ,n\right)$$

Likewise, by using the IVPFWG operator on the information obtained in $step 2$, we have,

$$y_i=IVPFW{G}_{\mathsf{\phi }}\left(L_1,L_2,\dots ,L_n\right)={\otimes }_{\tau =1}^n{\left(L_{\tau }\right)}^{{\mathsf{\phi }}_{\tau }},y_i\left(i=\mathrm{1,2},3,\dots ,n\right)$$

Similarly, by using the IVPFOWG operator on the information obtained in $step 2$, we have,

$$y_i=IVPFOW{G}_{\mathsf{\phi }}\left(L_1,L_2,\dots ,L_n\right)={\otimes }_{\tau =1}^n{\left(L_{\mathsf{\rho }\left(\tau \right)}\right)}^{{\mathsf{\phi }}_{\tau }},y_i\left(i=\mathrm{1,2},3,\dots ,n\right).$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}4.$ Calculate the scores of the alternatives $L_i$ by using $\mathrm{Z}\left(y_i\right)=\frac{\widehat{\mathfrak{u}}-\widehat{\mathfrak{f}}-\widehat{\mathfrak{y}}+\widehat{\mathfrak{w}}-\widehat{\mathfrak{h}}-\widehat{\mathfrak{z}}}{2}+\frac{\widehat{\mathfrak{u}}-\widehat{\mathfrak{y}}}{3}.$$\mathit{S}\mathit{t}\mathit{e}\mathit{p}5$. Using the score value, rank all the alternatives and select the best one.

5.1. Implementation of IVPFS Operators to Solve MCDM Problems

Effective stock management is a significant challenge in the current industrial landscape. A corporation cannot achieve its manufacturing objectives without proper stock management. Hence, efficient inventory management is the primary step towards achieving high production levels. Insufficient inventory of raw materials may disrupt the entire manufacturing process, leading to significant financial losses for the business. To monitor all its stock, a food corporation primarily engaged in manufacturing four food categories, namely “soft drinks,” “vegetable oils,” “jam,” and “bakery goods,” needs to make stock reordering decisions based on three factors: “cost price,” “storage facilities,” and “staleness level” for the ingredients in its inventory. The weight vector of these factors is represented as $q={\left(\mathrm{0.3,0.2,0.5}\right)}^T$. The suggested alternatives are rated in terms of IVPFNs, and analyzed under these three factors.

Step 1:

Table 1. Decision matrix representing the opinion of the decision maker about the physical situation in terms of IVPF.

	${\mathit{Q}}_1$	${\mathit{Q}}_2$	${\mathit{Q}}_3$
$L_1$	([0.10,0.20],[0.30,0.30],[0.40,0.50])	([0.20,0.30],[0.30,0.40],[0.00,0.10])	([0.30,0.60],[0.00,0.00],[0.10,0.20])
$L_2$	([0.20,0.25],[0.10,0.15],[0.65,0.70])	([0.40,0.50],[0.15,0.20],[0.01,0.10])	([0.50,0.60],[0.13,0.20],[0.01,0.11])
$L_3$	([0.20,0.20],[0.20,0.40],[0.30,0.35])	([0.10,0.20],[0.23,0.40],[0.10,0.15])	([0.20,0.30],[0.20,0.25],[0.10,0.20])
$L_4$	([0.10,0.20],[0.30,0.40],[0.10,0.20])	([0.30,0.35],[0.10,0.30],[0.20,0.30])	([0.01,0.20],[0.20,0.25],[0.40,0.45])

describes the evaluation of the attributes on the basis of the above information given by the decision maker in the form of a matrix whose entries are IVPFNs.

Step 2: The application of $\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ (7) to the values obtained in Step 1, gives the normalized matrix presented in the following

Table 2. Normalized matrix representing the opinion of the decision maker about the physical situation in terms of IVPF.

	${\mathit{Q}}_1$	${\mathit{Q}}_2$	${\mathit{Q}}_3$
$L_1$	([0.40,0.50],[0.30,0.30],[0.10,0.20])	([0.20,0.30],[0.30,0.40],[0.00,0.10])	([0.30,0.60],[0.00,0.00],[0.10,0.20])
$L_2$	([0.65,0.70],[0.10,0.15],[0.20,0.25])	([0.40,0.50],[0.15,0.20],[0.01,0.10])	([0.50,0.60],[0.13,0.20],[0.01,0.11])
$L_3$	([0.30,0.35],[0.20,0.40],[0.20,0.20])	([0.10,0.20],[0.23,0.40],[0.10,0.15])	([0.20,0.30],[0.20,0.25],[0.10,0.20])
$L_4$	([0.10,0.20],[0.30,0.40],[0.10,0.20])	([0.30,0.35],[0.10,0.30],[0.20,0.30])	([0.01,0.20],[0.20,0.25],[0.40,0.45])

.

Step 3: Applying the weight vector $q={\left(\mathrm{0.3,0.2,0.5}\right)}^T$ and the newly defined operators to the normalized decision matrix, give us the overall preference of all 4 alternatives $L_i\left(i=\mathrm{1,2},\mathrm{3,4}\right)$ in the following

Table 3. Preference table of the alternatives under IVPFWA (IVPFWG) operators.

	IVPFWA	IVPFWG
$L_1$	([0.31,0.52],[0.0,0.0],[0.0,0.17])	([0.30,0.49],[0.16,0.19],[0.08,0.18])
$L_2$	([0.53,0.62],[0.12,0.18],[0.02,0.14])	([0.51,0.60],[0.13,0.19],[0.07,0.15])
$L_3$	([0.21,0.29],[0.21,0.32],[0.12,0.19])	([0.19,0.29],[0.21,0.33],[0.13,0.19])
$L_4$	([0.10,0.23],[0.19,0.29],[0.23,0.33])	([0.03,0.22],[0.21,0.31],[0.28,0.35])

and

Table 4. Preference table of the alternatives under IVPFOWG (IVPFOWG) operators.

	IVPFOWA	IVPFOWG
$L_1$	([0.28,0.43],[0.00,0.00],[0.00,0.14])	([0.26,0.41],[0.22,0.27],[0.05,0.15])
$L_2$	([0.51,0.59],[0.13,0.18],[0.01,0.12])	([0.47,0.56],[0.13,0.19],[0.05,0.13])
$L_3$	([0.18,0.26],[0.21,0.34],[0.11,0.17])	([0.15,0.25],[0.21,0.36],[0.12,0.17])
$L_4$	([0.10,0.23],[0.17,0.29],[0.24,0.33])	([0.04,0.23],[0.19,0.25],[0.29,0.36])

.

Step 4: The score values of the alternatives are obtained in the following

Table 5. Score values of the alternatives.

	IVPFWA	IVPFWG	IVPFOWA	IVPFOWG
$L_1$	0.34	0.163	0.378	0.047
$L_2$	0.52	0.432	0.488	0.405
$L_3$	−0.14	−0.170	−0.172	−0.215
$L_4$	−0.39	−0.533	−0.397	−0.493

, by applying the score function.

Step 5: From Table 5, we note that $Z\left(L_2\right)>Z\left(L_1\right)>Z\left(L_3\right)>Z\left(L_4\right),$ and we have $L_2\succ L_1\succ L_3\succ L_4$. Consequently, $L_2$ is the most suitable option.

5.2. Comparative Analysis

A comparative analysis is established in order to show the validity and feasibility of the proposed technique in the subsequent discussion.

Table 6. Comparison table of newly defined techniques to existing techniques.

Techniques	Preference order	Optimal Alternative
IVIFWA [8]	$L_2\succ L_1\succ L_3\succ L_4$	$L_2$
IVIFWG [7]	$L_2\succ L_1\succ L_3\succ L_4$	$L_2$
IVPFWA	$L_2\succ L_1\succ L_3\succ L_4$	$L_2$
IVPFOWA	$L_2\succ L_1\succ L_3\succ L_4$	$L_2$
IVPFWG	$L_2\succ L_1\succ L_3\succ L_4$	$L_2$
IVPFWOG	$L_2\succ L_1\succ L_3\succ L_4$	$L_2$

interprets the comparison between the newly defined strategies of this article, and the existing techniques, namely, an IVIFWA operator and an IVIFWG operator. It is quite evident that the mechanism presented in this article is extremely productive, because all these methods produce the same rankings. It is also important to note that aggregation operators under IFS knowledge have a few restrictions and are unable to give the appropriate information that is available regarding the MADM problem under consideration, whereas the proposed aggregation operators consider the quantity of information represented by the extent of positive membership, neutral membership, and negative membership, as well as the credibility of the information conveyed by the refusal membership. Moreover, the concept of IVPFS is a powerful extension of IVIFS, as it contains a larger quantity of data than does IVIFS. Thus, IVIFS aggregation operators developed by Wei [7] and Wang et al. [8] become the special case of IVPFS aggregation operators.

6. Conclusions

In this paper, an improved score function has been designed in order to overcome the shortcomings of the existing score function to solve MCDM problems in the framework of an IVPF environment. The concepts of IVPFWA operators, IVPFOWA operators, IVPFWG operators, and IVPFOWG operators, have been presented. Moreover, an algorithm has been formulated in this article to solve MCDM problems by using these newly defined strategies. It has been shown that stock management reduces stock loss and misleading records, by keeping stock levels in check.

The primary focus of future efforts will be on creating a full-featured decision analysis aid based on IV aggregation operators, with the goal of increasing its usefulness and applicability in the real world. In addition, the methods laid out in this article can be used to successfully deal with water purification on commercial scale projects, and evaluation of construction projects in the construction sector. These efforts will make it possible to inexpensively and efficiently solve a wide range of significant MADM problems. Finally, a comparative analysis has been established in order to show the validity and feasibility of proposed techniques with existing methods.