Uncertainty evaluations through interval-valued Pythagorean hesitant fuzzy Archimedean aggregation operators in multicriteria decision making

Abstract

In many cases, use of Pythagorean hesitant fuzzy sets may not be sufficient to characterize uncertain information associated with decision making problems. From that view point the concept of interval-valued Pythagorean hesitant fuzzy sets are introduced in this paper. Considering the flexibility with the general parameters, Archimedean $t$ -conorms and $t$ -norms are applied to develop several operational laws in interval-valued Pythagorean hesitant fuzzy environment. Some characteristics of the developed operators are presented. The newly developed operators are used to derive a methodology for solving multicriteria decision making problems with interval-valued Pythagorean hesitant fuzzy information. Finally, two illustrative examples are provided to establish the validity of the proposed approach and are compared with the existing technique to exhibit its flexibility and effectiveness.

Keywords

Multicriteria decision making interval-valued Pythagorean fuzzy set Pythagorean hesitant fuzzy set Archimedean t-conorm and t-norm weighted averaging and geometric operators

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1. Introduction

Human perceptions are frequently involved with indeterminacy and indecisiveness. Most of the time, it becomes difficult for the decision makers (DMs) to exert their opinion using a single crisp value. Under this situation, fuzzy sets [1] came into account. Several variants of fuzzy sets, viz., intuitionistic fuzzy sets (IFSs) [2, 3, 4], Pythagorean fuzzy sets (PFSs) [5, 6], etc. appeared, thereafter, as the extensions of fuzzy sets and been implemented successfully in solving multi-criteria decision making (MCDM) [7, 8, 9, 10] problems. PFS is more general than IFS due to the fact that PFS consists membership degree along with non-membership degree having their square sum less than or equals to 1; whereas, the sum of membership and non-membership degrees is less than or equals to 1 in IFS. So, in terms of flexibility and dealing with uncertainties, PFS can express uncertain information more effectively. For example, if a DM provides 0.7, as membership value and 0.6, as non-membership value, PFS can successfully deal with such values. But IFS fails to consider those values. For this useful characteristics, PFSs are applied to solve real life decision making problems, e.g., pattern recognition [11], supplier selection [12], transportation problem [13], risk evaluation [14, 15], and other emerging areas. Following the concept of interval-valued fuzzy sets, Peng and Yang [16] introduced the idea of interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs), where the membership and non-membership values of an element to a given set are represented by the subintervals of [0, 1]. With the use of IVPFS, DMs can evaluate the input data in a more convenient way than IFS as well as PFS.

Sometimes, in many MCDM situations, DMs face difficulties in assigning membership value corresponding to some element; rather they prefer to express their opinions using a set of possible values. To address such situations, Torra and Narukawa [17] and Torra [18] proposed another generalization of fuzzy sets, called hesitant fuzzy sets (HFSs). It offers the DMs flexibility to assign the membership degree using several possible crisp numbers lying in between [0, 1]. In many MCDM problems HFSs [19, 20, 21, 22, 23, 24] are applied efficiently. Combining the idea of HFS and PFS, Wei et al. [25] defined Pythagorean hesitant fuzzy (PHF) set (PHFS). PHFS is found to be very much useful in modelling uncertainties associated with real life problems.

Inspired by the concept of IVPFS, PHFSs have been extended in this article to generate interval-valued PHF (IVPHF) set (IVPHFS). It possesses greater capability of capturing uncertainties than existing variants of fuzzy sets. It is to be noted here that, if lower and upper limits of the intervals coincide with each other, an IVPHFS becomes PHFS. Again, if each membership and non-membership degrees of the elements of IVPHFS are expressed by single intervals, IVPHFS reduces to IVPFS. Therefore, it can be concluded that IVPHFS is a more generalised version of the other variants of fuzzy sets as described above.

Moreover, Archimedean $t$ -norm and $t$ -conorm (A $t$ -N& $t$ -CN) satisfy all the properties of $t$ -norm ( $t$ -N) and $t$ -conorm ( $t$ -CN) to aggregate input arguments. Algebraic, Einstein, Hamacher, Frank classes are widely used A $t$ -N& $t$ -CNs for developing aggregation operators. Motivated by the aforementioned discussions, in this article various weighted averaging (WA) and weighted geometric (WG) aggregation operators are developed using A $t$ -N& $t$ -CN based operations on IVPHF numbers (IVPHFNs). Following those operations, IVPHF WA (IVPHFWA), IVPHF Einstein WA (IVPHFEWA), IVPHF Hamacher WA (IVPHFHWA) and IVPHF Frank WA (IVPHFFWA) operators, and their WG variants are presented. Several desirable properties of those developed operators are also investigated. Using those operators, an approach for solving MCDM problems under the IVPHF environment is presented to show the efficiency and usefulness of the proposed operators. Literature review on the allied topics is performed in the next section.

2. Literature review

It is well known that PFS [5] is one of the most useful tools to resolve ambiguous information of MCDM problems. Zhang and Xu [26] provided standard arithmetic operations on PFSs. For aggregating PFSs, Yager [6] proposed a series of WA and WG aggregation operators. Several aggregation operators [27, 28] and methods [29, 30] for solving MCDM problems are developed on PFSs by numerous researchers.

After the development of IVPFS [16], an accuracy function for ranking the IVPF numbers (IVPFNs) was developed by Garg [31], considering the hesitancy degree of IVPFNs for solving MCDM problems. Rahman et al. [32] introduced a class of IVPF geometric aggregation operators for IVPFNs, viz. IVPF WG, ordered WG, hybrid geometric operators. Biswas and Sarkar [33] introduced similarity measures based on point operators for IVPFSs. Technique for order of preference by similarity to ideal solution (TOPSIS) method was introduced by Garg [34] under IVPF environment. Further, Garg [35] proposed some new IVPF aggregation operators on the basis of novel exponential operational laws of IVPFS. Generalised IVPF aggregation operators are introduced by Rahman and Abdullah [36] for IVPF multicriteria group decision making (MCGDM) problems. An IVPF extended TOPSIS method was developed by Yu et al. [37] for sustainable supplier selection under MCGDM context. Tang et al. [38] proposed IVPF MCDM approach based on Muirhead mean and applied it on green supplier selection. Some induced IVPF aggregation operators are developed by Rahman et al. [39] for tackling uncertainties in MCDM situation.

Following the concepts of HFS and PFSs, PHFS [25] has now become another emerging area of research. An application to PHFS with incomplete weight information was proposed by Khan et al. [40] for solving group decision making problems. Wei et al. [25] developed PHF Hamacher WA (PHFHWA) and PHF Hamacher WG (PHFHWG) aggregation functions to aggregate the PHF information. A new approach for PHF TOPSIS method was presented by Khan et al. [41] in the context of solving MCDM problems. Recently, Sarkar and Biswas [42] proposed A $t$ -N& $t$ -CN based operations [43] on PHF numbers (PHFNs) to construct two types of aggregation operators such as A $t$ -N& $t$ -CN-based PHF WA and WG operators. An extension of VIKOR method was developed by Khan et al. [44] for PHF MCDM context. Further, Khan et al. [45] introduced Choquet integral to PHFSs and generated a series of aggregation operators, viz., PHF Choquet integral averaging operator, PHF Choquet integral geometric operator, generalised PHF Choquet integral averaging operator and generalised PHF Choquet integral geometric operator. Khan et al. [46] also developed some hybrid aggregation operators to aggregate PHF data for MCGDM. From the literature survey, it had been found that the combination of IVPFS and PHFS had not been done yet. Thus it would be significant to develop a series of aggregation operators, viz., IVPHFWA, IVPHFEWA, IVPHFHWA, IVPHFFWA, IVPHF WG (IVPHFWG), IVPHF Einstein WG (IVPHFEWG), IVPHF Hamacher WG (IVPHFHWG) and IVPHF Frank WG (IVPHFFWG) operators based on A $t$ -N& $t$ -CN for evaluating IVPHF information. In addition, the novelty of this research work is to deliver an approach for solving MCDM problems under the IVPHF environment to show the applicability of the developed operators. Two illustrative examples are provided for the sake of application of the developed MCDM method.

3. Preliminaries

In this section some elementary definitions related to the development of A $t$ -N& $t$ -CN based operators are briefly discussed.

3.1 IVPFS

Peng and Yang [16] introduced the concept of IVPFSs, which is presented as follows:

Definition 1. Let $X$ be a fixed set. An IVPFS, $\tilde{P}$ in $X$ is given by

$\displaystyle\tilde{P}=\left\{\left\langle x,\left[{\mu}^{l}_{\tilde{P}}\left(% x\right),{\mu}^{u}_{\tilde{P}}\left(x\right)\right],\left[{\nu}^{l}_{\tilde{P}% }\left(x\right),{\nu}^{u}_{\tilde{P}}\left(x\right)\right]\right\rangle\left|x% \in X\right.\right\},$

where two closed intervals $[{\mu}^{l}_{\tilde{P}}\left(x\right),{\mu}^{u}_{\tilde{P}}\left(x\right)]$ and $[{\nu}^{l}_{\tilde{P}}\left(x\right),{\nu}^{u}_{\tilde{P}}\left(x\right)]$ are subintervals of $[0,1]$ , denote the degree of membership and degree of non-membership of the element, $x\in X$ to the set $\tilde{P}$ , respectively, with the condition $0\leqslant{\left({\mu}^{u}_{\tilde{P}}\left(x\right)\right)}^{2}+{\left({\nu}^% {u}_{\tilde{P}}\left(x\right)\right)}^{2}\leqslant 1$ .

For computational convenience, Peng and Yang [16] used the notation, $\tilde{p}=\left(\left[{\mu}^{l},{\mu}^{u}\right],\left[{\nu}^{l},{\nu}^{u}% \right]\right)$ to represent an IVPFN.

3.2 PHFS

Inspired by the idea of PFSs and HFSs, Wei et al. [25] elaborated PFS to PHFS, which is structured with a set of several possible PFNs, symbolically defined as follows:

Definition 2. Let $X$ be a universe of discourse. A PHFS on $X$ is defined as:

$\displaystyle\hat{A}=\left\{\left\langle x,h_{\hat{A}}\left(x\right)\right% \rangle\left|x\in X\right.\right\},$

where $h_{\hat{A}}\left(x\right)$ is a set of possible PFNs.

For convenience, Wei et al. [25] called $\hat{a}=h_{\hat{A}}\left(x\right)=\bigcup\nolimits_{\left(\gamma,\eta\right)% \in h_{\hat{A}}\left(x\right)}{\left\{\left(\gamma,\eta\right)\right\}}$ as a PHFN, and it is denoted by $\hat{a}=h_{\hat{A}}=\left(\mu,\nu\right)$ .

For ordering of PHFNs, the score and accuracy functions of PHFNs are presented [25] as follows:

Let $\hat{a}=\left(\mu,\nu\right)$ be any PHFN, the score function, $S\left(\hat{a}\right)$ of $\hat{a}$ be presented as follows:

$\displaystyle S\left(\hat{a}\right)=\frac{1}{2}\left(1+\frac{1}{l_{h}}\sum% \nolimits_{\left(\gamma,\eta\right)\in\left(\mu,\nu\right)}{\left({\gamma}^{2}% -{\eta}^{2}\right)}\right),$

and the accuracy function, $A\left(\hat{a}\right)$ of $\hat{a}$ be expressed as follows:

$\displaystyle A\left(\hat{a}\right)=\frac{1}{l_{h}}\sum\nolimits_{\left(\gamma% ,\eta\right)\in\left(\mu,\nu\right)}{\left({\gamma}^{2}+{\eta}^{2}\right)},$

where $l_{h}$ are the number of elements in $h$ .

Definition 3. For two PHFNs, ${\hat{a}}_{i}=\left({\mu}_{i},{\nu}_{i}\right)$ $\left(i=1,2\right)$ , the ranking can be done in the following manners [25]:

$\bullet$
If $S\left({\hat{a}}_{1}\right)>S\left({\hat{a}}_{2}\right)$ , then ${\hat{a}}_{1}>{\hat{a}}_{2}$ ;
$\bullet$
If $S\left({\hat{a}}_{1}\right)=S\left({\hat{a}}_{2}\right)$ , then

(1)
If $A\left({\hat{a}}_{1}\right)>A\left({\hat{a}}_{2}\right)$ , then ${\hat{a}}_{1}>{\hat{a}}_{2}$ ;
(2)
If $A\left({\hat{a}}_{1}\right)=A\left({\hat{a}}_{2}\right)$ , then ${\hat{a}}_{1}={\hat{a}}_{2}$ .

3.3 At-N&t-CN operations

Definition 4. Satisfying the following properties, a function $T:\left[0,1\right]\times\left[0,1\right]\to\left[0,1\right]$ is known as a $t$ -N [43]:

(i)
$T\left(a,1\right)=a$ for all $a$ .
(ii)
$T\left(b,d\right)\leqslant T\left(b^{},d^{}\right)$ for $b\leqslant b^{}$ and $d\leqslant d^{}$
(iii)
$T\left(a,b\right)=T\left(b,a\right)$ for all $a$ and $b$ .
(iv)
$T\left(a,T\left(b,d\right)\right)=T\left(T\left(a,b\right),d\right)$ for all $a, b$ and $d$ .
(v)
$T$ is a continuous function.
(vi)
$T\left(a,a\right)<a$ .

Definition 5. A function $S:\left[0,1\right]\times\left[0,1\right]\to\left[0,1\right]$ is called a $t$ -CN if the following criteria are satisfied [43]:

(i)
$S\left(a,0\right)=a\$ for all $a$ .
(ii)
$S\left(b,d\right)\leqslant S\left(b^{},d^{}\right)$ when $b\leqslant b^{}$ and $d\leqslant d^{}$
(iii)
$S\left(a,b\right)=S\left(b,a\right)$ for all $a$ and for all $b$ .
(iv)
$S\left(a,S\left(b,d\right)\right)=S\left(S\left(a,b\right),d\right)$ for all $a, b$ and $d$ .
(v)
$S$ is a continuous function.
(vi)
$S\left(a,a\right)>a$ .

It is well known that A $t$ -N& $t$ -CN operations are expressed using increasing or decreasing generators as follows [47]:

Definition 6. An Archimedean $t$ -N (A $t$ -N), $T$ is formulated using a decreasing function, $f$ as

$\displaystyle T\left(a,b\right)=f^{\left(-1\right)}\left(f\left(a\right)+f% \left(b\right)\right).$ (1)

Similarly, using increasing function, $g$ , an Archimedean $t$ -CN (A $t$ -CN) $S$ is represented as

$\displaystyle S\left(a,b\right)=g^{\left(-1\right)}\left(g\left(a\right)+g% \left(b\right)\right)\ \text{for all}\ a,b\in\left[0,\ 1\right].$ (2)

The generators have the relationship $g\left(t\right)=f\left(1-t\right)$ for all $a,b,t\in\left[0,\ 1\right]$ .

Several $t$ -Ns and $t$ -CNs are derived by Klement [47] using different forms of increasing and decreasing functions which are furnished below:

(1)
For $f\left(t\right)=-\text{log}t$ , A $t$ -N& $t$ -CN is reduced to algebraic $t$ -N and $t$ -CN [48], and are defined as $S^{A}\left(a,b\right)=a+b-ab$ , $T^{A}\left(a,b\right)=ab$ .
(2)
When $f\left(t\right)={\text{log}\left(\frac{2-t}{t}\right)\ }$ , $g\left(t\right)={\text{log}\left(\frac{2-\left(1-t\right)}{1-t}\right)\ }$ , $f^{-1}\left(t\right)=\frac{2}{e^{t}+1}$ , $g^{-1}\left(t\right)=1-\frac{2}{e^{t}+1}$ , A $t$ -N& $t$ -CN is reduced to Einstein $t$ -N and $t$ -CN [48] and are defined as $S^{E}\left(a,b\right)=\frac{a+b}{1+ab}$ , $T^{E}\left(a,b\right)=\frac{ab}{1+\left(1-a\right)\left(1-b\right)}$ .
(3)
Let $f\left(t\right)={\text{log}\left(\frac{\psi+\left(1-\psi\right)t}{t}\right)\ }$ , $\psi>0$ , $g\left(t\right)={\text{log}\left(\frac{\psi+\left(1-\psi\right)\left(1-t\right% )}{1-t}\right)\ }$ , $f^{-1}\left(t\right)=\frac{\psi}{e^{t}+\psi-1}$ , $g^{-1}\left(t\right)=1-\frac{\psi}{e^{t}+\psi-1}$ , then A $t$ -N& $t$ -CN called Hamacher $t$ -N and $t$ -CN [48] and are presented as $S^{H}_{\psi}\left(a,b\right)=\frac{a+b-ab-\left(1-\psi\right)ab}{1-\left(1-% \psi\right)ab}$ , $T^{H}_{\psi}\left(a,b\right)=\frac{ab}{\psi+\left(1-\psi\right)\left(a+b-ab% \right)}$ , $\psi>0$ .

When $\psi=1$ , the Hamacher $t$ -N and $t$ -CN reduce to algebraic $t$ -N and $t$ -CN, respectively. Again, for $\psi=2$ , the Hamacher $t$ -N and $t$ -CN reduce to Einstein $t$ -N and $t$ -CN, respectively.
(4)
For $f\left(t\right)={\text{log}\left(\frac{\tau-1}{{\tau}^{t}-1}\right)\ }$ , $\tau>1$ , $g\left(t\right)={\text{log}\left(\frac{\tau-1}{{\tau}^{1-t}-1}\right)\ }$ , $f^{-1}\left(t\right)=\frac{{\text{log}\left(\frac{\tau-1+e^{t}}{e^{t}}\right)% \ }}{{\text{log}\tau\ }}$ , $g^{-1}\left(t\right)=1-\frac{{\text{log}\left(\frac{\tau-1+e^{t}}{e^{t}}\right% )\ }}{{\text{log}\tau\ }}$ , an A $t$ -N& $t$ -CN are named as Frank $t$ -N and $t$ -CN [48], respectively, and are defined as $S^{F}_{\tau}\left(a,b\right)=1-{{\text{log}}_{\tau}\left(1+\frac{\left({\tau}^% {1-a}-1\right)\left({\tau}^{1-b}-1\right)}{\tau-1}\right)\ }$ , $T^{F}_{\tau}\left(a,b\right)={{\text{log}}_{\tau}\left(1+\frac{\left({\tau}^{a% }-1\right)\left({\tau}^{b}-1\right)}{\tau-1}\right)\ }$ , $\tau>1$ . Especially, if $\tau\to 1$ , ${\mathop{\text{lim}}_{\tau\to 1}f\left(t\right)\ }=-{\text{log}\ t}$ .

3.4 At-N&t-CN operations on PHF environment

Sarkar and Biswas [42] recently introduced A $t$ -N& $t$ -CN operations on PHF environment, which are defined as follows:

Definition 7. Let $\hat{a}=\left(\mu,\nu\right)$ and ${\hat{a}}_{i}=\left({\mu}_{i},{\nu}_{i}\right)$ $\left(i=1,2\right)$ be three arbitrary PHFNs, and $\lambda$ be any positive number. Then the operational laws for the PHFNs based on A $t$ -N& $t$ -CN are defined as follows:

Let $\hat{a}=\left(\mu,\nu\right)$ and ${\hat{a}}_{i}=\left({\mu}_{i},{\nu}_{i}\right)\ \left(i=1,2\right)$ be three arbitrary PHFNs and $\lambda$ be any positive number. Then the operational laws for the PHFNs based on A $t$ -N& $t$ -CN are defined as follows:

(1)
${\hat{a}}_{1}{\oplus}_{A}{\hat{a}}_{2}=\bigcup_{\begin{subarray}{c}\left({% \gamma}_{i},{\eta}_{i}\right)\in\left({\mu}_{i},{\nu}_{i}\right)\\ i=1,2\end{subarray}}{\left\{\left(\sqrt{S\left({{\gamma}_{1}}^{2},{{\gamma}_{2% }}^{2}\right)},\sqrt{T\left({{\eta}_{1}}^{2},{{\eta}_{2}}^{2}\right)}\right)% \right\}}$

$\hskip 36.988583pt=\bigcup_{\begin{subarray}{c}\left({\gamma}_{i},{\eta}_{i}% \right)\in\left({\mu}_{i},{\nu}_{i}\right)\\ i=1,2\end{subarray}}{\left\{\left(\sqrt{g^{-1}\left(g\left({{\gamma}_{1}}^{2}% \right)+g\left({{\gamma}_{2}}^{2}\right)\right)},\sqrt{f^{-1}\left(f\left({{% \eta}_{1}}^{2}\right)+f\left({{\eta}_{2}}^{2}\right)\right)}\right)\right\}};$
(2)
${\hat{a}}_{1}{\otimes}_{A}{\hat{a}}_{2}=\bigcup_{\begin{subarray}{c}\left({% \gamma}_{i},{\eta}_{i}\right)\in\left({\mu}_{i},{\nu}_{i}\right)\\ i=1,2\end{subarray}}{\left\{\left(\sqrt{T\left({{\gamma}_{1}}^{2},{{\gamma}_{2% }}^{2}\right)},\sqrt{S\left({{\eta}_{1}}^{2},{{\eta}_{2}}^{2}\right)}\right)% \right\}}$

$\hskip 36.988583pt=\bigcup_{\begin{subarray}{c}\left({\gamma}_{i},{\eta}_{i}% \right)\in\left({\mu}_{i},{\nu}_{i}\right)\\ i=1,2\end{subarray}}{\left\{\left(\sqrt{f^{-1}\left(f\left({{\gamma}_{1}}^{2}% \right)+f\left({{\gamma}_{2}}^{2}\right)\right)},\sqrt{g^{-1}\left(g\left({{% \eta}_{1}}^{2}\right)+g\left({{\eta}_{1}}^{2}\right)\right)}\right)\right\};}$
(3)
$\lambda\hat{a}=\bigcup\nolimits_{\left(\gamma,\eta\right)\in\left(\mu,\nu% \right)}{\left\{\left(\sqrt{g^{-1}\left(\lambda g\left({\gamma}^{2}\right)% \right)},\sqrt{f^{-1}\left(\lambda f\left({\eta}^{2}\right)\right)}\right)% \right\}};$
(4)
${\hat{a}}^{\lambda}=\bigcup\nolimits_{\left(\gamma,\eta\right)\in\left(\mu,\nu% \right)}{\left\{\left(\sqrt{f^{-1}\left(\lambda f\left({\gamma}^{2}\right)% \right)},\sqrt{g^{-1}\left(\lambda g\left({\eta}^{2}\right)\right)}\right)% \right\}}.$

Based on the above concepts, the notion of IVPHFSs is introduced in the following section.
4. Development of IVPHFS

Sometimes, it becomes inadequate to describe an uncertain situation using PHF information. To tackle this type of situation, the concept of IVPHFS is introduced. In this section, PHFS is extended with the use of interval numbers to construct IVPHFS. Also, score and accuracy functions on them are defined. Furthermore, some operations based on A $t$ -N& $t$ -CN are proposed in this section.

4.1 IVPHFS

Definition 8. Let $X$ be a universe of discourse, an IVPHFS, $\tilde{K}$ on $X$ is expressed as:

$\displaystyle\tilde{K}{=}\left\{\langle x,h_{\tilde{K}}(x)\rangle|x\in X\right\},$ (3)

in which $h_{\tilde{K}}(x)=\{([{\gamma}^{l}(x)$ , ${\gamma}^{u}(x)]$ , $[{\eta}^{l}(x)$ , ${\eta}^{u}(x)])\}$ is a set containing some IVPFNs, where $[{\gamma}^{l}x)$ , ${\gamma}^{u}(x)]$ , $[{\eta}^{l}(x)$ , ${\eta}^{u}(x)]\subseteq[0,1]$ satisfies the condition $0\leqslant{({({\gamma}^{u}(x))}^{+})}^{2}+{({({\eta}^{u}(x))}^{{+}})}^{2}{% \leqslant}{1}$ , ${\left({\gamma}^{u}\left(x\right)\right)}^{{+}}{=}max\left\{{\gamma}^{u}\left(% x\right)\right\}$ and ${\left({\eta}^{u}\left(x\right)\right)}^{{+}}{=}max\left\{{\eta}^{u}\left(x% \right)\right\}$ , for all $x{\in}X$ .

Thus, IVPHFS represents a HFS whose membership degrees are expressed by several IVPFNs.

For convenience, $h_{\tilde{K}}=\left\{\left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{% \eta}^{l}{,\ }{\eta}^{u}\right]\right)\right\}$ is named as an IVPHFN and is denoted by $\tilde{k}$ . The set of all IVPHFNs is given by $\tilde{K}$ .

To illustrate the above definition, the following example is presented.

Example 1. Let the possible assessment values of an element $x{\in}X$ are represented by the IVPFNs, $([0.1,0.3]$ , $[0.3,0.4])$ , $([0.5,0.6]$ , $[0.4,0.5])$ and $([0.7,0.8]$ , $[0.4,0.6])$ , where $[0.1,0.3]$ , $[0.5,0.6]$ and $[0.7,0.8]$ indicate the possible degree of membership and $[0.3,0.4]$ , $[0.4,0.5]$ and $[0.4,0.6]$ indicate the possible degree of non-membership for the element $x{\in}X$ . Then the IVPHFN is represented as an element of the set $\tilde{K}$ as $\tilde{k}=\{(\left[0.1,0.3\right]$ , $\left[0.3,0.4\right]),\left(\left[0.5,0.6\right],\left[0.4,0.5\right]\right),% \left(\left[0.7,0.8\right],\left[0.4,0.6\right]\right)\}$ in which ${\left({\gamma}^{u}\right)}^{{+}}=\max\left\{{\gamma}^{u}\left(x\right)\right% \}={\max\left\{0.3,\ 0.6,\ 0.8\right\}\ }=0.8$ , ${\left({\eta}^{u}\right)}^{{+}}{=}\max\left\{{\eta}^{u}\left(x\right)\right\}{% =}{{\max}\left\{0.4,\ 0.5,\ 0.6\right\}=\ }{0.6}$ , satisfying the condition $0\leqslant\left(0.8\right)^{2}+\left(0.6\right)^{2}\leqslant{1}$ .

To compare any two IVPHFNs, the score and accuracy functions are presented as follows:

Definition 9. For an IVPHFN, $\tilde{k}$ , the score function, $S\left(\tilde{k}\right)$ is described as:

$\displaystyle S\left(\tilde{k}\right)=\frac{1}{2}\left(1+\frac{1}{2}\left(% \frac{1}{\left|\tilde{k}\right|}\sum\nolimits_{\left(\left[{\gamma}^{l}{,\ }{% \gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k}% }{\left({\left({\gamma}^{l}\right)}^{2}+{\left({\gamma}^{u}\right)}^{2}-{\left% ({\eta}^{l}\right)}^{2}-{\left({\eta}^{u}\right)}^{2}\right)}\right)\right),$ (4)

and the accuracy function, $A\left(\tilde{k}\right)$ is expressed as:

$\displaystyle A\left(\tilde{k}\right)=\frac{1}{2}\left(\frac{1}{\left|\tilde{k% }\right|}\sum\nolimits_{\left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left% [{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k}}{\left({\left({\gamma}^% {l}\right)}^{2}+{\left({\gamma}^{u}\right)}^{2}+{\left({\eta}^{l}\right)}^{2}+% {\left({\eta}^{u}\right)}^{2}\right)}\right),$ (5)

where $\left|\tilde{k}\right|$ is the number of IVPFNs in $\tilde{k}$ .

Using the score and accuracy functions, $S$ and $A$ , respectively, any two IVPHFNs can be compared via the following rules as shown below:

Definition 10. Let ${\tilde{k}}_{i}$ $\left(i=1,2\right)$ be any two IVPHFNs, then

(i)

If $S\left({\tilde{k}}_{1}\right)>S\left({\tilde{k}}_{2}\right)$ then ${\tilde{k}}_{1}>{\tilde{k}}_{2}$ ;

(ii)

If $S\left({\tilde{k}}_{1}\right)=S\left({\tilde{k}}_{2}\right)$ then

$\bullet$

${\tilde{k}}_{1}>{\tilde{k}}_{2}$ for $A\left({\tilde{k}}_{1}\right)>A\left({\tilde{k}}_{2}\right)$ ;

\bullet

${\tilde{k}}_{1}={\tilde{k}}_{2}$ for $A\left({\tilde{k}}_{1}\right)=A\left({\tilde{k}}_{2}\right)$ .

Thus from the above ranking process, it is clear that the rank of the IVPHFNs are first performed based on the score values. If those are found as equal, then the rank is made based on the value of the accuracy function.

4.2 IVPHF operations based on At-N&t-CN

On the basis of A $t$ -N& $t$ -CN operations, several operations on IVPHFNs are proposed as follows:

Definition 11. Let ${\tilde{k}}_{i}$ $\left(i=1,2\right)$ and $\tilde{k}$ be any three IVPHFNs, and $\lambda>0$ be any scalar. Now, utilising A $t$ -N& $t$ -CNs, the basic algebraic operations of IVPHFNs, viz., addition ${\oplus}_{A}$ and multiplication ${\otimes}_{A}$ , are defined as follows.

(1)
Addition: ${\tilde{k}}_{1}{\oplus}_{A}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[% {\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{k}}_{i}\\ i=1,2\end{subarray}}{\left\{\left(\left[\sqrt{S\left({\left({\gamma}^{l}_{1}% \right)}^{2},{\left({\gamma}^{l}_{2}\right)}^{2}\right)},\sqrt{S\left({\left({% \gamma}^{u}_{1}\right)}^{2},{\left({\gamma}^{u}_{2}\right)}^{2}\right)}\right]% ,\right.\right.}$

$\hskip 85.358268pt\left.\left.\left[\sqrt{T\left({\left({\eta}^{l}_{1}\right)}% ^{2},{\left({\eta}^{l}_{2}\right)}^{2}\right)},\sqrt{T\left({\left({\eta}^{u}_% {1}\right)}^{2},{\left({\eta}^{u}_{2}\right)}^{2}\right)}\right]\right)\right\};$

$=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{% \gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{% \tilde{k}}_{i}\\ i=1,2\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}\left(g\left({\left({% \gamma}^{l}_{1}\right)}^{2}\right)+g{\left({\gamma}^{l}_{2}\right)}^{2}\right)% },\sqrt{g^{-1}\left(g\left({\left({\gamma}^{u}_{1}\right)}^{2}\right)+g\left({% \left({\gamma}^{u}_{2}\right)}^{2}\right)\right)}\right],\right.\right.}$

$\left.\left.\left[\sqrt{f^{-1}\left(f\left({\left({\eta}^{l}_{1}\right)}^{2}% \right)+f\left({\left({\eta}^{l}_{2}\right)}^{2}\right)\right)},\sqrt{f^{-1}% \left(f\left({\left({\eta}^{u}_{1}\right)}^{2}\right)+f\left({\left({\eta}^{u}% _{2}\right)}^{2}\right)\right)}\right]\right)\right\};$
(2)
Multiplication: ${\tilde{k}}_{1}{\otimes}_{A}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[% {\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\\ \in{\tilde{k}}_{i},i=1,2\end{subarray}}{\left\{\left(\left[\sqrt{T\left({\left% ({\gamma}^{l}_{1}\right)}^{2},{\left({\gamma}^{l}_{2}\right)}^{2}\right)},% \sqrt{T\left({\left({\gamma}^{u}_{1}\right)}^{2},{\left({\gamma}^{u}_{2}\right% )}^{2}\right)}\right],\right.\right.}$

$\hskip 113.811024pt\left.\left.\left[\sqrt{S\left({\left({\eta}^{l}_{1}\right)% }^{2},{\left({\eta}^{l}_{2}\right)}^{2}\right)},\sqrt{S\left({\left({\eta}^{u}% _{1}\right)}^{2},{\left({\eta}^{u}_{2}\right)}^{2}\right)}\right]\right)\right\};$

$=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{% \gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{% \tilde{k}}_{i},\\ i=1,2\end{subarray}}{\left\{\left(\left[\sqrt{f^{-1}\left(f\left({\left({% \gamma}^{l}_{1}\right)}^{2}\right)+f\left({\left({\gamma}^{l}_{2}\right)}^{2}% \right)\right)},\sqrt{f^{-1}\left(f\left({\left({\gamma}^{u}_{1}\right)}^{2}% \right)+f\left({\left({\gamma}^{u}_{2}\right)}^{2}\right)\right)}\right]\right% .\right.},$

$\left.\left.\left[\sqrt{g^{-1}\left(g\left({\left({\eta}^{l}_{1}\right)}^{2}% \right)+g\left({\left({\eta}^{l}_{2}\right)}^{2}\right)\right)},\sqrt{g^{-1}% \left(g\left({\left({\eta}^{u}_{1}\right)}^{2}\right)+g\left({\left({\eta}^{u}% _{2}\right)}^{2}\right)\right)}\right]\right)\right\};$
(3)
Scalar multiplication: $\lambda\tilde{k}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,\ }{% \gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k}% }{\left\{\left(\left[\sqrt{g^{-1}\left(\lambda g\left({{\gamma}^{l}}^{2}\right% )\right)},\sqrt{g^{-1}\left(\lambda g\left({{\gamma}^{u}}^{2}\right)\right)}% \right],\right.\right.}$

$\hskip 119.501575pt\left.\left.\left[\sqrt{f^{-1}\left(\lambda f\left({\left({% \eta}^{l}\right)}^{2}\right)\right)},\sqrt{f^{-1}\left(\lambda f\left({\left({% \eta}^{u}\right)}^{2}\right)\right)}\right]\right)\right\};$
(4)
Exponentiation: ${\tilde{k}}^{{\lambda}}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,% \ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}% \tilde{k}}{\left\{\left(\left[\sqrt{f^{-1}\left(\lambda f\left({\left({\gamma}% ^{l}\right)}^{2}\right)\right)},\sqrt{f^{-1}\left(\lambda f\left({\left({% \gamma}^{u}\right)}^{2}\right)\right)}\right],\right.\right.}$

$\hskip 91.048819pt\left.\left.\left[\sqrt{g^{-1}\left(\lambda g\left({\left({% \eta}^{l}\right)}^{2}\right)\right)},\sqrt{g^{-1}\left(\lambda g\left({\left({% \eta}^{u}\right)}^{2}\right)\right)}\right]\right)\right\}.$

The above operations are defined based on several increasing and decreasing operators. Now, choosing specific decreasing generating functions, some special types of aggregation operators are obtained, which are presented in the following manners:

$\bullet$
Algebraic ${\bm{t}}$ -N and ${\bm{t}}$ -CN: Considering $f\left(t\right)=-{\text{log}\ t}$ , operational laws for IVPHFNs based on Algebraic $t$ -N and $t$ -CN, are described as follows:

(1)
${\tilde{k}}_{1}\oplus{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[% {\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{k}}_{i},\\ i=1,2\end{subarray}}$

$\left\{\left(\left[\sqrt{\left({\gamma}^{l}_{1}\right)^{2}+\left(\gamma^{l}_{2% }\right)^{2}-\left(\gamma^{l}_{1}\right)^{2}\left(\gamma^{l}_{2}\right)^{2}},% \sqrt{\left(\gamma^{u}_{1}\right)^{2}+\left(\gamma^{u}_{2}\right)^{2}-\left(% \gamma^{u}_{1}\right)^{2}\left(\gamma^{u}_{2}\right)^{2}}\right],\left[\eta^{l% }_{1}\eta^{l}_{2},\eta^{u}_{1}\eta^{u}_{2}\!\right]\!\right)\!\right\}$
(2)
${\tilde{k}}_{1}\otimes{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i},i=1,2}$

$\left\{\left(\left[{\gamma}^{l}_{1}{\gamma}^{l}_{2},{\gamma}^{u}_{1}{\gamma}^{% u}_{2}\right],\left[\sqrt{{\left({\eta}^{l}_{1}\right)}^{2}+{\left({\eta}^{l}_% {2}\right)}^{2}-{\left({\eta}^{l}_{1}\right)}^{2}{\left({\eta}^{l}_{2}\right)}% ^{2}},\sqrt{{\left({\eta}^{u}_{1}\right)}^{2}+{\left({\eta}^{u}_{2}\right)}^{2% }-{\left({\eta}^{u}_{1}\right)}^{2}{\left({\eta}^{u}_{2}\right)}^{2}}\right]% \right)\right\};$
(3)
$\lambda\tilde{k}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l},{\gamma}^% {u}\right],\left[{\eta}^{l},{\eta}^{u}\right]\right)\in\tilde{k}}\left\{\left(% \left[\sqrt{1-{\left(1-{\left({\gamma}^{l}\right)}^{2}\right)}^{\lambda}},% \sqrt{1-{\left(1-{\left({\gamma}^{u}\right)}^{2}\right)}^{\lambda}}\right],% \left[{\left({\eta}^{l}\right)}^{\lambda},{\left({\eta}^{u}\right)}^{\lambda}% \right]\right)\right\},$
(4)
${\tilde{k}}^{\lambda}{=}\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,% \ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}% \tilde{k}}\left\{\left(\left[{\left({\gamma}^{l}\right)}^{\lambda},{\left({% \gamma}^{u}\right)}^{\lambda}\right],\left[\sqrt{1-{\left(1-{\left({\eta}^{l}% \right)}^{2}\right)}^{\lambda}},\sqrt{1-{\left(1-{\left({\eta}^{u}\right)}^{2}% \right)}^{\lambda}}\right]\right)\right\}.$

$\bullet$
Einstein ${\bm{t}}$ -N and ${\bm{t}}$ -CN: Considering $f\left(t\right)={\text{log}\left(\frac{2-t}{t}\right)}$ , operational laws for IVPHFNs based on Einstein $t$ -N and $t$ -CN, are described as follows:

(1)
${\tilde{k}}_{1}{\oplus}_{E}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{\left(% \left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}% _{i}\right]\right){\in}{\tilde{k}}_{i},i=1,2}$

$\left\{\left(\left[\sqrt{\frac{{{\gamma}^{l}_{1}}^{2}{+}{{\gamma}^{l}_{2}}^{2}% }{{1+}{{\gamma}^{l}_{1}}^{2}{{\gamma}^{l}_{2}}^{2}}},\sqrt{\frac{{{\gamma}^{u}% _{1}}^{2}{+}{{\gamma}^{u}_{2}}^{2}}{{1+}{{\gamma}^{u}_{1}}^{2}{{\gamma}^{u}_{2% }}^{2}}}\right],\left[\frac{{\eta}^{l}_{1}{\eta}^{l}_{2}}{\sqrt{{1+}\left({1-}% {{\eta}^{l}_{1}}^{2}\right)\left({1-}{{\eta}^{l}_{2}}^{2}\right)}},\frac{{\eta% }^{u}_{1}{\eta}^{u}_{2}}{\sqrt{{1+}\left({1-}{{\eta}^{u}_{1}}^{2}\right)\left(% {1-}{{\eta}^{u}_{2}}^{2}\right)}}\right]\right)\right\};$
(2)
${\tilde{k}}_{1}{\otimes}_{E}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{\left(% \left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}% _{i}\right]\right){\in}{\tilde{k}}_{i},i=1,2}{\left\{\left(\left[\frac{{\gamma% }^{l}_{1}{\gamma}^{l}_{2}}{\sqrt{{1+}\left({1-}{\left({\gamma}^{l}_{1}\right)}% ^{2}\right)\left({1-}{\left({\gamma}^{l}_{2}\right)}^{2}\right)}},\frac{{% \gamma}^{u}_{1}{\gamma}^{u}_{2}}{\sqrt{{1+}\left({1-}{\left({\gamma}^{u}_{1}% \right)}^{2}\right)\left({1-}{\left({\gamma}^{u}_{2}\right)}^{2}\right)}}% \right],\right.\right.}$

$\hskip 51.214961pt\left.\left.\left[\sqrt{\frac{{\left({\eta}^{l}_{1}\right)}^% {2}{+}{\left({\eta}^{l}_{2}\right)}^{2}}{{1+}{\left({\eta}^{l}_{1}\right)}^{2}% {\left({\eta}^{l}_{2}\right)}^{2}}},\sqrt{\frac{{\left({\eta}^{u}_{1}\right)}^% {2}{+}{\left({\eta}^{u}_{2}\right)}^{2}}{{1+}{\left({\eta}^{u}_{1}\right)}^{2}% {\left({\eta}^{u}_{2}\right)}^{2}}}\right]\right)\right\};$
(3)
$\lambda\tilde{k}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,\ }{% \gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k}% }{\left\{\left(\left[\sqrt{\frac{{\left({1+}{\left({\gamma}^{l}\right)}^{2}% \right)}^{{\lambda}}{-}{\left({1-}{\left({\gamma}^{l}\right)}^{2}\right)}^{{% \lambda}}}{{\left({1+}{\left({\gamma}^{l}\right)}^{2}\right)}^{{\lambda}}{+}{% \left({1-}{\left({\gamma}^{l}\right)}^{2}\right)}^{{\lambda}}}},\sqrt{\frac{{% \left({1+}{\left({\gamma}^{u}\right)}^{2}\right)}^{{\lambda}}{-}{\left({1-}{% \left({\gamma}^{u}\right)}^{2}\right)}^{{\lambda}}}{{\left({1+}{\left({\gamma}% ^{u}\right)}^{2}\right)}^{{\lambda}}{+}{\left({1-}{\left({\gamma}^{u}\right)}^% {2}\right)}^{{\lambda}}}}\right],\right.\right.}$

$\hskip 28.452756pt\left.\left.\left[\frac{\sqrt{{2}}{\left({\eta}^{l}\right)}^% {{\lambda}}}{\sqrt{{\left({2-}{\left({\eta}^{l}\right)}^{2}\right)}^{{\lambda}% }{+}{\left({\left({\eta}^{l}\right)}^{2}\right)}^{{\lambda}}}},\frac{\sqrt{{2}% }{\left({\eta}^{u}\right)}^{{\lambda}}}{\sqrt{{\left({2-}{\left({\eta}^{u}% \right)}^{2}\right)}^{{\lambda}}{+}{\left({\left({\eta}^{u}\right)}^{2}\right)% }^{{\lambda}}}}\right]\right)\right\},$
(4)
${\tilde{k}}^{{\lambda}}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,% \ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}% \tilde{k}}{\left\{\left(\left[\frac{\sqrt{{2}}{\left({\gamma}^{l}\right)}^{{% \lambda}}}{\sqrt{{\left({2-}{\left({\gamma}^{l}\right)}^{2}\right)}^{{\lambda}% }{+}{\left({\left({\gamma}^{l}\right)}^{2}\right)}^{{\lambda}}}},\frac{\sqrt{{% 2}}{\left({\gamma}^{u}\right)}^{{\lambda}}}{\sqrt{{\left({2-}{\left({\gamma}^{% u}\right)}^{2}\right)}^{{\lambda}}{+}{\left({\left({\gamma}^{u}\right)}^{2}% \right)}^{{\lambda}}}}\right],\right.\right.}$

$\hskip 28.452756pt\left.\left.\left[\sqrt{\frac{{\left({1+}{\left({\eta}^{l}% \right)}^{2}\right)}^{{\lambda}}{-}{\left({1-}{\left({\eta}^{l}\right)}^{2}% \right)}^{{\lambda}}}{{\left({1+}{\left({\eta}^{l}\right)}^{2}\right)}^{{% \lambda}}{+}{\left({1-}{\left({\eta}^{l}\right)}^{2}\right)}^{{\lambda}}}},% \sqrt{\frac{{\left({1+}{\left({\eta}^{u}\right)}^{2}\right)}^{{\lambda}}{-}{% \left({1-}{\left({\eta}^{u}\right)}^{2}\right)}^{{\lambda}}}{{\left({1+}{\left% ({\eta}^{u}\right)}^{2}\right)}^{{\lambda}}{+}{\left({1-}{\left({\eta}^{u}% \right)}^{2}\right)}^{{\lambda}}}}\right]\right)\right\}.$

$\bullet$
Hamacher ${\bm{t}}$ -N and ${\bm{t}}$ -CN: When $f\left(t\right)={\text{log}\left(\frac{\psi+\left(1-\psi\right)t}{t}\right)}$ , $\psi>0$ , then the following operations for IVPHFNs, based on Hamacher $t$ -N and $t$ -CN, are defined as

(1)
${\tilde{k}}_{1}{\oplus}_{H}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{\left(% \left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}% _{i}\right]\right){\in}{\tilde{k}}_{i},i=1,2}{\left\{\left(\left[\sqrt{\frac{{% \left({\gamma}^{l}_{1}\right)}^{2}+{\left({\gamma}^{l}_{2}\right)}^{2}-{\left(% {\gamma}^{l}_{1}\right)}^{2}{\left({\gamma}^{l}_{2}\right)}^{2}-\left(1-\psi% \right){\left({\gamma}^{l}_{1}\right)}^{2}{\left({\gamma}^{l}_{2}\right)}^{2}}% {1-\left(1-\psi\right){\left({\gamma}^{l}_{1}\right)}^{2}{\left({\gamma}^{l}_{% 2}\right)}^{2}}},\right.\right.\right.}$

$\hskip 51.214961pt\left.\sqrt{\frac{{\left({\gamma}^{u}_{1}\right)}^{2}+{\left% ({\gamma}^{u}_{2}\right)}^{2}-{\left({\gamma}^{u}_{1}\right)}^{2}{\left({% \gamma}^{u}_{2}\right)}^{2}-\left(1-\psi\right){\left({\gamma}^{u}_{1}\right)}% ^{2}{\left({\gamma}^{u}_{2}\right)}^{2}}{1-\left(1-\psi\right){\left({\gamma}^% {u}_{1}\right)}^{2}{\left({\gamma}^{u}_{2}\right)}^{2}}}\right],$

$\left.\left.\left[\frac{{\eta}^{l}_{1}{\eta}^{l}_{2}}{\sqrt{\psi{+}\left({1-}% \psi\right)\left({\left({\eta}^{l}_{1}\right)}^{2}+{\left({\eta}^{l}_{2}\right% )}^{2}-{\left({\eta}^{l}_{1}\right)}^{2}{\left({\eta}^{l}_{2}\right)}^{2}% \right)}},\frac{{\eta}^{u}_{1}{\eta}^{u}_{2}}{\sqrt{\psi{+}\left({1-}\psi% \right)\left({\left({\eta}^{u}_{1}\right)}^{2}+{\left({\eta}^{u}_{2}\right)}^{% 2}-{\left({\eta}^{u}_{1}\right)}^{2}{\left({\eta}^{u}_{2}\right)}^{2}\right)}}% \right]\right)\right\};$
(2)
${\tilde{k}}_{1}{\otimes}_{H}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[% {\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{k}}_{i}\\ i=1,2\end{subarray}}$

$\hskip 34.143307pt\left\{\left(\left[\frac{{\gamma}^{l}_{1}{\gamma}^{l}_{2}}{% \sqrt{\psi{+}\left({1-}\psi\right)\left({\left({\gamma}^{l}_{1}\right)}^{2}+{% \left({\gamma}^{l}_{2}\right)}^{2}-{\left({\gamma}^{l}_{1}\right)}^{2}{\left({% \gamma}^{l}_{2}\right)}^{2}\right)}},\frac{{\gamma}^{u}_{1}{\gamma}^{u}_{2}}{% \sqrt{\psi{+}\left({1-}\psi\right)\left({\left({\gamma}^{u}_{1}\right)}^{2}+{% \left({\gamma}^{u}_{2}\right)}^{2}-{\left({\gamma}^{u}_{1}\right)}^{2}{\left({% \gamma}^{u}_{2}\right)}^{2}\right)}}\right],\right.\right.$

$\hskip 34.143307pt\left[\sqrt{\frac{{\left({\eta}^{l}_{1}\right)}^{2}+{\left({% \eta}^{l}_{2}\right)}^{2}-{\left({\eta}^{l}_{1}\right)}^{2}{\left({\eta}^{l}_{% 2}\right)}^{2}-\left(1-\psi\right){\left({\eta}^{l}_{1}\right)}^{2}{\left({% \eta}^{l}_{2}\right)}^{2}}{1-\left(1-\psi\right){\left({\eta}^{l}_{1}\right)}^% {2}{\left({\eta}^{l}_{2}\right)}^{2}}},\right.\left.\left.\left.\sqrt{\frac{{% \left({\eta}^{u}_{1}\right)}^{2}+{\left({\eta}^{u}_{2}\right)}^{2}-{\left({% \eta}^{u}_{1}\right)}^{2}{\left({\eta}^{u}_{2}\right)}^{2}-\left(1-\psi\right)% {\left({\eta}^{u}_{1}\right)}^{2}{\left({\eta}^{u}_{2}\right)}^{2}}{1-\left(1-% \psi\right){\left({\eta}^{u}_{1}\right)}^{2}{\left({\eta}^{u}_{2}\right)}^{2}}% }\right]\right)\right\};$
(3)
$\lambda\tilde{k}=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}{,\ }{\gamma}^{u}\right]\right.,\\ \left.\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k}% \end{subarray}}{\left\{\left(\left[\sqrt{\frac{{\left(1+\left(\psi-1\right){% \left({\gamma}^{l}\right)}^{2}\right)}^{\lambda}-{\left(1-{{\gamma}^{l}}^{2}% \right)}^{\lambda}}{{\left(1+\left(\psi-1\right){{\gamma}^{l}}^{2}\right)}^{% \lambda}+\left(\psi-1\right){\left(1-{{\gamma}^{l}}^{2}\right)}^{\lambda}}},% \sqrt{\frac{{\left(1+\left(\psi-1\right){\left({\gamma}^{u}\right)}^{2}\right)% }^{\lambda}-{\left(1-{\left({\gamma}^{u}\right)}^{2}\right)}^{\lambda}}{{\left% (1+\left(\psi-1\right){\left({\gamma}^{u}\right)}^{2}\right)}^{\lambda}+\left(% \psi-1\right){\left(1-{\left({\gamma}^{u}\right)}^{2}\right)}^{\lambda}}}% \right],\right.\right.}$

$\hskip 34.143307pt\left.\left.\left[\frac{\sqrt{\psi}{\left({\eta}^{l}\right)}% ^{\lambda}}{\sqrt{{\left(1+\left(\psi-1\right)\left(1-{\left({\eta}^{l}\right)% }^{2}\right)\!\right)}^{\lambda}\!+\!\left(\psi-1\right){\left({\eta}^{l}% \right)}^{2\lambda}}},\frac{\sqrt{\psi}{\left({\eta}^{u}\right)}^{\lambda}}{% \sqrt{{\left(1+\left(\psi-1\right)\left(1-{\left({\eta}^{u}\right)}^{2}\right)% \!\right)}^{\lambda}\!+\!\left(\psi-1\right){\left({\eta}^{u}\right)}^{2% \lambda}}}\!\right]\!\right)\!\!\right\};$
(4)
${\tilde{k}}^{{\lambda}}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,% \ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}% \tilde{k}}{\left\{\left(\left[\frac{\sqrt{\psi}{\left({\gamma}^{l}\right)}^{% \lambda}}{\sqrt{{\left(1+\left(\psi-1\right)\left(1-{{\gamma}^{l}}^{2}\right)% \right)}^{\lambda}+\left(\psi-1\right){{\gamma}^{l}}^{2\lambda}}},\frac{\sqrt{% \psi}{\left({\gamma}^{u}\right)}^{\lambda}}{\sqrt{{\left(1+\left(\psi-1\right)% \left(1-{{\gamma}^{u}}^{2}\right)\right)}^{\lambda}+\left(\psi-1\right){{% \gamma}^{u}}^{2\lambda}}}\right],\right.\right.}$

$\hskip 34.143307pt\left.\left.\left[\sqrt{\frac{{\left(1+\left(\psi-1\right){% \left({\eta}^{l}\right)}^{2}\right)}^{\lambda}\!-\!{\left(1-{\left({\eta}^{l}% \right)}^{2}\right)}^{\lambda}}{{\left(1+\left(\psi\!-\!1\right){\left({\eta}^% {l}\right)}^{2}\right)}^{\lambda}\!+\!\left(\psi-1\right){\left(1-{\left({\eta% }^{l}\right)}^{2}\right)}^{\lambda}}},\sqrt{\frac{{\left(1+\left(\psi-1\right)% {\left({\eta}^{u}\right)}^{2}\right)}^{\lambda}-{\left(1-{\left({\eta}^{u}% \right)}^{2}\right)}^{\lambda}}{{\left(1+\left(\psi-1\right){\left({\eta}^{u}% \right)}^{2}\right)}^{\lambda}\!+\!\left(\psi-1\right){\left(1-{\left({\eta}^{% u}\right)}^{2}\right)}^{\lambda}}}\right]\!\right)\!\!\right\}.$

For $\psi=1$ the above four Hamacher operational laws are reduced to algebraic operational laws and for ${\psi}{=2}$ those four Hamacher operational laws are reduced into Einstein operational laws.

Example 2. Let ${\tilde{k}}_{1}=\{([0.3,0.5],[0.2,0.4]),([0.7,0.8],[0.4,0.6])\}$ and $\tilde{k}_{2}=\{([0.3,0.4],[0.2,0.6]),([0.1,0.3]$ , $[0.2,0.5]),([0.4,0.7],[0.3,0.6])\}$ be two IVPHFNs, and consider $\psi=$ 2 and $\lambda=$ 3. The mathematical operations based on Hamacher $t$ -N and $t$ -CN are presented as follows:

(1)
${\tilde{k}}_{1}{\oplus}_{H}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[% {\eta}^{l}_{1},{\eta}^{u}_{1}\right]\right)\in\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right),\left(\left[0.7,% 0.8\right],\left[0.4,0.6\right]\right)\right\},\\ \left(\left[{\gamma}^{l}_{2},{\gamma}^{u}_{2}\right],\left[{\eta}^{l}_{2},{% \eta}^{u}_{2}\right]\right)\in\\ \{\left(\left[0.3,0.4\right],\left[0.2,0.6\right]\right),\left(\left[0.1,0.3% \right],\left[0.2,0.5\right]\right),\\ \left(\left[0.4,0.7\right],\left[0.3,0.6\right]\right)\}\end{subarray}}$

$\hskip 39.833858pt\left\{\left(\left[\sqrt{\frac{{{\gamma}^{l}_{1}}^{2}+{{% \gamma}^{l}_{2}}^{2}-{{\gamma}^{l}_{1}}^{2}{{\gamma}^{l}_{2}}^{2}+{{\gamma}^{l% }_{1}}^{2}{{\gamma}^{l}_{2}}^{2}}{1+{{\gamma}^{l}_{1}}^{2}{{\gamma}^{l}_{2}}^{% 2}}},\sqrt{\frac{{{\gamma}^{u}_{1}}^{2}+{{\gamma}^{u}_{2}}^{2}-{{\gamma}^{u}_{% 1}}^{2}{{\gamma}^{u}_{2}}^{2}+{{\gamma}^{u}_{1}}^{2}{{\gamma}^{u}_{2}}^{2}}{1+% {{\gamma}^{u}_{1}}^{2}{{\gamma}^{u}_{2}}^{2}}}\right],\right.\right.$

$\hskip 39.833858pt\left.\left.\left[\frac{{\eta}^{l}_{1}{\eta}^{l}_{2}}{\sqrt{% 2{-}\left({{\eta}^{l}_{1}}^{2}+{{\eta}^{l}_{2}}^{2}-{{\eta}^{l}_{1}}^{2}{{\eta% }^{l}_{2}}^{2}\right)}},\frac{{\eta}^{u}_{1}{\eta}^{u}_{2}}{\sqrt{2{-}\left({{% \eta}^{u}_{1}}^{2}+{{\eta}^{u}_{2}}^{2}-{{\eta}^{u}_{1}}^{2}{{\eta}^{u}_{2}}^{% 2}\right)}}\right]\right)\right\}$

$\hskip 39.833858pt=\left\{\left(\left[0.4146,0.6083\right],\left[0.0400,0.2400% \right]\right),\left(\left[0.3148,0.5635\right],\left[0.0400,0.2000\right]% \right),\right.$

$\hskip 39.833858pt\left(\left[0.4854,0.7858\right],\left[0.0600,0.2400\right]% \right),\left(\left[0.7321,0.8352\right],\left[0.0800,0.3600\right]\right),$

$\hskip 39.833858pt\left(\left[0.7036,0.8200\right],\left[0.0800,0.3000\right]% \right),\left(\left[0.7560,0.9035\right],\left[0.1200,0.3600\right]\right)\}$
(2)
${\tilde{k}}_{1}{\otimes}_{H}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[% {\eta}^{l}_{1},{\eta}^{u}_{1}\right]\right){\in}\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right),\left(\left[0.7,% 0.8\right],\left[0.4,0.6\right]\right)\right\},\\ \left(\left[{\gamma}^{l}_{2},{\gamma}^{u}_{2}\right],\left[{\eta}^{l}_{2},{% \eta}^{u}_{2}\right]\right){\in}\\ \left\{\left(\left[0.3,0.4\right],\left[0.2,0.6\right]\right),\left(\left[0.1,% 0.3\right],\left[0.2,0.5\right]\right)\right.,\\ \left.\left(\left[0.4,0.7\right],\left[0.3,0.6\right]\right)\right\}\end{subarray}}$

$\hskip 42.679134pt\left\{\left(\left[\frac{{\gamma}^{l}_{1}{\gamma}^{l}_{2}}{% \sqrt{2{-}\left({{\gamma}^{l}_{1}}^{2}+{{\gamma}^{l}_{2}}^{2}-{{\gamma}^{l}_{1% }}^{2}{{\gamma}^{l}_{2}}^{2}\right)}},\frac{{\gamma}^{u}_{1}{\gamma}^{u}_{2}}{% \sqrt{2{-}\left({{\gamma}^{u}_{1}}^{2}+{{\gamma}^{u}_{2}}^{2}-{{\gamma}^{u}_{1% }}^{2}{{\gamma}^{u}_{2}}^{2}\right)}}\right],\right.\right.$

$\hskip 42.679134pt\left.\left.\left[\sqrt{\frac{{{\eta}^{l}_{1}}^{2}+{{\eta}^{% l}_{2}}^{2}-{{\eta}^{l}_{1}}^{2}{{\eta}^{l}_{2}}^{2}+{{\eta}^{l}_{1}}^{2}{{% \eta}^{l}_{2}}^{2}}{1+{{\eta}^{l}_{1}}^{2}{{\eta}^{l}_{2}}^{2}}},\sqrt{\frac{{% {\eta}^{u}_{1}}^{2}+{{\eta}^{u}_{2}}^{2}-{{\eta}^{u}_{1}}^{2}{{\eta}^{u}_{2}}^% {2}+{{\eta}^{u}_{1}}^{2}{{\eta}^{u}_{2}}^{2}}{1+{{\eta}^{u}_{1}}^{2}{{\eta}^{u% }_{2}}^{2}}}\right]\right)\right\}$

$\hskip 42.679134pt=\left\{\left(\left[0.0900,0.2000\right],\left[0.2800,0.6800% \right]\right),\left(\left[0.0300,0.1500\right],\left[0.2800,0.6083\right]% \right),\right.$

$\hskip 42.679134pt\left(\left[0.1200,0.3500\right],\left[0.3555,0.6800\right]% \right),\left(\left[0.2100,0.3200\right],\left[0.4400,0.7684\right]\right),$

$\hskip 42.679134pt\left(\left[0.0700,0.2400\right],\left[0.4400,0.7211\right]% \right),\left(\left[0.2800,0.5600\right],\left[0.4854,0.7684\right]\right)\}$
(3)
${\tilde{k}}_{1}=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[{\eta}^{l}_{1},{\eta}^{u}_{1}% \right]\right)\in\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right)\right.\\ \left.\left(\left[0.7,0.8\right],\left[0.4,0.6\right]\right)\right\}\end{subarray}}$

$\hskip 19.916929pt\left\{\left(\left[\sqrt{\frac{{\left(1+{{\gamma}^{l}}^{2}% \right)}^{3}-{\left(1-{{\gamma}^{l}}^{2}\right)}^{3}}{{\left(1+{{\gamma}^{l}}^% {2}\right)}^{3}+{\left(1-{{\gamma}^{l}}^{2}\right)}^{3}}},\sqrt{\frac{{\left(1% +{{\gamma}^{u}}^{2}\right)}^{3}-{\left(1-{{\gamma}^{u}}^{2}\right)}^{3}}{{% \left(1+{{\gamma}^{u}}^{2}\right)}^{3}+{\left(1-{{\gamma}^{u}}^{2}\right)}^{3}% }}\right],\left[\frac{\sqrt{2}{{\eta}^{l}}^{3}}{\sqrt{{\left(1+\left(1-{{\eta}% ^{l}}^{2}\right)\right)}^{3}+{{\eta}^{l}}^{6}}},\frac{\sqrt{2}{{\eta}^{u}}^{3}% }{\sqrt{{\left(1+\left(1-{{\eta}^{u}}^{2}\right)\right)}^{3}+{{\eta}^{u}}^{6}}% }\right]\right)\right\}$

$=\left\{\left(\left[0.4964,0.7603\right],\left[0.0080,0.0640\right]\right),% \left(\left[0.9313,0.9764\right],\left[0.0640,0.2160\right]\right)\right\}$
(4)
${{\tilde{k}}_{1}}^{3}=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(% \left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[{\eta}^{l}_{1},{\eta}^{u}% _{1}\right]\right){\in}\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right)\right.\\ \left.\left(\left[0.7,0.8\right],\left[0.4,0.6\right]\right)\right\}\end{subarray}}$

$\hskip 19.916929pt\left\{\left(\left[\frac{\sqrt{2}{{\gamma}^{l}}^{3}}{\sqrt{{% \left(1+\left(1-{{\gamma}^{l}}^{2}\right)\right)}^{3}+{{\gamma}^{l}}^{6}}},% \frac{\sqrt{2}{{\gamma}^{u}}^{3}}{\sqrt{{\left(1+\left(1-{{\gamma}^{u}}^{2}% \right)\right)}^{3}+{{\gamma}^{u}}^{6}}}\right],\left[\sqrt{\frac{{\left(1+{{% \eta}^{l}}^{2}\right)}^{3}-{\left(1-{{\eta}^{l}}^{2}\right)}^{3}}{{\left(1+{{% \eta}^{l}}^{2}\right)}^{3}+{\left(1-{{\eta}^{l}}^{2}\right)}^{3}}},\sqrt{\frac% {{\left(1+{{\eta}^{u}}^{2}\right)}^{3}-{\left(1-{{\eta}^{u}}^{2}\right)}^{3}}{% {\left(1+{{\eta}^{u}}^{2}\right)}^{3}+{\left(1-{{\eta}^{u}}^{2}\right)}^{3}}}% \right]\right)\right\}$

$\hskip 19.916929pt=\left\{\left(\left[0.0270,0.1250\right],\left[0.3395,0.6382% \right]\right),\left(\left[0.3430,0.5120\right],\left[0.6382,0.8590\right]% \right)\right\}.$

$\bullet$
Frank ${\bm{t}}$ -N and ${\bm{t}}$ -CN: Considering $f\left(t\right)={\text{log}\left(\frac{\tau-1}{{\tau}^{t}-1}\right)}$ , $\tau>1$ , operational laws for IVPHFNs based on Frank $t$ -N and $t$ -CN are described as follows:

(1)
${\tilde{k}}_{1}{\oplus}_{F}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[% {\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\\ {\in}{\tilde{k}}_{i}\\ i=1,2\end{subarray}}$

$\left\{\left(\left[{\left(1-{{\text{log}}_{\tau}\left(1+\frac{\left({\tau}^{1-% {{\gamma}^{l}_{1}}^{2}}-1\right)\left({\tau}^{1-{{\gamma}^{l}_{2}}^{2}}-1% \right)}{\tau-1}\right)\ }\right)}^{\frac{1}{2}},{\left(1-{{\text{log}}_{\tau}% \left(1+\frac{\left({\tau}^{1-{{\gamma}^{u}_{1}}^{2}}-1\right)\left({\tau}^{1-% {{\gamma}^{u}_{2}}^{2}}-1\right)}{\tau-1}\right)\ }\right)}^{\frac{1}{2}}% \right],\right.\right.$

$\left.\left.\left[\left({\text{log}}_{\tau}\left(1+\frac{\left({\tau}^{{{\eta}% ^{l}_{1}}^{2}}-1\right)\left({\tau}^{{{\eta}^{l}_{2}}^{2}}-1\right)}{\tau-1}% \right)\ \right)^{\frac{1}{2}},{\left({{\text{log}}_{\tau}\left(1+\frac{\left(% {\tau}^{{{\eta}^{u}_{1}}^{2}}-1\right)\left({\tau}^{{{\eta}^{u}_{2}}^{2}}-1% \right)}{\tau-1}\right)\ }\right)}^{\frac{1}{2}}\right]\right)\right\};$
(2)
${\tilde{k}}_{1}{\otimes}_{F}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[% {\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{k}}_{i}\\ i=1,2\end{subarray}}{\left\{\left(\left[\sqrt{{{\text{log}}_{\tau}\left(1+% \frac{\left({\tau}^{{{\gamma}^{l}_{1}}^{2}}-1\right)\left({\tau}^{{{\gamma}^{l% }_{2}}^{2}}-1\right)}{\tau-1}\right)\ }},\right.\right.\right.}$

$\left.\sqrt{{{\text{log}}_{\tau}\left(1+\frac{\left({\tau}^{{\left({\gamma}^{u% }_{1}\right)}^{2}}-1\right)\left({\tau}^{{\left({\gamma}^{u}_{2}\right)}^{2}}-% 1\right)}{\tau-1}\right)\ }}\right],\left[\sqrt{1-{{\text{log}}_{\tau}\left(1+% \frac{\left({\tau}^{1-{\left({\eta}^{l}_{1}\right)}^{2}}-1\right)\left({\tau}^% {1-{\left({\eta}^{l}_{2}\right)}^{2}}-1\right)}{\tau-1}\right)\ }},\right.$

$\left.\left.\left.\sqrt{1-{{\text{log}}_{\tau}\left(1+\frac{\left({\tau}^{1-{% \left({\eta}^{u}_{1}\right)}^{2}}-1\right)\left({\tau}^{1-{\left({\eta}^{u}_{2% }\right)}^{2}}-1\right)}{\tau-1}\right)\ }}\right]\right)\right\};$
(3)
$\lambda\tilde{k}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,\ }{% \gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k}% }{\left\{\left(\left[\sqrt{1-{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}^{% 1-{\left({\gamma}^{l}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^% {\lambda-1}}\right)\ }},\sqrt{1-{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau% }^{1-{\left({\gamma}^{u}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right% )}^{\lambda-1}}\right)\ }}\right],\right.\right.}$

$\left.\left.\left[\sqrt{{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}^{{% \left({\eta}^{l}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^{% \lambda-1}}\right)\ }},\sqrt{{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}^{% {\left({\eta}^{u}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^{% \lambda-1}}\right)\ }}\right]\right)\right\};$
(4)
${\tilde{k}}^{\lambda}=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}{,\ }% {\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){\in}\tilde{k% }}{\left\{\left(\left[\sqrt{{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}^{{% \left({\gamma}^{l}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^{% \lambda-1}}\right)\ }},\sqrt{{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}^{% {\left({\gamma}^{u}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^{% \lambda-1}}\right)\ }}\right],\right.\right.}$

$\left.\left.\left[\sqrt{1-{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}^{1-{% \left({\eta}^{l}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^{% \lambda-1}}\right)\ }},\sqrt{1-{{\text{log}}_{\tau}\left(1+\frac{{\left({\tau}% ^{1-{\left({\eta}^{u}\right)}^{2}}-1\right)}^{\lambda}}{{\left(\tau-1\right)}^% {\lambda-1}}\right)\ }}\right]\right)\right\}.$

For $\tau\to 1$ the above four Frank operational laws are reduced to algebraic operations.

It is to be noted here that, for each of the cases (3) and (4), $\lambda>0$ is considered.

Example 3. Let ${\tilde{k}}_{1}{=}\left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right% ),\left(\left[0.7,0.8\right],\left[0.4,0.6\right]\right)\right\}$ and ${\tilde{k}}_{2}{=}\left\{\left(\left[0.3,0.4\right],\left[0.2,0.6\right]\right% ),\left(\left[0.1,0.3\right]\right.\right.$ , $\left.\left.\left[0.2,0.5\right]\right),\left(\left[0.4,0.7\right],\left[0.3,0% .6\right]\right)\right\}$ be two IVPHFNs and consider the Frank parameter as $\tau=$ 2 and the scalar value as $\lambda=$ 3. The mathematical operations based on Frank $t$ -N and $t$ -CN are presented as follows:

(1)
${\tilde{k}}_{1}{\oplus}_{F}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[% {\eta}^{l}_{1},{\eta}^{u}_{1}\right]\right){\in}\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right),\right.\\ \left.\left(\left[0.7,0.8\right],\left[0.4,0.6\right]\right)\right\},\\ \left(\left[{\gamma}^{l}_{2},{\gamma}^{u}_{2}\right],\left[{\eta}^{l}_{2},{% \eta}^{u}_{2}\right]\right){\in}\\ \left\{\left(\left[0.3,0.4\right],\left[0.2,0.6\right]\right)\right.,\\ \left(\left[0.1,0.3\right],\left[0.2,0.5\right]\right),\\ \left.\left(\left[0.4,0.7\right],\left[0.3,0.6\right]\right)\right\}% \end{subarray}}\left\{\left(\left[{\left(1-{{\text{log}}_{2}\left(1+\left(2^{1% -{{\gamma}^{l}_{1}}^{2}}-1\right)\left(2^{1-{{\gamma}^{l}_{2}}^{2}}-1\right)% \right)\ }\right)}^{\frac{1}{2}},\right.\right.\right.$

$\hskip 19.916929pt\left.{\left(1-{{\text{log}}_{2}\left(1+\left(2^{1-{{\gamma}% ^{u}_{1}}^{2}}-1\right)\left(2^{1-{{\gamma}^{u}_{2}}^{2}}-1\right)\right)\ }% \right)}^{\frac{1}{2}}\right],\left[{\left({{\text{log}}_{2}\left(1+\left(2^{{% {\eta}^{l}_{1}}^{2}}-1\right)\left(2^{{{\eta}^{l}_{2}}^{2}}-1\right)\right)\ }% \right)}^{\frac{1}{2}},\right.$

$\hskip 19.916929pt\left.\left.\left.{\left({{\text{log}}_{2}\left(1+\left(2^{{% {\eta}^{u}_{1}}^{2}}-1\right)\left(2^{{{\eta}^{u}_{2}}^{2}}-1\right)\right)\ }% \right)}^{\frac{1}{2}}\right]\right)\right\}$

$\hskip 19.916929pt=\left\{\left(\left[0.4172,0.6151\right],\left[0.0338,0.2172% \right]\right),\left(\left[0.3152,0.5679\right],\left[0.0338,0.1779\right]% \right),\right.$

$\hskip 19.916929pt\left(\left[0.4890,0.7960\right],\left[0.0511,0.2172\right]% \right),\left(\left[0.7368,0.8417\right],\left[0.0689,0.3339\right]\right),$

$\hskip 19.916929pt\left.\left(\left[0.7042,0.8241\right],\left[0.0689,0.2745% \right]\right),\left(\left[0.7636,0.9145\right],\left[0.1042,0.3339\right]% \right)\right\}$
(2)
${\tilde{k}}_{1}{\otimes}_{F}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{% \begin{subarray}{c}\left(\left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[% {\eta}^{l}_{1},{\eta}^{u}_{1}\right]\right)\in\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right),\left(\left[0.7,% 0.8\right],\left[0.4,0.6\right]\right)\right\},\\ \left(\left[{\gamma}^{l}_{2},{\gamma}^{u}_{2}\right],\left[{\eta}^{l}_{2},{% \eta}^{u}_{2}\right]\right)\in\\ \left\{\left(\left[0.3,0.4\right],\left[0.2,0.6\right]\right),\left(\left[0.1,% 0.3\right],\left[0.2,0.5\right]\right)\right.\\ \left.\left(\left[0.4,0.7\right],\left[0.3,0.6\right]\right)\right\}% \end{subarray}}{\left\{\left(\left[\sqrt{{{\text{log}}_{2}\left(1+\left(2^{{{% \gamma}^{l}_{1}}^{2}}-1\right)\left(2^{{{\gamma}^{l}_{2}}^{2}}-1\right)\right)% \ }},\right.\right.\right.}$

$\hskip 19.916929pt\left.\sqrt{{{\text{log}}_{2}\left(1+\left(2^{{{\gamma}^{u}_% {1}}^{2}}-1\right)\left(2^{{{\gamma}^{u}_{2}}^{2}}-1\right)\right)\ }}\right],% \left[\sqrt{1-{{\text{log}}_{2}\left(1+\left(2^{1-{{\eta}^{l}_{1}}^{2}}-1% \right)\left(2^{1-{{\eta}^{l}_{2}}^{2}}-1\right)\right)\ }},\right.$

$\hskip 19.916929pt\left.\left.\left.\sqrt{1-{{\text{log}}_{2}\left(1+\left(2^{% 1-{{\eta}^{u}_{1}}^{2}}-1\right)\left(2^{1-{{\eta}^{u}_{2}}^{2}}-1\right)% \right)\ }}\right]\right)\right\}$

$\hskip 19.916929pt=\left\{\left(\left[0.0772,0.1779\right],\left[0.2808,0.6876% \right]\right),\left(\left[0.0254,0.1322\right],\left[0.2808,0.6151\right]% \right),\right.$

$\hskip 19.916929pt\left(\left[0.1042,0.3262\right],\left[0.3569,0.6876\right]% \right),\left(\left[0.1926,0.3025\right],\left[0.4419,0.7801\right]\right),$

$\hskip 19.916929pt\left.\left(\left[0.0637,0.2257\right],\left[0.4419,0.7312% \right]\right),\left(\left[0.2586,0.5420\right],\left[0.4890,0.7801\right]% \right)\right\}$
(3)
$3{\tilde{k}}_{1}=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[{\eta}^{l}_{1},{\eta}^{u}_{1}% \right]\right){\in}\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right),\left(\left[0.7,% 0.8\right],\left[0.4,0.6\right]\right)\right\}\end{subarray}}{\left\{\left(% \left[\sqrt{1-{{\text{log}}_{2}\left(1+\frac{{\left(2^{1-{{\gamma}^{l}}^{2}}-1% \right)}^{3}}{{\left(2-1\right)}^{3-1}}\right)\ }}\right.,\right.\right.}$

$\hskip 19.916929pt\left.\sqrt{1-{{\text{log}}_{2}\left(1+\frac{{\left(2^{1-{{% \gamma}^{u}}^{2}}-1\right)}^{3}}{{\left(2-1\right)}^{3-1}}\right)\ }}\right],% \left.\left.\left[\sqrt{{{\text{log}}_{2}\left(1+\frac{{\left(2^{{{\eta}^{l}}^% {2}}-1\right)}^{3}}{{\left(2-1\right)}^{3-1}}\right)\ }},\sqrt{{{\text{log}}_{% 2}\left(1+\frac{{\left(2^{{{\eta}^{u}}^{2}}-1\right)}^{3}}{{\left(2-1\right)}^% {3-1}}\right)\ }}\right]\right)\right\}$

$\hskip 19.916929pt=\left\{\left(\left[0.6511,0.7312\right],\left[0.1277,0.2586% \right]\right),\left(\left[0.8327,0.8877\right],\left[0.2586,0.3957\right]% \right)\right\}$
(4)
${{\tilde{k}}_{1}}^{3}=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(% \left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[{\eta}^{l}_{1},{\eta}^{u}% _{1}\right]\right){\in}\\ \left\{\left(\left[0.3,0.5\right],\left[0.2,0.4\right]\right),\left(\left[0.7,% 0.8\right],\left[0.4,0.6\right]\right)\right\}\end{subarray}}{\left\{\left(% \left[\sqrt{{{\text{log}}_{\tau}\left(1+\frac{{\left(2^{{{\gamma}^{l}}^{2}}-1% \right)}^{3}}{{\left(2-1\right)}^{3-1}}\right)\ }},\right.\right.\right.}$

$\hskip 19.916929pt\left.\left.\left.\sqrt{{{\text{log}}_{2}\left(1+\frac{{% \left(2^{{{\gamma}^{u}}^{2}}-1\right)}^{3}}{{\left(2-1\right)}^{3-1}}\right)\ % }}\right],\left[\sqrt{1-{{\text{log}}_{2}\left(1+\frac{{\left(2^{1-{{\eta}^{l}% }^{2}}-1\right)}^{3}}{{\left(2-1\right)}^{3-1}}\right)\ }},\sqrt{1-{{\text{log% }}_{2}\left(1+\frac{{\left(2^{1-{{\eta}^{u}}^{2}}-1\right)}^{3}}{{\left(2-1% \right)}^{3-1}}\right)\ }}\right]\right)\right\}$

$\hskip 19.916929pt=\left\{\left(\left[0.1926,0.3262\right],\left[0.6233,0.6876% \right]\right),\left(\left[0.4675,0.5420\right],\left[0.6876,0.7801\right]% \right)\right\}.$

The arithmetic operations on IVPHFNs based on different commonly used A $t$ -N& $t$ -CNs are presented above. Now, based on those operations, IVPHF Archimedean aggregation operators are derived in the next section.
5. IVPHF archimedean aggregation operators

In this section some IVPHF Archimedean averaging operators and geometric operators are proposed using the operational rules based on A $t$ -N& $t$ -CN. Also, some special cases using specific generating functions are presented. Moreover, several important properties of those operators are also discussed in this section.

5.1 IVPHF archimedean averaging operators

To aggregate IVPHFNs, Archimedean operation-based IVPHF averaging aggregation operators are introduced through the following definition.

Definition 12. For any collection of IVPHFNs, ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ and weight vector, $\omega=\left({\omega}_{1},{\omega}_{2},\ldots,{\omega}_{n}\right)$ , an A $t$ -N& $t$ -CN-based IVPHFWA (AIVPHFWA) operator is defined by a mapping, $\textit{AIVPHFWA:K}^{n}\to K$ such that

$\displaystyle\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)={{\oplus}_{A}}^{n}_{\ i=1}\left({\omega}_{i}{\tilde{k}}_% {i}\right),$

where ${\omega}_{i}\in\left[0,1\right]$ and $\sum^{n}_{i=1}{{\omega}_{i}}=1$ .

Theorem 1. The aggregated value of any collections of IVPHFNs, ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ , using AIVPHFWA operator is also an IVPHFN and is given by

$\displaystyle\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle\qquad=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{% \gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{% \tilde{k}}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}g\left({{\gamma}^{l}_{i}}^{2}\right)}\right)},% \sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({{\gamma}^{u}_{i}% }^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{-1}\left(\sum\nolimits^{n}_% {i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}% _{i}\right)}^{2}\right)}\right)}\right]\right)\right\}.$

Proof: For $n=2$ ,

$\displaystyle{\omega}_{1}{\tilde{k}}_{1}=\mathop{\bigcup}\nolimits_{\left(% \left[{\gamma}^{l}_{1},{\gamma}^{u}_{1}\right],\left[{\eta}^{l}_{1},{\eta}^{u}% _{1}\right]\right){\in}{\tilde{k}}_{1}}{\left\{\left(\left[\sqrt{g^{-1}\left({% \omega}_{1}g\left({\left({\gamma}^{l}_{1}\right)}^{2}\right)\right)},\sqrt{g^{% -1}\left({\omega}_{1}g\left({\left({\gamma}^{u}_{1}\right)}^{2}\right)\right)}% \right],\right.\right.}$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{-1}\left({\omega}_{1}f\left% ({\left({\eta}^{l}_{1}\right)}^{2}\right)\right)},\sqrt{f^{-1}\left({\omega}_{% 1}f\left({\left({\eta}^{u}_{1}\right)}^{2}\right)\right)}\right]\right)\right\}.$ $\displaystyle{\omega}_{2}{\tilde{k}}_{2}=\mathop{\bigcup}\nolimits_{\left(% \left[{\gamma}^{l}_{2},{\gamma}^{u}_{2}\right],\left[{\eta}^{l}_{2},{\eta}^{u}% _{2}\right]\right){\in}{\tilde{k}}_{{2}}}{\left\{\left(\left[\sqrt{g^{-1}\left% ({\omega}_{2}g\left({\left({\gamma}^{l}_{2}\right)}^{2}\right)\right)},\sqrt{g% ^{-1}\left({\omega}_{2}g\left({\left({\gamma}^{u}_{2}\right)}^{2}\right)\right% )}\right],\right.\right.}$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{-1}\left({\omega}_{2}f\left% ({\left({\eta}^{l}_{2}\right)}^{2}\right)\right)},\sqrt{f^{-1}\left({\omega}_{% 2}f\left({\left({\eta}^{u}_{2}\right)}^{2}\right)\right)}\right]\right)\right\}.$

Now,

$\displaystyle{\omega}_{1}{\tilde{k}}_{1}{\oplus}_{A}{\omega}_{2}{\tilde{k}}_{2}=$ $\displaystyle\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}^% {u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{\tilde{k}}% _{i},i=1,2}{\left\{\left(\left[\sqrt{g^{-1}\left(g\left(g^{-1}\left({\omega}_{% 1}g\left({\left({\gamma}^{l}_{1}\right)}^{2}\right)\right)\right)+g\left(g^{-1% }\left({\omega}_{2}g\left({\left({\gamma}^{l}_{2}\right)}^{2}\right)\right)% \right)\right)},\right.\right.\right.}$ $\displaystyle\qquad\quad\left.\sqrt{g^{-1}\left(g\left(g^{-1}\left({\omega}_{1% }g\left({\left({\gamma}^{u}_{1}\right)}^{2}\right)\right)\right)+g\left(g^{-1}% \left({\omega}_{2}g\left({\left({\gamma}^{u}_{2}\right)}^{2}\right)\right)% \right)\right)}\right],$ $\displaystyle\qquad\quad\left[\sqrt{f^{-1}\left(f\left(f^{-1}\left({\omega}_{1% }f\left({\left({\eta}^{l}_{1}\right)}^{2}\right)\right)\right)+f\left(f^{-1}% \left({\omega}_{2}f\left({\left({\eta}^{l}_{2}\right)}^{2}\right)\right)\right% )\right)},\right.$ $\displaystyle\qquad\quad\left.\left.\left.\sqrt{f^{-1}\left(f\left(f^{-1}\left% ({\omega}_{1}f\left({\left({\eta}^{u}_{1}\right)}^{2}\right)\right)\right)+f% \left(f^{-1}\left({\omega}_{2}f\left({\left({\eta}^{u}_{2}\right)}^{2}\right)% \right)\right)\right)}\right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right)\\ \in{\tilde{k}}_{i},i=1,2\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}\left({% \omega}_{1}g\left({\left({\gamma}^{l}_{1}\right)}^{2}\right)+{\omega}_{2}g% \left({\left({\gamma}^{l}_{2}\right)}^{2}\right)\right)},\right.\right.\right.% }\left.\sqrt{y^{-1}\left({\omega}_{1}g\left({\left({\gamma}^{u}_{1}\right)}^{2% }\right)+{\omega}_{2}g\left({\left({\gamma}^{u}_{2}\right)}^{2}\right)\right)}% \right],$ $\displaystyle\qquad\quad\left[\sqrt{f^{-1}\left({\omega}_{1}f\left({\left({% \eta}^{l}_{1}\right)}^{2}\right)+{\omega}_{2}f\left({\left({\eta}^{l}_{2}% \right)}^{2}\right)\right)},\left.\left.\left.\sqrt{f^{-1}\left({\omega}_{1}f% \left({\left({\eta}^{u}_{1}\right)}^{2}\right)+{\omega}_{2}f\left({\left({\eta% }^{u}_{2}\right)}^{2}\right)\right)}\right]\right)\right\}\right.$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{\tilde{k}% }_{i},i=1,2}{\left\{\left(\left[\sqrt{g^{{-}{1}}\left(\sum\nolimits^{{2}}_{i{=% 1}}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{g^{{-}{1}}\left(\sum\nolimits^{{2}}_{i{=1}}{{\omega}_{i}g\left({\left({% \gamma}^{u}_{i}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{{-}{1}}\left(\sum\nolimits^% {{2}}_{i{=1}}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}% \right)},\sqrt{f^{{-}{1}}\left(\sum\nolimits^{{2}}_{i{=1}}{{\omega}_{i}f\left(% {\left({\eta}^{u}_{i}\right)}^{2}\right)}\right)}\right]\right)\right\}$

i.e., the theorem holds for $n=2$ .

Suppose that theorem is true for $n=m$ , i.e.,

$\displaystyle\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{m}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{\tilde{k}% }_{i},i=1,2,\ldots,m}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum\nolimits^{m}_{% i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{g^{-1}\left(\sum\nolimits^{m}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{% u}_{i}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\qquad\left.\left.\left[\sqrt{f^{-1}\left(\int\nolimits^{m}_{i=1}% {{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},\sqrt{f^% {-1}\left(\sum\nolimits^{m}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}% \right)}^{2}\right)}\right)}\right]\right)\right\}.$

Now, when $n=m+1$ ,

$\displaystyle\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{m},{\tilde{k}}_{m+1}\right)=\textit{AIVPHFWA}\left({\tilde{k}}_{1}% ,{\tilde{k}}_{2},\ldots,{\tilde{k}}_{m}\right){\oplus}_{A}{\omega}_{m+1}{% \tilde{k}}_{m+1}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{\tilde{k}% }_{i},i=1,2,\ldots,m}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum\nolimits^{m}_{% i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{g^{-1}\left(\sum\nolimits^{m}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{% u}_{i}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\quad\left.\left.\left[\sqrt{f^{-1}\left(\int\nolimits^{m}_{i=1}{% {\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},\sqrt{f^{% -1}\left(\sum\nolimits^{m}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}% \right)}^{2}\right)}\right)}\right]\right)\right\}{\oplus}_{A}$ $\displaystyle\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{m+1},{\gamma% }^{u}_{m+1}\right],\left[{\eta}^{l}_{m+1},{\eta}^{u}_{m+1}\right]\right){\in}{% \tilde{k}}_{m+1}}{\left\{\left(\left[\sqrt{g^{-1}\left({\omega}_{m+1}g\left({% \left({\gamma}^{l}_{m+1}\right)}^{2}\right)\right)},\sqrt{g^{-1}\left({\omega}% _{m+1}g\left({\left({\gamma}^{u}_{m+1}\right)}^{2}\right)\right)}\right],% \right.\right.}$ $\displaystyle\left.\left.\left[\sqrt{f^{-1}\left({\omega}_{m+1}f\left({\left({% \eta}^{l}_{m+1}\right)}^{2}\right)\right)},\sqrt{f^{-1}\left({\omega}_{m+1}f% \left({\left({\eta}^{u}_{m+1}\right)}^{2}\right)\right)}\right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i}\\ i=1,2,\ldots,m,m+1\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}\left(g\left(% g^{-1}\left(\sum\nolimits^{m}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}% \right)}^{2}\right)}\right)\right)+g\left(g^{-1}\left({\omega}_{k+1}g\left({% \left({\gamma}^{l}_{m+1}\right)}^{2}\right)\right)\right)\right)},\right.% \right.\right.}$ $\displaystyle\quad\left.\sqrt{g^{-1}\left(g\left(g^{-1}\left(\sum\nolimits^{m}% _{i=1}{{\omega}_{i}g\left({\left({\gamma}^{u}_{i}\right)}^{2}\right)}\right)% \right)+g\left(g^{-1}\left({\omega}_{m+1}g\left({\left({\gamma}^{u}_{m+1}% \right)}^{2}\right)\right)\right)\right)}\right],$ $\displaystyle\quad\left[\sqrt{f^{-1}\left(f\left(f^{-1}\left(\sum\nolimits^{m}% _{i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)% \right)+f\left(f^{-1}\left({\omega}_{m+1}f\left({\left({\eta}^{l}_{m+1}\right)% }^{2}\right)\right)\right)\right)},\right.$ $\displaystyle\quad\left.\left.\left.\sqrt{f^{-1}\left(f\left(f^{-1}\left(\sum% \nolimits^{m}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}\right)}^{2}\right% )}\right)\right)+f\left(f^{-1}\left({\omega}_{m+1}f\left({\left({\eta}^{u}_{m+% 1}\right)}^{2}\right)\right)\right)\right)}\right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,m,m+1}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum\nolimits% ^{m}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}+{% \omega}_{m+1}g\left({\left({\gamma}^{l}_{m+1}\right)}^{2}\right)\right)},% \right.\right.\right.}$ $\displaystyle\quad\left.\sqrt{g^{-1}\left(\sum\nolimits^{m}_{i=1}{{\omega}_{i}% g\left({{\gamma}^{u}_{i}}^{2}\right)}+{\omega}_{m+1}g\left({{\gamma}^{u}_{m+1}% }^{2}\right)\right)}\right],\left[\sqrt{f^{-1}\left(\sum\nolimits^{m}_{i=1}{{% \omega}_{i}f\left({{\eta}^{l}_{i}}^{2}\right)}+{\omega}_{m+1}f\left({{\eta}^{l% }_{m+1}}^{2}\right)\right)},\right.$ $\displaystyle\quad\left.\left.\left.\sqrt{f^{-1}\left(\sum\nolimits^{m}_{i=1}{% {\omega}_{i}f\left({\left({\eta}^{u}_{i}\right)}^{2}\right)}+{\omega}_{m+1}f% \left({\left({\eta}^{u}_{m+1}\right)}^{2}\right)\right)}\right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right)\in{\tilde{k}% }_{i},i=1,2,\ldots,m,m+1}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum\nolimits^{% m+1}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}% \right)},\sqrt{g^{-1}\left(\sum\nolimits^{m+1}_{i=1}{{\omega}_{i}g\left({\left% ({\gamma}^{u}_{i}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\quad\left.\left.\left[\sqrt{f^{-1}\left(\sum\nolimits^{m+1}_{i=1% }{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},\sqrt{f% ^{-1}\left(\sum\nolimits^{m+1}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}% \right)}^{2}\right)}\right)}\right]\right)\right\}.$

Hence, the above is true for $n{=}m{+1}$ , also. Thus, the theorem is true for all integers.

This completes the proof of the theorem.

Considering specific decreasing generating functions, different forms of AIVPHFWA operator can be generated, which are shown in the following manners.

$\bullet$
Algebraic ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWA: For $f\left(t\right)=-{\text{log}t\ }$ , the AIVPHFWA operator is reduced to the IVPHFWA operator and is described as:

$\displaystyle\textit{IVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\sqrt{1-\prod\nolimits^{n}_{i=1}{{% \left(1-{{\gamma}^{l}_{i}}^{2}\right)}^{{\omega}_{i}},}}\sqrt{1-\prod\nolimits% ^{n}_{i=1}{{\left(1-{{\gamma}^{u}_{i}}^{2}\right)}^{{\omega}_{i}}}}\right],% \right.\right.}$ $\displaystyle\left.\left.\left[\prod\nolimits^{n}_{i=1}{{\left({\eta}^{l}_{i}% \right)}^{{\omega}_{i}}},\prod\nolimits^{n}_{i=1}{{\left({\eta}^{u}_{i}\right)% }^{{\omega}_{i}}}\right]\right)\right\}.$ (6)
$\bullet$
Einstein ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWA: For $f\left(t\right)={\text{log}\left(\frac{2-t}{t}\right)}$ , the AIVPHFWA operator is reduced into IVPHFEWA operator and is presented as:

$\displaystyle\textit{IVPHFEWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i},\\ i=1,2,\ldots,n\end{subarray}}{\left\{\left(\left[\frac{\sqrt{\prod\nolimits^{n% }_{i=1}{{\left(1+{\left({\gamma}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}-}% \prod\nolimits^{n}_{i=1}{{\left(1-{\left({\gamma}^{l}_{i}\right)}^{2}\right)}^% {{\omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+{\left({\gamma}^{l}_% {i}\right)}^{2}\right)}^{{\omega}_{i}}+}\prod\nolimits^{n}_{i=1}{{\left(1-{% \left({\gamma}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}},\right.\right.% \right.}$ $\displaystyle\quad\left.\left.\left.\frac{\sqrt{\prod\nolimits^{n}_{i=1}{{% \left(1+{\left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}-}\prod% \nolimits^{n}_{i=1}{{\left(1-{\left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+{\left({\gamma}^{u}_{i% }\right)}^{2}\right)}^{{\omega}_{i}}+}\prod\nolimits^{n}_{i=1}{{\left(1-{\left% ({\gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}\right],\right.\right.$ $\displaystyle\left.\left.\left[\frac{\sqrt{2}\prod\nolimits^{n}_{i=1}{{\left({% \eta}^{l}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(2% -{\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}+}\prod\nolimits^{n}_% {i=1}{{\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}},\frac% {\sqrt{2}\prod\nolimits^{n}_{i=1}{{\left({\eta}^{u}_{i}\right)}^{{\omega}_{i}}% }}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(2-{\left({\eta}^{u}_{i}\right)}^{2}% \right)}^{{\omega}_{i}}+}\prod\nolimits^{n}_{i=1}{{\left({\left({\eta}^{u}_{i}% \right)}^{2}\right)}^{{\omega}_{i}}}}}\right]\right)\right\}.$ (7)
$\bullet$
Hamacher ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWA: For $f\left(t\right)={\text{log}\left(\frac{\psi+\left(1-\psi\right)t}{t}\right)\ }$ , $\psi>0$ , the AIVPHFWA operator is reduced into the IVPHFHWA operator which is described as:

$\displaystyle\textit{IVPHFHWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}{\!\left\{\!\left(\left[\frac{\sqrt{\prod\nolimits^{n}_% {i=1}{{\left(1+\left(\psi-1\right){\left({\gamma}^{l}_{i}\right)}^{2}\right)}^% {{\omega}_{i}}}-\prod\nolimits^{n}_{i=1}{{\left(1-{\left({\gamma}^{l}_{i}% \right)}^{2}\right)}^{{\omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1% +\left(\psi-1\right){\left({\gamma}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}% }+\left(\psi-1\right)\prod\nolimits^{n}_{i=1}{{\left(1-{\left({\gamma}^{l}_{i}% \right)}^{2}\right)}^{{\omega}_{i}}}}},\right.\right.\right.}$ $\displaystyle\quad\left.\frac{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+\left(% \psi-1\right){\left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}-\prod% \nolimits^{n}_{i=1}{{\left(1-{\left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+\left(\psi-1\right){% \left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}+\left(\psi-1\right)% \prod\nolimits^{n}_{i=1}{{\left(1-{\left({\gamma}^{u}_{i}\right)}^{2}\right)}^% {{\omega}_{i}}}}}\right],$ $\displaystyle\quad\left[\frac{\sqrt{\psi}\prod\nolimits^{n}_{i=1}{{\left({\eta% }^{l}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+% \left(\psi-1\right)\left(1-{\left({\eta}^{l}_{i}\right)}^{2}\right)\right)}^{{% \omega}_{i}}+}\left(\psi-1\right)\prod\nolimits^{n}_{i=1}{{\left({\left({\eta}% ^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}\right.,$ $\displaystyle\left.\left.\left.\frac{\sqrt{\psi}\prod\nolimits^{n}_{i=1}{{% \left({\eta}^{u}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{% \left(1+\left(\psi-1\right)\left(1-{\left({\eta}^{u}_{i}\right)}^{2}\right)% \right)}^{{\omega}_{i}}+}\left(\psi-1\right)\prod\nolimits^{n}_{i=1}{{\left({% \left({\eta}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}\right]\right)\right\}.$ (8)

For $\psi=1$ and $\psi=2$ , IVPHFHWA operator is converted into IVPHFWA and IVPHFEWA operators, respectively. So, IVPHFHWA operator is said to be a generalised version of IVPHFWA and IVPHFEWA operators.
$\bullet$
Frank ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWA: For $f\left(t\right)={\text{log}\left(\frac{\tau-1}{{\tau}^{t}-1}\right)\ }$ , $\tau>1$ , the AIVPHFWA operator is reduced to IVPHFFWA operator in the form of

$\displaystyle\textit{IVPHFFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i},\\ i=1,2,\ldots,n\end{subarray}}{\left\{\left(\left[\sqrt{1-\frac{{\text{log}% \left(1+\prod\nolimits^{n}_{i=1}{{\left({\tau}^{1-{\left({\gamma}^{l}_{i}% \right)}^{2}}-1\right)}^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}},% \right.\right.\right.}$ $\displaystyle\quad\left.\sqrt{1-\frac{{\text{log}\left(1+\prod\nolimits^{n}_{i% =1}{{\left({\tau}^{1-{\left({\gamma}^{u}_{i}\right)}^{2}}-1\right)}^{{\omega}_% {i}}}\right)\ }}{{\text{log}\ \tau\ }}}\right],$ $\displaystyle\quad\left.\left.\left[\sqrt{\frac{{\text{log}\left(1+\prod% \nolimits^{n}_{i=1}{{\left({\tau}^{{\left({\eta}^{l}_{i}\right)}^{2}}-1\right)% }^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}},\sqrt{\frac{{\text{log}% \left(1+\prod\nolimits^{n}_{i=1}{{\left({\tau}^{{\left({\eta}^{u}_{i}\right)}^% {2}}-1\right)}^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}}\right]\right)% \right\}.$

For $\tau\to 1$ , IVPHFFWA operator is reduced to IVPHFWA operator. So, IVPHFFWA operator may be considered as a generalisation of IVPHFWA operator.

Some important properties of the proposed AIVPHFWA aggregation operators are presented below.

Theorem 2. (Boundary) Let ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ be a collections of IVPHFNs, and for all $i{=1,2,}{\ldots}{,}n$ , let

$\displaystyle{\gamma}^{l}_{\textit{min}}=\textit{min}\left\{{\gamma}^{l}_{i_{% \textit{min}}}|{\gamma}^{l}_{i_{\textit{min}}}={\textit{min}}_{\left[{\gamma}^% {l}_{i},{\gamma}^{u}_{i}\right]{\in}{\tilde{h}}_{i}}\left\{{\gamma}^{l}_{i}% \right\}\right\},$ $\displaystyle{\gamma}^{u}_{\textit{max}}=\textit{max}\left\{{\gamma}^{u}_{i_{% \textit{max}}}|{\gamma}^{u}_{i_{\textit{max}}}={\textit{max}}_{\left[{\gamma}^% {l}_{i},{\gamma}^{u}_{i}\right]{\in}{\tilde{h}}_{i}}\left\{{\gamma}^{u}_{i}% \right\}\right\},$ $\displaystyle{\eta}^{l}_{\textit{max}}=\textit{max}\left\{{\eta}^{l}_{i_{% \textit{max}}}|{\eta}^{l}_{i_{\textit{max}}}={\textit{max}}_{\left[{\eta}^{l}_% {i},{\eta}^{u}_{i}\right]{\in}{\tilde{h}}_{i}}\left\{{\eta}^{l}_{i}\right\}% \right\},$ $\displaystyle{\eta}^{u}_{\textit{min}}=\textit{min}\left\{{\eta}^{u}_{i_{% \textit{min}}}|{\eta}^{u}_{i_{\textit{min}}}={\textit{min}}_{\left[{\eta}^{l}_% {i},{\eta}^{u}_{i}\right]{\in}{\tilde{h}}_{i}}\left\{{\eta}^{u}_{i}\right\}% \right\},$

and also ${\gamma}^{l}_{\textit{max}}$ , ${\gamma}^{u}_{\textit{min}}$ , ${\eta}^{l}_{min}$ and ${\eta}^{u}_{\textit{max}}$ convey the alike meanings as above. Then

$\displaystyle{\tilde{k}}_{-}\leqslant\textit{AIVPHFWA}\left({\tilde{k}}_{1},{% \tilde{k}}_{2},\ldots,{\tilde{k}}_{n}\right)\leqslant{\tilde{k}}_{+},$ (9)

where ${\tilde{k}}_{-}=\left(\left[{\gamma}^{l}_{\textit{min}},{\gamma}^{u}_{\textit{% min}}\right],\left[{\eta}^{l}_{\textit{max}},{\eta}^{u}_{\textit{max}}\right]\right)$ and ${\tilde{k}}_{+}=\left(\left[{\gamma}^{l}_{\textit{max}},{\gamma}^{u}_{\textit{% max}}\right],\left[{\eta}^{l}_{\textit{min}},{\eta}^{u}_{\textit{min}}\right]\right)$ .

Proof: For any $i{=1,2,}{\ldots}{,}n$ , ${\gamma}^{l}_{\textit{min}}\leqslant{\gamma}^{l}_{i}\leqslant{\gamma}^{l}_{% \textit{max}}$ and ${\gamma}^{u}_{\textit{min}}\leqslant{\gamma}^{u}_{i}\leqslant{\gamma}^{u}_{% \textit{max}}$ .

i.e., ${\left({\gamma}^{l}_{\textit{min}}\right)}^{2}\leqslant{\left({\gamma}^{l}_{i}% \right)}^{2}\leqslant{\left({\gamma}^{l}_{\textit{max}}\right)}^{2}$ and ${\left({\gamma}^{u}_{\textit{min}}\right)}^{2}\leqslant{\left({\gamma}^{u}_{i}% \right)}^{2}\leqslant{\left({\gamma}^{u}_{\textit{max}}\right)}^{2}$ .

Since $g\left(t\right)$ $\left(t\in[0,1]\right)$ is a monotonic increasing function,

$\displaystyle g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({% \gamma}^{l}_{\textit{min}}\right)}^{2}\right)}\right)\leqslant g^{-1}\left(% \sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}% \right)}\right)\leqslant g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g% \left({\left({\gamma}^{l}_{\textit{max}}\right)}^{2}\right)}\right),$

which implies that ${\left({\gamma}^{l}_{\textit{min}}\right)}^{2}\leqslant g^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}% \right)}\right)\leqslant{\left({\gamma}^{l}_{\textit{max}}\right)}^{2}$ .

$\displaystyle\text{So},{\gamma}^{l}_{\textit{min}}\leqslant\sqrt{g^{-1}\left(% \sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}% \right)}\right)}\leqslant{\gamma}^{l}_{\textit{max}}.$ (10)

$\displaystyle\text{Similarly},{\gamma}^{u}_{\textit{min}}\leqslant\sqrt{g^{-1}% \left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{u}_{i}\right% )}^{2}\right)}\right)}\leqslant{\gamma}^{u}_{\textit{max}}.$ (11)

Now, for any $i{=1,2,}{\ldots}{,}n$ , ${\left({\eta}^{l}_{\textit{min}}\right)}^{2}\leqslant{\left({\eta}^{l}_{i}% \right)}^{2}\leqslant{\left({\eta}^{l}_{\textit{max}}\right)}^{2}$ .

Since $f\left(t\right)$ $\left(t\in[0,1]\right)$ is a decreasing function,

$\displaystyle f^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({% \eta}^{l}_{\textit{min}}\right)}^{2}\right)}\right)\leqslant f^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right% )}\right)\leqslant f^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({% \left({\eta}^{l}_{\textit{max}}\right)}^{2}\right)}\right),$

which implies that ${\left({\eta}^{l}_{\textit{min}}\right)}^{2}\leqslant f^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right% )}\right)\leqslant{\left({\eta}^{l}_{\textit{max}}\right)}^{2}$ .

$\displaystyle\text{So},{\eta}^{l}_{\textit{min}}\leqslant\sqrt{f^{-1}\left(% \sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}% \right)}\right)}\leqslant{\eta}^{l}_{\textit{max}}.$ (12)

$\displaystyle\text{Similarly},{\eta}^{u}_{\textit{min}}\leqslant\sqrt{f^{-1}% \left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}\right)}% ^{2}\right)}\right)}\leqslant{\eta}^{u}_{\textit{max}}.$ (13)

From Eqs (10) and (12),

$\displaystyle{\gamma}^{l}_{\textit{min}}-{\eta}^{l}_{\textit{max}}\leqslant% \sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{% l}_{i}\right)}^{2}\right)}\right)}-\sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)}\leqslant{% \gamma}^{l}_{\textit{max}}-{\eta}^{l}_{\textit{min}}.$

From Eqs (11) and (13),

$\displaystyle{\gamma}^{u}_{\textit{min}}-{\eta}^{u}_{\textit{max}}\leqslant% \sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{% u}_{i}\right)}^{2}\right)}\right)}-\sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}f\left({\left({\eta}^{u}_{i}\right)}^{2}\right)}\right)}\leqslant{% \gamma}^{u}_{\textit{max}}-{\eta}^{u}_{\textit{min}}.$

Then, $S\left({\tilde{k}}_{-}\right)\leqslant S\left(\textit{AIVPHFWA}\left({\tilde{k% }}_{1},{\tilde{k}}_{2},\ldots,{\tilde{k}}_{n}\right)\right)\leqslant S\left({% \tilde{k}}_{+}\right)$ .

Therefore, ${\tilde{k}}_{-}\leqslant\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2}% ,\ldots,{\tilde{k}}_{n}\right)\leqslant{\tilde{k}}_{+}$ .

Theorem 3. Let ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ be a collection of IVPHFNs, ${\omega}_{i}\in\left[0,1\right]$ $\left(i=1,2,\ldots,n\right)$ be their corresponding weight vectors, and $\sum^{n}_{i=1}{{\omega}_{i}}=1$ . If $\tilde{k}$ be an IVPHFN, then

$\displaystyle\textit{AIVPHFWA}\left({\tilde{k}}_{1}{\oplus}_{A}\tilde{k},{% \tilde{k}}_{2}{\oplus}_{A}\tilde{k},\ldots,{\tilde{k}}_{n}{\oplus}_{A}\tilde{k% }\right)=\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{\tilde% {k}}_{n}\right){\oplus}_{A}\tilde{k}.$ (14)

Proof.

It is clear that ${\tilde{k}}_{i}{\oplus}_{A}\tilde{k}=$

$\displaystyle\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{\gamma% }^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]% \right){\in}{\tilde{k}}_{i}\\ \ \ \ i=1,2,\ldots,n,\\ \left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{% u}\right]\right){\in}\tilde{k}\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}% \left(g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)+g\left({\left({\gamma}% ^{l}\right)}^{2}\right)\right)},\sqrt{g^{-1}\left(g\left({\left({\gamma}^{u}_{% i}\right)}^{2}\right)+g\left({\left({\gamma}^{u}\right)}^{2}\right)\right)}% \right]\right.\right.},$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{-1}\left(f\left({\left({% \eta}^{l}_{i}\right)}^{2}\right)+f\left({\left({\eta}^{l}\right)}^{2}\right)% \right)},\sqrt{f^{-1}\left(f\left({\left({\eta}^{u}_{i}\right)}^{2}\right)+f% \left({\left({\eta}^{u}\right)}^{2}\right)\right)}\right]\right)\right\}.$

Let $\textit{AIVPHFWA}\left({\tilde{k}}_{1}{\oplus}_{A}\tilde{k},{\tilde{k}}_{2}{% \oplus}_{A}\tilde{k},\ldots,{\tilde{k}}_{n}{\oplus}_{A}\tilde{k}\right)$

$\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i}\\ \ \ i=1,2,\ldots,n\\ \left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{% u}\right]\right){\in}\tilde{k}\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}% \left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left(g^{-1}\left(g\left({\left({% \gamma}^{l}_{i}\right)}^{2}\right)+g\left({\left({\gamma}^{l}\right)}^{2}% \right)\right)\right)}\right)},\right.\right.\right.}$ $\displaystyle\qquad\quad\left.\sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}g\left(g^{-1}\left(g\left({\left({\gamma}^{u}_{i}\right)}^{2}\right% )+g\left({\left({\gamma}^{u}\right)}^{2}\right)\right)\right)}\right)}\right],$ $\displaystyle\qquad\quad\left[\sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}f\left(f^{-1}\left(f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)+% f\left({\left({\eta}^{l}\right)}^{2}\right)\right)\right)}\right)},\right.$ $\displaystyle\qquad\quad\left.\left.\left.\sqrt{f^{-1}\left(\sum\nolimits^{n}_% {i=1}{{\omega}_{i}f\left(f^{-1}\left(f\left({\left({\eta}^{u}_{i}\right)}^{2}% \right)+f\left({\left({\eta}^{u}\right)}^{2}\right)\right)\right)}\right)}% \right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i}\\ ,\ \ \ i=1,2,\ldots,n,\\ \left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{% u}\right]\right){\in}\tilde{k}\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}% \left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}\left(g\left({\left({\gamma}^{l}_{i}% \right)}^{2}\right)+g\left({\left({\gamma}^{l}\right)}^{2}\right)\right)}% \right)},\right.\right.\right.}$ $\displaystyle\qquad\quad\left.\sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}\left(g\left({\left({\gamma}^{u}_{i}\right)}^{2}\right)+g\left({% \left({\gamma}^{u}\right)}^{2}\right)\right)}\right)}\right],$ $\displaystyle\qquad\quad\left[\sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}\left(f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)+f\left({\left% ({\eta}^{l}\right)}^{2}\right)\right)}\right)},\!\!\right.\left.\left.\left.% \sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}\left(f\left({\left({% \eta}^{u}_{i}\right)}^{2}\right)+f\left({\left({\eta}^{u}\right)}^{2}\right)% \right)}\right)}\right]\!\right)\!\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i}\\ \ \ i=1,2,\ldots,n\\ \left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{% u}\right]\right){\in}\tilde{k}\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}% \left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}}g\left({\left({\gamma}^{l}_{i}% \right)}^{2}\right)+g\left({\left({\gamma}^{l}\right)}^{2}\right)\right)},% \right.\right.\right.}$ $\displaystyle\qquad\quad\left.\sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}}g\left({\left({\gamma}^{u}_{i}\right)}^{2}\right)+g\left({\left({% \gamma}^{u}\right)}^{2}\right)\right)}\right],\left[\sqrt{f^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}}f\left({\left({\eta}^{l}_{i}\right)}^{2}% \right)+f\left({\left({\eta}^{l}\right)}^{2}\right)\right)},\right.$ $\displaystyle\qquad\quad\left.\left.\left.\sqrt{f^{-1}\left(\sum\nolimits^{n}_% {i=1}{{\omega}_{i}}f\left({\left({\eta}^{u}_{i}\right)}^{2}\right)+f\left({% \left({\eta}^{u}\right)}^{2}\right)\right)}\right]\right)\right\}.$

Now, $\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{\tilde{k}}_{n}% \right){\oplus}_{A}\tilde{k}$

$\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum\nolimits^{n}% _{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{% u}_{i}\right)}^{2}\right)}\right)}\right]\right.\right.},$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{-1}\left(\sum\nolimits^{n}_% {i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}% _{i}\right)}^{2}\right)}\right)}\right]\right)\right\}{\oplus}_{A}\left\{\left% (\left[{\gamma}^{l},\ {\gamma}^{u}\right],\left[{\eta}^{l},\ {\eta}^{u}\right]% \right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i}\\ \ \ i=1,2,\ldots,n,\\ \left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{% u}\right]\right){\in}\tilde{k}\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}% \left(g\left(g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({% \gamma}^{l}_{i}\right)}^{2}\right)}\right)\right)+g\left({\left({\gamma}^{l}% \right)}^{2}\right)\right)}\right.\right.\right.},$ $\displaystyle\qquad\quad\left.\sqrt{g^{-1}\left(g\left(g^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{u}_{i}\right)}^{2}% \right)}\right)\right)+g\left({\left({\gamma}^{u}\right)}^{2}\right)\right)}% \right],$ $\displaystyle\qquad\quad\left[\sqrt{f^{-1}\left(f\left(f^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right% )}\right)\right)+f\left({\left({\eta}^{l}\right)}^{2}\right)\right)},\right.$ $\displaystyle\qquad\quad\left.\left.\left.\sqrt{f^{-1}\left(f\left(f^{-1}\left% (\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}\right)}^{2}% \right)}\right)\right)+f\left({\left({\eta}^{u}\right)}^{2}\right)\right)}% \right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}_{i},{\gamma}^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}% \right]\right){\in}{\tilde{k}}_{i}\\ \ \ \ i=1,2,\ldots,n\\ \left(\left[{\gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{% u}\right]\right){\in}\tilde{k}\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}% \left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right% )}^{2}\right)}+g\left({\left({\gamma}^{l}\right)}^{2}\right)\right)},\right.% \right.\right.}$ $\displaystyle\qquad\quad\left.\sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{% \omega}_{i}g\left({\left({\gamma}^{u}_{i}\right)}^{2}\right)}+g\left({\left({% \gamma}^{u}\right)}^{2}\right)\right)}\right],\left[\sqrt{f^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right% )}+f\left({\left({\eta}^{l}\right)}^{2}\right)\right)},\right.$ $\displaystyle\qquad\quad\left.\left.\left.\left.\sqrt{f^{-1}\left(\sum% \nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}\right)}^{2}\right% )}+f\left({\left({\eta}^{u}\right)}^{2}\right)\right)}\right]\right)\right\}% \right..$

Hence the theorem.

Theorem 4. (Idempotency) If all ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ are equal, and let ${\tilde{k}}_{i}=\left\{\left(\left[{\gamma}^{l},\ {\gamma}^{u}\right],\left[{% \eta}^{l},\ {\eta}^{u}\right]\right)\right\}$ for all $\left(i=1,2,\ldots,n\right)$ , then

$\displaystyle\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)=\tilde{k}=\left\{\left(\left[{\gamma}^{l},\ {\gamma}^{u}% \right],\left[{\eta}^{l},\ {\eta}^{u}\right]\right)\right\}.$ (15)

Proof: Here, $\textit{AIVPHFWA}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{\tilde{k}}_{n}\right)$

$\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\sqrt{g^{-1}\left(\sum\nolimits^{n}% _{i=1}{{\omega}_{i}g\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\gamma}^{% u}_{i}\right)}^{2}\right)}\right)}\right]\right.\right.},$ $\displaystyle\qquad\left.\left.\left[\sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}% {{\omega}_{i}f\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},\sqrt{f^% {-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\eta}^{u}_{i}% \right)}^{2}\right)}\right)}\right]\right)\right\}.$

Now, since ${\tilde{k}}_{i}=\tilde{k}=\left\{\left(\left[{\gamma}^{l},{\gamma}^{u}\right],% \left[{\eta}^{l},{\eta}^{u}\right]\right)\right\}$ for all $\left(i=1,2,\ldots,n\right)$ , then

${\gamma}^{l}_{i}={\gamma}^{l}$ , ${\gamma}^{u}_{i}={\gamma}^{u}$ , ${\eta}^{l}_{i}={\eta}^{l}$ and ${\eta}^{u}_{i}={\eta}^{u}$ for all $\left(i=1,2,\ldots,n\right)$ . Therefore,

$\displaystyle\textit{AIVPHFWA}\left(\tilde{k},\tilde{k},\ldots,\tilde{k}\right)$ $\displaystyle\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{\gamma% }^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]\right){% \in}\tilde{k}\\ i=1,2,\ldots,n\end{subarray}}{\left\{\left(\left[\sqrt{g^{-1}\left(g\left({% \left({\gamma}^{l}\right)}^{2}\right)\sum\nolimits^{n}_{i=1}{{\omega}_{i}}% \right)},\sqrt{g^{-1}\left(g\left({\left({\gamma}^{u}\right)}^{2}\right)\sum% \nolimits^{n}_{i=1}{{\omega}_{i}}\right)}\right]\right.\right.},$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{f^{-1}\left(f\left({\left({% \eta}^{l}\right)}^{2}\right)\sum\nolimits^{n}_{i=1}{{\omega}_{i}}\right)},% \sqrt{f^{-1}\left(f\left({\left({\eta}^{u}\right)}^{2}\right)\sum\nolimits^{n}% _{i=1}{{\omega}_{i}}\right)}\right]\right)\right\}$ $\displaystyle=\mathop{\bigcup}\nolimits_{\begin{subarray}{c}\left(\left[{% \gamma}^{l}{,\ }{\gamma}^{u}\right],\left[{\eta}^{l}{,\ }{\eta}^{u}\right]% \right){\in}\tilde{k}\\ i=1,2,\ldots,n\end{subarray}}{\left\{\left(\left[{\gamma}^{l},{\gamma}^{u}% \right],\left[{\eta}^{l},{\eta}^{u}\right]\right)\right\}}=\left\{\left(\left[% {\gamma}^{l},{\gamma}^{u}\right],\left[{\eta}^{l},{\eta}^{u}\right]\right)% \right\}.$

Hence the theorem.
5.2 IVPHF Archimedean geometric operators

In the following, some IVPHF Archimedean geometric operators based on the Archimedean operations of IVPHFNs is proposed.

Definition 13. For any collection of IVPHFNs, ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ and weight vector, $\omega=\left({\omega}_{1},{\omega}_{2},\ldots,{\omega}_{n}\right)$ , an A $t$ -N& $t$ -CN-based IVPHFWG (AIVPHFWG) operator is defined by a mapping, $AIVPHFWG:\ K^{n}\to K$ such that

$\displaystyle\textit{AIVPHFWG}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)={{\otimes}_{A}}^{n}_{i=1}\left({{\tilde{k}}_{i}}^{{% \omega}_{i}}\right),$

where ${\omega}_{i}\in\left[0,1\right]$ and $\sum^{n}_{i=1}{{\omega}_{i}}=1$ .

Theorem 5. The aggregated value of any collections of IVPHFNs, ${\tilde{k}}_{i}$ $\left(i=1,2,\ldots,n\right)$ using AIVPHFWG operator is also an IVPHFN, and is given by

$\displaystyle\textit{AIVPHFWG}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\sqrt{f^{-1}\left(\sum\nolimits^{n}% _{i=1}{{\omega}_{i}f\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{f^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}f\left({\left({\gamma}^{% u}_{i}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\qquad\quad\left.\left.\left[\sqrt{g^{-1}\left(\sum\nolimits^{n}_% {i=1}{{\omega}_{i}g\left({\left({\eta}^{l}_{i}\right)}^{2}\right)}\right)},% \sqrt{g^{-1}\left(\sum\nolimits^{n}_{i=1}{{\omega}_{i}g\left({\left({\eta}^{u}% _{i}\right)}^{2}\right)}\right)}\right]\right)\right\}.$ (16)

Proof. Proof is same as the proof of Theorem 1.

It is worthy to mention here that for different forms of the decreasing generator, $f$ , several forms of familiar IVPHFWG operators can be obtained, which are described below.

Algebraic ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWG: If $f\left(t\right)=-{\text{log}\ t}$ , then the AIVPHFWG operator reduces to the IVPHF WG (IVPHFWG) operator, and is defined as:

$\displaystyle\textit{IVPHFWG}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\prod\nolimits^{n}_{i=1}{{\left({% \gamma}^{l}_{i}\right)}^{{\omega}_{i}}},\prod\nolimits^{n}_{i=1}{{\left({% \gamma}^{u}_{i}\right)}^{{\omega}_{i}}}\right],\right.\right.}$ $\displaystyle\qquad\left.\left.\left[\sqrt{1-\prod\nolimits^{n}_{i=1}{{\left(1% -{\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}},\sqrt{1-\prod% \nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{u}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}\right]\right)\right\}.$ (17)

Einstein ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWG: If $f\left(t\right)={\text{log}\left(\frac{2-t}{t}\right)}$ is considered, then AIVPHFWG operator reduces to IVPHF Einstein WG (IVPHFEWG) operator as:

$\displaystyle\textit{IVPHFEWG}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)=$ $\displaystyle\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}^% {u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{k% }}_{i},i=1,2,\ldots,n}{\left\{\left(\left[\frac{\sqrt{2}\prod\nolimits^{n}_{i=% 1}{{\left({\gamma}^{l}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{% i=1}{{\left(2-{\left({\gamma}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}+}% \prod\nolimits^{n}_{i=1}{{\left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}},\right.\right.\right.}$ $\displaystyle\qquad\left.\frac{\sqrt{2}\prod\nolimits^{n}_{i=1}{{\left({\gamma% }^{u}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(2-{% \left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}+}\prod\nolimits^{n}_% {i=1}{{\left({\left({\gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}% \right],$ $\displaystyle\qquad\left[\frac{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+{\left(% {\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}-}\prod\nolimits^{n}_{i=1}{{% \left(1-{\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}{\sqrt{% \prod\nolimits^{n}_{i=1}{{\left(1+{\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}+}\prod\nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{l}_{i}\right)}^% {2}\right)}^{{\omega}_{i}}}}},\right.$ $\displaystyle\qquad\left.\left.\left.\frac{\sqrt{\prod\nolimits^{n}_{i=1}{{% \left(1+{\left({\eta}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}-}\prod% \nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{u}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+{\left({\eta}^{u}_{i}% \right)}^{2}\right)}^{{\omega}_{i}}+}\prod\nolimits^{n}_{i=1}{{\left(1-{\left(% {\eta}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}\right]\right)\right\}.$ (18)

$\bullet$

Hamacher ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWG: When $f\left(t\right)={\text{log}\left(\frac{\psi+\left(1-\psi\right)t}{t}\right)}$ , $\psi>0$ , AIVPHFWG operator reduces to IVPHF Hamacher WG (IVPHFHWG) operator in the form of

$\displaystyle\textit{IVPHFHWG}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}$ $\displaystyle\qquad{\left\{\left(\left[\frac{\sqrt{\psi}\prod\nolimits^{n}_{i=% 1}{{\left({\gamma}^{l}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{% i=1}{{\left(1+\left(\psi-1\right)\left(1-{\left({\gamma}^{l}_{i}\right)}^{2}% \right)\right)}^{{\omega}_{i}}+}\left(\psi-1\right)\prod\nolimits^{n}_{i=1}{{% \left({\left({\gamma}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}},\right.% \right.\right.}$ $\displaystyle\quad\left.\frac{\sqrt{\psi}\prod\nolimits^{n}_{i=1}{{\left({% \gamma}^{u}_{i}\right)}^{{\omega}_{i}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left% (1+\left(\psi-1\right)\left(1-{\left({\gamma}^{u}_{i}\right)}^{2}\right)\right% )}^{{\omega}_{i}}+}\left(\psi-1\right)\prod\nolimits^{n}_{i=1}{{\left({\left({% \gamma}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}}}\right],$ $\displaystyle\quad\left[\frac{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+\left(% \psi-1\right){\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}-\prod% \nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+\left(\psi-1\right){% \left({\eta}^{l}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}+\left(\psi-1\right)% \prod\nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{l}_{i}\right)}^{2}\right)}^{{% \omega}_{i}}}}}\right.$ $\displaystyle\qquad\left.\left.\left.\frac{\sqrt{\prod\nolimits^{n}_{i=1}{{% \left(1+\left(\psi-1\right){\left({\eta}^{u}_{i}\right)}^{2}\right)}^{{\omega}% _{i}}}-\prod\nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{u}_{i}\right)}^{2}% \right)}^{{\omega}_{i}}}}}{\sqrt{\prod\nolimits^{n}_{i=1}{{\left(1+\left(\psi-% 1\right){\left({\eta}^{u}_{i}\right)}^{2}\right)}^{{\omega}_{i}}}+\left(\psi-1% \right)\sum\nolimits^{n}_{i=1}{{\left(1-{\left({\eta}^{u}_{i}\right)}^{2}% \right)}^{{\omega}_{i}}}}}\right]\right)\right\}.$ (19)

For $\psi=1$ and $\psi=2$ , IVPHFHWG operator is turned into IVPHFWG and IVPHFEWG operators, respectively. So, IVPHFHWG operator becomes a generalised version of IVPHFWG and IVPHFEWG operators.

\bullet

Frank ${\bm{t}}$ -N and ${\bm{t}}$ -CN Operations on AIVPHFWG: When $f\left(t\right)={\text{log}\left(\frac{\tau-1}{{\tau}^{t}-1}\right)}$ , $\tau>1$ , AIVPHFWG operator reduces to IVPHF Frank WG (IVPHFFWG) operator; thus, IVPHFFWG operator is defined as:

$\displaystyle\textit{IVPHFFWG}\left({\tilde{k}}_{1},{\tilde{k}}_{2},\ldots,{% \tilde{k}}_{n}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{i},{\gamma}% ^{u}_{i}\right],\left[{\eta}^{l}_{i},{\eta}^{u}_{i}\right]\right){\in}{\tilde{% k}}_{i},i=1,2,\ldots,n}$ $\displaystyle\quad{\left\{\left(\left[\sqrt{\frac{{\text{log}\left(1+\prod% \nolimits^{n}_{i=1}{{\left({\tau}^{{\left({\gamma}^{l}_{i}\right)}^{2}}-1% \right)}^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}},\sqrt{\frac{{\text{% log}\left(1+\prod\nolimits^{n}_{i=1}{{\left({\tau}^{{\left({\gamma}^{u}_{i}% \right)}^{2}}-1\right)}^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}}% \right],\right.\right.}$ $\displaystyle\qquad\left.\left.\left[\sqrt{1\!-\!\frac{{\text{log}\left(1+% \prod\nolimits^{n}_{i=1}{{\left({\tau}^{1-{\left({\eta}^{l}_{i}\right)}^{2}}-1% \right)}^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}},\sqrt{1\!-\!\frac{{% \text{log}\left(1+\prod\nolimits^{n}_{i=1}{{\left({\tau}^{1-{\left({\eta}^{u}_% {i}\right)}^{2}}-1\right)}^{{\omega}_{i}}}\right)\ }}{{\text{log}\ \tau\ }}}% \right]\!\right)\!\!\right\}.$ (20)

For $\tau\to 1$ , IVPHFFWG operator is converted into IVPHFWG operator. So, IVPHFFWG operator is considered as a generalisation of IVPHFWG operator.

It is to be mentioned here that AIVPHFWG operator possesses same properties as like AIVPHFWA operator as discussed above. The proofs are also the same as that of AIVPHFWA operator. So, the proofs are skipped here.

6. MCDM method based on IVPHF information

In the present section, an approach to MCDM using IVPHF information is developed by utilising the proposed AIVPHFWA and AIVPHFWG operators.

Let $A=\left\{A_{1},A_{2},\ldots,A_{m}\right\}$ and $C=\left\{C_{1},C_{2},\ldots,C_{n}\right\}$ be a set of alternatives and a collection of criteria, respectively, such that weight vector of the criteria is given by $\omega=\left({\omega}_{1},{\omega}_{2},\ldots,{\omega}_{n}\right)$ with ${\omega}_{j}\in\left[0,1\right]$ for $j=1,2,\ldots,n$ , and $\sum\nolimits^{n}_{j=1}{{\omega}_{j}}=1$ . The IVPHF decision matrix $\tilde{D}{=}{({\tilde{d}}_{ij})}_{m{\times}n}$ is constructed only when, all the performance scores of the alternatives are obtained. In solving MCDM problems, the cost criteria are required to transform into benefit criteria using the following method. Thus the decision matrix, $\tilde{D}{=}{({\tilde{d}}_{ij})}_{m{\times}n}$ is converted into a normalised decision matrix, $\tilde{R}{=}{\left({\tilde{r}}_{ij}\right)}_{m{\times}n}$ by the following way:

$\displaystyle{\tilde{r}}_{ij}=\left\{\begin{array}[]{ll}{\tilde{d}}_{ij}&\text% {for\ benefit\ }\text{criteria}\ C_{j}\\ {\tilde{d}}^{c}_{ij}&\text{for\ cost\ }\text{criteria}\ C_{j}\end{array}\right.$ (21)

$i{=1,2,}{\ldots}{,}m$ and $j{=1,2,}{\ldots}{,}n$ . Here ${\tilde{d}}^{c}_{ij}$ denotes the complement of ${\tilde{d}}_{ij}$ .

Now, the newly defined AIVPHFWA (or AIVPHFWG) operator is utilized to develop an approach for solving MCDM problems under IVPHF environment. The whole process is described through the following steps:

Step 1: Transform the decision matrix, $\tilde{D}{=}{\left({\tilde{d}}_{ij}\right)}_{m{\times}n}$ into the normalised form, $\tilde{R}{=}{\left({\tilde{r}}_{ij}\right)}_{m{\times}n}$ using Eq. (21).

Step 2: Aggregate the IVPHFNs, ${\tilde{r}}_{ij}$ for each alternative, $A_{i}$ using AIVPHFWA (or AIVPHFWG) operator for a suitable weight vector, $w$ and the parameters, $\psi,\tau$ as follows:

$\displaystyle{\tilde{r}}^{A}_{i}=\textit{AIVPHFWA}\left({\tilde{r}}_{i1},{% \tilde{r}}_{i2},\ldots,{\tilde{r}}_{in}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{ij},{\gamma% }^{u}_{ij}\right],\left[{\eta}^{l}_{ij},{\eta}^{u}_{ij}\right]\right){\in}{% \tilde{r}}_{ij},j=1,2,\ldots,n}$ $\displaystyle\quad{\left\{\left(\left[\sqrt{g^{{-}{1}}\left(\sum\nolimits^{n}_% {j{=1}}{{\omega}_{j}g\left({\left({\gamma}^{l}_{ij}\right)}^{2}\right)}\right)% },\sqrt{g^{{-}{1}}\left(\sum\nolimits^{n}_{j{=1}}{{\omega}_{j}g\left({\left({% \gamma}^{u}_{ij}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\quad\left.\left.\left[\sqrt{f^{{-}{1}}\left(\sum\nolimits^{n}_{j% {=1}}{{\omega}_{j}f\left({\left({\eta}^{l}_{ij}\right)}^{2}\right)}\right)},% \sqrt{f^{{-}{1}}\left(\sum\nolimits^{n}_{i{=1}}{{\omega}_{j}f\left({\left({% \eta}^{u}_{ij}\right)}^{2}\right)}\right)}\right]\right)\right\}$ (22)

$\displaystyle{\tilde{r}}^{G}_{i}=\textit{AIVPHFWG}\left({\tilde{r}}_{i1},{% \tilde{r}}_{i2},\ldots,{\tilde{r}}_{in}\right)$ $\displaystyle=\mathop{\bigcup}\nolimits_{\left(\left[{\gamma}^{l}_{ij},{\gamma% }^{u}_{ij}\right],\left[{\eta}^{l}_{ij},{\eta}^{u}_{ij}\right]\right)\in{% \tilde{r}}_{ij},j=1,2,\ldots,n}$ $\displaystyle\quad{\left\{\left(\left[\sqrt{f^{{-}{1}}\left(\sum\nolimits^{n}_% {j{=1}}{{\omega}_{j}f\left({\left({\gamma}^{l}_{ij}\right)}^{2}\right)}\right)% },\sqrt{f^{{-}{1}}\left(\sum\nolimits^{n}_{j{=1}}{{\omega}_{j}f\left({\left({% \gamma}^{u}_{ij}\right)}^{2}\right)}\right)}\right],\right.\right.}$ $\displaystyle\quad\left.\left.\left[\sqrt{g^{{-}{1}}\left(\sum\nolimits^{n}_{j% {=1}}{{\omega}_{j}g\left({\left({\eta}^{l}_{ij}\right)}^{2}\right)}\right)},% \sqrt{g^{{-}{1}}\left(\sum\nolimits^{n}_{i{=1}}{{\omega}_{j}g\left({\left({% \eta}^{u}_{ij}\right)}^{2}\right)}\right)}\right]\right)\right\}$ (23) $\displaystyle\quad i=1,\ 2,\ldots,m;j=1,\ 2,\ldots,n.$

Step 3: Utilizing the score function as mentioned in Definition 2, the rank of all alternatives is evaluated.

The above method is validated through the following illustrative examples.

7. Illustrative examples

To establish application potentiality of the proposed approach, two illustrative examples are considered and solved in this section.

7.1 Example 1

At first, a revised problem (adapted from Wei et al. [25]) relating to green supply chain management (GSCM) is considered and solved under IVPHF context.

It is well known that the choice of suitable green supplier is a key factor of GSCM. Due to the fact that the entire supply chain depends upon the quality of the suppliers; and the ecological performance of industrial companies, organizations is directly or indirectly balanced by the suppliers’ characteristics, so, in the view of sociological or environmental impact of the suppliers, appropriate green supplier assessment has now become an emerging research topic. In this section, the proposed methodology is applied to GSCM with IVPHF data for supplier evaluation and choosing most potential supplier.

The problem is discussed in the following manner:

In a GSCM five green suppliers are available as alternatives which are given by the set, $A=\left\{A_{i}|i=1,2,3,4,5\right\}$ .

The four criteria on which the suppliers are evaluated by the experts are given by

$\bullet$
$C_{1}$ : The product quality factors.
$\bullet$
$C_{2}$ : Environmental factors.
$\bullet$
$C_{3}$ : Delivery factors.
$\bullet$
$C_{4}$ : Price factors.

The criteria weight vector is considered as $w={\left(0.4,0.1,0.2,0.3\right)}^{T}$ .

The DM provided judgement values on the alternatives considering the above-mentioned criteria by using IVPHFNs, and the resulting IVPHF decision matrix is presented in Table 1.

Table 1
IVPHF decision matrix

$C_{1}$ $C_{2}$

$A_{1}$ $\left\{\left(\left[.1,.2\right],\left[.3,.6\right]\right),\left(\left[.2,.3% \right],\left[.4,.7\right]\right)\right\}$ $\left\{\begin{array}[]{c}\left(\left[.3,.4\right],\left[.6,.9\right]\right),% \left(\left[.3,.5\right],\left[.4,.8\right]\right),\\ \left(\left[.2,.7\right],\left[.2,.6\right]\right)\end{array}\right\}$

$A_{2}$ $\left\{\left(\left[.2,.4\right],\left[.3,.5\right]\right),\left(\left[.5,.6% \right],\left[.4,.7\right]\right)\right\}$ $\left\{\left(\left[.2,.5\right],\left[.6,.7\right]\right),\left(\left[.3,.6% \right],\left[.4,.8\right]\right)\right\}$

$A_{3}$ $\left\{\left(\left[.6,.9\right],\left[.1,.2\right]\right),\left(\left[.7,.8% \right],\left[.2,.3\right]\right)\right\}$ $\left\{\left(\left[.4,.6\right],\left[.2,.4\right]\right),\left(\left[.5,.7% \right],\left[.3,.4\right]\right)\right\}$

$A_{4}$ $\left\{\left(\left[.6,.7\right],\left[.3,.5\right]\right),\left(\left[.7,.8% \right],\left[.3,.4\right]\right)\right\}$ $\left\{\left(\left[.4,.5\right],\left[.2,.5\right]\right),\left(\left[.5,.6% \right],\left[.4,.5\right]\right)\right\}$

$A_{5}$ $\left\{\left(\left[.3,.7\right],\left[.3,.4\right]\right),\left(\left[.6,.8% \right],\left[.4,.6\right]\right)\right\}$ $\left\{\left(\left[.3,.5\right],\left[.3,.6\right]\right),\left(\left[.3,.7% \right],\left[.4,.7\right]\right)\right\}$

$C_{3}$ $C_{4}$

$A_{1}$ $\left\{\left(\left[.2,.5\right],\left[.5,.8\right]\right),\left(\left[.2,.6% \right],\left[.3,.8\right]\right)\right\}$ $\left\{\left(\left[.2,.6\right],\left[.3,.8\right]\right),\left(\left[.2,.7% \right],\left[.4,.6\right]\right)\right\}$

$A_{2}$ $\left\{\left(\left[.2,.5\right],\left[.3,.7\right]\right),\left(\left[.3,.5% \right],\left[.6,.7\right]\right)\right\}$ $\left\{\left(\left[.2,.5\right],\left[.4,.7\right]\right),\left(\left[.4,.7% \right],\left[.5,.6\right]\right)\right\}$

$A_{3}$ $\left\{\begin{array}[]{c}\left(\left[.5,.6\right],\left[.1,.2\right]\right),% \left(\left[.7,.8\right],\left[.4,.6\right]\right),\\ \left(\left[.8,.9\right],\left[.1,.2\right]\right)\end{array}\right\}$ $\left\{\left(\left[.4,.7\right],\left[.3,.5\right]\right),\left(\left[.6,.9% \right],\left[.2,.4\right]\right)\right\}$

$A_{4}$ $\left\{\left(\left[.4,.5\right],\left[.2,.3\right]\right),\left(\left[.5,.6% \right],\left[.4,.5\right]\right)\right\}$ $\left\{\begin{array}[]{c}\left(\left[.4,.6\right],\left[.2,.6\right]\right),% \left(\left[.3,.6\right],\left[.3,.7\right]\right),\\ \left(\left[.5,.8\right],\left[.4,.6\right]\right)\end{array}\right\}$

$A_{5}$ $\left\{\left(\left[.2,.6\right],\left[.4,.6\right]\right),\left(\left[.3,.7% \right],\left[.4,.7\right]\right)\right\}$ $\left\{\left(\left[.3,.6\right],\left[.4,.6\right]\right),\left(\left[.5,.6% \right],\left[.3,.7\right]\right)\right\}$

Now, the developed method is applied to find the best supplier in the decision making context. The steps of the methodology are presented below:

Step 1: All the criteria as shown in Table 1, being the benefit criteria, the decision matrix need not be normalised further.

Step 2: Utilising IVPHFHWA operator to aggregate IVPHFNs for each alternative, $A_{i}$ , considering associated weight vector, $w={\left(0.4,0.1,0.2,0.3\right)}^{T}$ , and the value of the parameter, $\psi=2$ , the aggregated IVPHFNs are calculated. For simplicity of presentation, only the aggregated IVPHFNs corresponding to the alternative, $A_{2}$ is presented below.

$\displaystyle r^{A}_{2}=\textit{IVPHFHWA}\left({\tilde{r}}_{21},{\tilde{r}}_{2% 2},{\tilde{r}}_{23},{\tilde{r}}_{24}\right)$ $\displaystyle=\left\{\left(\left[{0.2000},{0.4631}\right],\left[{0.3520},{0.61% 55}\right]\right),\left(\left[{0.2762},{\ 0.5414}\right],\left[{0.3774},{0.586% 7}\right]\right),\left(\left[{0.2237},{0.4631}\right],\right.\right.$ $\displaystyle\quad\left.\left[{0.4061},{0.6155}\right]\right),\left(\left[{0.2% 937},{0.5414}\right],\left[\text{0.4346},{0.5867}\right]\right),\left(\left[{0% .2122},{0.4754}\right],\left[{0.3370},{0.6248}\right]\right),$ $\displaystyle\quad\left(\left[{0.2851},{0.5515}\right],\left[{0.3614},{0.5957}% \right]\right),\left(\left[{0.2346},{0.4754}\right],\left[{0.3891},{0.6248}% \right]\right),\left(\left[{0.3021},{0.5515}\right],\right.$ $\displaystyle\quad\left.\left[{0.4167},{0.5957}\right]\right),\left(\left[{0.3% 543},{\ 0.5431}\right],\left[{0.39450.7000},\right]\right),\left(\left[{0.4015% },{0.6081}\right],\left[{0.4224},{0.6693}\right]\right),$ $\displaystyle\quad\left(\left[{0.3679},{0.5431}\right],\left[{0.4538},{0.7000}% \right]\right),\left(\left[{0.4134},{0.6081}\right],\left[{0.4849},{0.6693}% \right]\right),\left(\left[{0.3612},{0.5532}\right],\right.$ $\displaystyle\quad\left.\left[{0.3780},{0.7098}\right]\right),\left(\left[{0.4% 075},{0.6166}\right],\left[{0.4049},{0.6790}\right]\right),\left(\left[{0.3746% },{0.5532}\right],\left[{0.4353},\text{0.7098}\right]\right),$ $\displaystyle\quad\left(\left[{0.4193},{0.6166}\right],\left[{0.4654},{0.6790}% \right]\right)\}$

Similarly, other values of ${\tilde{r}}^{A}_{i}$ ${(i=1,2,\ldots,5)}$ are calculated.

Step 3: The score values of each alternatives $A_{i}$ ${(i=1,2,\ldots,5)}$ are calculated and is presented in Table 2. The alternatives are ranked based on the score values.

Table 2
Score values for $\psi=$ 2

Operator Parameter $S\left({\tilde{r}}^{A}_{1}\right)$ $S\left({\tilde{r}}^{A}_{2}\right)$ $S\left({\tilde{r}}^{A}_{3}\right)$ $S\left({\tilde{r}}^{A}_{4}\right)$ $S\left({\tilde{r}}^{A}_{5}\right)$

IVPHFHWA $\psi=$ 2 0.4153 0.4545 0.7240 0.6049 0.5259

Figure 1.
Variation of the score values using IVPHFHWA operator.

In accordance with the score values, the alternatives are ranked, and the ordering of alternatives are obtained as $A_{3}\succ A_{4}\succ A_{5}\succ A_{2}\succ A_{1}$ . Therefore, the best alternative is identified as $A_{3}$ .
7.1.1 Sensitivity analysis

	$C_{1}$	$C_{2}$
$A_{1}$	$\left\{\left(\left[.1,.2\right],\left[.3,.6\right]\right),\left(\left[.2,.3% \right],\left[.4,.7\right]\right)\right\}$	$\left\{\begin{array}[]{c}\left(\left[.3,.4\right],\left[.6,.9\right]\right),% \left(\left[.3,.5\right],\left[.4,.8\right]\right),\\ \left(\left[.2,.7\right],\left[.2,.6\right]\right)\end{array}\right\}$
$A_{2}$	$\left\{\left(\left[.2,.4\right],\left[.3,.5\right]\right),\left(\left[.5,.6% \right],\left[.4,.7\right]\right)\right\}$	$\left\{\left(\left[.2,.5\right],\left[.6,.7\right]\right),\left(\left[.3,.6% \right],\left[.4,.8\right]\right)\right\}$
$A_{3}$	$\left\{\left(\left[.6,.9\right],\left[.1,.2\right]\right),\left(\left[.7,.8% \right],\left[.2,.3\right]\right)\right\}$	$\left\{\left(\left[.4,.6\right],\left[.2,.4\right]\right),\left(\left[.5,.7% \right],\left[.3,.4\right]\right)\right\}$
$A_{4}$	$\left\{\left(\left[.6,.7\right],\left[.3,.5\right]\right),\left(\left[.7,.8% \right],\left[.3,.4\right]\right)\right\}$	$\left\{\left(\left[.4,.5\right],\left[.2,.5\right]\right),\left(\left[.5,.6% \right],\left[.4,.5\right]\right)\right\}$
$A_{5}$	$\left\{\left(\left[.3,.7\right],\left[.3,.4\right]\right),\left(\left[.6,.8% \right],\left[.4,.6\right]\right)\right\}$	$\left\{\left(\left[.3,.5\right],\left[.3,.6\right]\right),\left(\left[.3,.7% \right],\left[.4,.7\right]\right)\right\}$
	$C_{3}$	$C_{4}$
$A_{1}$	$\left\{\left(\left[.2,.5\right],\left[.5,.8\right]\right),\left(\left[.2,.6% \right],\left[.3,.8\right]\right)\right\}$	$\left\{\left(\left[.2,.6\right],\left[.3,.8\right]\right),\left(\left[.2,.7% \right],\left[.4,.6\right]\right)\right\}$
$A_{2}$	$\left\{\left(\left[.2,.5\right],\left[.3,.7\right]\right),\left(\left[.3,.5% \right],\left[.6,.7\right]\right)\right\}$	$\left\{\left(\left[.2,.5\right],\left[.4,.7\right]\right),\left(\left[.4,.7% \right],\left[.5,.6\right]\right)\right\}$
$A_{3}$	$\left\{\begin{array}[]{c}\left(\left[.5,.6\right],\left[.1,.2\right]\right),% \left(\left[.7,.8\right],\left[.4,.6\right]\right),\\ \left(\left[.8,.9\right],\left[.1,.2\right]\right)\end{array}\right\}$	$\left\{\left(\left[.4,.7\right],\left[.3,.5\right]\right),\left(\left[.6,.9% \right],\left[.2,.4\right]\right)\right\}$
$A_{4}$	$\left\{\left(\left[.4,.5\right],\left[.2,.3\right]\right),\left(\left[.5,.6% \right],\left[.4,.5\right]\right)\right\}$	$\left\{\begin{array}[]{c}\left(\left[.4,.6\right],\left[.2,.6\right]\right),% \left(\left[.3,.6\right],\left[.3,.7\right]\right),\\ \left(\left[.5,.8\right],\left[.4,.6\right]\right)\end{array}\right\}$
$A_{5}$	$\left\{\left(\left[.2,.6\right],\left[.4,.6\right]\right),\left(\left[.3,.7% \right],\left[.4,.7\right]\right)\right\}$	$\left\{\left(\left[.3,.6\right],\left[.4,.6\right]\right),\left(\left[.5,.6% \right],\left[.3,.7\right]\right)\right\}$

Operator	Parameter	$S\left({\tilde{r}}^{A}_{1}\right)$	$S\left({\tilde{r}}^{A}_{2}\right)$	$S\left({\tilde{r}}^{A}_{3}\right)$	$S\left({\tilde{r}}^{A}_{4}\right)$	$S\left({\tilde{r}}^{A}_{5}\right)$
IVPHFHWA	$\psi=$ 2	0.4153	0.4545	0.7240	0.6049	0.5259

Now, based on the DM’s preferences, the parameter, $\psi$ can presume different values. To observe the variation of the ranking of the five alternatives based on the parameter, $\psi$ , the values between 0 to 50 are assigned. The achieved score values of the five alternatives are shown in Figs 1 and 2.

Figure 2.

Variation of the score values using IVPHFHWG operator.

From Fig. 1, it is observed that, using IVPHFHWA operator, the ordering of the alternatives does not change, but the score value of the alternatives decreases monotonically.

In a similar manner, IVPHFHWG operator is used on the given example, and the score value of alternatives are calculated by varying the parameter, $\psi$ between 0 to 50. The achieved results are presented in Fig. 2. It is to be noted that the ordering of alternatives does not change as like using IVPHFHWA operator. But the score value of the alternatives increases monotonically.

Figure 3.

Variation of the score values using IVPHFFWA operator.

Now, if IVPHFFWA and IVPHFFWG operators are used, individually, instead of using IVPHFHWA or IVPHFHWG operators for aggregating the attribute values of the alternatives, then the score values are presented in the Figs 3 and 4, respectively. As like above cases, similar observations are viewed through the figures corresponding to averaging and geometric operators.

It is worthy to mention here that no changes in the orderings of the alternatives are found while making the decision using different aggregation operators. Thus it stands that the proposed methodology possesses a strong consistency.

7.1.2 Comparison with existing method [25]

In the context of solving MCDM problems, Wei et al. [25] used PHFHWA and PHFHWG operators. It is to be noted here that the orderings of alternatives achieved by Wei et al. [25] are the same as like the proposed method. In the method developed by Wei et al. [25], PHF values are used. But in the proposed method, DM can evaluate the problem more adequately using IVPHFNs. So, the proposed method is advantageous for considering DM’s flexibility for making proper assessment in real-life MCDM contexts. Further, the existing Hamacher aggregation operators [25] for PHFNs become a particular case of the A $t$ -N& $t$ -CN based proposed operators. Moreover, the superiority of the proposed method is reflected by comparing the differences between two consecutive score values of the alternatives, which are ranked using the proposed method and the method described by Wei et al. [25]. The comparisons are presented through Figs 5 and 6.

Table 3
IVPHF decision matrix

	$C_{1}$	$C_{2}$	$C_{3}$
$A_{1}$	$\left\{\left(\left[0.4,0.5\right],\left[0.3,0.4\right]\right)\right\}$	$\left\{\begin{array}[]{c}\left(\left[0.4,0.6\right],\left[0.2,0.4\right]\right% ),\\ \left[0.5,0.7\right],\left[0.3,0.5\right]\end{array}\right\}$	$\left\{\begin{array}[]{c}\left(\left[0.1,0.2\right],\left[0.8,0.9\right]\right% ),\\ \left(\left[0.1,0.3\right],\left[0.5,0.6\right]\right)\end{array}\right\}$
$A_{2}$	$\left\{\begin{array}[]{c}\left(\left[0.6,0.7\right],\left[0.2,0.3\right]\right% ),\\ \left(\left[0.75,0.9\right],\left[0.1,0.3\right]\right)\end{array}\right\}$	$\left\{\left(\left[0.6,0.7\right],\left[0.2,0.3\right]\right)\right\}$	$\left\{\begin{array}[]{c}\left(\left[0.4,0.7\right],\left[0.1,0.2\right]\right% ),\\ \left(\left[0.7,0.9\right],\left[0.2,0.3\right]\right)\end{array}\right\}$
$A_{3}$	$\left\{\left(\left[0.3,0.6\right],\left[0.3,0.4\right]\right)\right\}$	$\left\{\begin{array}[]{c}\left(\left[0.5,0.6\right],\left[0.3,0.4\right]\right% ),\\ \left(\left[0.6,0.7\right],\left[0.4,0.5\right]\right)\end{array}\right\}$	$\left\{\left(\left[0.5,0.6\right],\left[0.1,0.3\right]\right)\right\}$
$A_{4}$	$\left\{\begin{array}[]{c}\left(\left[0.7,0.8\right],\left[0.1,0.2\right]\right% ),\\ \left(\left[0.6,0.8\right],\left[0.2,0.3\right]\right)\end{array}\right\}$	$\left\{\left(\left[0.6,0.7\right],\left[0.1,0.3\right]\right)\right\}$	$\left\{\begin{array}[]{c}\left(\left[0.3,0.4\right],\left[0.1,0.2\right]\right% ),\\ \ \left(\left[0.6,0.7\right],\left[0.1,0.25\right]\right)\end{array}\right\}$

Figure 4.

Variation of the score values using IVPHFFWG operator.

Figure 5.

Difference between two consecutive score values of the ranked alternatives using IVPHFHWA and PHFHWA operators.

Figure 6.

Difference between two consecutive score values of the ranked alternatives using IVPHFHWG and PHFHWG operators.

The above figures show that the difference between any two score values of the consecutively ranked alternatives increases, significantly, in the proposed method. So the rank of the alternatives can be identified in a better way than the existing method. Thus in comparison with the existing methods, the proposed methodology contains better efficiency in ranking the alternatives.

7.2 Example 2

Another problem is considered in this section to show the applicability and efficiency of the proposed methodology, more clearly. The problem related to investment of funds in an appropriate company. It is collected from a research article published by Garg [31] and revised under IVPHF environment. There are four investment companies, $A_{i}\left(i=1,2,3,4\right)$ satisfying three criteria, $C_{j}\left(j=1,2,3\right)$ with the weight vector, $w={\left(0.35,0.25,0.40\right)}^{T}$ . To choose the best investment company, the DM evaluates the alternatives based on the above criteria using IVPHFNs, and their evaluation values are listed in the following Table 3.

Using IVPHFHWA (Considering Hamacher parameter $\psi=$ 0.5), the score values are obtained as $S\left(A_{1}\right)=$ 0.4338, $S\left(A_{2}\right)=$ 0.7047, $S\left(A_{3}\right)=$ 0.5721, $S\left(A_{4}\right)=$ 0.6465.

So, the rank of the alternatives is $A_{2}\succ A_{4}\succ A_{3}\succ A_{1}$ .

Further, using IVPHFHWG operator (Considering Hamacher parameter $=$ 10), the score values are achieved as $S\left(A_{1}\right)=$ 0.3812, $S\left(A_{2}\right)=$ 0.6226, $S\left(A_{3}\right)=$ 0.5468, $S\left(A_{4}\right)=$ 0.5638.

So, the rank of the alternatives becomes $A_{2}\succ A_{4}\succ A_{3}\succ A_{1}$ .

Figure 7.

Variation of the score values using IVPHFHWA operator.

Figure 8.

Variation of the score values using IVPHFHWG operator.

7.2.1 Sensitivity analysis

Changing the Hamacher parameter, $\psi$ in $\left(0,10\right]$ , and using IVPHFHWA and IVPHFHWG operators, the corresponding score values are presented through the Figs 7 and 8, respectively. From Fig. 7, it is clear that the ranking of the alternatives remains unchanged using IVPHFHWA operator for $\psi\in\left(0,10\right]$ . But, Fig. 8 discloses that the ranking of the alternatives differs for some values of $\psi$ using IVPHFHWG operator. The ranking results are listed below.

Using IVPHFHWA operator, the ranking of alternative becomes $A_{2}\succ A_{3}\succ A_{4}\succ A_{1}$ for $\psi\in\left(0,10\right]$ . Thus, the best alternative remains the same as $A_{2}$ throughout the given range.

Besides, using IVPHFHWG operator, the rankings of alternatives become

$A_{3}\succ A_{2}\succ A_{4}\succ A_{1}$ for $\psi\in\left(0,0.1089\right)$ ;

$A_{2}\succ A_{3}\succ A_{4}\succ A_{1}$ for $\psi\in\left(0.1089,1.5762\right)$ ; and

$A_{2}\succ A_{4}\succ A_{3}\succ A_{1}$ for $\psi\in(1.5762,10]$ .

Hence the best alternative is either $A_{2}$ or $A_{3}$ according to the choice of the parameter.

Figure 9.

The differences of overall values of the consecutively ranked alternatives.

7.2.2 Comparison with other Method [31]

Using the Method developed by Garg [31], the ranking of the alternatives is found as $A_{2}\succ A_{4}\succ A_{3}\succ A_{1}$ ; which is also acquired by the proposed operators by choosing particular value of the Hamacher parameter, $\psi$ . Hence, in dealing real life MCDM problems, the proposed method is justified. Moreover, from the view point of the ranking results for different values of Hamacher parameter, it can be concluded that, by changing the values of the Hamacher parameter, $\psi$ , according to the needs of the DMs, exact decisions can be taken. Further, in Fig. 9, the differences of overall values of the consecutively ranked alternatives are presented to compare with the existing method [31]. It is figured out from the figure that, using the proposed operators, the differences of overall values of the consecutively ranked alternatives are considerably increased than the existing method [31]. This shows that the proposed method is more effective than the existing method [31] in order to accomplish the ranking results.

8. Conclusions

In this paper, it has been shown that the proposed A $t$ -N& $t$ -CN based aggregation operators for IVPHFNs are more flexible than the existing operators due to the presence of the parameters, $\psi,\tau$ . Depending on the preferences of the DMs, different values of the parameter can be chosen. As a consequence, several classes of aggregation operators can be generated. Thus the proposed method is capable enough to capture preferences of the DMs in MCDM problems in more flexible way. Through the illustrative examples, it has been established the fact that the proposed method not only capture the existing Hamacher operation based aggregation operators for PHFNs (Wei et al. [25]), but also extends the scope of using aggregation operators in IVPHF environments. The superiority of the proposed method is established by comparing the differences between any two score values of the consecutively ranked alternatives through the proposed method and the existing methods [25, 31]. The comparison shows that all the differences have significantly increased. Hence ordering of the alternatives using the proposed methodology is more powerful. In future, the proposed operators may be extended to other domains, viz., $q$ -rung orthopair fuzzy [49], bipolar fuzzy [50], cubic fuzzy [51, 52], Pythagorean cubic fuzzy [53], cubic bipolar fuzzy [54], hesitant Pythagorean fuzzy [55] and other environments to capture uncertainties in more efficient ways. However, it is hoped that the proposed method would open up new direction for resolving uncertainties associated with real life MCDM problems.

Footnotes

Acknowledgments

The authors remain grateful to the learned reviewers for their helpful comments and suggestions to improve the quality of the manuscript.

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Uncertainty evaluations through interval-valued Pythagorean hesitant fuzzy Archimedean aggregation operators in multicriteria decision making

Abstract

Keywords

1. Introduction

2. Literature review

3. Preliminaries

3.1 IVPFS

3.2 PHFS

∙ If S ⁢ ( a ^ 1 ) > S ⁢ ( a ^ 2 ) , then a ^ 1 > a ^ 2 ; ∙ If S ⁢ ( a ^ 1 ) = S ⁢ ( a ^ 2 ) , then (1) If A ⁢ ( a ^ 1 ) > A ⁢ ( a ^ 2 ) , then a ^ 1 > a ^ 2 ; (2) If A ⁢ ( a ^ 1 ) = A ⁢ ( a ^ 2 ) , then a ^ 1 = a ^ 2 . 3.3 At-N&t-CN operations

4.1 IVPHFS

5.1 IVPHF archimedean averaging operators

7.1 Example 1

Table 3 IVPHF decision matrix

8. Conclusions

Footnotes

Acknowledgments

References

Table 3
IVPHF decision matrix