In this part of the work, we propose some IVIF-Yager power operators using Yager’s t norm and s norm [57], and power operators [58].

Definition 3.1

[57]. For any two real numbers \(\eta\) and \(\varsigma ,\) the Yager’s t norm (\(\otimes\)) is defined as.

$$_(\eta ,\mathrm\varsigma )\mathrm=\mathrm\eta \otimes \varsigma \mathrm=\mathrm\mathrm\left(1-\mathrm^+^\right)}^^\!\left/ \!_\right.},\mathrm0\right),\mathrm\varpi \mathrm>\mathrm0.$$

(20)

where \(\varpi \mathrm\to \mathrm\infty ,\) the Yager’s t norm \(_(\eta ,\mathrm\varsigma )\) approaches to \(\mathrm(\eta ,\mathrm\varsigma ).\) Also, if \(\varpi\) is equal to unity, then the Yager’s t norm \(_(\eta ,\mathrm\varsigma )\) becomes Lukasiewicz’ t norm.

Definition 3.2

[57]. For any two real numbers \(\eta\) and \(\varsigma ,\), the Yager’s s norm (or t conorm) (\(\oplus\)) is defined as.

$$_(\eta ,\mathrm\varsigma )\mathrm=\mathrm\eta \oplus \varsigma \mathrm=\mathrm\mathrm\left(^+^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\varpi \mathrm>\mathrm0.$$

(21)

When \(\varpi \mathrm\to \mathrm\infty ,\) the Yager’s s norm \(_(\eta ,\mathrm\varsigma )\) approaches to \(\mathrm(\eta ,\mathrm\varsigma ).\) In addition, if \(\varpi\) is equal to unity, then the Yager’s s norm \(_(\eta ,\mathrm\varsigma )\) becomes Lukasiewicz’ t conorm.

Definition 3.3

[58]. For a set of real numbers \((_,\mathrm_,\mathrm...,\mathrm_),\) the power averaging (PA) operator is given by.

$$PA\left(_,\mathrm_,\mathrm...,\mathrm_\right)=\frac=1}^\left(1\mathrm+\mathrmT\left(_\right)\right)\mathrm_}=1}^\left(1\mathrm+\mathrmT\left(_\right)\right)}.$$

(22)

Definition 3.4

[58]. Let \((_,\mathrm_,\mathrm...,\mathrm_)\) be a set of real numbers. The power geometric (PG) operator is given by.

$$PG\left(_,\mathrm_,\mathrm...,\mathrm_\right)=\prod_=1}^_^+\mathrmT(_))}=1}^(1\mathrm+\mathrmT(_))}}.$$

(23)

Here, \(T(_)\mathrm=\mathrm\sum_i\mathrm=1\\ i\mathrm\ne \mathrmj\end}^Sup\left(_,\mathrm_\right)\) signifies total support of \(_\) from all the values excluding itself and \(Sup(_,\mathrm_)\mathrm=\mathrm1-\mathrm\mathfrak(_,\mathrm_)\) indicates the support measure between \(_\) and \(_,\) where \(Sup(_,\mathrm_)\mathrm\in \mathrm[0,\mathrm1],Sup\left(_,\mathrm_\right)\mathrm=\mathrmSup(_,\mathrm_),Sup(_,\mathrm_)\mathrm\ge \mathrmSup(_,\mathrm_)\) if \(|_\mathrm-\mathrm_|\mathrm\ge \mathrm|_-\mathrm_|.\)

In accordance with the advantages of power operators and Yager’s triangular norms, this study presents some new AOs, which are the IVIF-Yager power weighted averaging (IVIFYPWA), the IVIF-Yager power weighted geometric (IVIFYPWG), the IVIF-Yager power ordered weighted averaging (IVIFYPOWA) and the IVIF-Yager power ordered weighted geometric (IVIFYPOWG) AOs, respectively.

Based on Definition 3.1–3.3, we discuss the following definition under IVIFS context, where ‘Z’ be the set of all IVIFNs.

Definition 3.5.

Suppose \(_=\mathrm\langle [_^,\mathrm_^],[_^,\mathrm_^]\mathrm\rangle \mathrm(i\mathrm=1,\mathrm2,\mathrm...,m)\) be a collection of IVIFNs and \(_\) be weight of an IVIFN \(_,\) whose values lie between 0 and 1 and their sum is equal to unity. The function \(IVIFYPWA\mathrm:\mathrm^\mathrm\to Z\) is an IVIFYPWA AO, satisfying.

$$IVIFYPWA\left(_,\mathrm_,\mathrm...,\mathrm_\right)=\stackrel=1}}\mathrm_\mathrm_.$$

(24)

Here, \(_\mathrm=\mathrm\frac_\mathrm(1+\mathrmT(_))}^_\mathrm\left(1+\mathrmT\left(_\right)\right)}\) represents the significance degree of IVIFN, whose sum of the values is 1. The term \(T\left(_\right)=\sum_i\mathrm=1\\ i\mathrm\ne \mathrmj\end}^_\mathrmSup\left(_,\mathrm_\right)\) and \(Sup(_,\mathrm_)\mathrm=\mathrm1-\mathrmd\left(_,\mathrm_\right)\) is support measure, wherein \(d\left(_,\mathrm_\right)\) is given by Eq. (3).

Theorem 3.1.

The aggregated grade using the IVIFYPWA AO is also an IVIFN and given by.

$$IVIFYPWA(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\stackrel=1}}\mathrm_\mathrm_=\mathrm\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right),\mathrm\varpi \mathrm>\mathrm0,$$

(25)

where \(_\mathrm=\mathrm\frac_\mathrm(1+\mathrmT(_))}^_\mathrm(1+\mathrmT(_))}.\)

Proof.

With the use of principle of mathematical induction, we can prove the theorem.

Step 1. For m = 1 in Eq. (25), we have

$$IVIFYPWA(_)=\mathrm_\mathrm_=\left(\begin\left[\mathrm\left(_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right).$$

Thus, Eq. (22) holds for m = 1.

Step 2. For m = 2 in Eq. (25), we have

$$_\mathrm_\mathrm+\mathrm_\mathrm_=\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right).$$

Therefore,

$$IVIFYPWA(_,\mathrm_)=\mathrm_\mathrm_\mathrm\oplus \mathrm_\mathrm_=\mathrm\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right).$$

Thus, Eq. (22) holds for m = 2.

Step 3. Consider Eq. (22) is true for \(m\mathrm=\mathrms,\) i.e.,

$$IVIFPPWA(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\stackrel}=1}}\mathrm_\mathrm_=\mathrm\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right).$$

Step 4. Now, for m = s + 1, we have

$$IVIFPPWA(_,\mathrm_,\mathrm...,\mathrm_,\mathrm_)=\mathrm\stackrel=1}}\mathrm_\mathrm_\oplus \mathrm_\mathrm_+1}=\mathrm\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right)\oplus \left(\begin\left[\mathrm\left(_+1}\mathrm_s+1}^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(_+1}\mathrm_s+1}^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right).$$

This implies that

$$IVIFPPWA(_,\mathrm_,\mathrm...,\mathrm_,\mathrm_)=\mathrm\stackrel+1}=1}}\mathrm_\mathrm_=\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right).$$

Consequently, Eq. (22) holds for m = s + 1, hence, the proof.

Definition 3.6.

Let \(_=\mathrm\langle [_^,\mathrm_^],[_^,\mathrm_^]\mathrm\rangle \mathrm(i\mathrm=1,\mathrm2,\mathrm...,m)\) be a collection of IVIFNs and \(_\) be weight of an IVIFN \(_,\) whose values lie between 0 and 1 and their sum is equal to unity. The function \(IVIFYPWG\mathrm:\mathrm^\mathrm\to Z\) is an IVIFYPWG AO, satisfying.

$$IVIFYPWG\left(_,\mathrm_,\mathrm...,\mathrm_\right)=\stackrel=1}}\mathrm_^_}.$$

(26)

The term \(T(_)\mathrm=\mathrm\sum_i\mathrm=1\\ i\mathrm\ne \mathrmj\end}^_\mathrmSup(_,\mathrm_)\) and \(Sup(_,\mathrm_)\mathrm=\mathrm1-\mathrmd(_,\mathrm_)\) is support measure, wherein \(d(_,\mathrm_)\) is given by Eq. (3).

Theorem 3.2.

The aggregated grade using the IVIFYPWG AO is also an IVIFN and given by.

$$IVIFYPWG(_,\mathrm_,\mathrm...,\mathrm_)=IVIFYPWG(_,\mathrm_,\mathrm...,\mathrm_)==\mathrm\left(\begin\mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right],\\ \left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right]\end\right),\mathrm\varpi \mathrm>\mathrm0.$$

(27)

Proof.

It is obvious from Theorem 3.1.

Definition 3.7.

Let \(_=\mathrm\langle \left[_^,\mathrm_^\right],\left[_^,\mathrm_^\right]\mathrm\rangle \mathrm\left(i\mathrm=1,\mathrm2,\mathrm...,m\right)\) be a collection of IVIFNs and \(_\) be weight of an IVIFN \(_,\) whose values lie between 0 and 1 and their sum is equal to unity. The function \(IVIFYPOWA\mathrm:\mathrm^\mathrm\to Z\) is an IVIFYPOWA AO, satisfying.

$$IVIFYPOWA\left(_,\mathrm_,\mathrm...,\mathrm_\right)=\mathrm\stackrel=1}}\mathrm_\mathrm_.$$

(28)

where \((\sigma (1),\mathrm\sigma (2),\mathrm...,\mathrm\sigma (m))\) is the permutation of \((i\mathrm=1,\mathrm2,\mathrm...,\mathrmm)\) such that \(_\mathrm\ge \mathrm_,\mathrm\forall \mathrmi.\)

Theorem 3.3.

The aggregated rating using the IVIFYPOWA is an IVIFN and presented as.

$$IVIFYPOWA(_,\mathrm_,\mathrm...,\mathrm_)=\left(\begin\left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\\ \mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right]\end\right),\mathrm\varpi \mathrm>\mathrm0.$$

(29)

Proof.

Follow as Theorem 3.1.

Definition 3.8.

Let \(_=\mathrm\langle [_^,\mathrm_^],[_^,\mathrm_^]\mathrm\rangle \mathrm(i\mathrm=1,\mathrm2,\mathrm...,m)\) be a collection of IVIFNs and \(_\) be weight of an IVIFN \(_,\) whose values lie between 0 and 1 and their sum is equal to unity. The function \(IVIFYPOWG\mathrm:\mathrm^\mathrm\to Z\) is an IVIFYPOWG AO, satisfying.

$$IVIFYPOWG(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\stackrel=1}}\mathrm_^_},$$

(30)

where \((\sigma (1),\mathrm\sigma (2),\mathrm...,\mathrm\sigma (m))\) is the permutation of \((i\mathrm=1,\mathrm2,\mathrm...,\mathrmm)\) such that \(_\mathrm\ge \mathrm_,\mathrm\forall \mathrmi.\)

Theorem 3.4.

The aggregated grade by means of an IVIFYPOWG AO is an IVIFN and is given by.

$$IVIFYPOWG(_,\mathrm_,\mathrm...,\mathrm_)=\left(\begin\mathrm\left[\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right),\mathrm\mathrm\left(\left(1-\mathrm\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.}\right),\mathrm0\right)\right],\\ \left[\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right),\mathrm\mathrm\left(\sum_=1}^_\mathrm_^\right)}^\right)}^^\!\left/ \!_\right.},\mathrm1\right)\right],\end\right),\mathrm\varpi \mathrm>\mathrm0.$$

(31)

Proof.

Follow as Theorem 3.1.

Taking into consideration Theorems 3.1–3.4, we discuss the subsequent properties of the developed IVIFYPWA, IVIFYPWG, IVIFYPOWA, and IVIFYPOWG operators:

1. Idempotency: If all IVIFNs are equal, i.e., \(_\mathrm=\mathrm\gamma ,\mathrm\forall \mathrmi,\) then

\(IVIFYPWA(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\gamma ,\) \(IVIFYPWG(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\gamma ,\)

\(IVIFYPOWA(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\gamma\), and \(IVIFYPOWG(_,\mathrm_,\mathrm...,\mathrm_)=\mathrm\gamma .\)

2. Boundary: If \(^=\mathrm\langle [\mathrm_^,\mathrm\mathrm_^],[\mathrm_^,\mathrm\mathrm\mathrm_^]\mathrm\rangle\) and \(^=\mathrm([\mathrm_^,\mathrm\mathrm_^],\) \([\mathrm_^,\mathrm\mathrm\mathrm_^]),\) then

\(^\le IVIFYPWA(_,\mathrm_,\mathrm...,\mathrm_)\mathrm\le \mathrm^,\) \(^\le IVIFYPWG(_,\mathrm_,\mathrm...,\mathrm_)\mathrm\le \mathrm^,\)

\(^\le IVIFYPOWA(_,\mathrm_,\mathrm...,\mathrm_)\mathrm\le \mathrm^\) and \(^\le IVIFYPOWG(_,\mathrm_,\mathrm...,\mathrm_)\mathrm\le \mathrm^.\)

3. Monotonicity: Suppose \(_=\mathrm\langle [_^,\mathrm_^],[_^,\mathrm_^]\mathrm\rangle\) and \(_}=\mathrm\langle [_^},\mathrm_^}],[_^},\mathrm_^}]\mathrm\rangle\) be the collections of IVIFNs, where \(i\mathrm=1,\mathrm2,\mathrm...,\mathrmm.\) If \(_^}\mathrm\le \mathrm_^,_^}\mathrm\le \mathrm_^,_^}\ge \mathrm_^\) and \(_^}\mathrm\ge \mathrm_^,\) then

\(IVIFYPWA(_},_},\mathrm...,_})\mathrm\le \mathrmIVIFYPWA(_,\mathrm_,\mathrm...,\mathrm_),\)

\(IVIFYPWG(_},_},\mathrm...,\mathrm_})\mathrm\le \mathrmIVIFYPWG(_,\mathrm_,\mathrm...,\mathrm_),\)

\(IVIFYPOWA(_},_},\mathrm...,_})\mathrm\le \mathrmIVIFYPOWA(_,\mathrm_,\mathrm...,\mathrm_),\)

and

\(IVIFYPOWG(_},_},\mathrm...,\mathrm_})\mathrm\le \mathrmIVIFYPOWG(_,\mathrm_,\mathrm...,\mathrm_),\)