Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator and their application to group decision making

Abstract

The objective of the present work is divided into two folds. Firstly, interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator has been introduced along with their several properties, namely idempotency, boundedness and monotonicity. Secondly, we apply the proposed operator to deal with multi-attribute group decision-making problem under Pythagorean fuzzy information. For this, we construct an algorithm for multi-attribute group decision making. At the last, we construct a numerical example for multi-attribute group decision making. The main advantage of using the proposed operator is that this operator provides more accurate and precise results is compared to the existing methods.

Introduction

Multi-criteria group decision making is one of the successful processes for finding the optimal alternative from all the feasible alternatives according to some criteria or attributes. Traditionally, it has been generally assumed that all the information that access the alternative in terms of criteria and their corresponding weights are expressed in the form of crisp numbers. But most of the decisions in the real-life situations are taken in the environment where the goals and constraints are generally imprecise or vague in nature. In order to handle the uncertainties and fuzziness intuitionistic fuzzy set [1] theory is one of the successful extensions of the fuzzy set theory [2], which is characterized by the degree of membership and degree of non-membership has been presented. After the successful and positive applications of intuitionistic fuzzy set, aggregation operators become more interesting topic for research. Thus, many scholars in [3,4,5,6,7,8,9,10,11,12,13,14,15,16] developed several aggregation operators for group decision making using intuitionistic fuzzy information.

However, there are many cases where the decision maker may provide the degree of membership and nonmembership of a particular attribute in such a way that their sum is greater than one. To solve these types of problems, Yager [17, 18] introduced the concept of another set called Pythagorean fuzzy set. Pythagorean fuzzy set is more powerful tool to solve uncertain problems. Like intuitionistic fuzzy aggregation operators, Pythagorean fuzzy aggregation operators are also become an interesting and important area for research, after the advent of Pythagorean fuzzy set theory. Several researchers in [19,20,21,22,23,24,25,26,27,28] introduced many aggregation operators for decision using Pythagorean fuzzy information.

But, in some real decision-making problems, due to insufficiency in available information, it may be difficult for decision makers to exactly quantify their opinions with a crisp number, but they can be represented by an interval number within [0, 1]. Therefore, it is so important to present the idea of interval-valued Pythagorean fuzzy sets, which permit the membership degrees and non- membership degrees to a given set to have an interval value. Thus in [29] Peng and Yang introduced the concept of interval-valued Pythagorean fuzzy set. Rahman et al. [30,31,32,33] introduced many aggregation operators using interval-valued Pythagorean fuzzy numbers and applied them to multi-attribute group decision making.

Thus, keeping the advantages of these operators, in this paper, we introduce the notion of interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging operator. Moreover, we introduce some of their basic properties such as idempotency, boundedness and monotonicity. This motivation comes from [32], in which the authors introduced the notion of IVPFEWA operator and IVPFEOWA operator and applied them to group decision making. But in this paper we introduce the notion of IVPFEHWA operator, which is the generalization of the above mention operators.

The remainder of this paper is structured as follows. In Sect. "Preliminaries", we give some basic definitions and results which will be used in our later sections. In Sect. "Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator", we introduce the notion of interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging operator. In Sect. "An approach to multiple attribute group decision-making problems based on intervalvalued Pythagorean fuzzy information", we apply the proposed operator to multi-attribute group decision-making problem with Pythagorean fuzzy information. In Sect. "Illustrative example", we develop a numerical example. In Sect. "Conclusion", we have conclusion.

Preliminaries

Definition 1

[17, 18] Let K be a fixed set, then a Pythagorean fuzzy set can be defined as:

$$\begin{aligned} P=\{\langle {k,u_P ( k ),v_P (k)} \rangle |k\in K\}, \end{aligned}$$

(1)

where $u_P (k):P\rightarrow [{0,1}],v_P (k):K\rightarrow [{0,1}]$ are called membership function and non-membership function, respectively, with condition $0\le ({u_P (k)} )^{2}+({v_P (k)})^{2}\le 1,$ for all $k\in K$.

Let

$$\begin{aligned} \pi _P (k)=\sqrt{1-u_P^2 (k)-v_P^2 (k)}. \end{aligned}$$

(2)

Then, it is called the Pythagorean fuzzy index of $k\in K$, with condition $0\le \pi _P (k)\le 1,$ for every $k\in K$.

Definition 2

[29] Let K be a fixed set, then an interval-valued Pythagorean fuzzy set can be defined as:

$$\begin{aligned} I=\{{\langle {k,u_I (k),v_I (k)}\rangle |k\in K}\}, \end{aligned}$$

(3)

where

$$\begin{aligned} u_I ( k)=[ {u_I^a ( k),u_I^b ( k)}]\subset [ {0,1} ], \end{aligned}$$

(4)

and

$$\begin{aligned} v_I ( k)=[ {v_I^a ( k),v_I^b ( k)}]\subset [ {0,1}]. \end{aligned}$$

(5)

Also

$$\begin{aligned} u_I^a ( k)= & {} \inf ( {u_I ( k)}) ,\end{aligned}$$

(6)

$$\begin{aligned} u_I^b ( k)= & {} {\sup }( {u_I ( k)}), \end{aligned}$$

(7)

$$\begin{aligned} v_I^a ( k)= & {} {\inf }( {v_I ( k)}), \end{aligned}$$

(8)

$$\begin{aligned} v_I^b ( k)= & {} {\sup }( {v_I ( k)}), \end{aligned}$$

(9)

and

$$\begin{aligned} 0\le \left( {u_I^b ( k)}\right) ^{2}+\left( {v_I^b ( k)}\right) ^{2}\le 1. \end{aligned}$$

(10)

If

$$\begin{aligned} \pi _I ( k)=\left[ {\pi _I^a ( k),\pi _I^b ( k)}\right] , \hbox { for all }k\in K. \end{aligned}$$

(11)

Then, it is called the interval-valued Pythagorean fuzzy index of k to I, where

$$\begin{aligned} \pi _I^a ( k)=\sqrt{1-\left( {u_I^b ( k)}\right) ^{2}-\left( {v_I^b ( k)}\right) ^{2}}, \end{aligned}$$

(12)

and

$$\begin{aligned} \pi _I^b ( k)=\sqrt{1-\left( {u_I^a ( k)}\right) ^{2}-\left( {v_I^a ( k)}\right) ^{2}}. \end{aligned}$$

(13)

Definition 3

[29] Let $\lambda =( {[ {u_\lambda ,v_\lambda }],[ {x_\lambda ,y_\lambda }]})$ be an IVPFN, then the score function and accuracy function of $\lambda$ can be defined as follows, respectively:

$$\begin{aligned} s(\lambda )=\frac{1}{2}\left[ {\left( {u_\lambda }\right) ^{2} +\left( {v_\lambda }\right) ^{2}-\left( {x_\lambda }\right) ^{2} -\left( {y_\lambda }\right) ^{2}}\right] , \end{aligned}$$

(14)

and

$$\begin{aligned} h( \lambda )=\frac{1}{2}\left[ {\left( {u_\lambda }\right) ^{2} +\left( {v_\lambda }\right) ^{2}+\left( {x_\lambda }\right) ^{2}\ +\left( {y_\lambda }\right) ^{2}}\right] . \end{aligned}$$

(15)

If $\lambda _1$ and $\lambda _2$ are two IVPFNs, then

• 1. If $s(\lambda _1 )\prec s(\lambda _2 ),$ then $\lambda _1 \prec \lambda _2$.

• 2. If $s( {\lambda _1 })=s( {\lambda _2 }),$ then we have the following three conditions.

  • 1) If $h( {\lambda _1 })=h( {\lambda _2 }),$ then $\lambda _1 =\lambda _2$.

  • 2) If $h( {\lambda _1 })\prec h( {\lambda _2 }),$ then $\lambda _1 \prec \lambda _2$.

  • 3) If $h(\lambda _1 )\succ h(\lambda _2 ),$ then $\lambda _1 \succ \lambda _2$.

Definition 4

[32] Let $\lambda =( {[ {u,v}],[ {x,y}]}),\lambda _1 =( [ {u_1 ,v_1 }],[ {x_1 ,y_1 }]),\lambda _2 =( {[ {u_2 ,v_2 }],[ {x_2 ,y_2 }]})$ are three IVPFNs, and $\delta \succ 0$, then some Einstein operations for $\lambda ,\lambda _1 ,\lambda _2$ can be defined as follows:

• 1.
  $$\begin{aligned} \lambda _1 \oplus _\varepsilon \lambda _2= & {} \left( \left[ {\frac{\sqrt{u_1^2 +u_2^2 }}{\sqrt{1+u_1^2 u_2^2 }},\frac{\sqrt{v_1^2 +v_2^2 }}{\sqrt{1+v_1^2 v_2^2 }}}\right] \right. ,\\&\left. \left[ \frac{x_1 x_2 }{\sqrt{1+\left( {1-x_1^2 }\right) \left( {1-x_2^2 }\right) }},\right. \right. \\&\left. \left. \frac{y_1 y_2 }{\sqrt{1+\left( {1-y_1^2 }\right) \left( {1-y_2^2 }\right) }}\right] \right) \end{aligned}$$
• 2.
  $$\begin{aligned} \lambda _1 \otimes _\varepsilon \lambda _2= & {} \left( \left[ \frac{u_1 u_2 }{\sqrt{1+\left( {1-u_1^2 }\right) \left( {1-u_2^2 }\right) }}\right. \right. ,\\&\left. \left. \frac{v_1 v_2 }{\sqrt{1+\left( {1-v_1^2 }\right) \left( {1-v_2^2 }\right) }}\right] \right. ,\\&\left. \left[ {\frac{\sqrt{x_1^2 +x_2^2 }}{\sqrt{1+x_1^2 x_2^2 }},\frac{\sqrt{y_1^2 +y_2^2 }}{\sqrt{1+y_1^2 y_2^2 }}}\right] \right) \end{aligned}$$
• 3.
  $$\begin{aligned} \delta \lambda= & {} \left( \left[ \frac{\sqrt{( {1+u^{2}})^{\delta }-( {1-u^{2}})^{\delta }}}{\sqrt{( {1+u^{2}})^{\delta }+( {1-u^{2}})^{\delta }}}\right. \right. ,\\&\left. \left. \frac{\sqrt{( {1+v^{2}})^{\delta }-( {1-v^{2}})^{\delta }}}{\sqrt{( {1+v^{2}})^{\delta }+( {1-v^{2}})^{\delta }}}\right] \right. ,\\&\left. \left[ {\frac{\sqrt{2( {x^{2}})^{\delta }}}{\sqrt{( {2-x^{2}})^{\delta }+( {x^{2}})^{\delta }}},\frac{\sqrt{2( {y^{2}})^{\delta }}}{\sqrt{( {2-y^{2}})^{\delta }+( {y^{2}})^{\delta }}}}\right] \right) \end{aligned}$$
• 4.
  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2( {u^{2}})^{\delta }}}{\sqrt{( {2-u^{2}})^{\delta }+( {u^{2}})^{\delta }}},\frac{\sqrt{2( {v^{2}})^{\delta }}}{\sqrt{( {2-v^{2}})^{\delta }+( {v^{2}})^{\delta }}}}\right] \right. ,\\&\left. \left[ \frac{\sqrt{( {1+x^{2}})^{\delta }-( {1-x^{2}})^{\delta }}}{\sqrt{( {1+x^{2}})^{\delta }+( {1-x^{2}})^{\delta }}},\right. \right. \\&\left. \left. \frac{\sqrt{( {1+y^{2}})^{\delta }-( {1-y^{2}})^{\delta }}}{\sqrt{( {1+y^{2}})^{\delta }+( {1-y^{2}})^{\delta }}}\right] \right) \end{aligned}$$

Definition 5

[32] Let $\lambda _j =( {[ {u_j ,v_j }],[ {x_j ,y_j }]})( j=1,2,3,...,n)$ be the collection of IVPFVs, then IVPFEWA operator can be defined as:

$$\begin{aligned}&\hbox {IVPFEWA}_w ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,..., \lambda _n }) \nonumber \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\lambda _j }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-u_{\lambda _j }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\lambda _j }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\lambda _j }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\lambda _j }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-v_{\lambda _j }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\lambda _j }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-v_{\lambda _j }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^n \left( {x_{\lambda _j }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-x_{\lambda _j }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {x_{\lambda _j }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^n \left( {y_{\lambda _j }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-y_{\lambda _j }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {y_{\lambda _j }^2 }\right) ^{w_j }}}}\right] } \\ \end{array} }}\right) ,\nonumber \\ \end{aligned}$$

(16)

where $w=( w_1 ,w_2 ,w_3 ,...,w_n )^{T}$ is the weighted vector of $\lambda _j ( j=1,2,3,...,n)$, such that $w_j \in [ {0,1}]$ and $\sum \nolimits _{j=1}^n w_j =1.$

Definition 6

[32] Let $\lambda _j ( {j=1,2,3,...,n})$ be a collection of IVPFVs, then IVPFEOWA operator can be defined as:

$$\begin{aligned}&\hbox {IVPFEOWA}_w ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n }) \nonumber \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-u_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-v_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-v_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j}}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^n \left( {x_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-x_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {x_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j}}}, \frac{\sqrt{2\prod \limits _{j=1}^n \left( {y_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-y_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {y_{\lambda _{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] } \\ \end{array} }}\right) ,\nonumber \\ \end{aligned}$$

(17)

where $( {\sigma ( 1),\sigma ( 2),\sigma ( 3),...,\sigma ( n)})$ is a permutation of $( 1,2,3,...,n)$ such that $\sigma ( j)\le \sigma ( {j-1})$ for all jand $w=( {w_1 ,w_2 ,w_3 ,...,w_n })^{T}$ is the weighted vector of $\lambda _{\sigma ( j)} (j=1,2,3,...,n)$ such that $w_j \in [ {0,1}]$ and $\sum \nolimits _{j=1}^n w_j =1.$

Interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator

In this section, we introduce the notion of interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator. We also discuss some desirable properties such as idempotency, boundedness and monotonicity.

Definition 7

An interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging operator of dimension n is a mapping $\hbox {IVPFEHWA}:\Theta ^{n}\rightarrow \Theta ,$which has associated vector$w=( {w_1 ,w_2 ,w_3 ,...,w_n })^{T}$, such that $w_j \in [ {0,1}]$ and $\sum \nolimits _{j=1}^n w_j =1.$ Furthermore

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n }) \nonumber \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^n \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^n \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\!\right] } \\ \end{array} }}\!\right) \!,\nonumber \\ \end{aligned}$$

(18)

where $\dot{\lambda }_{\sigma ( j)}$ is the $j\mathrm{{th}}$ largest of the weighted interval-valued Pythagorean fuzzy values, $\dot{\lambda }_{\sigma ( j)} ( {\dot{\lambda }_{\sigma ( j)} =n\omega _j \lambda _j }). \quad \omega =( {\omega _1 ,\omega _2 ,\omega _3,...,\omega _n })^{T}$ is the weighted vector of $\lambda _j (j=1,2,3,...,n)$ such that $\omega _j \in [ {0,1}]$, $\sum \nolimits _{j=1}^n \omega _j =1,$ and n is the balancing coefficient, which plays a role of balance. If the vector $w=( {w_1 ,w_2 ,w_3 ,...,w_n })^{T}$ approaches to $\left( {\frac{1}{n}, \frac{1}{n},\frac{1}{n},...,\frac{1}{n}}\right) ^{T},$ then the vector $(n\omega _1 \lambda _1 ,n\omega _2 \lambda _2 ,...,n\omega _n \lambda _n )^{T}$ approaches to $( \lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n )^{T}.$

Theorem 1

Let $\lambda ,\lambda _1 ,\lambda _2$ be the three interval-valued Pythagorean fuzzy numbers and $\delta ,\delta _1 ,\delta _2 \succ 0,$ then the following conditions always hold:

• 1. $\lambda _1 \oplus _\varepsilon \lambda _2 =\lambda _2 \oplus _\varepsilon \lambda _1$,

• 2. $\lambda _1 \otimes _\varepsilon \lambda _2 =\lambda _2 \otimes _\varepsilon \lambda _1$,

• 3. $\delta ( {\lambda _1 \oplus _\varepsilon \lambda _2 })=\delta \lambda _1 \oplus _\varepsilon \delta \lambda _2$,

• 4. $( {\lambda _1 \otimes _\varepsilon \lambda _2 })^{\delta }=( {\lambda _1 })^{\delta }\otimes _\varepsilon ( {\lambda _2 })^{\delta }$,

• 5. $\delta _1 ( \lambda )\oplus _\varepsilon \delta _2 ( \lambda )=( {\delta _1 \oplus _\varepsilon \delta _2 })\lambda$,

• 6. $( \lambda )^{\delta _1 }\otimes _\varepsilon ( \lambda )^{\delta _2 }=\lambda ^{( {\delta _1 \otimes _\varepsilon \delta _2 })}$.

Proof

The proof is trivial, so it is omitted here.

Theorem 2

Let $\lambda _j =( {[ {u_j ,v_j }],[ {x_j ,y_j }]})( {j=1,2,3,...,n})$ be a collection of IVPFVs, then their aggregated value using the IVPFEHWA operator is also an IVPFV, and

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n }) \nonumber \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^n \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^n \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] } \\ \end{array} }}\!\right) \!,\nonumber \\ \end{aligned}$$

(19)

where $\dot{\lambda }_{\sigma ( j)}$ is the $j\mathrm{{th}}$ largest of the weighted interval-valued Pythagorean fuzzy values, $\dot{\lambda }_{\sigma ( j)} ( {\dot{\lambda }_{\sigma ( j)} =n\omega _j \lambda _j }), \quad w=( w_1 ,w_2 ,w_2 ,\ldots ,w_n )^{T}$ is the weighted vector of IVPFEHWA, such that $w_j \in [ {0,1}]$, $\sum \nolimits _{j=1}^n w_j =1. \quad \omega =( {\omega _1 ,\omega _2 ,\omega _2 ,\ldots ,\omega _n })^{T}$ is the weighted vector of $\lambda _j ( j=1,2,3,\ldots ,n)$ such that $\omega _j \in [ {0,1}]$, $\sum \nolimits _{j=1}^n \omega _j =1,$ and n is the balancing coefficient, which plays a role of balance. If the vector $w=( w_1 ,w_2 ,w_2 ,\ldots ,w_n )^{T}$ approaches $\left( {\frac{1}{n},\frac{1}{n},\frac{1}{n},\ldots ,\frac{1}{n}}\right) ^{T},$ then the vector $( nw\omega \lambda _1 , n\omega _2 \lambda _2 ,\ldots ,n\omega _n \lambda _n )^{T}$ approaches$( {\lambda _1 ,\lambda _2 ,\lambda _3 ,\ldots ,\lambda _n })^{T}.$

Proof

We can prove this theorem by mathematical induction on n.

For $n=2$

$$\begin{aligned} w_1 \dot{\lambda }_1= & {} \left( \left[ \frac{\sqrt{\left( {1+u_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }-\left( {1-u_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}{\sqrt{\left( {1+u_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }+\left( {1-u_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}\right. \right. ,\\&\left. \frac{\sqrt{\left( {1+v_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }-\left( {1-v_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}{\sqrt{\left( {1+v_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }+\left( {1-v_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}\right] ,\\&\left[ \frac{\sqrt{2\left( {x_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}{\sqrt{\left( {2-x_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }+\left( {x_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}},\right. \nonumber \\&\left. \left. \frac{\sqrt{2\left( {y_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}{\sqrt{\left( {2-y_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }+\left( {y_{\dot{\lambda }_1 }^2 }\right) ^{w_1 }}}\right] \right) \end{aligned}$$

and

$$\begin{aligned} w_2 \dot{\lambda }_2= & {} \left( \left[ \frac{\sqrt{\left( {1+u_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }-\left( {1-u_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}}{\sqrt{\left( {1+u_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }+\left( {1-u_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}}\right. \right. ,\\&\left. \frac{\sqrt{\left( {1+v_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }-\left( {1-v_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}}{\sqrt{\left( {1+v_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }+\left( {1-v_{\dot{\lambda }_2 }^2}\right) ^{w_2 }}}\right] ,\\&\left[ \frac{\sqrt{2\left( {x_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}}{\sqrt{\left( {2-x_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }+\left( {x_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}},\right. \nonumber \\&\left. \left. \frac{\sqrt{2\left( {y_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}}{\sqrt{\left( {2-y_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }+\left( {y_{\dot{\lambda }_2 }^2 }\right) ^{w_2 }}}\right] \right) \end{aligned}$$

Then

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 }) \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^2 \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^2 \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^2 \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^2 \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^2 \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^2 \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^2 \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^2 \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^2 \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^2 \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^2 \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^2 \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^2 \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^2 \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] } \\ \end{array} }}\right) . \\ \end{aligned}$$

Thus, the result is true for $n = 2$, now we assume that Eq. (19) holds for $n = k$. Thus

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _k }) \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^k \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^k \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^k \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^k \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^k \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^k \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] } \\ \end{array} }}\right) . \\ \end{aligned}$$

If Eq. (19) holds for $n=k$, then we show that Eq. (19) holds for $n=k+1.$ Thus

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _{k+1} }) \nonumber \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^k \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^k \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^k \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^k \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^k \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^k \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^k \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}}\!\right] } \\ \end{array} }}\!\right) \nonumber \\&\qquad \oplus _\varepsilon \left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\left( {1+u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }-\left( {1-u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}}{\sqrt{\left( {1+u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {1-u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}},\frac{\sqrt{\left( {1+v_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }-\left( {1-v_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}}{\sqrt{\left( {1+v_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {1-v_{\dot{\lambda }_{k+1} }^2 }\right) ^{wk+1}}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\left( {x_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}}{\sqrt{\left( {2-x_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {x_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}},\frac{\sqrt{2\left( {y_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}}{\sqrt{\left( {2-y_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {y_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }}}}\right] } \\ \end{array} }}\right) .\nonumber \\ \end{aligned}$$

(20)

Let

$$\begin{aligned} t_1= & {} \sqrt{\prod \limits _{j=1}^k \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^k \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ t_2= & {} \sqrt{\prod \limits _{j=1}^k \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ p_1= & {} \sqrt{\prod \limits _{j=1}^k \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^k \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ p_2= & {} \sqrt{\prod \limits _{j=1}^k \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ m_1= & {} \sqrt{\left( {1+u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }-\left( {1-u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }} \\ m_2= & {} \sqrt{\left( {1+u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {1-u_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }} \\ a_1= & {} \sqrt{\left( {1+v_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }-\left( {1-v_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }} \\ a_2= & {} \sqrt{\left( {1+v_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {1-v_{\dot{\lambda }_{k+1} }^2 }\right) ^{wk+1}} \\ r_2= & {} \sqrt{\prod \limits _{j=1}^k \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ r_1= & {} \sqrt{2\prod \limits _{j=1}^k \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }},s_1 =\sqrt{2\prod \limits _{j=1}^k \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ s_2= & {} \sqrt{\prod \limits _{j=1}^k \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^k \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }} \\ b_2= & {} \sqrt{\left( {2-x_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {x_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }} \\ b_1= & {} \sqrt{2\left( {x_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }},c_1 =\sqrt{2\left( {y_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }} \\ c_2= & {} \sqrt{\left( {2-y_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }+\left( {y_{\dot{\lambda }_{k+1} }^2 }\right) ^{w_{k+1} }} \\ \end{aligned}$$

Now putting these values in Eq. (20), we have

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _{k+1} }) \nonumber \\&\quad =\left( {\left[ {\frac{t_1 }{t_2 },\frac{p_1 }{p_2 }}\right] , \left[ {\frac{r_1 }{r_2},\frac{s_1 }{s_2 }}\right] }\right) \oplus _\varepsilon \left( {\left[ {\frac{m_1 }{m_2},\frac{a_1 }{a_2 }}\right] ,\left[ {\frac{b_1 }{b_2 },\frac{c_1 }{c_2 }}\right] }\right) \nonumber \\&\quad =\left( \left[ {\frac{\sqrt{\left( {\frac{t_1 }{t_2 }}\right) ^{2}+\left( {\frac{m_1 }{m_2 }}\right) ^{2}}}{\sqrt{1+\left( {\frac{t_1 }{t_2 }}\right) ^{2}\left( {\frac{m_1 }{m_2 }}\right) ^{2}}},\frac{\sqrt{\left( {\frac{p_1 }{p_2 }}\right) ^{2}+\left( {\frac{a_1 }{a_2 }}\right) ^{2}}}{\sqrt{1+\left( {\frac{p_1 }{p_2 }}\right) ^{2}\left( {\frac{a_1 }{a_2 }}\right) ^{2}}}}\right] ,\right. \nonumber \\&\qquad \left. \left[ \frac{\left( {\frac{r_1 }{r_2 }}\right) \left( {\frac{b_1 }{b_2 }}\right) }{\sqrt{1+\left( {1-\left( {\frac{r_1 }{r_2 }}\right) ^{2}}\right) \left( {1-\left( {\frac{b_1 }{b_2 }}\right) ^{2}}\right) }}\right. \right. ,\nonumber \\&\qquad \left. \left. \frac{\left( {\frac{s_1 }{s_2 }}\right) \left( {\frac{c_1 }{c_2 }}\right) }{\sqrt{1+\left( {1-\left( {\frac{s_1 }{s_2 }}\right) }\right) ^{2}\left( {1-\left( {\frac{c_1 }{c_2 }}\right) }\right) ^{2}}}\right] \right) \nonumber \\&\quad =\left( \left[ {\frac{\sqrt{\left( {t_1 m_2 }\right) ^{2}+\left( {t_2 m_1 }\right) ^{2}}}{\sqrt{\left( {t_2 m_2 }\right) ^{2}+\left( {t_1 m_1 }\right) ^{2}}},\frac{\sqrt{\left( {p_1 a_2 }\right) ^{2}+\left( {a_1 p_2 }\right) ^{2}}}{\sqrt{\left( {p_2 a_2 }\right) ^{2}+\left( {p_1 a_1 } \right) ^{2}}}}\right] ,\right. \nonumber \\&\quad \left[ \frac{r_1 b_1 }{\sqrt{2r_2^2 b_2^2 +r_1^2 b_1^2 -r_2^2 b_1^2 -r_1^2 b_2^2 }},\right. \nonumber \\&\qquad \left. \left. \frac{s_1 c_1 }{\sqrt{2s_2^2 c_2^2 +s_1^2 c_1^2 -s_2^2 c_1^2 -s_1^2 c_2^2 }}\right] \right) . \end{aligned}$$

(21)

Again putting the values of $( {t_1 m_2 })^{2}+( {t_2 m_1 })^{2},( {t_2 m_2 })^{2}+( {t_1 m_1 })^{2},( {p_1 a_2 })^{2}+( {a_1 p_2 })^{2},( {p_2 a_2 })^{2}+( {p_1 a_1 })^{2},r_1 b_1$, $2r_2^2 b_2^2 +r_1^2 b_1^2 -r_2^2 b_1^2 -r_1^2 b_2^2 ,s_1 c_1 ,2s_2^2 c_2^2 +s_1^2 c_1^2 -s_2^2 c_1^2 -s_1^2 c_2^2 ,$ in Eq. (21), then

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _{k+1} }) \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^{_{k+1} } \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_{_j } }-\prod \limits _{j=1}^{_{k+1} } \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^{_{k+1} } \left( {1+u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^{_{k+1} } \left( {1-u_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^{_{k+1} } \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }-\prod \limits _{j=1}^{_{k+1} } \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^{_{k+1} } \left( {1+v_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^{_{k+1} } \left( {1-v_{\dot{\lambda }_{\sigma (j)} }^2}\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^{_{k+1} } \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^{_{k+1} } \left( {2-x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^{_{k+1} } \left( {x_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^{_{k+1} } \left( {y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^{_{k+1} } \left( {2-y_{\dot{\lambda }_{\sigma (j)} }^2 }\right) ^{w_j }+\prod \limits _{j=1}^{_{k+1} } \left( {y_{\dot{\lambda }_{\sigma (j)} }^2}\right) ^{w_j }}}}\right] } \\ \end{array} }}\right) . \\ \end{aligned}$$

Hence, Eq. (19) holds for $n=k+1$. Thus, Eq. (19) holds for all n.

Remark 1

In the following, let us look $\delta \lambda$ and $\lambda ^{\delta }$ some special cases of $\delta$ and $\lambda$.

• 1. If $\lambda =( {[ {u,v}],[ {x,y}]})=( {[ {1,1}],[ {0,0}]})$ i. e,. $u=v=1$ and $u=v=1$, then

  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-u^{2}}\right) ^{\delta }+\left( {u^{2}}\right) ^{\delta }}},\frac{\sqrt{2\left( {v^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-v^{2}}\right) ^{\delta }+\left( {v^{2}}\right) ^{\delta }}}}\right] \right. ,\\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{\delta }-\left( {1-x^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+x^{2}}\right) ^{\delta }+\left( {1-x^{2}}\right) ^{\delta }}}\right. ,\\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{\delta }-\left( {1-y^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+y^{2}}\right) ^{\delta } +\left( {1-y^{2}}\right) ^{\delta }}}\right] \right) \\= & {} \left( \left[ {\frac{\sqrt{2(1)^{\delta }}}{\sqrt{({2-1})^{\delta }+(1)^{\delta }}}, \frac{\sqrt{2(1)^{\delta }}}{\sqrt{({2-1})^{\delta } +(1)^{\delta }}}}\right] \right. ,\\&\left. \left[ {\frac{\sqrt{\left( {1+0}\right) ^{\delta }-({1-0})^{\delta }}}{\sqrt{\left( {1+0}\right) ^{\delta }+({1-0})^{\delta }}},\frac{\sqrt{\left( {1+0}\right) ^{\delta }-({1-0})^{\delta }}}{\sqrt{\left( {1+0}\right) ^{\delta } +({1-0})^{\delta }}}}\right] \right) \\= & {} ( {[ {1,1}],[ {0,0}]}). \end{aligned}$$

  Thus $\lambda ^{\delta }=( {[ {1,1}],[ {0,0}]})$ and $\delta \lambda =( {[ {0,0}],[ {1,1}]}).$

• 2. If $\lambda =( {[ {u,v}],[ {x,y}]})=( {[ {0,0}],[ {1,1}]})$ i. e,. $u=v=0$ and $x=y=1,$ then

  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-u^{2}}\right) ^{\delta }+\left( {u^{2}}\right) ^{\delta }}},\frac{\sqrt{2\left( {v^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-v^{2}}\right) ^{\delta }+\left( {v^{2}}\right) ^{\delta }}}}\right] \right. ,\\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{\delta }-\left( {1-x^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+x^{2}}\right) ^{\delta }+\left( {1-x^{2}}\right) ^{\delta }}},\right. \\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{\delta }-\left( {1-y^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+y^{2}}\right) ^{\delta } +\left( {1-y^{2}}\right) ^{\delta }}}\right] \right) \\= & {} \left( \left[ {\frac{\sqrt{2(0)^{\delta }}}{\sqrt{({2-0})^{\delta }+(0)^{\delta }}},\frac{\sqrt{2(0)^{\delta }}}{\sqrt{({2-0})^{\delta }+(0)^{\delta }}}}\right] \right. ,\\&\left. \left[ {\frac{\sqrt{({1+1})^{\delta }-({1-1})^{\delta }}}{\sqrt{({1+1})^{\delta }+({1-1})^{\delta }}},\frac{\sqrt{({1+1})^{\delta }-({1-1})^{\delta }}}{\sqrt{({1+1})^{\delta } +({1-1})^{\delta }}}}\right] \right) \\= & {} ( {[ {0,0}],[ {1,1}]}). \end{aligned}$$

  Thus $\lambda ^{\delta }=( {[ {0,0}],[ {1,1}]})$ and $\delta \lambda =( {[ {1,1}],[ {0,0}]}).$

• 3. If $\lambda =( {[ {u,v}],[ {x,y}]})=( {[ {0,0}],[ {0,0}]})$ i. e,. $u=v=0$ and $x=y=0,$ then

  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-u^{2}}\right) ^{\delta }+\left( {u^{2}}\right) ^{\delta }}},\frac{\sqrt{2\left( {v^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-v^{2}}\right) ^{\delta }+\left( {v^{2}}\right) ^{\delta }}}}\right] \right. ,\\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{\delta }-\left( {1-x^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+x^{2}}\right) ^{\delta }+\left( {1-x^{2}}\right) ^{\delta }}},\right. \\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{\delta }-\left( {1-y^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+y^{2}}\right) ^{\delta } +\left( {1-y^{2}}\right) ^{\delta }}}\right] \right) \\= & {} \left( \left[ {\frac{\sqrt{2( 0)^{\delta }}}{\sqrt{( {2-0})^{\delta }+( 0)^{\delta }}},\frac{\sqrt{2( 0)^{\delta }}}{\sqrt{( {2-0})^{\delta }+( 0)^{\delta }}}}\right] \right. ,\\&\left. \left[ {\frac{\sqrt{( {1+0})^{\delta }-( {1-0})^{\delta }}}{\sqrt{( {1+0})^{\delta }+( {1-0})^{\delta }}},\frac{\sqrt{( {1+0})^{\delta }-( {1-0})^{\delta }}}{\sqrt{( {1+0})^{\delta }+({1-0})^{\delta }}}}\right] \right) \\= & {} ( {[ {0,0}],[ {0,0}]}). \end{aligned}$$

  Thus $\lambda ^{\delta }=( {[ {0,0}],[ {0,0}]})$ and $\delta \lambda =( {[ {0,0}],[ {0,0}]}).$

• 4. If $\delta \rightarrow 0$ and $0\le u,v,x,y\le 1,$ then

  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-u^{2}}\right) ^{\delta }+\left( {u^{2}}\right) ^{\delta }}},\frac{\sqrt{2\left( {v^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-v^{2}}\right) ^{\delta }+\left( {v^{2}}\right) ^{\delta }}}}\right] ,\right. \\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{\delta }-\left( {1-x^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+x^{2}}\right) ^{\delta }+\left( {1-x^{2}}\right) ^{\delta }}},\right. \\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{\delta }-\left( {1-y^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+y^{2}}\right) ^{\delta } +\left( {1-y^{2}}\right) ^{\delta }}}\right] \right) \\= & {} \left( \left[ {\frac{\sqrt{2( {u^{2}})^{0}}}{\sqrt{( {2-u^{2}})^{0}+( {u^{2}})^{0}}},\frac{\sqrt{2( {v^{2}})^{0}}}{\sqrt{( {2-v^{2}})^{0}+( {v^{2}})^{0}}}}\right] ,\right. \\&\left[ \frac{\sqrt{( {1+x^{2}})^{0}-( {1-x_1^2 })^{0}}}{\sqrt{( {1+x^{2}})^{0}+( {1-x^{2}})^{0}}},\right. \nonumber \\&\left. \left. \frac{\sqrt{( {1+y^{2}})^{0}-( {1-y^{2}})^{0}}}{\sqrt{({1+y^{2}})^{0} +( {1-y^{2}})^{0}}}\right] \right) \\= & {} ( {[ {1,1}],[ {0,0}]}). \end{aligned}$$

  Thus $\lambda ^{\delta }=( {[ {1,1}],[ {0,0}]})$ and $\delta \lambda =( {[ {0,0}],[ {1,1}]}).$

• 5. If $\delta \rightarrow +\infty$ and $0\le u,v,x,y\le 1,$ then

  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-u^{2}}\right) ^{\delta }+\left( {u^{2}}\right) ^{\delta }}},\frac{\sqrt{2\left( {v^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-v^{2}}\right) ^{\delta }+\left( {v^{2}}\right) ^{\delta }}}}\right] ,\right. \\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{\delta }-\left( {1-x^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+x^{2}}\right) ^{\delta }+\left( {1-x^{2}}\right) ^{\delta }}},\right. \\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{\delta }-\left( {1-y^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+y^{2}}\right) ^{\delta } +\left( {1-y^{2}}\right) ^{\delta }}}\right] \right) \\= & {} \left( \left[ {\frac{\sqrt{2( {u^{2}})^{\infty }}}{\sqrt{( {2-u^{2}})^{\infty }+( {u^{2}})^{\infty }}},\frac{\sqrt{2( {v^{2}})^{\infty }}}{\sqrt{( {2-v^{2}})^{\infty }+( {u^{2}})^{\infty }}}}\right] ,\right. \\&\left[ \frac{\sqrt{( {1+x^{2}})^{\infty }-( {1-x^{2}})^{\infty }}}{\sqrt{( {1+x^{2}})^{\infty }+( {1-x^{2}})^{\infty }}},\right. \nonumber \\&\left. \left. \frac{\sqrt{( {1+y^{2}})^{\infty }-( {1-y^{2}})^{\infty }}}{\sqrt{( {1+y^{2}})^{\infty } +({1-y^{2}})^{\infty }}}\right] \right) =({[{[{0,0}],1,1}]}). \end{aligned}$$

  Thus, $\lambda ^{\delta }=( {[ {0,0}],[ {1,1}]})$ and $\delta \lambda =( {[ {1,1}],[ {0,0}]}).$

• 6. If $\delta =1$ and $0\le u,v,x,y\le 1,$ then

  $$\begin{aligned} \lambda ^{\delta }= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-u^{2}}\right) ^{\delta }+\left( {u^{2}}\right) ^{\delta }}},\frac{\sqrt{2\left( {v^{2}}\right) ^{\delta }}}{\sqrt{\left( {2-v^{2}}\right) ^{\delta }+\left( {v^{2}}\right) ^{\delta }}}}\right] ,\right. \\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{\delta }-\left( {1-x^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+x^{2}}\right) ^{\delta }+\left( {1-x^{2}}\right) ^{\delta }}},\right. \\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{\delta }-\left( {1-y^{2}}\right) ^{\delta }}}{\sqrt{\left( {1+y^{2}}\right) ^{\delta } +\left( {1-y^{2}}\right) ^{\delta }}}\right] \right) \\= & {} \left( \left[ {\frac{\sqrt{2\left( {u^{2}}\right) ^{1}}}{\sqrt{\left( {2-u^{2}}\right) ^{1}+\left( {u^{2}}\right) ^{1}}},\frac{\sqrt{2\left( {v^{2}}\right) ^{1}}}{\sqrt{\left( {2-v^{2}}\right) ^{1}+\left( {u^{2}}\right) ^{1}}}}\right] ,\right. \\&\left[ \frac{\sqrt{\left( {1+x^{2}}\right) ^{1}-\left( {1-x^{2}}\right) ^{1}}}{\sqrt{\left( {1+x^{2}}\right) ^{1}+\left( {1-x^{2}}\right) ^{1}}},\right. \nonumber \\&\left. \left. \frac{\sqrt{\left( {1+y^{2}}\right) ^{1}-\left( {1-y^{2}}\right) ^{1}}}{\sqrt{\left( {1+y^{2}}\right) ^{1}+\left( {1-y^{2}}\right) ^{1}}}\right] \right) =\lambda . \end{aligned}$$

  Thus, $\lambda ^{\delta }=\lambda$ and $\delta \lambda =\lambda .$

Lemma 1

[6] Let $\lambda _j \succ 0,w_j \succ 0( {j=1,2,3,...,n})$ and $\mathop \sum \nolimits _{j=1}^n w_j =1,$ then

$$\begin{aligned} \prod \limits _{j=1}^n ( {\lambda _j })^{w_j }\le \sum \limits _{j=1}^n w_j \lambda _j, \end{aligned}$$

(22)

where the equality holds if and only if $\lambda _1 =\lambda _2 =\cdots =\lambda _n .$

Theorem 3

Let $\lambda _j =( {[ {u_j ,v_j }],[ {x_j ,y_j }]})( {j=1,2,3,...,n})$ be a collection of IVPFVs,  where the $w=( w_1 ,w_2 ,w_3 ,...,w_n )^{T}$ is the weighted vector of IVPFEHWA and IVPFHWA, such that $w_j \in [ {0,1}]$ and $\mathop \sum \nolimits _{j=1}^n w_j =1. \quad \omega =( \omega _1 ,\omega _2 ,\omega _2 ,...,\omega _n )^{T}$is the weighted vector of $\lambda _j ( {j=1,2,3,...,n})$ such that$\omega _j \in [ {0,1}]$, $\mathop \sum \nolimits _{j=1}^n \omega _j =1,$ then

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n })\nonumber \\&\quad \le \hbox {IVPFHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n }). \end{aligned}$$

(23)

Proof

Straight forward.

Theorem 4

Idempotency: If $\dot{\lambda }_{\sigma ( j)} =\dot{\lambda }$ for all $j( j=1,2,3,...,n),$ where $\lambda =( {[ {u,v}],[ {x,y}]}),$ then

$$\begin{aligned} \hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n })=\dot{\lambda }. \end{aligned}$$

(24)

Proof

Since $\dot{\lambda }_{\sigma ( j)} =\dot{\lambda }$ for all j, then we have

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n }) \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+u_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-u_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}},\frac{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }-\prod \limits _{j=1}^n \left( {1-v_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {1+v_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {1-v_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\prod \limits _{j=1}^n \left( {x_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-x_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {x_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}},\frac{\sqrt{2\prod \limits _{j=1}^n \left( {y_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}}{\sqrt{\prod \limits _{j=1}^n \left( {2-y_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }+\prod \limits _{j=1}^n \left( {y_{\dot{\lambda }_{\sigma \left( j\right) } }^2 }\right) ^{w_j }}}}\right] } \\ \end{array} }}\right) \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\left( {1+u_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }-\left( {1-u_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}}{\sqrt{\left( {1+u_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }+\left( {1-u_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}},\frac{\sqrt{\left( {1+v_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }-\left( {1-v_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}}{\sqrt{\left( {1+v_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }+\left( {1-v_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}}}\right] ,} \\ {\left[ {\frac{\sqrt{2\left( {x_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}}{\sqrt{\left( {2-x_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }+\left( {x_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}},\frac{\sqrt{2\left( {y_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}}{\sqrt{\left( {2-y_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }+\left( {y_{\dot{\lambda }}^2 }\right) ^{\sum \limits _{j=1}^n w_j }}}}\right] } \\ \end{array} }}\right) \\&\quad =\left( {{\begin{array}{l} {\left[ {\frac{\sqrt{\left( {1+u_{\dot{\lambda }}^2 }\right) -\left( {1-u_{\dot{\lambda }}^2 }\right) }}{\sqrt{\left( {1+u_{\dot{\lambda }}^2 }\right) +\left( {1-u_{\dot{\lambda }}^2 }\right) }},\frac{\sqrt{\left( {1+v_{\dot{\lambda }}^2 }\right) -\left( {1-v_{\dot{\lambda }}^2 }\right) }}{\sqrt{\left( {1+v_{\dot{\lambda }}^2 }\right) +\left( {1-v_{\dot{\lambda }}^2 }\right) }}}\right] ,} \\ {\left[ {\frac{\sqrt{2\left( {x_{\dot{\lambda }}^2 }\right) }}{\sqrt{\left( {2-x_{\dot{\lambda }}^2 }\right) +\left( {x_{\dot{\lambda }}^2 }\right) }},\frac{\sqrt{2\left( {y_{\dot{\lambda }}^2 }\right) }}{\sqrt{\left( {2-y_{\dot{\lambda }}^2 }\right) +\left( {y_{\dot{\lambda }}^2 }\right) }}}\right] } \\ \end{array} }}\right) =\dot{\lambda }. \\ \end{aligned}$$

The proof is completed.

Theorem 5

Boundedness: Let $\lambda _j =( {[ {u_{\lambda _j } ,v_{\lambda _j } }],[ {x_{\lambda _j } ,y_{\lambda _j } }]})( {j=1,2,3,...,n})$ be a collection of IVPFNs, then

$$\begin{aligned} \dot{\lambda }_{\min }\le & {} \hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n })\le \dot{\lambda }_{\max }, \end{aligned}$$

(25)

$$\begin{aligned} \dot{\lambda }_{\max }= & {} \mathop {\max }\limits _j( {\dot{\lambda }_{\sigma ( j)} }), \end{aligned}$$

(26)

$$\begin{aligned} \dot{\lambda }_{\min }= & {} \mathop {\min }\limits _j( {\dot{\lambda }_{\sigma ( j)} }). \end{aligned}$$

(27)

Proof

Proof is easy so it is omitted here.

Theorem 6

Monotonicity: If $\lambda _j \le \lambda _j^{*}$ for all $j( j=1,2,3,...,n),$ then

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n })\nonumber \\&\quad \le \hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1^{*} ,\lambda _2^{*} ,\lambda _3^{*} ,...,\lambda _n^{*} }). \end{aligned}$$

(28)

Proof

As we know that.

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n })\nonumber \\&\quad =w_1 \dot{\lambda }_{\sigma ( 1)} \oplus _\varepsilon w_2 \dot{\lambda }_{\sigma ( 2)} \oplus _\varepsilon w_3 \dot{\lambda }_{\sigma ( 3)} \oplus _\varepsilon \cdots \oplus _\varepsilon w_n \dot{\lambda }_{\sigma ( n)},\nonumber \\ \end{aligned}$$

(29)

and

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1^{*} ,\lambda _2^{*} ,\lambda _3^{*} ,...,\lambda _n^{*} })\nonumber \\&\quad =w_1 \dot{\lambda }_{\sigma ( 1)}^{*} \oplus _\varepsilon w_2 \dot{\lambda }_{\sigma ( 2)}^{*} \oplus _\varepsilon w_3 \dot{\lambda }_{\sigma ( 3)}^{*} \oplus _\varepsilon \cdots \oplus _\varepsilon w_n \dot{\lambda }_{\sigma ( n)}^{*}.\nonumber \\ \end{aligned}$$

(30)

Since $\lambda _j \le \lambda _j^{*}$ for all j, thus Eq. (28) always holds.

Theorem 7

Interval-valued Pythagorean fuzzy Einstein weighted averaging operator is a special case of the interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging operator.

Proof

Let $\omega =\left( {\frac{1}{n},\frac{1}{n},\frac{1}{n},..., \frac{1}{n},}\right) ^{T},$ then we have

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n })\\&\quad =w_1 \dot{\lambda }_{\sigma ( 1)} \oplus _\varepsilon w_2 \dot{\lambda }_{\sigma ( 2)} \oplus _\varepsilon \cdots \oplus _\varepsilon w_n \dot{\lambda }_{\sigma ( n)} \\&\quad =\frac{1}{n}( {\dot{\lambda }_{\sigma ( 1)} \oplus _\varepsilon \dot{\lambda }_{\sigma ( 2)} \oplus _\varepsilon \cdots \oplus _\varepsilon \dot{\lambda }_{\sigma ( n)} }) \\&\quad =\frac{1}{n}( {n\omega _1 \lambda _1 \oplus _\varepsilon n\omega _2 \lambda _2 \oplus _\varepsilon \cdots \oplus _\varepsilon n\omega _n \lambda _n }) \\&\quad =\omega _1 \lambda _1 \oplus _\varepsilon \omega _2 \lambda _2 \oplus _\varepsilon \cdots \oplus _\varepsilon \omega _n \lambda _n \\&\quad =\hbox {IVPFEWA}_w ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,...,\lambda _n }) . \end{aligned}$$

The proof is completed.

Theorem 8

Interval-valued Pythagorean fuzzy Einstein ordered weighted averaging operator is a special case of the interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging operator.

Proof

Let $w=\left( {\frac{1}{n},\frac{1}{n},\frac{1}{n},..., \frac{1}{n},}\right) ^{T},$ and $\dot{\lambda }_{\sigma ( j)} =\lambda _{\sigma ( j)} ,$ then we have

$$\begin{aligned}&\hbox {IVPFEHWA}_{\omega ,w} ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,\ldots ,\lambda _n })\\&\quad =w_1 \dot{\lambda }_{\sigma ( 1)} \oplus _\varepsilon w_2 \dot{\lambda }_{\sigma ( 2)} \oplus _\varepsilon \cdots \oplus _\varepsilon w_n \dot{\lambda }_{\sigma ( n)} \\&\quad =w_1 \lambda _{\sigma ( 1)} \oplus _\varepsilon w_2 \lambda _{\sigma ( 2)} \oplus _\varepsilon \cdots \oplus _\varepsilon w_n \lambda _{\sigma ( n)} \\&\quad =\hbox {IVPFEOWA}_w ( {\lambda _1 ,\lambda _2 ,\lambda _3 ,\ldots ,\lambda _n }). \end{aligned}$$

The proof completed.

An approach to multiple attribute group decision-making problems based on interval-valued Pythagorean fuzzy information

Algorithm Let $X=\{X_1 ,X_2 ,X_3 ,...,X_m \}$ be a finite set of m alternatives and $C=\{C_1 ,C_2 ,C_3 ,...,C_n \}$ be a finite set of n attributes. Suppose the grade of the alternatives$X_i (i=1,2,3,...,m)$on attribute$C_j (j=1,2,3,...,n)$ given by decision makers is interval-valued Pythagorean fuzzy numbers. Let $D=\{ {D_1 ,D_2 ,D_3 ,...,D_k }\}$ be the set of k decision makers, and let $w=( {w_1 ,w_2 ,w_3 ,...,w_n })^{T}$ be the weighted vector of the attributes $C_j ( {j=1,2,3,...,n}),$ such that $w_j \in [ {0,1}]$,$\sum \nolimits _{j=1}^n w_j =1,$ and let $\omega =( {\omega _1 ,\omega _2 ,\omega _3 ,...,\omega _k })^{T}$ be the weighted vector of the decision makers $D^{s}( {s=1,2,3,...,k}),$ such that $\omega _s \in [ {0,1}]$ and $\sum \nolimits _{s=1}^k \omega _s =1.$ Let $D=( {a_{ji} })=\langle {[ {u_{ji} ,v_{ji} }],[ {x_{ji} ,y_{ji} }]}\rangle (i=1,2,3,...,m,j=1,2,3,...,n)$ where $[ {u_{ji} ,v_{ji} }]$ indicates the interval degree that the alternative $X_i (i=1,2,3,...,m)$ satisfies the attribute $C_j ( {j=1,2,3,...,n})$ and $[ {x_{ji} ,y_{ji} }]$ indicates the interval degree that the alternative $X_i (i=1,2,3,...,m)$ does not satisfy the attribute $C_j ( j=1,2,3,...,n),$ And also $[ {u_{ji} ,v_{ji} }]\in [0,1],[ {x_{ji} ,y_{ji} }]\in [0,1]$ with condition $0\le ( {v_{ji} })^{2}+( {y_{ji} })^{2}\le 1,(i=1,2,3,...,m,j=1,2,3,...,n).$ This method has the following steps.

Step 1 :

Utilize the given information in the form of matrices, $D^{s}=\left[ {a_{ji}^{( s)} }\right] _{n\times m} ( {s=1,2,3,...,k})$.

Step 2 :

If the criteria have two types, such as benefit criteria and cost criteria, then the interval-valued Pythagorean fuzzy decision matrices, $D^{s}\!=\!\left[ {a_{ji}^{( s)} }\right] _{n\times m} ( {s=1,2,3,...,k})$ can be converted into the normalized interval-valued Pythagorean fuzzy decision matrices, $R^{s}=\left[ {r_{ji}^{( s)} }\right] _{_{n\times m} } ( {s=1,2,3,...,n})$, where

$$\begin{aligned} \quad \qquad r_{ji}^{( s)} =\left\{ {{\begin{array}{l} {a_{ji}^{( s)} ,\hbox { for benefit criteria }C_j } \\ {\bar{{a}}_{ji}^{( s)} ,\hbox { for cost criteria }C_j ,} \\ \end{array} }}\right. \left( {{\begin{array}{l} {j=1,2,3,...,n} \\ {i=1,2,3,...,m} \\ \end{array} }}\right) , \end{aligned}$$

and $\bar{{a}}_{ji}^{( s)}$ is the complement of $\alpha _{ji}^s .$ If all the criteria have the same type, then there is no need of normalization.

Step 3 :

Utilize the IVPFEWA operator to aggregate all the individual normalized interval-valued Pythagorean fuzzy decision matrices, $R^{s}=\left[ {r_{ji}^{( s)} }\right] _{n\times m} ( s=1,2,3,...,k)$ into a single interval-valued Pythagorean fuzzy decision-matrix, $R=[ {r_{ji} }]_{n\times m} ,$ where $r_{ji} =\langle {[ {u_{ji} ,v_{ji} }],[ {x_{ji} ,y_{ji} }]}\rangle .$

Step 4 :

In this step, we calculate $\dot{r}_{ji} =nw_j r_{ji}$.

Step 5 :

Calculate the scores function of $\dot{r}_{ji} (i=1,2,3,...,m,j=1,2,3,...,n).$ If there is no difference between two or more than two scores, then we must find out the accuracy degrees of the collective overall preference values.

Step 6 :

Utilize the IVPFEHWA operator to aggregate all preference values.

Step 7 :

Arrange the scores of the all alternatives in the form of descending order and select that alternative which has the highest score function.

Table 1 Interval-valued Pythagorean fuzzy decision matrix of $\hbox {D}^{1}$

Table 2 Interval-valued Pythagorean fuzzy decision matrix of $\hbox {D}^{2}$

Table 3 Interval-valued Pythagorean fuzzy decision matrix of $\hbox {D}^{3}$

Illustrative example

Suppose in Hazara University, the IT department wants to select a new information system for the purpose of the best productivity. After the first selection, there are only three $X_i ( {i=1,2,3})$ alternatives have been short listed. There are three experts $D^{s}( {s=1,2,3})$ from a group to act as decision makers, whose weight vector is $\omega =( {0.2,0.3,0.5})^{T}.$ There are many factors that must be considered while selecting the most suitable system, but here, we have consider only the following four criteria, whose weighted vector is $w=( {0.1,0.2,0.3,0.4})^{T}$

• 1. $C_1 :$ Costs of hardware.

• 2. $C_2 :$ Support of the organization.

• 3. $C_3 :$ Effort to transform from current systems.

• 4. $C_4 :$ Outsourcing software developer reliability,

where $C_1$, $C_3$, are cost type criteria and $C_2$, $C_4$ are benefit type criteria, i.e., the attributes have two types of criteria; thus, we must change the cost type criteria into benefit type criteria.

Step 1 :

Construct the decision-making matrices (Tables 1, 2 and 3).

Step 2 :

Construct the normalized decision making matrices (Tables 4, 5 and 6).

Table 4 Normalized Pythagorean fuzzy decision matrix $\hbox {R}^{1}$

Table 5 Normalized Pythagorean fuzzy decision matrix $\hbox {R}^{2}$

Table 6 Normalized Pythagorean fuzzy decision matrix $\hbox {R}^{3}$

Table 7 Collective interval-valued Pythagorean fuzzy decision matrix R

Step 3 :

Utilize the IVPFEWA operator to aggregate all the individual normalized interval-valued Pythagorean fuzzy decision matrices, $R^{s}=\left[ {r_{ji}^{( s)} }\right] _{n\times m}$ into a single interval-valued Pythagorean fuzzy decision matrix, $R=[ {r_{ji} }]_{n\times m}$ (Table 7).

Step 4 :

Calculate $\dot{\lambda }_{ji} =nw\lambda _{ji} .$

$$\begin{aligned} \dot{\lambda }_{11}= & {} ( {[ {0.262, 0.343}], [ {0.733, 0.897}]}),\\ \dot{\lambda }_{21}= & {} ( {[ {0.370, 0.534}], [ {0.523, 0.645}]}) \\ \dot{\lambda }_{31}= & {} ( {[ {0.385, 0.745}], [ {0.281, 0.513}]}),\\ \dot{\lambda }_{41}= & {} ( {[ {0.518, 0.788}], [ {0.201,0.329}]}) \\ \dot{\lambda }_{12}= & {} ( {[ {0.262, 0.424}], [ {0.742, 0.837}]}),\\ \dot{\lambda }_{22}= & {} ( {[ {0.315, 0.628}], [ {0.420, 0.757}]}) \\ \dot{\lambda }_{32}= & {} ( {[ {0.452, 0.665}], [ {0.307, 0.501}]}),\\ \dot{\lambda }_{42}= & {} ( {[ {0.593, 0.726}], [ {0.061, 0.359}]}) \\ \dot{\lambda }_{13}= & {} ( {[ {0.262, 0.382}], [ {0.605, 0.823}]}),\\ \dot{\lambda }_{23}= & {} ( {[ {0.370, 0.590}], [ {0.357, 0.672}]}) \\ \dot{\lambda }_{33}= & {} ( {[ {0.385, 0.745}], [ {0.136, 0.638}]}),\\ \dot{\lambda }_{43}= & {} ( {[ {0.518, 0.664}], [ {0.188, 0.374}]}) .\\ \end{aligned}$$

Step 5 :

Calculate the score functions (Table 8).

$$\begin{aligned} s( {\dot{\lambda }_{11} })= & {} -0.57,s( {\dot{\lambda }_{21} })=-0.13,s( {\dot{\lambda }_{31} })\\= & {} 0.18,s( {\dot{\lambda }_{41} })=0.37 \\ s( {\dot{\lambda }_{12} })= & {} -0.50,s( {\dot{\lambda }_{22} })=-0.12,s( {\dot{\lambda }_{32} })\\= & {} 0.15,s( {\dot{\lambda }_{42} })=0.37 \\ s( {\dot{\lambda }_{13} })= & {} -0.41,s( {\dot{\lambda }_{23} })=-0.04,s( {\dot{\lambda }_{33} })\\= & {} 0.13,s( {\dot{\lambda }_{43} })=0.26. \\ \end{aligned}$$

Table 8 Pythagorean fuzzy hybrid decision matrix R

Step 6 :

Utilize the IVPFEHWA aggregation operator to aggregate all preference values.

$$\begin{aligned} r_1= & {} ( {[ {0.354, 0.567}], [ {0.550, 0.674}]}) \\ r_2= & {} ( {[ {0.367, 0.581}], [ {0.422, 0.686}]}) \\ r_3= & {} ( {[ {0.354, 0.571}], [ {0.347, 0.695}]}). \\ \end{aligned}$$

Step 7 :

Calculate the score functions.

$$\begin{aligned} s( {r_1 })=-0.154,s( {r_1 })=-0.088,s( {r_1 })=-0.076. \end{aligned}$$

Step 8 :

Arrange the scores of the all alternatives in the form of descending order and select that alternative which has the highest score function.

$$\begin{aligned} s( {r_3 })\succ s( {r_2 })\succ s( {r_1 }) \end{aligned}$$

Thus, the best alternative is $X_3 .$

Conclusion

In this paper, we have developed the notion of interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator along with their some desirable properties such as idempotency, boundedness, and monotonicity. Actually interval-valued Pythagorean fuzzy Einstein weighted averaging aggregation operator weights only the Pythagorean fuzzy arguments and interval-valued Pythagorean fuzzy Einstein ordered weighted averaging aggregation operator weights only the ordered positions of the Pythagorean fuzzy arguments instead of weighting the Pythagorean fuzzy arguments themselves. To overcome these limitations, we have introduced an interval-valued Pythagorean fuzzy Einstein hybrid weighted averaging aggregation operator, which weights both the given Pythagorean fuzzy value and its ordered position. Finally, the proposed operator has been applied to decision-making problems to show the validity, practicality and effectiveness of the new approach.