Extension of Einstein geometric operators to multi-attribute decision making under q-rung orthopair fuzzy information

Abstract

Aggregation operators are mathematical functions and essential tools of unifying the several inputs into single valuable output. The purpose of this paper is to analyze the aggregation operators (AOs) under the q-rung orthopair fuzzy environment with the help of Einstein norms operations. This paper presents AOs, namely, q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG), q-rung orthopair fuzzy Einstein ordered weighted geometric (q-ROFEOWG), generalized q-rung orthopair fuzzy Einstein weighted geometric (Gq-ROFEWG), generalized q-rung orthopair fuzzy Einstein ordered weighted geometric (Gq-ROFEOWG) operators. Some properties of these operators are explained. An algorithmic model to deal with multi-attribute decision making problems in q-rung orthopair fuzzy(q-ROF) environment using generalized q-ROF Einstein weighted geometric operator is established. These operators can remunerate for the possible asymmetric roles of the attributes that represent the problem. At the end, to prove the validity and feasibility of the proposed model, we give applications for the selection of location of thermal power station and selection of best cardiac surgeon. The comparison analysis with other existing operators shows the reliability of our work.

1 Introduction

Multi-attribute decision making (MADM) performs an important role in finding an excellent alternative from all the appropriate alternatives depending upon certain parameters or attributes. Mostly, the approach of different alternatives and the corresponding weights for various attributes are given in crisp values. But many decisions in real-life problems, for which the objectives and conditions are vague and unclear. To handle such situations, Zadeh (1965) gave the theory of fuzzy sets (FSs) in which the satisfaction of an attribute is represented in terms of membership degree (MD) between zero and one. FSs have become one of the emerging areas in contemporary technologies of information processing (Chen and Chen 2014; Chen et al. 2009; Chen and Niou 2011). To represent the dissatisfaction degree (nonmembership degree) independently, Atanassov (1986) introduced intuitionistic fuzzy sets (IFSs) which are characterized by both the MD $\mu$ and the nonmembership degree (NMD) $\nu$ in [0, 1], respectively such that the summation of these two values is less than or equal to 1. An IFS has various applications in different fields of life (Chen and Cheng 2016; Chen et al. 2016). Yager(2013a; b) introduced Pythagorean fuzzy sets (PFSs) to broaden the space of IFSs and replaced the condition $\mu +\nu \le 1$ with $\mu ^2+\nu ^2\le 1$. PFSs are more applicable to deal the vagueness as compared to IFSs. Moreover, Yager (2016) proposed new notion of q-rung orthopair fuzzy sets (q-ROFSs) in which $\mu ^q+\nu ^q\le 1$. In decision making problems, q-ROFSs are more capable than IFSs and PFSs, as shown in Fig. 1.

For solving MADM issues and for a single decision in case of collective information, AOs have a key role. The cue of weighted AOs was presented by Yager (1988). The applications of novel geometric AOs for IFSs in MADM were presented by Xu and Yager (2006). The notion of intuitionistic fuzzy AOs was given by Xu (2007). The induced geometric AOs for IFSs were studied by Wei (2010). Wang and Liu (2011) discussed the intuitionistic fuzzy Einstein weighted geometric (IFEWG) operators. Yager (2013a, b) studied both averaging and geometric operators under Pythagorean fuzzy (PF) environment. Peng and Yang (2016; 2016) studied the fundamental properties of PF AOs. Zeng et al. (2016) developed a hybrid method for PF MADM. Garg (2017) developed the generalized PF Einstein weighted geometric AOs. Rahman et al. (2017) discussed the PF Einstein weighted geometric AOs with multiple-attribute group decision making applications. Akram et al. (2019a, b) discussed the Pythagorean Dombi fuzzy AOs. Shahzadi and Akram (2020) introduced the PF Yager AOs. Liu and Wang (2018) expressed q-ROF weighted AOs. Dombi AOs under q-ROF data were handled by Jana et al. (2019). The idea of q-rung orthopair fuzzy power Maclaurin symmetric mean operators was studied by Liu et al. (2018a, b). Some q-rung orthopair fuzzy Bonferroni mean operators were discussed by Liu and Liu (2018). Garg and Chen (2020) introduced the neutrality AOs of q-ROFSs. For other terminologies not discussed in the paper, the readers are referred to (Akram and Adeel 2018; Akram and Ali 2019, 2020; Akram et al. 2019a, b, 2020a, b; Akram and Bashir 2020; Akram and Shahzadi 2020a, b; Bai et al. 2018; Garg and Rani 2019; Joshi and Gegov 2019; Khan et al. 2019; Liu et al. 2018a, b; Peng and Yang 2016; Peng and Yuan 2016; Peng et al. 2018; Wang and Liu 2012; Wei 2017; Wei et al. 2017; Wei and Lu 2018a, b; Yager 1994; Zhang et al. 2014).

The motivations of this article are outlined as follows:

• 1. The judgement of a perfect alternative in a q-ROF environment is a laborious MADM problem. In existing techniques, assessment information is characterized by IF and PF numbers which promote to do work in q-rung orthopair fuzzy numbers (q-ROFNs).

• 2. q-ROFNs demonstrate extraordinary execution in providing vague, reliable, and inexact assessment information.

• 3. Taking into account that Einstein AOs are a straight forward, however ground-breaking, approach for solving decision making issues, this article, in general, aims to define Einstein AOs in the q-ROF context to tackle difficult problems of choice.

• 4. Einstein AOs make the decision results more precise and exact when applied to real-life MADM based on the q-ROF environment.

• 5. The proposed operators overcome the restrictions of existing operators.

The main contributions of this article are:

• 1. The idea of Einstein AOs is extended to q-ROFNs and properties of these operators are discussed.

• 2. An algorithm is developed to handle complex realistic problems with q-ROF data.

• 3. Two MADM problems are discussed using proposed operators.

• 4. At the end, the benefits and characteristics of these operators are discussed by comparison analysis.

The remaining paper is as follows: In Sect. 2, we recalled some basic definitions. In Sect. 3, Einstein operational laws for q-ROFNs are promoted. In Sect. 4 and Sect. 5, q-ROFEWG and q-ROFEOWG operators are proposed, respectively and related results to them. In Sect. 6, the idea of $\hbox {G}q$-ROFEWG and $\hbox {G}q$-ROFEOWG operators is handled. In Sect. 7, an algorithm for our new model is proposed and two MADM problems, one of which is the selection of location for thermal power station and other one is selection of best cardiac surgeon under these operators are discussed. Section 8 provides the comparison analysis of our model with generalized Pythagorean fuzzy Einstein weighted geometric (GPFEWG) (Garg 2017), intuitionistic fuzzy Einstein weighted geometric (IFEWG) (Wang and Liu 2011), q-rung orthopair fuzzy generalized Maclaurin symmetric mean (q-ROFGMSM) Liu and Wang (2020) operators. In Sect. 9, we have concluded results related to our proposed model.

Fig. 1

Comparison of q-ROFNs, PFNs and IFNs spaces

2 Preliminaries

Definition 1

(Yager 2016) A q-ROFS ${\mathfrak {F}}$ on non-empty set ${\mathcal {X}}$ is given by

$$\begin{aligned} {\mathfrak {F}}=\{\langle x, \mu _{\mathfrak {F}}(x), \nu _{\mathfrak {F}}(x)\rangle \}, \end{aligned}$$

where $\mu _{\mathfrak {F}}: {\mathcal {X}}\rightarrow [0,1]$ and $\nu _{\mathfrak {F}}: {\mathcal {X}}\rightarrow [0,1]$ show the MD and NMD of an element $x\in {\mathcal {X}}$, respectively. $\varpi _{\mathfrak {F}}(x)=\root q \of {1-(\mu _{\mathfrak {F}}(x))^q+(\nu _{\mathfrak {F}}(x))^q}$ is indeterminacy degree of an element. For easiness, ${\mathfrak {F}}=\{\langle x, \mu _{\mathfrak {F}}(x), \nu _{\mathfrak {F}}(x)\rangle \},$ called q-ROFN represented by ${\mathfrak {F}}=\langle \mu _{\mathfrak {F}}, \nu _{\mathfrak {F}}\rangle .$

Definition 2

(Yager 2016) Consider two q-ROFNs ${\mathfrak {F}}_1=\langle \mu _{{\mathfrak {F}}_1}, \nu _{{\mathfrak {F}}_1}\rangle$ and ${\mathfrak {F}}_2=\langle \mu _{{\mathfrak {F}}_2}, \nu _{{\mathfrak {F}}_2}\rangle$. The operational laws on q-ROFNs are

• 1. ${\mathfrak {F}}_1\oplus {\mathfrak {F}}_2=\Big \langle \root q \of {\mu ^q_{{\mathfrak {F}}_1}+\mu ^q_{{\mathfrak {F}}_2}-\mu ^q_{{\mathfrak {F}}_1}\mu ^q_{{\mathfrak {F}}_2}}, \nu _{{\mathfrak {F}}_1}\nu _{{\mathfrak {F}}_2}\Big \rangle$,

• 2. ${\mathfrak {F}}_1\otimes {\mathfrak {F}}_2=\Big \langle \mu _{{\mathfrak {F}}_1}\mu _{{\mathfrak {F}}_2}, \root q \of {\nu ^q_{{\mathfrak {F}}_1}+\nu ^q_{{\mathfrak {F}}_2}-\nu ^q_{{\mathfrak {F}}_1}\nu ^2_{{\mathfrak {F}}_2}}\Big \rangle$,

• 3. $\gamma {\mathfrak {F}}_1=\Big \langle \root q \of {1-(1-\mu ^q_{{\mathfrak {F}}_1})^\gamma }, \nu _{{\mathfrak {F}}_1}^\gamma \Big \rangle ,$

• 4. ${\mathfrak {F}}_1^\gamma =\Big \langle \mu _{{\mathfrak {F}}_1}^\gamma , \root q \of {1-(1-\nu ^q_{{\mathfrak {F}}_1})^\gamma }\Big \rangle ,$ where $\gamma >0.$

Definition 3

(Yager 2016) Consider a q-ROFN ${\mathfrak {F}}=\langle \mu _{{\mathfrak {F}}}, \nu _{{\mathfrak {F}}}\rangle$. The score ${\mathcal {S}}({\mathfrak {F}})$ and accuracy functions ${\mathcal {A}}({\mathfrak {F}})$ of ${\mathfrak {F}}$ are

$$\begin{aligned}&{\mathcal {S}}({\mathfrak {F}})=\mu ^q_{{\mathfrak {F}}}-\nu ^q_{{\mathfrak {F}}},\quad {\text {where}}\,{\mathcal {S}}({\mathfrak {F}})\in [-1,1], \\&{\mathcal {A}}({\mathfrak {F}})=\mu ^q_{{\mathfrak {F}}}+\nu ^q_{{\mathfrak {F}}},\quad {\text {where}}\,{\mathcal {A}}({\mathfrak {F}})\in [0,1]. \end{aligned}$$

Definition 4

(Yager 2016) Consider two q-ROFNs ${\mathfrak {F}}_1=\langle \mu _{{\mathfrak {F}}_1}, \nu _{{\mathfrak {F}}_1}\rangle$ and ${\mathfrak {F}}_2=\langle \mu _{{\mathfrak {F}}_2}, \nu _{{\mathfrak {F}}_2}\rangle$. Then

• 1. If ${\mathcal {S}}({\mathfrak {F}}_1)<{\mathcal {S}}({\mathfrak {F}}_2)$, then ${\mathfrak {F}}_1\prec {\mathfrak {F}}_2$,

• 2. If ${\mathcal {S}}({\mathfrak {F}}_1)>{\mathcal {S}}({\mathfrak {F}}_2)$, then ${\mathfrak {F}}_1\succ {\mathfrak {F}}_2$,

• 3. If ${\mathcal {S}}({\mathfrak {F}}_1)={\mathcal {S}}({\mathfrak {F}}_2)$, then

  • a. If ${\mathcal {A}}({\mathfrak {F}}_1)<{\mathcal {A}}({\mathfrak {F}}_2)$, then ${\mathfrak {F}}_1\prec {\mathfrak {F}}_2$,

  • b. If ${\mathcal {A}}({\mathfrak {F}}_1)>{\mathcal {A}}({\mathfrak {F}}_2)$, then ${\mathfrak {F}}_1\succ {\mathfrak {F}}_2$,

  • c. If ${\mathcal {A}}({\mathfrak {F}}_1)={\mathcal {A}}({\mathfrak {F}}_2)$, then ${\mathfrak {F}}_1\sim {\mathfrak {F}}_2$.

Definition 5

(Garg 2016) The Einstein sum $\oplus _\epsilon$ and Einstein product $\otimes _\epsilon$ under the PF environment are defined as

$$\begin{aligned}&S_\epsilon (x,y)=\sqrt{\frac{x^2+y^2}{1+x^2._\epsilon y^2}},\\&T_\epsilon (x,y)=\frac{x._\epsilon y}{\sqrt{1+(1-x^2)._\epsilon (1-y^2)}}. \end{aligned}$$

3 q-ROFNs under Einstein operational law

The Einstein operations for q-ROFNs are defined as follows:

Definition 6

Let ${\mathfrak {F}}=\langle \mu , \nu \rangle$, ${\mathfrak {F}}_1=\langle \mu _1, \nu _1\rangle$, ${\mathfrak {F}}_2=\langle \mu _2, \nu _2\rangle$ be q-ROFNs and $\gamma > 0$, then

• (i)  $\overline{{\mathfrak {F}}}=\langle \nu _{\mathfrak {F}},\,\mu _{\mathfrak {F}}\rangle$

• (ii)  ${\mathfrak {F}}_1\wedge _\epsilon {\mathfrak {F}}_2=\langle \min \{\mu _1, \mu _2\},\,\max \{\nu _1, \nu _2\}\rangle$

• (iii)  ${\mathfrak {F}}_1\vee _\epsilon {\mathfrak {F}}_2=\langle \max \{\mu _1, \mu _2\},\,\min \{\nu _1, \nu _2\}\rangle$

• (iv)  ${\mathfrak {F}}_1\oplus _\epsilon {\mathfrak {F}}_2=\bigg \langle \root q \of {\frac{\mu _{1}^q+\mu _{2}^q}{1+\mu _{1}^q._\epsilon \mu _{2}^q}},\, \frac{\nu _{1}._\epsilon \nu _{2}}{\root q \of {1+(1-\nu _{1}^q).\epsilon (1-\nu _{2}^q)}}\bigg \rangle$

• (v)  ${\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2=\bigg \langle \frac{\mu _{1}._\epsilon \mu _{2}}{\root q \of {1+(1-\mu _{1}^q).\epsilon (1-\mu _{2}^q)}},\,\root q \of {\frac{\nu _{1}^q+\nu _{2}^q}{1+\nu _{1}^q._\epsilon \nu _{2}^q}}\bigg \rangle$

• (vi)  $\gamma ._\epsilon {\mathfrak {F}}=\bigg \langle \root q \of {\frac{(1+\mu ^q)^\gamma -(1-\mu ^q)^\gamma }{(1+\mu ^q)^\gamma +(1-\mu ^q)^\gamma }},\, \frac{\root q \of {2}\nu ^\gamma }{\root q \of {(2-\nu ^q)^\gamma +(\nu ^q)^\gamma }}\bigg \rangle$

• (vii)  ${\mathfrak {F}}^\gamma =\bigg \langle \frac{\root q \of {2}\mu ^\gamma }{\root q \of {(2-\mu ^q)^\gamma +(\mu ^q)^\gamma }},\, \root q \of {\frac{(1+\nu ^q)^\gamma -(1-\nu ^q)^\gamma }{(1+\nu ^q)^\gamma +(1-\nu ^q)^\gamma }}\bigg \rangle$

Theorem 1

Let ${\mathfrak {F}}=\langle \mu _{{\mathfrak {F}}}, \nu _{{\mathfrak {F}}}\rangle$, ${\mathfrak {F}}_1=\langle \mu _{1}, \nu _{1}\rangle$ and ${\mathfrak {F}}_2=\langle \mu _{2}, \nu _{2}\rangle$ be three q-ROFNs; then, ${\mathfrak {F}}_3={\mathfrak {F}}^\gamma$ and ${\mathfrak {F}}_4={\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2$ are also q-ROFNs.

Proof

Since $\gamma >0$ and ${\mathfrak {F}}$ be a q-ROFN, therefore, $0\le \mu _{{\mathfrak {F}}}(x)\le 1$, $0\le \nu _{{\mathfrak {F}}}(x)\le 1$, and $0\le (\mu _{{\mathfrak {F}}}(x))^q+(\nu _{{\mathfrak {F}}}(x))^q\le 1$, then $1- (\mu _{{\mathfrak {F}}}(x))^q\ge (\nu _{{\mathfrak {F}}}(x))^q\ge 0$, $1- (\nu _{{\mathfrak {F}}}(x))^q\ge (\mu _{{\mathfrak {F}}}(x))^q\ge 0$, and $(1- (\mu _{{\mathfrak {F}}}(x))^q)^\gamma \ge (\nu _{{\mathfrak {F}}}(x))^q$, then, we have

$$\begin{aligned} \frac{\root q \of {2}(\mu _{\mathfrak {F}}(x))^\gamma }{\root q \of {(2-(\mu _{\mathfrak {F}}(x))^q)^\gamma +(\mu _{\mathfrak {F}}(x)^q)^\gamma }}\le & {} \frac{\root q \of {2}(\mu _{\mathfrak {F}}(x))^\gamma }{\root q \of {(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma +((\mu _{\mathfrak {F}}(x))^q)^\gamma }} \end{aligned}$$

and

$$\begin{aligned} \root q \of {\frac{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma -(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma +(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }}\le & {} \root q \of {\frac{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma -(\mu _{\mathfrak {F}}(x))^q)^\gamma }{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma +(\mu _{\mathfrak {F}}(x))^q)^\gamma }}. \end{aligned}$$

Thus,

$$\begin{aligned}&\left( \frac{\root q \of {2}(\mu _{\mathfrak {F}}(x))^\gamma }{\root q \of {(2 -(\mu _{\mathfrak {F}}(x))^q)^\gamma +(\mu _{\mathfrak {F}}(x)^q)^\gamma }}\right) ^q \\&\quad +\left( \root q \of {\frac{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma -(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma +(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }} \right) ^q\le 1. \end{aligned}$$

Furthermore,

$$\begin{aligned}&\left( \frac{\root q \of {2}(\mu _{\mathfrak {F}}(x))^\gamma }{\root q \of {(2-(\mu _{\mathfrak {F}}(x))^q)^\gamma +(\mu _{\mathfrak {F}}(x)^q)^\gamma }}\right) ^q \\&\quad +\left( \root q \of {\frac{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma -(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma +(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }}\right) ^q= 0 \end{aligned}$$

iff $\mu _{{\mathfrak {F}}}(x)=\nu _{{\mathfrak {F}}}(x)=0$ and

$$\begin{aligned}&\left( \frac{\root q \of {2}(\mu _{\mathfrak {F}}(x))^\gamma }{\root q \of {(2-(\mu _{\mathfrak {F}}(x))^q)^\gamma +(\mu _{\mathfrak {F}}(x)^q)^\gamma }}\right) ^q \\&\quad +\left( \root q \of {\frac{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma -(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }{(1+(\nu _{\mathfrak {F}}(x))^q)^\gamma +(1-(\nu _{\mathfrak {F}}(x))^q)^\gamma }}\right) ^q= 1 \end{aligned}$$

iff $(\mu _{{\mathfrak {F}}}(x))^q+(\nu _{{\mathfrak {F}}}(x))^q=1.$

Thus, ${\mathfrak {F}}_3={\mathfrak {F}}^\gamma$ is a q-ROFN for $\gamma > 0.$ $\square$

Theorem 2

Let $\gamma , \gamma _1, \gamma _2\ge 0,$ then

$\mathrm{(i)}\,{\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2={\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2$

$\mathrm{(ii)}\,{\mathfrak {F}}_1\oplus _\epsilon {\mathfrak {F}}_2={\mathfrak {F}}_2\oplus _\epsilon {\mathfrak {F}}_1$

$\mathrm{(iii)}\,({\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2)^\gamma ={\mathfrak {F}}_1^\gamma \otimes _\epsilon {\mathfrak {F}}_2^\gamma$

$\mathrm{(iv)}\,\gamma ._\epsilon ({\mathfrak {F}}_1\oplus _\epsilon {\mathfrak {F}}_2)=\gamma ._\epsilon {\mathfrak {F}}_1\oplus _\epsilon \gamma ._\epsilon {\mathfrak {F}}_2$

$\mathrm{(v)}\,{\mathfrak {F}}^{\gamma _1}\otimes _\epsilon {\mathfrak {F}}^{\gamma _2}={\mathfrak {F}}^{({\gamma _1}+{\gamma _2})}$

$\mathrm{(vi)}\,\gamma _1._\epsilon {\mathfrak {F}}\oplus _\epsilon \gamma _2._\epsilon {\mathfrak {F}}=(\gamma _1+\gamma _2)._\epsilon {\mathfrak {F}}$

$\mathrm{(vii)}\,({\mathfrak {F}}^{\gamma _1})^{\gamma _2}=({\mathfrak {F}})^{\gamma _{1}._\epsilon \gamma _2}$

$\mathrm{(viii)}\,\gamma _{1}._\epsilon (\gamma _2._\epsilon {\mathfrak {F}})=(\gamma _{1}._\epsilon \gamma _2)._\epsilon {\mathfrak {F}}$

Proof

We prove (i), (iii), (v) and similarly others can be verified.

(i)

$$\begin{aligned} {\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2 & = {} \bigg \langle \frac{\mu _{1}._\epsilon \mu _{2}}{\root q \of {1+(1-\mu _{1}^q).\epsilon (1-\mu _{2}^q)}}, \root q \of {\frac{\nu _{1}^q+\nu _{2}^q}{1+\nu _{1}^q._\epsilon \nu _{2}^q}}\bigg \rangle \\ & = {} \bigg \langle \frac{\mu _{2}._\epsilon \mu _{1}}{\root q \of {1+(1-\mu _{2}^q).\epsilon (1-\mu _{1}^q)}}, \root q \of {\frac{\nu _{2}^q+\nu _{1}^q}{1+\nu _{2}^q._\epsilon \nu _{1}^q}}\bigg \rangle \\ & = {} {\mathfrak {F}}_2\otimes _\epsilon {\mathfrak {F}}_1. \end{aligned}$$

(iii)

$$\begin{aligned} {\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2 & = {} \bigg \langle \frac{\mu _{1}._\epsilon \mu _{2}}{\root q \of {1+(1-\mu _{1}^q).\epsilon (1-\mu _{2}^q)}}, \root q \of {\frac{\nu _{1}^q+\nu _{2}^q}{1+\nu _{1}^q._\epsilon \nu _{2}^q}}\bigg \rangle , \end{aligned}$$

is equivalent to

$$\begin{aligned} {\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2 & = {} \bigg \langle \frac{\root q \of {2}\mu _{1}._\epsilon \mu _{2}}{\root q \of {(2-\mu _{1}^q).\epsilon (2-\mu _{2}^q)+\mu _1^q._\epsilon \mu _2^q}},\\&\root q \of {\frac{(1+\nu _{1}^q)._\epsilon (1+\nu _{2}^q)-(1-\nu _{1}^q)._\epsilon (1-\nu _{2}^q)}{(1+\nu _{1}^q)._\epsilon (1+\nu _{2}^q)+(1-\nu _{1}^q)._\epsilon (1-\nu _{2}^q)}}\bigg \rangle . \end{aligned}$$

Take ${\mathfrak {a}}=(1+\nu _{1}^q)._\epsilon (1+\nu _{2}^q)$, ${\mathfrak {b}}=(1-\nu _{1}^q)._\epsilon (1-\nu _{2}^q)$, ${\mathfrak {c}}=\mu _1^q._\epsilon \mu _2^q$, and ${\mathfrak {d}}=(2-\mu _{1}^q).\epsilon (2-\mu _{2}^q),$ then

$$\begin{aligned} {\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2 & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}}}{\root q \of {{\mathfrak {d}}+{\mathfrak {c}}}}, \root q \of {\frac{{\mathfrak {a}}-{\mathfrak {b}}}{{\mathfrak {a}}+{\mathfrak {b}}}}\bigg \rangle . \end{aligned}$$

By the Einstein q-ROF law,

$$\begin{aligned}&({\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2)^\gamma \\ {} & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}}}{\root q \of {{\mathfrak {d}}+{\mathfrak {c}}}}, \root q \of {\frac{{\mathfrak {a}}-{\mathfrak {b}}}{{\mathfrak {a}}+{\mathfrak {b}}}}\bigg \rangle ^\gamma \\ & = {} \bigg \langle \frac{\root q \of {2}.(\frac{\root q \of {2{\mathfrak {c}}}}{\root q \of {{\mathfrak {d}}+{\mathfrak {c}}}})^\gamma }{\root q \of {(2-\frac{2{\mathfrak {c}}}{{\mathfrak {d}}+{\mathfrak {c}}})^\gamma +(\frac{2{\mathfrak {c}}}{{\mathfrak {d}}+{\mathfrak {c}}})^\gamma }},\\&\root q \of {\frac{(1+\frac{{\mathfrak {a}}-{\mathfrak {b}}}{{\mathfrak {a}}+{\mathfrak {b}}})^\gamma -(1-\frac{{\mathfrak {a}}-{\mathfrak {b}}}{{\mathfrak {a}}+{\mathfrak {b}}})^\gamma }{(1+\frac{{\mathfrak {a}}-{\mathfrak {b}}}{{\mathfrak {a}}+{\mathfrak {b}}})^\gamma +(1-\frac{{\mathfrak {a}}-{\mathfrak {b}}}{{\mathfrak {a}}+{\mathfrak {b}}})^\gamma }}\bigg \rangle \\ & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}^\gamma }}{\root q \of {{\mathfrak {d}}^\gamma +{\mathfrak {c}}^\gamma }}, \root q \of {\frac{{\mathfrak {a}}^\gamma -{\mathfrak {b}}^\gamma }{{\mathfrak {a}}^\gamma +{\mathfrak {b}}^\gamma }}\bigg \rangle \\ & = {} \bigg \langle \frac{\root q \of {2}\mu _{1}^\gamma ._\epsilon \mu _{2}^\gamma }{\root q \of {(2-\mu _{1}^q)^\gamma .\epsilon (2-\mu _{2}^q)^\gamma +(\mu _1^q)^\gamma ._\epsilon (\mu _2^q)^\gamma }}, \\&\root q \of {\frac{(1+\nu _{1}^q)^\gamma ._\epsilon (1+\nu _{2}^q)^\gamma -(1-\nu _{1}^q)^\gamma ._\epsilon (1-\nu _{2}^q)^\gamma }{(1+\nu _{1}^q)^\gamma ._\epsilon (1+\nu _{2}^q)^\gamma +(1-\nu _{1}^q)^\gamma ._\epsilon (1-\nu _{2}^q)^\gamma }}\bigg \rangle . \end{aligned}$$

On the other hand,

$$\begin{aligned} {{\mathfrak {F}}_1}^\gamma & = {} \bigg \langle \frac{\root q \of {2}\mu _{1}^\gamma }{\root q \of {(2-\mu _{1}^q)^\gamma +(\mu _{1}^q)^\gamma }}, \root q \of {\frac{(1+\nu _{1}^q)^\gamma -(1-\nu _{1}^q)^\gamma }{(1+\nu _{1}^q)^\gamma +(1-\nu _{1}^q)^\gamma }}\bigg \rangle \\ & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_1}}{\root q \of {{\mathfrak {d}}_1+{\mathfrak {c}}_1}}, \root q \of {\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}}\bigg \rangle , \end{aligned}$$

and

$$\begin{aligned} {{\mathfrak {F}}_2}^\gamma & = {} \bigg \langle \frac{\root q \of {2}\mu _{2}^\gamma }{\root q \of {(2-\mu _{2}^q)^\gamma +(\mu _{2}^q)^\gamma }}, \root q \of {\frac{(1+\nu _{2}^q)^\gamma -(1-\nu _{2}^q)^\gamma }{(1+\nu _{2}^q)^\gamma +(1-\nu _{2}^q)^\gamma }}\bigg \rangle \\ & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_2}}{\root q \of {{\mathfrak {d}}_2+{\mathfrak {c}}_2}}, \root q \of {\frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}\bigg \rangle , \end{aligned}$$

where ${\mathfrak {a}}_1=(1+\nu _{1}^q)^\gamma$, ${\mathfrak {b}}_1=(1-\nu _{1}^q)^\gamma$, ${\mathfrak {c}}_1=(\mu _{1}^q)^\gamma$, ${\mathfrak {d}}_1=(2-\mu _{1}^q)^\gamma$, ${\mathfrak {a}}_2=(1+\nu _{2}^q)^\gamma$, ${\mathfrak {b}}_2=(1-\nu _{2}^q)^\gamma$, ${\mathfrak {c}}_2=(\mu _{2}^q)^\gamma$, ${\mathfrak {d}}_2=(2-\mu _{2}^q)^\gamma$, therefore,

$$\begin{aligned} {{\mathfrak {F}}_1}^\gamma \otimes _\epsilon {{\mathfrak {F}}_2}^\gamma & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_1}}{\root q \of {{\mathfrak {d}}_1+{\mathfrak {c}}_1}}, \root q \of {\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}}\bigg \rangle \otimes _\epsilon \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_2}}{\root q \of {{\mathfrak {d}}_2+{\mathfrak {c}}_2}}, \root q \of {\frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}\bigg \rangle \\ & = {} \bigg \langle \frac{2^{\frac{2}{q}}\root q \of {\frac{{\mathfrak {c}}_1._\epsilon {\mathfrak {c}}_2}{({\mathfrak {d}}_1+{\mathfrak {c}}_1)._\epsilon ({\mathfrak {d}}_2+{\mathfrak {c}}_2)}}}{\root q \of {1+(1-\frac{2{\mathfrak {c}}_1}{{\mathfrak {d}}_1+{\mathfrak {c}}_1})._\epsilon (1-\frac{2{\mathfrak {c}}_2}{{\mathfrak {d}}_2+{\mathfrak {c}}_2})}},\\&\root q \of {\frac{\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}+\frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}{1+\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}._\epsilon \frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}}\bigg \rangle \\ & = {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_1._\epsilon {\mathfrak {c}}_2}}{\root q \of {{\mathfrak {d}}_1._\epsilon {\mathfrak {d}}_2+{\mathfrak {c}}_1._\epsilon {\mathfrak {c}}_2}}, \root q \of {\frac{{\mathfrak {a}}_1._\epsilon {\mathfrak {a}}_2-{\mathfrak {b}}_1._\epsilon {\mathfrak {b}}_2}{{\mathfrak {a}}_1._\epsilon {\mathfrak {a}}_2+{\mathfrak {b}}_1._\epsilon {\mathfrak {b}}_2}}\bigg \rangle \\ & = {} \bigg \langle \frac{\root q \of {2}\mu _{1}^\gamma ._\epsilon \mu _{2}^\gamma }{\root q \of {(2-\mu _{1}^q)^\gamma .\epsilon (2-\mu _{2}^q)^\gamma +(\mu _1^q)^\gamma ._\epsilon (\mu _2^q)^\gamma }}, \\&\root q \of {\frac{(1+\nu _{1}^q)^\gamma ._\epsilon (1+\nu _{2}^q)^\gamma -(1-\nu _{1}^q)^\gamma ._\epsilon (1-\nu _{2}^q)^\gamma }{(1+\nu _{1}^q)^\gamma ._\epsilon (1+\nu _{2}^q)^\gamma +(1-\nu _{1}^q)^\gamma ._\epsilon (1-\nu _{2}^q)^\gamma }}\bigg \rangle . \end{aligned}$$

Hence, $({\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2)^\gamma ={{\mathfrak {F}}_1}^\gamma \otimes _\epsilon {{\mathfrak {F}}_2}^\gamma .$

(v) For $\gamma _1, \gamma _2 > 0.$

$$\begin{aligned} {\mathfrak {F}}^{\gamma _1}= & {} \bigg \langle \frac{\root q \of {2}\mu ^{\gamma _1}}{\root q \of {(2-\mu ^q)^{\gamma _1}+(\mu ^q)^{\gamma _1}}}, \root q \of {\frac{(1+\nu ^q)^{\gamma _1}-(1-\nu ^q)^{\gamma _1}}{(1+\nu ^q)^{\gamma _1}+(1-\nu ^q)^{\gamma _1}}}\bigg \rangle \\= & {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_1}}{\root q \of {{\mathfrak {d}}_1+{\mathfrak {c}}_1}}, \root q \of {\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}}\bigg \rangle .\\ {\mathfrak {F}}^{\gamma _2}= & {} \bigg \langle \frac{\root q \of {2}\mu ^{\gamma _2}}{\root q \of {(2-\mu ^q)^{\gamma _2}+(\mu ^q)^{\gamma _2}}}, \root q \of {\frac{(1+\nu ^q)^{\gamma _2}-(1-\nu ^q)^{\gamma _2}}{(1+\nu ^q)^{\gamma _2}+(1-\nu ^q)^{\gamma _2}}}\bigg \rangle \\= & {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_2}}{\root q \of {{\mathfrak {d}}_2+{\mathfrak {c}}_2}}, \root q \of {\frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}\bigg \rangle , \end{aligned}$$

where ${\mathfrak {a}}_{\mathfrak {j}}=(1+\nu ^q)^{\gamma _{\mathfrak {j}}}$, ${\mathfrak {b}}_{\mathfrak {j}}=(1-\nu ^q)^{\gamma _{\mathfrak {j}}}$, ${\mathfrak {c}}_{\mathfrak {j}}=(\mu ^q)^{\gamma _{\mathfrak {j}}}$, ${\mathfrak {d}}_{\mathfrak {j}}=(2-\mu ^q)^{\gamma _{\mathfrak {j}}}$, for ${\mathfrak {j}}=1, 2.$

$$\begin{aligned} {\mathfrak {F}}^{\gamma _1}\otimes _\epsilon {\mathfrak {F}}^{\gamma _2}= & {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_1}}{\root q \of {{\mathfrak {d}}_1+{\mathfrak {c}}_1}}, \root q \of {\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}}\bigg \rangle \otimes _\epsilon \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_2}}{\root q \of {{\mathfrak {d}}_2+{\mathfrak {c}}_2}}, \root q \of {\frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}\bigg \rangle \\= & {} \bigg \langle \frac{2^{\frac{2}{q}}\root q \of {\frac{{\mathfrak {c}}_1._\epsilon {\mathfrak {c}}_2}{({\mathfrak {d}}_1+{\mathfrak {c}}_1)._\epsilon ({\mathfrak {d}}_2+{\mathfrak {c}}_2)}}}{\root q \of {1+(1-\frac{2{\mathfrak {c}}_1}{{\mathfrak {d}}_1+{\mathfrak {c}}_1})._\epsilon (1-\frac{2{\mathfrak {c}}_2}{{\mathfrak {d}}_2+{\mathfrak {c}}_2})}},\\&\root q \of {\frac{\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}+\frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}{1+\frac{{\mathfrak {a}}_1-{\mathfrak {b}}_1}{{\mathfrak {a}}_1+{\mathfrak {b}}_1}._\epsilon \frac{{\mathfrak {a}}_2-{\mathfrak {b}}_2}{{\mathfrak {a}}_2+{\mathfrak {b}}_2}}}\bigg \rangle \\= & {} \bigg \langle \frac{\root q \of {2{\mathfrak {c}}_1._\epsilon {\mathfrak {c}}_2}}{\root q \of {{\mathfrak {d}}_1._\epsilon {\mathfrak {d}}_2+{\mathfrak {c}}_1._\epsilon {\mathfrak {c}}_2}}, \root q \of {\frac{{\mathfrak {a}}_1._\epsilon {\mathfrak {a}}_2-{\mathfrak {b}}_1._\epsilon {\mathfrak {b}}_2}{{\mathfrak {a}}_1._\epsilon {\mathfrak {a}}_2+{\mathfrak {b}}_1._\epsilon {\mathfrak {b}}_2}}\bigg \rangle \\= & {} \bigg \langle \frac{\root q \of {2}\mu ^{\gamma _1+\gamma _2}}{\root q \of {(2-\mu ^q)^{\gamma _1+\gamma _2}+(\mu _1^q)^{\gamma _1+\gamma _2}}}, \\&\root q \of {\frac{(1+\nu ^q)^{\gamma _1+\gamma _2}-(1-\nu ^q)^{\gamma _1+\gamma _2}}{(1+\nu ^q)^{\gamma _1+\gamma _2}+(1-\nu ^q)^{\gamma _1+\gamma _2}}}\bigg \rangle \\= & {} {\mathfrak {F}}^{(\gamma _1+\gamma _2)}. \end{aligned}$$

Hence, ${\mathfrak {F}}^{\gamma _1}\otimes _\epsilon {\mathfrak {F}}^{\gamma _2}={\mathfrak {F}}^{(\gamma _1+\gamma _2)}.$ $\square$

Theorem 3

Let ${\mathfrak {F}}_1=\langle \mu _1, \nu _1\rangle$, ${\mathfrak {F}}_2=\langle \mu _2, \nu _2\rangle$ be two q-ROFNs, then

$\mathrm{(i)}\,{\mathfrak {F}}_{1}^c\wedge _\epsilon {\mathfrak {F}}_{2}^c=({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{2})^c$

$\mathrm{(ii)}\,{\mathfrak {F}}_{1}^c\vee _\epsilon {\mathfrak {F}}_{2}^c=({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{2})^c$

$\mathrm{(iii)}\,{\mathfrak {F}}_{1}^c\oplus _\epsilon {\mathfrak {F}}_{2}^c=({\mathfrak {F}}_{1}\otimes _\epsilon {\mathfrak {F}}_{2})^c$

$\mathrm{(iv)}\,{\mathfrak {F}}_{1}^c\otimes _\epsilon {\mathfrak {F}}_{2}^c=({\mathfrak {F}}_{1}\oplus _\epsilon {\mathfrak {F}}_{2})^c$

$\mathrm{(v)}\,({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{2})\oplus _\epsilon ({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{2})={\mathfrak {F}}_{1}\oplus _\epsilon {\mathfrak {F}}_{2}$

$\mathrm{(vi)}\,({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{2})\otimes _\epsilon ({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{2})={\mathfrak {F}}_{1}\otimes _\epsilon {\mathfrak {F}}_{2}$

Proof

It is obvious. $\square$

Theorem 4

Let ${\mathfrak {F}}_1=\langle \mu _1, \nu _1\rangle$, ${\mathfrak {F}}_2=\langle \mu _2, \nu _2\rangle$ and ${\mathfrak {F}}_3=\langle \mu _3, \nu _3\rangle$ be three q-ROFNs, then

$\mathrm{(i)}\,({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{2})\wedge _\epsilon {\mathfrak {F}}_3=({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{3})\vee _\epsilon ({\mathfrak {F}}_{2}\wedge _\epsilon {\mathfrak {F}}_{3})$

$\mathrm{(ii)}\,({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{2})\vee _\epsilon {\mathfrak {F}}_3=({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{3})\wedge _\epsilon ({\mathfrak {F}}_{2}\vee _\epsilon {\mathfrak {F}}_{3})$

$\mathrm{(iii)}\,({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{2})\oplus _\epsilon {\mathfrak {F}}_3=({\mathfrak {F}}_{1}\oplus _\epsilon {\mathfrak {F}}_{3})\vee _\epsilon ({\mathfrak {F}}_{2}\oplus _\epsilon {\mathfrak {F}}_{3})$

$\mathrm{(iv)}\,({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{2})\oplus _\epsilon {\mathfrak {F}}_3=({\mathfrak {F}}_{1}\oplus _\epsilon {\mathfrak {F}}_{3})\wedge _\epsilon ({\mathfrak {F}}_{2}\oplus _\epsilon {\mathfrak {F}}_{3})$

$\mathrm{(v)}\,({\mathfrak {F}}_{1}\vee _\epsilon {\mathfrak {F}}_{2})\otimes _\epsilon {\mathfrak {F}}_3=({\mathfrak {F}}_{1}\otimes _\epsilon {\mathfrak {F}}_{3})\vee _\epsilon ({\mathfrak {F}}_{2}\otimes _\epsilon {\mathfrak {F}}_{3})$

$\mathrm{(vi)}\,({\mathfrak {F}}_{1}\wedge _\epsilon {\mathfrak {F}}_{2})\otimes _\epsilon {\mathfrak {F}}_3=({\mathfrak {F}}_{1}\otimes _\epsilon {\mathfrak {F}}_{3})\wedge _\epsilon ({\mathfrak {F}}_{2}\otimes _\epsilon {\mathfrak {F}}_{3})$

Proof

It is obvious. $\square$

4 q-Rung orthopair fuzzy Einstein weighted geometric operators

The Einstein weighted geometric operators under q-ROF environment are defined here.

Definition 7

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle ({\mathfrak {j}}=1, 2,\ldots ,{\mathfrak {s}})$ be a collection of q-ROFNs and $\varphi _{\mathfrak {j}}$ is the weight vector (WV) of ${\mathfrak {F}}_{\mathfrak {j}}$ with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \varphi _{\mathfrak {j}}=1$, then q-ROFEWG operator is a mapping ${\mathcal {Q}}^{\mathfrak {s}}\rightarrow {\mathcal {Q}}$ such that

$$\begin{aligned} q-\hbox {ROFEWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})={{\mathfrak {F}}_1}^{\varphi _1}\otimes _\epsilon {{\mathfrak {F}}_2}^{\varphi _2}\otimes _\epsilon \ldots \otimes _\epsilon {{\mathfrak {F}}_{\mathfrak {s}}}^{\varphi _{\mathfrak {s}}}. \end{aligned}$$

(1)

If $\varphi _{\mathfrak {j}}=1/{\mathfrak {s}}$, $\forall \,{\mathfrak {j}}$, then q-ROFEWG operator becomes q-ROFWG operator

$$\begin{aligned} q-\hbox {ROFWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=( {\mathfrak {F}}_1\otimes _\epsilon {\mathfrak {F}}_2\otimes _\epsilon \ldots \otimes _\epsilon {\mathfrak {F}}_{\mathfrak {s}})^\frac{1}{{\mathfrak {s}}}. \end{aligned}$$

Theorem 5

Let ${\mathfrak {F}}_{\mathfrak {j}}=(\mu _{\mathfrak {j}}, \nu _{\mathfrak {j}})\in q$-ROFNs, then aggregated value using Equation 1is

$$\begin{aligned}&q-\hbox {ROFEWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}) \nonumber \\= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}, \nonumber \\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1- \nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

(2)

Proof

We prove Eq. 2 by mathematical induction.

When ${\mathfrak {s}}=2$,

$$\begin{aligned} q-ROFEWG({\mathfrak {F}}_1, {\mathfrak {F}}_2)={{\mathfrak {F}}_1}^{\varphi _1}\otimes _\epsilon {{\mathfrak {F}}_2}^{\varphi _2}. \end{aligned}$$

From Theorem 1, both ${{\mathfrak {F}}_1}^{\varphi _1}$ and ${{\mathfrak {F}}_2}^{\varphi _2}$ are q-ROFNs and value of ${{\mathfrak {F}}_1}^{\varphi _1}\otimes _\epsilon {{\mathfrak {F}}_2}^{\varphi _2}$ is a q-ROFN. Using (vii) in Definition 6,

$$\begin{aligned} {{\mathfrak {F}}_1}^{\varphi _1}= & {} \bigg \langle \frac{\root q \of {2}\mu _{1}^{\varphi _1}}{\root q \of {(2-\mu _{1}^q)^{\varphi _1}+(\mu _{1}^q)^{\varphi _1}}}, \root q \of {\frac{(1+\nu _{1}^q)^{\varphi _1}-(1-\nu _{1}^q)^{\varphi _1}}{(1+\nu _{1}^q)^{\varphi _1}+(1-\nu _{1}^q)^{\varphi _1}}}\bigg \rangle .\\ {{\mathfrak {F}}_2}^{\varphi _2}= & {} \bigg \langle \frac{\root q \of {2}\mu _{2}^{\varphi _2}}{\root q \of {(2-\mu _{2}^q)^{\varphi _2}+(\mu _{2}^q)^{\varphi _2}}}, \root q \of {\frac{(1+\nu _{2}^q)^{\varphi _2}-(1-\nu _{2}^q)^{\varphi _2}}{(1+\nu _{2}^q)^{\varphi _2}+(1-\nu _{2}^q)^{\varphi _2}}}\bigg \rangle . \end{aligned}$$

Then

$$\begin{aligned} q-\hbox {ROFEWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2)= & {} {{\mathfrak {F}}_1}^{\varphi _1}\otimes _\epsilon {{\mathfrak {F}}_2}^{\varphi _2} \\= & {} \Bigg \langle \frac{\left( \frac{\root q \of {2}\mu _{1}^{\varphi _1}}{\root q \of {(2-\mu _{1}^q)^{\varphi _1}+(\mu _{1}^q)^{\varphi _1}}}\right) ._\epsilon \left( \frac{\root q \of {2}\mu _{2}^{\varphi _2}}{\root q \of {(2-\mu _{2}^q)^{\varphi _2}+(\mu _{2}^q)^{\varphi _2}}}\right) }{\root q \of {1+\left( 1-\frac{2\mu _{1}^{q\varphi _1}}{(2-\mu _{1}^q)^{\varphi _1}+(\mu _{1}^q)^{\varphi _1}}\right) ._\epsilon \left( 1-\frac{2\mu _{2}^{q\varphi _2}}{(2-\mu _{2}^q)^{\varphi _2}+(\mu _{2}^q)^{\varphi _2}}\right) }}, \\&\root q \of {\frac{\frac{(1+\nu _{1}^q)^{\varphi _1}-(1-\nu _{1}^q)^{\varphi _1}}{(1+\nu _{1}^q)^{\varphi _1}+(1-\nu _{1}^q)^{\varphi _1}}+ \frac{(1+\nu _{2}^q)^{\varphi _2}-(1-\nu _{2}^q)^{\varphi _2}}{(1+\nu _{2}^q)^{\varphi _2}+(1-\nu _{2}^q)^{\varphi _2}}}{1+ \left( \frac{(1+\nu _{1}^q)^{\varphi _1}-(1-\nu _{1}^q)^{\varphi _1}}{(1+\nu _{1}^q)^{\varphi _1}+(1-\nu _{1}^q)^{\varphi _1}}\right) ._\epsilon \left( \frac{(1+\nu _{2}^q)^{\varphi _2}-(1-\nu _{2}^q)^{\varphi _2}}{(1+\nu _{2}^q)^{\varphi _2}+(1-\nu _{2}^q)^{\varphi _2}}\right) }}\Bigg \rangle \\= & {} \Bigg \langle \frac{\root q \of {2}\mu _{1}^{\varphi _1}._\epsilon \mu _{2}^{\varphi _2}}{\root q \of {(2-\mu _{1}^q)^{\varphi _1}.\epsilon (2-\mu _{2}^q)^{\varphi _2}+(\mu _1^q)^{\varphi _1}._\epsilon (\mu _2^q)^{\varphi _2}}}, \\&\root q \of {\frac{(1+\nu _{1}^q)^{\varphi _1}._\epsilon (1+\nu _{2}^q)^{\varphi _2}-(1-\nu _{1}^q)^{\varphi _1}._\epsilon (1-\nu _{2}^q)^{\varphi _2}}{(1+\nu _{1}^q)^{\varphi _1}._\epsilon (1+\nu _{2}^q)^{\varphi _2}+(1-\nu _{1}^q)^{\varphi _1}._\epsilon (1-\nu _{2}^q)^{\varphi _2}}}\Bigg \rangle . \end{aligned}$$

Thus, result holds when ${\mathfrak {s}}=2$.

Suppose result holds for ${\mathfrak {s}}=k$,

$$\begin{aligned} q-\hbox {ROFEWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_k)= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^k\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^k(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^k(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}},\\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^k(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^k(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^k(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^k(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

Now for ${\mathfrak {s}}=k+1$, $q-\hbox {ROFEWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{k+1})$

$$\begin{aligned}= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^k\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^k(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^k(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}, \\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^k(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^k(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^k(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^k(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle \otimes _\epsilon \\&\Bigg \langle \frac{\root q \of {2}\mu _{k+1}^{\varphi _{k+1}}}{\root q \of {(2-\mu _{k+1}^q)^{\varphi _{k+1}}+(\mu _{k+1}^q)^{\varphi _{k+1}}}},\\&\root q \of {\frac{(1+\nu _{k+1}^q)^{\varphi _{k+1}}-(1-\nu _{k+1}^q)^{\varphi _{k+1}}}{(1+\nu _{k+1}^q)^{\varphi _{k+1}}+(1-\nu _{k+1}^q)^{\varphi _{k+1}}}}\Bigg \rangle \\ {}= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{k+1}\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{k+1}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{k+1}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}},\\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{k+1}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{k+1}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{k+1}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{k+1}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

Thus, result is true for ${\mathfrak {s}}=k+1$. Hence, Equation 2 holds, $\forall$ ${\mathfrak {s}}$. $\square$

Lemma 1

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle , \varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}{\varphi _{\mathfrak {j}}}=1$, then

$$\begin{aligned} \prod \limits _{{\mathfrak {j}}=1}^s{\mathfrak {F}}_{\mathfrak {j}}^{\varphi _{\mathfrak {j}}}\le \sum \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}{\varphi _{\mathfrak {j}}{\mathfrak {F}}_{\mathfrak {j}}}, \end{aligned}$$

equality holds iff ${\mathfrak {F}}_1={\mathfrak {F}}_2=\ldots ={\mathfrak {F}}_{\mathfrak {s}}$.

Theorem 6

If ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in$ q-ROFNs, then

$$\begin{aligned} q-ROFEWG({\mathfrak {F}}_1, {\mathfrak {F}}_2, \ldots , {\mathfrak {F}}_{\mathfrak {s}})\in q-ROFN. \end{aligned}$$

Proof

Since ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in$ q-ROFNs, so $0\le \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\le 1$ and $0\le \mu _{{\mathfrak {j}}}^q+\nu _{{\mathfrak {j}}}^q\le 1$. Therefore,

$$\begin{aligned} \frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}= & {} 1-\frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1- \nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\nonumber \\&\le 1-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1- \nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}\le 1. \end{aligned}$$

(3)

Also, $(1+\nu _{{\mathfrak {j}}}^q)\ge (1-\nu _{{\mathfrak {j}}}^q)\Rightarrow \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)\ge 0.$Therefore,

$$\begin{aligned} \frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\ge 0. \end{aligned}$$

Thus, $0\le \nu _{q-\hbox {ROFEWG}}\le 1.$

Moreover,

$$\begin{aligned} \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{{\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\le & {} \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\\\le & {} \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}\le 1. \end{aligned}$$

Also,

$$\begin{aligned} \prod \limits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}\ge 0\Leftrightarrow \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{{\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\ge 0. \end{aligned}$$

Thus, $0\le \mu _{q-\hbox {ROFEWG}}\le 1.$ Moreover,

$$\begin{aligned}&\mu _{q-\hbox {ROFEWG}}^q+\nu _{q-\hbox {ROFEWG}}^q\\&\quad = \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{{\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\\&\quad \,\,\,\,+ \frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\\&\quad \le \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\\&\quad \,\,\,\, +1- \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\\&\quad =1. \end{aligned}$$

Hence, q-$\hbox {ROFEWG}\,\in [0,1]$. Therefore,

$$\begin{aligned} q-\hbox {ROFEWG} ({\mathfrak {F}}_1, {\mathfrak {F}}_2, \ldots , {\mathfrak {F}}_{\mathfrak {s}})\in q-\hbox {ROFN}. \end{aligned}$$

Corollary 1

The q-ROFEWG and q-ROFWG operators have the relationship:

q-ROFEWG$({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})\ge$ q-ROFWG$({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}).$

Proof

Let q-ROFEWG $({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=(\mu _{\mathfrak {F}}^\beta , \nu _{\mathfrak {F}}^\beta )={\mathfrak {F}}^\beta$ and q-ROFWG $({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=(\mu _{\mathfrak {F}}, \nu _{\mathfrak {F}})={\mathfrak {F}}$. Since $\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}\le \sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}=2,$ then from 3, we get

$$\begin{aligned} \root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\le & {} \root q \of {1-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\\\Leftrightarrow \nu _{\mathfrak {F}}^\beta\le & {} \nu _{\mathfrak {F}}, \end{aligned}$$

equality holds iff $\nu _1=\nu _2=\ldots =\nu _{\mathfrak {s}}$.

Also,

$$\begin{aligned} \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{{\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\ge & {} \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{{\sum \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}\varphi _{\mathfrak {j}}(2-\mu _{{\mathfrak {j}}}^q)}+ \sum \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}\varphi _{\mathfrak {j}}\mu _{{\mathfrak {j}}}^q}\\ {}\ge & {} \prod \limits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}} (\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}. \end{aligned}$$

This implies

$$\begin{aligned} \root q \of {\frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}} (\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{{\prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\ge & {} \prod \limits _{{\mathfrak {j}}=1}^{{\mathfrak {s}}}\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}\\\Rightarrow&\mu _{\mathfrak {F}}^\beta \ge \mu _{\mathfrak {F}}, \end{aligned}$$

equality holds iff $\mu _1=\mu _2=\ldots =\mu _{\mathfrak {s}}$.

Thus,

$$\begin{aligned} {\mathcal {S}}({\mathfrak {F}}^\beta )=(\mu _{\mathfrak {F}}^\beta )^q-(\nu _{\mathfrak {F}}^\beta )^q\ge (\mu _{\mathfrak {F}})^q-(\nu _{\mathfrak {F}})^q={\mathcal {S}}({\mathfrak {F}}). \end{aligned}$$

If ${\mathcal {S}}({\mathfrak {F}}^\beta )>{\mathcal {S}}({\mathfrak {F}}),$ then q-ROFEWG$({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})>$ q-ROFWA$({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}).$ If ${\mathcal {S}}({\mathfrak {F}}^\beta )={\mathcal {S}}({\mathfrak {F}}),$ that is, $(\mu _{\mathfrak {F}}^\beta )^q-(\nu _{\mathfrak {F}}^\beta )^q=(\mu _{\mathfrak {F}})^q-(\nu _{\mathfrak {F}})^q,$ then by condition $\mu _{\mathfrak {F}}^\beta \ge \mu _{\mathfrak {F}}$ and $\nu _{\mathfrak {F}}^\beta \le \nu _{\mathfrak {F}}$; thus, the accuracy function ${\mathcal {A}}({\mathfrak {F}}^\beta )=(\mu _{\mathfrak {F}}^\beta )^q-(\nu _{\mathfrak {F}}^\beta )^q=(\mu _{\mathfrak {F}})^q-(\nu _{\mathfrak {F}})^q={\mathcal {A}}({\mathfrak {F}}).$

Thus

$$\begin{aligned} q-\hbox {ROFEWG}({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=q-\hbox {ROFWG}({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}). \end{aligned}$$

Hence,

$$\begin{aligned} q-\hbox {ROFEWG}({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})\ge q-\hbox {ROFWG}({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}), \end{aligned}$$

equality holds iff ${\mathfrak {F}}_1={\mathfrak {F}}_2=\ldots ={\mathfrak {F}}_{\mathfrak {s}}$. $\square$

Example 1

Let ${\mathfrak {F}}_1=(0.7, 0.6)$, ${\mathfrak {F}}_2=(0.8,0.4)$, ${\mathfrak {F}}_3=(0.5,0.7)$ and ${\mathfrak {F}}_4=(0.7,0.5)$ be four q-ROFNs and $\varphi =(0.3, 0.2,0.3,0.2)^T$, take $q=3$, then

$$\begin{aligned} q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4)= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{4}\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{4}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{4}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}},\\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{4}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}} -\prod \nolimits _{{\mathfrak {j}}=1}^{4}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{4}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{4}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle \\= & {} \Bigg \langle \frac{\root 3 \of {2}\times 0.65}{\root 3 \of {1.68+0.27}}, \root 3 \of {\frac{1.20-0.79}{1.20+0.79}}\Bigg \rangle \\ {}= & {} \langle 0.66, 0.59 \rangle . \end{aligned}$$

Now,

$$\begin{aligned} q-\hbox {ROFWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4)= & {} \Bigg \langle \prod \limits _{{\mathfrak {j}}=1}^4(\mu _{\mathfrak {j}})^{\varphi _{\mathfrak {j}}}, \root q \of {1-\prod \limits _{{\mathfrak {j}}=1}^4(1-\nu _{\mathfrak {j}}^q)^{\varphi _{\mathfrak {j}}}}\Bigg \rangle \\= & {} \langle 0.65, 0.59 \rangle . \end{aligned}$$

$\Rightarrow q$-$\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4)\ge q$-ROFWG$({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4).$

Proposition 1

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in q$-ROFNs and $\varphi _{\mathfrak {j}}$ is the weight of ${\mathfrak {F}}_{\mathfrak {j}}$, such that $\varphi _{\mathfrak {j}}\in [0,1]$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}=1$.

(i) Idempotency: If ${\mathfrak {F}}_{\mathfrak {j}}={\mathfrak {F}}_o=\langle \mu _o, \nu _o\rangle$ for all ${\mathfrak {j}}$, then

$$\begin{aligned} q-ROFEWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})={\mathfrak {F}}_o. \end{aligned}$$

Proof

As ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _o,\nu _o\rangle \in q$-ROFNs, $\forall \,{\mathfrak {j}}$, then

$$\begin{aligned}&q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})\\&\quad = \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\mu _{o}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{o}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{o}^q)^{\varphi _{\mathfrak {j}}}}},\\&\quad \root q \of {\frac{\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{o}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{o}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+ \nu _{o}^q)^{\varphi _{\mathfrak {j}}}+\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{o}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle \\&\quad =\Bigg \langle \frac{\root q \of {2}\mu _{o}^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}}{\root q \of {(2-\nu _{o}^q)^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}+ (\mu _{o}^q)^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}}},\\&\quad \root q \of {\frac{(1+\nu _{o}^q)^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}- (1-\nu _{o}^q)^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}}{(1+\nu _{o}^q)^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}+ (1-\nu _{o}^q)^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}}}\Bigg \rangle \\&\quad =\langle \mu _o,\nu _o\rangle . \end{aligned}$$

$\square$

(ii) Boundedness: Let ${\mathfrak {F}}^-=(\hbox {min}_{\mathfrak {j}}(\mu _{\mathfrak {j}}),\hbox {max}_{\mathfrak {j}}(\nu _{\mathfrak {j}}))$, ${\mathfrak {F}}^+=(\hbox {max}_{\mathfrak {j}}(\mu _{\mathfrak {j}}),\hbox {min}_{\mathfrak {j}}(\nu _{\mathfrak {j}}))$, then

$$\begin{aligned} {\mathfrak {F}}^-\le q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})\le {\mathfrak {F}}^+. \end{aligned}$$

Proof

Consider $f(x)=\frac{1-x}{1+x}, x\in [0,1]$, then $f'(x)=-\frac{2}{(1+x)^2}<0$, so f(x) is a decreasing function (DF). As $\nu _{{\mathfrak {j}},min}^q\le \nu _{\mathfrak {j}}^q\le \nu _{{\mathfrak {j}},max}^q, \forall \,{\mathfrak {j}}=1, 2, \ldots , {\mathfrak {s}},$ then $f(\nu _{{\mathfrak {j}},max}^q)\le f(\nu _{\mathfrak {j}}^q)\le f(\nu _{{\mathfrak {j}},min}^q), \forall \,{\mathfrak {j}}$, that is, $\frac{1-\nu _{{\mathfrak {j}},max}^q}{1+\nu _{{\mathfrak {j}},max}^q}\le \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\le \frac{1-\nu _{{\mathfrak {j}},min}^q}{1+\nu _{{\mathfrak {j}},min}^q}$, $\forall \,{\mathfrak {j}}$. Let $\varphi _{\mathfrak {j}}\in [0,1]$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}=1$, then

$$\begin{aligned}&\left( \frac{1-\nu _{{\mathfrak {j}},\hbox {max}}^q}{1+\nu _{{\mathfrak {j}},\hbox {max}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\\&\quad \le \left( \frac{1-\nu _{{\mathfrak {j}},\hbox {min}}^q}{1+\nu _{{\mathfrak {j}},\hbox {min}}^q}\right) ^{\varphi _{\mathfrak {j}}} \\&\quad \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{{\mathfrak {j}},max}^q}{1+\nu _{{\mathfrak {j}},max}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \left( \frac{1-\nu _{{\mathfrak {j}},\hbox {min}}^q}{1+\nu _{{\mathfrak {j}},\hbox {min}}^q}\right) ^{\varphi _{\mathfrak {j}}} \\&\quad \Leftrightarrow \left( \frac{1-\nu _{{\mathfrak {j}},\hbox {max}}^q}{1+\nu _{{\mathfrak {j}},\hbox {max}}^q}\right) ^{\sum \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}\le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{1-\nu _{{\mathfrak {j}},min}^q}{1+\nu _{{\mathfrak {j}},min}^q}\right) ^{\sum \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}} \\&\quad \Leftrightarrow \left( \frac{1-\nu _{{\mathfrak {j}},\hbox {max}}^q}{1+\nu _{{\mathfrak {j}},\hbox {max}}^q}\right) \le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{1-\nu _{{\mathfrak {j}},\hbox {min}}^q}{1+\nu _{{\mathfrak {j}},\hbox {min}}^q}\right) \\&\quad \Leftrightarrow \left( \frac{2}{1+\nu _{{\mathfrak {j}},\hbox {max}}^q}\right) \le 1+\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{2}{1+\nu _{{\mathfrak {j}},\hbox {min}}^q}\right) \\&\quad \Leftrightarrow \left( \frac{1+\nu _{{\mathfrak {j}},\hbox {min}}^q}{2}\right) \le \frac{1}{1+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}}\le \left( \frac{1+\nu _{{\mathfrak {j}},\hbox {max}}^q}{2}\right) \\&\quad \Leftrightarrow \left( {1+\nu _{{\mathfrak {j}},\hbox {min}}^q}\right) \le \frac{2}{1+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}}\le \left( {1+\nu _{{\mathfrak {j}},\hbox {max}}^q}\right) \\&\quad \Leftrightarrow {\nu _{{\mathfrak {j}},\hbox {min}}^q}\le \frac{2}{1+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{1-\nu _{\mathfrak {j}}^q}{1+\nu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}}-1\le {\nu _{{\mathfrak {j}},\hbox {max}}^q} \\&\quad \Leftrightarrow {\nu _{{\mathfrak {j}},\hbox {min}}^q}\le \frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\le {\nu _{{\mathfrak {j}},\hbox {max}}^q} \end{aligned}$$

Thus,

$$\begin{aligned} {\nu _{{\mathfrak {j}},\hbox {min}}}\le \root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\le {\nu _{{\mathfrak {j}},\hbox {max}}}. \end{aligned}$$

(4)

Consider $g(y)=\frac{2-y}{y}, y\in (0,1]$, then $g'(y)=-\frac{2}{y^2}$, i.e., g(y) is a DF on (0, 1]. Since $\mu _{{\mathfrak {j}},\hbox {min}}^q\le \mu _{\mathfrak {j}}^q\le \mu _{{\mathfrak {j}},\hbox {max}}^q, \forall \,{\mathfrak {j}},$ then $g(\mu _{{\mathfrak {j}},\hbox {max}}^q)\le g(\mu _{\mathfrak {j}}^q)\le g(\mu _{{\mathfrak {j}},\hbox {min}}^q), \forall \,{\mathfrak {j}}$, i.e., $\frac{2-\mu _{{\mathfrak {j}},\hbox {max}}^q}{\mu _{{\mathfrak {j}},\hbox {max}}^q}\le \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\le \frac{2-\mu _{{\mathfrak {j}},\hbox {min}}^q}{\mu _{{\mathfrak {j}},\hbox {min}}^q}.$ Then,

$$\begin{aligned}&\left( \frac{2-\mu _{{\mathfrak {j}},max}^q}{\mu _{{\mathfrak {j}},\hbox {max}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{2-\mu _{{\mathfrak {j}},\hbox {min}}^q}{\mu _{{\mathfrak {j}},\hbox {min}}^q}\right) ^{\varphi _{\mathfrak {j}}} \\&\quad \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{{\mathfrak {j}},\hbox {max}}^q}{\mu _{{\mathfrak {j}},\hbox {max}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{{\mathfrak {j}},\hbox {min}}^q}{\mu _{{\mathfrak {j}},\hbox {min}}^q}\right) ^{\varphi _{\mathfrak {j}}} \\&\quad \Rightarrow \left( \frac{2-\mu _{{\mathfrak {j}},max}^q}{\mu _{{\mathfrak {j}},\hbox {max}}^q}\right) ^{\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}}\le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{2-\mu _{{\mathfrak {j}},\hbox {min}}^q}{\mu _{{\mathfrak {j}},\hbox {min}}^q}\right) ^{\sum \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}} \\&\quad \Rightarrow \left( \frac{2-\mu _{{\mathfrak {j}},max}^q}{\mu _{{\mathfrak {j}},\hbox {max}}^q}\right) \le \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{{\mathfrak {j}}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{2-\mu _{{\mathfrak {j}},min}^q}{\mu _{{\mathfrak {j}},\hbox {min}}^q}\right) \\&\quad \Rightarrow \left( \frac{2}{\mu _{{\mathfrak {j}},\hbox {max}}^q}\right) \le 1+\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}\le \left( \frac{2}{\mu _{{\mathfrak {j}},\hbox {min}}^q}\right) \\&\quad \Rightarrow \left( \frac{\mu _{{\mathfrak {j}},\hbox {min}}^q}{2}\right) \le \frac{1}{1+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}}\le \left( \frac{\mu _{{\mathfrak {j}},\hbox {max}}^q}{2}\right) \\&\quad \Rightarrow \left( {\mu _{{\mathfrak {j}},min}^q}\right) \le \frac{2}{1+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}}\le \left( {\mu _{{\mathfrak {j}},\hbox {max}}^q}\right) \\&\quad \Rightarrow {\mu _{{\mathfrak {j}},\hbox {min}}^q}\le \frac{2}{1+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{2-\mu _{\mathfrak {j}}^q}{\mu _{\mathfrak {j}}^q}\right) ^{\varphi _{\mathfrak {j}}}}\le {\mu _{{\mathfrak {j}},\hbox {max}}^q} \\&\quad \Rightarrow {\mu _{{\mathfrak {j}},\hbox {min}}^q}\le \frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}\le {\mu _{{\mathfrak {j}},\hbox {max}}^q} \end{aligned}$$

$$\begin{aligned} \Rightarrow {\mu _{{\mathfrak {j}},\hbox {min}}}\le \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\le {\mu _{{\mathfrak {j}},\hbox {max}}}. \end{aligned}$$

(5)

Let q-$\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})={\mathfrak {F}}=\langle \mu _{\mathfrak {F}}, \nu _{\mathfrak {F}}\rangle$, then from (4) and (5),

$$\begin{aligned} \mu _{\hbox {min}}\le \mu _{\mathfrak {F}}\le \mu _{\hbox {max}}, \,\,\nu _{\hbox {min}}\le \nu _{\mathfrak {F}}\le \nu _{\hbox {max}}, \end{aligned}$$

where $\mu _{\hbox {min}}=\min \limits _{\mathfrak {j}}\{\mu _{\mathfrak {j}}\}$, $\mu _{\hbox {max}}=\max \limits _{\mathfrak {j}}\{\mu _{\mathfrak {j}}\}$, $\nu _{\hbox {min}}=\min \limits _{\mathfrak {j}}\{\nu _{\mathfrak {j}}\}$, and $\nu _{\hbox {max}}=\max \limits _{\mathfrak {j}}\{\nu _{\mathfrak {j}}\}$. So, ${\mathcal {S}}({\mathfrak {F}})=\mu _{\mathfrak {F}}^q-\nu _{\mathfrak {F}}^q\le \mu _{\hbox {max}}^{q}-\nu _{\hbox {min}}^{q}={\mathcal {S}}({\mathfrak {F}}^+)$ and ${\mathcal {S}}({\mathfrak {F}})=\mu _{\mathfrak {F}}^q-\nu _{\mathfrak {F}}^q\ge \mu _{\hbox {min}}^{q}-\nu _{\hbox {max}}^{q}={\mathcal {S}}({\mathfrak {F}}^-).$ As $S({\mathfrak {F}})< S({\mathfrak {F}}^+)$ and $S({\mathfrak {F}})> S({\mathfrak {F}}^-)$. So

$$\begin{aligned} {\mathfrak {F}}^-\le q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})\le {\mathfrak {F}}^+. \end{aligned}$$

$\square$

(iii) Monotonicity: When ${\mathfrak {F}}_{\mathfrak {j}}\le {\mathcal {P}}_{\mathfrak {j}}, \forall \,{\mathfrak {j}}$, then

$$\begin{aligned} q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})\le q-\hbox {ROFEWG}({\mathcal {P}}_1, {\mathcal {P}}_2,\ldots , {\mathcal {P}}_{\mathfrak {s}}). \end{aligned}$$

Proof

It is similar to (ii), so we omit it. $\square$

(iv) If $\eta =\langle \mu _\eta , \nu _\eta \rangle$ is a q-ROFN, then

$q-\hbox {ROFEWG}(\eta \otimes _\epsilon {\mathfrak {F}}_1, \eta \otimes _\epsilon {\mathfrak {F}}_2,\ldots , \eta \otimes _\epsilon {\mathfrak {F}}_{\mathfrak {s}})=\eta \otimes q-ROFEWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}}).$

Proof

As $R_i$ and $\eta$ are q-ROFNs, $\forall \,{\mathfrak {j}}$,

$$\begin{aligned} \eta \otimes _\epsilon R_{\mathfrak {j}}= & {} \Bigg \langle \frac{\root q \of {2}\mu _{R_{\mathfrak {j}}}._\epsilon \mu _{\eta }}{\root q \of {(2-\mu _{R_{\mathfrak {j}}}^q)._\epsilon (2-\mu _{\eta }^q)+\mu _{R_{\mathfrak {j}}}^q._\epsilon \mu _{\eta }^q}},\\&\sqrt{\frac{(1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)-(1-\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1-\nu _{\eta }^q)}{(1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)+(1-\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1-\nu _{\eta }^q)}} \Bigg \rangle. \end{aligned}$$

Therefore, by Theorem 5, $q-\hbox {ROFEWG}(\eta \otimes _\epsilon {\mathfrak {F}}_1, \eta \otimes _\epsilon {\mathfrak {F}}_2,\ldots , \eta \otimes _\epsilon {\mathfrak {F}}_{\mathfrak {s}})$

$$\begin{aligned}= & {} \Bigg \langle \root q \of {\frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \alpha \right) ^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 2-\alpha \right) ^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \alpha \right) ^{\varphi _{\mathfrak {j}}}}},\\&\frac{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1+\beta \right) ^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1-\beta \right) ^{\varphi _{\mathfrak {j}}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1+\beta \right) ^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1-\beta \right) ^{\varphi _{\mathfrak {j}}}}} \Bigg \rangle \\= & {} \Bigg \langle \root q \of {\frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \mu _{R_{\mathfrak {j}}}^q\mu _{\eta }^q\right) ^{\varphi _{\mathfrak {j}}}}{\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( (2-\mu _{R_{\mathfrak {j}}}^q)._\epsilon (2-\mu _{\eta }^q)\right) ^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \mu _{R_{\mathfrak {j}}}^q\mu _{\eta }^q\right) ^{\varphi _{\mathfrak {j}}}}},\\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( (1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)\right) ^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( (1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)\right) ^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( (1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)\right) ^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( (1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)\right) ^{\varphi _{\mathfrak {j}}}}} \Bigg \rangle \\= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{R_{\mathfrak {j}}})^{\varphi _{\mathfrak {j}}}._\epsilon \mu _{\eta }}{\root q \of {\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{R_{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}._\epsilon (2-\mu _{\eta }^q)+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{R_{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}._\epsilon \mu _{\eta }^q}},\\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{R_{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}._\epsilon (1+\nu _{\eta }^q)- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{R_{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}._\epsilon (1+\nu _{\eta }^q)}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{R_{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}._\epsilon (1+\nu _{\eta }^q)+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{R_{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}._\epsilon (1+\nu _{\eta }^q)}}\Bigg \rangle \\= & {} \langle \mu _{\eta }, \nu _{\eta }\rangle \otimes q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})\\= & {} \eta \otimes q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}}). \end{aligned}$$

where

$$\begin{aligned}&\alpha =\frac{2\mu _{R_i}^q.\mu _{\beta }^q}{(2-\mu _{R_{\mathfrak {j}}}^q)._\epsilon (2-\mu _{\beta }^q.)+\mu _{R_i}^q.\mu _{\beta }^q},\\&\beta =\frac{(1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)- (1-\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1-\nu _{\eta }^q)}{(1+\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1+\nu _{\eta }^q)+(1-\nu _{R_{\mathfrak {j}}}^q)._\epsilon (1-\nu _{\eta }^q)}. \end{aligned}$$

Hence, $q-\hbox {ROFEWG}(\eta \otimes _\epsilon {\mathfrak {F}}_1, \eta \otimes _\epsilon {\mathfrak {F}}_2,\ldots , \eta \otimes _\epsilon {\mathfrak {F}}_{\mathfrak {s}})=\eta \otimes q-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}}).$ $\square$

5 q-Rung orthopair fuzzy Einstein ordered weighted geometric operators

Definition 8

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in$ q-ROFNs and $\varphi _{\mathfrak {j}}$ is the weight of ${\mathfrak {F}}_{\mathfrak {j}}$ with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \varphi _{\mathfrak {j}}=1$, then q-ROFEOWG operator is a mapping ${\mathcal {Q}}^{\mathfrak {s}}\rightarrow {\mathcal {Q}}$ such that

$$\begin{aligned} q-\hbox {ROFEOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})={\mathfrak {F}}_{\varrho (1)}^{\varphi _1}\otimes _\epsilon {\mathfrak {F}}_{\varrho (2)}^{\varphi _2}\otimes _\epsilon \ldots \otimes _\epsilon {\mathfrak {F}}_{\varrho ({\mathfrak {s}})}^{\varphi _{\mathfrak {s}}}, \end{aligned}$$

where $(\varrho (1), \varrho (2),\ldots ,\varrho ({\mathfrak {s}}))$ is the permutation of $({\mathfrak {j}}=1, 2, \ldots ,{\mathfrak {s}})$ such that ${\mathfrak {F}}_{\varrho ({\mathfrak {j}}-1)}\ge {\mathfrak {F}}_{\varrho ({\mathfrak {j}})}, \forall \,{\mathfrak {j}}=1, 2, \ldots ,{\mathfrak {s}}.$

Theorem 7

Let ${\mathfrak {F}}_{\mathfrak {j}}=(\mu _{\mathfrak {j}}, \nu _{\mathfrak {j}})\in q$-ROFNs, then aggregated value using q-ROFEOWG is a q-ROFN and

$$\begin{aligned}&q-ROFEOWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}) \nonumber \\&\quad = \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\mu _{\varrho ({\mathfrak {j}})}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{\varrho ({{\mathfrak {j}}})}^q)^{\varphi _{\mathfrak {j}}}}},\nonumber \\&\quad \root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+ \nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

(6)

Proof

Using the similar arguments as used in Theorem 5, we can prove it. $\square$

We give some properties without their proofs.

Corollary 2

The q-ROFEOWG and q-ROFOWG operators have the relation:

$$\begin{aligned} q-\hbox {ROFEOWG}({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})\ge q-\hbox {ROFOWG}({\mathfrak {F}}_1,{\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}}). \end{aligned}$$

Example 2

Let ${\mathfrak {F}}_1=(0.6, 0.7)$, ${\mathfrak {F}}_2=(0.8,0.7)$, ${\mathfrak {F}}_3=(0.6,0.9)$ and ${\mathfrak {F}}_4=(0.9,0.4)$ be four q-ROFNs and $\varphi =(0.3,0.3,0.2,0.2)^T$, take $q=3$. As

$$\begin{aligned}&{\mathcal {S}}({\mathfrak {F}}_1)=(0.6)^3-(0.7)^3=-0.13, \\&{\mathcal {S}}({\mathfrak {F}}_2)=(0.8)^3-(0.7)^3=0.17, \\&{\mathcal {S}}({\mathfrak {F}}_3)=(0.6)^3-(0.9)^3=-0.51, \\&{\mathcal {S}}({\mathfrak {F}}_4)=(0.9)^3-(0.4)^3=0.67. \end{aligned}$$

Since ${\mathcal {S}}({\mathfrak {F}}_4)>{\mathcal {S}}({\mathfrak {F}}_2)>{\mathcal {S}}({\mathfrak {F}}_1)>{\mathcal {S}}({\mathfrak {F}}_3),$ therefore

$$\begin{aligned}&{\mathfrak {F}}_{\varrho (1)}={\mathfrak {F}}_4=(0.9,0.4), \\&{\mathfrak {F}}_{\varrho (2)}={\mathfrak {F}}_2=(0.8,0.7), \\&{\mathfrak {F}}_{\varrho (3)}={\mathfrak {F}}_1=(0.6, 0.7), \\&{\mathfrak {F}}_{\varrho (4)}={\mathfrak {F}}_3=(0.6,0.9). \end{aligned}$$

By q-ROFEOWG operator, we get

$$\begin{aligned}&q-\hbox {ROFEOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4)\\&\quad = \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{4}\mu _{\varrho ({\mathfrak {j}})}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{4}(2-\mu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{4}(\mu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}},\\&\quad \root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{4}(1+\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{4}(1-\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{4}(1+\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{4}(1-\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle \\&\quad =\Bigg \langle \frac{\root 3 \of {2}\times 0.72}{\root 3 \of {1.55+0.37}}, \root 3 \of {\frac{1.32-0.61}{1.32+0.61}}\Bigg \rangle \\&\quad =\langle 0.73, 0.72\rangle . \end{aligned}$$

Now,

$$\begin{aligned}&q-\hbox {ROFOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4) \\ {}= & {} \Bigg \langle \prod \limits _{{\mathfrak {j}}=1}^4(\mu _{\varrho ({\mathfrak {j}})})^{\varphi _{\mathfrak {j}}}, \root q \of {1-\prod \limits _{{\mathfrak {j}}=1}^4(1-\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}\Bigg \rangle \\&\quad = \langle 0.72, 0.73\rangle . \end{aligned}$$

$\Rightarrow q$-$\hbox {ROFEOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4)> q$-$\hbox {ROFOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,{\mathfrak {F}}_3, {\mathfrak {F}}_4).$

Proposition 2

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in q$-ROFNs and $\varphi _{\mathfrak {j}}$ is the WV of ${\mathfrak {F}}_{\mathfrak {j}}$, such that $\varphi _{\mathfrak {j}}\in [0,1]$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}=1$.

(i) Idempotency: If ${\mathfrak {F}}_{\mathfrak {j}}={\mathfrak {F}}_o=\langle \mu _o, \nu _o\rangle$, $\forall \,{\mathfrak {j}}$, then

$$\begin{aligned} q-ROFEOWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})={\mathfrak {F}}_o. \end{aligned}$$

(ii) Boundedness: Let ${\mathfrak {F}}^-=(min_{\mathfrak {j}}(\mu _{\mathfrak {j}}),max_{\mathfrak {j}}(\nu _{\mathfrak {j}}))$, ${\mathfrak {F}}^+=(max_{\mathfrak {j}}(\mu _{\mathfrak {j}}),min_{\mathfrak {j}}(\nu _{\mathfrak {j}}))$, then

$$\begin{aligned} {\mathfrak {F}}^-\le q-ROFEOWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})\le {\mathfrak {F}}^+. \end{aligned}$$

(iii) Monotonicity: When ${\mathfrak {F}}_{\mathfrak {j}}\le {\mathcal {P}}_{\mathfrak {j}}, \forall \,{\mathfrak {j}}$, then

$$\begin{aligned} q-ROFEOWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}})\le q-ROFEOWG({\mathcal {P}}_1, {\mathcal {P}}_2,\ldots ,{\mathcal {P}}_{\mathfrak {s}}). \end{aligned}$$

(iv) Shift invariance: If $\eta =\langle \mu _\eta , \nu _\eta \rangle$ is a q-ROFN, then $q-ROFEOWG(\eta \otimes _\epsilon {\mathfrak {F}}_1, \eta \otimes _\epsilon {\mathfrak {F}}_2,\ldots , \eta \otimes _\epsilon {\mathfrak {F}}_{\mathfrak {s}})=\eta \otimes q-ROFEOWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots , {\mathfrak {F}}_{\mathfrak {s}}).$

6 Generalized q-rung orthopair fuzzy Einstein weighted geometric operators

Definition 9

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in$ q-ROFNs and $\varphi _{\mathfrak {j}}$ is the WV of ${\mathfrak {F}}_{\mathfrak {j}}$ with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \varphi _{\mathfrak {j}}=1$, then Gq-ROFEWG operator is a mapping ${\mathcal {Q}}^{\mathfrak {s}}\rightarrow {\mathcal {Q}}$ such that

$$\begin{aligned} Gq-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})= \frac{1}{\gamma }._\epsilon \left( \bigotimes \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\gamma ._\epsilon {\mathfrak {F}}_{\mathfrak {j}})^{\varphi _{\mathfrak {j}}}\right) , \end{aligned}$$

where $\gamma > 0$.

Particularly,

• If $\gamma =1$, then Gq-ROFEWG becomes q-ROFEWG.

• If $\varphi =(1/{\mathfrak {s}}, 1/{\mathfrak {s}},\ldots ,1/{\mathfrak {s}})$, then

  $\hbox {G}q$-ROFEWG$({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=\frac{1}{\gamma }._ \epsilon \left( \bigotimes \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}} (\gamma {\mathfrak {F}}_{\mathfrak {j}})^{1/{\mathfrak {s}}}\right) .$

Theorem 8

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in q$-ROFNs and $\varphi _{\mathfrak {j}}$ is the WV of ${\mathfrak {F}}_{\mathfrak {j}}$ with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \varphi _{\mathfrak {j}}=1$, then $\hbox {G}q$-ROFEWG operator is a q-ROFN and

$Gq-ROFEWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=$

$$\begin{aligned}&\Bigg \langle \root q \of {\frac{\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}+3\prod \nolimits _{{\mathfrak {j}}=1} ^{\mathfrak {s}}\{(1+ \mu _{{\mathfrak {j}}}^q)^\gamma -\omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{-\left( \prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{ {\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^\mathfrak { s}\{(1+ \mu _{{\mathfrak {j}}}^q)^\gamma -\omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}{\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}+3\prod \nolimits _{{\mathfrak {j}}=1} ^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma -\omega _\gamma \}^{ \varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{+\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{ {\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^\mathfrak { s}\{(1+ \mu _{{\mathfrak {j}}}^q)^\gamma -\omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}}, \\&\quad \frac{\root q \of {2}\Big \{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}\Big \}^{1/q\gamma }}{\root q \of {\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}+ 3\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{+\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma - (\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}}\Bigg \rangle . \end{aligned}$$

where $\omega _\gamma =(1-{\mu _j}^q)^\gamma$.

Proof

Since

$$\begin{aligned} \gamma ._\epsilon {\mathfrak {F}}_{\mathfrak {j}}= & {} \Bigg \langle \root q \of {\frac{(1+\mu _{\mathfrak {j}}^q)^\gamma -(1-\mu _{\mathfrak {j}}^q)^\gamma }{(1+\mu _{\mathfrak {j}}^q)^\gamma +(1-\mu _{\mathfrak {j}}^q)^\gamma }}, \frac{\root q \of {2}\nu _{\mathfrak {j}}^\gamma }{\root q \of {(2-\nu _{\mathfrak {j}}^q)^\gamma +(\nu _{\mathfrak {j}}^q)^\gamma }}\Bigg \rangle \end{aligned}$$

$\Rightarrow \bigotimes \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\gamma ._\epsilon {\mathfrak {F}}_{\mathfrak {j}})^{\varphi _{\mathfrak {j}}}=$

$$\begin{aligned}&\Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \root q \of {\frac{(1+ \mu _{\mathfrak {j}}^q)^\gamma -(1-\mu _{\mathfrak {j}}^q)^\gamma }{(1+\mu _{\mathfrak {j}}^q)^\gamma +(1-\mu _{\mathfrak {j}}^q)^\gamma }}\right) ^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 2-\frac{(1+\mu _{\mathfrak {j}}^q)^\gamma -(1-\mu _{\mathfrak {j}}^q)^\gamma }{(1+\mu _{\mathfrak {j}}^q)^\gamma +(1-\mu _{\mathfrak {j}}^q)^\gamma }\right) ^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \frac{(1+\mu _{\mathfrak {j}}^q)^\gamma -(1-\mu _{\mathfrak {j}}^q)^\gamma }{(1+ \mu _{\mathfrak {j}}^q)^\gamma +(1-\mu _{\mathfrak {j}}^q)^\gamma }\right) ^{\varphi _{\mathfrak {j}}}}}, \\&\quad \root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1+\frac{2\nu _{{\mathfrak {j}}}^{q\gamma }}{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + (\nu _{{\mathfrak {j}}}^q)^\gamma }\right) ^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1-\frac{2\nu _{{\mathfrak {j}}}^{q\gamma }}{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +(\nu _{{\mathfrak {j}}}^q)^\gamma }\right) ^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1+\frac{2\nu _{{\mathfrak {j}}}^{q\gamma }}{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +(\nu _{{\mathfrak {j}}}^q)^\gamma }\right) ^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( 1-\frac{2\nu _{{\mathfrak {j}}}^{q\gamma }}{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +(\nu _{{\mathfrak {j}}}^q)^\gamma }\right) ^{\varphi _{\mathfrak {j}}}}} \Bigg \rangle \\&\quad =\Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\left( \root q \of {(1+\mu _{{\mathfrak {j}}}^q)^\gamma -\omega _\gamma }\right) ^{\varphi _\mathfrak { j}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma +3\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}}, \\&\quad \root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma - (\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma - (\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

Therefore, $\frac{1}{\gamma }._\epsilon \left( \bigotimes \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\gamma ._\epsilon {\mathfrak {F}}_{\mathfrak {j}})^{\varphi _{\mathfrak {j}}}\right) =$

$$\begin{aligned}&\Bigg \langle \root q \of {\frac{\frac{\Bigg (1+\frac{2\prod \nolimits _{\mathfrak {j}}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}}^q)^\gamma - \omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{\mathfrak {j}}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{\mathfrak {j}}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}{-\Bigg (1-\frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{ {\mathfrak {j}}}^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma +3\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}}{\frac{\Bigg (1+\frac{2\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{ (1+\mu _{{\mathfrak {j}}}^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma +3\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}{+\Bigg (1-\frac{2\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{ {\mathfrak {j}}}^q)^\gamma -\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}} ^q)^\gamma +3\omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{(1+\mu _{{\mathfrak {j}}}^q)^\gamma - \omega _\gamma \big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}}}, \\&\quad \frac{\root q \of {2\Bigg (\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}}{\root q \of {\frac{\Bigg (2-\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}{+\Bigg (\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma +3(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\big \{{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma }\big \}^{\varphi _{\mathfrak {j}}}}\Bigg )^{1/\gamma }}}}\Bigg \rangle \\&\quad =\Bigg \langle \root q \of {\frac{\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}+3\prod \nolimits _{{\mathfrak {j}}=1} ^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma - \omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{-\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{ {\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^\mathfrak { s}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma - \omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}{\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}+3\prod \limits _{{\mathfrak {j}}=1} ^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma - \omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{+\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma + 3\omega _\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \limits _{{\mathfrak {j}}=1}^\mathfrak { s}\{(1+\mu _{{\mathfrak {j}}}^q)^\gamma - \omega _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}}, \\&\quad \frac{\root q \of {2}\Big \{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma -(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}\Big \}^{1/q\gamma }}{\root q \of {\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}+3\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma - (\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{+\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma + 3(\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{{\mathfrak {j}}}^q)^\gamma - (\nu _{{\mathfrak {j}}}^q)^\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }} }}\Bigg \rangle . \end{aligned}$$

When $\gamma =1$, then

$$\begin{aligned} Gq-\hbox {ROFEWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\mu _{{\mathfrak {j}}}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}},\\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{{\mathfrak {j}}}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

$\square$

6.1 Generalized q-rung orthopair fuzzy Einstein ordered weighted geometric operators

Definition 10

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in$ q-ROFNs and $\varphi _{\mathfrak {j}}$ is the WV of ${\mathfrak {F}}_{\mathfrak {j}}$ with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \varphi _{\mathfrak {j}}=1$, then Gq-ROFEOWG operator is a mapping ${\mathcal {Q}}^{\mathfrak {s}}\rightarrow {\mathcal {Q}}$ such that

$$\begin{aligned} Gq-\hbox {ROFEOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=\frac{1}{\gamma }._\epsilon \left( \bigotimes \limits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\gamma ._\epsilon {\mathfrak {F}}_{\mathfrak {j}})^{\varphi _{\mathfrak {j}}}\right) , \end{aligned}$$

where $\gamma > 0$.

Theorem 9

Let ${\mathfrak {F}}_{\mathfrak {j}}=\langle \mu _{\mathfrak {j}}, \nu _{\mathfrak {j}}\rangle \in q$-ROFNs and $\varphi _{\mathfrak {j}}$ is the WV of ${\mathfrak {F}}_{\mathfrak {j}}$ with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \varphi _{\mathfrak {j}}=1$, then $\hbox {G}q$-ROFEOWG operator is a q-ROFN and

$Gq-ROFEOWG({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})=$

$$\begin{aligned}&\Bigg \langle \root q \of {\frac{\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{ \rho ^+_\gamma + 3\rho _\gamma \}^{\varphi _{\mathfrak {j}}}+3\prod \nolimits _{{\mathfrak {j}}=1}^\mathfrak {s }\{(1+ \mu _{\varrho ({\mathfrak {j}})}^q)^\gamma -\rho _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{-\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{ \varrho ({\mathfrak {j}})}^q)^\gamma + 3\rho _\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \{(1+ \mu _{\varrho ({\mathfrak {j}})}^q)^\gamma -\rho _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}{\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(\rho ^+_\gamma \}^{ \varphi _{\mathfrak {j}}}+3\prod \nolimits _{{\mathfrak {j}}=1}^\mathfrak { s }\{\rho ^+_\gamma -(\rho _\gamma \}^{\varphi _{\mathfrak {j}} } \right) ^{1/\gamma }}{+\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(1+\mu _{ \varrho ({\mathfrak {j}})}^q)^\gamma + 3\rho _\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}} \{(1+ \mu _{\varrho ({\mathfrak {j}})}^q)^\gamma -\rho _\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}}, \\&\quad \frac{\root q \of {2}\Big \{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma + 3(\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma -(\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}\Big \}^{1/q\gamma }}{\root q \of {\frac{\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma + 3(\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}+ 3\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma -(\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}{+\left( \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma + 3(\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma \}^{\varphi _{\mathfrak {j}}}-\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\{(2-\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma - (\nu _{\varrho ({\mathfrak {j}})}^q)^\gamma \}^{\varphi _{\mathfrak {j}}} \right) ^{1/\gamma }}}}\Bigg \rangle . \end{aligned}$$

where $\rho _\gamma =(1-\mu _{\varrho ({\mathfrak {j}})}^q)^\gamma$ and $\rho ^+_\gamma =(1+\mu _{\varrho ({\mathfrak {j}})}^q)^\gamma$.

Proof

Using the similar arguments as used in Theorem 8, we can prove it. $\square$

When $\gamma =1$, then

$$\begin{aligned} Gq-\hbox {ROFEOWG}({\mathfrak {F}}_1, {\mathfrak {F}}_2,\ldots ,{\mathfrak {F}}_{\mathfrak {s}})= & {} \Bigg \langle \frac{\root q \of {2}\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\mu _{\varrho ({\mathfrak {j}})}^{\varphi _{\mathfrak {j}}}}{\root q \of {\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(2-\mu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}+ \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(\mu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}}, \\&\root q \of {\frac{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}- \prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1-\nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}{\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1+ \nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}+\prod \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}(1- \nu _{\varrho ({\mathfrak {j}})}^q)^{\varphi _{\mathfrak {j}}}}}\Bigg \rangle . \end{aligned}$$

7 MADM problems using q-ROF information

We discuss the MADM problems with q-ROF information using q-ROF Einstein geometric operators proposed in the preceding sections. The following suppositions or notations are used to handle the MADM problem for the competent selection of alternative affected by human decision making with q-ROF information. let ${\mathcal {L}}=\{{\mathcal {L}}_1, {\mathcal {L}}_2,\ldots , {\mathcal {L}}_m\}$ be a set of possible alternatives and $J=\{J_1, J_2,\ldots , J_{\mathfrak {s}}\}$ be a set of possible attributes chosen by the decision maker. Let $\varphi =\{\varphi _1, \varphi _2,\ldots , \varphi _{\mathfrak {s}}\}$ is the WV with $\varphi _{\mathfrak {j}}>0$ and $\sum \nolimits _{{\mathfrak {j}}=1}^{\mathfrak {s}}\varphi _{\mathfrak {j}}=1$. Suppose that $\tilde{{\mathcal {E}}}=(\mu _{l{\mathfrak {j}}},\nu _{l{\mathfrak {j}}})_{m\times {\mathfrak {s}}}$ is the q-ROF decision matrix (DM), where $\mu _{l{\mathfrak {j}}}$ and $\nu _{l{\mathfrak {j}}}$ are the satisfaction and dissatisfaction degrees of the alternative for the attribute, where $0\le \mu _{l{\mathfrak {j}}}^q+\nu _{l{\mathfrak {j}}}^q\le 1$.

For solving a MADM problem, we give the following Algorithm 1.

7.1 Suitable location for thermal power station

A Thermal Power Station (TPS) uses heat energy generated from burning coal to produce electrical energy. Such power stations are broadly used in the world. Certain TPSs are made to produce heat for industrial purposes, for district heating, or desalination of water and for generating electrical power. Therefore, it is necessary to select a location for a TPS, as it needs a massive capacity of land and position to bear the static and dynamic pressure during the whole process. Suppose that the government wants to plant a TPS to fulfill the requirements of electric power. Let ${\mathcal {L}}_1$, ${\mathcal {L}}_2$, ${\mathcal {L}}_3$ and ${\mathcal {L}}_4$ be possible locations for TPSs. Let ${\mathcal {K}}_1$=“Availability of coal”, ${\mathcal {K}}_2$=“Availability of water”, ${\mathcal {K}}_3$=“Transportation facilities” be the three criteria for the judgement of a location.

The framework for selection of location for a TPS is given in Fig. 2.

Fig. 2

The framework for location-selection of a TPS

1. The q-ROFDM is shown in Table 1.

Table 1 q-ROFDM

2. The weights assigned by decision maker are

$$\begin{aligned} \varphi _1=0.4, \varphi _2=0.3, \varphi _3=0.3, \text {and} \sum \limits _{{\mathfrak {j}}=1}^3\varphi _{\mathfrak {j}}=1. \end{aligned}$$

We use the $\hbox {G}q$-ROFEWG operator for the selection of TPS location.

Step 1. For performance values ${\mathcal {B}}_l$ of locations, use the $\hbox {G}q$-ROFEWG operator for $q=3, \gamma =1.$

$$\begin{aligned} {\mathcal {B}}_1= & {} (0.65, 0.34 ), \\ {\mathcal {B}}_2= & {} (0.51, 0.46), \\ {\mathcal {B}}_3= & {} (0.55, 0.40), \\ {\mathcal {B}}_4= & {} (0.49, 0.45). \end{aligned}$$

Step 2. Compute the scores ${\mathcal {S}}({\mathcal {B}}_l)$ of q-ROFNs ${\mathcal {B}}_l$ and rank the locations.

$$\begin{aligned} {\mathcal {S}}({\mathcal {B}}_1)= & {} 0.24, \\ {\mathcal {S}}({\mathcal {B}}_2)= & {} 0.04, \\ {\mathcal {S}}({\mathcal {B}}_3)= & {} 0.10, \\ {\mathcal {S}}({\mathcal {B}}_4)= & {} 0.03. \end{aligned}$$

The ranking of locations is

$$\begin{aligned} {\mathcal {L}}_1>{\mathcal {L}}_3>{\mathcal {L}}_2>{\mathcal {L}}_4. \end{aligned}$$

Step 3. Therefore, ${\mathcal {L}}_1$ is the suitable location for TPS.

7.2 Selection of cardiothoracic surgeon

A Cardiothoracic Surgeon (CS) is a medical doctor, who is trained in surgical procedures of heart, lungs and other organs in the chest. Surgery of the heart and chest are performed by CSs. Cardiologists work with surgeons to handle patients and determine whether the patient needs surgery. They also work together to treat irregular heart beat problems. Suppose that a heart patient wants to select a best CS for heart surgery. Let ${\mathcal {L}}_1$, ${\mathcal {L}}_2$, ${\mathcal {L}}_3$ and ${\mathcal {L}}_4$ be possible surgeons for heart surgery. Let ${\mathcal {K}}_1$=“Cardiac Surgeon’s Experience”, ${\mathcal {K}}_2$=“Hospital Quality”, ${\mathcal {K}}_3$=“Communication Style” be the three criteria for the judgement of a surgeon.

The framework for selection of best CS is given in Fig. 3.

Fig. 3

The framework for selection of best CS

1. The q-ROFDM is shown in Table 2.

Table 2 q-ROFDM

2. The weights assigned by decision maker are

$$\begin{aligned} \varphi _1=0.3, \varphi _2=0.4, \varphi _3=0.3, \text {and} \sum \limits _{{\mathfrak {j}}=1}^3\varphi _{\mathfrak {j}}=1. \end{aligned}$$

We use the $\hbox {G}q$-ROFEWG operator for the selection of best CS.

Step 1. For performance values ${\mathcal {B}}_l$ of CSs, use the $\hbox {G}q$-ROFEWG operator for $q=3, \gamma =1.$

$$\begin{aligned} {\mathcal {B}}_1= & {} (0.58, 0.55), \\ {\mathcal {B}}_2= & {} (0.70, 0.61), \\ {\mathcal {B}}_3= & {} (0.82, 0.41), \\ {\mathcal {B}}_4= & {} (0.71, 0.59). \end{aligned}$$

Step 2. Calculate the scores ${\mathcal {S}}({\mathcal {B}}_l)$ of q-ROFNs ${\mathcal {B}}_l$ and rank CSs.

$$\begin{aligned}&{\mathcal {S}}({\mathcal {B}}_1)=0.03, \\&{\mathcal {S}}({\mathcal {B}}_2)=0.12, \\&{\mathcal {S}}({\mathcal {B}}_3)=0.48, \\&{\mathcal {S}}({\mathcal {B}}_4)=0.15. \end{aligned}$$

The ranking is

$$\begin{aligned} {\mathcal {L}}_3>{\mathcal {L}}_4>{\mathcal {L}}_2>{\mathcal {L}}_1. \end{aligned}$$

Step 3. Therefore, ${\mathcal {L}}_3$ is the best CS.

8 Comparison analysis

This section provides a comparison analysis of proposed operators with others operators such as GPFEWG (Garg 2017), IFEWG (Wang and Liu 2011), q-ROFGMSM (Liu and Wang (2020) operators to show the efficacy of our model.

• 1. For Application 7.1, it is clear from Table 3 that the final rankings by applying the $\hbox {G}q$-ROFEWG, GPFEWG, IFEWG, q-ROFGMSM operators are ${\mathcal {L}}_1>{\mathcal {L}}_3>{\mathcal {L}}_2>{\mathcal {L}}_4$, respectively. However, the final score values are not same. So, the optimal decision, using all these operators are same. This shows that our model is applicable to resolve real life problems.

• 2. The reason behind our proposed model is that IFNs and PFNs can deal only those situations where $\mu +\nu \le 1$ and $\mu ^2+\nu ^2\le 1$, respectively but many problems where data exceed by $\mu ^2+\nu ^2\le 1$, then we need q-ROFS. To show feasibility and attractiveness, we have proposed another application.

• 3. As the ranking lists obtained from proposed approach and q-ROFGMSM operator are same. The q-ROFGMSM operator is a good approach to solve DM problems but proposed Einstein AOs are more flexible and easy approach.

Table 3 Comparison analysis for Application 7.1 with GPFEWG, IFEWG, q-ROFGMSM operators (suppose $q=3, \gamma =1$)

9 Conclusions and future directions

The notion of q-rung orthopair fuzzy model generalizes the PF model to describe complicated uncertain information more effectively. In this article, we have worked to the progress of MADM with the study of problems in q-ROF environment. For the utilization in decision making, the logical basis of AOs need to be carefully considered. The shortcomings of existing methods and beneficial characteristics of Einstein AOs motivate us to consider their ability to produce suitable combinations of q-ROFNs. Therefore, we have introduced geometric operators to construct q-ROF AOs that closely follow the motivation of Einstein operations. They include the q-ROFEWG, q-ROFEOWG, $\hbox {G}q$-ROFEWG and $\hbox {G}q$-ROFEOWG operators. The elementary characteristics of these operators are explained so that the experts can select the version that better fits their needs. We have utilized these operators to expand a number of strategies to address MADM problems. The comparison analysis of proposed operators with existing operators is done. Finally, practical examples for the selection of location for TPS and the selection of best cardiac surgeon are given. In these problems, we have applied the concept of $\hbox {G}q$-ROFEWG operator to summarize the information corresponding to each alternative. Then, we have derived appropriate results with the help of score functions. These operators allow us to assess the value of each alternative in a comparable fashion. Altogether they build up a procedure and make a case for the pertinence and adequacy of the proposed approach. In the future, we plan to extend our study to (i). q-rung picture fuzzy Einstein hybrid weighted operators. (ii). q-rung orthopair fuzzy soft Einstein hybrid weighted operators.