Protraction of Einstein operators for decision-making under q -rung orthopair fuzzy model

Abstract

An q-rung orthopair fuzzy set is a generalized structure that covers the modern extensions of fuzzy set, including intuitionistic fuzzy set and Pythagorean fuzzy set, with an adjustable parameter q that makes it flexible and adaptable to describe the inexact information in decision making. The condition of q-rung orthopair fuzzy set, i.e., sum of q^th power of membership degree and nonmembership degree is bounded by one, makes it highly competent and adequate to get over the limitations of existing models. The basic purpose of this study is to establish some aggregation operators under the q-rung orthopair fuzzy environment with Einstein norm operations. Motivated by innovative features of Einstein operators and dominant behavior of q-rung orthopair fuzzy set, some new aggregation operators, namely, q-rung orthopair fuzzy Einstein weighted averaging, q-rung orthopair fuzzy Einstein ordered weighted averaging, generalized q-rung orthopair fuzzy Einstein weighted averaging and generalized q-rung orthopair fuzzy Einstein ordered weighted averaging operators are defined. Furthermore, some properties related to proposed operators are presented. Moreover, multi-attribute decision making problems related to career selection, agriculture land selection and residential place selection are presented under these operators to show the capability and proficiency of this new idea. The comparison analysis with existing theories shows the superiorities of proposed model.

Keywords

Einstein operators averaging operators generalized weighted averaging operators

1 Introduction

Multi-attribute decision making (MADM) performs a significant part in finding an excellent alternative from all the appropriate alternatives depending upon certain attributes. Mostly, the approach of different alternatives and the corresponding weights for various attributes are given in crisp values. The crisp sets were incompetent to portray the inexact information as there exist many complicated MADM problems in real-life for which the objectives and conditions are vague and unclear. To handle such situations, fuzzy sets (FSs) were initiated by Zadeh [43] which were employed to deal vagueness in problems. Atanassov [7] broadened the literature on fuzzy set theory by introducing well versed intuitionistic fuzzy sets (IFSs) to dominate the deficiencies of FSs. The IFSs have both membership degree (MD) and nonmembership degree (NMD) with condition μ + ν ≤ 1 but there are many problems in which sum of MD and NMD exceeds by 1. To reduce such type of complications, Yager [38] delivered the idea of Pythagorean fuzzy sets (PFSs) with condition μ² + ν² ≤ 1 which make them eminent among the existing models. The existence of problem, where the condition of PFS is violated, motivated the researchers to put forward a more generalized model to exhibit the imprecise data regardless the strict conditions of IFSs and PFSs. Finally, the spadework of Yager [37] provides the grounds for the establishment of q-rung orthopair fuzzy sets (q-ROFSs) in which μ^q + ν^q ≤ 1. q-ROFSs have more capability to handle the decision making problems and they possess larger space of membership and nonmembership functions as compared to IFSs and PFSs, as shown in Fig. 1.

Fig. 1

Comparison of q-ROFNs, PFNs and IFNs spaces.

For obtaining a unique and best decision, the concept of aggregation operators (AOs) was introduced. Xu [35] developed some AOs under intuitionistic fuzzy (IF) environment that proved to be very fruitful in decision making. The concept of generalized AOs for IFS was developed by Zhao et al. [48]. Zhao and Wei [47] studied the Einstein hybrid AOs under IF environment. Wang and Liu [30] established some advanced IF AOs using Einstein operations to cumulate the data more accurately. The induced interval-valued IF Einstein ordered weighted average operators were developed by Cai et al. [9]. Fahmi et al. [10] developed the cubic fuzzy Einstein AOs. Rahman et al. [26] introduced Pythagorean fuzzy (PF) AOs and demonstrated the efficiency of these operators with the help of applications. Garg [11] studied the generalized PF Einstein weighted arithmetic AOs. Garg [12] also proposed the generalized PF Einstein weighted geometric AOs. Pythagorean Dombi fuzzy AOs with applications were discussed by Akram et al. [2]. Shahzadi et al. [29] proposed the decision making approach using PF Yager AOs. Liu and Wang [20] expressed q-rung orthopair fuzzy (q-ROF) weighted AOs. Liu et al. [19] initiated the idea of q-ROF Bonferroni mean operators. Wei et al. [32] proposed weighted Heronian mean AOs for q-ROFS. Multi-attribute group decision making (MAGDM) approach using q-ROF power Maclaurin AOs was developed by Liu et al. [18]. Jana et al. [14] discussed Dombi AOs under q-rung orthopair fuzzy numbers (q-ROFNs). Joshi and Gegov [16] studied the confidence levels q-ROF aggregation operators. Garg and Chen [13] introduced the neutrality AOs for q-ROFS. For more information on operators, the readers are referred to [1 , 49–54].

The motivations of this article are as follows:

q-ROFS is a highly proficient and well versed tool to portray the ambiguous information, whose flexible condition makes it superior than the traditional models. In fact, q-ROFS, possessing larger space, is more potent to handle decision making issues as compared to IFS and PFS.

Einstein AOs yield the more precise and accurate results when applied to real-life MADM problems. Therefore, we have employed the theoretical basis of Einstein operations to define the proposed operators that enjoy the accuracy and potential of Einstein AOs.

Due to the dominant behavior of q-ROFS, the proposed operators can be efficiently applied to IFSs, PFSs and other extensions of fuzzy sets by selecting a suitable value of q. In other words, the proposed operators have an edge over the existing approaches under the decision making environments.

The contributions of this article are as follows:

Predominantly, the main objective of this article is to propose q-ROF Einstein AOs, namely, q-rung orthopair fuzzy Einstein weighted averaging (q-ROFEWA) operator, q-rung orthopair fuzzy Einstein ordered weighted averaging (q-ROFEOWA) operator, generalized q-rung orthopair fuzzy Einstein weighted averaging (Gq-ROFEWA) operator and generalized q-rung orthopair fuzzy Einstein ordered weighted averaging (Gq-ROFEOWA) operator, for the evaluation of single choice for alternative among various choices by merging the skills of Einstein operators with the flexibility of q-ROFS.

Some fundamental properties along with some major results of the proposed operators are illustrated.

An algorithm, based on Gq-ROFEWA operator, is presented to deal with complicated realistic MADM problems under q-ROF data.

Three explanatory MADM problems, for the selection of career, agriculture land and residential place, indicate the potential of proposed algorithm.

A validity test is discussed for the approval and authenticity of proposed theory.

The comparative study of the proposed MADM technique with existing approaches shows the dominance of proposed AOs over the existing operators.

The remaining paper is as follows: Section 2 recalls the notion of q-ROFS and related score functions. Section 3 introduces Einstein operational laws for q-ROFNs. In Section 4, we study the q-ROFEWA and q-ROFEOWA operators, respectively and related properties to them. Section 5 handles the idea of Gq-ROFEWA and Gq-ROFEOWA operators, respectively and some properties of these operators. In Section 6, we propose the algorithm for our new model and discuss MADM problems, i.e., selection of best future career, selection of agricultural land and residential place under proposed operators. Section 7 provides the validity criteria to prove the consistency of proposed work. Section 8 discusses the comparison analysis of proposed theory with existing operators. In Section 9, we conclude results about proposed theory.

2 Preliminaries

Definition 2.1. [37] A q-ROFS $Z$ on non-empty set $X$ is given by $Z = {〈 x, μ_{Z} (x), ν_{Z} (x) 〉},$ where $μ_{Z} : X \to [0, 1]$ and $ν_{Z} : X \to [0, 1]$ show the MD and NMD of an element $x \in X$ , respectively. $ϖ_{Z} (x) = \sqrt[q]{1 - (μ_{Z} (x))^{q} + (ν_{Z} (x))^{q}}$ is indeterminacy degree of an element. For easiness, $Z = {〈 x, μ_{Z} (x), ν_{Z} (x) 〉},$ called q-ROFN is represented by $Z = 〈 μ_{Z}, ν_{Z} 〉 .$

Definition 2.2. [37] Consider two q-ROFNs $Z_{1} = 〈 μ_{Z_{1}}, ν_{Z_{1}} 〉$ and $Z_{2} = 〈 μ_{Z_{2}}, ν_{Z_{2}} 〉$ . The operational laws on q-ROFNs are

$Z_{1} \oplus Z_{2} = 〈 \sqrt[q]{μ_{Z_{1}}^{q} + μ_{Z_{2}}^{q} - μ_{Z_{1}}^{q} μ_{Z_{2}}^{q}}, ν_{Z_{1}} ν_{Z_{2}} 〉$ ,

$Z_{1} \otimes Z_{2} = 〈 μ_{Z_{1}} μ_{Z_{2}}, \sqrt[q]{ν_{Z_{1}}^{q} + ν_{Z_{2}}^{q} - ν_{Z_{1}}^{q} ν_{Z_{2}}^{2}} 〉$ ,

$ω Z_{1} = 〈 \sqrt[q]{1 - (1 - μ_{Z_{1}}^{q})^{ω}}, ν_{Z_{1}}^{ω} 〉,$ ω > 0,

$Z_{1}^{ω} = 〈 μ_{Z_{1}}^{ω}, \sqrt[q]{1 - (1 - ν_{Z_{1}}^{q})^{ω}} 〉,$ ω > 0 .

Definition 2.3. [37] Consider a q-ROFN $Z = 〈 μ_{Z}, ν_{Z} 〉$ . The score function $S$ and accuracy function $A$ of $Z$ are $S (Z) = μ_{q}^{Z} - ν_{q}^{Z}, where S (Z) \in [- 1, 1],$ $A (Z) = μ_{q}^{Z} + ν_{q}^{Z}, where A (Z) \in [0, 1] .$

Definition 2.4. [37] Consider two q-ROFNs $Z_{1} = 〈 μ_{Z_{1}}, ν_{Z_{1}} 〉$ and $Z_{2} = 〈 μ_{Z_{2}}, ν_{Z_{2}} 〉$ . Then

If $S (Z_{1}) < S (Z_{2})$ , then $Z_{1} ≺ Z_{2}$ ,

If $S (Z_{1}) > S (Z_{2})$ , then $Z_{1} ≻ Z_{2}$ ,

If $S (Z_{1}) = S (Z_{2})$ , then

If $A (Z_{1}) < A (Z_{2})$ , then $Z_{1} ≺ Z_{2}$ ,

If $A (Z_{1}) > A (Z_{2})$ , then $Z_{1} ≻ Z_{2}$ ,

If $A (Z_{1}) = A (Z_{2})$ , then $Z_{1} \sim Z_{2}$ .

Definition 2.5. [11] The Einstein sum ⊕_ε and Einstein product ⊗_ε under the PF environment are defined as $\begin{matrix} S_{ε} (x, y) & = & \sqrt{\frac{x^{2} + y^{2}}{1 + x^{2} ._{ε} y^{2}}}, \\ T_{ε} (x, y) & = & \frac{x ._{ε} y}{\sqrt{1 + (1 - x^{2}) ._{ε} (1 - y^{2})}} . \end{matrix}$

3 Einstein operational law of q-ROFNs

Definition 3.1. Let $Z = 〈 μ, ν 〉$ , $Z_{1} = 〈 μ_{1}, ν_{1} 〉$ and $Z_{2} = 〈 μ_{2}, ν_{2} 〉$ be three q-ROFNs and 0 < ω > 1, then

(i) $\bar{Z} = 〈 ν_{Z}, μ_{Z} 〉,$

(ii) $Z_{1} \land_{ε} Z_{2} = 〈 min {μ_{1}, μ_{2}}, max {ν_{1}, ν_{2}} 〉,$

(iii) $Z_{1} \lor_{ε} Z_{2} = 〈 max {μ_{1}, μ_{2}}, min {ν_{1}, ν_{2}} 〉,$

(iv) $Z_{1} \oplus_{ε} Z_{2} = 〈 \sqrt[q]{\frac{μ_{1}^{q} + μ_{2}^{q}}{1 + μ_{1}^{q} ._{ε} μ_{2}^{q}}}, \frac{ν_{1} ._{ε} ν_{2}}{\sqrt[q]{1 + (1 - ν_{1}^{q}) . ε (1 - ν_{2}^{q})}} 〉,$

(v) $Z_{1} \otimes_{ε} Z_{2} = 〈 \frac{μ_{1} ._{ε} μ_{2}}{\sqrt[q]{1 + (1 - μ_{1}^{q}) . ε (1 - μ_{2}^{q})}}, \sqrt[q]{\frac{ν_{1}^{q} + ν_{2}^{q}}{1 + ν_{1}^{q} ._{ε} ν_{2}^{q}}} 〉,$

(vi) $ω ._{ε} Z = 〈 \sqrt[q]{\frac{(1 + μ^{q})^{ω} - (1 - μ^{q})^{ω}}{(1 + μ^{q})^{ω} + (1 - μ^{q})^{ω}}}, \frac{\sqrt[q]{2} ν^{ω}}{\sqrt[q]{(2 - ν^{q})^{ω} + (ν^{q})^{ω}}} 〉,$

(vii) $Z^{ω} = 〈 \frac{\sqrt[q]{2} μ^{ω}}{\sqrt[q]{(2 - μ^{q})^{ω} + (μ^{q})^{ω}}}, \sqrt[q]{\frac{(1 + ν^{q})^{ω} - (1 - ν^{q})^{ω}}{(1 + ν^{q})^{ω} + (1 - ν^{q})^{ω}}} 〉 .$

Theorem 3.1. Let $Z = 〈 μ_{Z}, ν_{Z} 〉$ , $Z_{1} = 〈 μ_{1}, ν_{1} 〉$ and $Z_{2} = 〈 μ_{2}, ν_{2} 〉$ be three q-ROFNs, then $Z_{3} = Z_{1} \oplus_{ε} Z_{2}$ and $Z_{4} = ω ._{ε} Z$ are also q-ROFNs.

Proof. Since ω > 0 and $Z$ is a q-ROFN, therefore, $0 \leq μ_{Z} (x) \leq 1$ , $0 \leq ν_{Z} (x) \leq 1$ and $0 \leq (μ_{Z} (x))^{q} + (ν_{Z} (x))^{q} \leq 1$ , then $1 - (μ_{Z} (x))^{q} \geq (ν_{Z} (x))^{q} \geq 0$ , $1 - (ν_{Z} (x))^{q} \geq (μ_{Z} (x))^{q} \geq 0$ and $(1 - (μ_{Z} (x))^{q})^{ω} \geq (ν_{Z} (x))^{q}$ , then, we have $\begin{matrix} \sqrt[q]{\frac{(1 + (μ_{Z} (x))^{q})^{ω} - (1 - (μ_{Z} (x))^{q})^{ω}}{(1 + (μ_{Z} (x))^{q})^{ω} + (1 - (μ_{Z} (x))^{q})^{ω}}} \\ \leq \sqrt[q]{\frac{(1 + (μ_{Z} (x))^{q})^{ω} - (ν_{Z} (x))^{q})^{ω}}{(1 + (μ_{Z} (x))^{q})^{ω} + (ν_{Z} (x))^{q})^{ω}}}, \end{matrix}$ and $\begin{matrix} \frac{\sqrt[q]{2} (ν_{Z} (x))^{ω}}{\sqrt[q]{(2 - (ν_{Z} (x))^{q})^{ω} + (ν_{Z} (x)^{q})^{ω}}} \\ \leq \frac{\sqrt[q]{2} (ν_{Z} (x))^{ω}}{\sqrt[q]{(1 + (μ_{Z} (x))^{q})^{ω} + ((ν_{Z} (x))^{q})^{ω}}} . \end{matrix}$ Thus, $\begin{matrix} (\sqrt[q]{\frac{(1 + (μ_{Z} (x))^{q})^{ω} - (1 - (μ_{Z} (x))^{q})^{ω}}{(1 + (μ_{Z} (x))^{q})^{ω} + (1 - (μ_{Z} (x))^{q})^{ω}}})^{q} \\ + (\frac{\sqrt[q]{2} (ν_{Z} (x))^{ω}}{\sqrt[q]{(2 - (ν_{Z} (x))^{q})^{ω} + (ν_{Z} (x)^{q})^{ω}}})^{q} & \leq & 1 . \end{matrix}$ Furthermore, $\begin{matrix} (\sqrt[q]{\frac{(1 + (μ_{Z} (x))^{q})^{ω} - (1 - (μ_{Z} (x))^{q})^{ω}}{(1 + (μ_{Z} (x))^{q})^{ω} + (1 - (μ_{Z} (x))^{q})^{ω}}})^{q} \\ + (\frac{\sqrt[q]{2} (ν_{Z} (x))^{ω}}{\sqrt[q]{(2 - (ν_{Z} (x))^{q})^{ω} + (ν_{Z} (x)^{q})^{ω}}})^{q} & = & 0, \end{matrix}$ iff $μ_{Z} (x) = ν_{Z} (x) = 0$ and $\begin{matrix} (\sqrt[q]{\frac{(1 + (μ_{Z} (x))^{q})^{ω} - (1 - (μ_{Z} (x))^{q})^{ω}}{(1 + (μ_{Z} (x))^{q})^{ω} + (1 - (μ_{Z} (x))^{q})^{ω}}})^{q} + \\ (\frac{\sqrt[q]{2} (ν_{Z} (x))^{ω}}{\sqrt[q]{(2 - (ν_{Z} (x))^{q})^{ω} + (ν_{Z} (x)^{q})^{ω}}})^{q} & = & 1, \end{matrix}$ iff $(μ_{Z} (x))^{q} + (ν_{Z} (x))^{q} = 1 .$

Thus, $Z_{4} = ω ._{ε} Z$ is a q-ROFN for ω > 0 . □

Theorem 3.2. Let ω, ω₁, ω₂ ≥ 0, then $(i) Z_{1} \oplus_{ε} Z_{2} = Z_{2} \oplus_{ε} Z_{1},$

$(ii) Z_{1} \otimes_{ε} Z_{2} = Z_{1} \otimes_{ε} Z_{2},$

$(iii) ω ._{ε} (Z_{1} \oplus_{ε} Z_{2}) = ω ._{ε} Z_{1} \oplus_{ε} ω ._{ε} Z_{2},$

$(iv) (Z_{1} \otimes_{ε} Z_{2})^{ω} = Z_{1}^{ω} \otimes_{ε} Z_{2}^{ω},$

$(v) ω_{1} ._{ε} Z \oplus_{ε} ω_{2} ._{ε} Z = (ω_{1} + ω_{2}) ._{ε} Z,$

$(vi) Z^{ω_{1}} \otimes_{ε} Z^{ω_{2}} = Z^{(ω_{1} + ω_{2})} .$

Proof. (i) $\begin{matrix} Z_{1} \oplus_{ε} Z_{2} \\ = 〈 \sqrt[q]{\frac{μ_{1}^{q} + μ_{2}^{q}}{1 + μ_{1}^{q} ._{ε} μ_{2}^{q}}}, \frac{ν_{1} ._{ε} ν_{2}}{\sqrt[q]{1 + (1 - ν_{1}^{q}) . ε (1 - ν_{2}^{q})}} 〉 \\ = 〈 \sqrt[q]{\frac{μ_{2}^{q} + μ_{1}^{q}}{1 + μ_{2}^{q} ._{ε} μ_{1}^{q}}}, \frac{ν_{2} ._{ε} ν_{1}}{\sqrt[q]{1 + (1 - ν_{2}^{q}) . ε (1 - ν_{1}^{q})}} 〉 \\ = Z_{2} \oplus_{ε} Z_{1} . \end{matrix}$ (iii) $\begin{matrix} Z_{1} \oplus_{ε} Z_{2} \\ = 〈 \sqrt[q]{\frac{μ_{1}^{q} + μ_{2}^{q}}{1 + μ_{1}^{q} ._{ε} μ_{2}^{q}}}, \frac{ν_{1} ._{ε} ν_{2}}{\sqrt[q]{1 + (1 - ν_{1}^{q}) . ε (1 - ν_{2}^{q})}} 〉, \end{matrix}$ is equivalent to $\begin{matrix} Z_{1} \oplus_{ε} Z_{2} \\ = 〈 \sqrt[q]{\frac{(1 + μ_{1}^{q}) ._{ε} (1 + μ_{2}^{q}) - (1 - μ_{1}^{q}) ._{ε} (1 - μ_{2}^{q})}{(1 + μ_{1}^{q}) ._{ε} (1 + μ_{2}^{q}) + (1 - μ_{1}^{q}) ._{ε} (1 - μ_{2}^{q})}}, \\ \frac{\sqrt[q]{2} ν_{1} ._{ε} ν_{2}}{\sqrt[q]{(2 - ν_{1}^{q}) . ε (2 - ν_{2}^{q}) + ν_{1}^{q} ._{ε} ν_{2}^{q}}} 〉 . \end{matrix}$ Take $a = (1 + μ_{1}^{q}) ._{ε} (1 + μ_{2}^{q})$ , $b = (1 - μ_{1}^{q}) ._{ε} (1 - μ_{2}^{q})$ , $c = ν_{1}^{q} ._{ε} ν_{2}^{q}$ and $d = (2 - ν_{1}^{q}) . ε (2 - ν_{2}^{q}),$ then $\begin{matrix} Z_{1} \oplus_{ε} Z_{2} & = & 〈 \sqrt[q]{\frac{a - b}{a + b}}, \frac{\sqrt[q]{2 c}}{\sqrt[q]{d + c}} 〉 . \end{matrix}$ By the Einstein q-ROF law, $\begin{matrix} ω ._{ε} (Z_{1} \oplus_{ε} Z_{2}) \\ = ω ._{ε} 〈 \sqrt[q]{\frac{a - b}{a + b}}, \frac{\sqrt[q]{2 c}}{\sqrt[q]{d + c}} 〉 \\ = 〈 \sqrt[q]{\frac{(1 + \frac{a - b}{a + b})^{ω} - (1 - \frac{a - b}{a + b})^{ω}}{(1 + \frac{a - b}{a + b})^{ω} + (1 - \frac{a - b}{a + b})^{ω}}}, \\ \frac{\sqrt[q]{2} . (\frac{\sqrt[q]{2 c}}{\sqrt[q]{d + c}})^{ω}}{\sqrt[q]{(2 - \frac{2 c}{d + c})^{ω} + (\frac{2 c}{d + c})^{ω}}} 〉 \\ = 〈 \sqrt[q]{\frac{a^{ω} - b^{ω}}{a^{ω} + b^{ω}}}, \frac{\sqrt[q]{2 c^{ω}}}{\sqrt[q]{d^{ω} + c^{ω}}} 〉 \\ = 〈 \sqrt[q]{\frac{(1 + μ_{1}^{q})^{ω} ._{ε} (1 + μ_{2}^{q})^{ω} - (1 - μ_{1}^{q})^{ω} ._{ε} (1 - μ_{2}^{q})^{ω}}{(1 + μ_{1}^{q})^{ω} ._{ε} (1 + μ_{2}^{q})^{ω} + (1 - μ_{1}^{q})^{ω} ._{ε} (1 - μ_{2}^{q})^{ω}}}, \\ \frac{\sqrt[q]{2} ν_{1}^{ω} ._{ε} ν_{2}^{ω}}{\sqrt[q]{(2 - ν_{1}^{q})^{ω} . ε (2 - ν_{2}^{q})^{ω} + (ν_{1}^{q})^{ω} ._{ε} (ν_{2}^{q})^{ω}}} 〉 . \end{matrix}$ On the other hand, $\begin{matrix} ω ._{ε} Z_{1} \\ = 〈 \sqrt[q]{\frac{(1 + μ_{1}^{q})^{ω} - (1 - μ_{1}^{q})^{ω}}{(1 + μ_{1}^{q})^{ω} + (1 - μ_{1}^{q})^{ω}}}, \frac{\sqrt[q]{2} ν_{1}^{ω}}{\sqrt[q]{(2 - ν_{1}^{q})^{ω} + (ν_{1}^{q})^{ω}}} 〉 \\ = 〈 \sqrt[q]{\frac{a_{1} - b_{1}}{a_{1} + b_{1}}}, \frac{\sqrt[q]{2 c_{1}}}{\sqrt[q]{d_{1} + c_{1}}} 〉, \end{matrix}$ and $\begin{matrix} ω ._{ε} Z_{2} \\ = 〈 \sqrt[q]{\frac{(1 + μ_{2}^{q})^{ω} - (1 - μ_{2}^{q})^{ω}}{(1 + μ_{2}^{q})^{ω} + (1 - μ_{2}^{q})^{ω}}}, \frac{\sqrt[q]{2} ν_{2}^{ω}}{\sqrt[q]{(2 - ν_{2}^{q})^{ω} + (ν_{2}^{q})^{ω}}} 〉 \\ = 〈 \sqrt[q]{\frac{a_{2} - b_{2}}{a_{2} + b_{2}}}, \frac{\sqrt[q]{2 c_{2}}}{\sqrt[q]{d_{2} + c_{2}}} 〉, \end{matrix}$ where $a_{1} = (1 + μ_{1}^{q})^{ω}$ , $b_{1} = (1 - μ_{1}^{q})^{ω}$ , $c_{1} = (ν_{1}^{q})^{ω}$ , $d_{1} = (2 - ν_{1}^{q})^{ω}$ , $a_{2} = (1 + μ_{2}^{q})^{ω}$ , $b_{2} = (1 - μ_{2}^{q})^{ω}$ , $c_{2} = (ν_{2}^{q})^{ω}$ and $d_{2} = (2 - ν_{2}^{q})^{ω}$ , therefore,

$(ω ._{ε} Z_{1}) \oplus_{ε} (ω ._{ε} Z_{2})$ $\begin{matrix} = 〈 \sqrt[q]{\frac{a_{1} - b_{1}}{a_{1} + b_{1}}}, \frac{\sqrt[q]{2 c_{1}}}{\sqrt[q]{d_{1} + c_{1}}} 〉 \oplus_{ε} 〈 \sqrt[q]{\frac{a_{2} - b_{2}}{a_{2} + b_{2}}}, \frac{\sqrt[q]{2 c_{2}}}{\sqrt[q]{d_{2} + c_{2}}} 〉 \\ = 〈 \sqrt[q]{\frac{\frac{a_{1} - b_{1}}{a_{1} + b_{1}} + \frac{a_{2} - b_{2}}{a_{2} + b_{2}}}{1 + \frac{a_{1} - b_{1}}{a_{1} + b_{1}} ._{ε} \frac{a_{2} - b_{2}}{a_{2} + b_{2}}}}, \\ \frac{2^{\frac{2}{q}} \sqrt[q]{\frac{c_{1} ._{ε} c_{2}}{(d_{1} + c_{1}) ._{ε} (d_{2} + c_{2})}}}{\sqrt[q]{1 + (1 - \frac{2 c_{1}}{d_{1} + c_{1}}) ._{ε} (1 - \frac{2 c_{2}}{d_{2} + c_{2}})}} 〉 \\ = 〈 \sqrt[q]{\frac{a_{1} ._{ε} a_{2} - b_{1} ._{ε} b_{2}}{a_{1} ._{ε} a_{2} + b_{1} ._{ε} b_{2}}}, \frac{\sqrt[q]{2 c_{1} ._{ε} c_{2}}}{\sqrt[q]{d_{1} ._{ε} d_{2} + c_{1} ._{ε} c_{2}}} 〉 \\ = 〈 \sqrt[q]{\frac{(1 + μ_{1}^{q})^{ω} ._{ε} (1 + μ_{2}^{q})^{ω} - (1 - μ_{1}^{q})^{ω} ._{ε} (1 - μ_{2}^{q})^{ω}}{(1 + μ_{1}^{q})^{ω} ._{ε} (1 + μ_{2}^{q})^{ω} + (1 - μ_{1}^{q})^{ω} ._{ε} (1 - μ_{2}^{q})^{ω}}}, \\ \frac{\sqrt[q]{2} ν_{1}^{ω} ._{ε} ν_{2}^{ω}}{\sqrt[q]{(2 - ν_{1}^{q})^{ω} . ε (2 - ν_{2}^{q})^{ω} + (ν_{1}^{q})^{ω} ._{ε} (ν_{2}^{q})^{ω}}} 〉 . \end{matrix}$ Hence, $ω ._{ε} (Z_{1} \oplus_{ε} Z_{2}) = ω ._{ε} Z_{1} \oplus_{ε} ω ._{ε} Z_{2} .$

(v) For ω₁, ω₂ > 0 . $\begin{matrix} ω_{1} ._{ε} Z & = & 〈 \sqrt[q]{\frac{(1 + μ^{q})^{ω_{1}} - (1 - μ^{q})^{ω_{1}}}{(1 + μ^{q})^{ω_{1}} + (1 - μ^{q})^{ω_{1}}}}, \\ \frac{\sqrt[q]{2} ν^{ω_{1}}}{\sqrt[q]{(2 - ν^{q})^{ω_{1}} + (ν^{q})^{ω_{1}}}} 〉 \\ = & 〈 \sqrt[q]{\frac{a_{1} - b_{1}}{a_{1} + b_{1}}}, \frac{\sqrt[q]{2 c_{1}}}{\sqrt[q]{d_{1} + c_{1}}} 〉 . \\ ω_{2} ._{ε} Z & = & 〈 \sqrt[q]{\frac{(1 + μ^{q})^{ω_{2}} - (1 - μ^{q})^{ω_{2}}}{(1 + μ^{q})^{ω_{2}} + (1 - μ^{q})^{ω_{2}}}}, \\ \frac{\sqrt[q]{2} ν^{ω_{2}}}{\sqrt[q]{(2 - ν^{q})^{ω_{2}} + (ν^{q})^{ω_{2}}}} 〉 \\ = & 〈 \sqrt[q]{\frac{a_{2} - b_{2}}{a_{2} + b_{2}}}, \frac{\sqrt[q]{2 c_{2}}}{\sqrt[q]{d_{2} + c_{2}}} 〉, \end{matrix}$ where $a_{a} = (1 + μ^{q})^{ω_{a}}$ , $b_{a} = (1 - μ^{q})^{ω_{a}}$ , $c_{a} = (ν^{q})^{ω_{a}}$ and $d_{a} = (2 - ν^{q})^{ω_{a}}$ for $a = 1, 2 .$

$\begin{matrix} (ω_{1} ._{ε} Z) \oplus_{ε} (ω_{2} ._{ε} Z) \\ = 〈 \sqrt[q]{\frac{a_{1} - b_{1}}{a_{1} + b_{1}}}, \frac{\sqrt[q]{2 c_{1}}}{\sqrt[q]{d_{1} + c_{1}}} 〉 \oplus_{ε} \\ 〈 \sqrt[q]{\frac{a_{2} - b_{2}}{a_{2} + b_{2}}}, \frac{\sqrt[q]{2 c_{2}}}{\sqrt[q]{d_{2} + c_{2}}} 〉 \\ = 〈 \sqrt[q]{\frac{\frac{a_{1} - b_{1}}{a_{1} + b_{1}} + \frac{a_{2} - b_{2}}{a_{2} + b_{2}}}{1 + \frac{a_{1} - b_{1}}{a_{1} + b_{1}} ._{ε} \frac{a_{2} - b_{2}}{a_{2} + b_{2}}}}, \\ \frac{2^{\frac{2}{q}} \sqrt[q]{\frac{c_{1} ._{ε} c_{2}}{(d_{1} + c_{1}) ._{ε} (d_{2} + c_{2})}}}{\sqrt[q]{1 + (1 - \frac{2 c_{1}}{d_{1} + c_{1}}) ._{ε} (1 - \frac{2 c_{2}}{d_{2} + c_{2}})}} 〉 \\ = 〈 \sqrt[q]{\frac{a_{1} ._{ε} a_{2} - b_{1} ._{ε} b_{2}}{a_{1} ._{ε} a_{2} + b_{1} ._{ε} b_{2}}}, \frac{\sqrt[q]{2 c_{1} ._{ε} c_{2}}}{\sqrt[q]{d_{1} ._{ε} d_{2} + c_{1} ._{ε} c_{2}}} 〉 \\ = 〈 \sqrt[q]{\frac{(1 + μ^{q})^{ω_{1} + ω_{2}} - (1 - μ^{q})^{ω_{1} + ω_{2}}}{(1 + μ^{q})^{ω_{1} + ω_{2}} + (1 - μ^{q})^{ω_{1} + ω_{2}}}}, \\ \frac{\sqrt[q]{2} ν^{ω_{1} + ω_{2}}}{\sqrt[q]{(2 - ν^{q})^{ω_{1} + ω_{2}} + (ν_{1}^{q})^{ω_{1} + ω_{2}}}} 〉 \\ = (ω_{1} + ω_{2}) ._{ε} Z . \end{matrix}$ Hence, $ω_{1} ._{ε} Z \oplus_{ε} ω_{2} ._{ε} Z = (ω_{1} + ω_{2}) ._{ε} Z .$

Similarly, others can be verified. □

Theorem 3.3. Let $Z_{1} = 〈 μ_{1}, ν_{1} 〉$ and $Z_{2} = 〈 μ_{2}, ν_{2} 〉$ be two q-ROFNs, then

$(i) Z_{1}^{c} \land_{ε} Z_{2}^{c} = (Z_{1} \lor_{ε} Z_{2})^{c},$

$(ii) Z_{1}^{c} \lor_{ε} Z_{2}^{c} = (Z_{1} \land_{ε} Z_{2})^{c},$

$(iii) Z_{1}^{c} \oplus_{ε} Z_{2}^{c} = (Z_{1} \otimes_{ε} Z_{2})^{c},$

$(iv) Z_{1}^{c} \otimes_{ε} Z_{2}^{c} = (Z_{1} \oplus_{ε} Z_{2})^{c},$

$(v) (Z_{1} \lor_{ε} Z_{2}) \oplus_{ε} (Z_{1} \land_{ε} Z_{2}) = Z_{1} \oplus_{ε} Z_{2},$

$(vi) (Z_{1} \lor_{ε} Z_{2}) \otimes_{ε} (Z_{1} \land_{ε} Z_{2}) = Z_{1} \otimes_{ε} Z_{2} .$

Proof. It is obvious. □

Theorem 3.4. Let $Z_{1} = 〈 μ_{1}, ν_{1} 〉$ , $Z_{2} = 〈 μ_{2}, ν_{2} 〉$ and $Z_{3} = 〈 μ_{3}, ν_{3} 〉$ be three q-ROFNs, then

$(i) (Z_{1} \lor_{ε} Z_{2}) \land_{ε} Z_{3} = (Z_{1} \land_{ε} Z_{3}) \lor_{ε} (Z_{2} \land_{ε} Z_{3}),$

$(ii) (Z_{1} \land_{ε} Z_{2}) \lor_{ε} Z_{3} = (Z_{1} \lor_{ε} Z_{3}) \land_{ε} (Z_{2} \lor_{ε} Z_{3}),$

$(iii) (Z_{1} \lor_{ε} Z_{2}) \oplus_{ε} Z_{3} = (Z_{1} \oplus_{ε} Z_{3}) \lor_{ε} (Z_{2} \oplus_{ε} Z_{3}),$

$(iv) (Z_{1} \land_{ε} Z_{2}) \oplus_{ε} Z_{3} = (Z_{1} \oplus_{ε} Z_{3}) \land_{ε} (Z_{2} \oplus_{ε} Z_{3}),$

$(v) (Z_{1} \lor_{ε} Z_{2}) \otimes_{ε} Z_{3} = (Z_{1} \otimes_{ε} Z_{3}) \lor_{ε} (Z_{2} \otimes_{ε} Z_{3}),$

$(vi) (Z_{1} \land_{ε} Z_{2}) \otimes_{ε} Z_{3} = (Z_{1} \otimes_{ε} Z_{3}) \land_{ε} (Z_{2} \otimes_{ε} Z_{3}) .$

Proof. It is obvious. □

4 q-Rung orthopair fuzzy Einstein aggregation operators

4.1 q-rung orthopair fuzzy Einstein weighted averaging operators

Here, we define Einstein weighted averaging operators under q-ROF environment

Definition 4.1. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉 (a = 1, 2, \dots, c)$ be a family of q-ROFNs and $ϖ_{a}$ is the weight of $Z_{a}$ with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then q-ROFEWA operator is a mapping $Q^{c} \to Q$ such that

$q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $= ϖ_{1} ._{ε} Z_{1} \oplus_{ε} ϖ_{2} ._{ε} Z_{2} \oplus_{ε} \dots \oplus_{ε} ϖ_{c} ._{ε} Z_{c} .$ (1) If $ϖ_{a} = 1 / c$ , $\forall a$ , then q-ROFEWA operator becomes q-ROFA operator $q - ROFA (Z_{1}, Z_{2}, \dots, Z_{c}) = \frac{1}{c} (Z_{1} \oplus_{ε} Z_{2} \oplus_{ε} \dots \oplus_{ε} Z_{c}) .$

Theorem 4.1. Let $Z_{a} = (μ_{a}, ν_{a})$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs, then aggregated value by using Equation 1 is q-ROFN which is given by

$q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}} 〉 . \end{matrix}$ (2)

Proof. Use the mathematical induction to prove Equation 2.

When $c = 2$ , $q - ROFEWA (Z_{1}, Z_{2}) = ϖ_{1} ._{ε} Z_{1} \oplus_{ε} ϖ_{2} ._{ε} Z_{2} .$ By Theorem 3.1, both $ϖ_{1} ._{ε} Z_{1}$ and $ϖ_{2} ._{ε} Z_{2}$ are q-ROFNs and value of $ϖ_{1} ._{ε} Z_{1} \oplus_{ε} ϖ_{2} ._{ε} Z_{2}$ is a q-ROFN. By using (vi) in Definition 3.1, we have $\begin{matrix} ϖ_{1} ._{ε} Z_{1} & = & 〈 \sqrt[q]{\frac{(1 + μ_{1}^{q})^{ϖ_{1}} - (1 - μ_{1}^{q})^{ϖ_{1}}}{(1 + μ_{1}^{q})^{ϖ_{1}} + (1 - μ_{1}^{q})^{ϖ_{1}}}}, \\ \frac{\sqrt[q]{2} ν_{1}^{ϖ_{1}}}{\sqrt[q]{(2 - ν_{1}^{q})^{ϖ_{1}} + (ν_{1}^{q})^{ϖ_{1}}}} 〉 . \end{matrix}$ $\begin{matrix} ϖ_{2} ._{ε} Z_{2} & = & 〈 \sqrt[q]{\frac{(1 + μ_{2}^{q})^{ϖ_{2}} - (1 - μ_{2}^{q})^{ϖ_{2}}}{(1 + μ_{2}^{q})^{ϖ_{2}} + (1 - μ_{2}^{q})^{ϖ_{2}}}}, \\ \frac{\sqrt[q]{2} ν_{2}^{ϖ_{2}}}{\sqrt[q]{(2 - ν_{2}^{q})^{ϖ_{2}} + (ν_{2}^{q})^{ϖ_{2}}}} 〉 . \end{matrix}$ Then

$q - ROFEWA (Z_{1}, Z_{2})$ $\begin{matrix} = ϖ_{1} ._{ε} Z_{1} \oplus_{ε} ϖ_{2} ._{ε} Z_{2} \\ = 〈 \sqrt[q]{\frac{\frac{(1 + μ_{1}^{q})^{ϖ_{1}} - (1 - μ_{1}^{q})^{ϖ_{1}}}{(1 + μ_{1}^{q})^{ϖ_{1}} + (1 - μ_{1}^{q})^{ϖ_{1}}} + \frac{(1 + μ_{2}^{q})^{ϖ_{2}} - (1 - μ_{2}^{q})^{ϖ_{2}}}{(1 + μ_{2}^{q})^{ϖ_{2}} + (1 - μ_{2}^{q})^{ϖ_{2}}}}{1 + (\frac{(1 + μ_{1}^{q})^{ϖ_{1}} - (1 - μ_{1}^{q})^{ϖ_{1}}}{(1 + μ_{1}^{q})^{ϖ_{1}} + (1 - μ_{1}^{q})^{ϖ_{1}}}) ._{ε} (\frac{(1 + μ_{2}^{q})^{ϖ_{2}} - (1 - μ_{2}^{q})^{ϖ_{2}}}{(1 + μ_{2}^{q})^{ϖ_{2}} + (1 - μ_{2}^{q})^{ϖ_{2}}})}}, \\ \frac{(\frac{\sqrt[q]{2} ν_{1}^{ϖ_{1}}}{\sqrt[q]{(2 - ν_{1}^{q})^{ϖ_{1}} + (ν_{1}^{q})^{ϖ_{1}}}}) ._{ε} (\frac{\sqrt[q]{2} ν_{1}^{ϖ_{2}}}{\sqrt[q]{(2 - ν_{2}^{q})^{ϖ_{2}} + (ν_{2}^{q})^{ϖ_{2}}}})}{\sqrt[q]{1 + (1 - \frac{2 ν_{1}^{q ϖ_{1}}}{(2 - ν_{1}^{q})^{ϖ_{1}} + (ν_{1}^{q})^{ϖ_{1}}}) ._{ε} (1 - \frac{2 ν_{2}^{q ϖ_{2}}}{(2 - ν_{2}^{q})^{ϖ_{2}} + (ν_{2}^{q})^{ϖ_{2}}})}} 〉 \\ = 〈 \sqrt[q]{\frac{(1 + μ_{1}^{q})^{ϖ_{1}} ._{ε} (1 + μ_{2}^{q})^{ϖ_{2}} - (1 - μ_{1}^{q})^{ϖ_{1}} ._{ε} (1 - μ_{2}^{q})^{ϖ_{2}}}{(1 + μ_{1}^{q})^{ϖ_{1}} ._{ε} (1 + μ_{2}^{q})^{ϖ_{2}} + (1 - μ_{1}^{q})^{ϖ_{1}} ._{ε} (1 - μ_{2}^{q})^{ϖ_{2}}}}, \\ \frac{\sqrt[q]{2} ν_{1}^{ϖ_{1}} ._{ε} ν_{2}^{ϖ_{2}}}{\sqrt[q]{(2 - ν_{1}^{q})^{ϖ_{1}} . ε (2 - ν_{2}^{q})^{ϖ_{2}} + (ν_{1}^{q})^{ϖ_{1}} ._{ε} (ν_{2}^{q})^{ϖ_{2}}}} 〉 . \end{matrix}$ Thus, Equation 2 is true when $c = 2$ .

Suppose result holds for $c = k$ ,

$q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{k})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{k} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{k} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{k} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{k} (1 - μ_{a}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{k} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{k} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{k} (ν_{a}^{q})^{ϖ_{a}}}} 〉 . \end{matrix}$ Now for $c = k + 1$ ,

$q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{k + 1})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{k} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{k} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{k} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{k} (1 - μ_{a}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{k} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{k} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{k} (ν_{a}^{q})^{ϖ_{a}}}} 〉 \oplus_{ε} \\ 〈 \sqrt[q]{\frac{(1 + μ_{k + 1}^{q})^{ϖ_{k + 1}} - (1 - μ_{k + 1}^{q})^{ϖ_{k + 1}}}{(1 + μ_{k + 1}^{q})^{ϖ_{k + 1}} + (1 - μ_{k + 1}^{q})^{ϖ_{k + 1}}}}, \\ \frac{\sqrt[q]{2} ν_{k + 1}^{ϖ_{k + 1}}}{\sqrt[q]{(2 - ν_{k + 1}^{q})^{ϖ_{k + 1}} + (ν_{k + 1}^{q})^{ϖ_{k + 1}}}} 〉 \\ = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{k + 1} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{k + 1} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{k + 1} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{k + 1} (1 - μ_{a}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{k + 1} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{k + 1} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{k + 1} (ν_{a}^{q})^{ϖ_{a}}}} 〉 . \end{matrix}$ Thus, result is true for $c = k + 1$ . Hence, Equation 2 holds, ∀ $c$ . □

Lemma 4.1. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉, ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then $\prod_{a = 1}^{s} R_{a}^{ϖ_{a}} \leq \sum_{a = 1}^{c} ϖ_{a} R_{a},$ equality holds iff $R_{1} = R_{2} = \dots = R_{c}$ .

Theorem 4.2. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs, then q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c})$ is also a q-ROFN.

Proof. Since $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ ( $a = 1, 2, \dots, c$ ) represents a collection of q-ROFNs, so $0 \leq μ_{a}, ν_{a} \leq 1$ and $0 \leq μ_{a}^{q} + ν_{a}^{q} \leq 1$ . Therefore,

$\frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}$ $\begin{matrix} = & 1 - \frac{2 \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \\ \leq 1 - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}} \leq 1 . \end{matrix}$ (3) Also, $(1 + μ_{a}^{q}) \geq (1 - μ_{a}^{q}) \Rightarrow \prod_{a = 1}^{c} (1 + μ_{a}^{q}) - \prod_{a = 1}^{c} (1 - μ_{a}^{q}) \geq 0$ . Therefore, $\frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \geq 0 .$ Thus, 0 ≤ μ_q-ROFEWA ≤ 1 .

Moreover, $\begin{matrix} \frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}} \\ \leq \frac{2 \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \\ \leq \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}} \leq 1 . \end{matrix}$ Also, $\begin{matrix} \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}} \geq 0 \Leftrightarrow \\ \frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}} \geq 0 . \end{matrix}$ Thus, 0 ≤ ν_q-ROFEWA ≤ 1 . Moreover, $\begin{matrix} μ_{q - ROFEWA}^{q} + ν_{q - ROFEWA}^{q} \\ = \frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \\ + \frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}} \\ \leq 1 - \frac{2 \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \\ + \frac{2 \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \\ = 1 . \end{matrix}$ Hence, q-ROFEWA∈[0, 1]. Therefore, q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c})$ is a q-ROFN. □

Corollary 4.1. The q-ROFEWA and q-ROFWA operators have the relationship:

$q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c}) \leq q - ROFWA$ $(Z_{1}, Z_{2}, \dots, Z_{c}) .$

Proof. Let q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c}) = (μ_{Z}^{β}, ν_{Z}^{β}) = Z^{β}$ and q-ROFWA $(Z_{1}, Z_{2}, \dots, Z_{c}) = (μ_{Z}, ν_{Z}) = Z$ . Since $\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}} \leq \sum_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \sum_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}} = 2,$ then from 3, we get $\begin{matrix} \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}} \\ \leq \sqrt[q]{1 - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}, \\ \Leftrightarrow μ_{Z}^{β} \leq μ_{Z} \end{matrix}$ equality holds iff $μ_{1} = μ_{2} = \dots = μ_{c}$ .

Also, $\begin{matrix} \frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}} \\ \geq \frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\sum_{a = 1}^{c} ϖ_{a} (2 - ν_{a}^{q}) + \sum_{a = 1}^{c} ϖ_{a} ν_{a}^{q}} \\ \geq \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}} \\ \Rightarrow \sqrt[q]{\frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}} \geq \prod_{a = 1}^{c} ν_{a}^{ϖ_{a}} \\ \Rightarrow ν_{Z}^{β} \leq ν_{Z}, \end{matrix}$ equality holds iff $ν_{1} = ν_{2} = \dots = ν_{c}$ .

Thus, $S (Z^{β}) = (μ_{Z}^{β})^{q} - (ν_{Z}^{β})^{q} \leq (μ_{Z})^{q} - (ν_{Z})^{q} = S (Z) .$ If $S (Z^{β}) < S (Z),$ then

q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c}) <$ q-ROFWA $(Z_{1}, Z_{2}, \dots, Z_{c}) .$ If $S (Z^{β}) = S (Z),$ that is, $(μ_{Z}^{β})^{q} - (ν_{Z}^{β})^{q} = (μ_{Z})^{q} - (ν_{Z})^{q},$ then by condition $μ_{Z}^{β} \leq μ_{Z}$ and $ν_{Z}^{β} \geq ν_{Z}$ ; thus, the accuracy function $A (Z^{β}) = (μ_{Z}^{β})^{q} - (ν_{Z}^{β})^{q} = (μ_{Z})^{q} - (ν_{Z})^{q} = A (Z) .$ Thus

q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c}) =$ q-ROFWA $(Z_{1}, Z_{2}, \dots, Z_{c}) .$ Hence, q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c}) \leq$ q-ROFWA $(Z_{1}, Z_{2}, \dots, Z_{c}),$ equality holds iff $Z_{1} = Z_{2} = \dots = Z_{c}$ . □

Example 4.1. Let $Z_{1} = (0.8, 0.5)$ , $Z_{2} = (0.9, 0.4)$ , $Z_{3} = (0.6, 0.7)$ and $Z_{4} = (0.8, 0.7)$ be four q-ROFNs and ϖ = (0.4, 0.2, 0.2, 0.2) ^T, take q = 3, then $\begin{matrix} q - ROFEWA (Z_{1}, Z_{2}, Z_{3}, Z_{4}) \\ = 〈 \sqrt[q]{\frac{\prod_{a = 1}^{4} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{4} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{4} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{4} (1 - μ_{a}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{4} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{4} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{4} (ν_{a}^{q})^{ϖ_{a}}}} 〉 \\ = 〈 \sqrt[3]{\frac{1.49 - 0.48}{1.49 + 0.48}}, \frac{\sqrt[3]{2} \times 0.55}{\sqrt[3]{1.80 + 0.16}} 〉 \\ = 〈 0.80, 0.55 〉 . \end{matrix}$ Now, $\begin{matrix} q - ROFWA (Z_{1}, Z_{2}, Z_{3}, Z_{4}) \\ = 〈 \sqrt[q]{1 - \prod_{a = 1}^{4} (1 - μ_{a}^{q})^{ϖ_{a}}}, \\ \prod_{a = 1}^{4} (ν_{a})^{ϖ_{a}} 〉 \\ = 〈 0.80, 0.55 〉 . \end{matrix}$ ⇒q-ROFEWA $(Z_{1}, Z_{2}, Z_{3}, Z_{4}) \leq q$ -ROFWA $(Z_{1}, Z_{2}, Z_{3}, Z_{4}) .$

Proposition 4.1. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs and $ϖ_{a}$ is the weight of $Z_{a}$ , such that $ϖ_{a} \in [0, 1]$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ .

(i) Idempotency: If $Z_{a} = Z_{o} = 〈 μ_{o}, ν_{o} 〉$ $\forall a$ , then $q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c}) = Z_{o} .$ (ii) Boundedness: Let $Z^{-} = (\min_{a} (μ_{a}), \max_{a} (ν_{a}))$ and $Z^{+} = (\max_{a} (μ_{a}), \min_{a} (ν_{a}))$ , then $Z^{-} \leq q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c}) \leq Z^{+} .$ (iii) Monotonicity: When $Z_{a} \leq P_{a}, \forall a$ , then $\begin{matrix} q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c}) \\ \leq q - ROFEWA (P_{1}, P_{2}, \dots, P_{c}) . \end{matrix}$

Proof. (i). As $Z_{a} = 〈 μ_{o}, ν_{o} 〉$ represents a q-ROFNs, $\forall a$ , then

$q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{o}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{o}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{o}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{o}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} ν_{o}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - ν_{o}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{o}^{q})^{ϖ_{a}}}} 〉 \\ = & 〈 \sqrt[q]{\frac{(1 + μ_{o}^{q})^{\sum_{a = 1}^{c} ϖ_{a}} - (1 - μ_{o}^{q})^{\sum_{a = 1}^{c} ϖ_{a}}}{(1 + μ_{o}^{q})^{\sum_{a = 1}^{c} ϖ_{a}} + (1 - μ_{o}^{q})^{\sum_{a = 1}^{c} ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} ν_{o}^{\sum_{a = 1}^{c} ϖ_{a}}}{\sqrt[q]{(2 - ν_{o}^{q})^{\sum_{a = 1}^{c} ϖ_{a}} + (ν_{o}^{q})^{\sum_{a = 1}^{c} ϖ_{a}}}} 〉 \\ = & 〈 μ_{o}, ν_{o} 〉 . \end{matrix}$ (ii) 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 $μ_{a, \min}^{q} \leq μ_{a}^{q} \leq μ_{a, \max}^{q}, \forall a = 1, 2, \dots, c,$ then $f (μ_{a, \max}^{q}) \leq f (μ_{a}^{q}) \leq f (μ_{a, \min}^{q}), \forall a$ , that is, $\frac{1 - μ_{a, \max}^{q}}{1 + μ_{a, \max}^{q}} \leq \frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}} \leq \frac{1 - μ_{a, \min}^{q}}{1 + μ_{a, \min}^{q}}$ , $\forall a$ . Let $ϖ_{a} \in [0, 1]$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , we have $(\frac{1 - μ_{a, \max}^{q}}{1 + μ_{a, \max}^{q}})^{ϖ_{a}} \leq (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}} \leq (\frac{1 - μ_{a, \min}^{q}}{1 + μ_{a, \min}^{q}})^{ϖ_{a}}$ $\prod_{a = 1}^{c} (\frac{1 - μ_{a, \max}^{q}}{1 + μ_{a, \max}^{q}})^{ϖ_{a}} \leq \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}} \leq \prod_{a = 1}^{c} (\frac{1 - μ_{a, \min}^{q}}{1 + μ_{a, \min}^{q}})^{ϖ_{a}}$ $(\frac{1 - μ_{a, \max}^{q}}{1 + μ_{a, \max}^{q}})^{\sum_{a = 1}^{c} ϖ_{a}} \leq \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}} \leq (\frac{1 - μ_{a, \min}^{q}}{1 + μ_{a, \min}^{q}})^{\sum_{a = 1}^{c} ϖ_{a}}$ $\Leftrightarrow (\frac{1 - μ_{a, \max}^{q}}{1 + μ_{a, \max}^{q}}) \leq \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}} \leq (\frac{1 - μ_{a, \min}^{q}}{1 + μ_{a, \min}^{q}})$ $\Leftrightarrow (\frac{2}{1 + μ_{a, \max}^{q}}) \leq 1 + \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}} \leq (\frac{2}{1 + μ_{a, \min}^{q}})$ $\Leftrightarrow (\frac{1 + μ_{a, \min}^{q}}{2}) \leq \frac{1}{1 + \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}}} \leq (\frac{1 + μ_{a, \max}^{q}}{2})$ $\Leftrightarrow (1 + μ_{a, \min}^{q}) \leq \frac{2}{1 + \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}}} \leq (1 + μ_{a, \max}^{q})$ $\Leftrightarrow μ_{a, \min}^{q} \leq \frac{2}{1 + \prod_{a = 1}^{c} (\frac{1 - μ_{a}^{q}}{1 + μ_{a}^{q}})^{ϖ_{a}}} - 1 \leq μ_{a, \max}^{q}$ $\Leftrightarrow μ_{a, \min}^{q} \leq \frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}} \leq μ_{a, \max}^{q} .$ Thus, $μ_{a, \min} \leq \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}} \leq μ_{a, \max} .$ (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 $ν_{a, \min}^{q} \leq ν_{a}^{q} \leq ν_{a, \max}^{q}, \forall a,$ then $g (ν_{a, \max}^{q}) \leq g (ν_{a}^{q}) \leq g (ν_{a, \min}^{q}), \forall a$ , that is, $\frac{2 - ν_{a, \max}^{q}}{ν_{a, \max}^{q}} \leq \frac{2 - ν_{a}^{q}}{ν_{a}^{q}} \leq \frac{2 - ν_{a, \min}^{q}}{ν_{a, \min}^{q}}$ . Then, $(\frac{2 - ν_{a, \max}^{q}}{ν_{a, \max}^{q}})^{ϖ_{a}} \leq (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}} \leq (\frac{2 - ν_{a, \min}^{q}}{ν_{a, \min}^{q}})^{ϖ_{a}}$ $\prod_{a = 1}^{c} (\frac{2 - ν_{a, \max}^{q}}{ν_{a, \max}^{q}})^{ϖ_{a}} \leq \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}} \leq \prod_{a = 1}^{c} (\frac{2 - ν_{a, \min}^{q}}{ν_{a, \min}^{q}})^{ϖ_{a}}$ $\Rightarrow (\frac{2 - ν_{a, \max}^{q}}{ν_{a, \max}^{q}})^{\sum_{a = 1}^{c} ϖ_{a}} \leq \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}} \leq (\frac{2 - ν_{a, \min}^{q}}{ν_{a, \min}^{q}})^{\sum_{a = 1}^{c} ϖ_{a}}$ $\Rightarrow (\frac{2 - ν_{a, \max}^{q}}{ν_{a, \max}^{q}}) \leq \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}} \leq (\frac{2 - ν_{a, \min}^{q}}{ν_{a, \min}^{q}})$ $\Rightarrow (\frac{2}{ν_{a, \max}^{q}}) \leq 1 + \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}} \leq (\frac{2}{ν_{a, \min}^{q}})$ $\Rightarrow (\frac{ν_{a, \min}^{q}}{2}) \leq \frac{1}{1 + \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}}} \leq (\frac{ν_{a, \max}^{q}}{2})$ $\Rightarrow (ν_{a, \min}^{q}) \leq \frac{2}{1 + \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}}} \leq (ν_{a, \max}^{q})$ $\Rightarrow ν_{a, \min}^{q} \leq \frac{2}{1 + \prod_{a = 1}^{c} (\frac{2 - ν_{a}^{q}}{ν_{a}^{q}})^{ϖ_{a}}} \leq ν_{a, \max}^{q}$ $\Rightarrow ν_{a, \min}^{q} \leq \frac{2 \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}} \leq ν_{a, \max}^{q}$ $\Rightarrow ν_{a, \min} \leq \frac{\sqrt[q]{2} \prod_{a = 1}^{c} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}} \leq ν_{a, \max} .$ (5) Let q-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c}) = Z = 〈 μ_{Z}, ν_{Z} 〉$ , then from 4 and 5, $μ_{\min} \leq μ_{Z} \leq μ_{\max}, ν_{\min} \leq ν_{Z} \leq ν_{\max},$ where $μ_{\min} = min_{a} {μ_{a}}$ , $μ_{\max} = max_{a} {μ_{a}}$ , $ν_{\min} = min_{a} {ν_{a}}$ and $ν_{\max} = max_{a} {ν_{a}}$ . So, $S (Z) = μ_{Z}^{q} - ν_{Z}^{q} \leq μ_{\max}^{q} - ν_{\min}^{q} = S (Z^{+})$ and $S (Z) = μ_{Z}^{q} - ν_{Z}^{q} \geq μ_{\min}^{q} - ν_{\max}^{q} = S (Z^{-}) .$ As $S (Z) < S (Z^{+})$ and $S (Z) > S (Z^{-})$ . So $Z^{-} \leq q - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c}) \leq Z^{+} .$ (iii) It is similar to (ii), so we omit it. □

4.2 q-rung orthopair fuzzy Einstein ordered weighted averaging operators

Here, we define Einstein ordered weighted averaging operators under q-ROF environment.

Definition 4.2. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ be a family of q-ROFNs and $ϖ_{a}$ is weight of $Z_{a}$ with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then q-ROFEOWA operator is a mapping $Q^{c} \to Q$ such that

$q - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $= ϖ_{1} ._{ε} Z_{ϱ (1)} \oplus_{ε} ϖ_{2} ._{ε} Z_{ϱ (2)} \oplus_{ε} \dots \oplus_{ε} ϖ_{c} ._{ε} Z_{ϱ (c)},$ where $(ϱ (1), ϱ (2), \dots, ϱ (c))$ is the permutation of $(1, 2, \dots, c)$ such that $Z_{ϱ (a - 1)} \geq Z_{ϱ (a)}, \forall a = 1, 2, \dots, c .$

Theorem 4.3. Let $Z_{a} = (μ_{a}, ν_{a})$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs, then aggregated value by using q-ROFEOWA is a q-ROFN and

$q - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{ϱ (a)}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{ϱ (a)}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} ν_{ϱ (a)}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - ν_{ϱ (a)}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{ϱ (a)}^{q})^{ϖ_{a}}}} 〉 . \end{matrix}$ (6)

Proof. It is similar to Theorem 4.1. □

We give some properties without their proofs.

Corollary 4.2. The q-ROFEOWA and q-ROFOWA operators have the relation:

q-ROFEOWA $(Z_{1}, Z_{2}, \dots, Z_{c}) \leq$ q-ROFOWA $(Z_{1}, Z_{2}, \dots, Z_{c}) .$

Proof. It is similar to Corollary 4.1. □

Example 4.2. Let $Z_{1} = (0.6, 0.7)$ , $Z_{2} = (0.8, 0.7)$ , $Z_{3} = (0.6, 0.9)$ and $Z_{4} = (0.9, 0.4)$ be four q-ROFNs and ϖ = (0.3, 0.3, 0.2, 0.2) ^T, take q = 3. As $S (Z_{1}) = (0.6)^{3} - (0.7)^{3} = - 0.13,$ $S (Z_{2}) = (0.8)^{3} - (0.7)^{3} = 0.17,$ $S (Z_{3}) = (0.6)^{3} - (0.9)^{3} = - 0.51,$ $S (Z_{4}) = (0.9)^{3} - (0.4)^{3} = 0.67 .$ Since $S (Z_{4}) > S (Z_{2}) > S (Z_{1}) > S (Z_{3}),$ therefore $Z_{ϱ (1)} = Z_{4} = (0.9, 0.4),$ $Z_{ϱ (2)} = Z_{2} = (0.8, 0.7),$ $Z_{ϱ (3)} = Z_{1} = (0.6, 0.7),$ $Z_{ϱ (4)} = Z_{3} = (0.6, 0.9) .$ By utilizing q-ROFEOWA operator,

$q - ROFEOWA (Z_{1}, Z_{2}, Z_{3}, Z_{4})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{4} (1 + μ_{ϱ (a)}^{q})^{ϖ_{a}} - \prod_{a = 1}^{4} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{4} (1 + μ_{ϱ (a)}^{q})^{ϖ_{a}} + \prod_{a = 1}^{4} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{4} ν_{ϱ (a)}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{4} (2 - ν_{ϱ (a)}^{q})^{ϖ_{a}} + \prod_{a = 1}^{4} (ν_{ϱ (a)}^{q})^{ϖ_{a}}}} 〉 \\ = & 〈 \sqrt[3]{\frac{1.44 - 0.49}{1.44 + 0.49}}, \frac{\sqrt[3]{2} \times 0.66}{\sqrt[3]{1.62 + 0.66}} 〉 \\ = & 〈 0.79, 0.55 〉 . \end{matrix}$ Now, $\begin{matrix} q - ROFOWA (Z_{1}, Z_{2}, Z_{3}, Z_{4}) \\ = 〈 \sqrt[q]{1 - \prod_{a = 1}^{4} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}, \\ \prod_{a = 1}^{4} (ν_{ϱ (a)})^{ϖ_{a}} 〉 \\ = 〈 0.80, 0.66 〉 . \end{matrix}$ ⇒q-ROFEOWA $(Z_{1}, Z_{2}, Z_{3}, Z_{4}) < q$ -ROFOWA $(Z_{1}, Z_{2}, Z_{3}, Z_{4}) .$

Proposition 4.2. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs and $ϖ_{a}$ is the weight of $Z_{a}$ , such that $ϖ_{a} \in [0, 1]$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ .

(i) Idempotency: If $Z_{a} = Z_{o} = 〈 μ_{o}, ν_{o} 〉$ , $\forall a$ , then $q - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c}) = Z_{o} .$ (ii) Boundedness: Let $Z^{-} = (\min_{a} (μ_{a}), \max_{a} (ν_{a}))$ and $Z^{+} = (\max_{a} (μ_{a}), \min_{a} (ν_{a}))$ , then $Z^{-} \leq q - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c}) \leq Z^{+} .$ (iii) Monotonicity: When $Z_{a} \leq P_{a}, \forall a$ , then

$q - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c}) \leq q - ROFEOWA (P_{1}, P_{2}, \dots, P_{c}) .$

Proof. It is similar to Proposition 4.1. □

5 Generalized q-rung orthopair fuzzy Einstein aggregation operators

5.1 Generalized q-rung orthopair fuzzy Einstein weighted averaging operators

Definition 5.1. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ be a collection of q-ROFNs and $ϖ_{a}$ is weight of $Z_{a}$ with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then Gq-ROFEWA operator is a mapping $Q^{c} \to Q$ such that $Gq - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c}) = (\oplus_{a = 1}^{c} (ϖ_{a} ._{ε} Z_{a}^{ω}))^{\frac{1}{ω}},$ where ω > 0.

Particularly,

If ω = 1, then Gq-ROFEWA becomes q-ROFEWA.

If $ϖ = (1 / c, 1 / c, \dots, 1 / c)^{T}$ , then Gq-ROFEWA $(Z_{1}, Z_{2}, \dots, Z_{c}) = (\frac{1}{c} ._{ε} \oplus_{a = 1}^{c} Z_{a}^{ω})^{\frac{1}{ω}} .$

Theorem 5.1. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs and $ϖ_{a}$ is the weight of $Z_{a}$ with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then aggregated value by applying Gq-ROFEWA operator is a q-ROFN and

$Gq - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = 〈 \frac{\sqrt[q]{2} {\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}^{1 / q ω}}{\sqrt[q]{\frac{(\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}})^{1 / ω}}}}, \\ \sqrt[q]{\frac{\frac{(\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}{- (\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}}{\frac{(\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}}} 〉 . \end{matrix}$ where $β = (2 - μ_{a}^{q}), α = (1 + ν_{a}^{q}), τ = (1 - ν_{a}^{q}),$ $λ = (μ_{a}^{q}) .$

Proof. Since $\begin{matrix} Z_{a}^{ω} \\ = 〈 \frac{\sqrt[q]{2} μ_{a}^{ω}}{\sqrt[q]{(2 - μ_{a}^{q})^{ω} + (μ_{a}^{q})^{ω}}}, \sqrt[q]{\frac{(1 + ν_{a}^{q})^{ω} - (1 - ν_{a}^{q})^{ω}}{(1 + ν_{a}^{q})^{ω} + (1 - ν_{a}^{q})^{ω}}} 〉, \end{matrix}$ $\Rightarrow \oplus_{a = 1}^{c} ϖ_{a} ._{ε} Z_{a}^{ω}$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + \frac{2 λ^{ω}}{β^{ω} + λ^{ω}})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - \frac{2 λ^{ω}}{β^{ω} + λ^{ω}})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + \frac{2 λ^{ω}}{β^{ω} + λ^{ω}})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - \frac{2 λ^{ω}}{β^{ω} + λ^{ω}})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} (\sqrt[q]{\frac{α^{ω} - τ^{ω}}{α^{ω} + τ^{ω}}})^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - \frac{α^{ω} - τ^{ω}}{α^{ω} + τ^{ω}})^{ϖ_{a}} + \prod_{a = 1}^{c} (\frac{α^{ω} - τ^{ω}}{α^{ω} + τ^{ω}})^{ϖ_{a}}}} 〉 \\ = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} (\sqrt[q]{α^{ω} - τ^{ω}})^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}}} 〉 . \end{matrix}$ Therefore, $(\oplus_{a = 1}^{c} ϖ_{a} ._{ε} Z_{a}^{ω})^{1 / ω} = 〈 \frac{\sqrt[q]{2 (\frac{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}})^{1 / ω}}}{\sqrt[q]{\frac{(2 - \frac{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}})^{1 / ω}}{+ (\frac{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}})^{1 / ω}}}}, \sqrt[q]{\frac{\frac{(1 + \frac{2 \prod_{a}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}}{\prod_{a}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + \prod_{a}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}})^{1 / ω}}{- (1 - \frac{2 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}})^{1 / ω}}}{\frac{(1 + \frac{2 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}})^{1 / ω}}{+ (1 - \frac{2 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}}{\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}}})^{1 / ω}}}} 〉 = 〈 \frac{\sqrt[q]{2} {\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}}}^{1 / q ω}}{\sqrt[q]{\frac{(\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {β^{ω} + 3 λ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {β^{ω} - λ^{ω}}^{ϖ_{a}})^{1 / ω}}}}, \sqrt[q]{\frac{\frac{(\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}{- (\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}}{\frac{(\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {α^{ω} + 3 τ^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {α^{ω} - τ^{ω}}^{ϖ_{a}})^{1 / ω}}}} 〉 .$

When ω = 1, then $Gq - ROFEWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{a}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} ν_{a}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - ν_{a}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{a}^{q})^{ϖ_{a}}}} 〉 . \end{matrix}$ □

5.2 Generalized q-rung orthopair fuzzy Einstein ordered weighted averaging operators

Definition 5.2. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ be a family of q-ROFNs and $ϖ_{a}$ is weight of $Z_{a}$ with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then Gq-ROFEOWA operator is a mapping $Q^{c} \to Q$ such that $\begin{matrix} Gq - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c}) \\ = (\oplus_{a = 1}^{c} (ϖ_{a} ._{ε} Z_{ϱ (a)}^{ω}))^{\frac{1}{ω}}, \end{matrix}$ where ω > 0.

Theorem 5.2. Let $Z_{a} = 〈 μ_{a}, ν_{a} 〉$ $(a = 1, 2, \dots, c)$ be a family of q-ROFNs and $ϖ_{a}$ is the weight of $Z_{a}$ with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ , then aggregated value by applying Gq-ROFEOWA operator is a q-ROFN and $Gq - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = 〈 \frac{\sqrt[q]{2} {\prod_{a = 1}^{c} {{β_{1}}^{ω} + 3 {λ_{1}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{β_{1}}^{ω} - {λ_{1}}^{ω}}^{ϖ_{a}}}^{1 / q ω}}{\sqrt[q]{\frac{(\prod_{a = 1}^{c} {{β_{1}}^{ω} + 3 {λ_{1}}^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {{β_{1}}^{ω} - {λ_{1}}^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {{β_{1}}^{ω} + 3 {λ_{1}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{β_{1}}^{ω} - {λ_{1}}^{ω}}^{ϖ_{a}})^{1 / ω}}}}, \\ \sqrt[q]{\frac{\frac{(\prod_{a = 1}^{c} {{α_{1}}^{ω} + 3 {τ_{1}}^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {{α_{1}}^{ω} - {τ_{1}}^{ω}}^{ϖ_{a}})^{1 / ω}}{- (\prod_{a = 1}^{c} {{α_{1}}^{ω} + 3 {τ_{1}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{α_{1}}^{ω} - {τ_{1}}^{ω}}^{ϖ_{a}})^{1 / ω}}}{\frac{(\prod_{a = 1}^{c} {{α_{1}}^{ω} + 3 {τ_{1}}^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {{α_{1}}^{ω} - {τ_{1}}^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {{α_{1}}^{ω} + 3 {τ_{1}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{α_{1}}^{ω} - {τ_{1}}^{ω}}^{ϖ_{a}})^{1 / ω}}}} 〉 . \end{matrix}$ where $β_{1} = (2 - μ_{ϱ (a)}^{q}), α_{1} = (1 + ν_{ϱ (a)}^{q}), τ_{1} = (1 - ν_{ϱ (a)}^{q}), λ_{1} = (μ_{ϱ (a)}^{q}) .$

Proof. It is similar to Theorem 5.1. □

When ω = 1, then $Gq - ROFEOWA (Z_{1}, Z_{2}, \dots, Z_{c})$ $\begin{matrix} = & 〈 \sqrt[q]{\frac{\prod_{a = 1}^{c} (1 + μ_{ϱ (a)}^{q})^{ϖ_{a}} - \prod_{a = 1}^{c} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}{\prod_{a = 1}^{c} (1 + μ_{ϱ (a)}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (1 - μ_{ϱ (a)}^{q})^{ϖ_{a}}}}, \\ \frac{\sqrt[q]{2} \prod_{a = 1}^{c} ν_{ϱ (a)}^{ϖ_{a}}}{\sqrt[q]{\prod_{a = 1}^{c} (2 - ν_{ϱ (a)}^{q})^{ϖ_{a}} + \prod_{a = 1}^{c} (ν_{ϱ (a)}^{q})^{ϖ_{a}}}} 〉 . \end{matrix}$

6 MADM problems using q-ROF information

To handle a MADM problem under q-ROF environment, let $K = {K_{1}, K_{2}, \dots, K_{m}}$ be a set of possible alternatives and $J = {J_{1}, J_{2}, \dots, J_{c}}$ is a set of attributes selected by the decision maker. Let $ϖ = (ϖ_{1}, ϖ_{2}, \dots, ϖ_{c})^{T}$ be weight vector (WV) with $ϖ_{a} > 0$ and $\sum_{a = 1}^{c} ϖ_{a} = 1$ . Suppose that $\tilde{E} = (μ_{l a}, ν_{l a})_{m \times c}$ is the q-ROF decision matrix (q-ROFDM), where $μ_{l a}$ and $ν_{l a}$ are the MD and NMD of the alternative $K_{l}$ for the attribute J_t, where $0 \leq μ_{l a}^{q} + ν_{l a}^{q} \leq 1$ .

The following Algorithm 1 is used to solve the MADM problems with q-ROFN based on using Gq-ROFEWA operator.

Algorithm 1

Input:

Selection of suitable alternatives and attributes.

Use the q-ROFDM and Gq-ROFEWA operator $B_{l} = Gq - ROFEWA (K_{l 1}, K_{l 2}, \dots, K_{l c}) = 〈 \frac{\sqrt[q]{2} {\prod_{a = 1}^{c} {{χ_{1}}^{ω} + 3 {χ_{2}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{χ_{1}}^{ω} - {χ_{2}}^{ω}}^{ϖ_{a}}}^{1 / q ω}}{\sqrt[q]{\frac{(\prod_{a = 1}^{c} {{χ_{1}}^{ω} + 3 {χ_{2}}^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {{χ_{1}}^{ω} - {χ_{2}}^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {{χ_{1}}^{ω} + 3 {χ_{2}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{χ_{1}}^{ω} - {χ_{2}}^{ω}}^{ϖ_{a}})^{1 / ω}}}}, \sqrt[q]{\frac{\frac{(\prod_{a = 1}^{c} {{χ_{3}}^{ω} + 3 {χ_{4}}^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {{χ_{3}}^{ω} - {χ_{4}}^{ω}}^{ϖ_{a}})^{1 / ω}}{- (\prod_{a = 1}^{c} {{χ_{3}}^{ω} + 3 {χ_{4}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{χ_{3}}^{ω} - {χ_{4}}^{ω}}^{ϖ_{a}})^{1 / ω}}}{\frac{(\prod_{a = 1}^{c} {{χ_{3}}^{ω} + 3 {χ_{4}}^{ω}}^{ϖ_{a}} + 3 \prod_{a = 1}^{c} {{χ_{3}}^{ω} - {χ_{4}}^{ω}}^{ϖ_{a}})^{1 / ω}}{+ (\prod_{a = 1}^{c} {{χ_{3}}^{ω} + 3 {χ_{4}}^{ω}}^{ϖ_{a}} - \prod_{a = 1}^{c} {{χ_{3}}^{ω} - {χ_{4}}^{ω}}^{ϖ_{a}})^{1 / ω}}}} 〉 .$ $χ_{1} = (2 - μ_{l a}^{q}), χ_{2} = μ_{l a}^{q}, χ_{3} = (1 + ν_{l a}^{q}), χ_{4} = (1 - ν_{l a}^{q})$ for overall preference values $B_{l} (l = 1, 2, \dots, m)$ of the alternative $K_{l}$ .

Use the score degree $S (B_{l}) (l = 1, 2, \dots, m)$ for the ranking of alternatives. For equal score values, compute the accuracy degree $A (B_{l})$ and rank the alternatives according to these values.

Output: The alternative containing maximum score value will be the most preferable as decision.

6.1 Selection of best future career

Human physiological and psychological needs are to be satisfied. This is necessary for life and for continuation of life. The way we address our these needs is important in our lives. The occupation or the career we select must conform to our basic needs and then to our psychological satisfaction. Selection of career is a big question for our youth graduating from universities. Some are thinking of initiating a business, some have tendency to industrial work, others try to join service sector or agriculture. These selections depend upon tendency, aptitude, previous education and futuristic view of oneself.

It is observed that in most cases, students opt for wrong career and ultimately end up with dissatisfaction and loss of interest in their work which certainly lead to unproductive efforts. So the selection of career and profession is vital in life. Consider a decision making problem under professional aspect. Consider a person who has completed his academic degree. Now he wants to select that profession which is suitable for him. He has three choices.

$K_{1}$ : Teaching,

$K_{2}$ : Sales and Marketing,

$K_{3}$ : Self Employment.

Some factors that influence his decision are outlined as below:

$J_{1}$ : Package: It not only includes the salary but also the fringe benefits that an organization offer to an employee, e.g. vehicle, house, medical coverage etc.

$J_{2}$ : Self Respect: The occupation or the career where employees work with high self respect are highly productive. Self respect is among the basic human psychological needs.

$J_{3}$ : Education and Experience: Education and pervious experience are pivotal in selection of career. Therefore, one’s choice of profession is highly influenced by his education and experience.

1. The q-ROFDM is shown in Table 1.

Table 1
q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$

$K_{1}$ (0.5,0.7) (0.6,0.5) (0.7,0.5)

$K_{2}$ (0.7,0.6) (0.6,0.4) (0.5,0.4)

$K_{3}$ (0.8,0.5) (0.7,0.3) (0.8,0.4)

$\tilde{E}$	$J_{1}$	$J_{2}$	$J_{3}$
$K_{1}$	(0.5,0.7)	(0.6,0.5)	(0.7,0.5)
$K_{2}$	(0.7,0.6)	(0.6,0.4)	(0.5,0.4)
$K_{3}$	(0.8,0.5)	(0.7,0.3)	(0.8,0.4)

2. The weights assigned by decision maker are $\begin{matrix} ϖ_{1} = 0.4, ϖ_{2} = 0.3, ϖ_{3} = 0.3 \\ and \sum_{a = 1}^{3} ϖ_{a} = 1 . \end{matrix}$ We use the Gq-ROFEWA operator for the selection of best career.

Step 1. For performance values $B_{l}$ of professions, use the Gq-ROFEWA operator for q = 3, ω = 1 . $B_{1} = (0.60, 0.57),$ $B_{2} = (0.63, 0.47),$ $B_{3} = (0.78, 0.40) .$

Step 2. Compute the scores $S (B_{l})$ of q-ROFNs $B_{l}$ and rank the professions. $S (B_{1}) = 0.03,$ $S (B_{2}) = 0.15,$ $S (B_{3}) = 0.41 .$ The ranking of professions is $K_{3} > K_{2} > K_{1} .$ Step 3. Therefore, $K_{3}$ is the suitable profession for future.

6.2 Selection of agriculture land

Agriculture provides the bedrock for a country’s economy. Those areas where land supports agriculture are blessed ones. Selection of a suitable land for cultivation has been remained a basic question for farmers. Since there are lot of factors that influence the cultivation so it has been very difficult to select and focus a suitable piece of land for cultivation. Here, we are trying to develop a model that will help the interested quarters in selection of agriculture land. Suppose that a farmer wants to choose a agriculture land. Let ${L_{1}, L_{2}, L_{3}, L_{4}}$ be the set of four lands. There are some unique factors influencing the decision of farmer in selection. These factors are

$J_{1}$ : Support for Mechanized Farming: In mechanized farming, the cultivation is extensive and always depends upon use of machines. The availability of fuels, transportation and maintenance of machines to be used are important details to be considered. If our selected land does not support these, cultivation will be badly influenced.

$J_{2}$ : Support from Ecological System: Ecological system includes a broader spectrum than environment. It includes climate, fertility, effect of living organisms on the land or cultivation.

$J_{3}$ : Arability for Specific Crop: Every type of land is not suitable for every type of crop. This factor includes first selection of crop to be cultivated then analysis of land to be selected with respect to crop.

1. The q-ROFDM is shown in Table 2.

Table 2
q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$

$L_{1}$ (0.6,0.4) (0.8,0.7) (0.9,0.4)

$L_{2}$ (0.5,0.6) (0.9,0.4) (0.9,0.6)

$L_{3}$ (0.7,0.8) (0.8,0.2) (0.7,0.4)

$L_{4}$ (0.3,0.4) (0.5,0.6) (0.7,0.4)

$\tilde{E}$	$J_{1}$	$J_{2}$	$J_{3}$
$L_{1}$	(0.6,0.4)	(0.8,0.7)	(0.9,0.4)
$L_{2}$	(0.5,0.6)	(0.9,0.4)	(0.9,0.6)
$L_{3}$	(0.7,0.8)	(0.8,0.2)	(0.7,0.4)
$L_{4}$	(0.3,0.4)	(0.5,0.6)	(0.7,0.4)

2. The weights assigned by decision maker are $ϖ_{1} = 0.3, ϖ_{2} = 0.4, ϖ_{3} = 0.4 and \sum_{a = 1}^{3} ϖ_{a} = 1 .$ We use the Gq-ROFEWA operator for the selection of suitable agriculture land.

Step 1. For performance values $B_{l}$ of lands, use the Gq-ROFEWA operator for q = 3, ω = 1 . $B_{1} = (0.80, 0.50),$ $B_{2} = (0.84, 0.50),$ $B_{3} = (0.75, 0.38),$ $B_{4} = (0.54, 0.47) .$

Step 2. Calculate the scores $S (B_{l})$ of q-ROFNs $B_{l}$ and rank the lands. $S (B_{1}) = 0.39,$ $S (B_{2}) = 0.47,$ $S (B_{3}) = 0.37,$ $S (B_{4}) = 0.05 .$ The ranking of lands is $L_{2} > L_{1} > L_{3} > L_{4} .$ Step 3. Therefore, $L_{2}$ is the suitable agriculture land.

The whole procedure of this application is explained in Fig. 2.

Fig. 2

Flowchart for the selection of suitable agriculture land.

6.3 Selection of residential place

Suppose a group of Professors of Vidyasagar University [28] wants to construct their home in a certain place. They construct a team named Senapati Construction Limited (SCL) among them. The SCL visits four places Ashoke Nagar ( $K_{1}$ ), Judge Court ( $K_{2}$ ), Patna Bazar ( $K_{3}$ ) and Kshudiram Nagar ( $K_{4}$ ) which are situated within 10 kilometers of Vidyasagar University. The factors for choosing a suitable land for home construction are as follows:

$J_{1}$ : Life style and neighbors,

$J_{2}$ : Soil type,

$J_{3}$ : Size, shape, orientation, and slope of the block of land,

$J_{4}$ : Existing roads and access to essential services,

$J_{5}$ : Cost.

1. The q-ROFDM is shown in Table 3.

Table 3
q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$ $J_{4}$ $J_{5}$

$K_{1}$ (0.7, 0.3) (0.4, 0.6) (0.5, 0.5) (0.8, 0.2) (0.8, 0.4)

$K_{2}$ (0.5, 0.8) (0.8, 0.6) (0.4, 0.5) (0.7, 0.4) (0.6, 0.5)

$K_{3}$ (0.9, 0.6) (0.8, 0.1) (0.6, 0.4) (0.7, 0.5) (0.9, 0.3)

$K_{4}$ (0.6, 0.7) (0.8, 0.3) (0.7, 0.2) (0.5, 0.3) (0.7, 0.3)

$\tilde{E}$	$J_{1}$	$J_{2}$	$J_{3}$	$J_{4}$	$J_{5}$
$K_{1}$	(0.7, 0.3)	(0.4, 0.6)	(0.5, 0.5)	(0.8, 0.2)	(0.8, 0.4)
$K_{2}$	(0.5, 0.8)	(0.8, 0.6)	(0.4, 0.5)	(0.7, 0.4)	(0.6, 0.5)
$K_{3}$	(0.9, 0.6)	(0.8, 0.1)	(0.6, 0.4)	(0.7, 0.5)	(0.9, 0.3)
$K_{4}$	(0.6, 0.7)	(0.8, 0.3)	(0.7, 0.2)	(0.5, 0.3)	(0.7, 0.3)

2. The weights assigned by decision maker are $\begin{matrix} ϖ_{1} = 0.2, ϖ_{2} = 0.2, ϖ_{3} = 0.1, ϖ_{4} = 0.3, \\ ϖ_{5} = 0.2 and \sum_{a = 1}^{5} ϖ_{a} = 1 . \end{matrix}$ We use the Gq-ROFEWA operator for the selection of suitable residential place.

Step 1. For performance values $B_{l}$ of places, use the Gq-ROFEWA operator for q = 3, ω = 1 . $B_{1} = (0.69, 0.34),$ $B_{2} = (0.66, 0.54),$ $B_{3} = (0.83, 0.33),$ $B_{4} = (0.66, 0.36) .$

Step 2. Calculate the scores $S (B_{l})$ of q-ROFNs $B_{l}$ and rank the places. $S (B_{1}) = 0.29,$ $S (B_{2}) = 0.13,$ $S (B_{3}) = 0.54,$ $S (B_{4}) = 0.24 .$ The ranking of places is $K_{3} > K_{1} > K_{4} > K_{2} .$ Step 3. Therefore, $K_{3}$ is the suitable residential place.

The best alterative from [28] and our proposed approach is same. So, it shows that the results of proposed approach are same as the original results exist in reality. The speciality of this method is that here we can see the effect of different values of parameter ω and it is very short and easy approach to get the best alternative. The theory given in [28] holds only for μ^q + ν^q ≤ 1, where q = 3 and fails when cubic sum of MD and NMD is exceed by 1 but proposed theory covers all such shortcomings.

7 Validity test

For the validity and authenticity of MADM methods, Wang and Triantaphyllou [31] developed a testing criteria, given as follows:

Criterion 1. A MADM technique is valid if the most suitable alternative remains same on changing a non-optimal alternative with some other poor alternative, without disturbing the respective decision criteria.

Criterion 2. The transitive property should be followed by a valid MADM technique.

Criterion 3. The ranking result of alternatives should not change on splitting the problem into the smaller sub-problems and by utilizing the same MADM technique on sub-problems.

Now, we discuss the validity of our proposed MADM technique for Application 6.1 by testing the above criteria.

Validity test by criterion 1: If we replace the decision values of a non-optimal alternative $K_{1}$ by ${\tilde{K}}_{1}$ , then the new q-ROFDM is given in Table 4.

By applying the Gq-ROFEWA operator for ω = 1 and score function, the score values of alternatives are $S (\tilde{B_{1}}) = 0,$ $S (B_{2}) = 0.15,$ $S (B_{3}) = 0.41 .$ The ranking of professions is $K_{3} > K_{2} > K_{1},$ which is same as original ranking order and the best profession is $K_{3}$ . Thus, our presented MADM model fulfills the test criterion 1.

Validity test by criteria 2 and 3: For the validity of proposed algorithm using criteria 2 and 3, we split the problem into the smaller sub-problems ${K_{1}, K_{2}}$ , ${K_{1}, K_{3}}$ , ${K_{2}, K_{3}}$ . By utilizing the proposed technique, the ranking orders of alternatives in these sub-problems are $K_{2} > K_{1}$ , $K_{3} > K_{1}$ and $K_{3} > K_{2}$ , respectively. The combine ranking of alternatives is $K_{3} > K_{2} > K_{1}$ , which is same as that of the original ranking. Hence, the proposed MADM technique is authentic and proficient under criteria 2 and 3.

Table 4
Reconstructed q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$

${\tilde{K}}_{1}$ (0.4,0.7) (0.5,0.6) (0.6,0.6)

$K_{2}$ (0.7,0.6) (0.6,0.4) (0.5,0.4)

$K_{3}$ (0.8,0.5) (0.7,0.3) (0.8,0.4)

$\tilde{E}$	$J_{1}$	$J_{2}$	$J_{3}$
${\tilde{K}}_{1}$	(0.4,0.7)	(0.5,0.6)	(0.6,0.6)
$K_{2}$	(0.7,0.6)	(0.6,0.4)	(0.5,0.4)
$K_{3}$	(0.8,0.5)	(0.7,0.3)	(0.8,0.4)

8 Comparison analysis

We provide comparison analysis of the developed Einstein AOs with others operators such as GPFEWA [11], GPFEWG [12], PFYWA [29] and PFYWG [29] operators to show the validity of our proposed model.

For Application 6.1, it can be seen from Table 5 and Fig. 3 that the final rankings by applying the Gq-ROFEWA, GPFEWA, GPFEWG, PFYWA and PFYWG operators are $K_{3} > K_{2} > K_{1}$ , respectively.

However, the final score values are not same. It is clear that the optimal decision, using all these operators, is same. This shows that our model is applicable to resolve the real life MADM problems.

2. The reason behind our proposed model is that PFNs can deal only those problems where μ² + ν² ≤ 1. There exist many real life problems where data does not fulfill the condition μ² + ν² ≤ 1, then we need q-ROFS. To show feasibility and attractiveness, we have elaborated another application for the selection of agriculture land. GPFEWA, GPFEWG, PFYWA and PFYWG operators cannot solve the Application 6.2. That’s why we need our proposed method.

The q-ROF Einstein AOs are a more flexible and easy approach. The best alternative can be obtained by a short process. The results from the proposed theory are more accurate and closest to original results.

Table 5
Comparison analysis for Application 14 with existing operators (suppose q = 3, ω = 1)

Methods $S (B_{1})$ $S (B_{2})$ $S (B_{3})$ Ranking order

Gq-ROFEWA 0.03 0.15 0.41 $K_{3} > K_{2} > K_{1}$

GPFEWA -0.01 0.13 0.39 $K_{3} > K_{2} > K_{1}$

GPFEWG -0.01 0.09 0.38 $K_{3} > K_{2} > K_{1}$

PFYWA -0.20 0.10 0.31 $K_{3} > K_{2} > K_{1}$

PFYWG -0.11 0.05 0.27 $K_{3} > K_{2} > K_{1}$

Methods	$S (B_{1})$	$S (B_{2})$	$S (B_{3})$	Ranking order
Gq-ROFEWA	0.03	0.15	0.41	$K_{3} > K_{2} > K_{1}$
GPFEWA	-0.01	0.13	0.39	$K_{3} > K_{2} > K_{1}$
GPFEWG	-0.01	0.09	0.38	$K_{3} > K_{2} > K_{1}$
PFYWA	-0.20	0.10	0.31	$K_{3} > K_{2} > K_{1}$
PFYWG	-0.11	0.05	0.27	$K_{3} > K_{2} > K_{1}$

Fig. 3

Comparison with existing operators.

Advantages and limitations of proposed model: The proposed model is superior than IF and PF model because it contains the space of IF and PF models. The MADM approaches discussed in [11 , 29] are fail to handle the proposed Application 6.2 because 0.8 + 0.7 > 1 and 0 . 8² + 0 .7² > 1 but proposed approach covers all such situations. The results are more precise and accurate by using proposed model. However, there are some limitations of this model. It cannot applied in the situations where we take the parameters for the evaluation of anything. It means, this theory has the lack of parametrization property.

9 Conclusions and future directions

q-ROFSs have appeared as more useful tool to portray uncertainty in MADM problems as compared to IFSs and PFSs. The speciality of the q-ROFS is that it generalizes the constraint of PFS that square sum of MD and NMD is bounded by 1 with the sum of q^th power of MD and NMD is bounded by 1. In this article, new multi-skilled AOs, namely, q-ROFEWA, q-ROFEOWA, Gq-ROFEWA and Gq-ROFEOWA operators have been proposed that enjoys the proficiency of q-ROFS as well as the advanced skills of Einstein AOs. Some fundamental properties of these operators have been discussed. In short, the proposed generalized Einstein AOs under the q-ROF environment, being highly competent, overcome the shortcomings of the existing AOs.

Another goal of this study have been achieved by presenting a MADM strategy based on proposed operators to manipulate q-ROF data. Further, three MADM problems under these operators have been discussed, i.e., the selection of best future career, the selection of agriculture land and residential place. Moreover, the results of the elaborated applications have been compared with existing operators to exhibit the superiority and potency of proposed operators. For the approval of proposed theory, validity test have been discussed to highlight the reliability and authenticity of the proposed strategy.

In future, our aim is to merge the theory of Einstein operators with some well versed models to develop innovative AOs, including, (i) q-ROF soft Einstein weighted averaging operators; (ii) q-rung picture fuzzy Einstein weighted averaging operators; (iii) q-rung picture fuzzy Einstein weighted geometric operators.

Conflict of interest

The authors declare no conflicts of interest.

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Protraction of Einstein operators for decision-making under q -rung orthopair fuzzy model

Abstract

Keywords

1 Introduction

3 Einstein operational law of q-ROFNs

4 q-Rung orthopair fuzzy Einstein aggregation operators

4.1 q-rung orthopair fuzzy Einstein weighted averaging operators

5.1 Generalized q-rung orthopair fuzzy Einstein weighted averaging operators

5.2 Generalized q-rung orthopair fuzzy Einstein ordered weighted averaging operators

6 MADM problems using q-ROF information

6.1 Selection of best future career

Table 1 q-ROFDM E ˜ J 1 J 2 J 3 K 1 (0.5,0.7) (0.6,0.5) (0.7,0.5) K 2 (0.7,0.6) (0.6,0.4) (0.5,0.4) K 3 (0.8,0.5) (0.7,0.3) (0.8,0.4)

Table 2 q-ROFDM E ˜ J 1 J 2 J 3 L 1 (0.6,0.4) (0.8,0.7) (0.9,0.4) L 2 (0.5,0.6) (0.9,0.4) (0.9,0.6) L 3 (0.7,0.8) (0.8,0.2) (0.7,0.4) L 4 (0.3,0.4) (0.5,0.6) (0.7,0.4)

Table 3 q-ROFDM E ˜ J 1 J 2 J 3 J 4 J 5 K 1 (0.7, 0.3) (0.4, 0.6) (0.5, 0.5) (0.8, 0.2) (0.8, 0.4) K 2 (0.5, 0.8) (0.8, 0.6) (0.4, 0.5) (0.7, 0.4) (0.6, 0.5) K 3 (0.9, 0.6) (0.8, 0.1) (0.6, 0.4) (0.7, 0.5) (0.9, 0.3) K 4 (0.6, 0.7) (0.8, 0.3) (0.7, 0.2) (0.5, 0.3) (0.7, 0.3)

Table 4 Reconstructed q-ROFDM E ˜ J 1 J 2 J 3 K ˜ 1 (0.4,0.7) (0.5,0.6) (0.6,0.6) K 2 (0.7,0.6) (0.6,0.4) (0.5,0.4) K 3 (0.8,0.5) (0.7,0.3) (0.8,0.4)

Conflict of interest

References

Table 1
q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$

$K_{1}$ (0.5,0.7) (0.6,0.5) (0.7,0.5)

$K_{2}$ (0.7,0.6) (0.6,0.4) (0.5,0.4)

$K_{3}$ (0.8,0.5) (0.7,0.3) (0.8,0.4)

Table 2
q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$

$L_{1}$ (0.6,0.4) (0.8,0.7) (0.9,0.4)

$L_{2}$ (0.5,0.6) (0.9,0.4) (0.9,0.6)

$L_{3}$ (0.7,0.8) (0.8,0.2) (0.7,0.4)

$L_{4}$ (0.3,0.4) (0.5,0.6) (0.7,0.4)

Table 4
Reconstructed q-ROFDM

$\tilde{E}$ $J_{1}$ $J_{2}$ $J_{3}$

${\tilde{K}}_{1}$ (0.4,0.7) (0.5,0.6) (0.6,0.6)

$K_{2}$ (0.7,0.6) (0.6,0.4) (0.5,0.4)

$K_{3}$ (0.8,0.5) (0.7,0.3) (0.8,0.4)