Confidence levels under complex q-rung orthopair fuzzy aggregation operators and their applications

Abstract

The major contribution of this analysis is to analyze the confidence complex q-rung orthopair fuzzy weighted averaging (CCQROFWA) operator, confidence complex q-rung orthopair fuzzy ordered weighted averaging (CCQROFOWA) operator, confidence complex q-rung orthopair fuzzy weighted geometric (CCQROFWG) operator, and confidence complex q-rung orthopair fuzzy ordered weighted geometric (CCQROFOWG) operator and invented their feasible properties and related results. Future more, under the invented operators, we diagnosed the best crystalline solid from the family of crystalline solids with the help of the opinion of different experts in the environment of decision-making strategy. Finally, to demonstrate the feasibility and flexibility of the invented works, we explored the sensitivity analysis and graphically shown of the initiated works.

Keywords

Complex q-rung orthopair fuzzy sets aggregation operators confidence levels decision-making methods

1 Introduction

To employ the unit interval instead of the crisp set is very complex. Zadeh [1] successfully employed the unit interval instead of the crisp set in the shape of the invented fuzzy set (FS). FS includes the truth grade whose values belong to the unit interval. Furthermore, yes, and no are also important parts of genuine life troubles and they can provide many difficulties for an expert to make the best decision under the circumstances of FS, for this, Atanassov [2] invented the intuitionistic FS (IFS). The well-known prominence of IFS is of the form such that the sum of the truth and falsity grades is limited to the unit interval. Based on the beneficial shape of the IFS, they have gotten a lot of attraction from several persons [3 –5]. IFS has to deal only with such sort of data whose sum is limited to the unit interval. What happened, when the sum of the duplet is exceeded from the unit interval, for this, Yager [6] invented the rule that is the sum of the square of the duplet is limited to unit interval for Pythagorean FS (PFS). PFS has more capable and extensive power than the IFS and FS to control awkward and intricate information in guanine decision troubles [7 –11]. Yager [12] again enhanced the rule of PFS to the rule of q-rung orthopair fuzzy set (QROFS) that is the sum of the q-powers of the duplet is restricted to the unit interval. QROFS has more capable and extensive power than the PFS and IFS to control awkward and intricate information in guanine decision troubles [13 –23].

Ramot et al. [24] employed the unit disc instead of unit interval in the environment of complex FS (CFS), whose values are in the shape of complex terms whose real and unreal objects are belonging to the unit interval. Moreover, Alkouri and Salleh [25] improved the quality of the research work in the shape of elaborated complex IFS (CIFS). CIFS massive implementation in the circumstances of distinct regions [26 –32]. Some data can’t handle from CIFS if an expert gives such sort of data whose sum of the real parts (also for unreal parts) of the duplet is exceeded from unit interval. For this, Ullah et al. [33] invented the complex PFS (CPFS) with a well-known technique that is the sum of the square of the real parts (also for unreal parts) of the duplet is restricted to the unit interval. CPFS has very famous in the circumstances of different regions [34 –38]. But there are still several dilemmas, Liu et al. [39] managed with these dilemmas which occurred in the environment of CPFS in the shape of complex QROFS (CQROFS). The well-known characteristic of CQROFS is that the authors have changed only the square into q-powers in the region of CPFSs. CQROFS has more capable and extensive power than the CPFS and CIFS to control awkward and intricate information in guanine decision troubles [40 –48].

Moreover, these unexpected happenings are based on a complex q-rung orthopair fuzzy (CQROF) natural environment, the whole of the prevailing attempts does not integrate the knowledge degree in the knowledge merging action. The specialists in a MADM problem provide accomplishment of the options on the source of the revealed principles only, that is, the knowledge (called CLs) of specialists with the assessment items is not involved. So, it is essential to integrate the expertise of the viewer in the unique knowledge based on CQROF natural environment. Keeping the benefits of the above prevailing theories the main contribution of this analysis is discussed below.

To invent the CCQROFWA, CCQROFOWA, CCQROFWG, and CCQROFOWG operators.

To diagnose the best crystalline solid from the family of crystalline solids with the help of the opinion of different experts in the environment of decision-making strategy.

To demonstrate the feasibility and flexibility of the invented works, we explored the sensitivity analysis and graphically shown of the initiated works.

Under the following ways, we constructed our manuscript, section 2 covers some fundamental ideas like CQROFSs, weighted averaging (WA), and ordered weighted averaging (OWA). Section 3 covers the CCQROFWA, CCQROFOWA, CCQROFWG, and CCQROFOWG operators, and their certain essential properties are properly recognized. Section 4 covers a decision-making process by using the elaborated operators is to determine the best sort of crystal for guanine life usage. The conclusion of this script is debated in section 5.

2 Preliminaries

The major aims of this structure are to illustrate the CQROFSs, WA, and OWA. The mathematical terms $\overset{︷}{X_{UNI}}$ , $μ_{T ι f_{CI}}$ and $η_{T ι f_{CI}}$ , stated the universal set, truth grade, and falsity grade.

Definition 1. [39] A CQROFS ${\tilde{Ξ}}_{CI}$ is demonstrated by:

${\tilde{Ξ}}_{C I} = {(μ_{{\tilde{Ξ}}_{C I}} (\overset{⌣}{x}), η_{{\tilde{Ξ}}_{C I}} (\overset{⌣}{x})) : \overset{⌣}{x} \in \overset{︷}{X_{U N I}}}$ (1)

Where $μ_{{\tilde{Ξ}}_{CI}} = μ_{{\tilde{Ξ}}_{RP}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP}})}$ and $η_{{\tilde{Ξ}}_{CI}} = η_{{\tilde{Ξ}}_{RP}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}})}$ with the rules such that $0 ⩽ μ_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}} + η_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}} ⩽ 1$ and $0 ⩽ μ_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}} + η_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}} ⩽ 1$ . We fix that the term $L_{{\tilde{Ξ}}_{CI}} = L_{{\tilde{Ξ}}_{RP}} e^{i 2 π (L_{{\tilde{Ξ}}_{IP}})} = {(1 - μ_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}} - η_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - μ_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}} - η_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}})}^{\frac{1}{𝕢_{SC}}}}$ , shown neutral grade. CQROFN is invented in the shape of ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ .

Definition 2. [40] By using two CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2$ ,then

${\tilde{Ξ}}_{CI - 1} \oplus {\tilde{Ξ}}_{CI - 2} = (\begin{matrix} {(μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} + μ_{{\tilde{Ξ}}_{RP - 2}}^{𝕢_{SC}} - μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} μ_{{\tilde{Ξ}}_{RP - 2}}^{𝕢_{SC}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(μ_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}} + μ_{{\tilde{Ξ}}_{IP - 2}}^{𝕢_{SC}} - μ_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}} μ_{{\tilde{Ξ}}_{IP - 2}}^{𝕢_{SC}})}^{\frac{1}{𝕢_{SC}}}}, \\ η_{{\tilde{Ξ}}_{RP - 1}} η_{{\tilde{Ξ}}_{RP - 2}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - 1}} η_{{\tilde{Ξ}}_{IP - 2}})} \end{matrix})$ (2)

${\tilde{Ξ}}_{CI - 1} \otimes {\tilde{Ξ}}_{CI - 2} = (\begin{matrix} μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} μ_{{\tilde{Ξ}}_{RP - 2}}^{𝕢_{SC}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - 1}} μ_{{\tilde{Ξ}}_{IP - 2}})}, \\ {(η_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} + η_{{\tilde{Ξ}}_{RP - 2}}^{𝕢_{SC}} - η_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} η_{{\tilde{Ξ}}_{RP - 2}}^{𝕢_{SC}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(η_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}} + η_{{\tilde{Ξ}}_{IP - 2}}^{𝕢_{SC}} - η_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}} η_{{\tilde{Ξ}}_{IP - 2}}^{𝕢_{SC}})}^{\frac{1}{𝕢_{SC}}}} \end{matrix})$ (3)

$Φ_{SC} {\tilde{Ξ}}_{CI - 1} = ({(1 - {(1 - μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}})}^{Φ_{SC}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}})}^{Φ_{SC}})}^{\frac{1}{𝕢_{SC}}}}, η_{{\tilde{Ξ}}_{RP - 1}}^{Φ_{SC}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - 1}}^{Φ_{SC}})})$ (4)

${\tilde{Ξ}}_{CI - 1}^{Φ_{SC}} = (μ_{{\tilde{Ξ}}_{RP - 1}}^{Φ_{SC}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - 1}}^{Φ_{SC}})}, {(1 - {(1 - η_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}})}^{Φ_{SC}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - {(1 - η_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}})}^{Φ_{SC}})}^{\frac{1}{𝕢_{SC}}}})$ (5)

Definition 3. [41] By using two CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1$ , then score value (SV) and accuracy value (AV) are exhibited by:

$\begin{matrix} 𝕊_{SV} ({\tilde{Ξ}}_{CI - 1}) \\ = \frac{1}{2} (μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} - η_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} + μ_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}} - η_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}}) \end{matrix}$ (6)

$\begin{matrix} ℍ_{AV} ({\tilde{Ξ}}_{CI - 1}) \\ = \frac{1}{2} (μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} + η_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}} + μ_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}} + η_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}}) \end{matrix}$ (7)

For Equations (6) and (7), we illustrated some rules:

When $𝕊_{SV} ({\tilde{Ξ}}_{CI - 1}) > 𝕊_{SV} ({\tilde{Ξ}}_{CI - 2}) \Rightarrow {\tilde{Ξ}}_{CI - 1} > {\tilde{Ξ}}_{CI - 2}$ ;

When $𝕊_{SV} ({\tilde{Ξ}}_{CI - 1}) = 𝕊_{SV} ({\tilde{Ξ}}_{CI - 2}) \Rightarrow$

When $ℍ_{AV} ({\tilde{Ξ}}_{CI - 1}) > ℍ_{AV} ({\tilde{Ξ}}_{CI - 2}) \Rightarrow {\tilde{Ξ}}_{CI - 1} > {\tilde{Ξ}}_{CI - 2}$ ;

When $ℍ_{AV} ({\tilde{Ξ}}_{CI - 1}) = ℍ_{AV} ({\tilde{Ξ}}_{CI - 2}) \Rightarrow {\tilde{Ξ}}_{CI - 1} = {\tilde{Ξ}}_{CI - 2}$ .

Definition 4. [39] By using two CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2$ , …, $\overset{︷}{ϖ}$ , the CQROFWA operator and CQROFWG operator are demonstrated by:

$C Q R O F W A ({\tilde{Ξ}}_{C I - 1}, {\tilde{Ξ}}_{C I - 2}, \dots, {\tilde{Ξ}}_{C I - \overset{︷}{ϖ}}) = (\begin{matrix} {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{R P - i}}^{𝕢_{S C}})}^{ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{I P - i}}^{𝕢_{S C}})}^{ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}}}, \\ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{R P - i}}^{ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{I P - i}}^{ω^{W V - i}})} \end{matrix})$ (8)

$C Q R O F O W A ({\tilde{Ξ}}_{C I - 1}, {\tilde{Ξ}}_{C I - 2}, \dots, {\tilde{Ξ}}_{C I - \overset{︷}{ϖ}}) = (\begin{matrix} {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{R P - σ (i)}}^{𝕢_{S C}})}^{ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{I P - σ (i)}}^{𝕢_{S C}})}^{ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}}}, \\ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{R P - σ (i)}}^{ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{I P - σ (i)}}^{ω^{W V - i}})} \end{matrix})$ (9)

Where the weight vector is demonstrated by: ω^WV = {ω^WV-1, ω^WV-2, …, ω^{WV-
$\overset{︷}{ϖ}$}} with a restriction that is $\sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} = 1, ω^{WV - i} \in [0, 1]$ with the permutations σ (i) of (i = 1, 2, …, $\overset{︷}{ϖ}$ ) with restrictions σ (i - 1) ⩾ σ (i).

3 Aggregation operators for CQROFSs by using confidence levels

The major aims of this analysis are to initiate the CCQROFWA, CCQROFOWA, CCQROFWG, and CCQROFOWG operators. Many beneficial results are also demonstrated under the explored works.

Definition 5. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, the CCQROFWA operator is demonstrated by:

$\begin{matrix} CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = \sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} (Γ_{i} {\tilde{Ξ}}_{CI - i}) \\ = ω^{WV - 1} (Γ_{1} {\tilde{Ξ}}_{CI - 1}) \oplus ω^{WV - 2} (Γ_{2} {\tilde{Ξ}}_{CI - 2}) \\ \oplus \dots, \oplus ω^{WV - \overset{︷}{ϖ}} (Γ_{\overset{︷}{ϖ}} {\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}) \end{matrix}$ (10)

Where ω^WV ={ ω^WV-1, ω^WV-2, …, ω^{WV-
$\overset{︷}{ϖ}$} } with $\sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} = 1, ω^{WV - i} \in [0, 1]$ .

Theorem 1. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, then by using the Eq. (10), we obtained

$ω^{W V - 1} (Γ_{1} {\tilde{Ξ}}_{C I - 1}) = (\begin{matrix} {(1 - {(1 - μ_{{\tilde{Ξ}}_{R P - 1}}^{𝕢_{S C}})}^{Γ_{1} ω^{W V - 1}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{I P - 1}}^{𝕢_{S C}})}^{Γ_{1} ω^{W V - 1}})}^{\frac{1}{𝕢_{S C}}}}, \\ η_{{\tilde{Ξ}}_{R P - 1}}^{Γ_{1} ω^{W V - 1}} e^{i 2 π (η_{{\tilde{Ξ}}_{I P - 1}}^{Γ_{1} ω^{W V - 1}})} \end{matrix})$ (11)

Proof. Put $\overset{︷}{ϖ}$ = 2, in Equation (11), we obtain $\begin{matrix} ω^{WV - 1} (Γ_{1} {\tilde{Ξ}}_{CI - 1}) = (\begin{matrix} {(1 - {(1 - μ_{{\tilde{Ξ}}_{RP - 1}}^{𝕢_{SC}})}^{Γ_{1} ω^{WV - 1}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{IP - 1}}^{𝕢_{SC}})}^{Γ_{1} ω^{WV - 1}})}^{\frac{1}{𝕢_{SC}}}}, \\ η_{{\tilde{Ξ}}_{RP - 1}}^{Γ_{1} ω^{WV - 1}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - 1}}^{Γ_{1} ω^{WV - 1}})} \end{matrix}) \end{matrix}$ $\begin{matrix} ω^{WV - 2} (Γ_{2} {\tilde{Ξ}}_{CI - 2}) = (\begin{matrix} {(1 - {(1 - μ_{{\tilde{Ξ}}_{RP - 2}}^{𝕢_{SC}})}^{Γ_{2} ω^{WV - 2}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{IP - 2}}^{𝕢_{SC}})}^{Γ_{2} ω^{WV - 2}})}^{\frac{1}{𝕢_{SC}}}}, \\ η_{{\tilde{Ξ}}_{RP - 2}}^{Γ_{2} ω^{WV - 2}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - 2}}^{Γ_{2} ω^{WV - 2}})} \end{matrix}) \end{matrix}$ $C Q R O F W A (({\tilde{Ξ}}_{C I - 1}, Γ_{1}), ({\tilde{Ξ}}_{C I - 2}, Γ_{2})) = ω^{W V - 1} (Γ_{1} {\tilde{Ξ}}_{C I - 1}) \oplus ω^{W V - 2} (Γ_{2} {\tilde{Ξ}}_{C I - 2}) = (\begin{matrix} {(1 - \prod_{i = 1}^{2} {(1 - μ_{{\tilde{Ξ}}_{R P - i}}^{𝕢_{S C}})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - \prod_{i = 1}^{2} {(1 - μ_{{\tilde{Ξ}}_{I P - i}}^{𝕢_{S C}})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}}}, \\ \prod_{i = 1}^{2} η_{{\tilde{Ξ}}_{R P - i}}^{Γ_{i} ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{2} η_{{\tilde{Ξ}}_{I P - i}}^{Γ_{i} ω^{W V - i}})} \end{matrix})$

Equation (11) is held for $\overset{︷}{ϖ}$ = 2. Further, for $\overset{︷}{ϖ}$ = k is also held, such that $C C Q R O F W A (({\tilde{Ξ}}_{C I - 1}, Γ_{1}), ({\tilde{Ξ}}_{C I - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{C I - k}, Γ_{k})) = (\begin{matrix} {(1 - \prod_{i = 1}^{k} {(1 - μ_{{\tilde{Ξ}}_{R P - i}}^{�_{S C}})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{�_{S C}}} e^{i 2 π {(1 - \prod_{i = 1}^{k} {(1 - μ_{{\tilde{Ξ}}_{I P - i}}^{�_{S C}})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{�_{S C}}}}, \\ \prod_{i = 1}^{k} η_{{\tilde{Ξ}}_{R P - i}}^{Γ_{i} ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{k} η_{{\tilde{Ξ}}_{I P - i}}^{Γ_{i} ω^{W V - i}})} \end{matrix})$

Know we check for $\overset{︷}{ϖ}$ = k + 1, such that $C C Q R O F W A (({\tilde{Ξ}}_{C I - 1}, Γ_{1}), ({\tilde{Ξ}}_{C I - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{C I - k + 1}, Γ_{k + 1})) = C C Q R O F W A (({\tilde{Ξ}}_{C I - 1}, Γ_{1}), ({\tilde{Ξ}}_{C I - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{C I - k}, Γ_{k})) \oplus ω^{W V - k + 1} (Γ_{\overset{𝕢}{ϖ}} {\tilde{Ξ}}_{C I - k + 1}) = (\begin{matrix} {(1 - \prod_{i = 1}^{k} {(1 - μ_{{\tilde{Ξ}}_{R P - i}}^{𝕢_{S C}})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - \prod_{i = 1}^{k} {(1 - μ_{{\tilde{Ξ}}_{I P - i}}^{𝕢_{S C}})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}}}, \\ \prod_{i = 1}^{k} η_{{\tilde{Ξ}}_{R P - i}}^{Γ_{i} ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{k} η_{{\tilde{Ξ}}_{I P - i}}^{Γ_{i} ω^{W V - i}})} \end{matrix}) \oplus (\begin{matrix} {(1 - {(1 - μ_{{\tilde{Ξ}}_{R P - k + 1}}^{𝕢_{S C}})}^{Γ_{k + 1} ω^{W V - k + 1}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{I P - k + 1}}^{𝕢_{S C}})}^{Γ_{k + 1} ω^{W V - k + 1}})}^{\frac{1}{𝕢_{S C}}}}, \\ η_{{\tilde{Ξ}}_{R P - k + 1}}^{Γ_{k + 1} ω^{W V - k + 1}} e^{i 2 π (η_{{\tilde{Ξ}}_{I P - k + 1}}^{Γ_{k + 1} ω^{W V - k + 1}})} \end{matrix})$ $\begin{matrix} = (\begin{matrix} {(1 - \prod_{i = 1}^{k + 1} {(1 - μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - \prod_{i = 1}^{k + 1} {(1 - μ_{{\tilde{Ξ}}_{IP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}}}, \\ \prod_{i = 1}^{k + 1} η_{{\tilde{Ξ}}_{RP - i}}^{Γ_{i} ω^{WV - i}} e^{i 2 π (\prod_{i = 1}^{k + 1} η_{{\tilde{Ξ}}_{IP - i}}^{Γ_{i} ω^{WV - i}})} \end{matrix}) \end{matrix}$

Therefore, the result holds for all $\overset{︷}{ϖ}$ .

Example 1. For any four CQROFNs with their confidence degree such that ${\tilde{Ξ}}_{CI - 1} = ((0.7 e^{i 2 π (0.7)}, 0.6 e^{i 2 π (0.6)}), 0.6)$ , ${\tilde{Ξ}}_{CI - 2} = ((0.8 e^{i 2 π (0.8)}, 0.4 e^{i 2 π (0.4)}), 0.4)$ , ${\tilde{Ξ}}_{CI - 3} = ((0.8 e^{i 2 π (0.8)}, 0.8 e^{i 2 π (0.8)}), 0.8)$ and ${\tilde{Ξ}}_{CI - 4} = ((0.6 e^{i 2 π (0.6)}, 0.4 e^{i 2 π (0.4)}), 0.5)$ with weight vector ω^WV = (0.1, 0.4, 0.2, 0.3) for a parameter 𝕢_SC = 4, then by using the Equation (11), we obtain ${(1 - \prod_{i = 1}^{4} {(1 - μ_{{\tilde{Ξ}}_{R P - i}}^{4})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{4}} e^{i 2 π {(1 - \prod_{i = 1}^{4} {(1 - μ_{{\tilde{Ξ}}_{R P - i}}^{4})}^{Γ_{i} ω^{W V - i}})}^{\frac{1}{4}}} = {(\begin{matrix} 1 - {(1 - {0.7}^{4})}^{0.6 * 0.1} \times {(1 - {0.8}^{4})}^{0.4 * 0.4} \times \\ {(1 - {0.8}^{4})}^{0.8 * 0.2} \times {(1 - {0.6}^{4})}^{0.5 * 0.3} \end{matrix})}^{\frac{1}{4}} e^{i 2 π {(\begin{matrix} 1 - {(1 - {0.7}^{4})}^{0.6 * 0.1} \times {(1 - {0.8}^{4})}^{0.4 * 0.4} \times \\ {(1 - {0.8}^{4})}^{0.8 * 0.2} \times {(1 - {0.6}^{4})}^{0.5 * 0.3} \end{matrix})}^{\frac{1}{4}}} = (0.6568) e^{i 2 π (0.6568)}$

$\prod_{i = 1}^{4} η_{{\tilde{Ξ}}_{R P - i}}^{Γ_{i} ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{4} η_{{\tilde{Ξ}}_{I P - i}}^{Γ_{i} ω^{W V - i}})} = ({0.6}^{0.6 * 0.1} \times {0.4}^{0.4 * 0.4} \times {0.8}^{0.8 * 0.2} \times {0.4}^{0.5 * 0.3}) e^{i 2 π ({0.6}^{0.6 * 0.1} \times {0.4}^{0.4 * 0.4} \times {0.8}^{0.8 * 0.2} \times {0.4}^{0.5 * 0.3})} = (0.8082) e^{i 2 π (0.8082)}$

As shown above, we get $\begin{matrix} CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \\ ({\tilde{Ξ}}_{CI - 3}, Γ_{3}), ({\tilde{Ξ}}_{CI - 4}, Γ_{4})) \\ = ((0.6568) e^{i 2 π (0.6568)}, (0.8082) e^{i 2 π (0.8082)}) \end{matrix}$

Additionally, by using Equation (11), we elaborated the idea of Idempotency, boundedness, and monotonicity.

Property 1. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} = {\tilde{Ξ}}_{CI} = (μ_{{\tilde{Ξ}}_{RP}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP}})}, η_{{\tilde{Ξ}}_{RP}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}})})$ i.e., $μ_{{\tilde{Ξ}}_{RP}} = μ_{{\tilde{Ξ}}_{RP - i}}, μ_{{\tilde{Ξ}}_{IP}} =$ $μ_{{\tilde{Ξ}}_{IP - i}}, η_{{\tilde{Ξ}}_{RP}} = η_{{\tilde{Ξ}}_{RP - i}}, η_{{\tilde{Ξ}}_{IP}} = η_{{\tilde{Ξ}}_{IP - i}}$ , and Γ = Γ_i, then

$\begin{matrix} CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \\ \dots, ({\tilde{Ξ}}_{\begin{matrix} CI - \overset{︷}{ϖ} \end{matrix}}, Γ_{\begin{matrix} \overset{︷}{ϖ} \end{matrix}})) = Γ {\tilde{Ξ}}_{CI} \end{matrix}$ (14)

$\begin{matrix} CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) \end{matrix}$ $\begin{matrix} = (\begin{matrix} {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}})}^{Γ ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}})}^{Γ ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}}}, \\ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{RP}}^{Γ ω^{WV - i}} e^{i 2 π (\prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{IP}}^{Γ ω^{WV - i}})} \end{matrix}) \end{matrix}$ $\begin{matrix} = (\begin{matrix} {(1 - {(1 - μ_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}})}^{Γ \sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}})}^{Γ \sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}}}, \\ η_{{\tilde{Ξ}}_{RP}}^{Γ \sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}}^{Γ \sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i}})} \end{matrix}) \end{matrix}$ $\begin{matrix} = ({(1 - {(1 - μ_{{\tilde{Ξ}}_{RP}}^{𝕢_{SC}})}^{Γ})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - {(1 - μ_{{\tilde{Ξ}}_{IP}}^{𝕢_{SC}})}^{Γ})}^{\frac{1}{𝕢_{SC}}}}, η_{{\tilde{Ξ}}_{RP}}^{Γ} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}}^{Γ})}) = Γ {\tilde{Ξ}}_{CI} \end{matrix}$

Property 2. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i}^{-} = (min_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, max_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ and ${\tilde{Ξ}}_{CI - i}^{+} = (max_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, min_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ , then

$\begin{matrix} {\tilde{Ξ}}_{CI - i}^{-} ⩽ CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \\ \dots, ({\tilde{Ξ}}_{\begin{matrix} CI - \overset{︷}{ϖ} \end{matrix}}, Γ_{\begin{matrix} \overset{︷}{ϖ} \end{matrix}})) ⩽ {\tilde{Ξ}}_{CI - i}^{+} \end{matrix}$ (12)

Proof. As shown in the statement, we know that, by using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i}^{-} = (min_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, max_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ and ${\tilde{Ξ}}_{CI - i}^{+} = (max_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, min_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ , then,we know that $\begin{matrix} min_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}} ⩽ μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}} ⩽ max_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}} \Rightarrow 1 - min_{i} \\ μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}} ⩾ 1 - μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}} ⩾ 1 - max_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}} \end{matrix}$ $\begin{matrix} \Rightarrow \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - min_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{min_{i} Γ_{i} ω^{WV - i}} \\ ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}} \\ ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - max_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{max_{i} Γ_{i} ω^{WV - i}} \end{matrix}$ $\begin{matrix} \Rightarrow {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - min_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{min_{i} Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \\ ⩽ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \\ ⩽ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - max_{i} μ_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{max_{i} Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \end{matrix}$

Similarly, we obtained for the imaginary part of the truth degree, such that $\begin{matrix} \Rightarrow {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - min_{i} μ_{{\tilde{Ξ}}_{IP - i}}^{𝕢_{SC}})}^{min_{i} Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \\ ⩽ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{IP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \\ ⩽ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - max_{i} μ_{{\tilde{Ξ}}_{IP - i}}^{𝕢_{SC}})}^{max_{i} Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \end{matrix}$

Similarly, we obtained for real and unreal parts of the falsity degree, such that $\begin{matrix} \Rightarrow \prod_{i = 1}^{\overset{︷}{ϖ}} max_{i} η_{{\tilde{Ξ}}_{RP - i}}^{max_{i} Γ_{i} ω^{WV - i}} ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{RP - i}}^{Γ_{i} ω^{WV - i}} \\ ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} min_{i} η_{{\tilde{Ξ}}_{RP - i}}^{min_{i} Γ_{i} ω^{WV - i}} \end{matrix}$ $\begin{matrix} \Rightarrow \prod_{i = 1}^{\overset{︷}{ϖ}} max_{i} η_{{\tilde{Ξ}}_{IP}}^{max_{i} Γ_{i} ω^{WV - i}} ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{RP - i}}^{Γ_{i} ω^{WV - i}} \\ ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} min_{i} η_{{\tilde{Ξ}}_{RP - i}}^{min_{i} Γ_{i} ω^{WV - i}} \end{matrix}$

By using Equation (6), we get the result, such that $\begin{matrix} {\tilde{Ξ}}_{CI - i}^{-} ⩽ CCQROFWA \\ (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) \\ ⩽ {\tilde{Ξ}}_{CI - i}^{+} \end{matrix}$

Property 3. By using any two families of CQROFNs ${\tilde{Ξ}}_{CI - i}^{*} = (μ_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}}^{*})}, η_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}}^{*})})$ and ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} ⩽ {\tilde{Ξ}}_{CI - i}^{*}$ , i.e., $μ_{{\tilde{Ξ}}_{RP - i}} ⩽ μ_{{\tilde{Ξ}}_{RP - i}}^{*}, μ_{{\tilde{Ξ}}_{IP - i}} ⩽ μ_{{\tilde{Ξ}}_{IP - i}}^{*}, η_{{\tilde{Ξ}}_{RP - i}} ⩾ η_{{\tilde{Ξ}}_{RP - i}}^{*}$ , and $η_{{\tilde{Ξ}}_{IP - i}} ⩾ η_{{\tilde{Ξ}}_{IP - i}}^{*}$ then

$\begin{matrix} CCQROFWA \\ (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) \\ ⩽ CCQROFWA \\ (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}})) \end{matrix}$ (13)

Similarly, for the imaginary part, we get $\begin{matrix} \Rightarrow {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{IP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \\ ⩽ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{IP - i}}^{* 𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} \end{matrix}$

Similarly, we obtained for real and unreal parts of the falsity degree, such that

$\begin{matrix} \Rightarrow \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{RP - i}}^{Γ_{i} ω^{WV - i}} ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{RP - i}}^{* Γ_{i} ω^{WV - i}}, \\ \Rightarrow \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{IP - i}}^{Γ_{i} ω^{WV - i}} ⩾ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{IP - i}}^{* Γ_{i} ω^{WV - i}} \end{matrix}$

Then by using $CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = {\tilde{Ξ}}_{CI}, CCQROFWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}})) = {\tilde{Ξ}}_{CI}^{*}$ and Equation (6), we discuss the following axioms:

When $𝕊_{SV} ({\tilde{Ξ}}_{CI}^{*}) > 𝕊_{SV} ({\tilde{Ξ}}_{CI}) \Rightarrow {\tilde{Ξ}}_{CI}^{*} > {\tilde{Ξ}}_{CI}$ i.e., $CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) < CCQROFWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}}))$ ;

When $𝕊_{SV} ({\tilde{Ξ}}_{CI}^{*}) = 𝕊_{SV} ({\tilde{Ξ}}_{CI}) \Rightarrow {\tilde{Ξ}}_{CI}^{*} = {\tilde{Ξ}}_{CI}$ i.e., $CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = CCQROFWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}}))$ , then we use Equation (7), such that

When $ℍ_{AV} ({\tilde{Ξ}}_{CI}^{*}) > ℍ_{AV} ({\tilde{Ξ}}_{CI}) \Rightarrow {\tilde{Ξ}}_{CI}^{*} > {\tilde{Ξ}}_{CI}$ i.e., $CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) < CCQROFWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}}))$ ;

When $ℍ_{AV} ({\tilde{Ξ}}_{CI}^{*}) = ℍ_{AV} ({\tilde{Ξ}}_{CI}) \Rightarrow {\tilde{Ξ}}_{CI}^{*} = {\tilde{Ξ}}_{CI}$ i.e., $CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = CCQROFWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}}))$ .

As shown above, we get the result, such that $\begin{matrix} CCQROFWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ CCQROFWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), \\ ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}})) \end{matrix}$

Definition 6. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, the CCQROFOWA operator is demonstrated by:

$\begin{matrix} CCQROFOWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = \sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} (Γ_{i} {\tilde{Ξ}}_{CI - i}) = ω^{WV - 1} \\ (Γ_{σ (1)} {\tilde{Ξ}}_{CI - σ (1)}) \oplus ω^{WV - 2} (Γ_{σ (2)} {\tilde{Ξ}}_{CI - σ (2)}) \oplus \dots, \\ \oplus ω^{WV - \overset{︷}{ϖ}} (Γ_{σ (\overset{︷}{ϖ})} {\tilde{Ξ}}_{CI - σ (\overset{︷}{ϖ})}) \end{matrix}$ (15)

Theorem 2. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, then by using the Eq. (15), we obtained

$C Q R O F O W A (({\tilde{Ξ}}_{C I - 1}, Γ_{1}), ({\tilde{Ξ}}_{C I - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{C I - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = (\begin{matrix} {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{R P - σ (i)}}^{𝕢_{S C}})}^{Γ_{σ (i)} ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}} e^{i 2 π {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - μ_{{\tilde{Ξ}}_{I P - σ (i)}}^{𝕢_{S C}})}^{Γ_{σ (i)} ω^{W V - i}})}^{\frac{1}{𝕢_{S C}}}}, \\ \prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{R P - σ (i)}}^{Γ_{σ (i)} ω^{W V - i}} e^{i 2 π (\prod_{i = 1}^{\overset{︷}{ϖ}} η_{{\tilde{Ξ}}_{I P - σ (i)}}^{Γ_{σ (i)} ω^{W V - i}})} \end{matrix})$ (16)

Proof. Trivial.

Additionally, by using Equation (16), we elaborated the idea of Idempotency, boundedness, and monotonicity.

Property 4. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} = {\tilde{Ξ}}_{CI} = (μ_{{\tilde{Ξ}}_{RP}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP}})}, η_{{\tilde{Ξ}}_{RP}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}})})$ i.e., $μ_{{\tilde{Ξ}}_{RP}} = μ_{{\tilde{Ξ}}_{RP - i}}, μ_{{\tilde{Ξ}}_{IP}} = μ_{{\tilde{Ξ}}_{IP - i}}, η_{{\tilde{Ξ}}_{RP}} = η_{{\tilde{Ξ}}_{RP - i}}, η_{{\tilde{Ξ}}_{IP}} = η_{{\tilde{Ξ}}_{IP - i}}$ , and Γ = Γ_i, then

$\begin{matrix} CCQROFOWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = Γ {\tilde{Ξ}}_{CI} \end{matrix}$ (17)

Proof. Trivial.

Property 5. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i}^{-} = (min_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, max_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ and ${\tilde{Ξ}}_{CI - i}^{+} = (max_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, min_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ , then

$\begin{matrix} {\tilde{Ξ}}_{CI - i}^{-} ⩽ CCQROFOWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \\ \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ {\tilde{Ξ}}_{CI - i}^{+} \end{matrix}$ (18)

Proof. Trivial.

Property 6. For CQROFNs ${\tilde{Ξ}}_{CI - i}^{*} = (μ_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}}^{*})}, η_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}}^{*})})$ and

${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} ⩽ {\tilde{Ξ}}_{CI - i}^{*}$ , i.e., $μ_{{\tilde{Ξ}}_{RP - i}} ⩽ μ_{{\tilde{Ξ}}_{RP - i}}^{*}, μ_{{\tilde{Ξ}}_{IP - i}} ⩽ μ_{{\tilde{Ξ}}_{IP - i}}^{*}, η_{{\tilde{Ξ}}_{RP - i}} ⩾ η_{{\tilde{Ξ}}_{RP - i}}^{*}$ , and $η_{{\tilde{Ξ}}_{IP - i}} ⩾ η_{{\tilde{Ξ}}_{IP - i}}^{*}$ then

$\begin{matrix} CCQROFOWA (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ CCQROFOWA (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), \\ ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}})) \end{matrix}$ (19)

Proof. Trivial.

Definition 7. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, the CCQROFWG operator is demonstrated by:

$\begin{matrix} CCQROFWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = \prod_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} (Γ_{i} {\tilde{Ξ}}_{CI - i}) \\ = ω^{WV - 1} (Γ_{1} {\tilde{Ξ}}_{CI - 1}) \otimes ω^{WV - 2} (Γ_{2} {\tilde{Ξ}}_{CI - 2}) \\ \otimes, \dots, \otimes ω^{WV - \overset{︷}{ϖ}} (Γ_{\overset{︷}{ϖ}} {\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}) \end{matrix}$ (20)

Theorem 3. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, then by using the Eq. (20), we obtained

$\begin{matrix} CCQROFWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) \\ = (\begin{matrix} \prod_{i = 1}^{\overset{︷}{ϖ}} μ_{{\tilde{Ξ}}_{RP - i}}^{Γ_{i} ω^{WV - i}} e^{i 2 π (\prod_{i = 1}^{\overset{︷}{ϖ}} μ_{{\tilde{Ξ}}_{IP - i}}^{Γ_{i} ω^{WV - i}})}, \\ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - η_{{\tilde{Ξ}}_{RP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - η_{{\tilde{Ξ}}_{IP - i}}^{𝕢_{SC}})}^{Γ_{i} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}}} \end{matrix}) \end{matrix}$ (21)

Proof. Trivial.

Additionally, by using Equation (21), we elaborated the idea of Idempotency, boundedness, and monotonicity.

Property 7. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} = {\tilde{Ξ}}_{CI} = (μ_{{\tilde{Ξ}}_{RP}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP}})}, η_{{\tilde{Ξ}}_{RP}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}})})$ i.e., $μ_{{\tilde{Ξ}}_{RP}} = μ_{{\tilde{Ξ}}_{RP - i}}, μ_{{\tilde{Ξ}}_{IP}} = μ_{{\tilde{Ξ}}_{IP - i}}, η_{{\tilde{Ξ}}_{RP}} = η_{{\tilde{Ξ}}_{RP - i}}, η_{{\tilde{Ξ}}_{IP}} = η_{{\tilde{Ξ}}_{IP - i}}$ , and Γ = Γ_i, then

$\begin{matrix} CCQROFWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = Γ {\tilde{Ξ}}_{CI} \end{matrix}$ (22)

Proof. Trivial.

Property 8. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ ,and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i}^{-} = (min_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, max_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ and ${\tilde{Ξ}}_{CI - i}^{+} = (max_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, min_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ , then

$\begin{matrix} {\tilde{Ξ}}_{CI - i}^{-} ⩽ CCQROFWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, \\ Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ {\tilde{Ξ}}_{CI - i}^{+} \end{matrix}$ (23)

Proof. Trivial.

Property 9. For CQROFNs ${\tilde{Ξ}}_{CI - i}^{*} = (μ_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}}^{*})}, η_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}}^{*})})$ and ${\tilde{Ξ}}_{CI - i} =$

$(μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} ⩽ {\tilde{Ξ}}_{CI - i}^{*}$ , i.e., $μ_{{\tilde{Ξ}}_{RP - i}} ⩽ μ_{{\tilde{Ξ}}_{RP - i}}^{*}, μ_{{\tilde{Ξ}}_{IP - i}} ⩽ μ_{{\tilde{Ξ}}_{IP - i}}^{*}, η_{{\tilde{Ξ}}_{RP - i}} ⩾ η_{{\tilde{Ξ}}_{RP - i}}^{*}$ , and $η_{{\tilde{Ξ}}_{IP - i}} ⩾ η_{{\tilde{Ξ}}_{IP - i}}^{*}$ then

$\begin{matrix} CCQROFWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ CCQROFWG (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), \\ ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}})) \end{matrix}$ (24)

Proof. Trivial.

Definition 8. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, the CCQROFOWG operator is demonstrated by:

$\begin{matrix} CCQROFOWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = \prod_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} (Γ_{i} {\tilde{Ξ}}_{CI - i}) = \\ ω^{WV - 1} (Γ_{σ (1)} {\tilde{Ξ}}_{CI - σ (1)}) \otimes ω^{WV - 2} (Γ_{σ (2)} {\tilde{Ξ}}_{CI - σ (2)}) \\ \otimes, \dots, \otimes ω^{WV - \overset{︷}{ϖ}} (Γ_{σ (\overset{︷}{ϖ})} {\tilde{Ξ}}_{CI - σ (\overset{︷}{ϖ})}) \end{matrix}$ (25)

Theorem 4. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, then by using the Eq. (25), we obtained

$\begin{matrix} CCQROFOWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) \\ = (\begin{matrix} \prod_{i = 1}^{\overset{︷}{ϖ}} μ_{{\tilde{Ξ}}_{RP - σ (i)}}^{Γ_{σ (i)} ω^{WV - i}} e^{i 2 π (\prod_{i = 1}^{\overset{︷}{ϖ}} μ_{{\tilde{Ξ}}_{IP - σ (i)}}^{Γ_{σ (i)} ω^{WV - i}})}, \\ {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - η_{{\tilde{Ξ}}_{RP - σ (i)}}^{𝕢_{SC}})}^{Γ_{σ (i)} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}} e^{i 2 π {(1 - \prod_{i = 1}^{\overset{︷}{ϖ}} {(1 - η_{{\tilde{Ξ}}_{IP - σ (i)}}^{𝕢_{SC}})}^{Γ_{σ (i)} ω^{WV - i}})}^{\frac{1}{𝕢_{SC}}}} \end{matrix}) \end{matrix}$ (26)

Proof. Trivial.

Additionally, by using Equation (26), we elaborated the idea of Idempotency, boundedness, and monotonicity.

Property 10. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} = {\tilde{Ξ}}_{CI} = (μ_{{\tilde{Ξ}}_{RP}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP}})}, η_{{\tilde{Ξ}}_{RP}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP}})})$ i.e., $μ_{{\tilde{Ξ}}_{RP}} = μ_{{\tilde{Ξ}}_{RP - i}}, μ_{{\tilde{Ξ}}_{IP}} = μ_{{\tilde{Ξ}}_{IP - i}}, η_{{\tilde{Ξ}}_{RP}} = η_{{\tilde{Ξ}}_{RP - i}}, η_{{\tilde{Ξ}}_{IP}} = η_{{\tilde{Ξ}}_{IP - i}}$ , and Γ = Γ_i, then

$\begin{matrix} CCQROFOWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) = Γ {\tilde{Ξ}}_{CI} \end{matrix}$ (27)

Proof. Trivial.

Property 11. By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i}^{-} = (min_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, max_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ and ${\tilde{Ξ}}_{CI - i}^{+} = (max_{i} μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (max_{i} μ_{{\tilde{Ξ}}_{IP - i}})}, min_{i} η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (min_{i} η_{{\tilde{Ξ}}_{IP - i}})})$ , then

$\begin{matrix} {\tilde{Ξ}}_{CI - i}^{-} ⩽ CCQROFOWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, \\ Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ {\tilde{Ξ}}_{CI - i}^{+} \end{matrix}$ (28)

Proof. Trivial.

Property 12. For CQROFNs ${\tilde{Ξ}}_{CI - i}^{*} = (μ_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}}^{*})}, η_{{\tilde{Ξ}}_{RP - i}}^{*} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}}^{*})})$ and ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1, if ${\tilde{Ξ}}_{CI - i} ⩽ {\tilde{Ξ}}_{CI - i}^{*}$ , i.e., $μ_{{\tilde{Ξ}}_{RP - i}} ⩽ μ_{{\tilde{Ξ}}_{RP - i}}^{*}, μ_{{\tilde{Ξ}}_{IP - i}} ⩽ μ_{{\tilde{Ξ}}_{IP - i}}^{*}, η_{{\tilde{Ξ}}_{RP - i}} ⩾ η_{{\tilde{Ξ}}_{RP - i}}^{*},$ and $η_{{\tilde{Ξ}}_{IP - i}} ⩾ η_{{\tilde{Ξ}}_{IP - i}}^{*}$ then

$\begin{matrix} CCQROFOWG (({\tilde{Ξ}}_{CI - 1}, Γ_{1}), ({\tilde{Ξ}}_{CI - 2}, Γ_{2}), \dots, \\ ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}, Γ_{\overset{︷}{ϖ}})) ⩽ CCQROFOWG (({\tilde{Ξ}}_{CI - 1}^{*}, Γ_{1}), \\ ({\tilde{Ξ}}_{CI - 2}^{*}, Γ_{2}), \dots, ({\tilde{Ξ}}_{CI - \overset{︷}{ϖ}}^{*}, Γ_{\overset{︷}{ϖ}})) \end{matrix}$ (29)

Proof. Trivial.

4 Analysis of different sorts of Crystalline

The iotas in translucent solids are firmly bound to one another, either in a customary mathematical grid (glasslike solids, which incorporate metals and normal ice). By and large, solids are described by underlying inflexibility, and protection from changes of shape or item doesn’t stream to assume the state of its compartment, nor does it extend to fill the whole volume accessible to it like a gas does. In any case, there are wide varieties in the properties of strong materials utilized for designing purposes. The properties of materials rely upon their interatomic bonds. These equivalent bonds additionally direct the space between the design of particles in solids. The length, edges of chief tomahawks, and the point between the unit cells are called grid constants or cross-section boundaries. The main four main sorts of crystals are discussed below in the form of Table 1.

Table 1
Crystalline Solids: Melting and Boiling Points

Sort of crystal solids Formulas Melting point °C Boiling point °C

Ionic NaCl 801 1314

CaF ₂ 1418 1533

Metallic Hg -39 630

Na 371 883

Au 1064 2856

W 3410 5660

Covalent Network B 2076 3927

C (diomond) 3500 3930

Sio_2 1600 2230

Molecular H ₂ -259 -253

I ₂ 114 184

NH ₃ -78 -33

H ₂ O 0 100

Sort of crystal solids	Formulas	Melting point °C	Boiling point °C
Ionic	NaCl	801	1314
	CaF ₂	1418	1533
Metallic	Hg	-39	630
	Na	371	883
	Au	1064	2856
	W	3410	5660
Covalent Network	B	2076	3927
	C (diomond)	3500	3930
	Sio_2	1600	2230
Molecular	H ₂	-259	-253
	I ₂	114	184
	NH ₃	-78	-33
	H ₂ O	0	100

Some general advantages and disadvantages of the information in Table 1 are discussed in the form of Table 2.

Table 2

Stated the major classes of solids

Ionic solids	Molecular solids	Covalent solids	Metallic solids
Poor conductor of heat and electricity	Poor conductor of heat and electricity	Poor conductor of heat and electricity	Good conductor of heat and electricity
Relatively high melting point	Low melting point	High melting point	Melting points depend strongly on electron configuration
Hard but brittle	Soft	Very hard and brittle	Ductile and malleable
Relatively dense	Low density	Low density	Usually, high density
Dull surface	Dull surface	Dull surface	Lustrous

To determine the best crystal for guanine life usage, we choose the hypothetical sorts of information’s are resolved by using the investigated operators to determine the consistency and legitimacy of the elaborated operators. For, we have constructed the algorithm and by using the elaborated algorithm are illustrated some numerical examples.

4.1 Algorithm

The major contribution of this analysis is to invent the beneficial optimal from the group of alternatives. For this, the family of alternative and their attributes are considered in the shape of: ${T ι^{Al - 1}, T ι^{Al - 2}, \dots, T ι^{Al - \overset{︷}{ϖ}}}$ and ${T ι^{At - 1}, T ι^{At - 2}, \dots, T ι^{At - \overset{\overset{︷}{ϖ}}{m}}}$ , where ω^WV = {ω^WV-1, ω^WV-2, …, ω^{WV-
$\overset{︷}{ϖ}$}}, stated the weight vectors $\sum_{i = 1}^{\overset{︷}{ϖ}} ω^{WV - i} = 1, ω^{WV - i} \in [0, 1]$ . By using CQROFNs ${\tilde{Ξ}}_{CI - i} = (μ_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (μ_{{\tilde{Ξ}}_{IP - i}})}, η_{{\tilde{Ξ}}_{RP - i}} e^{i 2 π (η_{{\tilde{Ξ}}_{IP - i}})}), i = 1, 2, \dots, \overset{︷}{ϖ}$ , and Γ_i be the CL of ${\tilde{Ξ}}_{CI - i}$ with 0 ⩽ Γ_i ⩽ 1.

To investigate the capability of the invented works, we illustrated some cases:

Case 1: By utilizing the group of CQROFNs, we develop the choice lattice, which incorporates the CQROFNs.

Case 2: By utilizing Equations (11) and (21), we total the developed lattice.

Case 3: By utilizing Equation (6), we decide the score norms of the gathered convictions.

Case 4: Rank all choices and find the best one.

Case 5: Finished.

The geometrical expressions of the ionic, metallic, covalent, and molecular are discussed in the form of Fig. 1.

Fig. 1

Graphical expressions of the bonding of the different sorts of crystals.

Example 2. To utilize the elaborated approaches excellently, a decision-maker troubles with the selection of crystal solids, we choose the four alternatives such as ionic, metallic, covalent, and molecular whose expressions are in the form of { $T ι^{A ι - 1}$ , $T ι^{A ι - 2}$ , $T ι^{A ι - 3}$ , $T ι^{A ι - 4}$ } and their four attributes whose representations are in the form of { $T ι^{At - 1}$ , $T ι^{At - 2}$ , $T ι^{At - 3}$ , $T ι^{At - 4}$ } with weight vectors ω^WV = (0.3, 0.3, 0.3, 0.1) ^T, then to accurately determine the above mentions troubles, the cases of the elaborated algorithm are discussed below.

Case 1: By utilizing the group of CQROFNs, we develop the choice lattice, which incorporates the CQROFNs, invented in the type of Table 3.

Table 3

Invented the matrix

	$T ι^{At - 1}$	$T ι^{At - 2}$	$T ι^{At - 3}$	$T ι^{At - 4}$
$T ι^{Al - 1}$	$((\begin{matrix} 0.8 e^{i 2 π (0.9)}, \\ 0.7 e^{i 2 π (0.8)} \end{matrix}), 0.8)$	$((\begin{matrix} 0.81 e^{i 2 π (0.91)}, \\ 0.71 e^{i 2 π (0.81)} \end{matrix}), 0.81)$	$((\begin{matrix} 0.82 e^{i 2 π (0.92)}, \\ 0.72 e^{i 2 π (0.82)} \end{matrix}), 0.82)$	$((\begin{matrix} 0.83 e^{i 2 π (0.93)}, \\ 0.73 e^{i 2 π (0.83)} \end{matrix}), 0.83)$
$T ι^{Al - 2}$	$((\begin{matrix} 0.9 e^{i 2 π (0.8)}, \\ 0.4 e^{i 2 π (0.5)} \end{matrix}), 0.9)$	$((\begin{matrix} 0.91 e^{i 2 π (0.81)}, \\ 0.41 e^{i 2 π (0.51)} \end{matrix}), 0.91)$	$((\begin{matrix} 0.92 e^{i 2 π (0.82)}, \\ 0.42 e^{i 2 π (0.52)} \end{matrix}), 0.92)$	$((\begin{matrix} 0.93 e^{i 2 π (0.83)}, \\ 0.43 e^{i 2 π (0.53)} \end{matrix}), 0.93)$
$T ι^{Al - 3}$	$((\begin{matrix} 0.7 e^{i 2 π (0.5)}, \\ 0.5 e^{i 2 π (0.4)} \end{matrix}), 0.5)$	$((\begin{matrix} 0.71 e^{i 2 π (0.51)}, \\ 0.51 e^{i 2 π (0.41)} \end{matrix}), 0.51)$	$((\begin{matrix} 0.72 e^{i 2 π (0.52)}, \\ 0.52 e^{i 2 π (0.42)} \end{matrix}), 0.52)$	$((\begin{matrix} 0.73 e^{i 2 π (0.53)}, \\ 0.53 e^{i 2 π (0.43)} \end{matrix}), 0.53)$
$T ι^{Al - 4}$	$((\begin{matrix} 0.7 e^{i 2 π (0.9)}, \\ 0.3 e^{i 2 π (0.1)} \end{matrix}), 0.7)$	$((\begin{matrix} 0.71 e^{i 2 π (0.91)}, \\ 0.31 e^{i 2 π (0.11)} \end{matrix}), 0.71)$	$((\begin{matrix} 0.72 e^{i 2 π (0.92)}, \\ 0.32 e^{i 2 π (0.12)} \end{matrix}), 0.72)$	$((\begin{matrix} 0.73 e^{i 2 π (0.93)}, \\ 0.33 e^{i 2 π (0.13)} \end{matrix}), 0.73)$

Case 2: By using Equations (11) and (21), we aggregate the constructed matrix is discussed in the form of Table 4.

Table 4

Aggregated values of the information are in Table 3

Alternatives	CCQROFWA Operator	CCQROFWG Operator
$T ι^{Al - 1}$	(0.9999e^i2π(0.9999), 0.0009e^i2π(0.0014))	(0.0014e^i2π(0.002), 0.9997e^i2π(0.9999))
$T ι^{Al - 2}$	(0.9999e^i2π(0.9999), 0.0001e^i2π(0.0002))	(0.002e^i2π(0.0013), 0.9996e^i2π(0.9996))
$T ι^{Al - 3}$	(0.9997e^i2π(0.9996), 0.0007e^i2π(0.0004))	(0.0014e^i2π(0.0007), 0.9996e^i2π(0.9996))
$T ι^{Al - 4}$	(0.9997e^i2π(0.9999), 0.00004e^{i2π(0.000006)})	(0.001e^i2π(0.0021), 0.9996e^i2π(0.9995))

Case 3: By utilizing Equation (6), we decide the score norms of the gathered convictions, invented in the type of Table 5.

Table 5

By using Equation (6), we determine the score value of the information in Table 4

Alternatives	CCQROFWA Operator	CCQROFWG Operator
$T ι^{Al - 1}$	0.9996	-0.996
$T ι^{Al - 2}$	0.9996	-0.997
$T ι^{Al - 3}$	0.9977	-0.997
$T ι^{Al - 4}$	0.999	-0.997

Case 4: We diagnosed the beneficial item, invented in the type of Table 6.

Table 6

Ranking values of the information of Example 2

Methods	Ranking values
CCQROFWA Operator	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
CCQROFWG Operator	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 3} = T ι^{Al - 4}$

As shown above, we obtained the same result from two different operators, the best alternatives $T ι^{At - 1}$ , means that the best crystal is Ionic solid.

Example 3. (Case Study of Green Supply Chain Management)

Natural issues have developed and spread quicker in ongoing many years, such backwoods fires, about country by region and overall region, which are the main source of climate change also, overall warming. Also, ecological shortage and air and water contamination have genuine ramifications for greenery and human existence, with different illnesses, for example, ischemic cardiovascular sickness, cellular breakdown in the lungs. In any case, organizations take on the green idea of “killing two birds with one stone” because the green stock chains will limit natural contamination and assembling expenses and along these lines advance financial development. Manageability or the green production network identifies with the idea that supportable practices ought to be incorporated into the customary inventory network [48]. This includes the acquisition, request, plan, creation, gathering, dispersion, and end-of-life of the executives of the provider. Figure 2 shows an illustration of a green supplier chain management (GSCM) for a seller of bassinets is discussed in Ref. [48]. To utilize the elaborated approaches excellently, a decision-maker troubles with the selection of best GCSM, we choose the four alternatives such as Sustainablemanagement of the supply network invented in the shape of ${T ι^{Al - 1}, T ι^{Al - 2}, T ι^{Al - 3}, T ι^{Al - 4}}$ and their four attributes whose representations are in the form of ${T ι^{At - 1}, T ι^{At - 2}, T ι^{At - 3}, T ι^{At - 4}}$ with weight vectors ω^WV = (0.3, 0.3, 0.3, 0.1), then to accurately determine the above mentions troubles, the cases of the elaborated algorithm are discussed below. Then the updated data is given in Table 7.

Fig. 2

Geometrical expressions of the information of Table 5.

Table 7

Stated the matrix

	$T ι^{At - 1}$	$T ι^{At - 2}$	$T ι^{At - 3}$	$T ι^{At - 4}$
$T ι^{Al - 1}$	$((\begin{matrix} 0.8 e^{i 2 π (0.9)}, \\ 0.1 e^{i 2 π (0.0)} \end{matrix}), 0.8)$	$((\begin{matrix} 0.81 e^{i 2 π (0.91)}, \\ 0.11 e^{i 2 π (0.01)} \end{matrix}), 0.81)$	$((\begin{matrix} 0.82 e^{i 2 π (0.92)}, \\ 0.12 e^{i 2 π (0.02)} \end{matrix}), 0.82)$	$((\begin{matrix} 0.83 e^{i 2 π (0.93)}, \\ 0.13 e^{i 2 π (0.03)} \end{matrix}), 0.83)$
$T ι^{Al - 2}$	$((\begin{matrix} 0.9 e^{i 2 π (0.8)}, \\ 0.0 e^{i 2 π (0.1)} \end{matrix}), 0.9)$	$((\begin{matrix} 0.91 e^{i 2 π (0.81)}, \\ 0.01 e^{i 2 π (0.11)} \end{matrix}), 0.91)$	$((\begin{matrix} 0.92 e^{i 2 π (0.82)}, \\ 0.02 e^{i 2 π (0.12)} \end{matrix}), 0.92)$	$((\begin{matrix} 0.93 e^{i 2 π (0.83)}, \\ 0.03 e^{i 2 π (0.13)} \end{matrix}), 0.93)$
$T ι^{Al - 3}$	$((\begin{matrix} 0.7 e^{i 2 π (0.5)}, \\ 0.2 e^{i 2 π (0.3)} \end{matrix}), 0.5)$	$((\begin{matrix} 0.71 e^{i 2 π (0.51)}, \\ 0.21 e^{i 2 π (0.31)} \end{matrix}), 0.51)$	$((\begin{matrix} 0.72 e^{i 2 π (0.52)}, \\ 0.22 e^{i 2 π (0.32)} \end{matrix}), 0.52)$	$((\begin{matrix} 0.73 e^{i 2 π (0.53)}, \\ 0.23 e^{i 2 π (0.33)} \end{matrix}), 0.53)$
$T ι^{Al - 4}$	$((\begin{matrix} 0.7 e^{i 2 π (0.9)}, \\ 0.1 e^{i 2 π (0.0)} \end{matrix}), 0.7)$	$((\begin{matrix} 0.71 e^{i 2 π (0.91)}, \\ 0.11 e^{i 2 π (0.01)} \end{matrix}), 0.71)$	$((\begin{matrix} 0.72 e^{i 2 π (0.92)}, \\ 0.12 e^{i 2 π (0.02)} \end{matrix}), 0.72)$	$((\begin{matrix} 0.73 e^{i 2 π (0.93)}, \\ 0.13 e^{i 2 π (0.03)} \end{matrix}), 0.73)$

The final values are invented in Table 8.

Table 8

By using Equation (6), we determine the score value of the information in Table 7

Alternatives	CCQROFWA Operator	CCQROFWG Operator
$T ι^{Al - 1}$	1	-0.9971
$T ι^{Al - 2}$	1	-0.9972
$T ι^{Al - 3}$	0.9977	-0.9973
$T ι^{Al - 4}$	0.9998	-0.9973

Rank all options and discover the top one by using the information in Table 8, which is discussed in the form of Table 9.

Table 9

Ranking values of information of Example 3

Methods	Ranking Values
CCQROFWA Operator	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
CCQROFWG Operator	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 3} = T ι^{Al - 4}$

As shown above, we obtained the same result from two different operators, the best alternatives $T ι^{Al - 1}$ , means that the best Sustainable management of the supply network.

The graphical expressions of the information in Table 5 are explained in the form of Fig. 2.

Example 4. To utilize the elaborated approaches excellently, a decision-maker troubles with the selection of crystal solids, we choose the four alternatives such as ionic, metallic, covalent, and molecular whose expressions are in the form of ${T ι^{Al - 1}, T ι^{Al - 2}, T ι^{Al - 3}, T ι^{Al - 4}}$ and their four attributes whose representations are in the form of ${T ι^{At - 1}, T ι^{At - 2}, T ι^{At - 3}, T ι^{At - 4}}$ with weight vectors ω^WV = (0.3, 0.3, 0.3, 0.1), then to accurately determine the above mentions troubles, the cases of the elaborated algorithm are discussed below. The updated data is given in Table 10.

Table 10

Stated the matrix

	$T ι^{At - 1}$	$T ι^{At - 2}$	$T ι^{At - 3}$	$T ι^{At - 4}$
$T ι^{Al - 1}$	$((\begin{matrix} 0.6 e^{i 2 π (0.7)}, \\ 0.1 e^{i 2 π (0.0)} \end{matrix}), 0.8)$	$((\begin{matrix} 0.61 e^{i 2 π (0.71)}, \\ 0.11 e^{i 2 π (0.01)} \end{matrix}), 0.81)$	$((\begin{matrix} 0.62 e^{i 2 π (0.72)}, \\ 0.12 e^{i 2 π (0.02)} \end{matrix}), 0.82)$	$((\begin{matrix} 0.63 e^{i 2 π (0.73)}, \\ 0.13 e^{i 2 π (0.03)} \end{matrix}), 0.83)$
$T ι^{Al - 2}$	$((\begin{matrix} 0.5 e^{i 2 π (0.8)}, \\ 0.0 e^{i 2 π (0.1)} \end{matrix}), 0.9)$	$((\begin{matrix} 0.51 e^{i 2 π (0.81)}, \\ 0.01 e^{i 2 π (0.11)} \end{matrix}), 0.91)$	$((\begin{matrix} 0.52 e^{i 2 π (0.82)}, \\ 0.02 e^{i 2 π (0.12)} \end{matrix}), 0.92)$	$((\begin{matrix} 0.53 e^{i 2 π (0.83)}, \\ 0.03 e^{i 2 π (0.13)} \end{matrix}), 0.93)$
$T ι^{Al - 3}$	$((\begin{matrix} 0.4 e^{i 2 π (0.3)}, \\ 0.2 e^{i 2 π (0.3)} \end{matrix}), 0.5)$	$((\begin{matrix} 0.41 e^{i 2 π (0.31)}, \\ 0.21 e^{i 2 π (0.31)} \end{matrix}), 0.51)$	$((\begin{matrix} 0.42 e^{i 2 π (0.32)}, \\ 0.22 e^{i 2 π (0.32)} \end{matrix}), 0.52)$	$((\begin{matrix} 0.43 e^{i 2 π (0.33)}, \\ 0.23 e^{i 2 π (0.33)} \end{matrix}), 0.53)$
$T ι^{Al - 4}$	$((\begin{matrix} 0.2 e^{i 2 π (0.4)}, \\ 0.1 e^{i 2 π (0.0)} \end{matrix}), 0.7)$	$((\begin{matrix} 0.21 e^{i 2 π (0.41)}, \\ 0.11 e^{i 2 π (0.01)} \end{matrix}), 0.71)$	$((\begin{matrix} 0.22 e^{i 2 π (0.42)}, \\ 0.12 e^{i 2 π (0.02)} \end{matrix}), 0.72)$	$((\begin{matrix} 0.23 e^{i 2 π (0.43)}, \\ 0.13 e^{i 2 π (0.03)} \end{matrix}), 0.73)$

The data of Table 9 are invented in the shape of Fig. 3.

Fig. 3

Geometrical expressions of the information of Table 8.

The final values are illustrated in Table 11.

Table 11

By using Equation (6), we determine the score value of the information in Table 10

Alternatives	CCQROFWA Operator	CCQROFWG Operator
$T ι^{Al - 1}$	0.9999	-0.9971
$T ι^{Al - 2}$	0.9999	-0.9971
$T ι^{Al - 3}$	0.9988	-0.9982
$T ι^{Al - 4}$	0.999	-0.9976

For our convenience, the rank values are invented in Table 12.

Table 12

Ranking values of information of Example 4

Methods	Ranking Values
CCQROFWA Operator	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
CCQROFWG Operator	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 4} = T ι^{Al - 3}$

As shown above, we obtained the same result from two different operators, the best alternatives $T ι^{Al - 1}$ , means that the best crystal is Ionic solid.

The data of Table 11 are invented in the shape of Figure 4.

Fig. 4

Geometrical expressions of the information of Table 11.

4.2 Sensitivity analysis

Because of the researched CQROFWA, CQROFOWA, CQROFWG, and CQROFOWG, not set in stone the dependability and consistency of the created administrators with the assistance of similar examination by utilizing the data of Tables 3, 7, and 10 shown in 4.1. The information related to existing theories are followed: Joshi and Gegov [19] elaborated the CLs QROFSs and their AOs, Zeng et al. [46] developed the CLs for PFSs, Xu and Yager [47] investigated the geometric AOs for IFSs, Garg, and Rani [32] studied the geometric AOs for CIFSs, Akram et al. [36] investigated the Dombi AOs for CPFSs, and Ali and Mahmood [40] explored the Maclaurin symmetric mean (MSM) operators for CQROFSs. Then Table 13 stated the results of the data in Table 3, by using the values of parameters s = t = 1, n = 3.

Table 13
Supremacy of the elaborated and existing operators by using Example 1

Methods Operators Score values Ranking values

Joshi and Gegov [19] WA Cannot be Resolved Cannot be Resolved

WG Cannot be Resolved Cannot be Resolved

Zeng et al. [46] WA Cannot be Resolved Cannot be Resolved

WG Cannot be Resolved Cannot be Resolved

Xu and Yager [47] Empty Cannot be Resolved Cannot be Resolved

WG Cannot be Resolved Cannot be Resolved

Garg and Rani [32] Empty Cannot be Resolved Cannot be Resolved

WG Cannot be Resolved Cannot be Resolved

Akram et al. [36] Dombi AOs Cannot be Resolved Cannot be Resolved

Empty Cannot be Resolved Cannot be Resolved

Ali and Mahmood [40] MSM $T ι^{Al - 1} = 0.9873, T ι^{Al - 2} = 0.9873,$ $T ι^{Al - 3} = 0.9854, T ι^{Al - 4} = 0.987$ $T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$

Empty Cannot be Resolved Cannot be Resolved

Explored Operators WA $T ι^{Al - 1} = 0.9996, T ι^{Al - 2} = 0.9996,$ $T ι^{Al - 3} = 0.9977, T ι^{Al - 4} = 0.999$ $T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$

WG $T ι^{Al - 1} = - 0.996, T ι^{Al - 2} = - 0.997,$ $T ι^{Al - 3} = - 0.997, T ι^{Al - 4} = - 0.997$ $T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 3} = T ι^{Al - 4}$

Methods	Operators	Score values	Ranking values
Joshi and Gegov [19]	WA	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Zeng et al. [46]	WA	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Xu and Yager [47]	Empty	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Garg and Rani [32]	Empty	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Akram et al. [36]	Dombi AOs	Cannot be Resolved	Cannot be Resolved
	Empty	Cannot be Resolved	Cannot be Resolved
Ali and Mahmood [40]	MSM	$T ι^{Al - 1} = 0.9873, T ι^{Al - 2} = 0.9873,$ $T ι^{Al - 3} = 0.9854, T ι^{Al - 4} = 0.987$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	Empty	Cannot be Resolved	Cannot be Resolved
Explored Operators	WA	$T ι^{Al - 1} = 0.9996, T ι^{Al - 2} = 0.9996,$ $T ι^{Al - 3} = 0.9977, T ι^{Al - 4} = 0.999$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	WG	$T ι^{Al - 1} = - 0.996, T ι^{Al - 2} = - 0.997,$ $T ι^{Al - 3} = - 0.997, T ι^{Al - 4} = - 0.997$	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 3} = T ι^{Al - 4}$

From Table 13, the CQROF sort of information is only it by the work of Ali and Mahmood [40] and elaborated operators. Data in Table 13 are invented in the shape of Fig. 5.

Fig. 5

Geometrical expressions of the information of Table 13.

Similarly, Under Table 7, we get data in Table 14.

Table 14

Supremacy of the elaborated and existing operators by using Example 3

Methods	Operators	Score values	Ranking values
Joshi and Gegov [19]	WA	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Zeng et al. [46]	WA	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Xu and Yager [47]	Empty	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Garg and Rani [32]	Empty	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Akram et al. [36]	Dombi AOs	$T ι^{Al - 1} = 1, T ι^{Al - 2} = 1,$ $T ι^{Al - 3} = 0.9976, T ι^{Al - 4} = 0.9999$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	Empty	Cannot be Resolved	Cannot be Resolved
Ali and Mahmood [40]	MSM	$T ι^{Al - 1} = 1, T ι^{Al - 2} = 1,$ $T ι^{Al - 3} = 0.9973, T ι^{Al - 4} = 0.9994$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	Empty	Cannot be Resolved	Cannot be Resolved
Explored Operators	WA	$T ι^{Al - 1} = 1, T ι^{Al - 2} = 1,$ $T ι^{Al - 3} = 0.9977, T ι^{Al - 4} = 0.9998$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	WG	$T ι^{Al - 1} = - 0.9971, T ι^{Al - 2} = - 0.9972,$ $T ι^{Al - 3} = - 0.9973, T ι^{Al - 4} = - 0.9973$	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 3} = T ι^{Al - 4}$

Data in Table 14 are invented in the shape of Fig. 6.

Fig .6

Geometrical expressions of the information of Table 14.

Similarly, under Table 10, we get data in Table 15.

Table 15

Supremacy of the elaborated and existing operators by using Example 4

Methods	Operators	Score Values	Ranking Values
Joshi and Gegov [19]	WA	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Zeng et al. [46]	WA	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Xu and Yager [47]	Empty	Cannot be Resolved	Cannot be Resolved
	WG	Cannot be Resolved	Cannot be Resolved
Garg and Rani [32]	Empty	Cannot be Resolved	Cannot be Resolved
	WG	$T ι^{Al - 1} = - 0.9860, T ι^{Al - 2} = - 0.9860,$ $T ι^{Al - 3} = - 0.9871, T ι^{Al - 4} = - 0.9865$	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 4} = T ι^{Al - 3}$
Akram et al. [36]	Dombi AOs	$T ι^{Al - 1} = 0.9888, T ι^{Al - 2} = 0.9888,$ $T ι^{Al - 3} = 0.9877, T ι^{Al - 4} = 0.9881$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	Empty	Cannot be Resolved	Cannot be Resolved
Ali and Mahmood [40]	MSM	$T ι^{Al - 1} = 0.9876, T ι^{Al - 2} = 0.9876,$ $T ι^{Al - 3} = 0.9865, T ι^{Al - 4} = 0.9871$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	Empty	Cannot be Resolved	Cannot be Resolved
Explored Operators	WA	$T ι^{Al - 1} = 0.9999, T ι^{Al - 2} = 0.9999,$ $T ι^{Al - 3} = 0.9988, T ι^{Al - 4} = 0.999$	$T ι^{Al - 1} = T ι^{Al - 2} ⩾ T ι^{Al - 4} ⩾ T ι^{Al - 3}$
	WG	$T ι^{Al - 1} = - 0.9971, T ι^{Al - 2} = - 0.9971,$ $T ι^{Al - 3} = - 0.9982, T ι^{Al - 4} = - 0.9976$	$T ι^{Al - 1} ⩾ T ι^{Al - 2} = T ι^{Al - 4} = T ι^{Al - 3}$

From Table 14, the CQROF sort of information is only it by the work of Ali and Mahmood [40], Akram et al. [36], and elaborated operators.

The graphical expressions of the information in Table 15 are explained in the form of Fig. 7.

Fig. 7

Geometrical expressions of the information of Table 15.

From Table 15, the CQROF kind of data is just by crafted by Ali and Mahmood [40], Akram et al. [36], Garg and Rani [32], and explained administrators. Subsequently, the explained administrators dependent on CQROFSs are all the more impressive and more approve is contrasted with existing administrators [19 , 47]. Moreover, we fix three alternatives, and nine attributes are to investigate the beneficial optimal. The updated data is given in Table 16.

Table 16

Stated large data scale

	$T ι^{Al - 1}$	$T ι^{Al - 2}$	$T ι^{Al - 3}$
$T ι^{At - 1}$	$((\begin{matrix} 0.8 e^{i 2 π (0.9)}, \\ 0.7 e^{i 2 π (0.8)} \end{matrix}), 0.8)$	$((\begin{matrix} 0.9 e^{i 2 π (0.8)}, \\ 0.4 e^{i 2 π (0.5)} \end{matrix}), 0.9)$	$((\begin{matrix} 0.7 e^{i 2 π (0.5)}, \\ 0.5 e^{i 2 π (0.4)} \end{matrix}), 0.5)$
$T ι^{At - 2}$	$((\begin{matrix} 0.81 e^{i 2 π (0.91)}, \\ 0.71 e^{i 2 π (0.81)} \end{matrix}), 0.81)$	$((\begin{matrix} 0.91 e^{i 2 π (0.81)}, \\ 0.41 e^{i 2 π (0.51)} \end{matrix}), 0.91)$	$((\begin{matrix} 0.71 e^{i 2 π (0.51)}, \\ 0.51 e^{i 2 π (0.41)} \end{matrix}), 0.51)$
$T ι^{At - 3}$	$((\begin{matrix} 0.82 e^{i 2 π (0.92)}, \\ 0.72 e^{i 2 π (0.82)} \end{matrix}), 0.82)$	$((\begin{matrix} 0.92 e^{i 2 π (0.82)}, \\ 0.42 e^{i 2 π (0.52)} \end{matrix}), 0.92)$	$((\begin{matrix} 0.72 e^{i 2 π (0.52)}, \\ 0.52 e^{i 2 π (0.42)} \end{matrix}), 0.52)$
$T ι^{At - 4}$	$((\begin{matrix} 0.83 e^{i 2 π (0.93)}, \\ 0.73 e^{i 2 π (0.83)} \end{matrix}), 0.83)$	$((\begin{matrix} 0.93 e^{i 2 π (0.83)}, \\ 0.43 e^{i 2 π (0.53)} \end{matrix}), 0.93)$	$((\begin{matrix} 0.73 e^{i 2 π (0.53)}, \\ 0.53 e^{i 2 π (0.43)} \end{matrix}), 0.53)$
$T ι^{At - 5}$	$((\begin{matrix} 0.84 e^{i 2 π (0.94)}, \\ 0.74 e^{i 2 π (0.84)} \end{matrix}), 0.84)$	$((\begin{matrix} 0.94 e^{i 2 π (0.84)}, \\ 0.44 e^{i 2 π (0.54)} \end{matrix}), 0.94)$	$((\begin{matrix} 0.74 e^{i 2 π (0.54)}, \\ 0.54 e^{i 2 π (0.44)} \end{matrix}), 0.54)$
$T ι^{At - 6}$	$((\begin{matrix} 0.85 e^{i 2 π (0.95)}, \\ 0.75 e^{i 2 π (0.85)} \end{matrix}), 0.85)$	$((\begin{matrix} 0.95 e^{i 2 π (0.85)}, \\ 0.45 e^{i 2 π (0.55)} \end{matrix}), 0.95)$	$((\begin{matrix} 0.75 e^{i 2 π (0.55)}, \\ 0.55 e^{i 2 π (0.45)} \end{matrix}), 0.55)$
$T ι^{At - 7}$	$((\begin{matrix} 0.86 e^{i 2 π (0.96)}, \\ 0.76 e^{i 2 π (0.86)} \end{matrix}), 0.86)$	$((\begin{matrix} 0.96 e^{i 2 π (0.86)}, \\ 0.46 e^{i 2 π (0.56)} \end{matrix}), 0.96)$	$((\begin{matrix} 0.76 e^{i 2 π (0.56)}, \\ 0.56 e^{i 2 π (0.46)} \end{matrix}), 0.56)$
$T ι^{At - 8}$	$((\begin{matrix} 0.87 e^{i 2 π (0.97)}, \\ 0.77 e^{i 2 π (0.87)} \end{matrix}), 0.87)$	$((\begin{matrix} 0.97 e^{i 2 π (0.87)}, \\ 0.47 e^{i 2 π (0.57)} \end{matrix}), 0.97)$	$((\begin{matrix} 0.77 e^{i 2 π (0.57)}, \\ 0.57 e^{i 2 π (0.47)} \end{matrix}), 0.57)$
$T ι^{At - 9}$	$((\begin{matrix} 0.88 e^{i 2 π (0.98)}, \\ 0.78 e^{i 2 π (0.88)} \end{matrix}), 0.88)$	$((\begin{matrix} 0.98 e^{i 2 π (0.88)}, \\ 0.48 e^{i 2 π (0.58)} \end{matrix}), 0.98)$	$((\begin{matrix} 0.78 e^{i 2 π (0.58)}, \\ 0.58 e^{i 2 π (0.48)} \end{matrix}), 0.58)$

The final values are illustrated in Table 17.

Table 17

The stated score value of the information is in Table 16

Alternatives	CCQROFWA Operator	CCQROFWG Operator
$T ι^{Al - 1}$	0.04395	0.28969
$T ι^{Al - 2}$	0.02961	0.25769
$T ι^{Al - 3}$	0.0062	0.03845

For our convenience, the rank values are invented in Table 18.

Table 18

Stated the ranking values

Methods	Ranking values
CCQROFWA Operator	$T ι^{Al - 1} ⩾ T ι^{Al - 2} ⩾ T ι^{Al - 3}$
CCQROFWG Operator	$T ι^{Al - 1} ⩾ T ι^{Al - 2} ⩾ T ι^{Al - 3}$

As shown above, we obtained the same result from two different operators, the best alternatives $T ι^{Al - 1}$ , means that the best crystal is Ionic solid.

As displayed above, we established the positioning outcomes by utilizing the explained administrator and applied them by CQROFS kinds of data to find the adequacy and capability of the found methodologies. Furthermore, to work on the nature of the proposed approach, we contrast the found methodologies and some current methodologies [2 , 32].

Atanassov [2] explained the IFSs. In [2], the creators proposed specific functional laws by utilizing the IFSs and deciding the best ideal to show the strength of the expounded administrators, however the hypothesis proposed in [2] dependent on IFS is the extraordinary instance of the proposed administrators dependent on CQROFSs. In this way, on the off chance that we pick the proposed kinds of data, the hypothesis of Atanassov [2] can’t resolve it, because the explained approach is broader than the predominant thoughts in [2].

Aggregation administrators dependent on CIFSSs were explained by Ali et al. [32], which is the combination of the accumulation administrators with complex intuitionistic fluffy delicate sets. In [32], the creators joined the total administrators with CIFSSs and decide the best ideal to show the predominance of the explained administrators, however the hypothesis proposed in [32] dependent on CIFSS is the extraordinary instance of the proposed administrators dependent on complex q-rung orthopair fluffy sets. Subsequently, on the off chance that we pick the proposed sorts of data, the hypothesis of Ali et al. [32] can’t resolve it, on the grounds that the expounded approach is broader than the overall thoughts in [32].

Ullah et al. [33] explained the likeness measures (SM) in view of a CPFS. In [33], the creators consolidated the SM with a CPFS and decide the best ideal to show the strength of the expounded administrators, however the hypothesis proposed in [33] dependent on CPFS is the exceptional instance of the proposed administrators dependent on CQROFSs. In this way, on the off chance that we pick the proposed kinds of data, the hypothesis of Ullah et al. [33] can’t resolve it, on the grounds that the explained approach is broader than the common thoughts in [33].

From the above hypotheses, we acquired the outcome that is the explained administrator dependent on another CQROFS is broadly valuable and more prevailing to oversee off-kilter and conflicting data in certifiable issues.

5 Conclusion

The CIFS model and CPFS concept are additional appropriate techniques to convey knowledge based on the ambiguous and awkward natural environment in MADM dilemmas. But Liu et al. [39] indicated out that the CQROFS is additional broad than the CIFS and CPFS. It is also remarkable that as the q_SC expands the freedom of satisfactory orthopairs rises and thus creates specialist’s further choice in conveying their confidence about truth grade. Established on these improvements, certain aggregation operators (AOs) were intended by separate creators to add CQROFNs. But these remaining CQROF AOs are established by guessing specialists are certainly conversant with assessed items, that is, all specialists offered their evaluation of the various option at a consistent level of belief. This sort of circumstance is moderately accomplished in demonstrating genuine world challenges. For this, the current survey presents a sequence of CCQROFWA, CCQROFOWA, CCQROFWG, and CCQROFOWG operators are deliberated. Their several valuable properties are well recognized. These characterized operators are qualified to justify the genuine-life condition beyond noticeably with the support of specialists’ CLs thorough assessment and will remind you of the very much more genuine circumstances by using the CQROF environment to examine the best crystal solid for guanine life usage. Ultimately, a comprehensive dialogue has been conducted out to demonstrate the pertinence and dominance of accessible methods over the remaining ones.

Data availability statement

The data used in this manuscript are hypothetical and anyone can use them without prior permission by just citing this article.

Footnotes

Acknowledgment

The authors would like to thank the anonymous referees for their helpful comments for improving this paper.

Conflicts of interest

The authors declare no conflict of interest.

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