On Circular q-Rung Orthopair Fuzzy Sets with Dombi Aggregation Operators and Application to Symmetry Analysis in Artificial Intelligence

Abstract

Circular q-rung orthopair fuzzy sets (FSs) were recently considered as an extension of q-rung orthopair FSs (q-ROFSs), circular intuitionistic FSs (Cir-IFSs), and circular Pythagorean FSs (Cir-PFSs). However, they are only considered for some simple algebraic properties. In this paper, we advance the work on circular q-ROFSs (Cirq-ROFSs) in Dombi aggregation operators (AOs) with more mathematical properties of algebraic laws. These include the circular q-rung orthopair fuzzy (Cirq-ROF) Dombi weighted averaging (Cirq-ROFDWA), Cirq-ROF Dombi ordered weighted averaging (Cirq-ROFDOWA), Cirq-ROF Dombi weighted geometric (Cirq-ROFDWG), and Cirq-ROF Dombi ordered weighted geometric (Cirq-ROFDOWG) operators. Additionally, we present the properties of idempotency, monotonicity, and boundedness for the proposed operators. In the context of artificial intelligence, symmetry analysis plays a significant and efficient role that can refer to several aspects. Thus, to compute the major aspect, we identify the multi-attribute decision-making (MADM) technique based on the proposed operators for Cirq-ROF numbers (Cirq-ROFNs) to enhance the worth of the evaluated operators. Finally, we use some existing techniques for comparison to our results to show the validity and supremacy of the proposed method.

1. Introduction

Symmetry analysis in artificial intelligence [1] is a multi-dimensional technique that spans algorithm design, knowledge representation, data manipulation, and fairness selections [2]. Utilizing symmetry analysis methods can improve the capability, fairness, supremacy, and interpretability of artificial intelligence across many domains. Furthermore, these concepts have been utilized in many fields—for instance, machine learning, game theory, artificial intelligence, neural networks, data mining, and decision-making—under the consideration of the classical set theory [3]. Unfortunately, these procedures have lost a lot of information because of limited options with only zero and one. Therefore, a stronger idea is required to maintain more information. Zadeh [4] initiated a fuzzy set (FS) where the truth grade ${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ of an FS becomes ${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\in \left[\mathrm{0,1}\right]$, and this FS theory has been applied in various fields [5,6,7]. Furthermore, Atanassov [8] modified the FS theory by adding another grade of falsity ${\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ and initiated the idea of intuitionistic FS (IFS) with the condition that $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\le 1$. IFS has since been studied in different works [9,10,11]. However, in the presence of the pair $\left(\mathrm{0.6,0.5}\right)$, the idea of IFS has not worked prominently. Therefore, Yager and Abbasov [12] presented the Pythagorean FS (PFS) with the new condition that $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^2+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^2\le 1$, resulting in the PFS having different applications [13,14,15]. However, the PFS still fails in the presence of the pair $\left(\mathrm{0.9,0.8}\right)$ due to its limited features. Yager [16] then presented the idea of q-rung orthopair FSs (q-ROFSs) with a prominent and dominant condition of $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1$, $q_{SC}\ge 1$. These q-ROFSs have been applied in many areas [17,18,19]. Some specifications of q-ROFSs are listed in

Table 1. Specifications of q-ROFSs.

Parameters	Methods	Conditions	Mathematical Forms
$q_{SC}=1, {\mathcal{N}}_{{\mathcal{Q}}_{CQ}}=0$	Fuzzy sets (FSs)	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}=1$	Intuitionistic FSs	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}=2$	Pythagorean FSs	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^2+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^2\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}\ge 1$	q-Rung orthopair FSs	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$

.

Table 1 contains the mathematical forms of FSs, IFSs, PFSs, and q-ROFSs. The IFS theory and its applications have been widely used in many areas because the computed structure of an FS is very reliable and dominant due to its features. However, finding the radius between the truth and the falsity grade is very complicated because no one can do it yet. After a long assessment, we noticed that the idea of circular IFS (Cir-IFS) was presented by Atanassov [20], where the truth grade, the falsity grade, and the radius of a circle around the point of both grades are also the major parts of Cir-IFS. Furthermore, Bozyigit et al. [21] proposed the circular PFS (Cir-PFS) as an extension of Cir-IFS, in which Cir-PFS contains the truth grade, represented by ${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$, and the falsity grade is shown by ${\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ under the condition of $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^2+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^2\le 1$, with the radius ${\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\in \left[\mathrm{0,1}\right]$ of a circle around the point $\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)$ on the plane. Cir-IFSs and Cir-PFSs had some applications in [22,23,24,25]. Recently, Yusoff et al. [26] extended Cir-PFS to circular q-ROFSs (Cirq-ROFSs), which have a condition of $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1,q_{SC}\ge 1$ with a radius of a circle around the point $\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)$ with ${\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\in \left[\mathrm{0,1}\right]$, and then gave some simple algebraic properties. Since Cirq-ROFSs are a new form, there is no further research on Cirq-ROFSs in the literature. It is known that Dombi [27] initiated the new concept of t-norm and t-conorm in 1982, calling them the Dombi t-norm (DTN) and the Dombi t-conorm (DTCN). Based on the above DTN and DTCN, some kinds of operators have been derived by different scholars. Khan et al. [28] gave the Dombi operators based on PFSs. Jana et al. [29] initiated the Dombi operators for q-ROFSs and their applications. Du [30] evaluated Dombi aggregation operators (AOs) for q-ROFSs. Seikh and Mandal [31] examined Dombi operators for IFSs. In this paper, we advance the study of Cirq-ROFSs in Dombi AOs. These include the circular q-rung orthopair fuzzy (Cirq-ROF) Dombi weighted averaging (Cirq-ROFDWA), Cirq-ROF Dombi ordered weighted averaging (Cirq-ROFDOWA), Cirq-ROF Dombi weighted geometric (Cirq-ROFDWG), and Cirq-ROF Dombi ordered weighted geometric (Cirq-ROFDOWG) operators. We then consider the mathematical properties of these proposed operators, such as idempotency, monotonicity, and boundedness. Some specifications of the Cirq-ROFSs are listed in

Table 2. Specifications of Cirq-ROFSs.

Parameters	Methods	Conditions	Mathematical Form
$q_{SC}=1,{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}=0, {\mathcal{R}}_{{\mathcal{Q}}_{CQ}}=0$	Fuzzy sets (FSs)	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}=1,{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}=0$	Intuitionistic FSs (IFSs)	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}=1$	Circular IFSs	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\le 1,{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\in \left[\mathrm{0,1}\right]$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}=2,{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}=0$	Pythagorean FSs (PFSs)	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^2+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^2\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}=2$	Circular PFSs	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^2+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^2\le 1,{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\in \left[\mathrm{0,1}\right]$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}\ge 1,{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}=0$	q-Rung orthopair FSs (q-ROFSs)	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$
$q_{SC}\ge 1$	Circular q-ROFSs	$0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1,{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\in \left[\mathrm{0,1}\right]$	$\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$

.

The demonstration in Table 1 and Table 2 with different parameters and conditions shows the advantages and disadvantages of these extended types of FSs. Thus, the major contributions of this paper are listed below:

(1)

Identification of the novel technique of Cirq-ROFSs with their flexible properties, such as algebraic laws and Dombi laws.

(2)

Proposal of the Cirq-ROFDWA, Cirq-ROFDOWA, Cirq-ROFDWG, and Cirq-ROF-DOWG operators with their properties, such as idempotency, monotonicity, and boundedness.

(3)

Creation of the multi-attribute decision-making (MADM) technique based on the proposed operators for Cirq-ROF numbers (Cirq-ROFNs) with application to the symmetry analysis in artificial intelligence to compute its major aspects.

(4)

Use of some existing techniques for comparison to our results to show the validity and supremacy of the proposed method.

The remainder of this paper is as follows. In Section 2, we review the DTN and DTCN with q-ROFSs and their operational laws for any universal set. In Section 3, we identify the novel technique of Cirq-ROFSs and their flexible properties, such as algebraic laws and Dombi laws. In Section 4, we propose the Cirq-ROFDWA, Cirq-ROFDOWA, Cirq-ROFDWG, and Cirq-ROFDOWG operators with some basic properties. In the context of artificial intelligence, symmetry analysis can refer to several aspects. Therefore, in Section 5, to compute the major aspect, we create the MADM technique based on the initiated operators for Cirq-ROFNs to enhance the worth of the proposed operators. In Section 6, we illustrate some numerical examples. In Section 7, we use existing techniques for comparison to our results to show the validity and supremacy of the presented theory. Some concluding remarks are stated in Section 8.

2. Preliminaries

This section discusses the DTN and DTCN with q-ROFSs and their operational laws for any universal set.

Definition 1

([27]). Let ${\mathfrak{f}}_{RN},{\mathfrak{g}}_{RN}\in \left[\mathrm{0,1}\right]$ with  ${\sigma }_{SC}\ge 1$. Then

$$DOM\left({\mathfrak{f}}_{RN},{\mathfrak{g}}_{RN}\right)=\frac{1}{1+{\left({\left(\frac{1-{\mathfrak{f}}_{RN}}{{\mathfrak{f}}_{RN}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathfrak{g}}_{RN}}{{\mathfrak{g}}_{RN}}\right)}^{{\sigma }_{SC}}\right)}^{{\sigma }_{SC}}}$$

$$DO{M}^c\left({\mathfrak{f}}_{RN},{\mathfrak{g}}_{RN}\right)=1-\frac{1}{1+{\left({\left(\frac{{\mathfrak{f}}_{RN}}{1-{\mathfrak{f}}_{RN}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathfrak{g}}_{RN}}{1-{\mathfrak{g}}_{RN}}\right)}^{{\sigma }_{SC}}\right)}^{{\sigma }_{SC}}}$$

Thus, algebraic norms are special cases of the above norms. We next talk about the existing techniques, called q-ROFSs, and discuss their operational laws.

Definition 2

([16]). Let ${\mathcal{X}}_{UNI}$ be a universe of discourse. The q-ROFS is defined below:

$${\mathcal{Q}}_{CQ}=\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$$

where the truth grade is represented by ${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ and the falsity grade is shown by ${\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ with  $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1,q_{SC}\ge 1$. Further, the refusal grade is represented by ${\mathcal{R}}_{CQ}\left(\tilde{\mathcal{x}}\right)={\left(1-\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\right)\right)}^{1/q_{SC}},$ where the mathematical shape of q-ROFN is stated by ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$.

Definition 3

([16]). Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of q-ROFNs. Then

$${\mathcal{Q}}_1\oplus {\mathcal{Q}}_2=\left({\left({\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{N}}_{{\mathcal{Q}}_1}{\mathcal{N}}_{{\mathcal{Q}}_2}\right)$$

$${\mathcal{Q}}_1\otimes {\mathcal{Q}}_2=\left({\mathcal{M}}_{{\mathcal{Q}}_1}{\mathcal{M}}_{{\mathcal{Q}}_2},{\left({\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}}\right)$$

$${\varrho }_{SC}{\mathcal{Q}}_j=\left({\left(1-{\left(1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{N}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}}\right)$$

$${\mathcal{Q}}_j^{{\varrho }_{SC}}=\left({\mathcal{M}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}},{\left(1-{\left(1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}}\right)$$

Definition 4

([16]). Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of q-ROFNs. Then, the score value $SCO\left({\mathcal{Q}}_j\right)$ and the accuracy value $SCO\left({\mathcal{Q}}_j\right)$ are defined as:

$$SCO\left({\mathcal{Q}}_j\right)={\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}-{\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}\in \left[-\mathrm{1,1}\right]$$

$$ACC\left({\mathcal{Q}}_j\right)={\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}+{\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}\in \left[\mathrm{0,1}\right]$$

The score value and accuracy value have the following properties:

(1) 

If $SCO\left({\mathcal{Q}}_1\right)>SCO\left({\mathcal{Q}}_2\right)$, then  ${\mathcal{Q}}_1>{\mathcal{Q}}_2$.

(2) 

If $SCO\left({\mathcal{Q}}_1\right)<SCO\left({\mathcal{Q}}_2\right)$, then  ${\mathcal{Q}}_1<{\mathcal{Q}}_2$.

(3) 

$If SCO\left({\mathcal{Q}}_1\right)=SCO\left({\mathcal{Q}}_2\right), then we have:$

(i). 

If $ACC\left({\mathcal{Q}}_1\right)>ACC\left({\mathcal{Q}}_2\right)$, then  ${\mathcal{Q}}_1>{\mathcal{Q}}_2$.

(ii). 

If $ACC\left({\mathcal{Q}}_1\right)<ACC\left({\mathcal{Q}}_2\right)$, then  ${\mathcal{Q}}_1<{\mathcal{Q}}_2.$

3. Circular q-ROFSs with Their Dombi Operational Laws

In this section, we first review the definition of circular q-ROFSs (Cirq-ROFSs). We then consider the novel techniques of Cirq-ROFSs with their operational laws. Moreover, we also evaluate some Dombi operational laws for Cirq-ROFNs with some properties.

Definition 5

([26]). Let ${\mathcal{X}}_{UNI}$ be a universe of discourse. The Cirq-ROFS is defined as

$${\mathcal{Q}}_{CQ}=\left\{\left(\tilde{\mathcal{x}},\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\right)\right):\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}\right\}$$

where the truth grade is represented by ${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ and the falsity grade is shown by ${\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ with  $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1,q_{SC}\ge 1,$ and  ${\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ represents the radius of the circle around the point (${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right),{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$) on the plane. The refusal grade is presented by${\mathcal{R}}_{CQ}\left(\tilde{\mathcal{x}}\right)={\left(1-\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\right)\right)}^{1/q_{SC}}$, where the mathematical shape of the Cirq-ROFN is stated by ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n.$

We mention that Definition 5 is more general than the original Cirq-ROFS defined in Yusoff et al. [26]. In Defintion 5, we consider that the radius ${\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ depends on  $\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI}.$ In fact, when  ${\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)={\mathcal{R}}_{{\mathcal{Q}}_{CQ}}$ (a constant) for all  $\tilde{\mathcal{x}}\in {\mathcal{X}}_{UNI},$ it is the same Cirq-ROFS as defined in Yusoff et al. [26].

Definition 6

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then

$${\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2=\left({\left({\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{N}}_{{\mathcal{Q}}_1}{\mathcal{N}}_{{\mathcal{Q}}_2},{\left({\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}}\right);$$

$${\mathcal{Q}}_1{\oplus }^t{\mathcal{Q}}_2=\left({\left({\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{N}}_{{\mathcal{Q}}_1}{\mathcal{N}}_{{\mathcal{Q}}_2},{\mathcal{R}}_{{\mathcal{Q}}_1}{\mathcal{R}}_{{\mathcal{Q}}_2}\right);$$

$${\mathcal{Q}}_1{\otimes }^{tc}{\mathcal{Q}}_2=\left({\mathcal{M}}_{{\mathcal{Q}}_1}{\mathcal{M}}_{{\mathcal{Q}}_2},{\left({\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{R}}_{{\mathcal{Q}}_1}{\mathcal{R}}_{{\mathcal{Q}}_2}\right);$$

$${\mathcal{Q}}_1{\otimes }^t{\mathcal{Q}}_2=\left({\mathcal{M}}_{{\mathcal{Q}}_1}{\mathcal{M}}_{{\mathcal{Q}}_2},{\left({\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}},{\left({\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}+{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}\right)}^{\frac{1}{q_{SC}}}\right);$$

$${\varrho }_{SC}{\mathcal{Q}}_j^{tc}=\left({\left(1-{\left(1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{N}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}},{\left(1-{\left(1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}}\right);$$

$${\varrho }_{SC}{\mathcal{Q}}_j^t=\left({\left(1-{\left(1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{N}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}},{\mathcal{R}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}}\right);$$

$${\left({\mathcal{Q}}_j^{tc}\right)}^{{\varrho }_{SC}}=\left({\mathcal{M}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}},{\left(1-{\left(1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}},{\mathcal{R}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}}\right);$$

$${\left({\mathcal{Q}}_j^t\right)}^{{\varrho }_{SC}}=\left({\mathcal{M}}_{{\mathcal{Q}}_j}^{{\varrho }_{SC}},{\left(1-{\left(1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}},{\left(1-{\left(1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}\right)}^{{\varrho }_{SC}}\right)}^{\frac{1}{q_{SC}}}\right).$$

These defined equations in Definition 6 are the newly proposed basic operators for Cirq-ROFSs. We next propose some new Dombi operational laws for Cirq-ROFNs in Definition 7 as follows.

Definition 7

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then, Dombi operational laws are as follows:

$${\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right);$$

$${\mathcal{Q}}_1{\oplus }^t{\mathcal{Q}}_2=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\end{array}\right);$$

$${\mathcal{Q}}_1{\otimes }^{tc}{\mathcal{Q}}_2=\left(\begin{array}{c}{\left(1/1+{\left({\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$${\mathcal{Q}}_1{\otimes }^t{\mathcal{Q}}_2=\left(\begin{array}{c}{\left(1/1+{\left({\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$${\varrho }_{SC}{\mathcal{Q}}_1^{tc}=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$${\varrho }_{SC}{\mathcal{Q}}_1^t=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$${\left({\mathcal{Q}}_1^{tc}\right)}^{{\varrho }_{SC}}=\left(\begin{array}{c}{\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$${\left({\mathcal{Q}}_1^t\right)}^{{\varrho }_{SC}}=\left(\begin{array}{c}{\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

Theorem 1.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then, we have the following results:

(1) 

${\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2={\mathcal{Q}}_2{\oplus }^{tc}{\mathcal{Q}}_1$.

(2) 

${\mathcal{Q}}_1{\otimes }^{tc}{\mathcal{Q}}_2={\mathcal{Q}}_2{\otimes }^{tc}{\mathcal{Q}}_1$.

(3) 

${\varrho }_{SC}{\mathcal{Q}}_1{\oplus }^{tc}{\varrho }_{SC}{\mathcal{Q}}_2={\varrho }_{SC}\left({\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2\right)$.

(4) 

${\left({\mathcal{Q}}_1\right)}^{{\varrho }_{SC}}{\oplus }^{tc}{\left({\mathcal{Q}}_2\right)}^{{\varrho }_{SC}}={\left({\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2\right)}^{{\varrho }_{SC}}$.

(5) 

${\varrho }_{SC-1}{\mathcal{Q}}_1{\oplus }^{tc}{\varrho }_{SC-2}{\mathcal{Q}}_1=\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right)\left({\mathcal{Q}}_1\right)$.

(6) 

${\left({\mathcal{Q}}_1\right)}^{{\varrho }_{SC-1}}{\oplus }^{tc}{\left({\mathcal{Q}}_1\right)}^{{\varrho }_{SC-1}}={\left({\mathcal{Q}}_1\right)}^{\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right)}$.

Proof. 

(1)

Based on Definition 7, we obtain:

$$\begin{gathered} {\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)={\mathcal{Q}}_2{\oplus }^{tc}{\mathcal{Q}}_1. \end{gathered}$$

(2)

Omitted because it is similar to step 1.

(3)

Based on Definition 7, we obtain:

$$\begin{gathered} {\varrho }_{SC}{\mathcal{Q}}_1{\oplus }^{tc}{\varrho }_{SC}{\mathcal{Q}}_2 \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^{tc}\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC}\left({\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ ={\varrho }_{SC}\left({\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2\right). \end{gathered}$$

(4)

Omitted because it is similar to step 3.

(5)

Based on Definition 7, we obtain:

$$\begin{gathered} {\varrho }_{SC-1}{\mathcal{Q}}_1{\oplus }^{tc}{\varrho }_{SC-2}{\mathcal{Q}}_1 \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC-1}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC-1}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC-1}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^{tc}\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC-2}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC-2}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{SC-2}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left(\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right){\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left(\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right){\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left(\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right){\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right)\left({\mathcal{Q}}_1\right). \end{gathered}$$

(6)

Omitted because it is similar to step 5. □

Theorem 2.

Let  ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then

(1) 

${\mathcal{Q}}_1{\oplus }^t{\mathcal{Q}}_2={\mathcal{Q}}_2{\oplus }^t{\mathcal{Q}}_1$.

(2) 

${\mathcal{Q}}_1{\otimes }^t{\mathcal{Q}}_2={\mathcal{Q}}_2{\otimes }^t{\mathcal{Q}}_1$.

(3) 

${\varrho }_{SC}{\mathcal{Q}}_1{\oplus }^t{\varrho }_{SC}{\mathcal{Q}}_2={\varrho }_{SC}\left({\mathcal{Q}}_1{\oplus }^t{\mathcal{Q}}_2\right)$.

(4) 

${\left({\mathcal{Q}}_1\right)}^{{\varrho }_{SC}}{\oplus }^t{\left({\mathcal{Q}}_2\right)}^{{\varrho }_{SC}}={\left({\mathcal{Q}}_1{\oplus }^{tc}{\mathcal{Q}}_2\right)}^{{\varrho }_{SC}}$.

(5) 

${\varrho }_{SC-1}{\mathcal{Q}}_1{\oplus }^t{\varrho }_{SC-2}{\mathcal{Q}}_1=\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right)\left({\mathcal{Q}}_1\right)$.

(6) 

${\left({\mathcal{Q}}_1\right)}^{{\varrho }_{SC-1}}{\oplus }^t{\left({\mathcal{Q}}_1\right)}^{{\varrho }_{SC-1}}={\left({\mathcal{Q}}_1\right)}^{\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right)}$.

Proof. 

(1)

Based on Definition 7, we obtain:

$$\begin{gathered} {\mathcal{Q}}_1{\oplus }^t{\mathcal{Q}}_2=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)={\mathcal{Q}}_2{\oplus }^t{\mathcal{Q}}_1. \end{gathered}$$

(2)

Omitted because it is similar to step 1.

(3)

Based on Definition 7, we obtain:

$$\begin{gathered} {\varrho }_{SC}{\mathcal{Q}}_1{\oplus }^t{\varrho }_{SC}{\mathcal{Q}}_2 \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^t\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\varrho }_{SC}{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC}\left({\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC}\left({\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\varrho }_{SC}\left({\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}+{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ ={\varrho }_{SC}\left({\mathcal{Q}}_1{\oplus }^t{\mathcal{Q}}_2\right). \end{gathered}$$

(4)

Omitted because it is similar to step 3.

(5)

Based on Definition 7, we obtain:

$$\begin{gathered} {\varrho }_{SC-1}{\mathcal{Q}}_1{\oplus }^t{\varrho }_{SC-2}{\mathcal{Q}}_1 \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC-1}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC-1}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\varrho }_{SC-1}{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^t\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{SC-2}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{SC-2}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left({\varrho }_{SC-2}{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left(\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right){\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left(\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right){\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\left(1/\left(1+{\left(\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right){\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left({\varrho }_{SC-1}+{\varrho }_{SC-2}\right)\left({\mathcal{Q}}_1\right). \end{gathered}$$

(6)

Omitted because it is similar to step 5. □

4. Cirq-ROF Dombi Averaging/Geometric Aggregation Operators

This section introduces the Cirq-ROFDWA operator, Cirq-ROFDOWA operator, Cirq-ROFDWG operator, and Cirq-ROFDOWG operator. Further, we evaluate their properties of idempotency, monotonicity, and boundedness.

Definition 8.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then

$$Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\varrho }_1{\mathcal{Q}}_1^{tc}{\oplus }^{tc}{\varrho }_2{\mathcal{Q}}_2^{tc}{\oplus }^{tc}\dots {\oplus }^{tc}{\varrho }_n{\mathcal{Q}}_n^{tc}={{\oplus }^{tc}}_{j=1}^n\left({\varrho }_j{\mathcal{Q}}_j^{tc}\right);$$

$$Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\varrho }_1{\mathcal{Q}}_1^{tc}{\oplus }^t{\varrho }_2{\mathcal{Q}}_2^{tc}{\oplus }^t\dots {\oplus }^t{\varrho }_n{\mathcal{Q}}_n^{tc}={{\oplus }^t}_{j=1}^n\left({\varrho }_j{\mathcal{Q}}_j^{tc}\right).$$

These are signified as a Cirq-ROFDWA operator for both t-conorm and t-norm, where the weight vector is represented by  ${\varrho }_j\in \left[\mathrm{0,1}\right]$ with ${\oplus }_{j=1}^n{\varrho }_j=1$.

Theorem 3.

Let  ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then, we show that the final result of the Cirq-ROFDWA operator for t-conorm and t-norm is again a Cirq-ROFN:

$$Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\sum }_{j=1}^n{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\sum }_{j=1}^n{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\sum }_{j=1}^n{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right);$$

$$Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\sum }_{j=1}^n{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\sum }_{j=1}^n{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\sum }_{j=1}^n{\varrho }_j{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right).$$

Proof. 

In the presence of the induction of mathematics, we evaluate our results. To do so, first, we derive it for $n=2$.

$${\varrho }_1{\mathcal{Q}}_1^{tc}=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_1{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_1{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_1{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right),$$

and

$${\varrho }_2{\mathcal{Q}}_2^{tc}=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_2{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_2{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_2{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right).$$

Thus,

$$\begin{gathered} Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2\right)={\varrho }_1{\mathcal{Q}}_1^{tc}{\oplus }^{tc}{\varrho }_1{\mathcal{Q}}_1^{tc} \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_1{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_1{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_1{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_1}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^{tc}\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_2{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_2{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_2{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_2}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^2}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^2}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^2}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \end{gathered}$$

For our considered value, it has been done. For $n=m$, we have

$$Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_m\right)=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

Then, we derive our target for $n=m+1$.

$$\begin{gathered} Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_{m+1}\right)=Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_m\right){\oplus }^{tc}{\varrho }_{m+1}{\mathcal{Q}}_{m+1}^{tc} \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^{tc}{\varrho }_{m+1}{\mathcal{Q}}_{m+1}^{tc} \\ =\left(\begin{array}{c}{\left(\begin{array}{c}1-\\ \left(1/\left(\begin{array}{c}1+\\ {\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\end{array}\right)\right)\end{array}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(\begin{array}{c}1-\\ \left(1/\left(\begin{array}{c}1+\\ {\left({\displaystyle \sum _{j=1}^m}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\end{array}\right)\right)\end{array}\right)}^{\frac{1}{q_{SC}}}\end{array}\right){\oplus }^{tc}\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\varrho }_{m+1}{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_{m+1}}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_{m+1}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\varrho }_{m+1}{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_{m+1}}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_{m+1}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\varrho }_{m+1}{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_{m+1}}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_{m+1}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \\ =\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^{m+1}}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^{m+1}}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^{m+1}}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right) \end{gathered}$$

Hence, our results are proven for all non-negative integers. Similarly, we follow the same procedure to obtain our major target: $Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right).$ The proof is completed. □

Property 1.

Let  ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then

(1) 

(Idempotency) If ${\mathcal{Q}}_j=\mathcal{Q}=\left({\mathcal{M}}_{\mathcal{Q}},{\mathcal{N}}_{\mathcal{Q}},{\mathcal{R}}_{\mathcal{Q}}\right),j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q};$$

$$Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q}.$$

(2) 

(Monotonicity) If ${\mathcal{Q}}_j\le {\mathcal{Q}}_j^{\#},j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right);$$

$$Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right).$$

(3) 

(Boundedness) If ${{\mathcal{Q}}_j^{-}}^{tc}=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{-}}^t=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$ and  ${{\mathcal{Q}}_j^{+}}^{tc}=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{+}}^t=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$, then

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc};$$

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc}.$$

Proof. 

The proofs are straightforward. □

Definition 9.

Let  ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then,

$$Cirq-ROFDOW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\varrho }_1{\mathcal{Q}}_{o\left(1\right)}^{tc}{\oplus }^{tc}{\varrho }_2{\mathcal{Q}}_{o\left(2\right)}^{tc}{\oplus }^{tc}\dots {\oplus }^{tc}{\varrho }_n{\mathcal{Q}}_{o\left(n\right)}^{tc}={{\oplus }^{tc}}_{j=1}^n\left({\varrho }_j{\mathcal{Q}}_{o\left(j\right)}^{tc}\right);$$

$$Cirq-ROFDOW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\varrho }_1{\mathcal{Q}}_{o\left(1\right)}^t{\oplus }^t{\varrho }_2{\mathcal{Q}}_{o\left(2\right)}^t{\oplus }^t\dots {\oplus }^t{\varrho }_n{\mathcal{Q}}_{o\left(n\right)}^t={{\oplus }^t}_{j=1}^n\left({\varrho }_j{\mathcal{Q}}_{o\left(j\right)}^t\right)$$

These are signified as a Cirq-ROFDOWA operator for both t-conorm and t-norm, where the weight vector is represented by ${\varrho }_j\in \left[\mathrm{0,1}\right]$ with ${\oplus }_{j=1}^n{\varrho }_j=1$ with an order $o\left(j\right)\le o\left(j-1\right)$.

Theorem 4.

Let  ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be a family of Cirq-ROFNs. Then, we show that the final result of the Cirq-ROFDOWA operator for t-conorm and t-norm is again a Cirq-ROFN:

$$Cirq-ROFDOW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$$Cirq-ROFDOW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{1-{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

Proof. 

The proof is similar to Theorem 3. □

Property 2.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then

(1) 

(Idempotency) If ${\mathcal{Q}}_j=\mathcal{Q}=\left({\mathcal{M}}_{\mathcal{Q}},{\mathcal{N}}_{\mathcal{Q}},{\mathcal{R}}_{\mathcal{Q}}\right),j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDOW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q};$$

$$Cirq-ROFDOW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q}.$$

(2) 

(Monotonicity) If ${\mathcal{Q}}_j\le {\mathcal{Q}}_j^{\#},j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDOW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDOW{A}^{tc}\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right);$$

$$Cirq-ROFDOW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDOW{A}^t\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right).$$

(3) 

(Boundedness) If ${{\mathcal{Q}}_j^{-}}^{tc}=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{-}}^t=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$ and  ${{\mathcal{Q}}_j^{+}}^{tc}=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{+}}^t=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$, then

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDOW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc};$$

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDOW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc}.$$

Proof. 

The proofs are straightforward. □

Definition 10.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then

$$Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\left({\mathcal{Q}}_1^{tc}\right)}^{{\varrho }_1}{\otimes }^{tc}{\left({\mathcal{Q}}_2^{tc}\right)}^{{\varrho }_2}{\otimes }^{tc}\dots {\otimes }^{tc}{\left({\mathcal{Q}}_n^{tc}\right)}^{{\varrho }_n}={{\otimes }^{tc}}_{j=1}^n{\left({\mathcal{Q}}_j^{tc}\right)}^{{\varrho }_j};$$

$$Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\left({\mathcal{Q}}_1^t\right)}^{{\varrho }_1}{\otimes }^t{\left({\mathcal{Q}}_2^t\right)}^{{\varrho }_2}{\otimes }^t\dots {\otimes }^t{\left({\mathcal{Q}}_n^t\right)}^{{\varrho }_n}={{\otimes }^t}_{j=1}^n{\left({\mathcal{Q}}_j^t\right)}^{{\varrho }_j}.$$

 These are signified as a Cirq-ROFDWG operator for both t-conorm and t-norm, where the weight vector is represented by ${\varrho }_j\in \left[\mathrm{0,1}\right]$ with ${\oplus }_{j=1}^n{\varrho }_j=1$.

Theorem 5.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then, we show that the final result of the Cirq-ROFDWG operator for t-conorm and t-norm is again a Cirq-ROFN:

$$Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$$Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

Proof. 

The proof is similar to Theorem 3. □

Property 3.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then

(1) 

(Idempotency) If ${\mathcal{Q}}_j=\mathcal{Q}=\left({\mathcal{M}}_{\mathcal{Q}},{\mathcal{N}}_{\mathcal{Q}},{\mathcal{R}}_{\mathcal{Q}}\right),j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q};$$

$$Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q}.$$

(2) 

(Monotonicity) If ${\mathcal{Q}}_j\le {\mathcal{Q}}_j^{\#},j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right);$$

$$Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right).$$

(3) 

(Boundedness) If ${{\mathcal{Q}}_j^{-}}^{tc}=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{-}}^t=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$and${{\mathcal{Q}}_j^{+}}^{tc}=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{+}}^t=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$, then

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc};$$

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc}.$$

Proof. 

The proofs are straightforward. □

Definition 11.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then

$$Cirq-ROFDOW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\left({\mathcal{Q}}_{o\left(1\right)}^{tc}\right)}^{{\varrho }_1}{\otimes }^{tc}{\left({\mathcal{Q}}_{o\left(2\right)}^{tc}\right)}^{{\varrho }_2}{\otimes }^{tc}\dots {\otimes }^{tc}{\left({\mathcal{Q}}_{o\left(n\right)}^{tc}\right)}^{{\varrho }_n}={{\otimes }^{tc}}_{j=1}^n{\left({\mathcal{Q}}_{o\left(j\right)}^{tc}\right)}^{{\varrho }_j};$$

$$Cirq-ROFDOW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)={\left({\mathcal{Q}}_{o\left(1\right)}^t\right)}^{{\varrho }_1}{\otimes }^t{\left({\mathcal{Q}}_{o\left(2\right)}^t\right)}^{{\varrho }_2}{\otimes }^t\dots {\otimes }^t{\left({\mathcal{Q}}_{o\left(n\right)}^t\right)}^{{\varrho }_n}={{\otimes }^t}_{j=1}^n{\left({\mathcal{Q}}_{o\left(j\right)}^t\right)}^{{\varrho }_j}$$

These are signified as a Cirq-ROFDOWG operator for both t-conorm and t-norm, where the weight vector is represented by ${\varrho }_j\in \left[\mathrm{0,1}\right]$ with ${\oplus }_{j=1}^n{\varrho }_j=1$ with order $o\left(j\right)\le o\left(j-1\right)$.

Theorem 6.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then, we show that the final result of the Cirq-ROFDOWG operator for t-conorm and t-norm is again a Cirq-ROFN:

$$Cirq-ROFDOW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

$$Cirq-ROFDOW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\left(\begin{array}{c}{\left(1/1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{1-{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{{\mathcal{M}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{1-{\mathcal{N}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}},\\ {\left(1-\left(1/\left(1+{\left({\displaystyle \sum _{j=1}^n}{\varrho }_j{\left(\frac{{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}{1-{\mathcal{R}}_{{\mathcal{Q}}_{o\left(j\right)}}^{q_{SC}}}\right)}^{{\sigma }_{SC}}\right)}^{\frac{1}{{\sigma }_{SC}}}\right)\right)\right)}^{\frac{1}{q_{SC}}}\end{array}\right)$$

Proof. 

The proof is similar to Theorem 3. □

Property 4.

Let ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$ be family of Cirq-ROFNs. Then

(1) 

(Idempotency) If ${\mathcal{Q}}_j=\mathcal{Q}=\left({\mathcal{M}}_{\mathcal{Q}},{\mathcal{N}}_{\mathcal{Q}},{\mathcal{R}}_{\mathcal{Q}}\right),j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDOW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q};$$

$$Cirq-ROFDOW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)=\mathcal{Q}.$$

(2) 

(Monotonicity) If ${\mathcal{Q}}_j\le {\mathcal{Q}}_j^{\#},j=\mathrm{1,2},\dots ,n$, then

$$Cirq-ROFDOW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDOW{G}^{tc}\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right);$$

$$Cirq-ROFDOW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le Cirq-ROFDOW{G}^t\left({\mathcal{Q}}_1^{\#},{\mathcal{Q}}_2^{\#},\dots ,{\mathcal{Q}}_n^{\#}\right).$$

(3) 

(Boundedness) If ${{\mathcal{Q}}_j^{-}}^{tc}=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{-}}^t=\left(\begin{array}{c}\mathit{min}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{max}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$ and  ${{\mathcal{Q}}_j^{+}}^{tc}=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{min}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$,${{\mathcal{Q}}_j^{+}}^t=\left(\begin{array}{c}\mathit{max}\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right),\mathit{min}\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right),\\ \mathit{max}\left({\mathcal{R}}_{{\mathcal{Q}}_j}\right)\end{array}\right)$, then

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDOW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc};$$

$${{\mathcal{Q}}_j^{-}}^{tc}\le Cirq-ROFDOW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)\le {{\mathcal{Q}}_j^{+}}^{tc}.$$

Proof. 

The proofs are straightforward. □

The proposed techniques in this section can be extensions of the existing techniques based on FSs, IFSs, PFSs, q-ROFSs, Cir-FSs, Cir-IFSs, and Cir-PFSs. We next used these proposed operators to create a Cirq-ROF multi-attribute decision-making (MADM) technique.

5. The MADM Method Based on the Initiated Operators

In this section, we present the MADM technique based on the proposed operators, such as the Cirq-ROFDWA and Cirq-ROFDWG operators for both norms, called DTN and DTCN, to improve the validity of the initiated techniques. Consider ${\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n$ to be the collection of alternative information with the family of attributes for each alternative, such as ${\mathcal{Q}}_1^{AT},{\mathcal{Q}}_2^{AT},\dots ,{\mathcal{Q}}_m^{AT}$ with weight vectors such as ${\varrho }_j\in \left[\mathrm{0,1}\right]$, where the condition of the weight vector is ${\sum }_{j=1}^n{\varrho }_j=1$. Further, we assign Cirq-ROFNs to each attribute in every alternative. We know that the truth grade is represented by ${\mathcal{M}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ and the falsity grade is shown by ${\mathcal{N}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)$ with $0\le {\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\le 1,q_{SC}\ge 1,$ and a radius for the truth grade and falsity grade is ${\mathcal{R}}_{{\mathcal{Q}}_{CQ}}\left(\tilde{\mathcal{x}}\right)\in \left[\mathrm{0,1}\right]$. The refusal grade is ${\mathcal{R}}_{CQ}\left(\tilde{\mathcal{x}}\right)={\left(1-\left({\mathcal{M}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}+{\mathcal{N}}_{{\mathcal{Q}}_{CQ}}^{q_{SC}}\right)\right)}^{1/q_{SC}}$, where the Cirq-ROFN is presented by ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$. We present the major steps of the MADM technique for evaluating any kind of real-life problems as follows:

Step 1:

To compute the matrix, we use Cirq-ROFNs ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$. Further, we evaluate the normalization of the matrix by using the theory below:

$$N=\left\{\begin{array}{cc}\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right)& for benefit\\ \left({\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right)& for cost\end{array}\right..$$

The normalization in the matrix is not carried out if we have beneficial types of information.

Step 2:

To consider the normalization information, we evaluate the aggregated values with the help of the Cirq-ROFDWA operators of $Cirq-ROFDW{A}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)$ and $Cirq-ROFDW{A}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)$, and the Cirq-ROFDWG operators of $Cirq-ROFDW{G}^{tc}\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)$ and $Cirq-ROFDW{G}^t\left({\mathcal{Q}}_1,{\mathcal{Q}}_2,\dots ,{\mathcal{Q}}_n\right)$ for both the DTN and DTCN norms, which are presented in Theorems 3 and 5 in Section 4.

Step 3:

Calculate the score values based on the information of the score function.

$$SCO\left({\mathcal{Q}}_j\right)={\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}*\left({\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}-{\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}\right)\in \left[-\mathrm{1,1}\right];$$

$$ACC\left({\mathcal{Q}}_j\right)={\mathcal{R}}_{{\mathcal{Q}}_j}^{q_{SC}}*\left({\left({\mathcal{M}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}+{\left({\mathcal{N}}_{{\mathcal{Q}}_j}\right)}^{q_{SC}}\right)\in \left[\mathrm{0,1}\right].$$

Step 4:

Calculate the ranking of alternatives by using the score values and evaluate the best one.

We next used the above-created Cirq-ROF-MADM technique for the application of symmetry analysis in artificial intelligence.

6. Symmetry Analysis in Artificial Intelligence Based on the Proposed Cirq-ROF-MADM

In this section, we evaluate the supremacy and validity of the derived operators by resolving the problem of symmetry analysis in artificial intelligence under the presence of the initiated Cirq-ROF-MADM technique. Symmetry analysis plays a significant and efficient role in various fields, including machine learning, artificial intelligence, neural networks, and data mining, to cope with vague and complicated problems. In the context of artificial intelligence, symmetry analysis can refer to many aspects. In general, the following six aspects are important: ${\mathcal{Q}}_1$: algorithmic symmetry (optimization algorithm and machine learning models), ${\mathcal{Q}}_2:$ data symmetry (data augmentation), ${\mathcal{Q}}_3:$ knowledge representation (symbolic artificial intelligence), ${\mathcal{Q}}_4$: natural language processing (semantic symmetry), ${\mathcal{Q}}_5$: fairness and bias (algorithm fairness), and ${\mathcal{Q}}_6$: explainability (interpretability of models).

To evaluate the best decision among the above six, we select the following weight vectors: ${\left(\mathrm{0.2,0.2,0.2,0.4}\right)}^T$. For these weight vectors, we have the following attributes: growth analysis, social impact, political impact, and environmental impact. Finally, we present the major steps of the Cirq-ROF-MADM technique for evaluating the real-life problem as follows:

Step 1:

To compute the matrix as shown in Table 3, we use the Cirq-ROFNs ${\mathcal{Q}}_j=\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right),j=\mathrm{1,2},\dots ,n$. Further, we evaluate the normalization of the matrix by using the theory below:

$$N=\left\{\begin{array}{cc}\left({\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right)& for benefit\\ \left({\mathcal{N}}_{{\mathcal{Q}}_j},{\mathcal{M}}_{{\mathcal{Q}}_j},{\mathcal{R}}_{{\mathcal{Q}}_j}\right)& for cost\end{array}\right..$$

Further, the normalization in the matrix is not carried out if we have beneficial types of information, so the data in Table 3 do not need to be normalized.

Step 2:

To consider the normalization information, we evaluate the aggregated values with the help of the Cirq-ROFDWA and Cirq-ROFDWG operators, as shown in Table 4.

Table 3. Cirq-ROF decision matrix.

	${\mathcal{Q}}_1^{\mathit{A}\mathit{T}}$	${\mathcal{Q}}_2^{\mathit{A}\mathit{T}}$	${\mathcal{Q}}_3^{\mathit{A}\mathit{T}}$	${\mathcal{Q}}_4^{\mathit{A}\mathit{T}}$
${\mathcal{Q}}_1$	$\left(\mathrm{0.9,0.5,0.7}\right)$	$\left(\mathrm{0.9,0.5,0.7}\right)$	$\left(\mathrm{0.9,0.5,0.7}\right)$	$\left(\mathrm{0.9,0.5,0.7}\right)$
${\mathcal{Q}}_2$	$\left(\mathrm{0.8,0.7,0.3}\right)$	$\left(\mathrm{0.8,0.7,0.3}\right)$	$\left(\mathrm{0.8,0.7,0.3}\right)$	$\left(\mathrm{0.8,0.7,0.3}\right)$
${\mathcal{Q}}_3$	$\left(\mathrm{0.7,0.5,0.1}\right)$	$\left(\mathrm{0.7,0.5,0.1}\right)$	$\left(\mathrm{0.7,0.5,0.1}\right)$	$\left(\mathrm{0.7,0.5,0.1}\right)$
${\mathcal{Q}}_4$	$\left(\mathrm{0.6,0.3,0.5}\right)$	$\left(\mathrm{0.6,0.3,0.5}\right)$	$\left(\mathrm{0.6,0.3,0.5}\right)$	$\left(\mathrm{0.6,0.3,0.5}\right)$
${\mathcal{Q}}_5$	$\left(\mathrm{0.5,0.3,0.4}\right)$	$\left(\mathrm{0.5,0.3,0.4}\right)$	$\left(\mathrm{0.5,0.3,0.4}\right)$	$\left(\mathrm{0.5,0.3,0.4}\right)$
${\mathcal{Q}}_6$	$\left(\mathrm{0.9,0.7,0.3}\right)$	$\left(\mathrm{0.9,0.7,0.3}\right)$	$\left(\mathrm{0.9,0.7,0.3}\right)$	$\left(\mathrm{0.9,0.7,0.3}\right)$

Table 3. Cirq-ROF decision matrix.

	${\mathcal{Q}}_1^{\mathit{A}\mathit{T}}$	${\mathcal{Q}}_2^{\mathit{A}\mathit{T}}$	${\mathcal{Q}}_3^{\mathit{A}\mathit{T}}$	${\mathcal{Q}}_4^{\mathit{A}\mathit{T}}$
${\mathcal{Q}}_1$	$\left(\mathrm{0.9,0.5,0.7}\right)$	$\left(\mathrm{0.9,0.5,0.7}\right)$	$\left(\mathrm{0.9,0.5,0.7}\right)$	$\left(\mathrm{0.9,0.5,0.7}\right)$
${\mathcal{Q}}_2$	$\left(\mathrm{0.8,0.7,0.3}\right)$	$\left(\mathrm{0.8,0.7,0.3}\right)$	$\left(\mathrm{0.8,0.7,0.3}\right)$	$\left(\mathrm{0.8,0.7,0.3}\right)$
${\mathcal{Q}}_3$	$\left(\mathrm{0.7,0.5,0.1}\right)$	$\left(\mathrm{0.7,0.5,0.1}\right)$	$\left(\mathrm{0.7,0.5,0.1}\right)$	$\left(\mathrm{0.7,0.5,0.1}\right)$
${\mathcal{Q}}_4$	$\left(\mathrm{0.6,0.3,0.5}\right)$	$\left(\mathrm{0.6,0.3,0.5}\right)$	$\left(\mathrm{0.6,0.3,0.5}\right)$	$\left(\mathrm{0.6,0.3,0.5}\right)$
${\mathcal{Q}}_5$	$\left(\mathrm{0.5,0.3,0.4}\right)$	$\left(\mathrm{0.5,0.3,0.4}\right)$	$\left(\mathrm{0.5,0.3,0.4}\right)$	$\left(\mathrm{0.5,0.3,0.4}\right)$
${\mathcal{Q}}_6$	$\left(\mathrm{0.9,0.7,0.3}\right)$	$\left(\mathrm{0.9,0.7,0.3}\right)$	$\left(\mathrm{0.9,0.7,0.3}\right)$	$\left(\mathrm{0.9,0.7,0.3}\right)$

Table 4. Cirq-ROF aggregated matrix.

	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{A}}^{\mathit{t}\mathit{c}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{A}}^{\mathit{t}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{G}}^{\mathit{t}\mathit{c}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{G}}^{\mathit{t}}$
${\mathcal{Q}}_1$	$\left(\mathrm{0.9236,0.51,0.9493}\right)$	$\left(\mathrm{0.9236,0.51,0.5137}\right)$	$\left(\mathrm{0.91,0.9994,0.874}\right)$	$\left(\mathrm{0.91,0.9994,0.9994}\right)$
${\mathcal{Q}}_2$	$\left(\mathrm{0.9743,0.7118,0.9785}\right)$	$\left(\mathrm{0.9743,0.7118,0.7062}\right)$	$\left(\mathrm{0.8116,0.9915,0.7951}\right)$	$\left(\mathrm{0.8116,0.9915,0.992}\right)$
${\mathcal{Q}}_3$	$\left(\mathrm{0.9915,0.51,0.992}\right)$	$\left(\mathrm{0.9915,0.51,0.5097}\right)$	$\left(\mathrm{0.7116,0.9994,0.7062}\right)$	$\left(\mathrm{0.7116,0.9994,0.9994}\right)$
${\mathcal{Q}}_4$	$\left(\mathrm{0.9976,0.4087,0.9976}\right)$	$\left(\mathrm{0.9976,0.4087,0.4088}\right)$	$\left(\mathrm{0.6109,0.9999,0.6099}\right)$	$\left(\mathrm{0.6109,0.9999,0.9999}\right)$
${\mathcal{Q}}_5$	$\left(\mathrm{0.9976,0.6109,0.9976}\right)$	$\left(\mathrm{0.9976,0.6109,0.6096}\right)$	$\left(\mathrm{0.6109,0.0076,0.6096}\right)$	$\left(\mathrm{0.6109,0.0076,0.9976}\right)$
${\mathcal{Q}}_6$	$\left(\mathrm{0.9236,0.7116,0.9519}\right)$	$\left(\mathrm{0.9236,0.7116,0.7062}\right)$	$\left(\mathrm{0.91,0.9915,0.8692}\right)$	$\left(\mathrm{0.91,0.9915,0.992}\right)$

Table 4. Cirq-ROF aggregated matrix.

	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{A}}^{\mathit{t}\mathit{c}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{A}}^{\mathit{t}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{G}}^{\mathit{t}\mathit{c}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{G}}^{\mathit{t}}$
${\mathcal{Q}}_1$	$\left(\mathrm{0.9236,0.51,0.9493}\right)$	$\left(\mathrm{0.9236,0.51,0.5137}\right)$	$\left(\mathrm{0.91,0.9994,0.874}\right)$	$\left(\mathrm{0.91,0.9994,0.9994}\right)$
${\mathcal{Q}}_2$	$\left(\mathrm{0.9743,0.7118,0.9785}\right)$	$\left(\mathrm{0.9743,0.7118,0.7062}\right)$	$\left(\mathrm{0.8116,0.9915,0.7951}\right)$	$\left(\mathrm{0.8116,0.9915,0.992}\right)$
${\mathcal{Q}}_3$	$\left(\mathrm{0.9915,0.51,0.992}\right)$	$\left(\mathrm{0.9915,0.51,0.5097}\right)$	$\left(\mathrm{0.7116,0.9994,0.7062}\right)$	$\left(\mathrm{0.7116,0.9994,0.9994}\right)$
${\mathcal{Q}}_4$	$\left(\mathrm{0.9976,0.4087,0.9976}\right)$	$\left(\mathrm{0.9976,0.4087,0.4088}\right)$	$\left(\mathrm{0.6109,0.9999,0.6099}\right)$	$\left(\mathrm{0.6109,0.9999,0.9999}\right)$
${\mathcal{Q}}_5$	$\left(\mathrm{0.9976,0.6109,0.9976}\right)$	$\left(\mathrm{0.9976,0.6109,0.6096}\right)$	$\left(\mathrm{0.6109,0.0076,0.6096}\right)$	$\left(\mathrm{0.6109,0.0076,0.9976}\right)$
${\mathcal{Q}}_6$	$\left(\mathrm{0.9236,0.7116,0.9519}\right)$	$\left(\mathrm{0.9236,0.7116,0.7062}\right)$	$\left(\mathrm{0.91,0.9915,0.8692}\right)$	$\left(\mathrm{0.91,0.9915,0.992}\right)$

Step 3:

Calculate the score values based on the information in the score function, as shown in

Table 5. Representation of the score terms.

	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{A}}^{\mathit{t}\mathit{c}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{A}}^{\mathit{t}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{G}}^{\mathit{t}\mathit{c}}$	$\mathit{C}\mathit{i}\mathit{r}\mathit{q}-\mathit{R}\mathit{O}\mathit{F}\mathit{D}\mathit{W}{\mathit{G}}^{\mathit{t}}$
${\mathcal{Q}}_1$	$0.4985$	$0.2697$	$-0.459$	$-0.525$
${\mathcal{Q}}_2$	$0.73$	$0.5269$	$-0.593$	$-0.74$
${\mathcal{Q}}_3$	$0.9223$	$0.4739$	$-0.657$	$-0.929$
${\mathcal{Q}}_4$	$0.9775$	$0.4006$	$-0.598$	$-0.98$
${\mathcal{Q}}_5$	$0.9589$	$0.586$	$-0.586$	$-0.959$
${\mathcal{Q}}_6$	$0.4416$	$0.3276$	$-0.403$	$-0.46$

.

Step 4:

The score values are used to obtain the ranking results of all alternatives and evaluate the best one, as shown in

Table 6. Representation of ranking values.

Methods	Ranking Values	Most Optimal
$Cirq-ROFDW{A}^{tc}$	${\mathcal{Q}}_4>{\mathcal{Q}}_5>{\mathcal{Q}}_3>{\mathcal{Q}}_2>{\mathcal{Q}}_1>{\mathcal{Q}}_6$	${\mathcal{Q}}_4$
$Cirq-ROFDW{A}^t$	${\mathcal{Q}}_5>{\mathcal{Q}}_2>{\mathcal{Q}}_3>{\mathcal{Q}}_4>{\mathcal{Q}}_6>{\mathcal{Q}}_1$	${\mathcal{Q}}_5$
$Cirq-ROFDW{G}^{tc}$	${\mathcal{Q}}_1>{\mathcal{Q}}_6>{\mathcal{Q}}_5>{\mathcal{Q}}_2>{\mathcal{Q}}_4>{\mathcal{Q}}_3$	${\mathcal{Q}}_1$
$Cirq-ROFDW{G}^t$	${\mathcal{Q}}_6>{\mathcal{Q}}_1>{\mathcal{Q}}_2>{\mathcal{Q}}_3>{\mathcal{Q}}_5>{\mathcal{Q}}_4$	${\mathcal{Q}}_6$

.

According to the results in Table 6, the best choice is as follows: $Cirq-ROFDW{A}^{tc}$ chooses ${\mathcal{Q}}_4$, $Cirq-ROFDW{A}^t$ chooses ${\mathcal{Q}}_5$, $Cirq-ROFDW{G}^{tc}$ chooses ${\mathcal{Q}}_1,$ and $Cirq-ROFDW{G}^t$ chooses ${\mathcal{Q}}_6$.

7. Comparative Analysis

For the evaluation comparisons, in this section, we consider the data in Table 3 and try to evaluate them by using some prevailing techniques. To do so, we collected some existing techniques and tried to compare their ranking results with our initial ranking results to show the supremacy and validity of the derived theory. For comparison, we use the following existing techniques, such as that of Liu and Wang [17], who presented aggregation operators for q-ROFSs. The Dombi operators based on the PFSs were presented by Khan et al. [28]. Jana et al. [29] initiated the Dombi operators for q-ROFSs and their applications. Du [30] evaluated more about Dombi aggregation operators for q-ROFSs, and Seikh and Mandal [31] examined the Dombi operators for IFSs. Further, Akram et al. [32] proposed the fuzzy N-soft sets, Yang et al. [33] exposed the partitioned Bonferroni mean operators for complex q-rung orthopair uncertain linguistic sets, and Suri et al. [34] exposed the entropy measures for q-ROFSs. Finally, based on the data in Table 3, the comparative analysis is listed in

Table 7. Representation of comparative analysis.

Methods	Ranking Values	Best Optimal
Liu and Wang [17]	Limited features	Limited features
Khan et al. [28]	Limited features	Limited features
Jana et al. [29]	Limited features	Limited features
Du [30]	Limited features	Limited features
Seikh and Mandal [31]	Limited features	Limited features
Akram et al. [32]	Limited features	Limited features
Yang et al. [33]	Limited features	Limited features
Suri et al. [34]	Limited features	Limited features
$Cirq-ROFDW{A}^{tc}$	${\mathcal{Q}}_4>{\mathcal{Q}}_5>{\mathcal{Q}}_3>{\mathcal{Q}}_2>{\mathcal{Q}}_1>{\mathcal{Q}}_6$	${\mathcal{Q}}_4$
$Cirq-ROFDW{A}^t$	${\mathcal{Q}}_5>{\mathcal{Q}}_2>{\mathcal{Q}}_3>{\mathcal{Q}}_4>{\mathcal{Q}}_6>{\mathcal{Q}}_1$	${\mathcal{Q}}_5$
$Cirq-ROFDW{G}^{tc}$	${\mathcal{Q}}_1>{\mathcal{Q}}_6>{\mathcal{Q}}_5>{\mathcal{Q}}_2>{\mathcal{Q}}_4>{\mathcal{Q}}_3$	${\mathcal{Q}}_1$
$Cirq-ROFDW{G}^t$	${\mathcal{Q}}_6>{\mathcal{Q}}_1>{\mathcal{Q}}_2>{\mathcal{Q}}_3>{\mathcal{Q}}_5>{\mathcal{Q}}_4$	${\mathcal{Q}}_6$

.

Based on the initiated technique, we obtained the following best optimal features: ${\mathcal{Q}}_4,{\mathcal{Q}}_5,{\mathcal{Q}}_1$, and ${\mathcal{Q}}_6$. From the results in Table 7, we can see that these existing techniques have a problem with limited features, and so they failed to evaluate the data in Table 3. The weakness of the existing techniques by Liu and Wang [17], Khan et al. [28], Jana et al. [29], Du [30], Seikh and Mandal [31], Akram et al. [32], Yang et al. [33], and Suri et al. [34] are that they are operators, methods, and measures based on existing techniques of q-ROFSs without circular behavior. However, the proposed methods are computed based on a new way of Cirq-ROFSs that provides circular behavior. The advantages and disadvantages of the proposed methods and existing techniques are discussed as follows:

(1)

Seikh and Mandal [31] proposed the AOs for intuitionistic FSs (IFSs), and Khan et al. [28] proposed the AOs for Pythagorean FSs (PFSs), where the IFSs and PFSs contain the truth grade and the falsity grade but without circular behavior. However, our proposed AOs are based on Cirq-ROFSs, which are a circular extended version of IFSs and PFSs. This means that the methods of Seikh and Mandal [31] and Khan et al. [28] cannot handle the data in Table 3.

(2)

Akram et al. [32] proposed the AOs for fuzzy N-soft sets, which only contain the truth grade, so they fail to process the data in Table 3 under the Cirq-ROF environment.

(3)

Liu and Wang [17], Jana et al. [29], Du [30], Yang et al. [33], and Suri et al. [34] proposed the AOs for q-ROFSs, which are special cases of Cirq-ROFSs without circular behavior. This means that these existing methods can not deal with the Cirq-ROF data shown in Table 3.

8. Conclusions

Since circular q-rung orthopair fuzzy sets (Cirq-ROFSs) were recently proposed as an extension of FSs, IFSs, q-ROFSs, Cir-IFSs, and Cir-PFSs by Yusoff et al. [26] with no other research in the literature, we conduct an advanced study of Cirq-ROFSs with more properties. Cirq-ROFSs are good for depicting vagueness and fuzziness for life problems. In this paper, we continue working on Cirq-ROFSs by proposing Dombi aggregation operators (AOs) for Cirq-ROFSs with more operational laws. These include the Cirq-ROFDWA, Cirq-ROFDOWA, Cirq-ROFDWG, and Cirq-ROFDOWG operators. We also provide more properties, such as idempotency, monotonicity, and boundedness, for the proposed operators. We then construct the Cirq-ROF multi-attribute decision-making technique and apply it to the context of artificial intelligence in which symmetry analysis can compute the major aspect based on the proposed operators for Cirq-ROFNs. We use some existing techniques to compare our results to show the validity and supremacy of the proposed method. Up to now, no one had proposed the distances, similarity measures, or entropy based on Cirq-ROFSs. In the future, we will conduct these advanced studies on Cirq-ROFSs. We should also consider new extensions of Cirq-ROFSs to ball-extended T-spherical FSs and complex versions of them by utilizing some phase terms. Furthermore, we will discuss their applications in the field of artificial intelligence, machine learning, neural networks, green supply chain management, and decision-making to enhance the worth of the proposed theory.