# A Novel MAGDM Approach Based on Cubicq-Rung Orthopair Fuzzy Power Generalized Maclaurin Symmetric Mean Operator

Authors: Qaisar Khan; Harish Garg; Hizbullah Khattak; Abdulhakim A. Al-Babtain; I. Elbatal; Mohammed Elgarhy
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The cubicq-rung orthopair fuzzy set (Cq-ROFS) is a basic simplification of several fuzzy notions, including fuzzy set (FS), interval valued FS (INVFS), intuitionistic FS (INTFS), q-rung orthopair FS (q-ROFS), and INV q-rung orthopair FS (INVq-ROFS). By the degrees of INVq-ROFS and q-ROFS, Cq-ROFS exposes fuzzy judgement, and this is one of the advanced mathematical tools to handle more complicated assessment information in multiple attribute group decision-making (MAGDM) problems. In this article, firstly, encouraged from power average (PA) operator, generalized MSM operator, and generalized dual MSM operator, two novel aggregation operators (AOs) such as cubic q-rung orthopair fuzzy power generalized Maclaurin symmetric mean (Cq-ROFPWGMSM) operator and cubic q-rung orthopair fuzzy power generalized dual Maclaurin symmetric mean (Cq-ROFPWGDMSM) operator are initiated by merging PA operator with GMSM and GDMSM operators and some core characteristics of these AOs are investigated. Furthermore, some core cases with respect to different general parameters values are also investigated and found that some aggregation operators are special cases of the initiated AOs. Secondly, the weighted form of these newly initiated aggregation operators is also offered. Thirdly, two MAGDM models are offered based on Cq-OFPGWMSM and Cq-ROFPGWDMSM operators. Lastly, a numerical example about the selection of the best assistant professor of the year is taken as an application to show the effectiveness and capability of the initiated approaches. There is also a comparison with existing decision-making models.


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## Body

## 1. Introduction

The fuzzy set (FS) was proposed by Zadeh [1] as a means of expressing and relaying unpredictability and ambiguities. FS has caught the imagination of scholars everywhere around the globe, who might have analyzed its conceptual and technical properties since its conception. Among the most recent academic studies on the theory and applications of FSs include economic and business [2–4], genetic algorithms [5, 6], and supply chain management [7, 8]. Following the development of the concept of FS, a number of FS improvements were projected, including interval-valued FS [9], which revealed membership degree as a subset of 0,1, and Atanassov’s ITFS [10], which clarified membership degree (MED) and nonmembership degree (NOMD) as a single number in 0,1, with the total of these two degrees having to be less or equal to 1. As a result, ITFS explains uncertainty and unreliability in greater depth than FS. When the MED of such an entity is described as INVFS and FS, the situation becomes more appealing. The traditional ITFS is unable to manage such data in these conditions. Jun et al. [11] proposed the concept of cubic set (CS) to address the above-mentioned problem. The above-mentioned sets are special cases of CS. Mahmood et al. [12] established the concept of CNs and developed a set of weighted aggregation operators (AOs) to solve MADM difficulties. Fahmi et al. [13] intercepted Einstein AOs for CNs and apply these AOs to solve MADM problems under cubic information. Ayub et al. [14] have presented a set of cubic fuzzy Dombi AOs that have been implemented on Dombi t-norm and t-conorm and used to solve MADM issues in cubic fuzzy context. Kaur and Garg [15, 16] extended the idea of CS and introduced the concept of ITF cubic set, anticipated various AOs, and used these AOs to deal with MADM problems under cubic information. The shortcomings of these fuzzy structures are clear, even though research on them has been quite successful. These structures are incapable of dealing with situations in which the total of maximum lower bound membership and maximum upper bound membership and total membership and nonmembership degrees exceeds one. Abbas et al. [17] developed the notion of cubic Pythagorean fuzzy set (CPyTFS), which consists of interval valued Pythagorean FS (IVNPyTFS) [18] and Pythagorean FS (PyTFS), launched certain aggregation operators, and applied these aggregation operators to deal with MADM problems under Pythagorean cubic fuzzy information to overcome this shortcoming of these fuzzy structures. Talukdar and Dutta [19] defined some similarity measures for CPyTFS and apply them to deal with MADM problems under PyTF information. Fahmi et al. [20] defined some CPyTF linguistic Maclaurin symmetric mean (CPyTFLMSM) operators and apply them to solve MADM problems under CPyTF linguistic environment. Naeem et al. [21] redefine CPyTFMSM operators and apply them to solve MADM problems with CPyTFL information. Likewise, CITFSs and CPyTFSs are incapable of dealing with situations in which the total of squares of maximum lower bound membership and maximum upper bound membership and the total of squares of membership and nonmembership degrees exceed one. To deal with such situations, Wang et al. [22] proposed the concept of cubic q-rung orthopair fuzzy set (Cq-ROFS) which consist of q-rung orthopair fuzzy set (q-ROFS) [23] and interval valued q-ROFS [24]. They also proposed some Cq-ROF power AOs and Cq-ROF power Muirhead mean operators and apply them to deal with MAGDM problems under Cq-ROF information. Cq-ROFS is the latest generalization of several fuzzy concepts and can deal with more complex information in MAGDM process. Later, Zhang et al. [25] introduced Cq-ROF Heronian mean operator and apply them to solve MADM problems under Cq-ROF environment. Garg et al. [26] initiated the concept of Cq-ROF linguistic set, developed some basic operational laws, and based on these initiated operational laws proposed some Muirhead mean operator and apply them to deal with MADM problem with Cq-ROF information.Aggregation operators (AOs) are an important part of the multiattribute group decision-making process. The AOs can have the ability to fuse many real numbers into a single number. The qualities of various AOs varies. Some AOs, such as Yager’s PA operator [27], can remove the negative effects of unpleasant data from final ranking results and have been expanded on by many researchers from around the world to figure out how to deal with various situations, such as Xu’s [28], who provided the ITF power AOs and implemented it to MAGDM, which can minimise the effects of inaccurate data. Some AOs, such as the BM [29], HM [30], MM [31], and MSM [32], took the link between input arguments into account. BOM and HOM can consider the relationship between two input arguments, but MSM and MM may consider the relationship between any numbers of input arguments. Wei et al. [33] initiated some q-ROF MSM operators and apply these AOs to deal with MADM problems under q-ROF information. Riaz et al. [34] generalized the concept of q-ROFS and initiated the idea of linear Diophantine fuzzy set and give its applications in MADM problems. Liu and Wang [35] proposed some q-ROF aggregation operators and apply them to solve MADM problems under q-ROF information. Liu et al. [36] and Khan et al. [37] initiated power Heronian mean and power Bonferroni mean to deal with 2-tuple and interval neutrosophic information. Wang et al. [38] and Gao et al. [39] initiated MM operators for PFSNs and q-ROFNs and apply them to solve MADM problems. Recently, Khan et al. [40] initiated power Heronian mean operators for T-SPFS and apply them to solve MADM problem under TSPF environment. Riaz et al. [41] generalized the concept of q-ROFS and linear Diophantine fuzzy set and proposed the idea of spherical linear Diophantine fuzzy set and give its applications in MADM problems. Hashmi et al. [41] further generalized the concept of spherical linear Diophantine fuzzy set and introduced spherical linear Diophantine fuzzy soft rough set and give its applications in MADM problems.It has been observed from above studies that yet no one has attempted to combine PA operator with generalized Maclaurin symmetric mean operator to deal with cubicq-rung orthopair fuzzy data. Therefore, we propose the following:
(1)
The PA operator is combined with generalized MSM and generalized dual MSM operators to initiate a hybrid aggregation operators Cq-ROFPGMSM and Cq-ROFPDMSM and is considerably more adaptable and superior to the prior aggregations operators in terms of a general parameter(2)
Fortunately, there are many MAGDM difficulties in which the characteristics are linked, and many existing AOs can only alleviate such scenarios when the attributes are in the form of real integers or other fuzzy structures(3)
There are presently no AOs in place to deal with MAGDM problems arising from cubicq-rung orthopair fuzzy data which have the capacity of removing bad effects of awkward data, can consider interrelationship among any number of input arguments, and can have the general parameter that makes the decision-making process more flexible. In response to this limitation, we combined PA operator with GMSM and GDMSM operators to address MAGDM problems utilizing cubic q-rung orthopair fuzzy informationThe following are the priorities and offerings of this effort because of important impacts from earlier studies:(1)
Developing innovative Cq-ROF power GMSM operators, Cq-ROF power GDMSM operators, and their weighted form(2)
Examining the commencing AOs’ basic features and exceptional cases(3)
Expecting the deployment of two MAGDM model on these commencing AOs(4)
Assessing selection of the best assistant professor of the year to show the practicality and effectiveness of the anticipated two MAGDM models(5)
Comparison of the launched two MAGDM models with existing MAGDM modelsThis article is designed in the subsequent way to attain these objectives. Part 2 familiarizes a variability of vital notions such as Cq-ROFSs, score and accuracy functions, PA, and MSM and DMSM operators. In part 3, we look at a few AOs such as Cq-ROFPGMSM and Cq-ROFPGDMSM operator and discussed its core properties and special cases with respect to the general parameter. Part 4 introduces the Cq-ROFPWGMSM and Cq-ROFPWGDMSM operators, as well as their basic properties. In part 5, we initiate two MAGDM model based on these AOs. In part 6, a numerical example about selection of assistant professor of the year is provided to verify the unassailability and compensations of the initiated approaches. Finally, in part 7, a brief conclusion is provided.


## 2. Preliminaries

For better realizing of this paper, in this part, we will seem back few basic ideas such as definitions, operational rules, distance measure, and score functions of Cq-ROFNs, GMSM, and GDMSM operators.


### 2.1. The Cq-ROFS

Definition 1 (see [23]).
Assume that the domain of universe is represented byΛ and s∈Λ be any element. Then, a q-ROFS QR~ in the domain of universe is specified by
(1)QR~=s,me~QR~s,ne~QR~ss∈Λ,where me~QR~ and ne~QR~, respectively, express the positive grade and the negative grade of s in the q-ROFS QR~ such that 0≤me~QR~s,ne~QR~s≤1 with the axiom 0≤me~QR~sq+ne~QR~sq≤1q≥1.
The hesitancy gradeHS~QR~s of the element s in the q-ROPFS QR~ is delineated by HS~QR~s=1−me~QR~sq−ne~QR~sq, satisfying 0≤HS~QR~s≤1 and s∈Λ.
The pairme~QR~s,ne~QR~s was named q-ROFN by Liu and Wang [35] and is denoted qr~=me~,ne~ satisfying the condition 0≤me~,ne~≤1 with the axiom 0≤me~q+ne~q≤1q≥1.Definition 2 (see [24]).
Assume that the domain of universe is represented byΛ and s∈Λ be any element. Then, an IVq-ROFS IVQ~ in the domain of universe is specified by
(2)IVQ~=s,me~IVQ~s,ne~IVQ~ss∈Λ,where me~IVQ~, ne~IVQ~⊆0,1 are two interval values, respectively, expressing the positive grade and the negative grade of s in the IVq-ROFS IVQ~ such that satisfying the axiom 0≤Supme~IVQ~sq+Supne~IVQ~sq≤1q≥1.
The pairκ=me~IVQ~s,ne~IVQ~s was named IVq-ROFN by Joshi [24] satisfying the condition 0≤Supme~q+Supne~q≤1q≥1.Definition 3 (see [22]).
Assume that the domain of universe is represented byΛ and s∈Λ be any element. Then, a Cq-ROFS CQ~ in the domain of universe is specified by
(3)CQ~=s,IVQ~s,QR~ss∈Λ,where IVQ~=s,me~IVQ~s,ne~IVQ~ss∈Λ is an IVq-ROFS and QR~=s,me~QR~s,ne~QR~ss∈Λ, is a q-ROFS classified on Λ.
For simplicity, the pairCq=a,b,c,d,e,f is said to be Cq-ROFN, where a,b,c,d and e,f respectively, represent IVq-ROFN and q-ROFN.Definition 4 (see [22]).
LetCq1=a1,b1,c1,d1,e1,f1,Cq2=a2,b2,c2,d2,e2,f2 and Cq3=a3,b3,c3,d3,e3,f3 be any three Cq-ROFNs. Then, the operational laws are specified as follows:
(1)
Cq2⊕Cq3=a2q+a3q−a2qa3q1/q,b2q+b3q−b2qb3q1/q,c2c3,d2d3,e2q+e3q−e2qe3q1/q,f2f3(2)
Cq2⊗Cq3=a2a3,b2b3,c2q+c3q−c2qc3q1/q,d2q+d3q−d2qd3q1/q,e2e3,f2q+f3q−f2qf3q1/q(3)
ζCq1=1−1−a1qζ1/q,1−1−b1qζ1/q,c1ζ,d1ζ1−1−e1qζ1/q,f1ζ,ζ>0(4)
Cq1ζ=a1ζ,b1ζ,1−1−c1qζ1/q,1−1−d1qζ1/qe1ζ,1−1−f1qζ1/q,ζ>0Definition 5 (see [22]).
LetCq1=a1,b1,c1,d1,e1,f1 be a Cq-ROFN. Then, the score and accuracy functions are explained as follows:
(4)SCR~Cq1=a1q+b1q−c1q−d1q2+e1q−f1q,(5)ACR~Cq1=a1q+b1q+c1q+d1q2+e1q+f1q.Further, Wang [22] et al. presented the comparison rules based on these scores and accuracy functions, which are stated below.LetCq1=a1,b1,c1,d1,e1,f1 and Cq2=a2,b2,c2,d2,e2,f2 be two Cq-ROFNs.
(1)
IfSCR~Cq1>SCR~Cq2, then Cq1>Cq2(2)
IfSCR~Cq1=SCR~Cq2, then
(i)
IfACR~Cq1>ACR~Cq2, then Cq1>Cq2(ii)
IfACR~Cq1=ACR~Cq2, then Cq1=Cq2Now, the definition of distance measure among two Cq-ROFNs presented by Wang et al. [22] is given below.Definition 6 (see [22]).
LetCq1=a1,b1,c1,d1,e1,f1 and Cq2=a2,b2,c2,d2,e2,f2 be two Cq-ROFNs; then, the distance measure among these Cq-ROFNs is specified as follows:
(6)DE~Cq1,Cq2=16a1q−a2q+b1q−b2q+c1q−c2q+d1q−d2q+e1q−e2q+f1q−f2q.


### 2.2. PA Operator

Definition 7 (see [27]).
LetIll=1,2,⋯,p be a set of positive crisp numbers. Then, the PA operator is specified as
(7)PAI1,I2,⋯,Ip=⊕l=1p1+TIlIl⊕l=1p1+TIl,where TIl=⊕m=1,l≠mpSptIl,Im, SptIl,Im indicates the support for Il from Im executing the axioms.
(1)0≤SptIl,Im≤1, (2) SptIl,Im=SptIm,Il, and (3) SptIg,Ih≤SptIl,Im, if Il,Im≥Ig,Ih.


### 2.3. The GMSM and GDMSM Operators

Definition 8 (see [32]).
The GMSM operator is described as(8)GMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s=⊕1≤l1<l2<⋯<lk≤s⊗m=1kgh~lmγmCsk1/γ1+γ2,+⋯+γk,where γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,sl1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (8) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient.
Certainly, the GMSM operators have the following characteristics [32].
(1)
Ifgh~l=gh~l=1,2,⋯,s,GMSMk,γ1,γ2,⋯,γkgh~,gh~,⋯,gh~=gh~(2)
Ifgh~l≤ghll=1,2,⋯,s,GMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤GMSMk,γ1,γ2,⋯,γkgh1,gh2,⋯,ghs(3)
Ifminlgh~l≤GMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤maxlgh~lDefinition 9 (see [32]).
The GDMSM operator is described as(9)GDMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s=1γ1+γ2+⋯+γk⊗1≤l1<l2<⋯<lk≤s⊕m=1kγmgh~lm1/Csk,where γ1,γ2,⋯,γk≥0,k is a parameter and k=1,2,⋯,s,l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (9) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient.
Certainly, the GMSM operators have the following characteristics [32].
(1)
Ifgh~l=gh~l=1,2,⋯,s,GDMSMk,γ1,γ2,⋯,γkgh~,gh~,⋯,gh~=gh~(2)
Ifgh~l≤ghll=1,2,⋯,s,GDMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤GDMSMk,γ1,γ2,⋯,γkgh1,gh2,⋯,ghs(3)
Ifminlgh~l≤GDMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤maxlgh~l


## 2.1. The Cq-ROFS

Definition 1 (see [23]).
Assume that the domain of universe is represented byΛ and s∈Λ be any element. Then, a q-ROFS QR~ in the domain of universe is specified by
(1)QR~=s,me~QR~s,ne~QR~ss∈Λ,where me~QR~ and ne~QR~, respectively, express the positive grade and the negative grade of s in the q-ROFS QR~ such that 0≤me~QR~s,ne~QR~s≤1 with the axiom 0≤me~QR~sq+ne~QR~sq≤1q≥1.
The hesitancy gradeHS~QR~s of the element s in the q-ROPFS QR~ is delineated by HS~QR~s=1−me~QR~sq−ne~QR~sq, satisfying 0≤HS~QR~s≤1 and s∈Λ.
The pairme~QR~s,ne~QR~s was named q-ROFN by Liu and Wang [35] and is denoted qr~=me~,ne~ satisfying the condition 0≤me~,ne~≤1 with the axiom 0≤me~q+ne~q≤1q≥1.Definition 2 (see [24]).
Assume that the domain of universe is represented byΛ and s∈Λ be any element. Then, an IVq-ROFS IVQ~ in the domain of universe is specified by
(2)IVQ~=s,me~IVQ~s,ne~IVQ~ss∈Λ,where me~IVQ~, ne~IVQ~⊆0,1 are two interval values, respectively, expressing the positive grade and the negative grade of s in the IVq-ROFS IVQ~ such that satisfying the axiom 0≤Supme~IVQ~sq+Supne~IVQ~sq≤1q≥1.
The pairκ=me~IVQ~s,ne~IVQ~s was named IVq-ROFN by Joshi [24] satisfying the condition 0≤Supme~q+Supne~q≤1q≥1.Definition 3 (see [22]).
Assume that the domain of universe is represented byΛ and s∈Λ be any element. Then, a Cq-ROFS CQ~ in the domain of universe is specified by
(3)CQ~=s,IVQ~s,QR~ss∈Λ,where IVQ~=s,me~IVQ~s,ne~IVQ~ss∈Λ is an IVq-ROFS and QR~=s,me~QR~s,ne~QR~ss∈Λ, is a q-ROFS classified on Λ.
For simplicity, the pairCq=a,b,c,d,e,f is said to be Cq-ROFN, where a,b,c,d and e,f respectively, represent IVq-ROFN and q-ROFN.Definition 4 (see [22]).
LetCq1=a1,b1,c1,d1,e1,f1,Cq2=a2,b2,c2,d2,e2,f2 and Cq3=a3,b3,c3,d3,e3,f3 be any three Cq-ROFNs. Then, the operational laws are specified as follows:
(1)
Cq2⊕Cq3=a2q+a3q−a2qa3q1/q,b2q+b3q−b2qb3q1/q,c2c3,d2d3,e2q+e3q−e2qe3q1/q,f2f3(2)
Cq2⊗Cq3=a2a3,b2b3,c2q+c3q−c2qc3q1/q,d2q+d3q−d2qd3q1/q,e2e3,f2q+f3q−f2qf3q1/q(3)
ζCq1=1−1−a1qζ1/q,1−1−b1qζ1/q,c1ζ,d1ζ1−1−e1qζ1/q,f1ζ,ζ>0(4)
Cq1ζ=a1ζ,b1ζ,1−1−c1qζ1/q,1−1−d1qζ1/qe1ζ,1−1−f1qζ1/q,ζ>0Definition 5 (see [22]).
LetCq1=a1,b1,c1,d1,e1,f1 be a Cq-ROFN. Then, the score and accuracy functions are explained as follows:
(4)SCR~Cq1=a1q+b1q−c1q−d1q2+e1q−f1q,(5)ACR~Cq1=a1q+b1q+c1q+d1q2+e1q+f1q.Further, Wang [22] et al. presented the comparison rules based on these scores and accuracy functions, which are stated below.LetCq1=a1,b1,c1,d1,e1,f1 and Cq2=a2,b2,c2,d2,e2,f2 be two Cq-ROFNs.
(1)
IfSCR~Cq1>SCR~Cq2, then Cq1>Cq2(2)
IfSCR~Cq1=SCR~Cq2, then
(i)
IfACR~Cq1>ACR~Cq2, then Cq1>Cq2(ii)
IfACR~Cq1=ACR~Cq2, then Cq1=Cq2Now, the definition of distance measure among two Cq-ROFNs presented by Wang et al. [22] is given below.Definition 6 (see [22]).
LetCq1=a1,b1,c1,d1,e1,f1 and Cq2=a2,b2,c2,d2,e2,f2 be two Cq-ROFNs; then, the distance measure among these Cq-ROFNs is specified as follows:
(6)DE~Cq1,Cq2=16a1q−a2q+b1q−b2q+c1q−c2q+d1q−d2q+e1q−e2q+f1q−f2q.


## 2.2. PA Operator

Definition 7 (see [27]).
LetIll=1,2,⋯,p be a set of positive crisp numbers. Then, the PA operator is specified as
(7)PAI1,I2,⋯,Ip=⊕l=1p1+TIlIl⊕l=1p1+TIl,where TIl=⊕m=1,l≠mpSptIl,Im, SptIl,Im indicates the support for Il from Im executing the axioms.
(1)0≤SptIl,Im≤1, (2) SptIl,Im=SptIm,Il, and (3) SptIg,Ih≤SptIl,Im, if Il,Im≥Ig,Ih.


## 2.3. The GMSM and GDMSM Operators

Definition 8 (see [32]).
The GMSM operator is described as(8)GMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s=⊕1≤l1<l2<⋯<lk≤s⊗m=1kgh~lmγmCsk1/γ1+γ2,+⋯+γk,where γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,sl1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (8) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient.
Certainly, the GMSM operators have the following characteristics [32].
(1)
Ifgh~l=gh~l=1,2,⋯,s,GMSMk,γ1,γ2,⋯,γkgh~,gh~,⋯,gh~=gh~(2)
Ifgh~l≤ghll=1,2,⋯,s,GMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤GMSMk,γ1,γ2,⋯,γkgh1,gh2,⋯,ghs(3)
Ifminlgh~l≤GMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤maxlgh~lDefinition 9 (see [32]).
The GDMSM operator is described as(9)GDMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s=1γ1+γ2+⋯+γk⊗1≤l1<l2<⋯<lk≤s⊕m=1kγmgh~lm1/Csk,where γ1,γ2,⋯,γk≥0,k is a parameter and k=1,2,⋯,s,l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (9) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient.
Certainly, the GMSM operators have the following characteristics [32].
(1)
Ifgh~l=gh~l=1,2,⋯,s,GDMSMk,γ1,γ2,⋯,γkgh~,gh~,⋯,gh~=gh~(2)
Ifgh~l≤ghll=1,2,⋯,s,GDMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤GDMSMk,γ1,γ2,⋯,γkgh1,gh2,⋯,ghs(3)
Ifminlgh~l≤GDMSMk,γ1,γ2,⋯,γkgh~1,gh~2,⋯,gh~s≤maxlgh~l


## 3. The Cq-ROFPGMSM and Cq-ROFPGDMSM Operators

In this part, we propose Cq-ROFGMSM and Cq-ROFGDMSM operators by combining PA operator with GMSM and GDMSM operators. Furthermore, some key properties and special cases with respect to generalized parameters of these newly initiated AOs are investigated.


### 3.1. The Cq-ROFPGMSM Operator

In this subpart, the Cq-ROFGMSM operators are initiated; some key properties and special cases with respect to the parameters are investigated.Definition 10.
LetCq1,Cq2,⋯ and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. The Cq-ROFPGMSM operator of the Cq-ROFNs Cq1,Cq2,⋯, and Cqn is describes as
(10)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,Cqs=⊕1≤l1<l2<⋯<lk≤s⊗j=1ks1+TCqlj/⊕z=1s1+TCqzCqljγjCsk1/γ1+γ2+⋯+γk,where Üz=1+TCqz/∑z=1s1+TCqz,∑z=1sÜz=1,TCqj=∏z=1z≠jsSptCqz,Cqj, is the support degree for Cqz from Cqj, which fulfil the following conditions:
(1)
SptCqz,Cqj∈0,1(2)
SptCqz,Cqj=SptCqj,Cqz(3)
IfDE~Cqz,Cqj≤DE~Cqs,Cqt, then SptCqz,Cqj≥SptCqs,Cqt, where DE~Cqz,Cqj express the distance measure among two Cq-ROFNs described in Definition 6. γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,s, l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (10) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient
In order to write Equation (10) in an easy way, we can assume
(11)öz=1+TCqz∑z=1s1+TCqz.
Subsequently, we identifyö1,ö2,⋯,ös as the power weight vector. As a result, Equation (10) can be written in an easy way as follows:
(12)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=⊕1≤l1<l2<⋯<lk≤s⊗j=1knöljCqljγjCsk1/γ1+γ2+⋯+γk.Theorem 11.
LetCq1,Cq2,⋯, and Cqn be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. Then, the aggregated value utilizing Equation (12) is still Cq-ROFN, even
(13)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q.Proof.
Based on the operational laws of Cq-ROFNs, we have(14)söljCqlj=1−1−aljqsölj1/q,1−1−bljqsölj1/q,cljsölj,dljsölj,1−1−eljqsölj1/q,fljsölj,söljCqljγj=1−1−aljqsölj1/qγj,1−1−bljqsölj1/qγj,1−1−cljsöljqγj1/q,1−1−dljsöljqγj1/q,1−1−eljqsölj1/qγj,1−1−fljsöljqγj1/q.
Therefore,(15)∏j=1ksöljCqljγj=∏j=1k1−1−aljqsölj1/qγj,∏j=1k1−1−bljqsölj1/qγj,1−∏j=1k1−cljsöljqγj1/q,1−∏j=1k1−dljsöljqγj1/q,∏j=1k1−1−eljqsölj1/qγj,1−∏j=1k1−fljsöljqγj1/q.
Further,(16)∏1≤l1<l2<⋯<lk≤s∏j=1ksöljCqljγj=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/q.
So,(17)∏1≤l1<l2<⋯<lk≤s∏j=1ksöljCqljγjCsk=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöijqγj1/q1/Csk,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/q1/Csk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/q1/Csk.
Therefore,(18)∏11≤l1<l2<⋯<lk≤s∏j=1ksöijCqljγjCsk1/γ1+γ2+⋯+γk=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöijqγj1/Csk1/γ1+γ2+⋯+γk1/q,Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q.Example 1.
Letcq1=0.5,0.6,0.7,0.8,0.55,0.75,cq2=0.4,0.6,0.2,0.4,0.5,0.3, cq3=0.65,0.75,0.3,0.35,0.75,0.35, and cq4=0.85,0.95,0.1,0.15,0.95,0.15 be any four Cq-ROFNs. Then, by utilizing Cq‐ROFPGMSMk,γ1,γ2,⋯,γk to get the comprehensive value, the following steps should followed:
Step 1: discover the support degreesSupCqd,Cqxd,x=1,2,⋯,4 by utilizing Equation (6), and then, we can have
(19)Supcq1,cq2=Supcq2,cq1=0.7866,Supcq1,cq3=Supcq3,cq1=0.7041,Supcq1,cq4=Supcq4,cq1=0.4849,Supcq2,cq3=Supcq3,cq2=0.8718Supcq2,cq4=Supcq4,cq2=0.6641,Supcq3,cq4=Supcq4,cq3=0.7808.
Step 2: discover the power weight vectorTcqdd=1,2,3,4, by utilizing Equation (11). So we can have
(20)Tcq1=Supcq1,cq2+Supcq1,cq3+Supcq1,cq4=0.7866+0.7041+0.4849=1.9757,Tcq2=2.3225,Tcq3=2.3567,Tcq4=1.9298,ö1=1+TCq11+TCq1+1+TCq2+1+TCq2+1+TCq2=0.2365,ö2=0.2640,ö3=0.2668,ö4=0.2328.
Step 3: discover the overall Cq-ROFNcq=a,b,c,d,e,f by utilizing Equation (13); we have (assume q=3,γ1=1,γ2=2)
(21)Cq‐ROFPGMSM2,1,2Cq1,Cq2,Cq3,Cq4=1−∏1≤l1<l2≤41−∏j=121−1−alj34öljγj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−blj34öljγj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−clj4ölj3γj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−dlj4ölj3γj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−elj34öljγj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−flj4ölj3γj1/61/31/3.
Hence, we can have the overall Cq-ROFNCq=0.7625,0.8426,0.3153,0.4136,0.8213,0.3771.Theorem 12 (idempotency).
LetCq=a,b,c,d,e,f and Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be two sets of Cq-ROFNs. When Cq=Cqii=1,2,⋯,n, we have
(22)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=Cq.Proof.
WhenCq=Cqll=1,2,⋯,s, we have s1+TCqlj/∑z=1s1+TCqz=s1+TCq/∑z=1s1+TCq=1. So, from Theorem 11, we have
(23)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöijqγj1/Csk1/γ1+γ2+⋯+γk1/q=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1/Csk1/γ1+γ2+⋯+γk1/q=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1/Csk1/γ1+γ2+⋯+γk1/q,=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1kaq×γj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1kbq×γj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1keq×γj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1/Csk1/γ1+γ2+⋯+γk1/q=1−∏1≤l1<l2<⋯<lk≤s1−aq×γ1,γ2,⋯,γk1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−bq×γ1,γ2,⋯,γk1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−1−cqγ1,γ2,⋯,γk1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−1−dqγ1,γ2,⋯,γk1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−eq×γ1,γ2,⋯,γk1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−1−fqγ1,γ2,⋯,γk1/Csk1/γ1+γ2+⋯+γk1/q=1−1−aq×γ1,γ2,⋯,γk1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1,γ2,⋯,γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,1−1−eq×γ1,γ2,⋯,γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q=aγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk,bγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk,1−1−cqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,1−1−dqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,eγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk,1−1−fqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q=a,b,c,d,e,f=Cq.Theorem 13 (boundedness).
LetCql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be a set of Cq-ROFNs, χ=minCq1,Cq2,⋯,Cqs and α=maxCq1,Cq2,⋯,Cqs, then
(24)χ≤Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.Proof.
Since,χ≤Cql≤α, then
(25)∑1≤l1<l2<⋯<lk≤s∏j=1ksÜljχljγjCsk1/γ1+γ2+⋯+γk≤∑1≤l1<l2<⋯<lk≤s∏j=1ksöljCqljγjCsk1/γ1+γ2+⋯+γk≤∑1≤l1<l2<⋯<lk≤s∏j=1ksöljαljγjCsk1/γ1+γ2+⋯+γk.
Therefore,(26)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkχ1,χ2,⋯,χs≤Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤Cq‐ROFPGMSMk,γ1,γ2,⋯,γkα1,α2,⋯,αs.
Hence,(27)χ≤Cq‐ROFPGHAMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.
The Cq-ROFPGMSM operator does not satisfy the property of monotonicity.
In addition, we can acquire some novel aggregation operators by giving distinct values for the parameters as follows:(1)
Whenk=2 and γ1=γ2=p, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power Bonferroni mean (Cq-ROFPBM) operator (when p=r), as shown below(28)Cq−ROFPGMSM2,p,pCq1,Cq2,⋯,Cqs=∑1≤l1<l2≤s∏j=12söljCqljpCs21/2p=∑1≤l1<l2≤ssöl1Cql1p⊗söl2Cql2pss−1/21/2p2∑1≤l1<l2≤nsölCqlp⊗söjCqjpss−11/2p=1ss−1∑l,j=1l≠jssölCqlp⊗söjCqjp1/2p=Cq−ROFPBMp,pCq1,Cq2,⋯,Cqs=1−∏l=j=1l≠js1−1−1−alqsölp1−1−ajqsöjp1/ss−11/q1/2p,1−∏l=j=1l≠js1−1−1−blqsölp1−1−bjqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−clqsölp1−cjqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−dlqsölp1−djqsölp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−elqsölp1−1−ejqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−flqsöip1−fjqsöjp1/ss−11/2p1/q=1−∏l=j=1l≠js1−1−1−alqsölp1−1−ajqsöjp1/ss−11/q1/2p,1−∏l=j=1l≠js1−1−1−blqsölp1−1−bjqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−clqsölp1−cjqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−dlqsölp1−djqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−clqsölp1−cjqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−dlqsölp1−djqsölp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−elqsölp1−1−ejqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−flqsöip1−fjqsöjp1/ss−11/2p1/q.(2)
Whenk=3 and γ1=γ2=γ3=p, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power generalized Bonferroni mean (Cq-ROFPGBM) operator (when p=r=s), as shown below(29)Cq−ROFPGMSM2,p,pCq1,Cq2,⋯,Cqs=∑1≤l1<l2<l3≤s∏j=13söljCqljpCs31/3p=1/3!∑1,l2,l3=1l1≠l2≠l3ssöl1Cql1p⊗söl2Cql2p⊗söl3Cql3pss−1s−2/3!1/3p=∑i,j,l=1i≠j≠lssöiCqip⊗söjCqjp⊗sölCqlpss−1s−21/2p=1ss−1s−2∑i,j,l=1i≠j≠lssöiCqip⊗söjCqjp⊗sölCqlp1/2p=Cq−ROFPGBMp,p,pCq1,Cq2,⋯,Cqs=1−∏=j=l=1i≠j≠ls1−1−1−aiqsöip1−1−ajqsöjp1−1−alqsölp1/ss−1s−21/q1/3p,1−∏=j=l=1i≠j≠ls1−1−1−biqsöip1−1−bjqsöjp1−1−blqsölp1/ss−1s−21/q1/3p,1−1−∏=j=l=1i≠j≠ls1−1−ciqsöip1−cjqsöjp1−clqsölp1/ss−1s−21/3p1/q,1−1−∏=j=l=1i≠j≠ls1−1−diqsöip1−djqsöjp1−dlqsölp1/ss−1s−21/3p1/q,1−∏=j=l=1i≠j≠ls1−1−1−eiqsöip1−1−ejqsöjp1−1−elqsöjp1/ss−1s−21/q1/3p,1−1−∏=j=l=1i≠j≠ls1−1−fiqsöip1−fjqsöip1−flqsölp1/ss−1s−21/3p1/q.(3)
Whenγ1=γ2=⋯=γn=1, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power MSM (Cq-ROFPMSM) operator, as shown below.(30)Cq−ROFPGMSMk,1,,.⋯,1Cq1,Cq2,⋯,Cqs=∑1≤l1<l2<l3≤s∏j=1ksöljCqlj1Csk1/1+1+⋯+1=∑1≤l1<l2<l3≤s∏j=1ksöljCqljCs31/k=Cq−ROFPMSMkCq1,Cq2,⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1s1−1−aljqsölj1/Csk1/q1/k,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1s1−1−bljqsölj1/Csk1/q1/k,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöijq1/Csk1/k1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöijq1/Csk1/k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1s1−1−eljqsölj1/Csk1/q1/k,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöijq1/Csk1/k1/q.(4)
Whenγ1=γ2=⋯=γn=1/k, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power HAM (Cq-ROFPHAM) operator, as shown below.(31)Cq−ROFPGMSMk,1/k,1/k,.⋯,1/kCq1,Cq2,⋯,Cqs=⊕1≤l1<l2<⋯<lk≤s⊗j=1ksöljCqlj1/kCsk1/1/k+1/k+⋯+1/k=⊕1≤l1<l2<⋯<lk≤s⊗j=1ksöljCqlj1/kCs31=⊕1≤l1<l2<⋯<lk≤s⊗j=1ksöljCqlj1/kCs3=Cq−ROFPHAMkCq1,Cq2,⋯,Cqs=1−∑11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsÜlj1/k1/Csk1/q,1−∑11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsölj1/k1/Csk1/q,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöilq1/k1/q1/Csk,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöijq1/k1/q1/Csk,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöij1/k1/Csk1/q,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljq1/k1/q1/Csk.
In the next subpart, we present a dual form of the Cq-ROFPGMSM operators based on the Cq-ROFNs and dual MSM operator.


### 3.2. The Cq-ROFPGDMSM Operator

Definition 14.
LetCq1,Cq2,⋯, and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s,q≥1. Then, the dual form of the Cq-ROFPDMSM operator is described as follows:
(32)Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljs1+TCqlj/∑z=1s1+TCqz1/Csk,
whereŁz=1+TCqz/∑z=1s1+TCqz,∑z=1sŁz=1,TCqj=∏z=1z≠jsSptCqz,Cqj is the support degree for Cqz from Cqj, which fulfil the following conditions:
(1)
SptCqz,Cqj∈0,1(2)
SptCqz,Cqj=SptCqj,Cqz(3)
IfDE~Cqz,Cqj≤DE~Cqs,Cqt, then SptCqz,Cqj≥SptCqs,Cqt, where DE~Cqz,Cqjexpress the distance measure among two Cq-ROFNs described in Definition 6. γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,n, l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,n. The denominator Cnk in Equation (32) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient
In order to write Equation (32) in an easy way, we can assume
(33)Łz=1+TCqz∑z=1s1+TCqz.
Subsequently, we identifyÕ1,Õ2,⋯,Õn as the power weight vector. As a result, Equation (32) can be written in an easy way as follows:
(34)Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljsŁlj1/Csk.Theorem 15.
LetCq1,Cq2,⋯, and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. Then, the aggregated value utilizing Equation (34) is still Cq-ROFN, even
(35)Cq−ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk.Proof.
Since(36)CqljsŁlj=aljsŁlj,bljsŁlj,1−1−cljqsŁlj1/q,1−1−dljqsŁlj1/q,eljsŁlj,1−1−fljqsŁlj1/q,γjCqljsŁlj=1−1−aljsŁljqγj1/q,1−1−bljsŁljqγj1/q,1−1−cljqsŁlj1/qγj,1−1−dljqsŁlj1/qγj,1−1−eljsŁljqγj1/q,1−1−fljqsŁlj1/qγj.
Therefore,(37)∑j=1kγjCqljsŁlj=1−∏j=1k1−aljsŁljqγj1/q,1−∏j=1k1−bljsŁljqγj1/q,∏j=1k1−1−cljqsŁlj1/qγj,∏j=1k1−1−dljqsŁlj1/qγj,1−∏j=1k1−eljsŁljqγj1/q,∏j=1k1−1−fljqsŁlj1/qγj.
Further,(38)∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljsŁlj=∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/q.
So,(39)∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljsŁlj1/Csk=∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/q1/Csk,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/q1/Csk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/q1/Csk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fijqnŁljγj1/Cnk1/q.
Therefore,(40)1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∏j=1kγjCqljsŁlj1/Csk=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,(41)Cq−ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk.Example 2.
Consider the four Cq-ROFNs given in Example1. Then, by utilizing Cq‐ROFPGDMSMk,γ1,γ2,⋯,γk to get the comprehensive value. The following steps should be followed.
Steps 1 and 2 are the same as for Example1.
Step 3: discover the overall Cq-ROFNcq=a,b,c,d,e,f by utilizing Equation (35); we have (assume q=3,γ1=1,γ2=2)
(42)Cq−ROFPGDMSM2,1,2Cq1,Cq2,Cq3,Cq4=1−1−∏1≤l1<l2≤41−∏j=121−alj4Łlj3γj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−blj4Łlj3γj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−clj34Łljγj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−dlj34Łljγj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−elj4Łlj3γj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−flj34Łljγj1/61/31/3.
Hence, we can have the overall Cq-ROFNCq=0.6326,0.7426,0.5449,0.6475,0.7119,0.6048.
Now, we will discuss some basic properties of the Cq-ROFPGDMSM operators.Theorem 16 (idempotency).
LetCq=a,b,c,d,e,f and Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be two sets of Cq-ROFNs. When Cq=Cqll=1,2,⋯,s, we have
(43)Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=Cq.Proof.
Proof of Theorem16 is the same as that of Theorem 12.Theorem 17 (boundedness).
LetCql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be a set of Cq-ROFNs, χ=minCq1,Cq2,⋯,Cqs and α=maxCq1,Cq2,⋯,Cqs, then
(44)χ≤Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.Proof.
Proof of Theorem17 is the same as that of Theorem 13.
In addition, we can acquire some novel aggregation operators by giving distinct values for the parameters as follows:(1)
Whenk=2 and γ1=γ2=p, the developed Cq-ROFPGDMSM operator degenerates into the cubic q-rung orthopair fuzzy power geometric Bonferroni mean (Cq-ROFPGBM) operator (when p=r), as shown below(45)Cq−ROFPGDMSM2,p,pCq1,Cq2,⋯,Cqn=12p∏1≤l1<l2≤s∑j=1kpCqljsŁlj1/Cs2=12p∏1≤l1<l2≤spCql1sŁl1⊕pCql2sŁl21/ss−1=12p∏l,j=1l≠jspCqlsŁl⊕pCqjsŁj1/ss−1=Cq−ROFPGBMp,pCq1,Cq2,⋯,Cqs=1−1−∏l=j=1l≠js1−1−alqsŁlp1−ajqsŁjp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−blqsŁlp1−bjqsŁjp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−clqsŁlp1−1−cjqsŁjp1/ss−11/q1/2p,1−∏l=j=1l≠js1−1−1−dlqsŁlp1−1−djqsŁjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−elqsŁlp1−ejqsŁjp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−flqsŁlp1−1−fjqsŁjp1/ss−11/q1/2p.(2)
Whenk=3 and γ1=γ2=γ3=p, the developed Cq-ROFPGDMSM operator degenerates into the cubic q-rung orthopair fuzzy power generalized geometric Bonferroni mean (Cq-ROFPGGBM) operator when (p=r=s), as shown below(46)Cq−ROFPGDMSM3,p,p,pCq1,Cq2,⋯,Cqs=13p∏1≤l1<l2<l3≤s∑j=13pCqljsŁlj1/Cn3=13p⊗l1,l2,l3=1l1≠l2≠l3spCql1sŁl1⊕pCql2sŁl2⊕Cql3sŁl31/3!3!/ss−1s−2=13p⊗l,m,n=1l≠m≠nnpCqlsŁl⊗pCqmsŁj⊗pCqnsŁn1/ss−1s−2=Cq−ROFPGGBMp,p,pCq1,Cq2,⋯,Cqs=1−1−∏l=m=n=1,l≠m≠ns1−1−alqsŁlp1−amqsŁmp1−anqsŁnp1/ss−1s−21/3p1/q,1−1−∏l=m=n=1,l≠m≠ns1−1−alqsÕlp1−bmqsŁmp1−bnqsŁnp1/ss−1s−21/3p1/q,1−∏l=m=n=1,l≠m≠ns1−1−1−clqsŁip1−1−cmqmŁjp1−1−cnqsŁlp1/ss−1s−21/q1/3p,1−∏l=m=n=1,l≠m≠ns1−1−1−dlqsŁip1−1−dmqmŁjp1−1−dnqsŁlp1/ss−1s−21/q1/3p,1−1−∏l=m=n=1,l≠m≠ns1−1−elqsŁlp1−emqsŁmp1−enqsŁnp1/ss−1s−21/3p1/q,1−∏l=m=n=1,l≠m≠ns1−1−1−flqsŁip1−1−fmqmŁjp1−1−fnqsŁlp1/ss−1s−21/q1/3p.(3)
Whenγ1=γ2=⋯=γn=1, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power MSM (Cq-ROFPDMSM) operator, as shown below.(47)Cq−ROFPGDMSMk,1,,.⋯,1Cq1,Cq2,⋯,Cqs=11+1+⋯+1∏1≤l1<l2<l3≤s∑j=1k1CqljsŁlj1/Csk=1k∏1≤l1<l2<l3≤s∑j=1kCqljsŁlj1/Cs3=Cq−ROFPDMSMkCq1,Cq2,⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljq1/Csk1/k1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljq1/Csk1/k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁll1/Csk1/q1/k,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁlj1/Csk1/q1/k,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljq1/Csk1/k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁlj1/Csk1/q1/k.(4)
Whenγ1=γ2=⋯=γn=1/k, the developed Cq-ROFPGDMSM operator degenerates into the cubic q-rung orthopair fuzzy power HAM (Cq-ROFPDHAM) operator, as shown below.(48)Cq−ROFPGDMSMk,1/k,1/k,.⋯,1/kCq1,Cq2,⋯,Cqn=11/k+1/k+⋯+1/k∏1≤l1<l2<⋯<lk≤s∑j=1k1kCqljsŁlj1/Csk=∏1≤l1<l2<⋯<lk≤s∑j=1k1kCqljsŁlj1/Csk=Cq−ROFPDHAMkCq1,Cq2,⋯,Cqs=∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljq1/k1/q1/Csk,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljq1/k1/q1/Csk,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁlj1/k1/Csk1/q,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁlj1/k1/Csk1/q,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljq1/k1/q1/Csk,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁlj1/k1/Csk1/q.


## 3.1. The Cq-ROFPGMSM Operator

In this subpart, the Cq-ROFGMSM operators are initiated; some key properties and special cases with respect to the parameters are investigated.Definition 10.
LetCq1,Cq2,⋯ and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. The Cq-ROFPGMSM operator of the Cq-ROFNs Cq1,Cq2,⋯, and Cqn is describes as
(10)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,Cqs=⊕1≤l1<l2<⋯<lk≤s⊗j=1ks1+TCqlj/⊕z=1s1+TCqzCqljγjCsk1/γ1+γ2+⋯+γk,where Üz=1+TCqz/∑z=1s1+TCqz,∑z=1sÜz=1,TCqj=∏z=1z≠jsSptCqz,Cqj, is the support degree for Cqz from Cqj, which fulfil the following conditions:
(1)
SptCqz,Cqj∈0,1(2)
SptCqz,Cqj=SptCqj,Cqz(3)
IfDE~Cqz,Cqj≤DE~Cqs,Cqt, then SptCqz,Cqj≥SptCqs,Cqt, where DE~Cqz,Cqj express the distance measure among two Cq-ROFNs described in Definition 6. γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,s, l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (10) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient
In order to write Equation (10) in an easy way, we can assume
(11)öz=1+TCqz∑z=1s1+TCqz.
Subsequently, we identifyö1,ö2,⋯,ös as the power weight vector. As a result, Equation (10) can be written in an easy way as follows:
(12)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=⊕1≤l1<l2<⋯<lk≤s⊗j=1knöljCqljγjCsk1/γ1+γ2+⋯+γk.Theorem 11.
LetCq1,Cq2,⋯, and Cqn be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. Then, the aggregated value utilizing Equation (12) is still Cq-ROFN, even
(13)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q.Proof.
Based on the operational laws of Cq-ROFNs, we have(14)söljCqlj=1−1−aljqsölj1/q,1−1−bljqsölj1/q,cljsölj,dljsölj,1−1−eljqsölj1/q,fljsölj,söljCqljγj=1−1−aljqsölj1/qγj,1−1−bljqsölj1/qγj,1−1−cljsöljqγj1/q,1−1−dljsöljqγj1/q,1−1−eljqsölj1/qγj,1−1−fljsöljqγj1/q.
Therefore,(15)∏j=1ksöljCqljγj=∏j=1k1−1−aljqsölj1/qγj,∏j=1k1−1−bljqsölj1/qγj,1−∏j=1k1−cljsöljqγj1/q,1−∏j=1k1−dljsöljqγj1/q,∏j=1k1−1−eljqsölj1/qγj,1−∏j=1k1−fljsöljqγj1/q.
Further,(16)∏1≤l1<l2<⋯<lk≤s∏j=1ksöljCqljγj=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/q.
So,(17)∏1≤l1<l2<⋯<lk≤s∏j=1ksöljCqljγjCsk=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöijqγj1/q1/Csk,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/q1/Csk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/q1/Csk.
Therefore,(18)∏11≤l1<l2<⋯<lk≤s∏j=1ksöijCqljγjCsk1/γ1+γ2+⋯+γk=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöijqγj1/Csk1/γ1+γ2+⋯+γk1/q,Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q.Example 1.
Letcq1=0.5,0.6,0.7,0.8,0.55,0.75,cq2=0.4,0.6,0.2,0.4,0.5,0.3, cq3=0.65,0.75,0.3,0.35,0.75,0.35, and cq4=0.85,0.95,0.1,0.15,0.95,0.15 be any four Cq-ROFNs. Then, by utilizing Cq‐ROFPGMSMk,γ1,γ2,⋯,γk to get the comprehensive value, the following steps should followed:
Step 1: discover the support degreesSupCqd,Cqxd,x=1,2,⋯,4 by utilizing Equation (6), and then, we can have
(19)Supcq1,cq2=Supcq2,cq1=0.7866,Supcq1,cq3=Supcq3,cq1=0.7041,Supcq1,cq4=Supcq4,cq1=0.4849,Supcq2,cq3=Supcq3,cq2=0.8718Supcq2,cq4=Supcq4,cq2=0.6641,Supcq3,cq4=Supcq4,cq3=0.7808.
Step 2: discover the power weight vectorTcqdd=1,2,3,4, by utilizing Equation (11). So we can have
(20)Tcq1=Supcq1,cq2+Supcq1,cq3+Supcq1,cq4=0.7866+0.7041+0.4849=1.9757,Tcq2=2.3225,Tcq3=2.3567,Tcq4=1.9298,ö1=1+TCq11+TCq1+1+TCq2+1+TCq2+1+TCq2=0.2365,ö2=0.2640,ö3=0.2668,ö4=0.2328.
Step 3: discover the overall Cq-ROFNcq=a,b,c,d,e,f by utilizing Equation (13); we have (assume q=3,γ1=1,γ2=2)
(21)Cq‐ROFPGMSM2,1,2Cq1,Cq2,Cq3,Cq4=1−∏1≤l1<l2≤41−∏j=121−1−alj34öljγj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−blj34öljγj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−clj4ölj3γj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−dlj4ölj3γj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−elj34öljγj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−flj4ölj3γj1/61/31/3.
Hence, we can have the overall Cq-ROFNCq=0.7625,0.8426,0.3153,0.4136,0.8213,0.3771.Theorem 12 (idempotency).
LetCq=a,b,c,d,e,f and Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be two sets of Cq-ROFNs. When Cq=Cqii=1,2,⋯,n, we have
(22)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=Cq.Proof.
WhenCq=Cqll=1,2,⋯,s, we have s1+TCqlj/∑z=1s1+TCqz=s1+TCq/∑z=1s1+TCq=1. So, from Theorem 11, we have
(23)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöijqγj1/Csk1/γ1+γ2+⋯+γk1/q=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1/Csk1/γ1+γ2+⋯+γk1/q=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk=1−∏11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eqγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1/Csk1/γ1+γ2+⋯+γk1/q,=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1kaq×γj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1kbq×γj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1keq×γj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1/Csk1/γ1+γ2+⋯+γk1/q=1−∏1≤l1<l2<⋯<lk≤s1−aq×γ1,γ2,⋯,γk1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−bq×γ1,γ2,⋯,γk1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−1−cqγ1,γ2,⋯,γk1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−1−dqγ1,γ2,⋯,γk1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−eq×γ1,γ2,⋯,γk1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−1−fqγ1,γ2,⋯,γk1/Csk1/γ1+γ2+⋯+γk1/q=1−1−aq×γ1,γ2,⋯,γk1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1,γ2,⋯,γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,1−1−eq×γ1,γ2,⋯,γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q=aγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk,bγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk,1−1−cqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,1−1−dqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q,eγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk,1−1−fqγ1,γ2,⋯,γk1/γ1+γ2+⋯+γk1/q=a,b,c,d,e,f=Cq.Theorem 13 (boundedness).
LetCql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be a set of Cq-ROFNs, χ=minCq1,Cq2,⋯,Cqs and α=maxCq1,Cq2,⋯,Cqs, then
(24)χ≤Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.Proof.
Since,χ≤Cql≤α, then
(25)∑1≤l1<l2<⋯<lk≤s∏j=1ksÜljχljγjCsk1/γ1+γ2+⋯+γk≤∑1≤l1<l2<⋯<lk≤s∏j=1ksöljCqljγjCsk1/γ1+γ2+⋯+γk≤∑1≤l1<l2<⋯<lk≤s∏j=1ksöljαljγjCsk1/γ1+γ2+⋯+γk.
Therefore,(26)Cq‐ROFPGMSMk,γ1,γ2,⋯,γkχ1,χ2,⋯,χs≤Cq‐ROFPGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤Cq‐ROFPGMSMk,γ1,γ2,⋯,γkα1,α2,⋯,αs.
Hence,(27)χ≤Cq‐ROFPGHAMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.
The Cq-ROFPGMSM operator does not satisfy the property of monotonicity.
In addition, we can acquire some novel aggregation operators by giving distinct values for the parameters as follows:(1)
Whenk=2 and γ1=γ2=p, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power Bonferroni mean (Cq-ROFPBM) operator (when p=r), as shown below(28)Cq−ROFPGMSM2,p,pCq1,Cq2,⋯,Cqs=∑1≤l1<l2≤s∏j=12söljCqljpCs21/2p=∑1≤l1<l2≤ssöl1Cql1p⊗söl2Cql2pss−1/21/2p2∑1≤l1<l2≤nsölCqlp⊗söjCqjpss−11/2p=1ss−1∑l,j=1l≠jssölCqlp⊗söjCqjp1/2p=Cq−ROFPBMp,pCq1,Cq2,⋯,Cqs=1−∏l=j=1l≠js1−1−1−alqsölp1−1−ajqsöjp1/ss−11/q1/2p,1−∏l=j=1l≠js1−1−1−blqsölp1−1−bjqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−clqsölp1−cjqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−dlqsölp1−djqsölp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−elqsölp1−1−ejqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−flqsöip1−fjqsöjp1/ss−11/2p1/q=1−∏l=j=1l≠js1−1−1−alqsölp1−1−ajqsöjp1/ss−11/q1/2p,1−∏l=j=1l≠js1−1−1−blqsölp1−1−bjqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−clqsölp1−cjqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−dlqsölp1−djqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−clqsölp1−cjqsölp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−dlqsölp1−djqsölp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−elqsölp1−1−ejqsöjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−flqsöip1−fjqsöjp1/ss−11/2p1/q.(2)
Whenk=3 and γ1=γ2=γ3=p, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power generalized Bonferroni mean (Cq-ROFPGBM) operator (when p=r=s), as shown below(29)Cq−ROFPGMSM2,p,pCq1,Cq2,⋯,Cqs=∑1≤l1<l2<l3≤s∏j=13söljCqljpCs31/3p=1/3!∑1,l2,l3=1l1≠l2≠l3ssöl1Cql1p⊗söl2Cql2p⊗söl3Cql3pss−1s−2/3!1/3p=∑i,j,l=1i≠j≠lssöiCqip⊗söjCqjp⊗sölCqlpss−1s−21/2p=1ss−1s−2∑i,j,l=1i≠j≠lssöiCqip⊗söjCqjp⊗sölCqlp1/2p=Cq−ROFPGBMp,p,pCq1,Cq2,⋯,Cqs=1−∏=j=l=1i≠j≠ls1−1−1−aiqsöip1−1−ajqsöjp1−1−alqsölp1/ss−1s−21/q1/3p,1−∏=j=l=1i≠j≠ls1−1−1−biqsöip1−1−bjqsöjp1−1−blqsölp1/ss−1s−21/q1/3p,1−1−∏=j=l=1i≠j≠ls1−1−ciqsöip1−cjqsöjp1−clqsölp1/ss−1s−21/3p1/q,1−1−∏=j=l=1i≠j≠ls1−1−diqsöip1−djqsöjp1−dlqsölp1/ss−1s−21/3p1/q,1−∏=j=l=1i≠j≠ls1−1−1−eiqsöip1−1−ejqsöjp1−1−elqsöjp1/ss−1s−21/q1/3p,1−1−∏=j=l=1i≠j≠ls1−1−fiqsöip1−fjqsöip1−flqsölp1/ss−1s−21/3p1/q.(3)
Whenγ1=γ2=⋯=γn=1, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power MSM (Cq-ROFPMSM) operator, as shown below.(30)Cq−ROFPGMSMk,1,,.⋯,1Cq1,Cq2,⋯,Cqs=∑1≤l1<l2<l3≤s∏j=1ksöljCqlj1Csk1/1+1+⋯+1=∑1≤l1<l2<l3≤s∏j=1ksöljCqljCs31/k=Cq−ROFPMSMkCq1,Cq2,⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1s1−1−aljqsölj1/Csk1/q1/k,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1s1−1−bljqsölj1/Csk1/q1/k,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöijq1/Csk1/k1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöijq1/Csk1/k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1s1−1−eljqsölj1/Csk1/q1/k,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöijq1/Csk1/k1/q.(4)
Whenγ1=γ2=⋯=γn=1/k, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power HAM (Cq-ROFPHAM) operator, as shown below.(31)Cq−ROFPGMSMk,1/k,1/k,.⋯,1/kCq1,Cq2,⋯,Cqs=⊕1≤l1<l2<⋯<lk≤s⊗j=1ksöljCqlj1/kCsk1/1/k+1/k+⋯+1/k=⊕1≤l1<l2<⋯<lk≤s⊗j=1ksöljCqlj1/kCs31=⊕1≤l1<l2<⋯<lk≤s⊗j=1ksöljCqlj1/kCs3=Cq−ROFPHAMkCq1,Cq2,⋯,Cqs=1−∑11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsÜlj1/k1/Csk1/q,1−∑11≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsölj1/k1/Csk1/q,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsöilq1/k1/q1/Csk,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsöijq1/k1/q1/Csk,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsöij1/k1/Csk1/q,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsöljq1/k1/q1/Csk.
In the next subpart, we present a dual form of the Cq-ROFPGMSM operators based on the Cq-ROFNs and dual MSM operator.


## 3.2. The Cq-ROFPGDMSM Operator

Definition 14.
LetCq1,Cq2,⋯, and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s,q≥1. Then, the dual form of the Cq-ROFPDMSM operator is described as follows:
(32)Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljs1+TCqlj/∑z=1s1+TCqz1/Csk,
whereŁz=1+TCqz/∑z=1s1+TCqz,∑z=1sŁz=1,TCqj=∏z=1z≠jsSptCqz,Cqj is the support degree for Cqz from Cqj, which fulfil the following conditions:
(1)
SptCqz,Cqj∈0,1(2)
SptCqz,Cqj=SptCqj,Cqz(3)
IfDE~Cqz,Cqj≤DE~Cqs,Cqt, then SptCqz,Cqj≥SptCqs,Cqt, where DE~Cqz,Cqjexpress the distance measure among two Cq-ROFNs described in Definition 6. γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,n, l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,n. The denominator Cnk in Equation (32) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficient
In order to write Equation (32) in an easy way, we can assume
(33)Łz=1+TCqz∑z=1s1+TCqz.
Subsequently, we identifyÕ1,Õ2,⋯,Õn as the power weight vector. As a result, Equation (32) can be written in an easy way as follows:
(34)Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljsŁlj1/Csk.Theorem 15.
LetCq1,Cq2,⋯, and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. Then, the aggregated value utilizing Equation (34) is still Cq-ROFN, even
(35)Cq−ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk.Proof.
Since(36)CqljsŁlj=aljsŁlj,bljsŁlj,1−1−cljqsŁlj1/q,1−1−dljqsŁlj1/q,eljsŁlj,1−1−fljqsŁlj1/q,γjCqljsŁlj=1−1−aljsŁljqγj1/q,1−1−bljsŁljqγj1/q,1−1−cljqsŁlj1/qγj,1−1−dljqsŁlj1/qγj,1−1−eljsŁljqγj1/q,1−1−fljqsŁlj1/qγj.
Therefore,(37)∑j=1kγjCqljsŁlj=1−∏j=1k1−aljsŁljqγj1/q,1−∏j=1k1−bljsŁljqγj1/q,∏j=1k1−1−cljqsŁlj1/qγj,∏j=1k1−1−dljqsŁlj1/qγj,1−∏j=1k1−eljsŁljqγj1/q,∏j=1k1−1−fljqsŁlj1/qγj.
Further,(38)∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljsŁlj=∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/q.
So,(39)∏1≤l1<l2<⋯<lk≤s∑j=1kγjCqljsŁlj1/Csk=∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/q1/Csk,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/q1/Csk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/q1/Csk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fijqnŁljγj1/Cnk1/q.
Therefore,(40)1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∏j=1kγjCqljsŁlj1/Csk=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,(41)Cq−ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,.⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1/Csk1/q1/γ1+γ2+⋯+γk.Example 2.
Consider the four Cq-ROFNs given in Example1. Then, by utilizing Cq‐ROFPGDMSMk,γ1,γ2,⋯,γk to get the comprehensive value. The following steps should be followed.
Steps 1 and 2 are the same as for Example1.
Step 3: discover the overall Cq-ROFNcq=a,b,c,d,e,f by utilizing Equation (35); we have (assume q=3,γ1=1,γ2=2)
(42)Cq−ROFPGDMSM2,1,2Cq1,Cq2,Cq3,Cq4=1−1−∏1≤l1<l2≤41−∏j=121−alj4Łlj3γj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−blj4Łlj3γj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−clj34Łljγj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−dlj34Łljγj1/61/31/3,1−1−∏1≤l1<l2≤41−∏j=121−elj4Łlj3γj1/61/31/3,1−∏1≤l1<l2≤41−∏j=121−1−flj34Łljγj1/61/31/3.
Hence, we can have the overall Cq-ROFNCq=0.6326,0.7426,0.5449,0.6475,0.7119,0.6048.
Now, we will discuss some basic properties of the Cq-ROFPGDMSM operators.Theorem 16 (idempotency).
LetCq=a,b,c,d,e,f and Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be two sets of Cq-ROFNs. When Cq=Cqll=1,2,⋯,s, we have
(43)Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=Cq.Proof.
Proof of Theorem16 is the same as that of Theorem 12.Theorem 17 (boundedness).
LetCql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be a set of Cq-ROFNs, χ=minCq1,Cq2,⋯,Cqs and α=maxCq1,Cq2,⋯,Cqs, then
(44)χ≤Cq‐ROFPGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.Proof.
Proof of Theorem17 is the same as that of Theorem 13.
In addition, we can acquire some novel aggregation operators by giving distinct values for the parameters as follows:(1)
Whenk=2 and γ1=γ2=p, the developed Cq-ROFPGDMSM operator degenerates into the cubic q-rung orthopair fuzzy power geometric Bonferroni mean (Cq-ROFPGBM) operator (when p=r), as shown below(45)Cq−ROFPGDMSM2,p,pCq1,Cq2,⋯,Cqn=12p∏1≤l1<l2≤s∑j=1kpCqljsŁlj1/Cs2=12p∏1≤l1<l2≤spCql1sŁl1⊕pCql2sŁl21/ss−1=12p∏l,j=1l≠jspCqlsŁl⊕pCqjsŁj1/ss−1=Cq−ROFPGBMp,pCq1,Cq2,⋯,Cqs=1−1−∏l=j=1l≠js1−1−alqsŁlp1−ajqsŁjp1/ss−11/2p1/q,1−1−∏l=j=1l≠js1−1−blqsŁlp1−bjqsŁjp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−clqsŁlp1−1−cjqsŁjp1/ss−11/q1/2p,1−∏l=j=1l≠js1−1−1−dlqsŁlp1−1−djqsŁjp1/ss−11/q1/2p,1−1−∏l=j=1l≠js1−1−elqsŁlp1−ejqsŁjp1/ss−11/2p1/q,1−∏l=j=1l≠js1−1−1−flqsŁlp1−1−fjqsŁjp1/ss−11/q1/2p.(2)
Whenk=3 and γ1=γ2=γ3=p, the developed Cq-ROFPGDMSM operator degenerates into the cubic q-rung orthopair fuzzy power generalized geometric Bonferroni mean (Cq-ROFPGGBM) operator when (p=r=s), as shown below(46)Cq−ROFPGDMSM3,p,p,pCq1,Cq2,⋯,Cqs=13p∏1≤l1<l2<l3≤s∑j=13pCqljsŁlj1/Cn3=13p⊗l1,l2,l3=1l1≠l2≠l3spCql1sŁl1⊕pCql2sŁl2⊕Cql3sŁl31/3!3!/ss−1s−2=13p⊗l,m,n=1l≠m≠nnpCqlsŁl⊗pCqmsŁj⊗pCqnsŁn1/ss−1s−2=Cq−ROFPGGBMp,p,pCq1,Cq2,⋯,Cqs=1−1−∏l=m=n=1,l≠m≠ns1−1−alqsŁlp1−amqsŁmp1−anqsŁnp1/ss−1s−21/3p1/q,1−1−∏l=m=n=1,l≠m≠ns1−1−alqsÕlp1−bmqsŁmp1−bnqsŁnp1/ss−1s−21/3p1/q,1−∏l=m=n=1,l≠m≠ns1−1−1−clqsŁip1−1−cmqmŁjp1−1−cnqsŁlp1/ss−1s−21/q1/3p,1−∏l=m=n=1,l≠m≠ns1−1−1−dlqsŁip1−1−dmqmŁjp1−1−dnqsŁlp1/ss−1s−21/q1/3p,1−1−∏l=m=n=1,l≠m≠ns1−1−elqsŁlp1−emqsŁmp1−enqsŁnp1/ss−1s−21/3p1/q,1−∏l=m=n=1,l≠m≠ns1−1−1−flqsŁip1−1−fmqmŁjp1−1−fnqsŁlp1/ss−1s−21/q1/3p.(3)
Whenγ1=γ2=⋯=γn=1, the developed Cq-ROFPGMSM operator degenerates into the cubic q-rung orthopair fuzzy power MSM (Cq-ROFPDMSM) operator, as shown below.(47)Cq−ROFPGDMSMk,1,,.⋯,1Cq1,Cq2,⋯,Cqs=11+1+⋯+1∏1≤l1<l2<l3≤s∑j=1k1CqljsŁlj1/Csk=1k∏1≤l1<l2<l3≤s∑j=1kCqljsŁlj1/Cs3=Cq−ROFPDMSMkCq1,Cq2,⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljq1/Csk1/k1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljq1/Csk1/k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁll1/Csk1/q1/k,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁlj1/Csk1/q1/k,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljq1/Csk1/k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁlj1/Csk1/q1/k.(4)
Whenγ1=γ2=⋯=γn=1/k, the developed Cq-ROFPGDMSM operator degenerates into the cubic q-rung orthopair fuzzy power HAM (Cq-ROFPDHAM) operator, as shown below.(48)Cq−ROFPGDMSMk,1/k,1/k,.⋯,1/kCq1,Cq2,⋯,Cqn=11/k+1/k+⋯+1/k∏1≤l1<l2<⋯<lk≤s∑j=1k1kCqljsŁlj1/Csk=∏1≤l1<l2<⋯<lk≤s∑j=1k1kCqljsŁlj1/Csk=Cq−ROFPDHAMkCq1,Cq2,⋯,Cqs=∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljq1/k1/q1/Csk,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljq1/k1/q1/Csk,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁlj1/k1/Csk1/q,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁlj1/k1/Csk1/q,∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljq1/k1/q1/Csk,1−∑1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁlj1/k1/Csk1/q.


## 4. The Cq-ROFPWGMSM Operator and Cq-ROFPWGDMSM Operator

In Definitions10 and 14, they can deal with the situation when the importance degrees of all the input arguments are the same. But in real decision-making, sometimes attributes have their own importance, so to deal with such situation, we propose the weighted form of the proposed aggregation operators given in Definitions 10 and 14.
(2)
Proof of this part is easy, so it is omitted hereDefinition 18.
LetCql=al,bl,cl,dl,el,fl be a set of Cq-ROFNs, พ=พ1,พ2,⋯,พsT be the importance degree of Cql,พl∈0,1, such that ⊕ssพl=1; then, the Cq-ROFPWGMSM operator can be revealed as follows:
(49)Cq‐ROFPWGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=∑1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj∑j=1kŁljCqljγjCs−1k1/γ1+γ2+⋯+γk,when1≤k<s,⊗i=1kŁiCqiγi+1−พi/n−11/γ1+γ2+⋯+γk,whenk=s,
whereŁz=1+TCqz/⊕z=1s1+TCqz,∑z=1sŁz=1,TCqj=∏a=1z≠jsSptCqz,Cqj, is the support degree for Cqz from Cqj, which fulfil the following conditions:
(1)
SptCqz,Cqj∈0,1(2)
SptCqz,Cqj=SptCqj,Cqz(3)
IfDE~Cqz,Cqj≤DE~Cqs,Cqt, then SptCqz,Cqj≥SptCqs,Cqt, where DE~Cqz,Cqj express the distance measure among two Cq-ROFNs described in Definition 6. γ1,γ2,⋯,γk≥0,k, is a parameter and k=1,2,⋯,n, l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (49) expresses the binomial coefficient s!/k!s−k! and n is the balancing coefficientTheorem 19.
LetCq1,Cq2,⋯, and Cqs be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. Then, the aggregated value utilizing Equation (49) is still Cq-ROFN, even
(50)Cq−ROFPWGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsŁljqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsŁljqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsŁljqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,
or(51)Cq−ROFPWGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=∏j=1k1−1−alqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,∏j=1k1−1−blqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,1−∏j=1k1−clsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,1−∏j=1k1−dlsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,∏j=1k1−1−elqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,1−∏j=1k1−flsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,fork=s.Proof.
Based on the operational laws of Cq-ROFNs, we can prove Theorem19.
(1)
For1≤k<n, we can have(52)sŁljCqlj=1−1−aljqsŁlj1/q,1−1−bljqsŁlj1/q,cljsŁij,dljsŁij,1−1−eljqsŁlj1/q,fljsŁlj,sŁljCqljγj=1−1−aljqsŁij1/qγj,1−1−bljqsŁlj1/qγj,1−1−cljsŁijqγj1/q,1−1−dljsŁljqγj1/q,1−1−eljqsŁlj1/qγj,1−1−fljsŁljqγj1/q,∏j=1ksŁljCqljγj=∏j=1k1−1−aljqsŁlj1/qγj,∏j=1k1−1−bljqsŁlj1/qγj,1−∏j=1k1−cljsŁljqγj1/q,1−∏j=1k1−dljsŁljqγj1/q,∏j=1k1−1−eljqsŁlj1/qγj,1−∏j=1k1−fljsŁljqγj1/q.
So,(53)1−⊕j=1kพlj∏j=1ksŁljCqljγj=1−1−∏j=1k1−1−aljqsŁljγj1−⊕j=1kพlj1/q,1−1−∏j=1k1−1−bljqsŁljγj1−⊕j=1kพlj1/q,1−∏j=1k1−cljsŁljqγj1/q1−⊕j=1kพlj,1−∏j=1k1−dljsŁljqγj1/q1−⊕j=1kพlj,1−1−∏j=1k1−1−eljqsŁljγj1−⊕j=1kพlj1/q,1−∏j=1k1−fljsŁljqγj1/q1−⊕j=1kพlj.
Further,(54)∑1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj∏j=1ksŁljCqljγj=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsŁljγj1−⊕j=1kพlj1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsŁljγj1−⊕j=1kพlj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsŁljqγj1/q1−⊕j=1kพlj,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsŁljqγj1/q1−⊕j=1kพlj,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsŁljγj1−⊕j=1kพlj1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsŁljqγj1/q1−⊕j=1kพlj.
Therefore,(55)∑1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj∏j=1ksŁljCqljγjCs−1k=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsŁljqγj1/q1−⊕j=1kพlj/Cs−1k,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsŁljqγj1/q1−⊕j=1kพlj/Cs−1k,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q,∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsŁljqγj1/q1−⊕j=1kพlj/Cs−1k.
Hence,(56)∑1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj∏j=1ksŁljCqljγjCsk1/γ1+γ2+⋯+γk=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cljsŁljqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−dljsŁljqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eljqsŁljγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fljsŁljqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q.(2)
Fork=s, we can have(57)sŁlCql=1−1−alqsŁl1/q,1−1−blqsŁl1/q,clsŁl,dlsŁl,1−1−elqsŁl1/q,flsŁl,sŁlCqlγl+1−sพi/s−1=1−1−alqsŁl1/qγl+1−sพi/s−1,1−1−blqsŁl1/qγl+1−sพi/s−1,1−1−clsŁlqγl+1−sพi/s−11/q,1−1−dlsŁlqγl+1−sพi/s−11/q,1−1−elqsŁl1/qγl+1−sพi/s−1,1−1−flsŁlqγl+1−sพi/s−11/q,∏j=1ksŁlCqlγl+1−sพl/s−1=∏j=1k1−1−alqsŁl1/qγl+1−sพl/s−1,∏j=1k1−1−blqsŁl1/qγl+1−sพl/s−1,1−∏j=1k1−clsŁlqγl+1−sพl/s−11/q1−∏j=1k1−dlsŁlqγl+1−sพl/s−11/q,∏j=1k1−1−elqsŁl1/qγl+1−sพl/s−1,1−∏j=1k1−flsŁlqγl+1−sพl/s−11/q.
So,(58)∏j=1ksŁlCqlγl+1−sพi/s−11/γ1+γ2+⋯+γk=∏j=1k1−1−alqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,∏j=1k1−1−blqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,1−∏j=1k1−clsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,1−∏j=1k1−dlsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,∏j=1k1−1−elqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,1−∏j=1k1−flsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q.
This is the required proof.Theorem 20 (idempotency).
LetCq=a,b,c,d,e,f and Cql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be two sets of Cq-ROFNs. When Cq=Cqll=1,2,⋯,s, we have
(59)Cq‐ROFPGWMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=Cq.Proof.
WhenCq=Cqll=1,2,⋯,s, we have s1+TCqlj/∑z=1s1+TCqz=s1+TCq/∑z=1s1+TCq=1. So, from Theorem 19, we have the following.
(1)
For1≤k<s, we have(60)Cq−ROFPWGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−aqγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−bqγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−cqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏J=1K1−dqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−eqγj1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−fqγj1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,=1−∏1≤l1<l2<⋯<lk≤s1−aq×γ1+γ2+⋯+γk1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−bq×γ1+γ2+⋯+γk1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−1−cqγ1+γ2+⋯+γk1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−1−dqγ1+γ2+⋯+γk1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−eq×γ1+γ2+⋯+γk1−⊕j=1kพlj/Cs−1k1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−1−fqγ1+γ2+⋯+γk1−⊕j=1kพlj/Cs−1k1/γ1+γ2+⋯+γk1/q,=1−1−aq×γ1+γ2+⋯+γk1/Cs−1k⊕1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1+γ2+⋯+γk1/Cs−1k⊕1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1+γ2+⋯+γk1/Cs−1k⊕1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1+γ2+⋯+γk1/Cs−1k⊕1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj1/γ1+γ2+⋯+γk1/q,1−1−eq×γ1+γ2+⋯+γk1/Cs−1k⊕1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1+γ2+⋯+γk1/Cs−1k⊕1≤l1<l2<⋯<lk≤s1−⊕j=1kพlj1/γ1+γ2+⋯+γk1/q.=1−1−aq×γ1+γ2+⋯+γk1/Cs−1kCsk−⊕1≤l1<l2<⋯<lk≤s⊕j=1kพlj1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1+γ2+⋯+γk1/Cs−1kCsk−⊕1≤l1<l2<⋯<lk≤s⊕j=1kพlj1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1+γ2+⋯+γk1/Cs−1kCsk−⊕1≤l1<l2<⋯<lk≤s⊕j=1kพlj1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1+γ2+⋯+γk1/Cs−1kCsk−⊕1≤l1<l2<⋯<lk≤s⊕j=1kพlj1/γ1+γ2+⋯+γk1/q,1−1−eq×γ1+γ2+⋯+γk1/Cs−1kCsk−⊕1≤l1<l2<⋯<lk≤s⊕j=1kพlj1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1+γ2+⋯+γk1/Cs−1kCsk−⊕1≤l1<l2<⋯<lk≤s⊕j=1kพlj1/γ1+γ2+⋯+γk1/q.=1−1−aq×γ1+γ2+⋯+γk1/Cs−1kCsk−⊕ij=1kCs−1k−1พlj1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1+γ2+⋯+γk1/Cs−1kCnk−⊕lj=1kCs−1k−1พlj1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1+γ2+⋯+γk1/Cs−1kCsk−⊕lj=1kCs−1k−1พij1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1+γ2+⋯+γk1/Cs−1kCsk−⊕lj=1kCs−1k−1พlj1/γ1+γ2+⋯+γk1/q,1−1−eq×γ1+γ2+⋯+γk1/Cs−1kCsk−⊕lj=1kCs−1k−1พlj1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1+γ2+⋯+γk1/Cs−1kCsk−⊕lj=1kCs−1k−1พlj1/γ1+γ2+⋯+γk1/q=1−1−aq×γ1+γ2+⋯+γk1/Cs−1kCsk−Cs−1k−11/q1/γ1+γ2+⋯+γk,1−1−bq×γ1+γ2+⋯+γk1/Cs−1kCsk−Cs−1k−11/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1+γ2+⋯+γk1/Cs−1kCsk−Cs−1k−11/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1+γ2+⋯+γk1/Cs−1kCsk−Cs−1k−11/γ1+γ2+⋯+γk1/q,1−1−eq×γ1+γ2+⋯+γk1/Cs−1kCsk−Cs−1k−11/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1+γ2+⋯+γk1/Cs−1kCsk−Cs−1k−11/γ1+γ2+⋯+γk1/q;=1−1−aq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q1−1−eq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q=1−1−aq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−bq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−cqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q,1−1−1−1−dqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q1−1−eq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−1−1−fqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q=aq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,bq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−cqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q,1−1−dqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/qeq×γ1+γ2+⋯+γk1/q1/γ1+γ2+⋯+γk,1−1−fqγ1+γ2+⋯+γk1/γ1+γ2+⋯+γk1/q=a,b,c,d,e,f=Cq.Theorem 21 (boundedness).
LetCql=al,bl,cl,dl,el,fl,l=1,2,⋯,s be a set of Cq-ROFNs, χ=minCq1,Cq2,⋯,Cqn and α=maxCq1,Cq2,⋯,Cqn, then
(61)χ≤Cq‐ROFPGWMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs≤α.Proof.
Proof of Theorem21 is the same as that of Theorem 13.Definition 22.
LetCql=al,bl,cl,dl,el,fl be a set of Cq-ROFNs, ฝ=พ1,พ2,⋯,พsT be the importance degree of Cqi, พl∈0,1 such that ⊕l=1sพl=1; then, the Cq-ROFPWGDMSM operator can be revealed as follows:
(62)Cq‐ROFPWGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1γ1+γ2+⋯+γk∏1≤l1<l2<⋯<lk≤s∏j=1kγjCqljsŁlj1−⊕j=1kพlj1/Cs−1k,when1≤k<s,1γ1+γ2+⋯+γk∑j=1kγl+1−พls−1CqlsÕl,whenk=s,
whereŁz=1+TCqz/⊕z=1s1+TCqz,⊕z=1sŁz=1,TCqj=⊕z=1z≠jsSptCqz,Cqj, is the support degree for Cqz from Cqj, which fulfil the following conditions:
(1)
SptCqz,Cqj∈0,1(2)
SptCqz,Cqj=SptCqj,Cqz(3)
IfDE~Cqz,Cqj≤DE~Cqs,Cqt, then SptCqz,Cqj≥SptCqs,Cqt, where DE~Cqz,Cqj express the distance measure among two Cq-ROFNs described in Definition 6. γ1,γ2,⋯,γk≥0, k is a parameter and k=1,2,⋯,s, l1,l2,⋯,lk traverses all the k-tuple combination of 1,2,⋯,s. The denominator Csk in Equation (62) expresses the binomial coefficient s!/k!s−k! and s is the balancing coefficientTheorem 23.
LetCq1,Cq2,⋯, and Cqn be Cq-ROFNs, where Cql=al,bl,cl,dl,el,fll=1,2,⋯,sq≥1. Then, the aggregated value utilizing Equation (62) is still Cq-ROFN, even
(63)Cq−ROFPWGDMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−aljsŁljqγj1/q1−⊕j=1kพlj/Csk1/γ1+γ2+⋯+γk1/q,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−bljsŁljqγj1/q1−⊕j=1kพlj/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−cljqsŁljγj1−⊕j=1kพlj/Csk1/q1/γ1+γ2+⋯+γk,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−dljqsŁljγj1−⊕j=1kพlj/Csk1/q1/γ1+γ2+⋯+γk,1−1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−eljsŁljqγj1/q1−⊕j=1kพlj/Csk1/γ1+γ2+⋯+γk1/q,1−∏1≤l1<l2<⋯<lk≤s1−∏j=1k1−1−fljqsŁljγj1−⊕j=1kพlj/Csk1/q1/γ1+γ2+⋯+γk,for1≤k<s,
or(64)Cq−ROFPWGMSMk,γ1,γ2,⋯,γkCq1,Cq2,⋯,Cqs=1−∏j=1k1−alsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,1−∏j=1k1−blsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,∏j=1k1−1−clqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,∏j=1k1−1−dlqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,1−∏j=1k1−elsŁlqγl+1−sพi/s−11/γ1+γ2+⋯+γk1/q,∏j=1k1−1−flqsŁl1/qγl+1−sพi/s−11/γ1+γ2+⋯+γk,fork=s.Proof.
Proof of Theorem23 is the same as that of Theorem 19. So, it is omitted here.
Similar to Cq-ROFPGWMSM operator, Cq-ROFPGWDMSM has the properties of idempotency and boundedness.


## 5. A Novel Approach to MAGDM Established on the Initiated Operators

In this section, we pertain the afore-conferred Cq-ROFPGWMSM and Cq-ROFGWDMM operators to ascertain productive model for MAGDM under Cq-ROF environments. LetALE¯¯=ale¯¯1,ale¯¯2,⋯,ale¯¯m be the group of alternatives, the group of attributes is intimated by Cri¯¯=Cri¯¯1,Cri¯¯2,⋯,Cri¯¯n, the importance degree of the attributes is intimated by Wet¯¯=wet¯¯1,wet¯¯2,⋯,wet¯¯nT such that wet¯¯e∈0,1,∑e=1nwet¯¯e=1, and DMR¯¯=dmr¯¯1,dmr¯¯2,⋯,dmr¯¯a refer to the group of a decision-makers (DMR), with importance degree intimated by WGT¯¯=wgt¯¯1,wgt¯¯2,⋯,wgt¯¯aT such that wgt¯¯b∈0,1,∑b=1awgt¯¯=1. Assume that DMT¯¯b=Cqcebm×n is the decision matrix (DMT), where Cqceb=aceb,bceb,cceb,dceb,eceb,fceb takes the structure of Cq-ROFN, specified by DMR dmr¯¯b for alternative ale¯¯c with respect to the attribute Cri¯¯e.Then, turning on practical decision circumstances where the importance degrees of both DMRs and attributes are entirely identified in advance. As a result, in the following, we ascertain a MAGDM approach hinge on the anticipated Cq-ROFPGWMSM and Cq-ROFPGDWMSM operators. To do accordingly, now pursue the steps below.


### 5.1. Decision-Making Model Utilizing Cq-ROFPGWMSM

Step 1.
Firstly, the specified DMTsDMT¯¯b=Cqcebm×n should be distorted into uniform DMT DMT¯¯b=Cqcebm×n. We modify the cost-type attribute into benefit-type attribute utilizing the following formula.
(65)Cqceb=Cqceb=aceb,bceb,cceb,dceb,eceb,fcebforbenefit‐typeattributeCri¯¯ec=1,2,⋯,m,e=1,2,⋯,n,Cqcebc=cceb,dceb,aceb,bceb,fceb,ecebforbenefit‐typeattributeCri¯¯e.Step 2.
Ascertain the support degrees by the following formula:(66)SupCqceb,Cqcel=1−DE~HCqceb,Cqcel,b,l=1,2,⋯,a;c=1,2,⋯,m;e=1,2,⋯,n.,
which gratify the conditions for support function andD¯¯ECqceb,Cqcel is the distance measure stipulated in Definition 6.Step 3.
Ascertain the support degreeTCqceb that Cq-ROFN Cqceb accepts from other Cq-ROFNs Cqcell=1,2,⋯,a;l≠b, where
(67)TCqceb=∑l=1,l≠basupCqceb,Cqcel.Step 4.
Discover the weightsödec associated with the Cq-OFN Cqdec,(68)ödec=a1+TCqdec∑c=1a1+TCqdec,c=1,2,⋯,a,
whereödec≥0 and ∑c=1aödec=1..Step 5.
Aggregate all the individual decision matricesDMT¯¯b=Cqcebm×nb=1,2,⋯,a into group decision matrix DMT~=Cqcem×n by utilizing Cq-ROFPGWMSM operators, where
(69)Cqceb=Cq‐ROFPGMSMCqce1,Cqce2,⋯,Cqcea.Step 6.
Determine support degreesSupCqce,Cqcx by the following formula:
(70)SupCqce,Cqcx=1−DE~Cqce,Cqcx;c=1,2,⋯,m,e=1,2,⋯,n,e≠x,where DE~Cqce,Cqcx is distance measure given in Definition 6.Step 7.
Ascertain the support degreeTCqce that Cq-ROFNs Cqce collects from other Cq-ROFNs Cqcxx=1,2,⋯,n;e≠x, where
(71)TCqce=∑x=1,x≠enSupCqce,Cqcx.Step 8.
Ascertain weighting vectorÕcec=1,2,.⋯,m,e=1,2,⋯,n associated with Cqce,
(72)öce=n1+TCqce∑e=1n1+TCqce.Step 9.
Utilize Cq-ROFPGWMSM operator to aggregate all assessment valuesCqcec=1,2,.⋯,m,e=1,2,⋯,n into overall assessment value Cqcc=1,2,.⋯,m corresponding to the alternatives ale¯¯cc=1,2,⋯,m:(73)Cqc=Cq‐ROFPWMSMCqc1,Cqc2,⋯,Cqcn.Step 10.
Ascertain the scoresSCR~Cqd for the overall Cq-RFN of the alternatives ale¯¯dd=1,2,⋯,g by utilizing Definition 6.Step 11.
Rank all alternativesale¯¯dd=1,2,⋯,g and select the optimal one (s) with the ranking order Cqdd=1,2,⋯,g.


### 5.2. Decision-Making Model Utilizing Cq-ROFPGWDMSM Operator

Steps 1–4. These are the same as for Cq-ROFPGWMSM operator.Step 5. Aggregate all the individual decision matricesDMT¯¯b=Cqcebm×nb=1,2,⋯,a into group decision matrix DMT~=Cqcem×n by utilizing Cq-ROFPGWDMSM operators, where
(74)Cqceb=Cq‐ROFPGWDMSMCqce1,Cqce2,⋯,Cqcea.Step 6. Determine support degreesSupCqce,Cqcx by the following formula:
(75)SupCqce,Cqcx=1−DE~Cqce,Cqcx;c=1,2,⋯,m,e=1,2,⋯,n,e≠x,where DE~Cqce,Cqcx is distance measure given in Definition 5.Step 7. Ascertain the support degreeTCqce that Cq-ROFNs Cqce collects from other Cq-ROFNs Cqcxx=1,2,⋯,n;e≠x, where
(76)TCqce=∑x=1,x≠enSupCqce,Cqcx.Step 8. Ascertain weighting vectorÕcec=1,2,.⋯,m,e=1,2,⋯,n associated with Cqce,
(77)Łce=n1+TCqce∑e=1n1+TCqce.Step 9. Utilize Cq-ROFPGWDMSM operator to aggregate all assessment valuesCqcec=1,2,.⋯,m,e=1,2,⋯,n into overall assessment value Cqcc=1,2,.⋯,m corresponding to the alternatives ale¯¯cc=1,2,⋯,m:(78)Cqc=Cq‐ROFPWDMSMCqc1,Cqc2,⋯,Cqcn.Step 10. Ascertain the scoresSCR~Cqd for the overall Cq-RFN of the alternatives ale¯¯dd=1,2,⋯,g by utilizing Definition 6.Step 11. Rank all alternativesale¯¯dd=1,2,⋯,g and select the optimal one (s) with the ranking order Cqdd=1,2,⋯,g.


## 5.1. Decision-Making Model Utilizing Cq-ROFPGWMSM

Step 1.
Firstly, the specified DMTsDMT¯¯b=Cqcebm×n should be distorted into uniform DMT DMT¯¯b=Cqcebm×n. We modify the cost-type attribute into benefit-type attribute utilizing the following formula.
(65)Cqceb=Cqceb=aceb,bceb,cceb,dceb,eceb,fcebforbenefit‐typeattributeCri¯¯ec=1,2,⋯,m,e=1,2,⋯,n,Cqcebc=cceb,dceb,aceb,bceb,fceb,ecebforbenefit‐typeattributeCri¯¯e.Step 2.
Ascertain the support degrees by the following formula:(66)SupCqceb,Cqcel=1−DE~HCqceb,Cqcel,b,l=1,2,⋯,a;c=1,2,⋯,m;e=1,2,⋯,n.,
which gratify the conditions for support function andD¯¯ECqceb,Cqcel is the distance measure stipulated in Definition 6.Step 3.
Ascertain the support degreeTCqceb that Cq-ROFN Cqceb accepts from other Cq-ROFNs Cqcell=1,2,⋯,a;l≠b, where
(67)TCqceb=∑l=1,l≠basupCqceb,Cqcel.Step 4.
Discover the weightsödec associated with the Cq-OFN Cqdec,(68)ödec=a1+TCqdec∑c=1a1+TCqdec,c=1,2,⋯,a,
whereödec≥0 and ∑c=1aödec=1..Step 5.
Aggregate all the individual decision matricesDMT¯¯b=Cqcebm×nb=1,2,⋯,a into group decision matrix DMT~=Cqcem×n by utilizing Cq-ROFPGWMSM operators, where
(69)Cqceb=Cq‐ROFPGMSMCqce1,Cqce2,⋯,Cqcea.Step 6.
Determine support degreesSupCqce,Cqcx by the following formula:
(70)SupCqce,Cqcx=1−DE~Cqce,Cqcx;c=1,2,⋯,m,e=1,2,⋯,n,e≠x,where DE~Cqce,Cqcx is distance measure given in Definition 6.Step 7.
Ascertain the support degreeTCqce that Cq-ROFNs Cqce collects from other Cq-ROFNs Cqcxx=1,2,⋯,n;e≠x, where
(71)TCqce=∑x=1,x≠enSupCqce,Cqcx.Step 8.
Ascertain weighting vectorÕcec=1,2,.⋯,m,e=1,2,⋯,n associated with Cqce,
(72)öce=n1+TCqce∑e=1n1+TCqce.Step 9.
Utilize Cq-ROFPGWMSM operator to aggregate all assessment valuesCqcec=1,2,.⋯,m,e=1,2,⋯,n into overall assessment value Cqcc=1,2,.⋯,m corresponding to the alternatives ale¯¯cc=1,2,⋯,m:(73)Cqc=Cq‐ROFPWMSMCqc1,Cqc2,⋯,Cqcn.Step 10.
Ascertain the scoresSCR~Cqd for the overall Cq-RFN of the alternatives ale¯¯dd=1,2,⋯,g by utilizing Definition 6.Step 11.
Rank all alternativesale¯¯dd=1,2,⋯,g and select the optimal one (s) with the ranking order Cqdd=1,2,⋯,g.


## 5.2. Decision-Making Model Utilizing Cq-ROFPGWDMSM Operator

Steps 1–4. These are the same as for Cq-ROFPGWMSM operator.Step 5. Aggregate all the individual decision matricesDMT¯¯b=Cqcebm×nb=1,2,⋯,a into group decision matrix DMT~=Cqcem×n by utilizing Cq-ROFPGWDMSM operators, where
(74)Cqceb=Cq‐ROFPGWDMSMCqce1,Cqce2,⋯,Cqcea.Step 6. Determine support degreesSupCqce,Cqcx by the following formula:
(75)SupCqce,Cqcx=1−DE~Cqce,Cqcx;c=1,2,⋯,m,e=1,2,⋯,n,e≠x,where DE~Cqce,Cqcx is distance measure given in Definition 5.Step 7. Ascertain the support degreeTCqce that Cq-ROFNs Cqce collects from other Cq-ROFNs Cqcxx=1,2,⋯,n;e≠x, where
(76)TCqce=∑x=1,x≠enSupCqce,Cqcx.Step 8. Ascertain weighting vectorÕcec=1,2,.⋯,m,e=1,2,⋯,n associated with Cqce,
(77)Łce=n1+TCqce∑e=1n1+TCqce.Step 9. Utilize Cq-ROFPGWDMSM operator to aggregate all assessment valuesCqcec=1,2,.⋯,m,e=1,2,⋯,n into overall assessment value Cqcc=1,2,.⋯,m corresponding to the alternatives ale¯¯cc=1,2,⋯,m:(78)Cqc=Cq‐ROFPWDMSMCqc1,Cqc2,⋯,Cqcn.Step 10. Ascertain the scoresSCR~Cqd for the overall Cq-RFN of the alternatives ale¯¯dd=1,2,⋯,g by utilizing Definition 6.Step 11. Rank all alternativesale¯¯dd=1,2,⋯,g and select the optimal one (s) with the ranking order Cqdd=1,2,⋯,g.


## 6. Numerical Examples

In this part, we apply the intended approach based on the intended AOs in the case of a university in Pakistan, that is, International Islamic University Islamabad Pakistan. The Dean of the faculty of basic and applied sciences is mystified on how to choose the most excellent assistant professor at the end of the year. The intended approach based on intended AOs was pertained to the procedure of assistant professor (AP) selection of performance assessment of this university to help it evaluating their APs and investigate the efficiency of suggested approach based on these newly suggested AOs.The input data of the suggested approach were assembled by accomplishing semi-structured interviews with the head of departments (HODs) of each department of basic and applied sciences of the university. After initial screening, fixed APs Dr. Syed Salman HussianAale¯¯1, Dr. Tahir Mahmood ale¯¯2, Dr. M. Imran Shabbir ale¯¯3, Dr. Zafeer Saqib ale¯¯4, and Dr. Usman Nasir ale¯¯5 are picked for further assessment. A team of three experts E1,E2, and E3 is liable for the assessment of these five selected APs and the picking of the most excellent APs. Four attributes are taken into account consisting of teaching method As1, total number of publications As2, student evaluation As3, and personality As4. The experts provide their evaluation exploiting C-qROFNs and are shown in Tables 1–3.Table 1
Evaluation values provided by expertsE1.
Alternatives/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.350.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.35ale¯¯20.85,0.95,0.1,0.15,0.95,0.150.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.350.85,0.95,0.1,0.15,0.95,0.15ale¯¯20.85,0.95,0.1,0.15,0.95,0.150.85,0.95,0.1,0.15,0.95,0.150.45,0.55,0.45,0.55,0.55,0.550.45,0.55,0.45,0.55,0.55,0.55ale¯¯40.85,0.95,0.1,0.15,0.95,0.150.75,0.85,0.2,0.25,0.85,0.250.85,0.95,0.1,0.15,0.95,0.150.75,0.85,0.2,0.25,0.85,0.25ale¯¯50.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.350.65,0.75,0.3,0.35,0.75,0.350.65,0.75,0.3,0.35,0.75,0.35Table 2
Evaluation values provided by expertsE2.
Alternatives/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.65,0.75,0.3,0.35,0.75,0.350.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.350.85,0.95,0.1,0.15,0.95,0.15ale¯¯20.75,0.85,0.2,0.25,0.85,0.250.85,0.95,0.1,0.15,0.95,0.150.85,0.95,0.1,0.15,0.95,0.150.45,0.55,0.45,0.55,0.55,0.55ale¯¯30.65,0.75,0.3,0.35,0.75,0.350.65,0.75,0.2,0.25,0.75,0.250.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.35ale¯¯40.45,0.55,0.45,0.55,0.55,0.550.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.350.45,0.55,0.45,0.55,0.55,0.55ale¯¯50.3,0.35,0.65,0.75,0.35,0.750.75,0.85,0.2,0.25,0.85,0.250.45,0.55,0.45,0.55,0.55,0.550.45,0.55,0.45,0.55,0.55,0.55Table 3
Evaluation values provided by expertsE3.
Alternatives/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.85,0.95,0.1,0.15,0.95,0.150.65,0.75,0.3,0.35,0.75,0.350.45,0.55,0.45,0.55,0.55,0.550.75,0.85,0.2,0.25,0.85,0.25ale¯¯20.65,0.75,0.3,0.35,0.75,0.350.65,0.75,0.3,0.35,0.75,0.350.75,0.85,0.2,0.25,0.85,0.250.75,0.85,0.2,0.25,0.85,0.25ale¯¯30.75,0.85,0.2,0.25,0.85,0.250.75,0.85,0.2,0.25,0.85,0.250.85,0.95,0.1,0.15,0.95,0.150.75,0.85,0.2,0.25,0.85,0.25ale¯¯40.2,0.25,0.75,0.85,0.25,0.850.85,0.95,0.1,0.15,0.95,0.150.75,0.85,0.2,0.25,0.85,0.250.65,0.75,0.3,0.35,0.75,0.35ale¯¯50.85,0.95,0.1,0.15,0.95,0.150.45,0.55,0.45,0.55,0.55,0.550.75,0.85,0.2,0.25,0.85,0.250.85,0.95,0.10.15,0.95,0.15


### 6.1. Decision-Making Steps Utilizing Cq-ROFPGWDMSM Operator

Step 1.
Since all the attributes are of the same type, so there is no need to normalize it.Step 2.
Discover the support degreesSupCqdeb,Cqdel,d=1,2,⋯,5,e=1,2,3,c,l=1,2,⋯,4 by the consuming Formula (66). For easiness, we shall write Sdecd=1,2,⋯,5,e=1,2,3,c=1,2,⋯,4 in place of SupCqdec,Cqdel. We have
(79)S11,121=S12,111=0.8991,S11,131=S13,111=0.8791,S12,131=S13,131=0.7808,S21,221=S22,211=0.8816,S21,231=S23,211=0.7808,S22,221=S23,221=0.8991,S31,321=S32,311=0.7808,S31,331=S33,311=0.8816,S32,331=S33,321=0.8991,S41,421=S42,411=0.6131,S41,431=S43,411=0.3447,S42,431=S43,421=0.7315,S51,521=S52,511=0.5639,S51,531=S53,511=0.8816,S52,531=S53,521=0.4455,S11,123=S12,113=0.8991,S11,133=S13,113=0.7315,S12,133=S13,133=0.8324,S21,223=S22,213=0.7808,S21,233=S23,213=0.8991,S22,223=S23,223=0.8816,S31,323=S32,313=0.7315,S31,333=S33,313=0.6131,S32,333=S33,323=0.8816,S41,423=S42,413=0.7808,S41,433=S43,413=0.8816,S42,433=S43,423=0.8991,S51,523=S52,513=0.8324,S51,533=S53,513=0.8991,S52,533=S53,523=0.7315,S11,124=S12,114=0.7808,S11,134=S13,114=0.8991,S12,134=S13,134=0.8816,S21,224=S22,214=0.6131,S21,234=S23,214=0.8816,S22,224=S23,224=0.7315,S31,324=S32,314=0.8324,S31,334=S33,314=0.7315,S32,334=S33,324=0.8991,S41,424=S42,414=0.7315,S41,434=S43,414=0.8991,S42,434=S43,424=0.8324,S51,524=S52,514=0.8324,S51,534=S53,514=0.7808,S52,534=S53,524=0.6131.Step 3.
Discover the support degreeTCqdec by utilizing Formula (67). For easiness, we shall write Tdecc=1,2,..,4,d=1,2,.⋯,5,e=1,2,3 in place of TCqdec. We have
(80)T111=1.7783,T121=1.6799,T131=1.6599,T211=1.6624,T221=1.7808,T231=1.6799,T311=1.6624,T321=1.6799,T331=1.7808,T411=0.9578,T421=1.3447,T431=1.0762,T511=1.4455,T521=1.0095,T531=1.3272,T112=1.8991,T122=1.7983,T132=1.8991,T212=1.7808,T222=1.6624,T232=1.6799,T312=1.6746,T322=1.7044,T332=1.7930,T412=1.8816,T422=1.8816,T432=1.7633,T512=1.7315,T522=1.6306,T532=1.5639,T113=1.6306,T123=1.7315,T133=1.5639,T213=1.6799,T223=1.6624,T233=1.7808,T313=1.3447,T323=1.6131,T333=1.4948,T413=1.6624,T423=1.6799,T433=1.7808,T513=1.7315,T523=1.5639,T533=1.6306,T114=1.6799,T124=1.6624,T134=1.7808,T214=1.4948,T224=1.3447,T234=1.6131,T314=1.5639,T324=1.7315,T334=1.6306,T414=1.6306,T424=1.5639,T434=1.7315,T514=1.6131,T524=1.4455,T534=1.3939.Step 4.
Employ Formula (68) to discover the weights ödec; we have
(81)ö111=1.0253,ö121=0.9879,ö131=0.9867,ö211=0.9871,ö221=1.0257,ö231=0.9871,ö311=0.9871,ö321=0.9871,ö331=1.0257,ö411=0.9156,ö421=1.1174,ö431=0.9670,ö511=1.1659,ö521=0.9171,ö531=1.0457,ö112=1.0122,ö122=0.9755,ö132=1.0122,ö212=1.0257,ö222=0.9871,ö232=0.9871,ö312=0.9839,ö322=0.9942,ö332=1.0220,ö412=1.0122,ö422=1.0122,ö432=0.9755,ö512=1.0395,ö522=0.9990,ö532=0.9615,ö113=0.9990,ö123=1.0395,ö133=0.9615,ö213=0.9871,ö223=0.9871,ö233=1.0257,ö313=0.9307,ö323=1.0561,ö333=1.0132,ö413=0.9871,ö423=0.9871,ö433=1.0257,ö513=1.0395,ö523=0.9615,ö533=0.9990,ö114=0.9871,ö124=0.9871,ö134=1.0257,ö214=1.0132,ö224=0.9307,ö234=1.0561,ö314=0.9615,ö324=1.0395,ö334=0.9990,ö414=0.9990,ö424=0.9615,ö434=1.0395,ö514=1.0561,ö524=0.9736,ö534=0.9704.Step 5.
Employ Formula (69) to aggregate all the individual decision matrices DMT¯¯c=Cqdecm×nc=1,2,⋯,4 into group decision matrix DMT~=Cqdcm×n, which is given in Table 4q=2,γ1=γ2=1.Step 6.
Employ Formula (70) to discover the support degrees SupCqdc,Cqdxd=1,2,⋯,5,c,x=1,2,⋯,4. For easiness, we shall designate Sdc in place of SupCqdc,Cqdx. We have
(82)S12=S21=0.9254,S13=S31=0.8783,S14=S41=0.9835,S23=S32=0.9530,S24=S42=8514,S34=S43=0.8044,S12=S21=0.9937,S13=S31=0.9769,S14=S41=9462,S23=S32=0.9832,S24=S42=0.8736,S34=S43=0.8834,S12=S21=0.9842,S13=S31=0.8931,S14=S41=0.8407,S23=S32=0.8774,S24=S42=0.7712,S34=S43=0.8537,S12=S21=0.7294,S13=S31=0.7559,S14=S41=0.8841,S23=S32=0.9735,S24=S42=0.7960,S34=S43=0.8124,S12=S21=0.9481,S13=S31=0.9509,S14=S41=0.9770,S23=S32=0.9972,S24=S42=0.8654,S34=S43=0.8682.Step 7.
Employ Formula (71), to discover the support degree TCqdc. For easiness, we shall write Tdc in place of TCqdc. We have
(83)T11=2.7872,T12=2.7297,T13=2.6357,T14=2.6392,T21=2.9169,T22=2.8505,T23=2.8435,T24=2.7032,T31=2.7180,T32=2.6328,T33=2.6242,T34=2.4656,T41=2.3693,T42=2.4989,T43=2.5418,T44=2.4924,T51=2.8760,T52=2.8106,T53=2.8163,T54=2.7106.Step 8.
By employing Formula (72), to discover the weighting vector ödcd=1,2,.⋯,5,c=1,2,⋯,4, we have
(84)ö11=1.0241,ö12=1.0086,ö13=0.9832,ö14=0.9841,ö21=1.0231,ö22=1.0057,ö23=1.0039,ö24=0.9673,ö31=1.0299,ö32=1.0063,ö33=1.0039,ö34=0.9600,ö41=0.9694,ö42=1.0067,ö43=1.0191,ö44=1.0048,ö51=1.0191,ö52=1.0019,ö53=1.0034,ö54=0.9756.Step 9.
Employ Formula (73) to aggregate all assessment values ale¯¯dcd=1,2,.⋯,5,c=1,2,⋯,4 into overall assessment value Cqdd=1,2,.⋯,5 corresponding to the alternatives ale¯¯dd=1,2,⋯,5:(85)q=2,γ1=γ2=1ale¯¯1=0.4681,0.5503,0.6724,0.7035,0.5506,0.7035,ale¯¯2=0.4914,0.5794,0.6558,0.6895,0.5799,0.6893,ale¯¯3=0.4575,0.5402,0.6772,0.7127,0.5404,0.7127,ale¯¯4=0.4399,0.5199,0.7000,0.7363,0.5199,0.7363,ale¯¯5=0.4306,0.5085,0.7063,0.7410,0.5085,0.7390.Step 10.
By employing Definition6 to discover the scores SCR~ale¯¯d, we have
SCR~ale¯¯1=−0.4042,SCR~ale¯¯2=−0.3031,SCR~ale¯¯3=−0.4487,SCR~ale¯¯4=−0.5560,SCR~ale¯¯5=−0.5895.Step 11.
According to score values, the ranking order of the alternativesale¯¯dd=1,2,⋯,5 is ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5. According to the ranking order, the best alternative is ale¯¯2 (i.e., Dr. Tahir Mahmood), while the last one is ale¯¯5 (i.e., Dr. Usman Nasir).Table 4
Collective decision matrixDMT~ employing Cq-ROFPGWMSM operator.
Alternative/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯5ale¯¯10.5916,0.6899,0.4974,0.5405,0.6914,0.54050.5275,0.6159,0.5551,0.5935,0.6159,0.59350.4889,0.5750,0.5891,0.6368,0.5750,0.63680.5791,0.6761,0.5128,0.5538,0.6761,0.553811ale¯¯20.5975,0.6980,0.4935,0.5372,0.6980,0.53650.5916,0.6914,0.4975,0.5410,0.6914,0.54050.5791,0.6761,0.5128,0.5538,0.6761,0.55380.5568,0.6555,0.5367,0.5884,0.6555,0.5884ale¯¯30.5975,0.6980,0.4935,0.5365,0.6980,0.53650.5974,0.6979,0.4595,0.5067,0.6979,0.50670.5136,0.6057,0.5772,0.6305,0.6068,0.63050.4606,0.5448,0.6111,0.6619,0.5448,0.6619ale¯¯40.4023,0.4804,0.6852,0.7441,0.4804,0.74410.6159,0.7191,0.4665,0.5123,0.7191,0.51320.5975,0.6980,0.4935,0.5373,0.6980,0.53650.4889,0.5750,0.5891,0.6368,0.5750,0.6368ale¯¯50.5453,0.6398,0.5676,0.6179,0.6398,0.61790.4851,0.5709,0.5922,0.6394,0.57087,0.63940.4851,0.5709,0.5922,0.6394,0.5709,0.62600.5140,0.6052,0.5744,0.6221,0.6052,0.6221


### 6.2. Decision-Making Steps Utilizing Cq-ROFPGWDMSM Operator

Steps 1–4. These are the same as those for decision-making model utilizing Cq-ROFPGWMSM operator.Step 5. Employ Formula (74) to aggregate all the individual decision matrices DMT¯¯c=Cqdecm×nc=1,2,⋯,4 into group decision matrix DMT~=Cqdcm×n, which is given in Table 5. q=2,γ1=γ2=1.Table 5
Collective decision matrixDMT~employing Cq-ROFPGWDMSM operator.
Alternatives/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.8622,0.9180,0.1565,0.1937,0.9198,0.19370.8207,0.8774,0.2066,0.2445,0.8774,0.24450.8010,0.8599,0.2294,0.2787,0.8599,0.27870.8594,0.9181,0.1585,0.1965,0.9181,0.1965ale¯¯20.8712,0.9306,0.1427,0.1807,0.9306,0.18070.8622,0.9198,0.1565,0.1936,0.9198,0.19360.8594,0.9181,0.1585,0.1965,0.9181,0.19650.8503,0.9140,0.1707,0.2197,0.9140,0.2197ale¯¯30.7940,0.9306,0.1427,0.1807,0.9306,0.18070.8712,0.9306,0.1210,0.1589,0.9306,0.15890.8319,0.8955,0.1919,0.2430,0.8955,0.24300.7851,0.8445,0.24738,0.2992,0.8445,0.2992ale¯¯40.7738,0.8333,0.3134,0.3787,0.8333,0.37870.8774,0.9342,0.1313,0.1688,0.9342,0.16880.8712,0.9306,0.1427,0.1807,0.9306,0.18070.8010,0.8599,0.2294,0.2787,0.8599,0.2787ale¯¯50.8179,0.8800,0.2303,0.2815,0.8800,0.28150.7909,0.8494,0.2427,0.2925,0.8494,0.29250.7909,0.8494,0.2427,0.2925,0.8494,0.29250.8111,0.8705,0.2294,0.278015,0.8705,0.278015Step 6. Employ Formula (75) to discover the support degrees SupCqdc,Cqdxd=1,2,⋯,5,c,x=1,2,⋯,4. For easiness, we shall designate Sdc in place of SupCqdc,Cqdx. We have
(86)S12=S21=0.9525,S13=S31=0.9295,S14=S41=0.9977,S23=S32=0.9769,S24=S42=0.9286,S34=S43=0.9056,S12=S21=0.9884,S13=S31=0.9861,S14=S41=0.9771,S23=S32=0.9977,S24=S42=0.9648,S34=S43=0.9661,S12=S21=0.9751,S13=S31=0.9568,S14=S41=0.9209,S23=S32=0.9525,S24=S42=0.8992,S34=S43=0.9254,S12=S21=0.8602,S13=S31=0.8662,S14=S41=0.9483,S23=S32=0.9940,S24=S42=0.9129,S34=S43=0.9166,S12=S21=0.9721,S13=S31=0.9728,S14=S41=0.9919,S23=S32=1.0000,S24=S42=0.9383,S34=S43=0.9383.Step 7. Employ Formula (76), to discover the support degree TCqdc. For easiness, we shall write Tdc in place of TCqdc. We have
(87)T11=2.8797,T12=2.8581,T13=2.8120,T14=2.8319,T21=2.9516,T22=2.9509,T23=2.9498,T24=2.9080,T31=2.8529,T32=2.8269,T33=2.8347,T34=2.7456,T41=2.6747,T42=2.7672,T43=2.7769,T44=2.7779,T51=2.9368,T52=2.9104,T53=2.9111,T54=2.8685.Step 8. By employing Formula (77), to discover the weighting vector Õdcd=1,2,.⋯,5,c=1,2,⋯,4, we have
(88)Ł11=1.0089,Ł12=1.0033,Ł13=0.9913,Ł14=0.9965,Ł21=1.0029,Ł22=1.0027,Ł23=1.0025,Ł24=0.9919,Ł31=1.0099,Ł32=1.0031,Ł33=1.0052,Ł34=0.9818,Ł41=0.9801,Ł42=1.0048,Ł43=1.0074,Ł44=1.0077,Ł51=1.0077,Ł52=1.0009,Ł53=1.0011,Ł54=0.9902.Step 9. Employ Formula (73) to aggregate all assessment values ale¯¯dcd=1,2,.⋯,5,c=1,2,⋯,4 into overall assessment value Cqdd=1,2,.⋯,5 corresponding to the alternatives ale¯¯dd=1,2,⋯,5, q=2,γ1=γ2=1(89)ale¯¯1=0.8887,0.9292,0.1548,0.1885,0.9292,0.1885,ale¯¯2=0.9028,0.9445,0.1324,0.1670,0.9445,0.1670,ale¯¯3=0.8738,0.9310,0.1511,0.1889,0.9310,0.1889,ale¯¯4=0.8805,0.9217,0.1808,0.2218,0.9217,0.2218,ale¯¯5=0.8639,0.9053,0.1980,0.2399,0.9053,0.2399.Step 10. By employing Definition6 to discover the scores SCR~ale¯¯d, we have
(90)SCR~ale¯¯1=1.6246,SCR~ale¯¯2=1.6950,SCR~ale¯¯3=1.6168,SCR~ale¯¯4=1.5718,SCR~ale¯¯5=1.4965.Step 11. According to score values, the ranking order of the alternativesale¯¯dd=1,2,⋯,5 is ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5. According to the ranking order, the best alternative is (i.e., Dr. Tahir Mahmood), while the last one is ale¯¯5 (i.e., Dr. Usman Nasir).


### 6.3. Effect of Parameterq on Final Ranking Result

In this subpart, the influence of the parameterq is discussed on the final ranking results while utilizing Cq-ROFPGWMSM and Cq-ROFPGWDMSM operators. The score values and ranking orders are given in Table 6. From Table 6, one can observe distinct values of the parameter q, while the values of generalized parameters γ1=γ2=1 and k=2 are fixed, the obtained ranking orders are different. When the value of parameter q=25, while utilizing Cq-ROFPGWMSM operator the best alternative remains the same, but the worst alternative is ale¯¯4. One can also observe from Table 6, when the value of parameter q increases, the score values increase utilizing Cq-ROFPGWMSM operator. Similarly, when the value of parameter q increases, the score values decrease utilizing Cq-ROFPGWDMSM operator. So, the decision-makers choose the value of parameter q according to the actual need of the situation.Table 6
Effect of parameterq on the final ranking results utilizing Cq-ROFPGWMSM and Cq-ROFPGWDMSM operator.
ParameterScore value Cq-ROFPGWMSMScore value Cq-ROFPGWDMSMRanking order Cq-ROFPGWMSM and Cq-ROFPGWDMSMq=3SCR~ale¯¯1=−0.31159,SCR~ale¯¯2=−0.20941,SCR~ale¯¯3=−0.3605,SCR~ale¯¯4=−0.4766,SCR~ale¯¯5=−0.5062.SCR~ale¯¯1=1.5550,SCR~ale¯¯2=1.6360,SCR~ale¯¯3=1.5475,SCR~ale¯¯4=1.5113,SCR~ale¯¯5=1.4323.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.q=5SCR~ale¯¯1=−0.1986,SCR~ale¯¯2=−0.1021,SCR~ale¯¯3=−0.2459,SCR~ale¯¯4=−0.3669,SCR~ale¯¯5=−0.3831.SCR~ale¯¯1=1.3895,SCR~ale¯¯2=1.4957,SCR~ale¯¯3=1.3902,SCR~ale¯¯4=1.3677,SCR~ale¯¯5=1.2791.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.q=10SCR~ale¯¯1=−0.0725,SCR~ale¯¯2=−0.0022,SCR~ale¯¯3=−0.1045,SCR~ale¯¯4=−0.2100,SCR~ale¯¯5=−0.1980.SCR~ale¯¯1=1.0841,SCR~ale¯¯2=1.2357,SCR~ale¯¯3=1.1104,SCR~ale¯¯4=1.0901,SCR~ale¯¯5=0.9928.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.q=18SCR~ale¯¯1=−0.0129,SCR~ale¯¯2=0.0205,SCR~ale¯¯3=−0.0283,SCR~ale¯¯4=−0.0924,SCR~ale¯¯5=−0.0709.SCR~ale¯¯1=0.7950,SCR~ale¯¯2=0.9774,SCR~ale¯¯3=0.8473,SCR~ale¯¯4=0.8116,SCR~ale¯¯5=0.7171.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.q=25SCR~ale¯¯1=0.0008,SCR~ale¯¯2=0.0164,SCR~ale¯¯3=−0.0052,SCR~ale¯¯4=−0.0379,SCR~ale¯¯5=−0.0282.SCR~ale¯¯1=0.6339,SCR~ale¯¯2=0.8236,SCR~ale¯¯3=0.6978,SCR~ale¯¯4=0.6506,SCR~ale¯¯5=0.5604.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.q=50SCR~ale¯¯1=0.0008,SCR~ale¯¯2=0.0017,SCR~ale¯¯3=0.0011,SCR~ale¯¯4=0.0006,SCR~ale¯¯5=0.0002.SCR~ale¯¯1=0.3264,SCR~ale¯¯2=0.4961,SCR~ale¯¯3=0.4019,SCR~ale¯¯4=0.3325,SCR~ale¯¯5=0.2641.ale¯¯2>ale¯¯3>ale¯¯1>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.


### 6.4. Effect of the Generalized Parameter

In this subpart, the influence of the generalized parametersγ1,γ2 are discussed on the final ranking results while utilizing Cq-ROFPGWMSM and Cq-ROFPGWDMSM operators. The values of the parameter q=2,k=2 are fixed. The score values and ranking orders are given in Table 7. From Table 7, one can observe that for distinct values of the parameter γ1,γ2, the obtained ranking orders are different. When the value of the parameter γ1=15,γ2=25, while utilizing Cq-ROFPGWMSM operator, the best alternative remains the same, but the worst alternative is ale¯¯4. One can also observe from Table 7, when the value of the parameters γ1,γ2 increases, the score values increases utilizing Cq-ROFPGWMSM operator. Similarly, when the value of the parameters γ1,γ2 increases, the score values decrease utilizing Cq-ROFPGWDMSM operator. So, the decision-makers choose the value of the parameters γ1,γ2 according to the actual need of the situation.Table 7
Influence of the generalized parametersγ1,γ2 on final ranking results.
ParameterScore value Cq-ROFPGWMSMScore value Cq-ROFPGWDMSMRanking orderγ1=2,γ2=2SCR~ale¯¯1=0.6819,SCR~ale¯¯2=0.7760,SCR~ale¯¯3=0.6228,SCR~ale¯¯4=0.5136,SCR~ale¯¯5=0.4817.SCR~ale¯¯1=1.0694,SCR~ale¯¯2=1.2103,SCR~ale¯¯3=1.0667,SCR~ale¯¯4=0.9187,SCR~ale¯¯5=0.8065.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.γ1=4,γ2=5SCR~ale¯¯1=1.4374,SCR~ale¯¯2=1.4913,SCR~ale¯¯3=1.3867,SCR~ale¯¯4=1.3291,SCR~ale¯¯5=1.3019.SCR~ale¯¯1=0.1293,SCR~ale¯¯2=0.3069,SCR~ale¯¯3=0.1218,SCR~ale¯¯4=−0.0897,SCR~ale¯¯5=−0.2203.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.γ1=10,γ2=8SCR~ale¯¯1=1.7306,SCR~ale¯¯2=1.7588,SCR~ale¯¯3=1.7090,SCR~ale¯¯4=1.6619,SCR~ale¯¯5=1.6642.SCR~ale¯¯1=−0.5304,SCR~ale¯¯2=−0.3679,SCR~ale¯¯3=−0.5529,SCR~ale¯¯4=−0.8495,SCR~ale¯¯5=−0.8023.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.γ1=15,γ2=25SCR~ale¯¯1=1.8655,SCR~ale¯¯2=1.8823,SCR~ale¯¯3=1.8496,SCR~ale¯¯4=1.8312,SCR~ale¯¯5=1.8355.SCR~ale¯¯1=−1.0354,SCR~ale¯¯2=−0.7914,SCR~ale¯¯3=−0.9676,SCR~ale¯¯4=−1.2396,SCR~ale¯¯5=−1.2463.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯4>ale¯¯5.γ1=30,γ2=35SCR~ale¯¯1=1.9032,SCR~ale¯¯2=1.9186,SCR~ale¯¯3=1.9010,SCR~ale¯¯4=1.8843,SCR~ale¯¯5=1.8893.SCR~ale¯¯1=−1.1533,SCR~ale¯¯2=−0.2095,SCR~ale¯¯3=−0.2537,SCR~ale¯¯4=−1.4230,SCR~ale¯¯5=−1.3193.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯5>ale¯¯4.γ1=50,γ2=50SCR~ale¯¯1=1.9245,SCR~ale¯¯2=1.9389,SCR~ale¯¯3=1.9288,SCR~ale¯¯4=1.9163,SCR~ale¯¯5=1.9175.SCR~ale¯¯1=−0.3141,SCR~ale¯¯2=0.0656,SCR~ale¯¯3=−0.3119,SCR~ale¯¯4=−0.3697,SCR~ale¯¯5=−0.3531.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯5>ale¯¯4.


### 6.5. Effect of Parameterk on Final Ranking Results

In this subpart, the effect of parameterk is discussed on the final ranking results, where the values of the generalized parameters γ1=γ2=1 and q=2 are fixed. From Table 8, one can observe that for different values of parameter k, the obtained ranking orders are different. When the value of the parameter k=2, that is, when considering interrelationship among input arguments, the best alternatives is ale¯¯2.Table 8
Effect of parameterk on the final ranking results.
ParameterScore value Cq-ROFPGWMSMScore value Cq-ROFPGWDMSMRanking ordersk=2SCR~ale¯¯1=−0.4042,SCR~ale¯¯2=−0.3031,SCR~ale¯¯3=−0.4487,SCR~ale¯¯4=−0.5560,SCR~ale¯¯5=−0.5895.SCR~ale¯¯1=1.6246,SCR~ale¯¯2=1.6950,SCR~ale¯¯3=1.6168,SCR~ale¯¯4=1.5718,SCR~ale¯¯5=1.4965.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.k=3SCR~ale¯¯1=0.3027,SCR~ale¯¯2=0.3552,SCR~ale¯¯3=0.1937,SCR~ale¯¯4=−0.0768,SCR~ale¯¯5=0.0381.SCR~ale¯¯1=1.4176,SCR~ale¯¯2=1.4887,SCR~ale¯¯3=1.4236,SCR~ale¯¯4=1.3206,SCR~ale¯¯5=1.3970.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯5>ale¯¯4.


### 6.6. Comparison with Existing Approaches

In this subpart, we compare our initiated approach with some of the existing approaches such Wang et al. [22], Zhang et al. [25], and Abbas et al. [42]. From Table 9, one can notice that the ranking orders obtained from the existing approaches and the developed approach in this article are slightly different, although the best alternatives obtained from the existing approaches and the approaches initiated in this paper are the same. This shows that the proposed approaches are valid. The developed approaches, which are based on these newly AOs, have certain advantages over the existing developed approaches. The developed approaches based on these newly AOs can remove the effect of awkward data, can take interrelationship among any number of input arguments, at the same time, and have a general parameter, while the existing approaches can only have the capability of removing the effect of awkward data or can consider interrelationship between any two input arguments at a time. Due to the above characteristics, these newly initiated AOs can make the decision-making process more flexible. So, the proposed AOs are more practical and effective while dealing with Cq-ROF information.Table 9
Comparison with existing approaches.
ApproachScore valuesRanking orderCq-PWA operator [22]SCR~ale¯¯1=0.9298,SCR~ale¯¯2=1.2256,SCR~ale¯¯3=1.1570,SCR~ale¯¯4=1.0640,SCR~ale¯¯5=0.8413.ale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.Cq-HM operator [25]SCR~ale¯¯1=−1.3039,SCR~ale¯¯2=−1.0502,SCR~ale¯¯3=−1.0627,SCR~ale¯¯4=−1.1320,SCR~ale¯¯5=−1.2982.ale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.PhyWA [42]SCR~ale¯¯1=1.0372,SCR~ale¯¯2=1.3125,SCR~ale¯¯3=1.2453,SCR~ale¯¯4=1.1450,SCR~ale¯¯5=1.0721.ale¯¯2>ale¯¯3>ale¯¯4>ale¯¯5>ale¯¯1.In this articleSCR~ale¯¯1=−0.4042,SCR~ale¯¯2=−0.3031,SCR~ale¯¯3=−0.4487,SCR~ale¯¯4=−0.5560,SCR~ale¯¯5=−0.5895.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.In this articleSCR~ale¯¯1=1.6246,SCR~ale¯¯2=1.6950,SCR~ale¯¯3=1.6168,SCR~ale¯¯4=1.5718,SCR~ale¯¯5=1.4965.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.


## 6.1. Decision-Making Steps Utilizing Cq-ROFPGWDMSM Operator

Step 1.
Since all the attributes are of the same type, so there is no need to normalize it.Step 2.
Discover the support degreesSupCqdeb,Cqdel,d=1,2,⋯,5,e=1,2,3,c,l=1,2,⋯,4 by the consuming Formula (66). For easiness, we shall write Sdecd=1,2,⋯,5,e=1,2,3,c=1,2,⋯,4 in place of SupCqdec,Cqdel. We have
(79)S11,121=S12,111=0.8991,S11,131=S13,111=0.8791,S12,131=S13,131=0.7808,S21,221=S22,211=0.8816,S21,231=S23,211=0.7808,S22,221=S23,221=0.8991,S31,321=S32,311=0.7808,S31,331=S33,311=0.8816,S32,331=S33,321=0.8991,S41,421=S42,411=0.6131,S41,431=S43,411=0.3447,S42,431=S43,421=0.7315,S51,521=S52,511=0.5639,S51,531=S53,511=0.8816,S52,531=S53,521=0.4455,S11,123=S12,113=0.8991,S11,133=S13,113=0.7315,S12,133=S13,133=0.8324,S21,223=S22,213=0.7808,S21,233=S23,213=0.8991,S22,223=S23,223=0.8816,S31,323=S32,313=0.7315,S31,333=S33,313=0.6131,S32,333=S33,323=0.8816,S41,423=S42,413=0.7808,S41,433=S43,413=0.8816,S42,433=S43,423=0.8991,S51,523=S52,513=0.8324,S51,533=S53,513=0.8991,S52,533=S53,523=0.7315,S11,124=S12,114=0.7808,S11,134=S13,114=0.8991,S12,134=S13,134=0.8816,S21,224=S22,214=0.6131,S21,234=S23,214=0.8816,S22,224=S23,224=0.7315,S31,324=S32,314=0.8324,S31,334=S33,314=0.7315,S32,334=S33,324=0.8991,S41,424=S42,414=0.7315,S41,434=S43,414=0.8991,S42,434=S43,424=0.8324,S51,524=S52,514=0.8324,S51,534=S53,514=0.7808,S52,534=S53,524=0.6131.Step 3.
Discover the support degreeTCqdec by utilizing Formula (67). For easiness, we shall write Tdecc=1,2,..,4,d=1,2,.⋯,5,e=1,2,3 in place of TCqdec. We have
(80)T111=1.7783,T121=1.6799,T131=1.6599,T211=1.6624,T221=1.7808,T231=1.6799,T311=1.6624,T321=1.6799,T331=1.7808,T411=0.9578,T421=1.3447,T431=1.0762,T511=1.4455,T521=1.0095,T531=1.3272,T112=1.8991,T122=1.7983,T132=1.8991,T212=1.7808,T222=1.6624,T232=1.6799,T312=1.6746,T322=1.7044,T332=1.7930,T412=1.8816,T422=1.8816,T432=1.7633,T512=1.7315,T522=1.6306,T532=1.5639,T113=1.6306,T123=1.7315,T133=1.5639,T213=1.6799,T223=1.6624,T233=1.7808,T313=1.3447,T323=1.6131,T333=1.4948,T413=1.6624,T423=1.6799,T433=1.7808,T513=1.7315,T523=1.5639,T533=1.6306,T114=1.6799,T124=1.6624,T134=1.7808,T214=1.4948,T224=1.3447,T234=1.6131,T314=1.5639,T324=1.7315,T334=1.6306,T414=1.6306,T424=1.5639,T434=1.7315,T514=1.6131,T524=1.4455,T534=1.3939.Step 4.
Employ Formula (68) to discover the weights ödec; we have
(81)ö111=1.0253,ö121=0.9879,ö131=0.9867,ö211=0.9871,ö221=1.0257,ö231=0.9871,ö311=0.9871,ö321=0.9871,ö331=1.0257,ö411=0.9156,ö421=1.1174,ö431=0.9670,ö511=1.1659,ö521=0.9171,ö531=1.0457,ö112=1.0122,ö122=0.9755,ö132=1.0122,ö212=1.0257,ö222=0.9871,ö232=0.9871,ö312=0.9839,ö322=0.9942,ö332=1.0220,ö412=1.0122,ö422=1.0122,ö432=0.9755,ö512=1.0395,ö522=0.9990,ö532=0.9615,ö113=0.9990,ö123=1.0395,ö133=0.9615,ö213=0.9871,ö223=0.9871,ö233=1.0257,ö313=0.9307,ö323=1.0561,ö333=1.0132,ö413=0.9871,ö423=0.9871,ö433=1.0257,ö513=1.0395,ö523=0.9615,ö533=0.9990,ö114=0.9871,ö124=0.9871,ö134=1.0257,ö214=1.0132,ö224=0.9307,ö234=1.0561,ö314=0.9615,ö324=1.0395,ö334=0.9990,ö414=0.9990,ö424=0.9615,ö434=1.0395,ö514=1.0561,ö524=0.9736,ö534=0.9704.Step 5.
Employ Formula (69) to aggregate all the individual decision matrices DMT¯¯c=Cqdecm×nc=1,2,⋯,4 into group decision matrix DMT~=Cqdcm×n, which is given in Table 4q=2,γ1=γ2=1.Step 6.
Employ Formula (70) to discover the support degrees SupCqdc,Cqdxd=1,2,⋯,5,c,x=1,2,⋯,4. For easiness, we shall designate Sdc in place of SupCqdc,Cqdx. We have
(82)S12=S21=0.9254,S13=S31=0.8783,S14=S41=0.9835,S23=S32=0.9530,S24=S42=8514,S34=S43=0.8044,S12=S21=0.9937,S13=S31=0.9769,S14=S41=9462,S23=S32=0.9832,S24=S42=0.8736,S34=S43=0.8834,S12=S21=0.9842,S13=S31=0.8931,S14=S41=0.8407,S23=S32=0.8774,S24=S42=0.7712,S34=S43=0.8537,S12=S21=0.7294,S13=S31=0.7559,S14=S41=0.8841,S23=S32=0.9735,S24=S42=0.7960,S34=S43=0.8124,S12=S21=0.9481,S13=S31=0.9509,S14=S41=0.9770,S23=S32=0.9972,S24=S42=0.8654,S34=S43=0.8682.Step 7.
Employ Formula (71), to discover the support degree TCqdc. For easiness, we shall write Tdc in place of TCqdc. We have
(83)T11=2.7872,T12=2.7297,T13=2.6357,T14=2.6392,T21=2.9169,T22=2.8505,T23=2.8435,T24=2.7032,T31=2.7180,T32=2.6328,T33=2.6242,T34=2.4656,T41=2.3693,T42=2.4989,T43=2.5418,T44=2.4924,T51=2.8760,T52=2.8106,T53=2.8163,T54=2.7106.Step 8.
By employing Formula (72), to discover the weighting vector ödcd=1,2,.⋯,5,c=1,2,⋯,4, we have
(84)ö11=1.0241,ö12=1.0086,ö13=0.9832,ö14=0.9841,ö21=1.0231,ö22=1.0057,ö23=1.0039,ö24=0.9673,ö31=1.0299,ö32=1.0063,ö33=1.0039,ö34=0.9600,ö41=0.9694,ö42=1.0067,ö43=1.0191,ö44=1.0048,ö51=1.0191,ö52=1.0019,ö53=1.0034,ö54=0.9756.Step 9.
Employ Formula (73) to aggregate all assessment values ale¯¯dcd=1,2,.⋯,5,c=1,2,⋯,4 into overall assessment value Cqdd=1,2,.⋯,5 corresponding to the alternatives ale¯¯dd=1,2,⋯,5:(85)q=2,γ1=γ2=1ale¯¯1=0.4681,0.5503,0.6724,0.7035,0.5506,0.7035,ale¯¯2=0.4914,0.5794,0.6558,0.6895,0.5799,0.6893,ale¯¯3=0.4575,0.5402,0.6772,0.7127,0.5404,0.7127,ale¯¯4=0.4399,0.5199,0.7000,0.7363,0.5199,0.7363,ale¯¯5=0.4306,0.5085,0.7063,0.7410,0.5085,0.7390.Step 10.
By employing Definition6 to discover the scores SCR~ale¯¯d, we have
SCR~ale¯¯1=−0.4042,SCR~ale¯¯2=−0.3031,SCR~ale¯¯3=−0.4487,SCR~ale¯¯4=−0.5560,SCR~ale¯¯5=−0.5895.Step 11.
According to score values, the ranking order of the alternativesale¯¯dd=1,2,⋯,5 is ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5. According to the ranking order, the best alternative is ale¯¯2 (i.e., Dr. Tahir Mahmood), while the last one is ale¯¯5 (i.e., Dr. Usman Nasir).Table 4
Collective decision matrixDMT~ employing Cq-ROFPGWMSM operator.
Alternative/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯5ale¯¯10.5916,0.6899,0.4974,0.5405,0.6914,0.54050.5275,0.6159,0.5551,0.5935,0.6159,0.59350.4889,0.5750,0.5891,0.6368,0.5750,0.63680.5791,0.6761,0.5128,0.5538,0.6761,0.553811ale¯¯20.5975,0.6980,0.4935,0.5372,0.6980,0.53650.5916,0.6914,0.4975,0.5410,0.6914,0.54050.5791,0.6761,0.5128,0.5538,0.6761,0.55380.5568,0.6555,0.5367,0.5884,0.6555,0.5884ale¯¯30.5975,0.6980,0.4935,0.5365,0.6980,0.53650.5974,0.6979,0.4595,0.5067,0.6979,0.50670.5136,0.6057,0.5772,0.6305,0.6068,0.63050.4606,0.5448,0.6111,0.6619,0.5448,0.6619ale¯¯40.4023,0.4804,0.6852,0.7441,0.4804,0.74410.6159,0.7191,0.4665,0.5123,0.7191,0.51320.5975,0.6980,0.4935,0.5373,0.6980,0.53650.4889,0.5750,0.5891,0.6368,0.5750,0.6368ale¯¯50.5453,0.6398,0.5676,0.6179,0.6398,0.61790.4851,0.5709,0.5922,0.6394,0.57087,0.63940.4851,0.5709,0.5922,0.6394,0.5709,0.62600.5140,0.6052,0.5744,0.6221,0.6052,0.6221


## 6.2. Decision-Making Steps Utilizing Cq-ROFPGWDMSM Operator

Steps 1–4. These are the same as those for decision-making model utilizing Cq-ROFPGWMSM operator.Step 5. Employ Formula (74) to aggregate all the individual decision matrices DMT¯¯c=Cqdecm×nc=1,2,⋯,4 into group decision matrix DMT~=Cqdcm×n, which is given in Table 5. q=2,γ1=γ2=1.Table 5
Collective decision matrixDMT~employing Cq-ROFPGWDMSM operator.
Alternatives/attributesCri¯¯1Cri¯¯2Cri¯¯3Cri¯¯4ale¯¯10.8622,0.9180,0.1565,0.1937,0.9198,0.19370.8207,0.8774,0.2066,0.2445,0.8774,0.24450.8010,0.8599,0.2294,0.2787,0.8599,0.27870.8594,0.9181,0.1585,0.1965,0.9181,0.1965ale¯¯20.8712,0.9306,0.1427,0.1807,0.9306,0.18070.8622,0.9198,0.1565,0.1936,0.9198,0.19360.8594,0.9181,0.1585,0.1965,0.9181,0.19650.8503,0.9140,0.1707,0.2197,0.9140,0.2197ale¯¯30.7940,0.9306,0.1427,0.1807,0.9306,0.18070.8712,0.9306,0.1210,0.1589,0.9306,0.15890.8319,0.8955,0.1919,0.2430,0.8955,0.24300.7851,0.8445,0.24738,0.2992,0.8445,0.2992ale¯¯40.7738,0.8333,0.3134,0.3787,0.8333,0.37870.8774,0.9342,0.1313,0.1688,0.9342,0.16880.8712,0.9306,0.1427,0.1807,0.9306,0.18070.8010,0.8599,0.2294,0.2787,0.8599,0.2787ale¯¯50.8179,0.8800,0.2303,0.2815,0.8800,0.28150.7909,0.8494,0.2427,0.2925,0.8494,0.29250.7909,0.8494,0.2427,0.2925,0.8494,0.29250.8111,0.8705,0.2294,0.278015,0.8705,0.278015Step 6. Employ Formula (75) to discover the support degrees SupCqdc,Cqdxd=1,2,⋯,5,c,x=1,2,⋯,4. For easiness, we shall designate Sdc in place of SupCqdc,Cqdx. We have
(86)S12=S21=0.9525,S13=S31=0.9295,S14=S41=0.9977,S23=S32=0.9769,S24=S42=0.9286,S34=S43=0.9056,S12=S21=0.9884,S13=S31=0.9861,S14=S41=0.9771,S23=S32=0.9977,S24=S42=0.9648,S34=S43=0.9661,S12=S21=0.9751,S13=S31=0.9568,S14=S41=0.9209,S23=S32=0.9525,S24=S42=0.8992,S34=S43=0.9254,S12=S21=0.8602,S13=S31=0.8662,S14=S41=0.9483,S23=S32=0.9940,S24=S42=0.9129,S34=S43=0.9166,S12=S21=0.9721,S13=S31=0.9728,S14=S41=0.9919,S23=S32=1.0000,S24=S42=0.9383,S34=S43=0.9383.Step 7. Employ Formula (76), to discover the support degree TCqdc. For easiness, we shall write Tdc in place of TCqdc. We have
(87)T11=2.8797,T12=2.8581,T13=2.8120,T14=2.8319,T21=2.9516,T22=2.9509,T23=2.9498,T24=2.9080,T31=2.8529,T32=2.8269,T33=2.8347,T34=2.7456,T41=2.6747,T42=2.7672,T43=2.7769,T44=2.7779,T51=2.9368,T52=2.9104,T53=2.9111,T54=2.8685.Step 8. By employing Formula (77), to discover the weighting vector Õdcd=1,2,.⋯,5,c=1,2,⋯,4, we have
(88)Ł11=1.0089,Ł12=1.0033,Ł13=0.9913,Ł14=0.9965,Ł21=1.0029,Ł22=1.0027,Ł23=1.0025,Ł24=0.9919,Ł31=1.0099,Ł32=1.0031,Ł33=1.0052,Ł34=0.9818,Ł41=0.9801,Ł42=1.0048,Ł43=1.0074,Ł44=1.0077,Ł51=1.0077,Ł52=1.0009,Ł53=1.0011,Ł54=0.9902.Step 9. Employ Formula (73) to aggregate all assessment values ale¯¯dcd=1,2,.⋯,5,c=1,2,⋯,4 into overall assessment value Cqdd=1,2,.⋯,5 corresponding to the alternatives ale¯¯dd=1,2,⋯,5, q=2,γ1=γ2=1(89)ale¯¯1=0.8887,0.9292,0.1548,0.1885,0.9292,0.1885,ale¯¯2=0.9028,0.9445,0.1324,0.1670,0.9445,0.1670,ale¯¯3=0.8738,0.9310,0.1511,0.1889,0.9310,0.1889,ale¯¯4=0.8805,0.9217,0.1808,0.2218,0.9217,0.2218,ale¯¯5=0.8639,0.9053,0.1980,0.2399,0.9053,0.2399.Step 10. By employing Definition6 to discover the scores SCR~ale¯¯d, we have
(90)SCR~ale¯¯1=1.6246,SCR~ale¯¯2=1.6950,SCR~ale¯¯3=1.6168,SCR~ale¯¯4=1.5718,SCR~ale¯¯5=1.4965.Step 11. According to score values, the ranking order of the alternativesale¯¯dd=1,2,⋯,5 is ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5. According to the ranking order, the best alternative is (i.e., Dr. Tahir Mahmood), while the last one is ale¯¯5 (i.e., Dr. Usman Nasir).


## 6.3. Effect of Parameterq on Final Ranking Result

In this subpart, the influence of the parameterq is discussed on the final ranking results while utilizing Cq-ROFPGWMSM and Cq-ROFPGWDMSM operators. The score values and ranking orders are given in Table 6. From Table 6, one can observe distinct values of the parameter q, while the values of generalized parameters γ1=γ2=1 and k=2 are fixed, the obtained ranking orders are different. When the value of parameter q=25, while utilizing Cq-ROFPGWMSM operator the best alternative remains the same, but the worst alternative is ale¯¯4. One can also observe from Table 6, when the value of parameter q increases, the score values increase utilizing Cq-ROFPGWMSM operator. Similarly, when the value of parameter q increases, the score values decrease utilizing Cq-ROFPGWDMSM operator. So, the decision-makers choose the value of parameter q according to the actual need of the situation.Table 6
Effect of parameterq on the final ranking results utilizing Cq-ROFPGWMSM and Cq-ROFPGWDMSM operator.
ParameterScore value Cq-ROFPGWMSMScore value Cq-ROFPGWDMSMRanking order Cq-ROFPGWMSM and Cq-ROFPGWDMSMq=3SCR~ale¯¯1=−0.31159,SCR~ale¯¯2=−0.20941,SCR~ale¯¯3=−0.3605,SCR~ale¯¯4=−0.4766,SCR~ale¯¯5=−0.5062.SCR~ale¯¯1=1.5550,SCR~ale¯¯2=1.6360,SCR~ale¯¯3=1.5475,SCR~ale¯¯4=1.5113,SCR~ale¯¯5=1.4323.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.q=5SCR~ale¯¯1=−0.1986,SCR~ale¯¯2=−0.1021,SCR~ale¯¯3=−0.2459,SCR~ale¯¯4=−0.3669,SCR~ale¯¯5=−0.3831.SCR~ale¯¯1=1.3895,SCR~ale¯¯2=1.4957,SCR~ale¯¯3=1.3902,SCR~ale¯¯4=1.3677,SCR~ale¯¯5=1.2791.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.q=10SCR~ale¯¯1=−0.0725,SCR~ale¯¯2=−0.0022,SCR~ale¯¯3=−0.1045,SCR~ale¯¯4=−0.2100,SCR~ale¯¯5=−0.1980.SCR~ale¯¯1=1.0841,SCR~ale¯¯2=1.2357,SCR~ale¯¯3=1.1104,SCR~ale¯¯4=1.0901,SCR~ale¯¯5=0.9928.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.q=18SCR~ale¯¯1=−0.0129,SCR~ale¯¯2=0.0205,SCR~ale¯¯3=−0.0283,SCR~ale¯¯4=−0.0924,SCR~ale¯¯5=−0.0709.SCR~ale¯¯1=0.7950,SCR~ale¯¯2=0.9774,SCR~ale¯¯3=0.8473,SCR~ale¯¯4=0.8116,SCR~ale¯¯5=0.7171.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.q=25SCR~ale¯¯1=0.0008,SCR~ale¯¯2=0.0164,SCR~ale¯¯3=−0.0052,SCR~ale¯¯4=−0.0379,SCR~ale¯¯5=−0.0282.SCR~ale¯¯1=0.6339,SCR~ale¯¯2=0.8236,SCR~ale¯¯3=0.6978,SCR~ale¯¯4=0.6506,SCR~ale¯¯5=0.5604.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.q=50SCR~ale¯¯1=0.0008,SCR~ale¯¯2=0.0017,SCR~ale¯¯3=0.0011,SCR~ale¯¯4=0.0006,SCR~ale¯¯5=0.0002.SCR~ale¯¯1=0.3264,SCR~ale¯¯2=0.4961,SCR~ale¯¯3=0.4019,SCR~ale¯¯4=0.3325,SCR~ale¯¯5=0.2641.ale¯¯2>ale¯¯3>ale¯¯1>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.


## 6.4. Effect of the Generalized Parameter

In this subpart, the influence of the generalized parametersγ1,γ2 are discussed on the final ranking results while utilizing Cq-ROFPGWMSM and Cq-ROFPGWDMSM operators. The values of the parameter q=2,k=2 are fixed. The score values and ranking orders are given in Table 7. From Table 7, one can observe that for distinct values of the parameter γ1,γ2, the obtained ranking orders are different. When the value of the parameter γ1=15,γ2=25, while utilizing Cq-ROFPGWMSM operator, the best alternative remains the same, but the worst alternative is ale¯¯4. One can also observe from Table 7, when the value of the parameters γ1,γ2 increases, the score values increases utilizing Cq-ROFPGWMSM operator. Similarly, when the value of the parameters γ1,γ2 increases, the score values decrease utilizing Cq-ROFPGWDMSM operator. So, the decision-makers choose the value of the parameters γ1,γ2 according to the actual need of the situation.Table 7
Influence of the generalized parametersγ1,γ2 on final ranking results.
ParameterScore value Cq-ROFPGWMSMScore value Cq-ROFPGWDMSMRanking orderγ1=2,γ2=2SCR~ale¯¯1=0.6819,SCR~ale¯¯2=0.7760,SCR~ale¯¯3=0.6228,SCR~ale¯¯4=0.5136,SCR~ale¯¯5=0.4817.SCR~ale¯¯1=1.0694,SCR~ale¯¯2=1.2103,SCR~ale¯¯3=1.0667,SCR~ale¯¯4=0.9187,SCR~ale¯¯5=0.8065.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.γ1=4,γ2=5SCR~ale¯¯1=1.4374,SCR~ale¯¯2=1.4913,SCR~ale¯¯3=1.3867,SCR~ale¯¯4=1.3291,SCR~ale¯¯5=1.3019.SCR~ale¯¯1=0.1293,SCR~ale¯¯2=0.3069,SCR~ale¯¯3=0.1218,SCR~ale¯¯4=−0.0897,SCR~ale¯¯5=−0.2203.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.γ1=10,γ2=8SCR~ale¯¯1=1.7306,SCR~ale¯¯2=1.7588,SCR~ale¯¯3=1.7090,SCR~ale¯¯4=1.6619,SCR~ale¯¯5=1.6642.SCR~ale¯¯1=−0.5304,SCR~ale¯¯2=−0.3679,SCR~ale¯¯3=−0.5529,SCR~ale¯¯4=−0.8495,SCR~ale¯¯5=−0.8023.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.γ1=15,γ2=25SCR~ale¯¯1=1.8655,SCR~ale¯¯2=1.8823,SCR~ale¯¯3=1.8496,SCR~ale¯¯4=1.8312,SCR~ale¯¯5=1.8355.SCR~ale¯¯1=−1.0354,SCR~ale¯¯2=−0.7914,SCR~ale¯¯3=−0.9676,SCR~ale¯¯4=−1.2396,SCR~ale¯¯5=−1.2463.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯4>ale¯¯5.γ1=30,γ2=35SCR~ale¯¯1=1.9032,SCR~ale¯¯2=1.9186,SCR~ale¯¯3=1.9010,SCR~ale¯¯4=1.8843,SCR~ale¯¯5=1.8893.SCR~ale¯¯1=−1.1533,SCR~ale¯¯2=−0.2095,SCR~ale¯¯3=−0.2537,SCR~ale¯¯4=−1.4230,SCR~ale¯¯5=−1.3193.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯5>ale¯¯4.γ1=50,γ2=50SCR~ale¯¯1=1.9245,SCR~ale¯¯2=1.9389,SCR~ale¯¯3=1.9288,SCR~ale¯¯4=1.9163,SCR~ale¯¯5=1.9175.SCR~ale¯¯1=−0.3141,SCR~ale¯¯2=0.0656,SCR~ale¯¯3=−0.3119,SCR~ale¯¯4=−0.3697,SCR~ale¯¯5=−0.3531.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯5>ale¯¯4.


## 6.5. Effect of Parameterk on Final Ranking Results

In this subpart, the effect of parameterk is discussed on the final ranking results, where the values of the generalized parameters γ1=γ2=1 and q=2 are fixed. From Table 8, one can observe that for different values of parameter k, the obtained ranking orders are different. When the value of the parameter k=2, that is, when considering interrelationship among input arguments, the best alternatives is ale¯¯2.Table 8
Effect of parameterk on the final ranking results.
ParameterScore value Cq-ROFPGWMSMScore value Cq-ROFPGWDMSMRanking ordersk=2SCR~ale¯¯1=−0.4042,SCR~ale¯¯2=−0.3031,SCR~ale¯¯3=−0.4487,SCR~ale¯¯4=−0.5560,SCR~ale¯¯5=−0.5895.SCR~ale¯¯1=1.6246,SCR~ale¯¯2=1.6950,SCR~ale¯¯3=1.6168,SCR~ale¯¯4=1.5718,SCR~ale¯¯5=1.4965.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.andale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.k=3SCR~ale¯¯1=0.3027,SCR~ale¯¯2=0.3552,SCR~ale¯¯3=0.1937,SCR~ale¯¯4=−0.0768,SCR~ale¯¯5=0.0381.SCR~ale¯¯1=1.4176,SCR~ale¯¯2=1.4887,SCR~ale¯¯3=1.4236,SCR~ale¯¯4=1.3206,SCR~ale¯¯5=1.3970.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯5>ale¯¯4.andale¯¯2>ale¯¯3>ale¯¯1>ale¯¯5>ale¯¯4.


## 6.6. Comparison with Existing Approaches

In this subpart, we compare our initiated approach with some of the existing approaches such Wang et al. [22], Zhang et al. [25], and Abbas et al. [42]. From Table 9, one can notice that the ranking orders obtained from the existing approaches and the developed approach in this article are slightly different, although the best alternatives obtained from the existing approaches and the approaches initiated in this paper are the same. This shows that the proposed approaches are valid. The developed approaches, which are based on these newly AOs, have certain advantages over the existing developed approaches. The developed approaches based on these newly AOs can remove the effect of awkward data, can take interrelationship among any number of input arguments, at the same time, and have a general parameter, while the existing approaches can only have the capability of removing the effect of awkward data or can consider interrelationship between any two input arguments at a time. Due to the above characteristics, these newly initiated AOs can make the decision-making process more flexible. So, the proposed AOs are more practical and effective while dealing with Cq-ROF information.Table 9
Comparison with existing approaches.
ApproachScore valuesRanking orderCq-PWA operator [22]SCR~ale¯¯1=0.9298,SCR~ale¯¯2=1.2256,SCR~ale¯¯3=1.1570,SCR~ale¯¯4=1.0640,SCR~ale¯¯5=0.8413.ale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.Cq-HM operator [25]SCR~ale¯¯1=−1.3039,SCR~ale¯¯2=−1.0502,SCR~ale¯¯3=−1.0627,SCR~ale¯¯4=−1.1320,SCR~ale¯¯5=−1.2982.ale¯¯2>ale¯¯3>ale¯¯4>ale¯¯1>ale¯¯5.PhyWA [42]SCR~ale¯¯1=1.0372,SCR~ale¯¯2=1.3125,SCR~ale¯¯3=1.2453,SCR~ale¯¯4=1.1450,SCR~ale¯¯5=1.0721.ale¯¯2>ale¯¯3>ale¯¯4>ale¯¯5>ale¯¯1.In this articleSCR~ale¯¯1=−0.4042,SCR~ale¯¯2=−0.3031,SCR~ale¯¯3=−0.4487,SCR~ale¯¯4=−0.5560,SCR~ale¯¯5=−0.5895.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.In this articleSCR~ale¯¯1=1.6246,SCR~ale¯¯2=1.6950,SCR~ale¯¯3=1.6168,SCR~ale¯¯4=1.5718,SCR~ale¯¯5=1.4965.ale¯¯2>ale¯¯1>ale¯¯3>ale¯¯4>ale¯¯5.


## 7. Conclusion

The cubicq-rung orthopair fuzzy set (Cq-ROFS) is a fundamental simplification of numerous fuzzy concepts and one of the latest generalizations of q-ROFSs. Cq-ROFS exposes fuzzy judgement to the degrees of INVq-ROFS and q-ROFS, and it is one of the sophisticated mathematical tools for dealing with more intricate assessment information in multiple attribute group decision-making (MAGDM) situations. The article mainly introduced two types of hybrid AOs such as cubic q-rung orthopair fuzzy power generalized Maclaurin symmetric mean operator and cubic q-rung orthopair fuzzy power generalized dual Maclaurin symmetric mean operator. Then, a few of its core characteristics and some of the existing AOs are obtained from the proposed AOs by assigning different values to general parameters. Furthermore, weighted forms of the initiated AOs are also proposed. After that, two MAGDM models are offered based on Cq-OFPGWMSM and Cq-ROFPGWDMSM operators. Lastly, a numerical example about the selection of the best assistant professor of the year is taken as an application to show the effectiveness and capability of the initiated approaches. There is also a comparison with some existing decision-making models and discussed advantages of the proposed two MAGDM models over the existing MAGDM models for Cq-ROFS.In the future, we will explore these AOs for other generalizations of linear Diophantine fuzzy set [34], T-spherical fuzzy sets [40], spherical linear Diophantine fuzzy set [41], and spherical linear Diophantine fuzzy soft rough set [43] on different t-norm and t-conorm and apply these AOs to solve MADM and MAGDM problems in different fields.



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*Source: 9056605-2022-08-05.xml*