# Group Generalized q-Rung Orthopair Fuzzy Soft Sets: New Aggregation Operators and Their Applications

Authors: Khizar Hayat; Raja Aqib Shamim; Hussain AlSalman; Abdu Gumaei; Xiao-Peng Yang; Muhammad Azeem Akbar
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

In recent years, q-rung orthopair fuzzy sets have been appeared to deal with an increase in the value ofq>1, which allows obtaining membership and nonmembership grades from a larger area. Practically, it covers those membership and nonmembership grades, which are not in the range of intuitionistic fuzzy sets. The hybrid form of q-rung orthopair fuzzy sets with soft sets have emerged as a useful framework in fuzzy mathematics and decision-makings. In this paper, we presented group generalized q-rung orthopair fuzzy soft sets (GGq-ROFSSs) by using the combination of q-rung orthopair fuzzy soft sets and q-rung orthopair fuzzy sets. We investigated some basic operations on GGq-ROFSSs. Notably, we initiated new averaging and geometric aggregation operators on GGq-ROFSSs and investigated their underlying properties. A multicriteria decision-making (MCDM) framework is presented and validated through a numerical example. Finally, we showed the interconnection of our methodology with other existing methods.


---

## Body

## 1. Introduction

Zadeh originated the fuzzy set (FS) as an enlargement of the standard sets by the concept of inclusion of vague human judgements in computing situations [1]. The FS is indicated by the fuzzy information μ, which gives values from the unit close interval 0,1 for each prospector x∈X. The idea of the FS plays an important role in the domain of soft computing, which manages vagueness, robustness, and partial truth. In some real-world difficulties where humanoid though attains reliable and unreliable information, the FS may not be sufficient to deal with underlying uncertainties.In 1986, another shape of the FS called intuitionistic fuzzy sets (IFSs) was authorized by Atanassov, which provide a reliable gradeμx and unreliable grade νx for all x in the universe of discourse X. The IFSs are characterized by the sum μx+νx≤1 and the degree of indeterminacy πx=1−μx−νx [2]. Xu and Yager [3, 4] discussed the intuitionistic fuzzy value (IFV), which is an ordered pair of reliable and unreliable information for a component in the IFS on any x. Different rudiments of IFSs have been established such as aggregation operators [4], similarity and distance function [5, 6], and multicriteria decision-makings (MCDM) [7]. The aggregation operators are imperious in the MCDM process, which attains a shape of the measurable information by the accumulation of big data [8–10].The IFSs enhance FSs in a meaningful approach, which is more capable of overcoming uncertainties, sharpless boundaries caused by the hesitation, and lack of assurance in human cognition. Xu and Zhao [11] extended a meaningful and insightful view on the information synthesis for MCDM using IFSs. To deal with real-life cases of reliable and unreliable information, which do not satisfy inequality μx+νx≤1, Yager initiated Pythagorean fuzzy sets (PFSs) [12, 13] and q-rung orthopair fuzzy sets (q-ROFSs) [14], which are crucial generalizations of IFSs. The q-ROFSs possess overall anticipation of symmetry of reliable and unreliable information in a larger space [15], that is, μxq+νxq≤1. A q-ROFS appears as an IFS (PFS) when q=1q=2. The fundamental score function and operators for q-ROFSs were investigated by Liu and Wang [16]. Several basic properties of PFSs and q-ROFSs can be seen in the literature [17–21]. Notably, researchers around the planet check out hybrid MCDM methods of PFSs and q-ROFSs using TODIM [22], TOPSIS [23, 24], MULTIMOORA [25], MABAC method [26], aggregation operators [17, 27–33], entropy measures [34], and distance measures [35].A general parametrization model called soft set theory initiated by Molodtsov [36] has a great tendency to cop uncertainties. The soft set is free of inadequacy as it is a classical tool for coping parameters. It is further connected with usual mathematical operations on sets by Maji et al. [37] and Ali et al. [38]. A combination of soft sets and FSs known as fuzzy soft sets was introduced by Maji et al. [39], and it has been applied in various fields [40–46]. An extended form of fuzzy soft sets, known as intuitionistic fuzzy soft sets (IFSSs), was initiated by Maji et al. [47]. Recently, q-rung orthopair fuzzy soft sets (q-ROFSSs) have been introduced by Hamid et al. [48]. The model of q-ROFSS is a valuable tool to deal with vagueness by means of the label of parameters along with reliable and unreliable grades in the larger space [49]. Hussain et al. [50] presented MCDM techniques using averaging operators on q-ROFSSs. The generalized IFSSs (GIFSSs) were investigated by Agarwal et al. [51], and it possesses an important opinion with the model IFSS. A different scenario that overcomes the adequacies [52] of the original concept of GIFSSs was given by Feng et al. [53]. Both the ideas of GIFSSs were extended by several researchers Garg and Arora [54], Hayat et al. [55, 56], and Khan et al. [57, 58]. GGIFSSs produce a deep and meaningful insight in the MCDM problem by merging aggregation operators [56]. Another aspect of GGIFSS-based operators has been investigated by Hayat et al. [59], which handle information in a collected form. On the prospect of group-based GIFSSs (GGIFSSs) [56, 59], it is required to develop underlying operators, which can handle MCDM problems in different scenarios of combinations of information. More importantly, the extended space of q-ROFSs is the general form to deal with any implicit information.On this prospect, there is a huge capacity to exercise another view of GGIFSS aggregation operators because the q-ROFSs relays the ambiguous information in higher productive ways than the GGIFSSs. Another important and fundamental point is to develop a different study to GGIFSSs that aggregate information concerning attributes until final ranking appears. Thus, we developed the group-based generalized q-ROFSSs (GGq-ROFSSs) and new aggregation operators through entire components in GGq-ROFSSs. By motivations of the above discussion, the purpose and aim of this article are given as follows:(1)
To initiate a different form of aggregation operators for GGq-ROFSSs that do not abandon the importance of attributes initially and do not quickly fascinate alternatives(2)
To develop an internal mechanism that gently addresses the importance of parameters in aggregation operators for GGq-ROFSSs(3)
To address the higher range of reliable and unreliable information in GGq-ROFSSs for possible values ofq(4)
To develop the MCDM method for GGq-ROFSSs environmentIn Section2, we recall basic ideas of IFSs, PFSs, q-ROFSs, soft sets, and q-ROFSSs. In Sections 3 and 4, we discuss the notions of GGq-ROFSSs and their operations. In Section 5, we define new aggregations operators on GGq-ROFSSs. In Section 6, we give a new method of MCDM and a numerical example of real-life applications. Section 7 gives comparisons with other existing methods, and the last section concludes the paper.


## 2. Preliminaries

In this section, we will recall the concepts of IFSs, PFSs, q-ROFSs, soft sets, and q-ROFSSs. Throughout this section,X will represent the collection of alternatives.


### 2.1. Intuitionistic Fuzzy Sets and Pythagorean Fuzzy Sets

A FS is a mappingμ:X⟶0,1, where μ is membership grade for an element g∈X [1]. In several real-life situations, reliable and unreliable information rectifies the proper signification of uncertainties. FSs were not sufficient in such situations; therefore, the concept of IFS was introduced.Definition 1 (see [2]).
An IFSℱ is expressed as follows:(1)ℱ=g,μg,νg|g∈X.
With the functionsμ:X⟶0,1, ν:X⟶0,1 called reliable and unreliable grades of an element g of X under the condition that the following inequality holds:(2)0≤μg+νg≤1.
For an elementg∈X, μ,ν is called IF value (IFV) in ℱ. In an IFS, the hesitancy of an IFV μ,ν to ℱ is given by(3)πℱ=1−μg−νg.
The hesitancy of IFV is also called indeterminacy of theg∈X in ℱ.Definition 2 (see [12, 13]).
A PFSℱ is expressed as follows:(4)P=g,μg,νg|g∈X.
With the functionsμ:X⟶0,1, ν:X⟶0,1 called reliable and unreliable grades of an element g of X under the condition that the following inequality holds:(5)0≤μg2+νg2≤1.
For an elementg∈X, μ,ν is called PF value (PFV) in P. In an PFS, the hesitancy (or indeterminacy) of an PFV μ,ν to P is given by(6)πP=1−μg2−νg2.


### 2.2. q-Rung Orthopair Fuzzy Sets

In 2016, Yager extended the range of double-graded fuzzy models in higher space. It is defined asDefinition 3 (see [14]).
A q-RFSℛ is defined as(7)ℛ=g,μg,νg|g∈X.
With the functionsμ:X⟶0,1, ν:X⟶0,1 called reliable and unreliable grades of an element g of X under the condition that the following inequality holds:(8)0≤μgq+νgq≤1.
Particularly, the hesitancy degree for q-ROFS is given as(9)πℛ=1−μg2−νg2q.
The pairμ,ν is called q-rung orthopair fuzzy value (q-ROFV) for an object g∈X. Let ℛ1=g,μg,νg|g∈X and ℛ2=g,sg,tg|g∈X be two q-ROFSs; then,(10)ℛ1∪ℛ2=g,maxμg,sg,minνg,tg|g∈X,ℛ1∩ℛ2=g,minμg,sg,maxνg,tg|g∈X,ℛ1⊆ℛ2⇔μg≤sg,νg≤tg∀g∈X,ℛ1c=g,νg,μg|g∈X.
The study on q-ROFS is extended by Liu and Wang [16] in the following crucial notions:Definition 4 (see [16]).
Letb=μ,ν be q-ROFN; then, the score function is defined as follows:(11)Sliub=μq−νq.
This notion is effective when we have to transfer a q-ROFN to a real value in interval−1,1 and therefore we can compare two or more q-ROFNs on their score functions. Consider a special case when b1=μ1,ν1, b2=μ2,ν2 with the condition μi=νi∀i=1,2. Then, sb1=0=sb2; thus, Sliu has an inadequacy for such a case. Therefore, in this study, when Sliu fails, we will use the following.Definition 5 (see [32]).
Letb=μ,ν be q-ROFN; then,(12)Spxb=eμq−νqeμq−νq+1−12πq+μq−νq.Definition 6 (see [16]).
Let us take the collection of q-ROFNsbk′=μk′,νk′, k′=1,2,3,…,n and weight vector λ1,λ2,…,λn; then, the q-rung orthopair fuzzy weighted averaging operator (q-ROFWA) is indicated as(13)q−ROFWAb1,b2,…,bn=1−∏k′=1n1−μk′qλk′q,∏k′=1nνk′λk′.Definition 7 (see [16]).
Let us take the collection of q-ROFNsbk′=μk′,νk′, k′=1,2,3,…,n and weight vector λ1,λ2,…,λn; then, the q-rung orthopair fuzzy weighted geometric operator (q-ROFWG) is indicated as(14)q−ROFWGb1,b2,…,bn=∏k′=1nμk′λk′,1−∏k′=1n1−νk′qλk′q.
The q-ROFWA and q-ROFWG are effective to aggregate data involving large number of q-ROFNs to a single q-ROFN.


### 2.3. q-Rung Orthopair Fuzzy Soft Sets

Jointly the approach of soft sets [36] with q-ROFSs is known as the q-rung orthopair fuzzy soft set, which is handy in MCDM problems. The notion of soft set is described as follows.Definition 8 (see [36]).
LetX be a fixed set and PX be the set of all subsets of X. Consider the set of parameters E and A be the subset of E. Let us define a function S as(15)S:A⟶PX.
Then, pairS,A=g,Sg|g∈A,Sg∈PX is called the soft set.Definition 9 (see [49]).
LetX,E be a soft universe and A⊆E. Define a mapping Q:A⟶q−ROFSX; then, pair Q,A is called the q-rung orthopair fuzzy soft set (q-ROFSS) over X, where q−ROFSX denotes the collection of all q-ROFSs over X. The q-ROFSS Q,A can be described as(16)Q,A=e,ρ,μρ,νρ|e∈A,ρ∈X,orQ,A=e,ρμρ,νρ|e∈A,ρ∈X,where μρ:X⟶0,1 and νρ:X⟶0,1 are reliable and nonreliable grades of ρ fulfilling inequality 0≤μqρ+νqρ≤1. Take μij=μejρi, νij=νejρi, where i=1,2,…,m,j=1,2,…,n. Then, the tabular form of q-ROFSS Q,A is given in Table 1.
For simplicity, we will representQ,A by QA.Table 1
q-ROFSSQ,A.
X|Ae1e2⋯enρ1μ11,ν11μ12,ν12⋯μ1n,ν1nρ2μ21,ν21μ22,ν22⋯μ2n,ν2n⋮⋮⋮⋱⋮ρmμm1,νm1μm2,νm2⋯μmn,νmnDefinition 10 (see [49]).
Consider two q-ROFSSsQA1 and QA2′ on X. We say QA1⊆QA2′ if and only if(i)
A1⊆A2(ii)
Qe is q-ROF subset of Q′e for all e∈A1⊆A2Definition 11 (see [49]).
Consider two q-ROFSSsQA1′ and QA2″ on X. Then, QA=A1∪A2=QA1′∪QA2″ denotes the union of QA1′ and QA2″ on X, such that(17)Qe=Q′e,ife∈A1−A2,Q″e,ife∈A2−A1,Q′e∪Q″e,ife∈A1∩A2,where Q′e∪Q″e is the union of q-ROFSs Q′e,Q″e.Definition 12 (see [49]).
Consider two q-ROFSSsQA1′ and QA2″ on X. Then, QA=A1∩A2=QA1′∩QA2″ denotes the intersection of QA1′ and QA2″ on X, such that Qe=Q′e∩Q″e, where e∈A1∩A2≠∅, where Q′e∩Q″e is the intersection of q-ROFSs Q′e,Q″e.


## 2.1. Intuitionistic Fuzzy Sets and Pythagorean Fuzzy Sets

A FS is a mappingμ:X⟶0,1, where μ is membership grade for an element g∈X [1]. In several real-life situations, reliable and unreliable information rectifies the proper signification of uncertainties. FSs were not sufficient in such situations; therefore, the concept of IFS was introduced.Definition 1 (see [2]).
An IFSℱ is expressed as follows:(1)ℱ=g,μg,νg|g∈X.
With the functionsμ:X⟶0,1, ν:X⟶0,1 called reliable and unreliable grades of an element g of X under the condition that the following inequality holds:(2)0≤μg+νg≤1.
For an elementg∈X, μ,ν is called IF value (IFV) in ℱ. In an IFS, the hesitancy of an IFV μ,ν to ℱ is given by(3)πℱ=1−μg−νg.
The hesitancy of IFV is also called indeterminacy of theg∈X in ℱ.Definition 2 (see [12, 13]).
A PFSℱ is expressed as follows:(4)P=g,μg,νg|g∈X.
With the functionsμ:X⟶0,1, ν:X⟶0,1 called reliable and unreliable grades of an element g of X under the condition that the following inequality holds:(5)0≤μg2+νg2≤1.
For an elementg∈X, μ,ν is called PF value (PFV) in P. In an PFS, the hesitancy (or indeterminacy) of an PFV μ,ν to P is given by(6)πP=1−μg2−νg2.


## 2.2. q-Rung Orthopair Fuzzy Sets

In 2016, Yager extended the range of double-graded fuzzy models in higher space. It is defined asDefinition 3 (see [14]).
A q-RFSℛ is defined as(7)ℛ=g,μg,νg|g∈X.
With the functionsμ:X⟶0,1, ν:X⟶0,1 called reliable and unreliable grades of an element g of X under the condition that the following inequality holds:(8)0≤μgq+νgq≤1.
Particularly, the hesitancy degree for q-ROFS is given as(9)πℛ=1−μg2−νg2q.
The pairμ,ν is called q-rung orthopair fuzzy value (q-ROFV) for an object g∈X. Let ℛ1=g,μg,νg|g∈X and ℛ2=g,sg,tg|g∈X be two q-ROFSs; then,(10)ℛ1∪ℛ2=g,maxμg,sg,minνg,tg|g∈X,ℛ1∩ℛ2=g,minμg,sg,maxνg,tg|g∈X,ℛ1⊆ℛ2⇔μg≤sg,νg≤tg∀g∈X,ℛ1c=g,νg,μg|g∈X.
The study on q-ROFS is extended by Liu and Wang [16] in the following crucial notions:Definition 4 (see [16]).
Letb=μ,ν be q-ROFN; then, the score function is defined as follows:(11)Sliub=μq−νq.
This notion is effective when we have to transfer a q-ROFN to a real value in interval−1,1 and therefore we can compare two or more q-ROFNs on their score functions. Consider a special case when b1=μ1,ν1, b2=μ2,ν2 with the condition μi=νi∀i=1,2. Then, sb1=0=sb2; thus, Sliu has an inadequacy for such a case. Therefore, in this study, when Sliu fails, we will use the following.Definition 5 (see [32]).
Letb=μ,ν be q-ROFN; then,(12)Spxb=eμq−νqeμq−νq+1−12πq+μq−νq.Definition 6 (see [16]).
Let us take the collection of q-ROFNsbk′=μk′,νk′, k′=1,2,3,…,n and weight vector λ1,λ2,…,λn; then, the q-rung orthopair fuzzy weighted averaging operator (q-ROFWA) is indicated as(13)q−ROFWAb1,b2,…,bn=1−∏k′=1n1−μk′qλk′q,∏k′=1nνk′λk′.Definition 7 (see [16]).
Let us take the collection of q-ROFNsbk′=μk′,νk′, k′=1,2,3,…,n and weight vector λ1,λ2,…,λn; then, the q-rung orthopair fuzzy weighted geometric operator (q-ROFWG) is indicated as(14)q−ROFWGb1,b2,…,bn=∏k′=1nμk′λk′,1−∏k′=1n1−νk′qλk′q.
The q-ROFWA and q-ROFWG are effective to aggregate data involving large number of q-ROFNs to a single q-ROFN.


## 2.3. q-Rung Orthopair Fuzzy Soft Sets

Jointly the approach of soft sets [36] with q-ROFSs is known as the q-rung orthopair fuzzy soft set, which is handy in MCDM problems. The notion of soft set is described as follows.Definition 8 (see [36]).
LetX be a fixed set and PX be the set of all subsets of X. Consider the set of parameters E and A be the subset of E. Let us define a function S as(15)S:A⟶PX.
Then, pairS,A=g,Sg|g∈A,Sg∈PX is called the soft set.Definition 9 (see [49]).
LetX,E be a soft universe and A⊆E. Define a mapping Q:A⟶q−ROFSX; then, pair Q,A is called the q-rung orthopair fuzzy soft set (q-ROFSS) over X, where q−ROFSX denotes the collection of all q-ROFSs over X. The q-ROFSS Q,A can be described as(16)Q,A=e,ρ,μρ,νρ|e∈A,ρ∈X,orQ,A=e,ρμρ,νρ|e∈A,ρ∈X,where μρ:X⟶0,1 and νρ:X⟶0,1 are reliable and nonreliable grades of ρ fulfilling inequality 0≤μqρ+νqρ≤1. Take μij=μejρi, νij=νejρi, where i=1,2,…,m,j=1,2,…,n. Then, the tabular form of q-ROFSS Q,A is given in Table 1.
For simplicity, we will representQ,A by QA.Table 1
q-ROFSSQ,A.
X|Ae1e2⋯enρ1μ11,ν11μ12,ν12⋯μ1n,ν1nρ2μ21,ν21μ22,ν22⋯μ2n,ν2n⋮⋮⋮⋱⋮ρmμm1,νm1μm2,νm2⋯μmn,νmnDefinition 10 (see [49]).
Consider two q-ROFSSsQA1 and QA2′ on X. We say QA1⊆QA2′ if and only if(i)
A1⊆A2(ii)
Qe is q-ROF subset of Q′e for all e∈A1⊆A2Definition 11 (see [49]).
Consider two q-ROFSSsQA1′ and QA2″ on X. Then, QA=A1∪A2=QA1′∪QA2″ denotes the union of QA1′ and QA2″ on X, such that(17)Qe=Q′e,ife∈A1−A2,Q″e,ife∈A2−A1,Q′e∪Q″e,ife∈A1∩A2,where Q′e∪Q″e is the union of q-ROFSs Q′e,Q″e.Definition 12 (see [49]).
Consider two q-ROFSSsQA1′ and QA2″ on X. Then, QA=A1∩A2=QA1′∩QA2″ denotes the intersection of QA1′ and QA2″ on X, such that Qe=Q′e∩Q″e, where e∈A1∩A2≠∅, where Q′e∩Q″e is the intersection of q-ROFSs Q′e,Q″e.


## 3. Group Generalized q-Rung Orthopair Fuzzy Soft Sets

In this section, we will define generalized q-Rung orthopair fuzzy soft sets (Gq-ROFSSs) and group-based generalized q-Rung orthopair fuzzy soft sets (GGq-ROFSSs). First Gq-ROFSS is described as follows.Definition 13.
Consider a soft universeX,E and A contained in E. A triple Q,A,α is called Gq-ROFSS over X if Q,A is a q-ROFSS over X and α is a q-ROFS over A.
In Gq-ROFSS, only one extra opinionα appears, but in many real-life satisfactions, more than one crucial additional opinions are needed. Thus, we define a greater prospect of Gq-ROFSS as GGq-ROFSS as follows.Definition 14.
Consider a soft universeX,E and A contained in E. A triple Q,A,η is called GGq-ROFSS over X if Q,A is a q-ROFSS over X and η=α1,α2,…,αp, where α1,α2,…,αp are the parametrized q-ROFSs (Pq-ROFSs) over A. In other words, η is the group Pq-ROFSs considered by “p” number of senior experts/moderators.Remark 1.
Ifp=1 in GGq-ROFSSs, then it is Gq-ROFSS. Therefore, Gq-ROFSS is a special case of GGq-ROFSS, and thus, generally, we will focus on GGq-ROFSSs.
A broaden tabular form of GGq-ROFSS (in Definition14) is given in Table 2.
In Table2, the light gray part represents q-ROFSS and the brown part represents the group of q-ROFSs in GGq-ROFSS.Table 2
GGq-ROFSSQ,A,η.


## 4. Operations on Group-Based Generalized q-Rung Orthopair Fuzzy Soft Sets

In this section, we will define subset, union, intersection, and complement of GGq-ROFSSs.Definition 15.
Consider a soft universeX,E and A,ℬ contained in E. Let two GGq-ROFSSs Q1,A1,η1 and Q2,A2,η2 on X, where A⊆ℬ, η1=α1,α2,…,αp, η2=β1,β2,…,βp, respectively, are group Pq-ROFSs related to “p” number of senior experts/moderators. The η1 is group q-ROF subset of η1 if and only if sα1ek≤sβ1ek,sα2ek≤sβ2ek, …, sαpek≤sβpek, and tα1ek≤tβ1ek,tα2ek≤tβ2ek, …, tαpek≤tβpek and for each ek∈A. It is denoted as η1⋐η2.
In the prospect of Definition15, we introduce GGq-ROFS subsets.Definition 16.
Let two GGq-ROFSSsQ1,A1,η1 and Q2,A2,η2 on X, where A,ℬ⊂E. Then, Q1,A1,η1 is a GGq-ROFS subset of Q2,A2,η2 if(i)
Q1,A1⊆Q2,A2(ii)
η1⋐η2Definition 17.
Let two GGq-ROFSSsQ1,A1,η1 and Q2,A2,η2 on X, where A,ℬ⊂E such that A1∪A2=C and η1=α1,α2,…,αp, η2=β1,β2,…,βp. The extended union of Q1,A1,η1 and Q2,A2,η2 is investigated in the form of GGq-ROFSS. which is given as follows:(18)Q,C,η=Q1,A1,η1∪eQ2,A2,η2,such that(i)
Q,C=Q1,A1∪eQ2,A2.(ii)
For eache∈C, μγi′,νγi′i′=1,2,…,p is defined as(19)μγi′e=μαi′e,ife∈A1−A2;μβi′e,ife∈A2−A1;maxμαi′e,μβi′e,ife∈A1∩A2,νγi′e=ναi′e,ife∈A1−A2;νβi′e,ife∈A2−A1;minναi′e,νβi′e,ife∈A1∩A2,where η=γ1,γ2,…,γp.Definition 18.
Let two GGq-ROFSSs beQ1,A1,η1 and Q2,A2,η2 on X, where A,ℬ⊂E such that A1∩A2=C and η1=α1,α2,…,αp, η2=β1,β2,…,βp. The restricted intersection of Q1,A1,η1 and Q2,A2,η2 is investigated in the form of GGq-ROFSS, which is given as(20)Q,C,η=Q1,A1,η1∩rQ2,A2,η2,such that(i)
Q,C=Q1,A1∩rQ2,A2(ii)
For eache∈C, μγi′,νγi′i′=1,2,…,p is defined as μγi′e=minμαi′e,μβi′e and νγi′e=maxναi′e,νβi′e for all e∈A1∩A2, where η=γ1,γ2,…,γp
Now, we will take an example of GGq-ROFSS to clarify the above concepts.Example 1.
LetX=ρ1,ρ2,ρ3,ρ4,ρ5 be the universal set consisting of five different kinds of face masks available in market, and the set of attributes E is given as E=e1,e2,e3,e4,e5, where each ei, respectively, stands for affordable price, good fabrication, effective comfortable design, capable of stopping viruses and bacteria, and comfortable breathing while wearing the mask. Let the two buyers T1 and T2, respectively, have the following preferences while buying the face mask:(21)A=e1,e4,ℬ=e2,e4,e5⊂E.
The GGq-ROFSSsQ,A,η and Q′,ℬ,η′ for buyers T1 and T2 are interpreted in Tables 3 and 4, where two senior persons h1 and h2 provide their opinions on q-ROFSSs (given in the light gray parts of Tables 3 and 4). The extra inputs as the group of q-ROFSs of h1 and h2 are interpreted in the brown parts of both the tables. The union and intersection of Q,A,η and Q′,ℬ,η′ are computed in Tables 5 and 6 respectively.Table 3
GGq-ROFSSQ,A,η for T1.Table 4
GGq-ROFSSQ′,ℬ,η′ for T2.Table 5
Union of GGq-ROFSSsℛ,A∪ℬ,δ.Table 6
Intersection of GGq-ROFSSsℛ,A∩ℬ,δ.Definition 19.
Consider a soft universeX,E and A contained in E. Let GGq-ROFSS be Q,A,η on X where η=α1,α2,…,αp. The complement of Q,A,η is denoted by Qc,A,ηc, where Qc,A is the complement of q-ROFSS Q,A and ηc=α1c,α2c,…,αpc.Definition 20.
Let GGq-ROFSS beQ,A,η given in Table 2.(1)
If >μjk,νjk< = >1,0< and >si′k,ti′k< = >1,0< for all j=1,2,…,m, k=1,2,…,n and i′=1,2,…,p, then Q,A,η is called whole GGq-ROFSS. It is denoted by WA.(2)
If >μjk,νjk< = >0,1< and >si′k,ti′k< = >0,1< for all j=1,2,…,m, k=1,2,…,n and i′=1,2,…,p, then Q,A,η is called null GGq-ROFSS. It is denoted by NA.Proposition 1.
LetQAη=Q,A,η be a GGq-ROFSS over X. Then,(1)
QAη∪QAη=QAη, QAη∩QAη=QAη(2)
QAη∪WA=WA, QAη∩WA=QAη(3)
QAη∩NA=NA, QAη∪NA=QAη


## 5. Aggregation Operators on GGq-ROFSS

To attain substantial effect, there is an immense need to define better aggregation operators that accurately deal with all components of GGq-ROFSSs.


### 5.1. GWq-ROFW Operators

Definition 21.
Consider Definition14, where GGq-ROFSS Q,A,η is given in Table 2, where the light gray part represents q-ROFSS and the brown part describes the group of q-ROFSs of “p” number of senior moderators. Let μ0kj,ν0kj=μjk,νjk, j=1,2,…,n,k=1,2,…,m, that is to say μ0kj,ν0kj are q-ROFVs in the light gray part of Table 2. Moreover, μi′kj,νi′kj=si′k,ti′k, i′=1,2,…,p, that is to say μi′kj,νi′kj are q-ROFVs in the brown part of Table 2. Assume that i=0,1,2,…,p. A symbolization is given by(22)μikj,νikj=μjk,νjk,ifi=0;si′k,ti′k,ifi>0.
On the above fundamental and crucial symbolic notion of the GGq-ROFSS, we contemplate novel averaging aggregation operators.Definition 22.
Consider Definition14, where GGq-ROFSS Q,A,η is given in Table 2. Let Ψ=ϖ1,ϖ2,…,ϖnT be the weighted vector over A, such that ∑k=1nϖk=1 and ϖk>0. Also take weighted vector W=φ0,φ1,φ2,…,φpT such that ∑i=1pϖi=1 and ϖi>0, where φ1,…,φp are weights for the judgements of the “p” number of senior moderators and φ0 is the weight for each q-ROFV in the light gray part of Table 2. In other words, φ0 is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Assume i=0,1,2,…,p. Then, the generalized weighted q-Rung orthopair fuzzy averaging operator (GWq-ROFA) undertaken by GGq-ROFSS is contemplated as follows:(23)Θj=GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=q−ROFWAkq−ROFWAiμ01j,ν01j,μ11j,ν11j,…,μp1j,νp1jq−ROFWAiμ02j,ν02j,μ12j,ν12j,…,μp2j,νp2j⋯q−ROFWAiμ0nj,ν0nj,μ1nj,ν1nj,…,μpnj,νpnj=q−ROFWAkq−ROFWAiμj1,νj1,s11,t11,s21,t21,…,sp1j,νp1jq−ROFWAiμj2,νj2,s12,t12,s22,t22,…,sp2,tp2⋯q−ROFWAiμjn,νjn,s1n,t1n,s2n,t2n,…,spn,tpn,where q−ROFWAk is the q-rung orthopair fuzzy weighted averaging operator and it operates over the set of criteria, and q−ROFWAi is the q-rung orthopair fuzzy weighted averaging operator and it operates mutually over q-ROFSS and on the set of senior moderators.
The set of all GWq-ROFA operators form number of alternatives is indicated as Δ=Θ1,Θ2,…,Θm. The above novel GWq-ROFA operators are realistic instruments for linear and entire aggregations of q-ROFVs in GGq-ROFSS.
The GWq-ROFA operators have a specific way of incorporating each component of GGq-ROFSS. They entirely compel q-ROFVs in a linear way towards attributes until the final q-ROFV appears.Example 2.
Consider GGq-ROFSS indicated in Table4 in Example 20. Given q−ROFVρ1=0.8,0.8,0.8,0.6,0.8,0.8. We have to calculate the GWq-ROFA operator for ρ1. The q-ROFVs are depicted as(24)q−ROFWAe2μ11,ν11,s11,t11,s21,t21=0.8,0.8,0.4,0.4,0.8,0.2,q−ROFWAe4μ22,ν22,s12,t12,s22,t22=0.8,0.6,0.8,0.7,0.7,0.4,q−ROFWAe5μ23,ν23,s13,t13,s23,t23=0.8,0.8,0.7,0.7,0.6,0.6.
LetΨ=ϖ1,ϖ2,ϖ3T=0.3/e2,0.4/e4,0.3/e5T be the weighted vector over A. Also take weighted vector W=φ0,φ1,φ2T=0.40,0.25,0.35T for q-ROFSS and judgements of senior person/moderators. For q=4, we have(25)q−ROFWA0.8,0.8,0.4,0.4,0.8,0.2=0.7584,0.4141,q−ROFWA0.8,0.6,0.8,0.7,0.7,0.4=0.7719,0.5411,q−ROFWA0.8,0.8,0.7,0.7,0.6,0.6=0.7272,0.6996.
Now, GWq-ROFA is given byΘ1=GWq−ROFA0.8,0.8,0.8,0.6,0.8,0.8 = q−ROFWA0.7584,0.4141,0.7719,0.5411,0.7272,0.6996 = 0.7555,0.5394. Similarly, GWq-ROFA operators Θ2,Θ3,Θ4, and Θ5 can be obtained.Theorem 1.
Letajk=μjk,νjk and di′k=si′j,ti′j be the q-ROFVs in GGq-ROFSS in Table 2, where k=1,2,…,n, j=1,2,…,m and i′=1,2,…,p. If we consider i=0,1,2,…,p, then the GWq-ROFA operator is given by(26)GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=1−∏k=1n1−1−∏i=0p1−μikjqφiϖk,∏k=1n∏i=0pνikjqφiϖkq.Proof.
Assume thatp=1 and n=2. Take j=1; we apply mathematical induction on n. By the definition of GGq-ROFSS,(27)Θj=GWq−ROFWAμ11,ν11,μ12,ν12=IFWAjIWAiμ011,ν011,μ111,ν111IWAiμ021,ν021,μ121,ν121=IFWAjIWAiμ11,ν11,s11,t11IWAiμ12,ν12,s12,t12.
Now,IWAiμ11,ν11,s11,t11=1−1−μ11qφ0⋅1−s11qφ1q,ν11φ0⋅t11φ1, IWAiμ12,ν12,s12,t12=1−1−μ12qφ0⋅1−s12qφ1q,ν12φ0⋅t12φ1. Thus,(28)Θj=GWq−ROFWAμ11,ν11,μ12,ν12=IFWAj1−1−μ11qφ0⋅1−s11qφ1q,ν11φ0⋅t11φ1,1−1−μ12qφ0⋅1−s12qφ1q,ν12φ0⋅t12φ1=1−1−1−1−μ11qφ0⋅1−s11qφ1ϖ1⋅1−1−1−1−μ12qφ0⋅1−s12qφ1ϖ2q,ν11φ0⋅s11φ1ϖ1⋅ν12φ0⋅s12φ1ϖ2=1−1−1−1−μ011qφ0⋅1−μ111qφ1ϖ1⋅1−1−1−1−μ021qφ0⋅1−μ121qφ1ϖ2q,ν011φ0⋅ν111φ1ϖ1⋅ν021φ0⋅ν121φ1ϖ2=1−1−1−∏i=011−μi11qφiϖ1⋅1−1−1−∏i=011−μi21qφiϖ2q,∏i=01νi11φiϖ1⋅∏i=01νi21φiϖ2=1−∏k=121−1−∏i=011−μik1qφiϖkq,∏k=12∏i=01νik1qφiϖk.
Hence, the theorem is valid forn=2. Considering that this result is fine for n=n′ that is(29)GWq−ROFAμj1,νj1,μj2,νj2,…,μjn′,νjn′=1−∏k=1n′1−1−∏i=0p1−μikjqφiϖkq,∏k=1n′∏i=0pνikjqφiϖk.
Then, forn=n′+1, we have(30)GWq−ROFAμj1,νj1,μj2,νj2,…,μjn′+1,νjn′+1=1−∏k=1n′+11−1−∏i=0p1−μikjqφiϖkq,∏k=1n′+1∏i=0pνikjqφiϖk.
Hence, by mathematical induction, Theorem1 satisfy for all positive integer n.Theorem 2.
(idempotency) Ifμjk,νjk=μj,νj and μi′k,νi′k=μ,ν=μj,νj for all k=1,2,…,n, then GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=μ,ν.Proof.
Givenμjk,νjk=μ,ν and μi′k,νi′k=μ,ν for all i′=1,2,…,p and k=1,2,…,n.(31)Θj=GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=1−∏k=1n1−1−∏i=0p1−μikjqφiϖkq,∏k=1n∏i=0pνikjqφiϖk=1−∏k=1n1−1−1−μq∑i=0pφiϖkq,∏k=1nνq∑i=0pφiϖk=1−∏k=1n1−1−1−μqϖkq,∏k=1nνqϖk=1−1−1−1−μq∑k=1nϖkq,νq∑k=1nϖk=μ,ν,GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=μ,ν.Theorem 3.
(boundedness) Ifμ+,ν−=μikjmax,νikjmin and μ−,ν+=μikjmin,νikjmax for all j=1,2,…,m and i=0,1,2,…,p, then μ−,ν+≤GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn≤μ+,ν−.Proof.
Sinceμ−≤μikj≤μ+⇔μ−q≤μikjq≤μ+q⇔1−μ−q≥1−μikjq≥1−μ+q⇔∏i=0p1−μ−qφi ≥ ∏i=0p1−μikjqφi≥∏i=0p1−μ+qφi⇔1−μ−q∑i=0pφi≥∏i=0p1−μikjqφi≥1−μ+q∑i=0pφi⇔1−μ−q ≥ ∏i=0p1−μikjqφi≥1−μ+q⇔1−1−μ−q≤1−∏i=0p1−μikjqφi ≤ 1−1−μ+q⇔μ−q≤1−∏i=0p1−μikjqφi≤μ+q⇔1−μ−q≥1−1−∏i=0p1−μikjqφi ≥ 1−μ+q⇔∏k=1n1−μ−qϖk≥∏k=1n1−1−∏i=0p1−μikjqφiϖk≥∏k=1n1−μ+qϖk⇔1−μ−q∑k=1nϖk ≥ ∏k=1n1−1−∏i=0p1−μikjqφi≥1−μ+q∑k=1nϖk⇔1−μ−q ≥ ∏k=1n1−1−∏i=0p1−μikjqφiϖk≥1−μ+q⇔1−1−μ−q≤ 1−∏k=1n1−1−∏i=0p1−μikjqφiϖk≤1−1−μ+q⇔μ−q ≤ 1−∏k=1n1−1−∏i=0p1−μikjqφiϖk≤μ+q⇔μ− ≤ 1−∏k=1n1−1−∏i=0p1−μikjqφiϖkq≤μ+.
Similarly, nonmembership part is aggregated asμ+≤∏k=1n∏i=0pνikjqφiϖk≤μ−. This concluded the proof of the theorem, μ−,ν+≤GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn≤μ+,ν−.Theorem 4.
(monotonicity) Ifμikj,νikj and μ¯ikj,ν¯ikj for all i=0,1,2,…,p, j=1,2,…,m,k=1,2,…,n are two IFVs such that μikj,νikj≤μ¯ikj,ν¯ikj, then GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn≤GWq−ROFAμ¯j1,ν¯j1,μ¯j2,ν¯j2,…,μ¯jn,ν¯jn.Proof.
It can be concluded from Theorem3.Proposition 2.
LetQ,A,η be a GGq-ROFSS, given in Table 2. Then,(1)
Ifsi′k,ti′k=1,0 for all i′ and k, then Θj=1,0,1,0,…,1,0.(2)
Ifμ0kj,ν0kj=μjk,νjk=1,0 for all j and k, then Θj=1,0,1,0,…,1,0.(3)
Ifsi′k,ti′k=1,0 for all i′,k and μ0kj,ν0kj=μjk,νjk=1,0 for all j,k, then Θj=1,0,1,0,…,1,0.(4)
Ifsi′k,ti′k=0,1 for all i′,k and μ0kj,ν0kj=μjk,νjk=0,1 for all j,k then Θj=0,1,0,1,…,0,1.Definition 23.
Consider Definition14, where GGq-ROFSS Q,A,η is given in Table 2. Let Ψ=ϖ1,ϖ2,…,ϖnT be the weighted vector over A, such that ∑k=1nϖk=1 and ϖk>0. Also take weighted vector W=φ0,φ1,φ2,…,φpT such that ∑i=1pϖi=1 and ϖi>0, where φ1,…,φp are weights for the judgements of the “p” number of senior moderators and φ0 is the weight for each q-ROFV in the light gray part of Table 2. In other words, φ0 is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Assume i=0,1,2,…,p. Then, the generalized weighted q-Rung orthopair fuzzy geometric operator (GWq-ROFG) undertook by GGq-ROFSS is contemplated as follows:(32)Θj=GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn=q−ROFWGkq−ROFWGiμ01j,ν01j,μ11j,ν11j,…,μp1j,νp1jq−ROFWGiμ02j,ν02j,μ12j,ν12j,…,μp2j,νp2j⋯q−ROFWGiμ0nj,ν0nj,μ1nj,ν1nj,…,μpnj,νpnj=q−ROFWGkq−ROFWGiμj1,νj1,s11,t11,s21,t21,…,sp1j,νp1jq−ROFWGiμj2,νj2,s12,t12,s22,t22,…,sp2,tp2⋯q−ROFWGiμjn,νjn,s1n,t1n,s2n,t2n,…,spn,tpn,where q−ROFWGk is the q-rung orthopair fuzzy weighted geometric operator and it operates over the set of criteria, and q−ROFWGi is the q-rung orthopair fuzzy weighted geometric operator and it operates mutually over q-ROFSS and on the set of senior moderators.
The set of all GWq-ROFG operators form number of alternatives is indicated as Δ′=Θ1′,Θ2′,…,Θm′. The above novel GWq-ROFG operators are realistic instruments for linear and entire aggregations of q-ROFVs in GGq-ROFSS.
The GWq-ROFG operators have a specific way of incorporating each component of GGq-ROFSS. They entirely compel q-ROFVs in a linear way towards attributes until the final q-ROFV appears.Example 3.
Consider GGq-ROFSS indicated in Table4 in Example 3. Given q−ROFVρ2=0.5,0.3,0.5,0.7,0.4,0.3, we have to calculate the GWq-ROFG operator for ρ1. The q-ROFVs are depicted as(33)q−ROFWGe2μ21,ν21,s11,t11,s21,t21=0.5,0.3,0.4,0.4,0.8,0.2,q−ROFWGe4μ22,ν22,s12,t12,s22,t22=0.5,0.7,0.8,0.7,0.7,0.4,q−ROFWGe5μ23,ν23,s13,t13,s23,t23=0.4,0.3,0.7,0.7,0.6,0.6.
LetΨ=ϖ1,ϖ2,ϖ3T=0.3/e2,0.4/e4,0.3/e5T be the weighted vector over A. Also take weighted vector W=φ0,φ1,φ2T=0.40,0.25,0.35T for q-ROFSS and judgements of senior person/moderators. For q=4, we have(34)q−ROFWG0.5,0.3,0.4,0.4,0.8,0.2=0.6644,0.2797,q−ROFWG0.5,0.7,0.8,0.7,0.7,0.4=0.6880,0.5755,q−ROFWG0.4,0.3,0.7,0.7,0.6,0.6=0.5883,0.4726.
Now GWq-ROFG is given by(35)Θ2′=GWq−ROFG0.5,0.3,0.5,0.7,0.4,0.3=q−ROFWG0.6644,0.2797,0.6880,0.5755,0.5883,0.4726=0.6560,0.4369.
Similarly, GWq-ROFG operatorsΘ1′,Θ3′,Θ4′, and Θ5′ can be obtained.Theorem 5.
Letajk=μjk,νjk and di′k=si′j,ti′j be the q-ROFVs in GGq-ROFSS in Table 2, where k=1,2,…,n, j=1,2,…,m, and i′=1,2,…,p. If we consider i=0,1,2,…,p, then GWq-ROFG operator is given by(36)GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn=∏k=1n∏i=0pμikjqφiϖk,q1−∏k=1n1−1−∏i=0p1−νikjqφiϖk.Proof.
Same as the proof of Theorem1.Theorem 6.
(idempotency) Ifμjk,νjk=μj,νj and μi′k,νi′k=μ,ν=μj,νj for all k=1,2,…,n, then GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn=μ,ν.Proof.
Same as the proof of Theorem2.Theorem 7.
(boundedness) Ifμ+,ν−=μikjmax,νikjmin and μ−,ν+=μikjmin,νikjmax for all j=1,2,…,m and i=0,1,2,…,p, thenμ−,ν+≤GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn≤μ+,ν−.Proof.
Same as the proof of Theorem3.Theorem 8.
(monotonicity) Ifμikj,νikj and μ¯ikj,ν¯ikj for all i=0,1,2,…,p, j=1,2,…,m,k=1,2,…,n are two IFVs such that μikj,νikj≤μ¯ikj,ν¯ikj, then(37)GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn≤GWq−ROFGμ¯j1,ν¯j1,μ¯j2,ν¯j2,…,μ¯jn,ν¯jn.Proof.
It can be concluded from Theorem4.Proposition 3.
LetQ,A,η be a GGq-ROFSS, given in Table 2. Then,(1)
Ifsi′k,ti′k=0,1 for all i′ and k, then Θj′=0,1,0,1,…,0,1.(2)
Ifμ0kj,ν0kj=μjk,νjk=0,1 for all j and k, then Θj′=0,1,0,1,…,0,1.(3)
Ifsi′k,ti′k=0,1 for all i′,k and μ0kj,ν0kj=μjk,νjk=0,1 for all j,k then Θj′=0,1,0,1,…,0,1.(4)
Ifsi′k,ti′k=1,0 for all i′,k and μ0kj,ν0kj=μjk,νjk=1,0 for all j,k then Θj′=1,0,1,0,…,1,0.


## 5.1. GWq-ROFW Operators

Definition 21.
Consider Definition14, where GGq-ROFSS Q,A,η is given in Table 2, where the light gray part represents q-ROFSS and the brown part describes the group of q-ROFSs of “p” number of senior moderators. Let μ0kj,ν0kj=μjk,νjk, j=1,2,…,n,k=1,2,…,m, that is to say μ0kj,ν0kj are q-ROFVs in the light gray part of Table 2. Moreover, μi′kj,νi′kj=si′k,ti′k, i′=1,2,…,p, that is to say μi′kj,νi′kj are q-ROFVs in the brown part of Table 2. Assume that i=0,1,2,…,p. A symbolization is given by(22)μikj,νikj=μjk,νjk,ifi=0;si′k,ti′k,ifi>0.
On the above fundamental and crucial symbolic notion of the GGq-ROFSS, we contemplate novel averaging aggregation operators.Definition 22.
Consider Definition14, where GGq-ROFSS Q,A,η is given in Table 2. Let Ψ=ϖ1,ϖ2,…,ϖnT be the weighted vector over A, such that ∑k=1nϖk=1 and ϖk>0. Also take weighted vector W=φ0,φ1,φ2,…,φpT such that ∑i=1pϖi=1 and ϖi>0, where φ1,…,φp are weights for the judgements of the “p” number of senior moderators and φ0 is the weight for each q-ROFV in the light gray part of Table 2. In other words, φ0 is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Assume i=0,1,2,…,p. Then, the generalized weighted q-Rung orthopair fuzzy averaging operator (GWq-ROFA) undertaken by GGq-ROFSS is contemplated as follows:(23)Θj=GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=q−ROFWAkq−ROFWAiμ01j,ν01j,μ11j,ν11j,…,μp1j,νp1jq−ROFWAiμ02j,ν02j,μ12j,ν12j,…,μp2j,νp2j⋯q−ROFWAiμ0nj,ν0nj,μ1nj,ν1nj,…,μpnj,νpnj=q−ROFWAkq−ROFWAiμj1,νj1,s11,t11,s21,t21,…,sp1j,νp1jq−ROFWAiμj2,νj2,s12,t12,s22,t22,…,sp2,tp2⋯q−ROFWAiμjn,νjn,s1n,t1n,s2n,t2n,…,spn,tpn,where q−ROFWAk is the q-rung orthopair fuzzy weighted averaging operator and it operates over the set of criteria, and q−ROFWAi is the q-rung orthopair fuzzy weighted averaging operator and it operates mutually over q-ROFSS and on the set of senior moderators.
The set of all GWq-ROFA operators form number of alternatives is indicated as Δ=Θ1,Θ2,…,Θm. The above novel GWq-ROFA operators are realistic instruments for linear and entire aggregations of q-ROFVs in GGq-ROFSS.
The GWq-ROFA operators have a specific way of incorporating each component of GGq-ROFSS. They entirely compel q-ROFVs in a linear way towards attributes until the final q-ROFV appears.Example 2.
Consider GGq-ROFSS indicated in Table4 in Example 20. Given q−ROFVρ1=0.8,0.8,0.8,0.6,0.8,0.8. We have to calculate the GWq-ROFA operator for ρ1. The q-ROFVs are depicted as(24)q−ROFWAe2μ11,ν11,s11,t11,s21,t21=0.8,0.8,0.4,0.4,0.8,0.2,q−ROFWAe4μ22,ν22,s12,t12,s22,t22=0.8,0.6,0.8,0.7,0.7,0.4,q−ROFWAe5μ23,ν23,s13,t13,s23,t23=0.8,0.8,0.7,0.7,0.6,0.6.
LetΨ=ϖ1,ϖ2,ϖ3T=0.3/e2,0.4/e4,0.3/e5T be the weighted vector over A. Also take weighted vector W=φ0,φ1,φ2T=0.40,0.25,0.35T for q-ROFSS and judgements of senior person/moderators. For q=4, we have(25)q−ROFWA0.8,0.8,0.4,0.4,0.8,0.2=0.7584,0.4141,q−ROFWA0.8,0.6,0.8,0.7,0.7,0.4=0.7719,0.5411,q−ROFWA0.8,0.8,0.7,0.7,0.6,0.6=0.7272,0.6996.
Now, GWq-ROFA is given byΘ1=GWq−ROFA0.8,0.8,0.8,0.6,0.8,0.8 = q−ROFWA0.7584,0.4141,0.7719,0.5411,0.7272,0.6996 = 0.7555,0.5394. Similarly, GWq-ROFA operators Θ2,Θ3,Θ4, and Θ5 can be obtained.Theorem 1.
Letajk=μjk,νjk and di′k=si′j,ti′j be the q-ROFVs in GGq-ROFSS in Table 2, where k=1,2,…,n, j=1,2,…,m and i′=1,2,…,p. If we consider i=0,1,2,…,p, then the GWq-ROFA operator is given by(26)GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=1−∏k=1n1−1−∏i=0p1−μikjqφiϖk,∏k=1n∏i=0pνikjqφiϖkq.Proof.
Assume thatp=1 and n=2. Take j=1; we apply mathematical induction on n. By the definition of GGq-ROFSS,(27)Θj=GWq−ROFWAμ11,ν11,μ12,ν12=IFWAjIWAiμ011,ν011,μ111,ν111IWAiμ021,ν021,μ121,ν121=IFWAjIWAiμ11,ν11,s11,t11IWAiμ12,ν12,s12,t12.
Now,IWAiμ11,ν11,s11,t11=1−1−μ11qφ0⋅1−s11qφ1q,ν11φ0⋅t11φ1, IWAiμ12,ν12,s12,t12=1−1−μ12qφ0⋅1−s12qφ1q,ν12φ0⋅t12φ1. Thus,(28)Θj=GWq−ROFWAμ11,ν11,μ12,ν12=IFWAj1−1−μ11qφ0⋅1−s11qφ1q,ν11φ0⋅t11φ1,1−1−μ12qφ0⋅1−s12qφ1q,ν12φ0⋅t12φ1=1−1−1−1−μ11qφ0⋅1−s11qφ1ϖ1⋅1−1−1−1−μ12qφ0⋅1−s12qφ1ϖ2q,ν11φ0⋅s11φ1ϖ1⋅ν12φ0⋅s12φ1ϖ2=1−1−1−1−μ011qφ0⋅1−μ111qφ1ϖ1⋅1−1−1−1−μ021qφ0⋅1−μ121qφ1ϖ2q,ν011φ0⋅ν111φ1ϖ1⋅ν021φ0⋅ν121φ1ϖ2=1−1−1−∏i=011−μi11qφiϖ1⋅1−1−1−∏i=011−μi21qφiϖ2q,∏i=01νi11φiϖ1⋅∏i=01νi21φiϖ2=1−∏k=121−1−∏i=011−μik1qφiϖkq,∏k=12∏i=01νik1qφiϖk.
Hence, the theorem is valid forn=2. Considering that this result is fine for n=n′ that is(29)GWq−ROFAμj1,νj1,μj2,νj2,…,μjn′,νjn′=1−∏k=1n′1−1−∏i=0p1−μikjqφiϖkq,∏k=1n′∏i=0pνikjqφiϖk.
Then, forn=n′+1, we have(30)GWq−ROFAμj1,νj1,μj2,νj2,…,μjn′+1,νjn′+1=1−∏k=1n′+11−1−∏i=0p1−μikjqφiϖkq,∏k=1n′+1∏i=0pνikjqφiϖk.
Hence, by mathematical induction, Theorem1 satisfy for all positive integer n.Theorem 2.
(idempotency) Ifμjk,νjk=μj,νj and μi′k,νi′k=μ,ν=μj,νj for all k=1,2,…,n, then GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=μ,ν.Proof.
Givenμjk,νjk=μ,ν and μi′k,νi′k=μ,ν for all i′=1,2,…,p and k=1,2,…,n.(31)Θj=GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=1−∏k=1n1−1−∏i=0p1−μikjqφiϖkq,∏k=1n∏i=0pνikjqφiϖk=1−∏k=1n1−1−1−μq∑i=0pφiϖkq,∏k=1nνq∑i=0pφiϖk=1−∏k=1n1−1−1−μqϖkq,∏k=1nνqϖk=1−1−1−1−μq∑k=1nϖkq,νq∑k=1nϖk=μ,ν,GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn=μ,ν.Theorem 3.
(boundedness) Ifμ+,ν−=μikjmax,νikjmin and μ−,ν+=μikjmin,νikjmax for all j=1,2,…,m and i=0,1,2,…,p, then μ−,ν+≤GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn≤μ+,ν−.Proof.
Sinceμ−≤μikj≤μ+⇔μ−q≤μikjq≤μ+q⇔1−μ−q≥1−μikjq≥1−μ+q⇔∏i=0p1−μ−qφi ≥ ∏i=0p1−μikjqφi≥∏i=0p1−μ+qφi⇔1−μ−q∑i=0pφi≥∏i=0p1−μikjqφi≥1−μ+q∑i=0pφi⇔1−μ−q ≥ ∏i=0p1−μikjqφi≥1−μ+q⇔1−1−μ−q≤1−∏i=0p1−μikjqφi ≤ 1−1−μ+q⇔μ−q≤1−∏i=0p1−μikjqφi≤μ+q⇔1−μ−q≥1−1−∏i=0p1−μikjqφi ≥ 1−μ+q⇔∏k=1n1−μ−qϖk≥∏k=1n1−1−∏i=0p1−μikjqφiϖk≥∏k=1n1−μ+qϖk⇔1−μ−q∑k=1nϖk ≥ ∏k=1n1−1−∏i=0p1−μikjqφi≥1−μ+q∑k=1nϖk⇔1−μ−q ≥ ∏k=1n1−1−∏i=0p1−μikjqφiϖk≥1−μ+q⇔1−1−μ−q≤ 1−∏k=1n1−1−∏i=0p1−μikjqφiϖk≤1−1−μ+q⇔μ−q ≤ 1−∏k=1n1−1−∏i=0p1−μikjqφiϖk≤μ+q⇔μ− ≤ 1−∏k=1n1−1−∏i=0p1−μikjqφiϖkq≤μ+.
Similarly, nonmembership part is aggregated asμ+≤∏k=1n∏i=0pνikjqφiϖk≤μ−. This concluded the proof of the theorem, μ−,ν+≤GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn≤μ+,ν−.Theorem 4.
(monotonicity) Ifμikj,νikj and μ¯ikj,ν¯ikj for all i=0,1,2,…,p, j=1,2,…,m,k=1,2,…,n are two IFVs such that μikj,νikj≤μ¯ikj,ν¯ikj, then GWq−ROFAμj1,νj1,μj2,νj2,…,μjn,νjn≤GWq−ROFAμ¯j1,ν¯j1,μ¯j2,ν¯j2,…,μ¯jn,ν¯jn.Proof.
It can be concluded from Theorem3.Proposition 2.
LetQ,A,η be a GGq-ROFSS, given in Table 2. Then,(1)
Ifsi′k,ti′k=1,0 for all i′ and k, then Θj=1,0,1,0,…,1,0.(2)
Ifμ0kj,ν0kj=μjk,νjk=1,0 for all j and k, then Θj=1,0,1,0,…,1,0.(3)
Ifsi′k,ti′k=1,0 for all i′,k and μ0kj,ν0kj=μjk,νjk=1,0 for all j,k, then Θj=1,0,1,0,…,1,0.(4)
Ifsi′k,ti′k=0,1 for all i′,k and μ0kj,ν0kj=μjk,νjk=0,1 for all j,k then Θj=0,1,0,1,…,0,1.Definition 23.
Consider Definition14, where GGq-ROFSS Q,A,η is given in Table 2. Let Ψ=ϖ1,ϖ2,…,ϖnT be the weighted vector over A, such that ∑k=1nϖk=1 and ϖk>0. Also take weighted vector W=φ0,φ1,φ2,…,φpT such that ∑i=1pϖi=1 and ϖi>0, where φ1,…,φp are weights for the judgements of the “p” number of senior moderators and φ0 is the weight for each q-ROFV in the light gray part of Table 2. In other words, φ0 is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Assume i=0,1,2,…,p. Then, the generalized weighted q-Rung orthopair fuzzy geometric operator (GWq-ROFG) undertook by GGq-ROFSS is contemplated as follows:(32)Θj=GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn=q−ROFWGkq−ROFWGiμ01j,ν01j,μ11j,ν11j,…,μp1j,νp1jq−ROFWGiμ02j,ν02j,μ12j,ν12j,…,μp2j,νp2j⋯q−ROFWGiμ0nj,ν0nj,μ1nj,ν1nj,…,μpnj,νpnj=q−ROFWGkq−ROFWGiμj1,νj1,s11,t11,s21,t21,…,sp1j,νp1jq−ROFWGiμj2,νj2,s12,t12,s22,t22,…,sp2,tp2⋯q−ROFWGiμjn,νjn,s1n,t1n,s2n,t2n,…,spn,tpn,where q−ROFWGk is the q-rung orthopair fuzzy weighted geometric operator and it operates over the set of criteria, and q−ROFWGi is the q-rung orthopair fuzzy weighted geometric operator and it operates mutually over q-ROFSS and on the set of senior moderators.
The set of all GWq-ROFG operators form number of alternatives is indicated as Δ′=Θ1′,Θ2′,…,Θm′. The above novel GWq-ROFG operators are realistic instruments for linear and entire aggregations of q-ROFVs in GGq-ROFSS.
The GWq-ROFG operators have a specific way of incorporating each component of GGq-ROFSS. They entirely compel q-ROFVs in a linear way towards attributes until the final q-ROFV appears.Example 3.
Consider GGq-ROFSS indicated in Table4 in Example 3. Given q−ROFVρ2=0.5,0.3,0.5,0.7,0.4,0.3, we have to calculate the GWq-ROFG operator for ρ1. The q-ROFVs are depicted as(33)q−ROFWGe2μ21,ν21,s11,t11,s21,t21=0.5,0.3,0.4,0.4,0.8,0.2,q−ROFWGe4μ22,ν22,s12,t12,s22,t22=0.5,0.7,0.8,0.7,0.7,0.4,q−ROFWGe5μ23,ν23,s13,t13,s23,t23=0.4,0.3,0.7,0.7,0.6,0.6.
LetΨ=ϖ1,ϖ2,ϖ3T=0.3/e2,0.4/e4,0.3/e5T be the weighted vector over A. Also take weighted vector W=φ0,φ1,φ2T=0.40,0.25,0.35T for q-ROFSS and judgements of senior person/moderators. For q=4, we have(34)q−ROFWG0.5,0.3,0.4,0.4,0.8,0.2=0.6644,0.2797,q−ROFWG0.5,0.7,0.8,0.7,0.7,0.4=0.6880,0.5755,q−ROFWG0.4,0.3,0.7,0.7,0.6,0.6=0.5883,0.4726.
Now GWq-ROFG is given by(35)Θ2′=GWq−ROFG0.5,0.3,0.5,0.7,0.4,0.3=q−ROFWG0.6644,0.2797,0.6880,0.5755,0.5883,0.4726=0.6560,0.4369.
Similarly, GWq-ROFG operatorsΘ1′,Θ3′,Θ4′, and Θ5′ can be obtained.Theorem 5.
Letajk=μjk,νjk and di′k=si′j,ti′j be the q-ROFVs in GGq-ROFSS in Table 2, where k=1,2,…,n, j=1,2,…,m, and i′=1,2,…,p. If we consider i=0,1,2,…,p, then GWq-ROFG operator is given by(36)GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn=∏k=1n∏i=0pμikjqφiϖk,q1−∏k=1n1−1−∏i=0p1−νikjqφiϖk.Proof.
Same as the proof of Theorem1.Theorem 6.
(idempotency) Ifμjk,νjk=μj,νj and μi′k,νi′k=μ,ν=μj,νj for all k=1,2,…,n, then GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn=μ,ν.Proof.
Same as the proof of Theorem2.Theorem 7.
(boundedness) Ifμ+,ν−=μikjmax,νikjmin and μ−,ν+=μikjmin,νikjmax for all j=1,2,…,m and i=0,1,2,…,p, thenμ−,ν+≤GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn≤μ+,ν−.Proof.
Same as the proof of Theorem3.Theorem 8.
(monotonicity) Ifμikj,νikj and μ¯ikj,ν¯ikj for all i=0,1,2,…,p, j=1,2,…,m,k=1,2,…,n are two IFVs such that μikj,νikj≤μ¯ikj,ν¯ikj, then(37)GWq−ROFGμj1,νj1,μj2,νj2,…,μjn,νjn≤GWq−ROFGμ¯j1,ν¯j1,μ¯j2,ν¯j2,…,μ¯jn,ν¯jn.Proof.
It can be concluded from Theorem4.Proposition 3.
LetQ,A,η be a GGq-ROFSS, given in Table 2. Then,(1)
Ifsi′k,ti′k=0,1 for all i′ and k, then Θj′=0,1,0,1,…,0,1.(2)
Ifμ0kj,ν0kj=μjk,νjk=0,1 for all j and k, then Θj′=0,1,0,1,…,0,1.(3)
Ifsi′k,ti′k=0,1 for all i′,k and μ0kj,ν0kj=μjk,νjk=0,1 for all j,k then Θj′=0,1,0,1,…,0,1.(4)
Ifsi′k,ti′k=1,0 for all i′,k and μ0kj,ν0kj=μjk,νjk=1,0 for all j,k then Θj′=1,0,1,0,…,1,0.


## 6. Multicriteria Decision-Making Method

In this section, a methodology on proposed operators is introduced, and for application, a numerical application is investigated.


### 6.1. Methodology

LetX=ρ1,ρ2,…,ρm be the set of alternatives for a problem in which we have to choose the best alternative. Every alternative possesses the specific q-ROFVs of a set of criteria/attributes E. Let A⊆E. An expert’s committee give the q-ROFVs on set X comprising attributes in the form of q-ROFSS. The final examination of the q-ROFSS is provided by some senior experts/moderators in the form of q-ROFs over A=e1,e2,…,en. This forms GGq-ROFSS Q,A,η given in Table 2 in Definition 14. In real-life problem, usually, the importance of attributes is given in the form of a weighted vector. Let Ψ=φ1,φ2,…,φnT be the weighted vector over A, such that ∑k=1nφk=1,φk>0. Also take weighted vector W=ϖ0,ϖ1,ϖ2,…,ϖpT such that ∑i′=0pϖi=1,ϖi>0 where ϖ1,ϖ2,…,ϖp are the weights for the judgements of the “p” number of senior moderators, and ϖ0 is the weight for each q-ROFV in the light gray part of Table 2. In other words, ϖ0 is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Next, the normalization of the q-ROFVs in Q,A,η∀i,j,k is obtained by using the following equation:(38)μikj,νikj=μikje,νikje,ifeis benefit kind criteria,νikje,μikje,ifeis cost kind criteria.This designs a new GGq-ROFSSQ′,A,η′. Then, we compute GWq-ROFA or GWq-ROFG operators on Q′,A,η′ and calculate score function on each GWq-ROFA or GWq-ROFG. Finally, rank the alternatives on the real values of score functions.In order to evaluate most suitable alternative, we numerate the steps of the MCDM method in an algorithm as follows.Algorithm1: MCDM procedure.
(1)
Consider a set of alternatives sayX=ρ1,ρ2,…,ρs their attributes as a set E. A=e1,e2,…,en.(2)
Constituter number of committees of experts CM1,CM2,…,CMr comprising sets of attributes A1,A2,…,Ar respectively, where A1⊂E,A2⊂E,…,Ar⊂E.(3)
Obtainr number q-ROFSSs on the CM1,CM2,…,CMr respectively.(4)
Obtain curial opinions ofp number moderators on each q-ROFSSs. This formulate a GGq-ROFSSs Q1,A1,η1,Q2,A2,η2,…,Qr,Ar,ηr.(5)
Take unionQ1,A1,η1,Q2,A2,η2,…,Qr,Ar,ηr which is equal to Q,A,η, where A=∪j′=1rAj′=e1,e2,…,en(6)
NormalizeQ,A,η, which gives normalized GGq-ROFSS Q′,A,η′.(7)
Obtain the weighted vectorΨ=φ1,φ2,…,φnT over set A of attributes. Obtain the weighted vector W=ϖ0,ϖ1,ϖ2,…,ϖpT comprising q-ROFSS and opinions of moderators.(8)
Compute GGq-ROFA (or GGq-ROFG) operatorsΘjorΘj′j=1,2,…,m on normalized GGq-ROFSS Q′,A,η′.(9)
Get score functionSliu or Spx on each GGq-ROFA (or GGq-ROFG) operator Θj or Θj′j=1,2,…,m using Definition 4.(10)
Classify alternatives and foremost alternative is on the maximum score.This algorithm is shown in Figure1.Figure 1
A flowchart of our algorithm.


### 6.2. Numerical Example

An automation company wants to choose a director for their Research and Development (R and D) department from four applicantsρ1,ρ2,ρ3, and ρ4. The crucial attributes for evaluation areC1: personalityC2: self-confidenceC3: score obtained in a college degreeC4: administrative skillsC5: experienceC6: abilities in research areasIn order to evaluate most suitable candidate, two different committees of experts, namely,CM1 and CM2, are constituted by the company. The attribute specifications for CM1 and CM2 are A=C1,C2,C5 and ℬ=C3,C4,C6, respectively. The q-ROFV-based evaluations are q-ROFSSs Q,A (in the light gray part of Table 7) and Q′,ℬ (in the light gray part of Table 8). To finalize q-ROFV-based data, two senior experts provided their opinion as q-ROFSs on the q-ROFSSs (see the brown part of Tables 7 and 8). The following steps are adopted to finalize decision-making process:(1)
Take union on GGq-ROFSSsQ,A,η and Q′,ℬ,η′, and obtain GGq-ROFSS ℛ,E,γ in Table 9.(2)
TakeΨ=ϖ1,ϖ2,…,ϖ6T=0.2,0.3,0.1,0.2,0.1,0.1T be the weighted vector over E. Also take weighted vector W=φ0,φ1,φ2T=0.4,0.25,0.35T for q-ROFSS and judgements of senior experts.(3)
Calculate GWq-ROFA operators; we useq=4 and obtain Θ1=0.6490,0.5719,Θ2=0.7065,0.5555,Θ3=0.7431,0.4530, and Θ4=0.6464,0.6013.(4)
Calculate score function; we obtainSliuΘ1=0.0771,SliuΘ2=0.1510, SliuΘ3=0.2901, and SliuΘ4=0.0451.(5)
Final ranking of candidates is given byρ3>ρ2>ρ1>ρ4. Thus, ρ3 is the most appropriate candidate for the position of director in the company.Table 7
GGq-ROFSSsQ,A,η.Table 8
GGq-ROFSSsQ′,ℬ,η′.Table 9
GGq-ROFSSsℛ,E,γ.


## 6.1. Methodology

LetX=ρ1,ρ2,…,ρm be the set of alternatives for a problem in which we have to choose the best alternative. Every alternative possesses the specific q-ROFVs of a set of criteria/attributes E. Let A⊆E. An expert’s committee give the q-ROFVs on set X comprising attributes in the form of q-ROFSS. The final examination of the q-ROFSS is provided by some senior experts/moderators in the form of q-ROFs over A=e1,e2,…,en. This forms GGq-ROFSS Q,A,η given in Table 2 in Definition 14. In real-life problem, usually, the importance of attributes is given in the form of a weighted vector. Let Ψ=φ1,φ2,…,φnT be the weighted vector over A, such that ∑k=1nφk=1,φk>0. Also take weighted vector W=ϖ0,ϖ1,ϖ2,…,ϖpT such that ∑i′=0pϖi=1,ϖi>0 where ϖ1,ϖ2,…,ϖp are the weights for the judgements of the “p” number of senior moderators, and ϖ0 is the weight for each q-ROFV in the light gray part of Table 2. In other words, ϖ0 is the weight of whole data in q-ROFSS (see the light gray part of Table 2). Next, the normalization of the q-ROFVs in Q,A,η∀i,j,k is obtained by using the following equation:(38)μikj,νikj=μikje,νikje,ifeis benefit kind criteria,νikje,μikje,ifeis cost kind criteria.This designs a new GGq-ROFSSQ′,A,η′. Then, we compute GWq-ROFA or GWq-ROFG operators on Q′,A,η′ and calculate score function on each GWq-ROFA or GWq-ROFG. Finally, rank the alternatives on the real values of score functions.In order to evaluate most suitable alternative, we numerate the steps of the MCDM method in an algorithm as follows.Algorithm1: MCDM procedure.
(1)
Consider a set of alternatives sayX=ρ1,ρ2,…,ρs their attributes as a set E. A=e1,e2,…,en.(2)
Constituter number of committees of experts CM1,CM2,…,CMr comprising sets of attributes A1,A2,…,Ar respectively, where A1⊂E,A2⊂E,…,Ar⊂E.(3)
Obtainr number q-ROFSSs on the CM1,CM2,…,CMr respectively.(4)
Obtain curial opinions ofp number moderators on each q-ROFSSs. This formulate a GGq-ROFSSs Q1,A1,η1,Q2,A2,η2,…,Qr,Ar,ηr.(5)
Take unionQ1,A1,η1,Q2,A2,η2,…,Qr,Ar,ηr which is equal to Q,A,η, where A=∪j′=1rAj′=e1,e2,…,en(6)
NormalizeQ,A,η, which gives normalized GGq-ROFSS Q′,A,η′.(7)
Obtain the weighted vectorΨ=φ1,φ2,…,φnT over set A of attributes. Obtain the weighted vector W=ϖ0,ϖ1,ϖ2,…,ϖpT comprising q-ROFSS and opinions of moderators.(8)
Compute GGq-ROFA (or GGq-ROFG) operatorsΘjorΘj′j=1,2,…,m on normalized GGq-ROFSS Q′,A,η′.(9)
Get score functionSliu or Spx on each GGq-ROFA (or GGq-ROFG) operator Θj or Θj′j=1,2,…,m using Definition 4.(10)
Classify alternatives and foremost alternative is on the maximum score.This algorithm is shown in Figure1.Figure 1
A flowchart of our algorithm.


## 6.2. Numerical Example

An automation company wants to choose a director for their Research and Development (R and D) department from four applicantsρ1,ρ2,ρ3, and ρ4. The crucial attributes for evaluation areC1: personalityC2: self-confidenceC3: score obtained in a college degreeC4: administrative skillsC5: experienceC6: abilities in research areasIn order to evaluate most suitable candidate, two different committees of experts, namely,CM1 and CM2, are constituted by the company. The attribute specifications for CM1 and CM2 are A=C1,C2,C5 and ℬ=C3,C4,C6, respectively. The q-ROFV-based evaluations are q-ROFSSs Q,A (in the light gray part of Table 7) and Q′,ℬ (in the light gray part of Table 8). To finalize q-ROFV-based data, two senior experts provided their opinion as q-ROFSs on the q-ROFSSs (see the brown part of Tables 7 and 8). The following steps are adopted to finalize decision-making process:(1)
Take union on GGq-ROFSSsQ,A,η and Q′,ℬ,η′, and obtain GGq-ROFSS ℛ,E,γ in Table 9.(2)
TakeΨ=ϖ1,ϖ2,…,ϖ6T=0.2,0.3,0.1,0.2,0.1,0.1T be the weighted vector over E. Also take weighted vector W=φ0,φ1,φ2T=0.4,0.25,0.35T for q-ROFSS and judgements of senior experts.(3)
Calculate GWq-ROFA operators; we useq=4 and obtain Θ1=0.6490,0.5719,Θ2=0.7065,0.5555,Θ3=0.7431,0.4530, and Θ4=0.6464,0.6013.(4)
Calculate score function; we obtainSliuΘ1=0.0771,SliuΘ2=0.1510, SliuΘ3=0.2901, and SliuΘ4=0.0451.(5)
Final ranking of candidates is given byρ3>ρ2>ρ1>ρ4. Thus, ρ3 is the most appropriate candidate for the position of director in the company.Table 7
GGq-ROFSSsQ,A,η.Table 8
GGq-ROFSSsQ′,ℬ,η′.Table 9
GGq-ROFSSsℛ,E,γ.


## 7. Comparative Study

In this part of the article, we compare our method with existing frameworks. The primely advantage of new aggregation operators is to characterize a weighted vectorW=φ0,φ1,φ2,…,φpT on GGq-ROFSS, where φ1,φ2,…,φp are the weights on q-ROFSs of p number of moderators respectively, and φ0 is a weight for q-ROFSS in GGq-ROFSS. Since assessments of moderators on q-ROFSS are the crucial component in GGq-ROFSS, in many real-life problems, their weights should be greater than φ0. On the other hand, in some real-life situations, weight φ0 on q-ROFSS is also important, the fact that it denotes a perception on fundamental data (or q-ROFSS) given by a committee of experts on crucial judgements on alternatives. Notably, we consider following example.Example 4.
Consider GGq-ROFSS as depicted in Table6, where A=e1,e2,e3 is set of attributes with weighted vector Ψ=0.3/e1,0.4/e2,0.3/e3 and X=ρ1,ρ2,ρ3 being the set of alternatives. α1 is q-ROFS for moderator’s assessments on q-ROFSS in Table 10.
Letq=3 and a weighted vector W=0.5/w0,0.5/w1, where w0 is the weight for q-ROFSS and w1 is the weight for q-ROFS α1. By our new operators on GGq-ROFSSs, we obtain SliuΘ1=0.0233, SliuΘ2=0.0225, SliuΘ3=0.0235 and ρ3>ρ1>ρ2.
By the operators in Hayat et al. [56], we obtain Sρ1=0.7883,Sρ2=0.7646,Sρ3=0.7777 and ρ1>ρ3>ρ2.
By Feng et al. [54], Sρ1=0.5931,Sρ2=0.5398,Sρ3=0.5509,ρ1>ρ3>ρ2.
By the operators in Hayat et al. [55, 59], we obtain ρ1>ρ3>ρ2. A comparison for different methods on this example is shown in Table 11.
The main advantage our framework is that it aggregates information or data by concerning attributes until final ranking appears. It develop an internal mechanism that gently addresses the importance of parameters in aggregation operators for GGq-ROFSSs. The different components of the above methods are discussed in Table12.Table 10
GGq-ROFSSQ,A,η.Table 11
Comparison.
MethodActRankingProposed methodq=3ρ3>ρ1>ρ2Hayat et al. [55]q=1ρ3>ρ1>ρ2Hayat et al. [56]q=1ρ1>ρ3>ρ2Feng et al. [53]q=1ρ1>ρ3>ρ2Hayat et al. [59]q=1ρ3>ρ1>ρ2Table 12
Advantages.
Methods\approchesNumbers of extra inputsWeighted vectorsOperators typeDeficiencyHayat et al. [55]OneNoneAND operationDiminish the importance of reliable and unreliable grades on AND operationHayat et al. [56]More than oneOver attributesWA or WGNot efficient on the same IF values of a moderator on each attributeHayat et al. [59]More than oneTwo weighted vectors over attributes and moderators respectivelyWA or WGDo not deal with larger spaceq>1Feng et al. [53]OneConversion of extra input into weighted vectorWA or WGDiminish the importance of reliable and unreliable grades by the conversion of extra input into weighted vectorProposed methodMore than oneTwo weighted vectors over attributes and moderatorsWA or WGSeem no deficiency


## 8. Conclusions

In this article, we have presented GGq-ROFSS and investigated some basic operations. Mainly, we have initiated new averaging and geometric aggregation operators on GGq-ROFSSs and investigated the underlying properties of these operators. We have given a MCDM framework and its validation through a numerical example. Finally, we have given comparison of our methodology with other existing methods. In GGq-ROFSSs, expert’s judgements suggest the reliability of the evaluation of the alternatives on criteria. These judgements appear as generalization part in GGq-ROFSSs to fulfil and complete q-ROFSS-based data as some final examination. Therefore, for prospect MCDM, GGq-ROFSS, which indicates inμq+νq≤1 for some q≥1 suggests a better framework to deal with uncertainty.The method of computation of q-ROFS-based data from newly introduced ideas can sight wide applications for available data in machine learning, applied intelligence, and supply chain management. Although there are some methods in machine learning, artificial intelligence, and supply chain management, proposed operators’ beauty will be a notable sight in such domains. Our focus will be on these operators’ new rudiments and their supply chain management applications in future work. Furthermore, we will define GGq-ROFSS-based order operators, Shapley Choquet operators, and distance aggregation operator.



---

*Source: 5672097-2021-12-31.xml*