# Decision-Making Based on q-Rung Orthopair Fuzzy Soft Rough Sets

Authors: Yinyu Wang; Azmat Hussain; Tahir Mahmood; Muhammad Irfan Ali; Hecheng Wu; Yun Jin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Recently, Yager presented the new concept of q-rung orthopair fuzzy (q-ROF) set (q-ROFS) which emerged as the most significant generalization of Pythagorean fuzzy set (PFS). From the analysis of q-ROFS, it is clear that the rung q is the most significant feature of this notion. When the rung q increases, the orthopair adjusts in the boundary range which is needed. Thus, the input range of q-ROFS is more flexible, resilient, and suitable than the intuitionistic fuzzy set (IFS) and PFS. The aim of this manuscript is to investigate the hybrid concept of soft set (StS) and rough set with the notion of q-ROFS to obtain the new notion of q-ROF soft rough (q-ROFStR) set (q-ROFStRS). In addition, some averaging aggregation operators such as q-ROFStR weighted averaging (q-ROFStRWA), q-ROFStR ordered weighted averaging (q-ROFStROWA), and q-ROFStR hybrid averaging (q-ROFStRHA) operators are presented. Then, the basic desirable properties of these investigated averaging operators are discussed in detail. Moreover, we investigated the geometric aggregation operators, such as q-ROFStR weighted geometric (q-ROFStRWG), q-ROFStR ordered weighted geometric (q-ROFStROWG), and q-ROFStR hybrid geometric (q-ROFStRHG) operators, and proposed the basic desirable characteristics of the investigated geometric operators. The technique for multicriteria decision-making (MCDM) and the stepwise algorithm for decision-making by utilizing the proposed approaches are demonstrated clearly. Finally, a numerical example for the developed approach is presented and a comparative study of the investigated models with some existing methods is brought to light in detail which shows that the initiated models are more effective and useful than the existing methodologies.


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## Body

## 1. Introduction

Decision-making has always been a hot topic under consideration by the researchers. Multicriteria decision-making (MCDM) has a high prospective and discipline manner to improve and evaluate multiple conflicting criteria in all areas of decision-making. In this competitive environment, an enterprise needs the most accurate and rapid response to change the customer needs. So, MCDM has the ability to handle successfully the evaluation process of multiple contradictory criteria. For an intelligent decision, the experts analyze each and every character of an alternative and then they take the decision. For an intelligent and successful decision, the experts require careful preparation and analysis of each and every character for an alternative and then they can take a good decision if they are armed with all the data and information that they need. To handle this complexity, Zadeh [1] originated the dominant and pioneer concept of the fuzzy set. For each domain in the fuzzy set, a value is assigned from the unit interval and is called membership grade (MemG). From the inception of the fuzzy set, it has been generalized in different directions in which one of the most significant concepts is intuitionistic fuzzy (IF) set (IFS). Atanassov [2] initiated this dominant concept of IFS which is characterized by two mappings called MemG and nonmembership grade (NMemG). IFS is defined on the basis of restriction that the sum of MemG and NMemG must not exceed the unit interval 0,1. The notion of IFS appears as a hot research area after its origination. In literature, researchers used different techniques to handle the ranking with score or accuracy functions but all these techniques had some drawbacks. So, Ali et al. [3] initiated a graphical technique for ranking the IF values. Xu [4] investigated the series of aggregation operators such as IF weighted averaging (IFWA), IF ordered weighted averaging (IFOWA), and IF hybrid averaging (IFHA) operators under IF environment. The series of geometric operators, namely, IF weighted geometric (IFWG), IF ordered weighted geometric (IFOWG), and IF hybrid geometric (IFHG) operators, were presented by Xu and Yager [5]. Zhao et al. [6] initiated the idea of generalized IFWA, generalized IFOWA, and generalized IFHA operators by utilizing the IF information. Wang and Liu [7, 8] originated the concepts of geometric and averaging aggregation operators by utilizing Einstein operations. Zeng et al. [9] investigated a novel score function of intuitionistic fuzzy and then presented its application in decision-making. In a different scenario, the professional experts are restricted to provide their choices in the range of IFS. To cover this shortcoming, Yager [10] investigated the powerful paradigm of Pythagorean fuzzy (PF) set (PFS) in which the square sum of MemG and NMemG must lie between the real numbers 0and1. PFS relaxes and widens the boundary range by providing additional space to the decision-maker. Yager [11] originated the geometric and averaging aggregation operation by using PF information. Peng and Yang [12] initiated the concept of subtraction and division operators and proved some of its basic properties. Peng and Yang [13] investigated the notion of PF Choquet integral average and PF Choquet integral geometric operators. Garg [14, 15] proposed some PF Einstein averaging and PF Einstein geometric operators and presented their basic characteristics. Garg [16] investigated confidence PF weighted and ordered weighted averaging operators with their basic properties. Wei and Lu [17] originated the notions of PF power averaging and geometric operators and presented their desirable characteristics of these investigated operators. Wei [18] presented some interaction averaging and geometric operators by suing PF information. The concept of Hamacher operations for PF averaging and geometric operators was presented by Wu and Wei [19]. In the PF environment, the decision-makers are restricted to their boundary limitation and they cannot provide their preferred values freely. Due to these restrictions, some decisive information cannot be effectively handled by PFS.Recently, Yager [20] presented the new concept of q-rung orthopair fuzzy (q-ROF) set (q-ROFS) from which the most significant generalization of PFS emerged. In q-ROFS, the sum of qth power of MemG and qth power of NMemG must be confined to the unit interval 0,1 and, furthermore, when the rung q increases, then the range of orthopair satisfies the boundary restriction which is needed. Thus, the concept of q-ROFS is more useful and powerful than IFS and PFS because these are the special cases of q-ROFS. The basic properties of q-ROFS are proposed by Yager and Alajlan [21] and have been utilized in knowledge representation. Ali [22] proposed another view of q-ROFS by using the idea of orbits. Liu and Wang [23] proposed the concepts of q-ROF weighted averaging (q-ROFWA) and q-ROF weighted geometric (q-ROFWG). Liu and Liu [24] presented the combined study of Bonferroni mean (BM) operators with q-ROFS to investigate the q-ROF BM operators and also study q-ROF geometric BM operators with their desirable properties. Jana et al. [25] initiated the q-ROF Dombi averaging and geometric aggregation operators with their fundamental desirable characteristics. Wang et al. [26] investigated the combined concept of Muirhead means (MM) operators with q-ROFS to get the new aggregation operators that are q-ROF MM operators. Joshi and Gegov [27] initiated the concept of the confidence level of experts to the original information under q-ROF environment to propose some aggregation operators such as confidence q-ROFWA (Cq-ROFWA) and confidence q-ROFWG (Cq-ROFWG), Cq-ROFOWA, and Cq-ROFOWG operators. Yang et al. [28] presented the idea of q-RO normal fuzzy sets and defined the operational laws and score function for it. They also initiated some aggregation operators for the same concept that is q-RONFWA and q-RONFOWG. Furthermore, Hussain et al. [29] proposed hesitant q-ROFWA and hesitant q-ROFWG operators and discussed their desirable properties. Hussain et al. [30] proposed the generalized and group generalized averaging operation by using q-ROF information.The dominant theory of rough set was first proposed by Pawlak [31] which generalized the classical set theory to cope with the imprecise, vague, and uncertain information. By the definition of Pawlak rough set, a universal set is characterized by two approximation sets known as upper and lower approximations. The lower approximation consists of those alternatives which contain a subset and the upper approximation consists of those alternatives having nonempty intersection with a subset. Further equivalence relation plays a key role in Pawlak rough set for approximations but this condition too restricts the practical and theoretical aspects of rough set. So, researchers used the generalized structure by using the nonequivalence structure; for details, see [32–38]. From the inception, researchers used the hybrid study of rough set theory with different concepts. The hybrid study of rough set and IFS was proposed by Chakrabarty et al. [39] to obtain the notion of IF rough set (IFRS) and IFRS became the hot and progressive research area for the researchers; for details, see [40–43]. Zhou and Wu [44] proposed the combined study of rough set and IFS by using crisp and fuzzy approximation. Zhou and Wu [45] initiated the constructive and axiomatic approach under the IF rough environment. Hussain et al. [46] investigated the idea of rough PF ideals by using the algebraic structure of semigroups. Zeng [47] proposed a new MCDM technique based on probabilistic information by using the PF environment. Hussain et al. [48] proposed the notion of q-ROF rough set by utilizing fuzzy β-covering and fuzzy β-covering neighborhood. Molodtsov [49] originated the prominent and pioneer concept of soft set (StS) which generalized the classical set theory and is free from inherent complexity which the contemporary theories faced. It is observed that StS has a very close relation with fuzzy set and rough set. The StS theory is an essential concept and powerful mathematical tool for coping the uncertain, ambiguous, and imprecise data. Maji et al. [50, 51] proposed the hybrid notion of StS with fuzzy set and IFS to obtain fuzzy StS and IFStS which play a key role among these theories. Ali et al. [52] improved some existing definitions and operations in StS theory. The concept of generalized IFStS was proposed by Agarwal et al. [53]. Arora and Garg [54] presented the concept of IFSt weighted averaging (IFStWA) and IFSt weighted geometric (IFStWG) operators. Garg and Arora [55] proposed the notion of some power averaging and geometric aggregation operators by utilizing generalized IFStS. Arora [56] initiated the notion of IFStWA and IFStWG by using the Einstein operations. Feng et al. [57] improved some existing literature related to generalized IF StS and proposed some new operations for the developed concept. The combined study StS, rough set, and PFS were presented by Hussain et al. [58] to achieve the new concept of soft rough PFS and PF soft rough set. Riaz and Hashmi [59] presented the hybrid study of StS, rough set, PFS, and m-polar fuzzy set to get the new notion of Pythagorean m-polar fuzzy soft rough set. Hussain et al. [60] originated the hybrid structure of StS with q-ROFS to get the prominent notion of q-ROF soft (q-ROFSt) set (q-ROFStS) and proposed some aggregation operators such as q-ROFSt weighted averaging (q-ROFStWA), q-ROFSt ordered weighted averaging (q-ROFStOWA), and q-ROFSt hybrid averaging (q-ROFStHA).q-Rung orthopair fuzzy soft rough sets, a hybrid intelligent structure of soft sets, rough sets, and q-rung orthopair fuzzy sets are a powerful mathematical tool to deal with indeterminate, inconsistent, and incomplete information, which has caught the attention of the researchers. From the analysis, it is observed that aggregation operators have great importance in decision-making to aggregate the collective evaluated information of different sources into a single value. According to the best of our knowledge up till now, no application of the aggregation operators with the hybridization of q-ROFS with soft set and rough set is reported in q-ROF environment. Therefore, this motivates the current work of q-ROFSt rough study, and, further, we will investigate aggregation operators based on soft rough information that are q-ROFStRWA, q-ROFStROWA, q-ROFStRHA, q-ROFStRWG, q-ROFStROWG, and q-ROFStRHG operators.The design of the remaining sections of the manuscript is summarized as follows: Section2 consists of a brief study of the basic notions connecting the link with the coming sections. Section 3 is devoted to investigating the hybrid concept of StS and rough set with the notion of q-ROFS to obtain the new concept of q-ROFStRS. In Section 4, we presented the averaging aggregation operators such as q-ROFStRWA, q-ROFStROWA, and q-ROFStRHA. Furthermore, the basic desirable properties of investigated averaging operators that are Idempotency, Boundedness, Monotonicity, shift invariance, and Homogeneity are investigated in detail. Section 5 is devoted to the geometric aggregation operators such as q-ROFStRWG, q-ROFStROWG, and q-ROFStRHG. Moreover, the basic desirable characteristics of these investigated geometric operators that are Idempotency, Boundedness, Monotonicity, shift invariance, and Homogeneity are investigated in detail. In Section 6, the technique for MCDM and the stepwise algorithm for decision-making are demonstrated by utilizing the proposed approach. In Section 7, a numerical example for the developed approach is presented and a comparative study of the investigated models with some existing methods is given in detail which shows that the investigated models are more effective and useful than existing approaches. The final Section 8 consists of a conclusion of the manuscript.


## 2. Preliminaries

This section consists of some basic notions including IFS, PFS, q-ROFS, and q-ROFStS which will be helpful in on word sections.Definition 1 (see [2]).
Consider a universeY and an IFST on set Y denoted and defined as(1)T=s,βTs,ϑTs|s∈Y,in which βT,ϑT:Y⟶0,1 denotes the MemG and NMemG of an alternative s∈Y to the set T having the condition that 0≤βTs+ϑTs≤1.πT=1−βTs+ϑTs is called hesitancy degree of s∈Y.Definition 2 (see [10]).
Consider a universeY and a PFST on set Y defined and denoted as(2)T=s,βTs,ϑTs|s∈Y,in which βT,ϑT:Y⟶0,1 denotes the MemG and NMemG of an alternative s∈Y to the set T having the condition that 0≤βTs2+ϑTs2≤1.πT=1−βTs2−ϑTs2 is called hesitancy degree of s∈Y.Definition 3 (see [20]).
ConsiderY as a universe of discourse and a q-ROFST on set Y is an object of the form(3)T=s,βTs,ϑTsq|s∈Yandq≥1,in which βT,ϑT:Y⟶0,1 shows the MemG and NMemG of an alternative s∈Y to the set T having the condition that 0≤βTsq+ϑTsq≤1. The hesitancy degree is shown as πT=1−βTsq−ϑTsqq for each s∈Y.
Molodtsov [49] originated the prominent and pioneer concept of soft set (StS) which generalized the classical set theory and is free from inherent complexity which the contemporary theories faced. It is observed that StS has very close relation with fuzzy set and rough set. The StS theory is an essential concept and powerful mathematical tool for coping the uncertain, ambiguous and imprecise data which is defined as:Definition 4 (see [49]).
LetY be a fixed set and ℰ be set of parameters with ℬ⊆ℰ. Then the pair T,ℬ is said to be StS, where T is function given as T:ℬ⟶PY. PY denotes the collection of all subsets of Y.Definition 5 (see [50]).
LetT,ℬ a StS over Y with ℬ⊆ℰ. Then T∗,ℬ is known as fuzzy StS over Y, where T∗ is a function given as T∗:ℬ⟶FY. FY denotes the collection of all fuzzy subsets of Y and mathematically it is gives as(4)Tcj∗=si,βjsi|si∈Y.Definition 6 (see [60]).
Consider a universal setY. Let ℰ be set of parameters and ℬ⊆ℰ. Then a q-ROFStS is a pair T,ℬ over set Y and T is a mapping given as T:ℬ⟶q−ROFStSY in which q−ROFStSY contains the collection of all q-ROFSs. Then, the q-ROFStS is denoted and defined as(5)Tcjsi=si,βjsi,ϑjsiq|si∈Yandq≥1,in which βjsi,ϑjsi represents the MemG and NMemG of an alternative si∈Y to the set Tcj satisfying that 0≤βjsiq+ϑjsiq≤1.πT=1−βjsiq−ϑjsiqq is known to be hesitancy degree of si∈Y. For simplicity Tcjsi=si,βjsi,ϑjsiq is denoted as Tcj=βij,ϑij if there is no confusion and is called q-ROFSt number (q-ROFStN).
Considering two q-ROFStNs Tcj=β1j,ϑ1j for j=1,2, reference [20] defined the following operation are defined on them:(i)
Tc1∪Tc2=maxβ11,β12,minϑ11,ϑ12(ii)
Tc1∩Tc2=minβ11,β12,maxϑ11,ϑ12(iii)
Tc1⊕Tc2=β11q+β12q−β11qβ12qq,ϑ11ϑ12(iv)
Tc1⊕Tc2=β11β12,ϑ11q+ϑ12q−ϑ11qϑ12qq(v)
Tc1≤Tc2ifβ11≤β12,ϑ11≥ϑ12(vi)
Tc1c=ϑ11,β11 where Tc1c represents the complement of Tc1(vii)
αTc1=1−1−β11qαq,ϑ11αforα≥1(viii)
Tc1α=β11α,1−1−ϑ11qαqforα≥1


## 3. q-ROF Soft Rough Set (q-ROFStRS)

The concept ofStS theory, is the generalization of classical set theory and is free from inherent complexity which the contemporary theories faced. The StS theory and rough set theory are essential concepts and powerful mathematical tools for coping the uncertain, ambiguous, and imprecise data. Motivated from the combining study of soft rough set, this section is devoted to the hybridized study of q-ROFS with StS and rough set to obtain the new concept of q-ROFStRS. Some basic operations, a new score function, and some desirable characteristics of the proposed concept are investigated in detail.Definition 7.
LetT,ℰ be a q-ROFStS over Y. Any subset ℒ of Y×ℰ is set to a q-ROFSt relation from Ytoℰ and is defined as(6)ℒ=si,cj,βsi,cj,ϑsi,cj|si,cj∈Y×ℰ,where β:Y×ℰ⟶0,1andϑ:Y×ℰ⟶0,1 denote the MemG and NMemG with 0≤βsi,cjq+ϑsi,cjq≤1 for all si,cj∈Y×ℰ.
IfY=s1,s2,…,smandℰ=c1,c2,…,cn, then, q-ROFSt relation ℒ from Ytoℰ can be presented in Table 1.Table 1
q-ROFSt relation ℒ from Ytoℰ.
ℒc1c2⋯cns1βs1,c1,ϑs1,c1βs1,c2,ϑs1,c2⋯βs1,cn,ϑs1,cns2βs2,c1,ϑs2,c1βs2,c2,ϑs2,c2⋯βs2,cn,ϑs2,cn⋮⋮⋮⋱⋮smβsm,c1,ϑsm,c1βsm,c2,ϑsm,c2⋯βsm,cn,ϑsm,cnDefinition 8.
Consider a universalY, ℰ as the set of parameters and T,ℰ as a q-ROFStS. Let ℒ be an arbitrary q-ROFSt relation from set Y to ℰ. The pair T,ℰ,ℒ is said to be q-ROFSt approximation space. For any optimum decision normal object ℳ∈q−ROFSℰ, then the lower and upper approximation of ℳ with respect to approximation space T,ℰ,ℒ are denoted and defined as(7)ℒ¯ℳ=si,βj¯si,ϑj¯si|si∈Y,(8)ℒ¯ℳ=si,βj¯si,ϑj¯si|si∈Y,where βj¯si=∧cj∈ℰβℒsi,cj∧βℳcj, ϑj¯si=∨cj∈ℰϑℒsi,cj∨ϑℳcj and βj¯si=∨cj∈ℰβℒsi,cj∨βℳcj, ϑj¯si=∧cj∈ℰϑℒsi,cj∧ϑℳcj such that 0≤βj¯siq+ϑj¯siq≤1and0≤βj¯siq+ϑj¯siq≤1.
It is observed thatℒ¯ℳ and ℒ¯ℳ are two q-ROFSs in Y. Thus, the operators ℒ¯ℳ,ℒ¯ℳ:q−ROFStℰ⟶q−ROFStY are, respectively, known as lower and upper q-ROFStR approximation operators. Therefore, q-ROFStRS is a pair ℒℳ=ℒ¯ℳ,ℒ¯ℳ=si,βj¯si,ϑj¯si,βj¯si,ϑj¯si.
For simplicity, we can writeℒℳ=ℒ¯ℳ,ℒ¯ℳ=si,βj¯si,ϑj¯si,βj¯si,ϑj¯si as ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi=βij¯,ϑij¯,βij¯,ϑij¯ and call q-ROFStR number (q-ROFStRN), if there is no confusion.Remark 1.
(a)
Ifq=1 is fixed, then the developed q-ROFStR approximation operators reduce to IFStR approximation operators(b)
Ifq=2 is fixed, then the developed q-ROFStR approximation operators reduce to PFStR approximation operators
Consider the following example to better understand the concept of q-ROFStR approximation operators.Example 1.
Suppose a decision-makerZ buys a house, as given in set Y=s1,s2,s3,s4,s5 under consideration. Let the parameter set ℰ=c1,c2,c3,c4 where c1=beautiful, c2=size, c3=expensive, and c4=location. A decision-maker Z wants to purchase a house from the available houses which fulfill the utmost extent of given parameters. Consider the decision-maker Z presents the gorgeous of houses in form of q-ROFSt relation ℒ from set Ytoℰ and is given in Table 2.
Consider a decision-makerZ presents the optimum normal decision object ℳ which is a q-ROF subset over parameter set ℰ; that is,(9)ℳ=c1,0.9,0.2,c2,0.4,0.6,c3,0.8,0.4,c4,0.5,0.1.
Now, by using equations (7) and (8), we have(10)β1¯s1=0.4,ϑ1¯s1=0.6,β2¯s2=0.3,ϑ2¯s2=0.7,β3¯s3=0.2,ϑ3¯s3=0.9,β4¯s4=0.4,ϑ4¯s4=0.6,β5¯s5=0.3,ϑ5¯s5=0.7,β1¯s1=0.9,ϑ1¯s1=0.1,β2¯s2=0.9,ϑ2¯s2=0.1,β3¯s3=0.95,ϑ3¯s3=0.1β4¯s4=0.93,ϑ4¯s4=0.1,β5¯s5=0.9,ϑ5¯s5=0.1.
Now, to get the lower and upper q-ROFStR approximation operators,(11)ℒ¯ℳ=s1,0.7,0.6,s2,0.3,0.7,s3,0.2,0.9,s4,0.4,0.6,s5,0.3,0.7,ℒ¯ℳ=s1,0.9,0.1,s2,0.9,0.1,s3,0.95,0.1,s4,0.93,0.1,s5,0.9,0.1.
Therefore,(12)ℒℳ=ℒ¯ℳ,ℒ¯ℳ=s1,0.7,0.6,0.9,0.1,s2,0.3,0.7,0.9,0.1,s3,0.2,0.9,0.95,0.1,s4,0.4,0.6,0.93,0.1,s5,0.3,0.7,0.9,0.1.Table 2
q-ROFSt relation from set Ytoℰ for q≥3.
ℒc1c2c3c4s10.9,0.40.8,0.20.7,0.30.65,0.2s20.8,0.50.5,0.10.85,0.20.3,0.7s30.6,0.90.2,0.60.6,0.10.95,0.3s40.7,0.40.93,0.40.4,0.20.5,0.1s50.3,0.70.78,0.250.8,0.150.7,0.4Definition 9.
Considerℒcjℳ1=ℒcj¯ℳ1,ℒcj¯ℳ1 for j=1,2 are the two q-ROFStRNs. Then, the following operators are defined on them:(i)
ℒc1ℳ1∪ℒc2ℳ1=ℒc1¯ℳ1∪ℒc2¯ℳ1,ℒc1¯ℳ1∪ℒc2¯ℳ1(ii)
ℒc1ℳ1∩ℒc2ℳ1=ℒc1¯ℳ1∩ℒc2¯ℳ1,ℒc1¯ℳ1∩ℒc2¯ℳ1(iii)
ℒc1ℳ1⊕ℒc2ℳ1=ℒc1¯ℳ1⊕ℒc2¯ℳ1,ℒc1¯ℳ1⊕ℒc2¯ℳ1(iv)
ℒc1ℳ1⊗ℒc2ℳ1=ℒc1¯ℳ1⊗ℒc2¯ℳ1,ℒc1¯ℳ1⊗ℒc2¯ℳ1(v)
ℒc1ℳ1⊆ℒc2ℳ1=ℒc1¯ℳ1⊆ℒc2¯ℳ1andℒc1¯ℳ1⊆ℒc2¯ℳ1(vi)
αℒc1ℳ1=αℒc1¯ℳ1,αℒc1¯ℳ1 for α≥1(vii)
ℒc1ℳ1α=ℒc1¯ℳ1α,ℒc1¯ℳ1α for α≥1(viii)
ℒc1ℳ1c=ℒcj¯ℳ1c,ℒcj¯ℳ1c, where ℒcj¯ℳ1careℒcj¯ℳ1c, the complements of q-ROFStR approximation operators ℒcj¯ℳ1andℒcj¯ℳ1, that is, ℒcj¯ℳ1c=ϑij¯,βij¯(ix)
ℒℳ1=ℒℳ2 iff ℒ¯ℳ1=ℒ¯ℳ2andℒ¯ℳ1=ℒ¯ℳ2Definition 10.
Letℒc1ℳ1=ℒc1¯ℳ1,ℒc1¯ℳ1=β11¯,ϑ11¯,β11¯,ϑ11¯ be a q-ROFStRN. Then, the score function is given as(13)SCℒc1ℳ1=12β11¯q+β11¯q−ϑ11¯q−ϑ11¯q,SCℒc1ℳ1∈−1,1andq≥1.
The greater the score value, the greater the q-ROFStRN.Proposition 1.
LetT,ℰ,ℒ be q-ROFSt approximation space. For any two ℒℳ1=ℒ¯ℳ1,ℒ¯ℳ1 and ℒℳ2=ℒ¯ℳ2,ℒ¯ℳ2 q-ROFStRSs over a common universe set Y, then the following properties hold:(i)
∼∼ℒℳ1=ℳ1, where ∼ℒℳ1 is the complement of ℒℳ1, ℒℳ1∪ℒℳ2=ℒℳ2∪ℒℳ1,ℒℳ1∩ℒℳ2=ℒℳ2∩ℒℳ1(ii)
∼ℒℳ1∪ℒℳ2=∼ℒℳ1∩∼ℒℳ2(iii)
∼ℒℳ1∩ℒℳ2=∼ℒℳ1∪∼ℒℳ2(iv)
Ifℳ1⊆ℳ2, then ℒℳ1⊆ℒℳ2(v)
ℒℳ1∪ℳ2⊇ℒℳ1∪ℒℳ2(vi)
ℒℳ1∩ℳ2⊆ℒℳ1∩ℒℳ2


## 4. q-Rung Orthopair Fuzzy Soft Rough Averaging (q-ROFStRA) Aggregation Operator

This section is devoted to the analysis of q-ROFStR averaging aggregation operators such as q-ROFStRWA, q-ROFStROWA, and q-ROFStRHA operators. We will present the fundamental properties of these operators in detail.


### 4.1. q-Rung Orthopair Fuzzy Soft Rough Weighted Averaging (q-ROFStRWA) Operator

Definition 11.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRWA operator is defined as(14)q−ROFStRWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒcj¯ℳi,⊕j=1nvj⊕i=1mtiℒcj¯ℳi.
In view of the above definition, Theorem1 illustrates the aggregated result for q-ROFStRWA.Theorem 1.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1 and 0≤ti,vj≤1, respectively. Then, q-ROFStRWA operator is given as(15)q−ROFStRWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒcj¯ℳi,⊕j=1nvj⊕i=1mtiℒcj¯ℳi=1−∏j=1n∏i=1m1−βij¯qtivjq,∏j=1n∏i=1mϑij¯tivj,1−∏j=1n∏i=1m1−βij¯qtivjq,∏j=1n∏i=1mϑij¯tivj.Proof.
To get the required proof, we will use the method of mathematical induction.
As by operational law,(16)ℒc1ℳ1⊕ℒc1ℳ2=β11¯,ϑ11¯⊕β12¯,ϑ12¯,β11¯,ϑ11¯⊕β12¯,ϑ12¯=β11¯q+β12¯q+β11¯qβ12¯qq,ϑ11¯ϑ12¯,β11¯q+β12¯q+β11¯qβ12¯qq,ϑ11¯ϑ12¯,αℒc1ℳ1=1−1−β11¯qαq,ϑ11¯α,1−1−β11¯qαq,ϑ11¯α.
Suppose the result is true form=2andn=2; that is,(17)q−ROFStRWAℒcjℳi,ℒcjℳi=⊕j=12vj⊕i=12tiℒcj¯ℳi,⊕j=12vj⊕i=12tiℒcj¯ℳi.
Now, consider(18)q−ROFStRWAℒcjℳi,ℒcjℳi=⊕j=12vj⊕i=12tiℒcj¯ℳi,⊕j=12vj⊕i=12tiℒcj¯ℳi=v1t1ℒc1¯ℳ1⊕t2ℒc1¯ℳ2⊕v2t1ℒc2¯ℳ1⊕t2ℒc2¯ℳ2,v1t1ℒcj¯ℳ1⊕t2ℒcj¯ℳ2⊕v2t1ℒcj¯ℳ1⊕t2ℒcj¯ℳ2,q−ROFStRWAℒcjℳi,ℒcjℳi=1−∏j=12∏i=121−βij¯qtivjq,∏j=12∏i=12ϑij¯tivj,1−∏j=12∏i=121−βij¯qtivjq,∏j=12∏i=12ϑij¯tivj.
The result is true form=2andn=2.
Now, consider the result is forn=k1andm=k2:(19)q−ROFStRWAℒcjℳi,ℒcjℳi,…,ℒck1ℳk2=1−∏j=1k1∏i=1k21−βij¯qtivjq,∏j=1k1∏i=1k2ϑij¯tivj,1−∏j=1k1∏i=1k21−βij¯qtivjq,∏j=1k1∏i=1k2ϑij¯tivj.
Suppose the result holds forn=k1+1andm=k2+1, so we have(20)q−ROFStRWAℒcjℳi,ℒcjℳi,…,ℒck1ℳk2,ℒck1+1ℳk2+1=⊕j=1k1vj⊕i=1k2tiℒcj¯ℳi⊕vk1+1tk2+1ℒck1+1¯ℳk2+1,⊕j=1k1vj⊕i=1k2tiℒcj¯ℳi⊕vk1+1tk2+1ℒck1+1¯ℳk2+1=1−∏j=1k1+1∏i=1k2+11−βij¯qtivjq,∏j=1k1+1∏i=1k2+1ϑij¯tivj,1−∏j=1k1+1∏i=1k2+11−βij¯qtivjq,∏j=1k1+1∏i=1k2+1ϑij¯tivj.
This implies the result is true forn=k1+1andm=k2+1. Therefore, the result holds for all m,n≥1.
Since it is clear thatℒcj¯ℳiandℒcj¯ℳi are q-ROFNs, by Definition 7, we have that ⊕j=1nvj⊕i=1mtiℒcj¯ℳi and ⊕j=1nvj⊕i=1mtiℒcj¯ℳi are also q-ROFNs. Therefore, q−ROFStRWAℒc1ℳ1,…,ℒcnℳm is also a q-ROFStRN in approximation space T,ℰ,ℒ.Example 2.
LetY=s1,s2,s3 be a set and ℳ=c1,c2⊆ℰ a set of parameters with weight vector t=0.25,0.3,0.45T for sii=1,2,3 and v=0.55,0.45 for cjj=1,2. Then, q-ROFStRNs is given in Table 3:(21)q−ROFStRWAℒc1ℳ1,…,ℒcnℳm=⊕j=12vj⊕i=13tiℒcj¯ℳi,⊕j=12vj⊕i=13tiℒcj¯ℳi=1−1−0.930.251−0.730.251−0.2930.450.551−0.5530.251−0.9230.251−0.430.450.453,0.30.250.10.30.250.450.550.20.250.30.30.850.450.45,1−1−0.430.251−0.230.251−0.6530.450.551−0.7630.251−0.630.251−0.8830.450.453,0.40.250.750.30.150.450.550.140.250.30.30.120.450.45=0.831432,0.255487,0.72581,0.26258.
From the analysis of Theorem1, some related characteristics of q-ROFStRWA operator are given as follows:Table 3
Tabular representation ofℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi.
ℒc1c2s10.9,0.3,0.8,0.40.55,0.2,0.76,0.14s20.7,0.1,0.2,0.750.92,0.3,0.6,0.3s30.92,0.25,0.65,0.150.4,0.85,0.88,0.12Theorem 2.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1 and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStRWA operator:(i)
(Idempotency): ifℒcjℳi=PcNfor alli=1,2,…,mandj=1,2,…n, where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(22)q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒcjℳi−=minjminiℒcj¯ℳi,maxjmaxiℒcj¯ℳi and ℒcjℳi+=maxjmaxiℒcj¯ℳi,minjminiℒcj¯ℳi. Then,(23)ℒcjℳi−≤q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒcjℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(24)q−ROFStRWAPc1N1,Pc2N2,…,PcnNm≤q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(25)q−ROFStRWAℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(26)q−ROFStRWAλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 2.
(a)
Ifq=1, so, in this case, the developed q-ROFStRWA operator reduces to IFStRWA operator(b)
Ifq=2, so, in this case, the developed q-ROFStRWA operator reduces to PFStRWA operator(c)
If the soft parameterc1 is only one meansn=1, then, in this case, the developed q-ROFStRWA operator reduces to q-ROFRWA operator


### 4.2. q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Averaging (q-ROFStROWA) Operator

This subsection presents the detailed study of q-ROFStROWA operator and some of its desirable characteristics such as Idempotency, Boundedness, and Monotonicity. The basic advantage of q-ROFStROWG operator is to weight the ordered position of the q-ROFVs instead of weighting the values themselves.Definition 12.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStROWA operator is defined as(27)q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj¯ℳi.
In view of the above definition, the aggregated result for q-ROFStROWA is described in Theorem 3.Theorem 3.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStROWA operator is given as(28)q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj¯ℳi=1−∏j=1n∏i=1m1−βδij¯qtivjq,∏j=1n∏i=1mϑδij¯tivj,1−∏j=1n∏i=1m1−βδij¯qtivjq,∏j=1n∏i=1mϑδij¯tivj,where ℒδcjℳi=ℒδcj¯ℳi,ℒδcj¯ℳi denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi.Example 3.
Consider Table3 of Example 2, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi and the new ordered of tabular representation of ℒcjℳi through score function is given in Table 4.
Now,q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=⊕j=12vj⊕i=13tiℒδcj¯ℳi,⊕j=12vj⊕i=13tiℒδcj¯ℳi(29)q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=0.838595,0.261642,0.727318,0.255189.
From the analysis of Theorem3, the following desirable properties hold for q-ROFStROWA operator.Table 4
Tabular representation ofℒδcjℳi=ℒδcj¯ℳi,ℒδcj¯ℳi.
ℒc1c2s10.9,0.2,0.8,0.40.92,0.3,0.6,0.3s20.92,0.25,0.65,0.150.55,0.2,0.76,0.14s30.7,0.1,0.2,0.750.4,0.85,0.88,0.12Theorem 4.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStROWA operator:(i)
(Idempotency): ifℒδcjℳi=PcNfor alli=1,2,…,mandj=1,2,…,n, where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(30)q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒδcjℳi−=minjminiℒδcj¯ℳi,maxjmaxiℒδcj¯ℳi and ℒδcjℳi+=maxjmaxiℒδcj¯ℳi,minjminiℒδcj¯ℳi. Then,(31)ℒδcjℳi−≤q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒδcjℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(32)q−ROFStROWAPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(33)q−ROFStROWAℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(34)q−ROFStROWAλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 3.
(a)
Ifq=1, then the developed q-ROFStROWA operator reduces to IFStROWA operator(b)
Ifq=2, then the developed q-ROFStROWA operator reduces to PFStROWA operator(c)
If the soft parameterc1 is one meansn=1, then the developed q-ROFStROWA operator reduces to q-ROFROWA operator


### 4.3. q-Rung Orthopair Fuzzy Soft Rough Hybrid Averaging (q-ROFStRHA) Operator

From the analysis of Definitions11 and 12, it is clear that the q-ROFStRWA operator weights only the q-ROFVs, while q-ROFStROWA operator weights the ordered position of the q-ROFVs instead of weighting the values themselves. To overcome this limitation and motivated by the idea of combining weighted averaging and the ordered weighted averaging by using the combined notion of soft rough set, we present q-ROFStRHA operator, which weights both the given q-ROFV and its ordered position. The basic desirable properties of the developed operator are presented in detail.Definition 13.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider t=t1,t2,…,tmTandv=v1,v2,…,vnT as the associated weight vectors of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRHA operator is defined as(35)q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi.
From the above definition, the aggregated result for q-ROFStRHA operator is described in Theorem 5.Theorem 5.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRHA operator is given as(36)q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi=1−∏j=1n∏i=1m1−βδij∗¯qtivjq,∏j=1n∏i=1mϑδij∗¯tivj,1−∏j=1n∏i=1m1−βδij∗¯qtivjq,∏j=1n∏i=1mϑδij∗¯tivj,where ℒδcj∗ℳi=nklℒcj=nklℒcj¯ℳi,nklℒcj¯ℳi denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi and n represents the balancing coefficient.Example 4.
Consider Table3 of Example 2, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi with k=0.33,0.37,0.3Tandl=0.42,0.58T as the weight vectors of experts si and parameters cj. Consider t=0.36,0.34,0.3Tandv=0.55,0.45T as the associated weight vectors of experts si and parameters cj. The tabular representation of ℒδcj∗ℳi through operation law and score function is given in Tables 5 and 6. Now,(37)q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=⊕j=12vj⊕i=13tiℒδcj∗¯ℳi,⊕j=12vj⊕i=13tiℒδcj∗¯ℳi,q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=0.701609,0.129765,0.600425,0.122969.
From the analysis of Theorem5, the following characteristics hold in q-ROFStRHA operator.Table 5
Tabular representation by using operation law forℒδcj∗ℳi=nklℒcj=nklℒcj¯ℳi,nklℒcj¯ℳi.
ℒc1c2s10.7483,0.1247,0.6366,0.16630.4629,0.1148,0.6561,0.0804s20.5624,0.0466,0.1551,0.34970.8533,0.1931,0.5254,0.1931s30.7574,0.0945,0.4853,0.05670.3238,0.4437,0.7661,0.0626Table 6
Tabular representation after using score functionℒδcj∗ℳi=ℒδcj∗¯ℳi,ℒδcj∗¯ℳi.
ℒc1c2s10.7483,0.1247,0.6366,0.16630.8533,0.1931,0.5254,0.1931s20.7574,0.0945,0.4853,0.05670.3238,0.4437,0.7661,0.0626s30.5624,0.0466,0.1551,0.34970.4629,0.1148,0.6561,0.0804Theorem 6.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStRHA operator:(i)
(Idempotency): ifℒδcj∗ℳi=PcNfor alli=1,2,…,mandj=1,2,…,n where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(38)q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒδcj∗ℳi−=minjminiℒδcj∗¯ℳi,maxjmaxiℒδcj∗¯ℳi and ℒδcj∗ℳi+=maxjmaxiℒδcj∗¯ℳi,minjminiℒδcj∗¯ℳi. Then,(39)ℒδcj∗ℳi−≤q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒδcj∗ℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(40)q−ROFStRHAPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(41)q−ROFStRHAℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(42)q−ROFStRHAλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 4.
(a)
Ifq=1, then q-ROFStRHA operator reduces to IFStRHA operator(b)
Ifq=2, then q-ROFStRHA operator reduces to PFStRHA operator(c)
If soft parameterc1 is one meansn=1, then q-ROFStRHA operator reduces to q-ROFRHA operator(d)
Ifkl=1/n,1/n,…,1/nT, then the proposed q-ROFStRHA operator reduces to q-ROFStRWA operator(e)
Iftv=1/n,1/n,…,1/nT, then the proposed q-ROFStRHA operator reduces to q-ROFStROWA operator


## 4.1. q-Rung Orthopair Fuzzy Soft Rough Weighted Averaging (q-ROFStRWA) Operator

Definition 11.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRWA operator is defined as(14)q−ROFStRWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒcj¯ℳi,⊕j=1nvj⊕i=1mtiℒcj¯ℳi.
In view of the above definition, Theorem1 illustrates the aggregated result for q-ROFStRWA.Theorem 1.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1 and 0≤ti,vj≤1, respectively. Then, q-ROFStRWA operator is given as(15)q−ROFStRWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒcj¯ℳi,⊕j=1nvj⊕i=1mtiℒcj¯ℳi=1−∏j=1n∏i=1m1−βij¯qtivjq,∏j=1n∏i=1mϑij¯tivj,1−∏j=1n∏i=1m1−βij¯qtivjq,∏j=1n∏i=1mϑij¯tivj.Proof.
To get the required proof, we will use the method of mathematical induction.
As by operational law,(16)ℒc1ℳ1⊕ℒc1ℳ2=β11¯,ϑ11¯⊕β12¯,ϑ12¯,β11¯,ϑ11¯⊕β12¯,ϑ12¯=β11¯q+β12¯q+β11¯qβ12¯qq,ϑ11¯ϑ12¯,β11¯q+β12¯q+β11¯qβ12¯qq,ϑ11¯ϑ12¯,αℒc1ℳ1=1−1−β11¯qαq,ϑ11¯α,1−1−β11¯qαq,ϑ11¯α.
Suppose the result is true form=2andn=2; that is,(17)q−ROFStRWAℒcjℳi,ℒcjℳi=⊕j=12vj⊕i=12tiℒcj¯ℳi,⊕j=12vj⊕i=12tiℒcj¯ℳi.
Now, consider(18)q−ROFStRWAℒcjℳi,ℒcjℳi=⊕j=12vj⊕i=12tiℒcj¯ℳi,⊕j=12vj⊕i=12tiℒcj¯ℳi=v1t1ℒc1¯ℳ1⊕t2ℒc1¯ℳ2⊕v2t1ℒc2¯ℳ1⊕t2ℒc2¯ℳ2,v1t1ℒcj¯ℳ1⊕t2ℒcj¯ℳ2⊕v2t1ℒcj¯ℳ1⊕t2ℒcj¯ℳ2,q−ROFStRWAℒcjℳi,ℒcjℳi=1−∏j=12∏i=121−βij¯qtivjq,∏j=12∏i=12ϑij¯tivj,1−∏j=12∏i=121−βij¯qtivjq,∏j=12∏i=12ϑij¯tivj.
The result is true form=2andn=2.
Now, consider the result is forn=k1andm=k2:(19)q−ROFStRWAℒcjℳi,ℒcjℳi,…,ℒck1ℳk2=1−∏j=1k1∏i=1k21−βij¯qtivjq,∏j=1k1∏i=1k2ϑij¯tivj,1−∏j=1k1∏i=1k21−βij¯qtivjq,∏j=1k1∏i=1k2ϑij¯tivj.
Suppose the result holds forn=k1+1andm=k2+1, so we have(20)q−ROFStRWAℒcjℳi,ℒcjℳi,…,ℒck1ℳk2,ℒck1+1ℳk2+1=⊕j=1k1vj⊕i=1k2tiℒcj¯ℳi⊕vk1+1tk2+1ℒck1+1¯ℳk2+1,⊕j=1k1vj⊕i=1k2tiℒcj¯ℳi⊕vk1+1tk2+1ℒck1+1¯ℳk2+1=1−∏j=1k1+1∏i=1k2+11−βij¯qtivjq,∏j=1k1+1∏i=1k2+1ϑij¯tivj,1−∏j=1k1+1∏i=1k2+11−βij¯qtivjq,∏j=1k1+1∏i=1k2+1ϑij¯tivj.
This implies the result is true forn=k1+1andm=k2+1. Therefore, the result holds for all m,n≥1.
Since it is clear thatℒcj¯ℳiandℒcj¯ℳi are q-ROFNs, by Definition 7, we have that ⊕j=1nvj⊕i=1mtiℒcj¯ℳi and ⊕j=1nvj⊕i=1mtiℒcj¯ℳi are also q-ROFNs. Therefore, q−ROFStRWAℒc1ℳ1,…,ℒcnℳm is also a q-ROFStRN in approximation space T,ℰ,ℒ.Example 2.
LetY=s1,s2,s3 be a set and ℳ=c1,c2⊆ℰ a set of parameters with weight vector t=0.25,0.3,0.45T for sii=1,2,3 and v=0.55,0.45 for cjj=1,2. Then, q-ROFStRNs is given in Table 3:(21)q−ROFStRWAℒc1ℳ1,…,ℒcnℳm=⊕j=12vj⊕i=13tiℒcj¯ℳi,⊕j=12vj⊕i=13tiℒcj¯ℳi=1−1−0.930.251−0.730.251−0.2930.450.551−0.5530.251−0.9230.251−0.430.450.453,0.30.250.10.30.250.450.550.20.250.30.30.850.450.45,1−1−0.430.251−0.230.251−0.6530.450.551−0.7630.251−0.630.251−0.8830.450.453,0.40.250.750.30.150.450.550.140.250.30.30.120.450.45=0.831432,0.255487,0.72581,0.26258.
From the analysis of Theorem1, some related characteristics of q-ROFStRWA operator are given as follows:Table 3
Tabular representation ofℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi.
ℒc1c2s10.9,0.3,0.8,0.40.55,0.2,0.76,0.14s20.7,0.1,0.2,0.750.92,0.3,0.6,0.3s30.92,0.25,0.65,0.150.4,0.85,0.88,0.12Theorem 2.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1 and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStRWA operator:(i)
(Idempotency): ifℒcjℳi=PcNfor alli=1,2,…,mandj=1,2,…n, where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(22)q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒcjℳi−=minjminiℒcj¯ℳi,maxjmaxiℒcj¯ℳi and ℒcjℳi+=maxjmaxiℒcj¯ℳi,minjminiℒcj¯ℳi. Then,(23)ℒcjℳi−≤q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒcjℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(24)q−ROFStRWAPc1N1,Pc2N2,…,PcnNm≤q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(25)q−ROFStRWAℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(26)q−ROFStRWAλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 2.
(a)
Ifq=1, so, in this case, the developed q-ROFStRWA operator reduces to IFStRWA operator(b)
Ifq=2, so, in this case, the developed q-ROFStRWA operator reduces to PFStRWA operator(c)
If the soft parameterc1 is only one meansn=1, then, in this case, the developed q-ROFStRWA operator reduces to q-ROFRWA operator


## 4.2. q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Averaging (q-ROFStROWA) Operator

This subsection presents the detailed study of q-ROFStROWA operator and some of its desirable characteristics such as Idempotency, Boundedness, and Monotonicity. The basic advantage of q-ROFStROWG operator is to weight the ordered position of the q-ROFVs instead of weighting the values themselves.Definition 12.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStROWA operator is defined as(27)q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj¯ℳi.
In view of the above definition, the aggregated result for q-ROFStROWA is described in Theorem 3.Theorem 3.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStROWA operator is given as(28)q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj¯ℳi=1−∏j=1n∏i=1m1−βδij¯qtivjq,∏j=1n∏i=1mϑδij¯tivj,1−∏j=1n∏i=1m1−βδij¯qtivjq,∏j=1n∏i=1mϑδij¯tivj,where ℒδcjℳi=ℒδcj¯ℳi,ℒδcj¯ℳi denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi.Example 3.
Consider Table3 of Example 2, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi and the new ordered of tabular representation of ℒcjℳi through score function is given in Table 4.
Now,q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=⊕j=12vj⊕i=13tiℒδcj¯ℳi,⊕j=12vj⊕i=13tiℒδcj¯ℳi(29)q−ROFStROWAℒc1ℳ1,…,ℒcnℳm=0.838595,0.261642,0.727318,0.255189.
From the analysis of Theorem3, the following desirable properties hold for q-ROFStROWA operator.Table 4
Tabular representation ofℒδcjℳi=ℒδcj¯ℳi,ℒδcj¯ℳi.
ℒc1c2s10.9,0.2,0.8,0.40.92,0.3,0.6,0.3s20.92,0.25,0.65,0.150.55,0.2,0.76,0.14s30.7,0.1,0.2,0.750.4,0.85,0.88,0.12Theorem 4.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStROWA operator:(i)
(Idempotency): ifℒδcjℳi=PcNfor alli=1,2,…,mandj=1,2,…,n, where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(30)q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒδcjℳi−=minjminiℒδcj¯ℳi,maxjmaxiℒδcj¯ℳi and ℒδcjℳi+=maxjmaxiℒδcj¯ℳi,minjminiℒδcj¯ℳi. Then,(31)ℒδcjℳi−≤q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒδcjℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(32)q−ROFStROWAPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(33)q−ROFStROWAℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(34)q−ROFStROWAλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStROWAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 3.
(a)
Ifq=1, then the developed q-ROFStROWA operator reduces to IFStROWA operator(b)
Ifq=2, then the developed q-ROFStROWA operator reduces to PFStROWA operator(c)
If the soft parameterc1 is one meansn=1, then the developed q-ROFStROWA operator reduces to q-ROFROWA operator


## 4.3. q-Rung Orthopair Fuzzy Soft Rough Hybrid Averaging (q-ROFStRHA) Operator

From the analysis of Definitions11 and 12, it is clear that the q-ROFStRWA operator weights only the q-ROFVs, while q-ROFStROWA operator weights the ordered position of the q-ROFVs instead of weighting the values themselves. To overcome this limitation and motivated by the idea of combining weighted averaging and the ordered weighted averaging by using the combined notion of soft rough set, we present q-ROFStRHA operator, which weights both the given q-ROFV and its ordered position. The basic desirable properties of the developed operator are presented in detail.Definition 13.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT for experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider t=t1,t2,…,tmTandv=v1,v2,…,vnT as the associated weight vectors of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRHA operator is defined as(35)q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi.
From the above definition, the aggregated result for q-ROFStRHA operator is described in Theorem 5.Theorem 5.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRHA operator is given as(36)q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi,⊕j=1nvj⊕i=1mtiℒδcj∗¯ℳi=1−∏j=1n∏i=1m1−βδij∗¯qtivjq,∏j=1n∏i=1mϑδij∗¯tivj,1−∏j=1n∏i=1m1−βδij∗¯qtivjq,∏j=1n∏i=1mϑδij∗¯tivj,where ℒδcj∗ℳi=nklℒcj=nklℒcj¯ℳi,nklℒcj¯ℳi denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi and n represents the balancing coefficient.Example 4.
Consider Table3 of Example 2, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi with k=0.33,0.37,0.3Tandl=0.42,0.58T as the weight vectors of experts si and parameters cj. Consider t=0.36,0.34,0.3Tandv=0.55,0.45T as the associated weight vectors of experts si and parameters cj. The tabular representation of ℒδcj∗ℳi through operation law and score function is given in Tables 5 and 6. Now,(37)q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=⊕j=12vj⊕i=13tiℒδcj∗¯ℳi,⊕j=12vj⊕i=13tiℒδcj∗¯ℳi,q−ROFStRHAℒc1ℳ1,…,ℒcnℳm=0.701609,0.129765,0.600425,0.122969.
From the analysis of Theorem5, the following characteristics hold in q-ROFStRHA operator.Table 5
Tabular representation by using operation law forℒδcj∗ℳi=nklℒcj=nklℒcj¯ℳi,nklℒcj¯ℳi.
ℒc1c2s10.7483,0.1247,0.6366,0.16630.4629,0.1148,0.6561,0.0804s20.5624,0.0466,0.1551,0.34970.8533,0.1931,0.5254,0.1931s30.7574,0.0945,0.4853,0.05670.3238,0.4437,0.7661,0.0626Table 6
Tabular representation after using score functionℒδcj∗ℳi=ℒδcj∗¯ℳi,ℒδcj∗¯ℳi.
ℒc1c2s10.7483,0.1247,0.6366,0.16630.8533,0.1931,0.5254,0.1931s20.7574,0.0945,0.4853,0.05670.3238,0.4437,0.7661,0.0626s30.5624,0.0466,0.1551,0.34970.4629,0.1148,0.6561,0.0804Theorem 6.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStRHA operator:(i)
(Idempotency): ifℒδcj∗ℳi=PcNfor alli=1,2,…,mandj=1,2,…,n where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(38)q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒδcj∗ℳi−=minjminiℒδcj∗¯ℳi,maxjmaxiℒδcj∗¯ℳi and ℒδcj∗ℳi+=maxjmaxiℒδcj∗¯ℳi,minjminiℒδcj∗¯ℳi. Then,(39)ℒδcj∗ℳi−≤q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒδcj∗ℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(40)q−ROFStRHAPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(41)q−ROFStRHAℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(42)q−ROFStRHAλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRHAℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 4.
(a)
Ifq=1, then q-ROFStRHA operator reduces to IFStRHA operator(b)
Ifq=2, then q-ROFStRHA operator reduces to PFStRHA operator(c)
If soft parameterc1 is one meansn=1, then q-ROFStRHA operator reduces to q-ROFRHA operator(d)
Ifkl=1/n,1/n,…,1/nT, then the proposed q-ROFStRHA operator reduces to q-ROFStRWA operator(e)
Iftv=1/n,1/n,…,1/nT, then the proposed q-ROFStRHA operator reduces to q-ROFStROWA operator


## 5. q-Rung Orthopair Fuzzy Soft Rough Geometric (q-ROFStRG) Aggregation Operator

This section is devoted to the study of q-ROFStR geometric aggregation operators such as q-ROFStRWG, q-ROFStROWG, and q-ROFStRHG operators. We will present the fundamental properties of these operators in detail.


### 5.1. q-Rung Orthopair Fuzzy Soft Rough Weighted Geometric (q-ROFStRWG) Operator

Definition 14.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRWG operator is defined as(43)q−ROFStRWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒcj¯ℳitivj,⊕j=1n⊕i=1mℒcj¯ℳitivj.
Based on the above definition the aggregated result for q-ROFStRWG operator is given in Theorem 7.Theorem 7.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRWG operator is given as(44)q−ROFStRWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒcj¯ℳitivj,⊕j=1n⊕i=1mℒcj¯ℳitivj=∏j=1n∏i=1mβij¯tivj,1−∏j=1n∏i=1m1−ϑij¯qtivjq,∏j=1n∏i=1mβij¯tivj,1−∏j=1n∏i=1m1−ϑij¯qtivjq.Since it is clear thatℒcj¯ℳiandℒcj¯ℳi are q-ROFNs, by Definition 7, we have that ⊕j=1n⊕i=1mℒcj¯ℳitivj and ⊕j=1n⊕i=1mℒcj¯ℳitivj are also q-ROFNs. Therefore, q−ROFStRWGℒc1ℳ1,…,ℒcnℳm is also a q-ROFStRN in approximation space T,ℰ,ℒ.Example 5.
Consider Table3 of Example 2, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi. Then, the aggregated result for q−ROFStRWG is given as(45)q−ROFStRWGℒc1ℳ1,…,ℒcnℳm=⊕j=12⊕i=13ℒcj¯ℳitivj,⊕j=12⊕i=13ℒcj¯ℳitivj=0.90.250.70.30.920.450.550.550.250.920.30.40.450.45,1−1−0.330.251−0.130.251−0.2530.450.551−0.230.251−0.330.251−0.8530.450.453,0.80.250.20.30.650.450.550.760.250.60.30.880.450.45,1−1−0.430.251−0.7530.251−0.1530.450.551−0.1430.251−0.330.251−0.1230.450.453=0.715607,0.509925,0.573442,0.484819.
From the analysis of Theorem7, the following hold for q-ROFStRWG operator.Theorem 8.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStRWG operator:(i)
(Idempotency): ifℒcjℳi=PcNfor alli=1,2,…,mandj=1,2,…,n where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(46)q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒcjℳi−=minjminiℒcj¯ℳi,maxjmaxiℒcj¯ℳi and ℒcjℳi+=maxjmaxiℒcj¯ℳi,minjminiℒcj¯ℳi. Then,(47)ℒcjℳi−≤q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒcjℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(48)q−ROFStRWGPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(49)q−ROFStRWGℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(50)q−ROFStRWGλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 5.
(a)
Ifq=1, then q-ROFStRWG operator reduces to IFStRWG operator(b)
Ifq=2, then q-ROFStRWG operator reduces to PFStRWG operator(c)
If soft parameterc1 is one meansn=1, then q-ROFStRWG operator reduces to q-ROFRWG operator


### 5.2. q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Geometric (q-ROFStROWG) Operator

Here, we will put forward the detailed study of q-ROFStROWG operator and some of its fundamental properties. The q-ROFStROWG operator weights the ordered position of the q-ROFVs instead of weighting the values themselves.Definition 15.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStROWG operator is defined as(51)q−ROFStROWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj¯ℳitivj,⊕j=1n⊕i=1mℒδcj¯ℳitivj.
In view of the above definition, the aggregated result for q-ROFStROWG operator is described in Theorem 9.Theorem 9.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStROWG operator is given as(52)q−ROFStROWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj¯ℳitivj,⊕j=1n⊕i=1mℒδcj¯ℳitivj=∏j=1n∏i=1mβδij¯tivj,1−∏j=1n∏i=1m1−ϑδij¯qtivjq,∏j=1n∏i=1mβδij¯tivj,1−∏j=1n∏i=1m1−ϑδij¯qtivjq,where ℒδcjℳi=ℒδcj¯ℳi,ℒδcj¯ℳi denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs, ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi.Remark 6.
(a)
Ifq=1, then q-ROFStROWG operator reduces to IFStROWG operator(b)
Ifq=2, then q-ROFStROWG operator reduces to PFStROWG operator(c)
If the soft parameterc1 is one meansn=1, then q-ROFStROWG operator reduces to q-ROFROWG operator


### 5.3. q-Rung Orthopair Fuzzy Soft Rough Hybrid Geometric (q-ROFStRHG) Operator

From the analysis of Definitions14 and 15, it is clear that the q-ROFStRWG operator weights only the q-ROFVs, while q-ROFStROWG operator weights the ordered position of the q-ROFVs instead of weighting themselves. Motivated by the idea of combining weighted geometric and the ordered weighted geometric by using the combined notion of soft rough set, we present q-ROFStRHG operator, which weights both the given q-ROFV and its ordered position. The basic desirable properties of the developed operator are presented in detail.Definition 16.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRHG operator is defined as(53)q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj∗¯ℳitivj,⊕j=1n⊕i=1mℒδcj∗¯ℳitivj.
From the above definition, the aggregated value for q-ROFStRHG operator is described in Theorem 11.Theorem 10.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRHG operator is given as(54)q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj∗¯ℳitivj,⊕j=1n⊕i=1mℒδcj∗¯ℳitivj=∏j=1n∏i=1mβδij∗¯tivj,1−∏j=1n∏i=1m1−ϑδij∗¯qtivjq,∏j=1n∏i=1mβδij∗¯tivj,1−∏j=1n∏i=1m1−ϑδij∗¯qtivjq,where ℒδcj∗ℳi=ℒcjnkl=ℒcj¯ℳinkl,ℒcj¯ℳinkl denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi and n represents the balancing coefficient.Example 6.
Consider Tables2, 4, and 5 of Examples 2 and 4, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi with k=0.33,0.37,0.3Tandl=0.42,0.58T is the weight vectors of experts si and parameters cj. Consider t=0.36,0.34,0.3Tandv=0.55,0.45T as the associated weight vectors of experts si and parameters cj. Then, the aggregated result for ℒδcj∗ℳi is given as(55)q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=⊕j=12⊕i=13ℒδcj¯ℳitivj,⊕j=12⊕i=13ℒδcj¯ℳitivj,q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=0.602629,0.250898,0.479995,0.210973.
From the analysis of Theorem11, the properties of q-ROFStRHG operator are given below.Theorem 11.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRHG operator holds as follows:(i)
(Idempotency): ifℒδcj∗ℳi=PcNfor alli=1,2,…,mandj=1,2,…,n where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(56)q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒδcj∗ℳi−=minjminiℒδcj∗¯ℳi,maxjmaxiℒδcj∗¯ℳi and ℒδcj∗ℳi+=maxjmaxiℒδcj∗¯ℳi,minjminiℒδcj∗¯ℳi. Then,(57)ℒδcj∗ℳi−≤q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒδcj∗ℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(58)q−ROFStRHGPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(59)q−ROFStRHGℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(60)q−ROFStRHGλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 7.
(a)
Ifq=1, then q-ROFStRHG operator reduces to IFStRHG operator(b)
If the value of rungq=2, then q-ROFStRHG operator reduces to PFStRHG operator(c)
If the soft parameterc1 is one meansn=1, then q-ROFStRHG operator reduces to q-ROFRHG operator(d)
Ifkl=1/n,1/n,…,1/nT, then the proposed q-ROFStRHG operator reduces to q-ROFStRWG operator(e)
Iftv=1/n,1/n,…,1/nT, then the proposed q-ROFStRHG operator reduces to q-ROFStROWG operator


## 5.1. q-Rung Orthopair Fuzzy Soft Rough Weighted Geometric (q-ROFStRWG) Operator

Definition 14.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRWG operator is defined as(43)q−ROFStRWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒcj¯ℳitivj,⊕j=1n⊕i=1mℒcj¯ℳitivj.
Based on the above definition the aggregated result for q-ROFStRWG operator is given in Theorem 7.Theorem 7.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRWG operator is given as(44)q−ROFStRWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒcj¯ℳitivj,⊕j=1n⊕i=1mℒcj¯ℳitivj=∏j=1n∏i=1mβij¯tivj,1−∏j=1n∏i=1m1−ϑij¯qtivjq,∏j=1n∏i=1mβij¯tivj,1−∏j=1n∏i=1m1−ϑij¯qtivjq.Since it is clear thatℒcj¯ℳiandℒcj¯ℳi are q-ROFNs, by Definition 7, we have that ⊕j=1n⊕i=1mℒcj¯ℳitivj and ⊕j=1n⊕i=1mℒcj¯ℳitivj are also q-ROFNs. Therefore, q−ROFStRWGℒc1ℳ1,…,ℒcnℳm is also a q-ROFStRN in approximation space T,ℰ,ℒ.Example 5.
Consider Table3 of Example 2, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi. Then, the aggregated result for q−ROFStRWG is given as(45)q−ROFStRWGℒc1ℳ1,…,ℒcnℳm=⊕j=12⊕i=13ℒcj¯ℳitivj,⊕j=12⊕i=13ℒcj¯ℳitivj=0.90.250.70.30.920.450.550.550.250.920.30.40.450.45,1−1−0.330.251−0.130.251−0.2530.450.551−0.230.251−0.330.251−0.8530.450.453,0.80.250.20.30.650.450.550.760.250.60.30.880.450.45,1−1−0.430.251−0.7530.251−0.1530.450.551−0.1430.251−0.330.251−0.1230.450.453=0.715607,0.509925,0.573442,0.484819.
From the analysis of Theorem7, the following hold for q-ROFStRWG operator.Theorem 8.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, the following properties hold for q-ROFStRWG operator:(i)
(Idempotency): ifℒcjℳi=PcNfor alli=1,2,…,mandj=1,2,…,n where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(46)q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒcjℳi−=minjminiℒcj¯ℳi,maxjmaxiℒcj¯ℳi and ℒcjℳi+=maxjmaxiℒcj¯ℳi,minjminiℒcj¯ℳi. Then,(47)ℒcjℳi−≤q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒcjℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(48)q−ROFStRWGPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(49)q−ROFStRWGℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(50)q−ROFStRWGλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRWGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 5.
(a)
Ifq=1, then q-ROFStRWG operator reduces to IFStRWG operator(b)
Ifq=2, then q-ROFStRWG operator reduces to PFStRWG operator(c)
If soft parameterc1 is one meansn=1, then q-ROFStRWG operator reduces to q-ROFRWG operator


## 5.2. q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Geometric (q-ROFStROWG) Operator

Here, we will put forward the detailed study of q-ROFStROWG operator and some of its fundamental properties. The q-ROFStROWG operator weights the ordered position of the q-ROFVs instead of weighting the values themselves.Definition 15.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStROWG operator is defined as(51)q−ROFStROWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj¯ℳitivj,⊕j=1n⊕i=1mℒδcj¯ℳitivj.
In view of the above definition, the aggregated result for q-ROFStROWG operator is described in Theorem 9.Theorem 9.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStROWG operator is given as(52)q−ROFStROWGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj¯ℳitivj,⊕j=1n⊕i=1mℒδcj¯ℳitivj=∏j=1n∏i=1mβδij¯tivj,1−∏j=1n∏i=1m1−ϑδij¯qtivjq,∏j=1n∏i=1mβδij¯tivj,1−∏j=1n∏i=1m1−ϑδij¯qtivjq,where ℒδcjℳi=ℒδcj¯ℳi,ℒδcj¯ℳi denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs, ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi.Remark 6.
(a)
Ifq=1, then q-ROFStROWG operator reduces to IFStROWG operator(b)
Ifq=2, then q-ROFStROWG operator reduces to PFStROWG operator(c)
If the soft parameterc1 is one meansn=1, then q-ROFStROWG operator reduces to q-ROFROWG operator


## 5.3. q-Rung Orthopair Fuzzy Soft Rough Hybrid Geometric (q-ROFStRHG) Operator

From the analysis of Definitions14 and 15, it is clear that the q-ROFStRWG operator weights only the q-ROFVs, while q-ROFStROWG operator weights the ordered position of the q-ROFVs instead of weighting themselves. Motivated by the idea of combining weighted geometric and the ordered weighted geometric by using the combined notion of soft rough set, we present q-ROFStRHG operator, which weights both the given q-ROFV and its ordered position. The basic desirable properties of the developed operator are presented in detail.Definition 16.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The q-ROFStRHG operator is defined as(53)q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj∗¯ℳitivj,⊕j=1n⊕i=1mℒδcj∗¯ℳitivj.
From the above definition, the aggregated value for q-ROFStRHG operator is described in Theorem 11.Theorem 10.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRHG operator is given as(54)q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=⊕j=1n⊕i=1mℒδcj∗¯ℳitivj,⊕j=1n⊕i=1mℒδcj∗¯ℳitivj=∏j=1n∏i=1mβδij∗¯tivj,1−∏j=1n∏i=1m1−ϑδij∗¯qtivjq,∏j=1n∏i=1mβδij∗¯tivj,1−∏j=1n∏i=1m1−ϑδij∗¯qtivjq,where ℒδcj∗ℳi=ℒcjnkl=ℒcj¯ℳinkl,ℒcj¯ℳinkl denotes the largest value of the permutation from ith row and jth column of the collection i×j q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi and n represents the balancing coefficient.Example 6.
Consider Tables2, 4, and 5 of Examples 2 and 4, for the collection q-ROFStRNs ℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳi with k=0.33,0.37,0.3Tandl=0.42,0.58T is the weight vectors of experts si and parameters cj. Consider t=0.36,0.34,0.3Tandv=0.55,0.45T as the associated weight vectors of experts si and parameters cj. Then, the aggregated result for ℒδcj∗ℳi is given as(55)q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=⊕j=12⊕i=13ℒδcj¯ℳitivj,⊕j=12⊕i=13ℒδcj¯ℳitivj,q−ROFStRHGℒc1ℳ1,…,ℒcnℳm=0.602629,0.250898,0.479995,0.210973.
From the analysis of Theorem11, the properties of q-ROFStRHG operator are given below.Theorem 11.
Letℒcjℳi=ℒcj¯ℳi,ℒcj¯ℳii=1,2,…,m,j=1,2,…,n be the collection of q-ROFStRNs. Let the weight vectors k=k1,k2,…,kmTandl=l1,l2,…,lnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ki,lj≤1. Consider the associated weight vectors t=t1,t2,…,tmTandv=v1,v2,…,vnT of experts si and parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. Then, q-ROFStRHG operator holds as follows:(i)
(Idempotency): ifℒδcj∗ℳi=PcNfor alli=1,2,…,mandj=1,2,…,n where Pcℳ=Pc¯ℳ,Pc¯ℳ=b¯,d¯,b¯,d¯, then(56)q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm=Pcℳ.(ii)
(Boundedness): letℒδcj∗ℳi−=minjminiℒδcj∗¯ℳi,maxjmaxiℒδcj∗¯ℳi and ℒδcj∗ℳi+=maxjmaxiℒδcj∗¯ℳi,minjminiℒδcj∗¯ℳi. Then,(57)ℒδcj∗ℳi−≤q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm≤ℒδcj∗ℳi+.(iii)
(Monotonicity): letPcjNi=Pcj¯Ni,Pcj¯Nii=1,2,…,m,j=1,2,…,n be another collection of q-ROFStRNs such that Pcj¯Ni≤ℒcj¯ℳi and Pcj¯Ni≤ℒcj¯ℳi. Then,(58)q−ROFStRHGPc1ℳ1,Pc2ℳ2,…,Pcnℳm≤q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.(iv)
(Shift invariance): letPcN=Pc¯N,Pc¯N=b¯,d¯,b¯,d¯ be any other q−ROFStRN. Then,(59)q−ROFStRHGℒc1ℳ1⊕PcN,ℒc2ℳ2⊕PcN,…,ℒcnℳm⊕PcN=q−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm⊕PcN.(v)
(Homogeneity): for any real numberλ>0,(60)q−ROFStRHGλℒc1ℳ1,λℒc2ℳ2,…,λℒcnℳm=λq−ROFStRHGℒc1ℳ1,ℒc2ℳ2,…,ℒcnℳm.Remark 7.
(a)
Ifq=1, then q-ROFStRHG operator reduces to IFStRHG operator(b)
If the value of rungq=2, then q-ROFStRHG operator reduces to PFStRHG operator(c)
If the soft parameterc1 is one meansn=1, then q-ROFStRHG operator reduces to q-ROFRHG operator(d)
Ifkl=1/n,1/n,…,1/nT, then the proposed q-ROFStRHG operator reduces to q-ROFStRWG operator(e)
Iftv=1/n,1/n,…,1/nT, then the proposed q-ROFStRHG operator reduces to q-ROFStROWG operator


## 6. MCDM Based on Soft Rough Aggregation Operator by Using q-ROF Information

MCDM has a high potential and discipline process to improve and evaluate multiple conflicting criteria in all areas of decision-making. In this competitive environment, an enterprise needs a more accurate and more repaid response to change customer needs. So, MCDM has the ability to handle successfully the evaluation process of multiple contradictory criteria. For an intelligent decision, experts analyze each and every character of an alternative and, then, they take the decision. Further, we will present the model for MCDM and their basic steps of construction by utilizing the proposed aggregation operators under q-ROF soft rough information.Suppose thatY=s1,s2,s3,…,sp be the initial set of various alternatives and ℰ=c1,c2,c3,…,cn be the set of n parameters. Consider Z=z1,z2,z3,…,zm as the set of m professional experts of this area who present their assessment expertise for each alternative skk=1,2,…,p corresponding to n parameters. Let the weight vectors t=t1,t2,…,tmT of experts zi and v=v1,v2,…,vnT of parameters cj with ∑i=1mti=1, ∑j=1nvj=1, and 0≤ti,vj≤1, respectively. The professional experts express their preference evaluation for alternative sk with respect to parameter cj in the form of q-ROFStRNs. The collective preference information given by the professionals is managed in q-ROFStR decision matrix, which is M=ℒcjℳin×m where ℳ⊆ℰ. Further, using the proposed models aggregate, the preferred choices of experts are to get the aggregated results ℛkk=1,…,p for each alternative sk against their parameter cj. Finally, they utilize the score function on the aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯ and rank all the results in a specific order to get the most desirable option. The stepwise decision algorithm for the investigated operators:Step (i): the professional experts express their preference evaluation for alternativesk with respect to parameter cj in the form of q-ROFStRNs. Then, they collect the preference information given by the professionals and manage them in q-ROFStR decision matrix, which is M=ℒcjℳin×m where ℳ⊆ℰ.Step (ii): apply the presented aggregation operators of each decision matrixM=ℒcjℳin×m for each alternative skk=1,2,…,p against parameter cj to get the aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯.Step (iii): calculate the score value of aggregated resultsℛk=β¯,ϑ¯,β¯,ϑ¯ for each object sk.Step (iv): rank the score value ofℛk in a specific order to get the optimum option of professional experts.


## 7. Numerical Example

In this section, we will initiate an illustrative example to prove the quality and excellency of the developed operators. Let the Higher Education Commission (HEC) in Pakistan plans to introduce a selection board of four high potential and professional professorsZ=z1,z2,z3,z4 from home and abroad to select the most desirable applicant. Out of many applicants, three applicants Y=s1,s2,s3 were called for interviews. The interview mainly judges the applicants against some parameters ℳ=c1=academic level,c2=development potential,c3=professional ethics,c4=research productivity⊆ℰ. Let the weight vector t=0.3,0.28,0.24,0.18T for professional experts zii=1,…,4 and v=0.32,0.17,0.31,0.2T be the weight vectors for parameters cjj=1,2,3, respectively. The professional experts express their preference evaluation for candidate sk with respect to parameter cj in the form of q-ROFStRNs. Finally, follow the following steps by utilizing the proposed models to select the most desirable and suitable applicant sk.


### 7.1. Aggregation Results Rendered by the q-ROFStRWA Method

Step (i): the professional experts express their preference evaluation for alternativesk with respect to parameter cj in the form of q-ROFStRNs. Then, they collect the preference information given by the professionals and manage them in q-ROFStR decision matrix, which is M=ℒcjℳin×m where ℳ⊆ℰ which is given in Tables 7–9.Step (ii): apply the presented q-ROFStRWA aggregation operators on each decision matrix M=ℒcjℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯; that is,(61)ℛ1=0.640506,0.218382,0.600249,0.254431,ℛ2=0.607809,0.217229,0.730645,0.202415,ℛ3=0.606198,0.198551,0.649417,0.183108.Step (iii): calculate the score value of aggregated resultsℛk=β¯,ϑ¯,β¯,ϑ¯ for each object sk; that is,(62)SCℛ1=0.226074,SCℛ2=0.298025,SCℛ3=0.241341.Step (iv): rank the score value ofℛk in a specific order to get the optimum option of professional experts; that is,(63)SCℛ2>SCℛ3>SCℛ1.Table 7
q-ROFStR matrix for candidate s1.
c1c2c3c4z10.7,0.2,0.8,0.10.65,0.25,0.3,0.60.82,0.18,0.6,0.40.5,0.2,0.4,0.1z20.6,0.1,0.5,0.30.5,0.1,0.7,0.150.3,0.2,0.2,0.70.2,0.3,0.6,0.2z30.4,0.5,0.6,0.20.75,0.2,0.4,0.10.65,0.3,0.7,0.250.5,0.4,0.1,0.5z40.5,0.3,0.3,0.70.6,0.4,0.9,0.10.78,0.22,0.45,0.40.8,0.1,0.3,0.1Table 8
q-ROFStR matrix for candidate s2.
c1c2c3c4z10.6,0.3,0.9,0.10.2,0.4,0.6,0.10.5,0.2,0.9,0.10.6,0.2,0.7,0.2z20.4,0.25,0.3,0.50.5,0.15,0.7,0.30.77,0.1,0.6,0.350.4,0.3,0.5,0.1z30.3,0.6,0.65,0.250.66,0.2,0.8,0.170.8,0.15,0.55,0.20.7,0.1,0.3,0.6z40.5,0.15,0.55,0.20.8,0.1,0.4,0.50.62,0.3,0.9,0.10.2,0.4,0.5,0.3Table 9
q-ROFStR matrix for candidate s3.
c1c2c3c4z10.8,0.13,0.8,0.10.5,0.2,0.6,0.10.4,0.1,0.7,0.20.7,0.1,0.6,0.3z20.5,0.16,0.4,0.20.8,0.12,0.3,0.40.6,0.2,0.4,0.30.5,0.2,0.4,0.1z30.4,0.5,0.7,0.30.5,0.4,0.5,0.120.2,0.4,0.8,0.140.3,0.4,0.2,0.5z40.3,0.2,0.5,0.150.6,0.25,0.7,0.30.8,0.18,0.75,0.10.6,0.2,0.8,0.1Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


### 7.2. Aggregation Results Rendered by the q-ROFStRWG Method

Step (i): similar to above.Step (ii): apply the presented q-ROFStRWG aggregation operators on each decision matrix M=ℒcjℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯; that is,(64)ℛ1=0.540892,0.29117,0.446263,0.448329,ℛ2=0.50166,0.321735,0.598613,0.331553,ℛ3=0.491017,0.291113,0.545527,0.264822.Step (iii): calculate the score value of the proposed q-ROFStRWG aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯ for each object sk; that is,(65)SCℛ1=0.06616,SCℛ2=0.135502,SCℛ3=0.118744.Step (iv): rank the score value ofℛk in a specific order to get the optimum option of professional experts; that is,(66)SCℛ2>SCℛ3>SCℛ1.Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


### 7.3. Aggregation Results Rendered by the q-ROFStROWA Method

Step (i): similar to above.Step (ii):(67)ℛ1=0.657933,0.217542,0.621125,0.234448,ℛ2=0.612743,0.216413,0.744755,0.193923,ℛ3=0.621268,0.203392,0.677814,0.180796.Step (iii):SCℛ1=0.250625,SCℛ2=0.312857,SCℛ3=0.26844.Step (iv):SCℛ2>SCℛ3>SCℛ1.Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


### 7.4. Aggregation Results Rendered by the q-ROFStROWG Method

Step (i): similar to above.Step (ii):(68)ℛ1=0.562833,0.293072,0.473363,0.418793,ℛ2=0.517595,0.319441,0.614971,0.321818,ℛ3=0.49826,0.296096,0.583543,0.259781.Step (iii):SCℛ1=0.09287,SCℛ2=0.152658,SCℛ3=0.139459.Step (iv):SCℛ2>SCℛ3>SCℛ1⇒Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


### 7.5. Aggregation Results Rendered by the q-ROFStRHA Method

Step (i): similar to above.Step (ii): apply the presented q-ROFStRHA aggregation operators on each decision matrix M=ℒδcj∗ℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=βδ∗¯,ϑδ∗¯,βδ∗¯,ϑδ∗¯, with k=0.25,0.29,0.3,0.16Tandl=0.27,0.23,0.32,0.18T being the weight vectors of experts si and parameters cj. Then, the aggregated result for ℒδcj∗ℳi is given as(69)ℛ1=0.432025,0.669215,0.408108,0.691652,ℛ2=0.417457,0.662671,0.503394,0.6509,ℛ3=0.392067,0.655437,0.442525,0.635251.Step (iii):SCℛ1=−0.24099,SCℛ2=−0.18323,SCℛ3=−0.1955.Step (iv):SCℛ2>SCℛ3>SCℛ1.Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


### 7.6. Aggregation Results Rendered by the q-ROFStRHG Method

Step (i): similar to above.Step (ii): apply the presented q-ROFStRHG aggregation operators on each decision matrix M=ℒδcj∗ℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=βδ∗¯,ϑδ∗¯,βδ∗¯,ϑδ∗¯, with k=0.25,0.29,0.3,0.16Tandl=0.27,0.23,0.32,0.18T being the weight vectors of experts si and parameters cj. Then, the aggregated result for ℒδcj∗ℳi is given as(70)ℛ1=0.365745,0.709626,0.308693,0.751049,ℛ2=0.332518,0.713922,0.407739,0.706165,ℛ3=0.318639,0.693983,0.36813,0.673448.Step (iii):SCℛ1=−0.35133,SCℛ2=−0.30573,SCℛ3=−0.27871.Step (iv):SCℛ3>SCℛ2>SCℛ1.Therefore, from the above analysis, it is observed thats3 is a more suitable and desirable candidate against the given position. The higher the score value, the more optimist that value and the smaller the score value, the more pessimist that value. From the ranking results of the above proposed operators, it is clear that averaging operators’ results are more optimist than geometric operators.


### 7.7. Comparative Study

To present the applicability and efficiency of the developed approach with some other existing methods based on IFS, PFS, and q-ROFS methods by using the same illustrative example. A comparative study has been made based on different aggregation operators (see [4, 5, 23, 27, 28, 54, 60, 61]). For this purpose, different parameters of the above numerical example are aggregated by utilizing the proposed aggregation operators having weight vectors v=0.32,0.17,0.31,0.2T, and their collective aggregated decision matrix for each candidate sii=1,2,3 is given in Table 10. Now, by using the information of the evaluated matrix, a comparative study of the investigated models with some existing aggregation operators is presented in Table 11. From Table 11, it is observed that the methods presented in [4, 5, 23, 27, 28, 54, 60, 61] are only capable of solving the q-ROFV of the form β,ϑ but are not capable of solving the q-ROFStRV of the form β¯,ϑ¯,β¯,ϑ¯. Thus, from the existing methods, it is clear that these existing methods have a lake of rough information and they are not capable of solving and ranking the given illustrative example. Therefore, from this analysis, it is clear that the developed methods are more superior and capable than the existing methods.Table 10
Aggregated matrix for candidate.
s1s2s3z10.719,0.201,0.651,0.2080.538,0.256,0.849,0.1150.670,0.122,0.713,0.154z20.471,0.154,0.534,0.3200.602,0.179,0.550,0.2980.616,0.171,0.387,0.222z30.598,0.350,0.582,0.2290.678,0.226,0.629,0.2600.370,0.430,0.682,0.225z40.703,0.230,0.616,0.2870.605,0.211,0.733,0.2050.650,0.201,0.703,0.137Table 11
Comparative analysis of different methods.
MethodsScore valuesRankingℛ1ℛ2ℛ3IFWA [4]Unapproachable×IFStWA [54]Unapproachable×PFStWA [60]Unapproachable×q-ROFWA [23]Unapproachable×q-ROFHOWAGA [61]Unapproachable×q-ROFStWA [60]Unapproachable×Cq-ROFWA [27]Unapproachable×q-RONFWA [28]Unapproachable×q-ROFStRWA (proposed)0.226074,0.298025,0.241341ℛ2≽ℛ3≽ℛ1q-ROFStROWA (proposed)0.250625,0.312857,0.26844ℛ2≽ℛ3≽ℛ1q-ROFStRHA (proposed)−0.24099,−0.18323,−0.1955ℛ2≽ℛ3≽ℛ1IFWG [5]Unapproachable×IFStWG [54]Unapproachable×q-ROFWG [23]Unapproachable×Cq-ROFWG [27]Unapproachable×q-RONFWG [28]Unapproachable×q-ROFStRWG (proposed)0.06616,0.135502,0.118744ℛ2≽ℛ3≽ℛ1q-ROFStROWG (proposed)0.09287,0.152658,0.139459ℛ2≽ℛ3≽ℛ1q-ROFStRHG (proposed)−0.35133,−0.30573,−0.27871ℛ3≽ℛ2≽ℛ1


## 7.1. Aggregation Results Rendered by the q-ROFStRWA Method

Step (i): the professional experts express their preference evaluation for alternativesk with respect to parameter cj in the form of q-ROFStRNs. Then, they collect the preference information given by the professionals and manage them in q-ROFStR decision matrix, which is M=ℒcjℳin×m where ℳ⊆ℰ which is given in Tables 7–9.Step (ii): apply the presented q-ROFStRWA aggregation operators on each decision matrix M=ℒcjℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯; that is,(61)ℛ1=0.640506,0.218382,0.600249,0.254431,ℛ2=0.607809,0.217229,0.730645,0.202415,ℛ3=0.606198,0.198551,0.649417,0.183108.Step (iii): calculate the score value of aggregated resultsℛk=β¯,ϑ¯,β¯,ϑ¯ for each object sk; that is,(62)SCℛ1=0.226074,SCℛ2=0.298025,SCℛ3=0.241341.Step (iv): rank the score value ofℛk in a specific order to get the optimum option of professional experts; that is,(63)SCℛ2>SCℛ3>SCℛ1.Table 7
q-ROFStR matrix for candidate s1.
c1c2c3c4z10.7,0.2,0.8,0.10.65,0.25,0.3,0.60.82,0.18,0.6,0.40.5,0.2,0.4,0.1z20.6,0.1,0.5,0.30.5,0.1,0.7,0.150.3,0.2,0.2,0.70.2,0.3,0.6,0.2z30.4,0.5,0.6,0.20.75,0.2,0.4,0.10.65,0.3,0.7,0.250.5,0.4,0.1,0.5z40.5,0.3,0.3,0.70.6,0.4,0.9,0.10.78,0.22,0.45,0.40.8,0.1,0.3,0.1Table 8
q-ROFStR matrix for candidate s2.
c1c2c3c4z10.6,0.3,0.9,0.10.2,0.4,0.6,0.10.5,0.2,0.9,0.10.6,0.2,0.7,0.2z20.4,0.25,0.3,0.50.5,0.15,0.7,0.30.77,0.1,0.6,0.350.4,0.3,0.5,0.1z30.3,0.6,0.65,0.250.66,0.2,0.8,0.170.8,0.15,0.55,0.20.7,0.1,0.3,0.6z40.5,0.15,0.55,0.20.8,0.1,0.4,0.50.62,0.3,0.9,0.10.2,0.4,0.5,0.3Table 9
q-ROFStR matrix for candidate s3.
c1c2c3c4z10.8,0.13,0.8,0.10.5,0.2,0.6,0.10.4,0.1,0.7,0.20.7,0.1,0.6,0.3z20.5,0.16,0.4,0.20.8,0.12,0.3,0.40.6,0.2,0.4,0.30.5,0.2,0.4,0.1z30.4,0.5,0.7,0.30.5,0.4,0.5,0.120.2,0.4,0.8,0.140.3,0.4,0.2,0.5z40.3,0.2,0.5,0.150.6,0.25,0.7,0.30.8,0.18,0.75,0.10.6,0.2,0.8,0.1Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


## 7.2. Aggregation Results Rendered by the q-ROFStRWG Method

Step (i): similar to above.Step (ii): apply the presented q-ROFStRWG aggregation operators on each decision matrix M=ℒcjℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯; that is,(64)ℛ1=0.540892,0.29117,0.446263,0.448329,ℛ2=0.50166,0.321735,0.598613,0.331553,ℛ3=0.491017,0.291113,0.545527,0.264822.Step (iii): calculate the score value of the proposed q-ROFStRWG aggregated results ℛk=β¯,ϑ¯,β¯,ϑ¯ for each object sk; that is,(65)SCℛ1=0.06616,SCℛ2=0.135502,SCℛ3=0.118744.Step (iv): rank the score value ofℛk in a specific order to get the optimum option of professional experts; that is,(66)SCℛ2>SCℛ3>SCℛ1.Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


## 7.3. Aggregation Results Rendered by the q-ROFStROWA Method

Step (i): similar to above.Step (ii):(67)ℛ1=0.657933,0.217542,0.621125,0.234448,ℛ2=0.612743,0.216413,0.744755,0.193923,ℛ3=0.621268,0.203392,0.677814,0.180796.Step (iii):SCℛ1=0.250625,SCℛ2=0.312857,SCℛ3=0.26844.Step (iv):SCℛ2>SCℛ3>SCℛ1.Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


## 7.4. Aggregation Results Rendered by the q-ROFStROWG Method

Step (i): similar to above.Step (ii):(68)ℛ1=0.562833,0.293072,0.473363,0.418793,ℛ2=0.517595,0.319441,0.614971,0.321818,ℛ3=0.49826,0.296096,0.583543,0.259781.Step (iii):SCℛ1=0.09287,SCℛ2=0.152658,SCℛ3=0.139459.Step (iv):SCℛ2>SCℛ3>SCℛ1⇒Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


## 7.5. Aggregation Results Rendered by the q-ROFStRHA Method

Step (i): similar to above.Step (ii): apply the presented q-ROFStRHA aggregation operators on each decision matrix M=ℒδcj∗ℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=βδ∗¯,ϑδ∗¯,βδ∗¯,ϑδ∗¯, with k=0.25,0.29,0.3,0.16Tandl=0.27,0.23,0.32,0.18T being the weight vectors of experts si and parameters cj. Then, the aggregated result for ℒδcj∗ℳi is given as(69)ℛ1=0.432025,0.669215,0.408108,0.691652,ℛ2=0.417457,0.662671,0.503394,0.6509,ℛ3=0.392067,0.655437,0.442525,0.635251.Step (iii):SCℛ1=−0.24099,SCℛ2=−0.18323,SCℛ3=−0.1955.Step (iv):SCℛ2>SCℛ3>SCℛ1.Therefore, from the above analysis, it is observed thats2 is a more suitable and desirable candidate against the given position.


## 7.6. Aggregation Results Rendered by the q-ROFStRHG Method

Step (i): similar to above.Step (ii): apply the presented q-ROFStRHG aggregation operators on each decision matrix M=ℒδcj∗ℳin×m for each alternative skk=1,2,3 against parameter cj to get the aggregated results ℛk=βδ∗¯,ϑδ∗¯,βδ∗¯,ϑδ∗¯, with k=0.25,0.29,0.3,0.16Tandl=0.27,0.23,0.32,0.18T being the weight vectors of experts si and parameters cj. Then, the aggregated result for ℒδcj∗ℳi is given as(70)ℛ1=0.365745,0.709626,0.308693,0.751049,ℛ2=0.332518,0.713922,0.407739,0.706165,ℛ3=0.318639,0.693983,0.36813,0.673448.Step (iii):SCℛ1=−0.35133,SCℛ2=−0.30573,SCℛ3=−0.27871.Step (iv):SCℛ3>SCℛ2>SCℛ1.Therefore, from the above analysis, it is observed thats3 is a more suitable and desirable candidate against the given position. The higher the score value, the more optimist that value and the smaller the score value, the more pessimist that value. From the ranking results of the above proposed operators, it is clear that averaging operators’ results are more optimist than geometric operators.


## 7.7. Comparative Study

To present the applicability and efficiency of the developed approach with some other existing methods based on IFS, PFS, and q-ROFS methods by using the same illustrative example. A comparative study has been made based on different aggregation operators (see [4, 5, 23, 27, 28, 54, 60, 61]). For this purpose, different parameters of the above numerical example are aggregated by utilizing the proposed aggregation operators having weight vectors v=0.32,0.17,0.31,0.2T, and their collective aggregated decision matrix for each candidate sii=1,2,3 is given in Table 10. Now, by using the information of the evaluated matrix, a comparative study of the investigated models with some existing aggregation operators is presented in Table 11. From Table 11, it is observed that the methods presented in [4, 5, 23, 27, 28, 54, 60, 61] are only capable of solving the q-ROFV of the form β,ϑ but are not capable of solving the q-ROFStRV of the form β¯,ϑ¯,β¯,ϑ¯. Thus, from the existing methods, it is clear that these existing methods have a lake of rough information and they are not capable of solving and ranking the given illustrative example. Therefore, from this analysis, it is clear that the developed methods are more superior and capable than the existing methods.Table 10
Aggregated matrix for candidate.
s1s2s3z10.719,0.201,0.651,0.2080.538,0.256,0.849,0.1150.670,0.122,0.713,0.154z20.471,0.154,0.534,0.3200.602,0.179,0.550,0.2980.616,0.171,0.387,0.222z30.598,0.350,0.582,0.2290.678,0.226,0.629,0.2600.370,0.430,0.682,0.225z40.703,0.230,0.616,0.2870.605,0.211,0.733,0.2050.650,0.201,0.703,0.137Table 11
Comparative analysis of different methods.
MethodsScore valuesRankingℛ1ℛ2ℛ3IFWA [4]Unapproachable×IFStWA [54]Unapproachable×PFStWA [60]Unapproachable×q-ROFWA [23]Unapproachable×q-ROFHOWAGA [61]Unapproachable×q-ROFStWA [60]Unapproachable×Cq-ROFWA [27]Unapproachable×q-RONFWA [28]Unapproachable×q-ROFStRWA (proposed)0.226074,0.298025,0.241341ℛ2≽ℛ3≽ℛ1q-ROFStROWA (proposed)0.250625,0.312857,0.26844ℛ2≽ℛ3≽ℛ1q-ROFStRHA (proposed)−0.24099,−0.18323,−0.1955ℛ2≽ℛ3≽ℛ1IFWG [5]Unapproachable×IFStWG [54]Unapproachable×q-ROFWG [23]Unapproachable×Cq-ROFWG [27]Unapproachable×q-RONFWG [28]Unapproachable×q-ROFStRWG (proposed)0.06616,0.135502,0.118744ℛ2≽ℛ3≽ℛ1q-ROFStROWG (proposed)0.09287,0.152658,0.139459ℛ2≽ℛ3≽ℛ1q-ROFStRHG (proposed)−0.35133,−0.30573,−0.27871ℛ3≽ℛ2≽ℛ1


## 8. Conclusion

MCDM has a high potential and discipline process to improve and evaluate multiple conflicting criteria in all areas of decision-making. For an intelligent decision, the experts analyze each and every character of an alternative and then they take the decision. For an intelligent and successful decision, the experts require a careful preparation and analysis of each and every character for an alternative and then they can take a good decision if they are armed with all the data and information that they need. The dominant notions of fuzzy sets,StSs, and rough sets generalized the classical set theory to cope with uncertain information. Molodtsov investigated the pioneer notion of StS which is free from the inherent complexity which the contemporary theories faced. It is observed that StS has too close relation with fuzzy sets and rough sets. The StS theory is an effective mathematical tool for handling the uncertain, ambiguous, and imprecise data. The aim of our work is to investigate the hybrid concept of StS and rough set with the notion of q-ROFS to obtain the new notion of q-ROFStRS. In addition, some averaging aggregation operators such as q-ROFStRWA, q-ROFStROWA, and q-ROFStRHA are presented. Then, important properties of these investigated averaging operators are given in detail. Moreover, we investigated the geometric aggregation operators such as q-ROFStRWG, q-ROFStROWG, and q-ROFStRHG and proposed the basic desirable characteristics of investigated geometric operators. The technique for MCDM and stepwise algorithm for decision-making by utilizing the proposed approaches are demonstrated clearly. Finally, a numerical example for the developed approach is presented and a comparative study of the investigated models with some existing methods is brought to light in detail which shows that the proposed models are more effective and superior than existing approaches.


### 8.1. Future Work

In the future, we intend to further discuss the following topics:(i)
The investigation of q-ROF-entropy of soft rough sets(ii)
The investigation of the picture and spherical fuzzy information by using soft rough sets(iii)
Applying other decision-making methodology based q-ROFStRS to solve the MCDM problem(iv)
The discussions of other applied methods in information systems


## 8.1. Future Work

In the future, we intend to further discuss the following topics:(i)
The investigation of q-ROF-entropy of soft rough sets(ii)
The investigation of the picture and spherical fuzzy information by using soft rough sets(iii)
Applying other decision-making methodology based q-ROFStRS to solve the MCDM problem(iv)
The discussions of other applied methods in information systems



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*Source: 6671001-2020-12-22.xml*