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

College of Economic and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University Islamabad, Islamabad, Pakistan Department of Mathematics, Islamabad Model College for Boys, G-11/1, Islamabad, Pakistan Wuxi Vocational College of Science and Technology, Wuxi 214028, China


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]. e 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. e 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 0 and 1. 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. e 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. us, the concept of q-ROFS is more useful and powerful than IFS and PFS because these are the special cases of q-ROFS. e 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. ey 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.
e 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. e 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][33][34][35][36][37][38]. From the inception, researchers used the hybrid study of rough set theory with different concepts. e 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][41][42][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 (S t S) which generalized the classical set theory and is free from inherent complexity which the contemporary theories faced. It is observed that S t S has a very close relation with fuzzy set and rough set. e S t S 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 S t S with fuzzy set and IFS to obtain fuzzy S t S and IFS t S which play a key role among these theories. Ali et al. [52] improved some existing definitions and operations in S t S theory. e concept of generalized IFS t S was proposed by Agarwal et al. [53]. Arora and Garg [54] presented the concept of IFS t weighted averaging (IFS t WA) and IFS t weighted geometric (IFS t WG) operators. Garg and Arora [55] proposed the notion of some power averaging and geometric aggregation operators by utilizing generalized IFS t S. Arora [56] initiated the notion of IFS t WA and IFS t WG by using the Einstein operations. Feng et al. [57] improved some existing literature related to generalized IF S t S and proposed some new operations for the developed concept. e combined study S t S, 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 S t S, 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 S t S with q-ROFS to get the prominent notion of q-ROF soft (q-ROFS t ) set (q-ROFS t S) and proposed some aggregation operators such as q-ROFS t weighted averaging (q-ROFS t WA), q-ROFS t ordered weighted averaging (q-ROFS t OWA), and q-ROFS t hybrid averaging (q-ROFS t HA).
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. erefore, this motivates the current work of q-ROFS t rough study, and, further, we will investigate aggregation operators based on soft rough information that are q-ROFS t RWA, q-ROFS t ROWA, q-ROFS t RHA, q-ROFS t RWG, q-ROFS t ROWG, and q-ROFS t RHG operators. e design of the remaining sections of the manuscript is summarized as follows: Section 2 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 S t S and rough set with the notion of q-ROFS to obtain the new concept of q-ROFS t RS. In Section 4, we presented the averaging aggregation operators such as q-ROFS t RWA, q-ROFS t ROWA, and q-ROFS t RHA. 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-ROFS t RWG, q-ROFS t ROWG, and q-ROFS t RHG. 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. e final Section 8 consists of a conclusion of the manuscript.

Preliminaries
is section consists of some basic notions including IFS, PFS, q-ROFS, and q-ROFS t S which will be helpful in on word sections.
Definition 1 (see [2]). Consider a universe Y and an IFST on set Y denoted and defined as in which β T , ϑ T : Y ⟶ [0, 1] denotes the MemG and NMemG of an alternative s ∈ Y to the set T having the condition that 0 ≤ β T (s) + ϑ T (s) ≤ 1. π T � 1 − (β T (s) + ϑ T (s)) is called hesitancy degree of s ∈ Y.
Definition 2 (see [10]). Consider a universe Y and a PFST on set Y defined and denoted as in which β T , ϑ T : Y ⟶ [0, 1] denotes the MemG and NMemG of an alternative s ∈ Y to the set T having the condition that 0 ≤ (β T (s)) 2 + (ϑ T (s)) 2 ≤ 1.
Definition 3 (see [20]). Consider Y as a universe of discourse and a q-ROFST on set Y is an object of the form in which β T , ϑ T : Y ⟶ [0, 1] shows the MemG and NMemG of an alternative s ∈ Y to the set T having the condition that 0 ≤ (β T (s)) q + (ϑ T (s)) q ≤ 1. e hesitancy degree is shown as Molodtsov [49] originated the prominent and pioneer concept of soft set (S t S) which generalized the classical set theory and is free from inherent complexity which the contemporary theories faced. It is observed that S t S has very close relation with fuzzy set and rough set. e S t S theory is an essential concept and powerful mathematical tool for Mathematical Problems in Engineering coping the uncertain, ambiguous and imprecise data which is defined as: Definition 4 (see [49]). Let Y be a fixed set and E be set of parameters with B ⊆ E. en the pair (T, B) is said to be S t S, where T is function given as T: B ⟶ P(Y). P(Y) denotes the collection of all subsets of Y.
Definition 5 (see [50]). Let (T, B) a S t S over Y with B ⊆ E.
en (T * , B) is known as fuzzy S t S over Y, where T * is a function given as T * : B ⟶ F(Y). F(Y) denotes the collection of all fuzzy subsets of Y and mathematically it is gives as Definition 6 (see [60]). Consider a universal set Y. Let E be set of parameters and B ⊆ E. en a q-ROFS t S is a pair (T, B) over set Y and T is a mapping given as T: B ⟶ q − ROFS t S(Y) in which q − ROFS t S(Y) contains the collection of all q-ROFSs. en, the q-ROFS t S is denoted and defined as in which β j (s i ), ϑ j (s i ) represents the MemG and NMemG of an alternative s i ∈ Y to the set T c j satisfying that if there is no confusion and is called q-ROFS t number (q-ROFS t N). Considering two q-ROFS t Ns T c j � (β 1j , ϑ 1j ) for (j � 1, 2), reference [20] defined the following operation are defined on them:

q-ROF Soft Rough Set (q-ROFS t RS)
e concept of S t S theory, is the generalization of classical set theory and is free from inherent complexity which the contemporary theories faced. e S t S 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 S t S and rough set to obtain the new concept of q-ROFS t RS. Some basic operations, a new score function, and some desirable characteristics of the proposed concept are investigated in detail.
Definition 7. Let (T, E) be a q-ROFS t S over Y. Any subset L of Y × E is set to a q-ROFS t relation from Y to E and is defined as If Y � s 1 , s 2 , . . . , s m and E � c 1 , c 2 , . . . , c n , then, q-ROFS t relation L from Y to E can be presented in Table 1.
Definition 8. Consider a universal Y, E as the set of parameters and (T, E) as a q-ROFS t S. Let L be an arbitrary q-ROFS t relation from set Y to E. e pair (T, E, L) is said to be q-ROFS t approximation space. For any optimum decision normal object M ∈ q − ROFS(E), then the lower and upper approximation of M with respect to approximation space (T, E, L) are denoted and defined as It is observed that L(M) and L(M) are two q-ROFSs in Y.

Remark 1
(a) If q � 1 is fixed, then the developed q-ROFS t R approximation operators reduce to IFS t R approximation operators (b) If q � 2 is fixed, then the developed q-ROFS t R approximation operators reduce to PFS t R approximation operators Consider the following example to better understand the concept of q-ROFS t R approximation operators. 3 � expensive, and c 4 � location. A decisionmaker 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-ROFS t relation L from set Y to E and is given in Table 2.
Consider a decision-maker Z presents the optimum normal decision object M which is a q-ROF subset over parameter set E; that is, Now, by using equations (7) and (8), we have Now, to get the lower and upper q-ROFS t R approximation operators, for (j � 1, 2) are the two q-ROFS t RNs. en, the following operators are defined on them: en, the score function is given as e greater the score value, the greater the q-ROFS t RN.
) q-ROFS t RSs over a common universe set Y, then the following properties hold:

q-Rung Orthopair Fuzzy Soft Rough Averaging (q-ROFS t RA) Aggregation Operator
is section is devoted to the analysis of q-ROFS t R averaging aggregation operators such as q-ROFS t RWA, q-ROFS t ROWA, and q-ROFS t RHA operators. We will present the fundamental properties of these operators in detail.

q-Rung Orthopair Fuzzy Soft Rough Weighted Averaging
In view of the above definition, eorem 1 illustrates the aggregated result for q-ROFS t RWA.
Proof. To get the required proof, we will use the method of mathematical induction.
As by operational law,

Mathematical Problems in Engineering
Suppose the result is true for m � 2 and n � 2; that is, Now, consider e result is true for m � 2 and n � 2. Now, consider the result is for n � k 1 and m � k 2 : Suppose the result holds for n � k 1 + 1 and m � k 2 + 1, so we have is implies the result is true for n � k 1 + 1 and m � k 2 + 1. erefore, the result holds for all m, n ≥ 1.
Since it is clear that L c j (M i ) and L c j (M i ) are q-ROFNs, by Definition 7, we have that en, q-ROFS t RNs is given in Table 3: From the analysis of eorem 1, some related characteristics of q-ROFS t RWA operator are given as follows: en, the following properties hold for q-ROFS t RWA operator: (v) (Homogeneity): for any real number λ > 0,

Remark 2
(a) If q � 1, so, in this case, the developed q-ROFS t RWA operator reduces to IFS t RWA operator (b) If q � 2, so, in this case, the developed q-ROFS t RWA operator reduces to PFS t RWA operator is subsection presents the detailed study of q-ROFS t ROWA operator and some of its desirable characteristics such as Idempotency, Boundedness, and Monotonicity. e basic advantage of q-ROFS t ROWG operator is to weight the ordered position of the q-ROFVs instead of weighting the values themselves.
In view of the above definition, the aggregated result for q-ROFS t ROWA is described in eorem 3.
where L δc j (M i ) � (L δc j (M i ), L δc j (M i )) denotes the largest value of the permutation from ith row and jth column of Table 3 of Example 2, for the collection q-ROFS t RNs L c j (M i ) � (L c j (M i ), L c j (M i )) and the new ordered of tabular representation of L c j (M i ) through score function is given in Table 4. 0.838595, 0.261642), (0.727318, 0.255189)).

q-Rung Orthopair Fuzzy Soft Rough Geometric (q-ROFS t RG) Aggregation Operator
is section is devoted to the study of q-ROFS t R geometric aggregation operators such as q-ROFS t RWG, q-ROFS t ROWG, and q-ROFS t RHG operators. We will present the fundamental properties of these operators in detail.
Let the weight vectors t � (t 1 , t 2 , . . . , t m ) T and v � (v 1 , v 2 , . . . , v n ) T of experts s i and parameters c j with m i�1 t i � 1, n j�1 v j � 1, and 0 ≤ t i , v j ≤ 1, respectively. en, q-ROFS t RWG operator is given as Since it is clear that are also q-ROFNs. erefore,    q − ROFS t RWG(L c 1 (M 1 ), . . . , L c n (M m )) is also a q-ROFS t RN in approximation space (T, E, L). Table 3 of Example 2, for the collection

Example 5. Consider
. en, the aggregated result for q − ROFS t RWG is given as From the analysis of eorem 7, the following hold for q-ROFS t RWG operator.  , d), (b, d)), then (iii) (Monotonicity): let P c j (N i ) � (P c j (N i ), P c j (N i ))(i � 1, 2, . . . , m, j � 1, 2, . . . , n) be another collection of q-ROFS t RNs such that P c j (N i ) ≤ (v) (Homogeneity): for any real number λ > 0, Remark 5 (a) If q � 1, then q-ROFS t RWG operator reduces to IFS t RWG operator (b) If q � 2, then q-ROFS t RWG operator reduces to PFS t RWG operator (c) If soft parameter c 1 is one (means n � 1), then q-ROFS t RWG operator reduces to q-ROFRWG operator Mathematical Problems in Engineering 13

q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Geometric (q-ROFS t ROWG) Operator.
Here, we will put forward the detailed study of q-ROFS t ROWG operator and some of its fundamental properties. e q-ROFS t ROWG operator weights the ordered position of the q-ROFVs instead of weighting the values themselves. 1, 2, . . . , m, j � 1, 2, . . . , n) be the collection of q-ROFS t RNs.
Let the weight vectors t � (t 1 , t 2 , . . . , t m ) T and v � (v 1 , v 2 , . . . , v n ) T of experts s i and parameters c j with m i�1 t i � 1, n j�1 v j � 1, and 0 ≤ t i , v j ≤ 1, respectively. e q-ROFS t ROWG operator is defined as In view of the above definition, the aggregated result for q-ROFS t ROWG operator is described in eorem 9. 1, 2, . . . , m, j � 1, 2, . . . , n) be the collection of q-ROFS t RNs.
Let the weight vectors t � (t 1 , t 2 , . . . , t m ) T and v � (v 1 , v 2 , . . . , v n ) T of experts s i and parameters c j with m i�1 t i � 1, n j�1 v j � 1, and 0 ≤ t i , v j ≤ 1, respectively. en, q-ROFS t ROWG operator is given as where L δc j (M i ) � (L δc j (M i ), L δc j (M i )) denotes the largest value of the permutation from ith row and jth column of

Remark 6
(a) If q � 1, then q-ROFS t ROWG operator reduces to IFS t ROWG operator (b) If q � 2, then q-ROFS t ROWG operator reduces to PFS t ROWG operator (c) If the soft parameter c 1 is one (means n � 1), then q-ROFS t ROWG operator reduces to q-ROFROWG operator

q-Rung Orthopair Fuzzy Soft Rough Hybrid Geometric (q-ROFS t RHG)
Operator. From the analysis of Definitions 14 and 15, it is clear that the q-ROFS t RWG operator weights only the q-ROFVs, while q-ROFS t ROWG 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-ROFS t RHG operator, which weights both the given q-ROFV and its ordered position. e basic desirable properties of the developed operator are presented in detail.

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 that Y � s 1 , s 2 , s 3 , . . . , s p be the initial set of various alternatives and E � c 1 , c 2 , c 3 , . . . , c n be the set of n parameters. Consider Z � z 1 , z 2 , z 3 , . . . , z m as the set of m professional experts of this area who present their assessment expertise for each alternative s k (k � 1, 2, . . . , p) corresponding to n parameters. Let the weight vectors t � (t 1 , t 2 , . . . , t m ) T of experts z i and v � (v 1 , v 2 , . . . , v n ) T of parameters c j with m i�1 t i � 1, n j�1 v j � 1, and 0 ≤ t i , v j ≤ 1, respectively. e professional experts express their preference evaluation for alternative s k with respect to parameter c j in the form of q-ROFS t RNs.
e collective preference information given by the professionals is managed in where M ⊆ E. Further, using the proposed models aggregate, the preferred choices of experts are to get the aggregated results R k (k � 1, . . . , p) for each alternative s k against their parameter c j . Finally, they utilize the score function on the aggregated results R k � [(β, ϑ), (β, ϑ)] and rank all the results in a specific order to get the most desirable option. e stepwise decision algorithm for the investigated operators: Step (i): the professional experts express their preference evaluation for alternative s k with respect to parameter c j in the form of q-ROFS t RNs. en, they collect the preference information given by the professionals and manage them in q- Step (ii): apply the presented aggregation operators of each decision matrix M � [L c j (M i )] n×m for each alternative s k (k � 1, 2, . . . , p) against parameter c j to get the aggregated results R k � [(β, ϑ), (β, ϑ)].
Step (iv): rank the score value of R k in a specific order to get the optimum option of professional experts.

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 professors Z � z 1 , z 2 , z 3 , z 4 from home and abroad to select the most desirable applicant. Out of many applicants, three applicants Y � s 1 , s 2 , s 3 were called for interviews. e interview mainly judges the applicants against some parameters M � c 1 � academic level, c 2 � development potential, c 3 � professional ethics, c 4 � research productivity} ⊆ E. Let the weight vector t � (0.3, 0.28, 0.24, 0.18) T for professional experts z i (i � 1, . . . , 4) and v � (0.32, 0.17, 0.31, 0.2) T be the weight vectors for parameters c j (j � 1, 2, 3), respectively. e professional experts express their preference evaluation for candidate s k with respect to parameter c j in the form of q-ROFS t RNs. Finally, follow the following steps by utilizing the proposed models to select the most desirable and suitable applicant s k .

Aggregation Results Rendered by the q-ROFS t RWA Method
Step (i): the professional experts express their preference evaluation for alternative s k with respect to parameter c j in the form of q-ROFS t RNs. en, they collect the preference information given by the professionals and manage them in q-ROFS t R decision matrix, which is M � [L c j (M i )] n×m where M ⊆ E which is given in Tables 7-9. Step Step (iii): calculate the score value of aggregated results R k � [(β, ϑ), (β, ϑ)] for each object s k ; that is, Step (iv): rank the score value of R k in a specific order to get the optimum option of professional experts; that is, erefore, from the above analysis, it is observed that s 2 is a more suitable and desirable candidate against the given position.

Aggregation Results Rendered by the q-ROFS t RWG Method
Step (i): similar to above.
Step (64) Step (iii): calculate the score value of the proposed q-ROFS t RWG aggregated results R k � [(β, ϑ), (β, ϑ)] for each object s k ; that is, Step (iv): rank the score value of R k in a specific order to get the optimum option of professional experts; that is, erefore, from the above analysis, it is observed that s 2 is a more suitable and desirable candidate against the given position.
erefore, from the above analysis, it is observed that s 2 is a more suitable and desirable candidate against the given position.

Aggregation Results Rendered by the q-ROFS t ROWG Method
Step (i): similar to above.
Step (iv): SC(R 2 ) > SC(R 3 ) > SC(R 1 )⇒ erefore, from the above analysis, it is observed that s 2 is a more suitable and desirable candidate against the given position.

Aggregation Results Rendered by the q-ROFS t RHA Method
Step (i): similar to above.
Step (ii): apply the presented q-ROFS t RHA aggregation operators on each decision matrix M � [L * δc j (M i )] n×m for each alternative s k (k � 1, 2, 3) against parameter c j to get the aggregated results Step ( Step (iv): SC(R 2 ) > SC(R 3 ) > SC(R 1 ).
erefore, from the above analysis, it is observed that s 2 is a more suitable and desirable candidate against the given position.

Aggregation Results Rendered by the q-ROFS t RHG Method
Step (i): similar to above.
Step (ii): apply the presented q-ROFS t RHG aggregation operators on each decision matrix M � [L * δc j (M i )] n×m for each alternative s k (k � 1, 2, 3) against parameter c j to get the aggregated results (70) Step (iii): Step (iv): SC(R 3 ) > SC(R 2 ) > SC(R 1 ). erefore, from the above analysis, it is observed that s 3 is a more suitable and desirable candidate against the given position. e 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.

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, 0.2) T , and their collective aggregated decision matrix for each candidate s i (i � 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-ROFS t RV of the form [(β, ϑ), (β, ϑ)]. us, 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. erefore, from this analysis, it is clear that the developed methods are more superior and capable than the existing methods.

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. e dominant notions of fuzzy sets, S t Ss, and rough sets generalized the classical set theory to cope with uncertain information. Molodtsov investigated the pioneer notion of S t S which is free from the inherent complexity which the contemporary theories faced. It is observed that S t S has too close relation with fuzzy sets and rough sets. e S t S theory is an effective mathematical tool for handling the uncertain, ambiguous, and imprecise data. e aim of our work is to investigate the hybrid concept of S t S and rough set with the notion of q-ROFS to obtain the new notion of q-ROFS t RS. In addition, some averaging aggregation operators such as q-ROFS t RWA, q-ROFS t ROWA, and q-ROFS t RHA are presented. en, important properties of these investigated averaging operators are given in detail. Moreover, we investigated the geometric aggregation operators such as q-ROFS t RWG, q-ROFS t ROWG, and q-ROFS t RHG and proposed the basic desirable characteristics of investigated geometric operators. e 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.

Future Work.
In the future, we intend to further discuss the following topics: (i) e investigation of q-ROF-entropy of soft rough sets (ii) e investigation of the picture and spherical fuzzy information by using soft rough sets (iii) Applying other decision-making methodology based q-ROFS t RS to solve the MCDM problem (iv) e discussions of other applied methods in information systems
Data Availability e data used in this article are artificial and hypothetical, and anyone can use these data before prior permission by just citing this article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.