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 (S) and rough set with the notion of q-ROFS to obtain the new notion of q-ROF soft rough (q-ROFR) set (q-ROFRS). In addition, some averaging aggregation operators such as q-ROFR weighted averaging (q-ROFRWA), q-ROFR ordered weighted averaging (q-ROFROWA), and q-ROFR hybrid averaging (q-ROFRHA) 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-ROFR weighted geometric (q-ROFRWG), q-ROFR ordered weighted geometric (q-ROFROWG), and q-ROFR hybrid geometric (q-ROFRHG) 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.
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 . 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 . 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 power of MemG and power of NMemG must be confined to the unit interval and, furthermore, when the rung 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 -covering and fuzzy -covering neighborhood. Molodtsov [49] originated the prominent and pioneer concept of soft set (S) which generalized the classical set theory and is free from inherent complexity which the contemporary theories faced. It is observed that S has a very close relation with fuzzy set and rough set. The 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 with fuzzy set and IFS to obtain fuzzy S and IFS which play a key role among these theories. Ali et al. [52] improved some existing definitions and operations in S theory. The concept of generalized IFS was proposed by Agarwal et al. [53]. Arora and Garg [54] presented the concept of IF weighted averaging (IFWA) and IF weighted geometric (IFWG) operators. Garg and Arora [55] proposed the notion of some power averaging and geometric aggregation operators by utilizing generalized IFS. Arora [56] initiated the notion of IFWA and IFWG by using the Einstein operations. Feng et al. [57] improved some existing literature related to generalized IF S and proposed some new operations for the developed concept. The combined study 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, 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 with q-ROFS to get the prominent notion of q-ROF soft (q-ROF) set (q-ROFS) and proposed some aggregation operators such as q-ROF weighted averaging (q-ROFWA), q-ROF ordered weighted averaging (q-ROFOWA), and q-ROF hybrid averaging (q-ROFHA).
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-ROF rough study, and, further, we will investigate aggregation operators based on soft rough information that are q-ROFRWA, q-ROFROWA, q-ROFRHA, q-ROFRWG, q-ROFROWG, and q-ROFRHG operators.
The 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 and rough set with the notion of q-ROFS to obtain the new concept of q-ROFRS. In Section 4, we presented the averaging aggregation operators such as q-ROFRWA, q-ROFROWA, and q-ROFRHA. 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-ROFRWG, q-ROFROWG, and q-ROFRHG. 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-ROFS which will be helpful in on word sections.
Definition 1 (see [2]). Consider a universe and an IFS on set denoted and defined asin which denotes the MemG and NMemG of an alternative to the set having the condition that is called hesitancy degree of .
Definition 2 (see [10]). Consider a universe and a PFS on set defined and denoted asin which denotes the MemG and NMemG of an alternative to the set having the condition that is called hesitancy degree of .
Definition 3 (see [20]). Consider as a universe of discourse and a q-ROFS on set is an object of the formin which shows the MemG and NMemG of an alternative to the set having the condition that The hesitancy degree is shown as for each .
Molodtsov [49] originated the prominent and pioneer concept of soft set (S) which generalized the classical set theory and is free from inherent complexity which the contemporary theories faced. It is observed that S has very close relation with fuzzy set and rough set. The S 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]). Let be a fixed set and be set of parameters with . Then the pair is said to be S, where is function given as . denotes the collection of all subsets of .
Definition 5 (see [50]). Let a S over with . Then is known as fuzzy S over , where is a function given as . denotes the collection of all fuzzy subsets of and mathematically it is gives as
Definition 6 (see [60]). Consider a universal set . Let be set of parameters and . Then a q-ROFS is a pair over set and is a mapping given as in which contains the collection of all q-ROFSs. Then, the q-ROFS is denoted and defined asin which represents the MemG and NMemG of an alternative to the set satisfying that is known to be hesitancy degree of . For simplicity is denoted as if there is no confusion and is called q-ROF number (q-ROFN).
Considering two q-ROFNs for , reference [20] defined the following operation are defined on them:(i)(ii)(iii)(iv)(v)(vi) where represents the complement of (vii)(viii)
3. q-ROF Soft Rough Set (q-ROFRS)
The concept of S theory, is the generalization of classical set theory and is free from inherent complexity which the contemporary theories faced. The 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 and rough set to obtain the new concept of q-ROFRS. Some basic operations, a new score function, and some desirable characteristics of the proposed concept are investigated in detail.
Definition 7. Let be a q-ROFS over . Any subset of is set to a q-ROF relation from and is defined aswhere denote the MemG and NMemG with for all .
If , then, q-ROF relation from can be presented in Table 1.
Definition 8. Consider a universal , as the set of parameters and as a q-ROFS. Let be an arbitrary q-ROF relation from set to . The pair is said to be q-ROF approximation space. For any optimum decision normal object , then the lower and upper approximation of with respect to approximation space are denoted and defined aswhere , and , such that .
It is observed that and are two q-ROFSs in . Thus, the operators are, respectively, known as lower and upper q-ROFR approximation operators. Therefore, q-ROFRS is a pair .
For simplicity, we can write as and call q-ROFR number (q-ROFRN), if there is no confusion.
Remark 1. (a)If is fixed, then the developed q-ROFR approximation operators reduce to IFR approximation operators(b)If is fixed, then the developed q-ROFR approximation operators reduce to PFR approximation operatorsConsider the following example to better understand the concept of q-ROFR approximation operators.
Example 1. Suppose a decision-maker buys a house, as given in set under consideration. Let the parameter set where , , , and . A decision-maker wants to purchase a house from the available houses which fulfill the utmost extent of given parameters. Consider the decision-maker presents the gorgeous of houses in form of q-ROF relation from set and is given in Table 2.
Consider a decision-maker presents the optimum normal decision object which is a q-ROF subset over parameter set ; that is,Now, by using equations (7) and (8), we haveNow, to get the lower and upper q-ROFR approximation operators,Therefore,
Definition 9. Consider for are the two q-ROFRNs. Then, the following operators are defined on them:(i)(ii)(iii)(iv)(v)(vi) for (vii) for (viii), where , the complements of q-ROFR approximation operators , that is, (ix) iff
Definition 10. Let be a q-ROFRN. Then, the score function is given asThe greater the score value, the greater the q-ROFRN.
Proposition 1. Let be q-ROF approximation space. For any two and q-ROFRSs over a common universe set , then the following properties hold:(i), where is the complement of , (ii)(iii)(iv)If , then (v)(vi)
4. q-Rung Orthopair Fuzzy Soft Rough Averaging (q-ROFRA) Aggregation Operator
This section is devoted to the analysis of q-ROFR averaging aggregation operators such as q-ROFRWA, q-ROFROWA, and q-ROFRHA operators. We will present the fundamental properties of these operators in detail.
4.1. q-Rung Orthopair Fuzzy Soft Rough Weighted Averaging (q-ROFRWA) Operator
Definition 11. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , , and , respectively. The q-ROFRWA operator is defined asIn view of the above definition, Theorem 1 illustrates the aggregated result for q-ROFRWA.
Theorem 1. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , and , respectively. Then, q-ROFRWA operator is given as
Proof. To get the required proof, we will use the method of mathematical induction.
As by operational law,Suppose the result is true for ; that is,Now, considerThe result is true for .
Now, consider the result is for :Suppose the result holds for , so we haveThis implies the result is true for . Therefore, the result holds for all .
Since it is clear that are q-ROFNs, by Definition 7, we have that and are also q-ROFNs. Therefore, is also a q-ROFRN in approximation space .
Example 2. Let be a set and a set of parameters with weight vector for and for . Then, q-ROFRNs is given in Table 3:From the analysis of Theorem 1, some related characteristics of q-ROFRWA operator are given as follows:
Theorem 2. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , and , respectively. Then, the following properties hold for q-ROFRWA operator:(i)(Idempotency): if , where , then(ii)(Boundedness): let and . Then,(iii)(Monotonicity): let be another collection of q-ROFRNs such that and . Then,(iv)(Shift invariance): let be any other . Then,(v)(Homogeneity): for any real number ,
Remark 2. (a)If , so, in this case, the developed q-ROFRWA operator reduces to IFRWA operator(b)If , so, in this case, the developed q-ROFRWA operator reduces to PFRWA operator(c)If the soft parameter is only one , then, in this case, the developed q-ROFRWA operator reduces to q-ROFRWA operator
4.2. q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Averaging (q-ROFROWA) Operator
This subsection presents the detailed study of q-ROFROWA operator and some of its desirable characteristics such as Idempotency, Boundedness, and Monotonicity. The basic advantage of q-ROFROWG operator is to weight the ordered position of the q-ROFVs instead of weighting the values themselves.
Definition 12. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , , and , respectively. The q-ROFROWA operator is defined asIn view of the above definition, the aggregated result for q-ROFROWA is described in Theorem 3.
Theorem 3. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , , and , respectively. Then, q-ROFROWA operator is given aswhere denotes the largest value of the permutation from and of the collection q-ROFRNs .
Example 3. Consider Table 3 of Example 2, for the collection q-ROFRNs and the new ordered of tabular representation of through score function is given in Table 4.
Now, From the analysis of Theorem 3, the following desirable properties hold for q-ROFROWA operator.
Theorem 4. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , , and , respectively. Then, the following properties hold for q-ROFROWA operator:(i)(Idempotency): if , where , then(ii)(Boundedness): let and . Then,(iii)(Monotonicity): let be another collection of q-ROFRNs such that and . Then,(iv)(Shift invariance): let be any other . Then,(v)(Homogeneity): for any real number ,
Remark 3. (a)If , then the developed q-ROFROWA operator reduces to IFROWA operator(b)If , then the developed q-ROFROWA operator reduces to PFROWA operator(c)If the soft parameter is one , then the developed q-ROFROWA operator reduces to q-ROFROWA operator
4.3. q-Rung Orthopair Fuzzy Soft Rough Hybrid Averaging (q-ROFRHA) Operator
From the analysis of Definitions 11 and 12, it is clear that the q-ROFRWA operator weights only the q-ROFVs, while q-ROFROWA 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-ROFRHA 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. Let be the collection of q-ROFRNs. Let the weight vectors for experts and parameters with , , and . Consider as the associated weight vectors of experts and parameters with , , and , respectively. The q-ROFRHA operator is defined asFrom the above definition, the aggregated result for q-ROFRHA operator is described in Theorem 5.
Theorem 5. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and . Consider the associated weight vectors of experts and parameters with , , and , respectively. Then, q-ROFRHA operator is given aswhere denotes the largest value of the permutation from and of the collection q-ROFRNs and represents the balancing coefficient.
Example 4. Consider Table 3 of Example 2, for the collection q-ROFRNs with as the weight vectors of experts and parameters . Consider as the associated weight vectors of experts and parameters . The tabular representation of through operation law and score function is given in Tables 5 and 6. Now,From the analysis of Theorem 5, the following characteristics hold in q-ROFRHA operator.
Theorem 6. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and . Consider the associated weight vectors of experts and parameters with , , and , respectively. Then, the following properties hold for q-ROFRHA operator:(i)(Idempotency): if where , then(ii)(Boundedness): let and . Then,(iii)(Monotonicity): let be another collection of q-ROFRNs such that and . Then,(iv)(Shift invariance): let be any other . Then,(v)(Homogeneity): for any real number ,
Remark 4. (a)If , then q-ROFRHA operator reduces to IFRHA operator(b)If , then q-ROFRHA operator reduces to PFRHA operator(c)If soft parameter is one , then q-ROFRHA operator reduces to q-ROFRHA operator(d)If , then the proposed q-ROFRHA operator reduces to q-ROFRWA operator(e)If , then the proposed q-ROFRHA operator reduces to q-ROFROWA operator
5. q-Rung Orthopair Fuzzy Soft Rough Geometric (q-ROFRG) Aggregation Operator
This section is devoted to the study of q-ROFR geometric aggregation operators such as q-ROFRWG, q-ROFROWG, and q-ROFRHG operators. We will present the fundamental properties of these operators in detail.
5.1. q-Rung Orthopair Fuzzy Soft Rough Weighted Geometric (q-ROFRWG) Operator
Definition 14. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and , respectively. The q-ROFRWG operator is defined asBased on the above definition the aggregated result for q-ROFRWG operator is given in Theorem 7.
Theorem 7. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and , respectively. Then, q-ROFRWG operator is given as
Since it is clear that are q-ROFNs, by Definition 7, we have that and are also q-ROFNs. Therefore, is also a q-ROFRN in approximation space .
Example 5. Consider Table 3 of Example 2, for the collection q-ROFRNs . Then, the aggregated result for is given asFrom the analysis of Theorem 7, the following hold for q-ROFRWG operator.
Theorem 8. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and , respectively. Then, the following properties hold for q-ROFRWG operator:(i)(Idempotency): if where , then(ii)(Boundedness): let and . Then,(iii)(Monotonicity): let be another collection of q-ROFRNs such that and . Then,(iv)(Shift invariance): let be any other . Then,(v)(Homogeneity): for any real number ,
Remark 5. (a)If , then q-ROFRWG operator reduces to IFRWG operator(b)If , then q-ROFRWG operator reduces to PFRWG operator(c)If soft parameter is one , then q-ROFRWG operator reduces to q-ROFRWG operator
5.2. q-Rung Orthopair Fuzzy Soft Rough Ordered Weighted Geometric (q-ROFROWG) Operator
Here, we will put forward the detailed study of q-ROFROWG operator and some of its fundamental properties. The q-ROFROWG operator weights the ordered position of the q-ROFVs instead of weighting the values themselves.
Definition 15. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and , respectively. The q-ROFROWG operator is defined asIn view of the above definition, the aggregated result for q-ROFROWG operator is described in Theorem 9.
Theorem 9. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and , respectively. Then, q-ROFROWG operator is given aswhere denotes the largest value of the permutation from and of the collection q-ROFRNs, .
Remark 6. (a)If , then q-ROFROWG operator reduces to IFROWG operator(b)If , then q-ROFROWG operator reduces to PFROWG operator(c)If the soft parameter is one , then q-ROFROWG operator reduces to q-ROFROWG operator
5.3. q-Rung Orthopair Fuzzy Soft Rough Hybrid Geometric (q-ROFRHG) Operator
From the analysis of Definitions 14 and 15, it is clear that the q-ROFRWG operator weights only the q-ROFVs, while q-ROFROWG 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-ROFRHG 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. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and . Consider the associated weight vectors of experts and parameters with , , and , respectively. The q-ROFRHG operator is defined asFrom the above definition, the aggregated value for q-ROFRHG operator is described in Theorem 11.
Theorem 10. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and . Consider the associated weight vectors of experts and parameters with , , and , respectively. Then, q-ROFRHG operator is given aswhere denotes the largest value of the permutation from and of the collection q-ROFRNs and represents the balancing coefficient.
Example 6. Consider Tables 2, 4, and 5 of Examples 2 and 4, for the collection q-ROFRNs with is the weight vectors of experts and parameters . Consider as the associated weight vectors of experts and parameters . Then, the aggregated result for is given asFrom the analysis of Theorem 11, the properties of q-ROFRHG operator are given below.
Theorem 11. Let be the collection of q-ROFRNs. Let the weight vectors of experts and parameters with , , and . Consider the associated weight vectors of experts and parameters with , , and , respectively. Then, q-ROFRHG operator holds as follows:(i)(Idempotency): if where , then(ii)(Boundedness): let and . Then,(iii)(Monotonicity): let be another collection of q-ROFRNs such that and . Then,(iv)(Shift invariance): let be any other . Then,(v)(Homogeneity): for any real number ,
Remark 7. (a)If , then q-ROFRHG operator reduces to IFRHG operator(b)If the value of rung , then q-ROFRHG operator reduces to PFRHG operator(c)If the soft parameter is one , then q-ROFRHG operator reduces to q-ROFRHG operator(d)If , then the proposed q-ROFRHG operator reduces to q-ROFRWG operator(e)If , then the proposed q-ROFRHG operator reduces to q-ROFROWG 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 that be the initial set of various alternatives and be the set of parameters. Consider as the set of professional experts of this area who present their assessment expertise for each alternative corresponding to parameters. Let the weight vectors of experts and of parameters with , , and , respectively. The professional experts express their preference evaluation for alternative with respect to parameter in the form of q-ROFRNs. The collective preference information given by the professionals is managed in q-ROFR decision matrix, which is where . Further, using the proposed models aggregate, the preferred choices of experts are to get the aggregated results for each alternative against their parameter . Finally, they utilize the score function on the aggregated results 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 alternative with respect to parameter in the form of q-ROFRNs. Then, they collect the preference information given by the professionals and manage them in q-ROFR decision matrix, which is where . Step (ii): apply the presented aggregation operators of each decision matrix for each alternative against parameter to get the aggregated results . Step (iii): calculate the score value of aggregated results for each object . Step (iv): rank the score value of 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 professors from home and abroad to select the most desirable applicant. Out of many applicants, three applicants were called for interviews. The interview mainly judges the applicants against some parameters . Let the weight vector for professional experts and be the weight vectors for parameters , respectively. The professional experts express their preference evaluation for candidate with respect to parameter in the form of q-ROFRNs. Finally, follow the following steps by utilizing the proposed models to select the most desirable and suitable applicant .
7.1. Aggregation Results Rendered by the q-ROFRWA Method
Step (i): the professional experts express their preference evaluation for alternative with respect to parameter in the form of q-ROFRNs. Then, they collect the preference information given by the professionals and manage them in q-ROFR decision matrix, which is where which is given in Tables 7–9. Step (ii): apply the presented q-ROFRWA aggregation operators on each decision matrix for each alternative against parameter to get the aggregated results ; that is, Step (iii): calculate the score value of aggregated results for each object ; that is, Step (iv): rank the score value of in a specific order to get the optimum option of professional experts; that is,
Therefore, from the above analysis, it is observed that is a more suitable and desirable candidate against the given position.
7.2. Aggregation Results Rendered by the q-ROFRWG Method
Step (i): similar to above. Step (ii): apply the presented q-ROFRWG aggregation operators on each decision matrix for each alternative against parameter to get the aggregated results ; that is, Step (iii): calculate the score value of the proposed q-ROFRWG aggregated results for each object ; that is, Step (iv): rank the score value of in a specific order to get the optimum option of professional experts; that is,
Therefore, from the above analysis, it is observed that is a more suitable and desirable candidate against the given position.
7.3. Aggregation Results Rendered by the q-ROFROWA Method
Step (i): similar to above. Step (ii): Step (iii): Step (iv): .
Therefore, from the above analysis, it is observed that is a more suitable and desirable candidate against the given position.
7.4. Aggregation Results Rendered by the q-ROFROWG Method
Step (i): similar to above. Step (ii): Step (iii): Step (iv):
Therefore, from the above analysis, it is observed that is a more suitable and desirable candidate against the given position.
7.5. Aggregation Results Rendered by the q-ROFRHA Method
Step (i): similar to above. Step (ii): apply the presented q-ROFRHA aggregation operators on each decision matrix for each alternative against parameter to get the aggregated results , with being the weight vectors of experts and parameters . Then, the aggregated result for is given as Step (iii): Step (iv): .
Therefore, from the above analysis, it is observed that is a more suitable and desirable candidate against the given position.
7.6. Aggregation Results Rendered by the q-ROFRHG Method
Step (i): similar to above. Step (ii): apply the presented q-ROFRHG aggregation operators on each decision matrix for each alternative against parameter to get the aggregated results , with being the weight vectors of experts and parameters . Then, the aggregated result for is given as Step (iii): Step (iv): .
Therefore, from the above analysis, it is observed that 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 , and their collective aggregated decision matrix for each candidate 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 but are not capable of solving the q-ROFRV of the form . 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.
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, Ss, and rough sets generalized the classical set theory to cope with uncertain information. Molodtsov investigated the pioneer notion of S which is free from the inherent complexity which the contemporary theories faced. It is observed that S has too close relation with fuzzy sets and rough sets. The S 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 S and rough set with the notion of q-ROFS to obtain the new notion of q-ROFRS. In addition, some averaging aggregation operators such as q-ROFRWA, q-ROFROWA, and q-ROFRHA are presented. Then, important properties of these investigated averaging operators are given in detail. Moreover, we investigated the geometric aggregation operators such as q-ROFRWG, q-ROFROWG, and q-ROFRHG 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-ROFRS to solve the MCDM problem(iv)The discussions of other applied methods in information systems
Abbreviations
| IFS: | Intuitionistic fuzzy set |
| PFS: | Pythagorean fuzzy set |
| q-ROFS: | q-Rung orthopair fuzzy set |
| S: | Soft set |
| q-ROFRS: | q-ROF soft rough set |
| q-ROFRWA: | q-ROF soft rough weighted averaging |
| q-ROFROWA: | q-ROF soft rough ordered weighted averaging |
| q-ROFRHA: | q-ROF soft rough hybrid averaging |
| q-ROFRWG: | q-ROF soft rough weighted geometric |
| q-ROFROWG: | q-ROF soft rough ordered weighted geometric |
| q-ROFRHG: | q-ROF soft rough hybrid geometric |
| MCDM: | Multicriteria decision-making |
| MemG: | Membership grade |
| NMemG: | Nonmembership grade |
| IFWA: | IF weighted averaging |
| IFOWA: | IF ordered weighted averaging |
| IFHA: | IF hybrid averaging |
| IFWG: | IF weighted geometric |
| IFOWG: | IF ordered weighted geometric |
| IFHG: | IF hybrid geometric |
| q-ROFWA: | q-ROF weighted averaging |
| q-ROFWG: | q-ROF weighted geometric |
| IFS: | IF soft set |
| IFWA: | IF soft weighted averaging |
| IFWG: | IF soft weighted geometric |
| q-ROFS: | q-Rung orthopair fuzzy soft set |
| q-ROFWA: | q-ROF soft weighted averaging |
| q-ROFOWA: | q-ROF soft ordered weighted averaging |
| q-ROFHA: | q-ROF soft hybrid averaging |
| q-ROFN: | q-ROF soft number |
| q-ROFRN: | q-ROF soft rough number. |
Data Availability
The 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
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by Major Humanities and Social Sciences Research Projects in Zhejiang Universities (no. 2018QN058).