Cq-ROFRS: covering q-rung orthopair fuzzy rough sets and its application to multi-attribute decision-making process

Pythagorean fuzzy sets (briefly, PFSs) were created as an upgrade to intuitionistic fuzzy sets (briefly, IFSs) which helped to address some problems that IFSs couldn’t solve. The definition of q-rung orthopair fuzzy sets (briefly, q-ROFS) is then declared to generalize and solve PFS and IFS failures. Using the concept of PF β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood, Zhan et al. defined the description of the covering through the Pythagorean fuzzy rough set (briefly, CPFRS). Hussain et al. also developed the concept of q-ROF β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood to build the concept of covering through q-rung orthopair fuzzy rough sets (Cq-ROFRS). To enhance the results in Zhan et al.’s and Hussain et al.’s method and in a related context, the concept of PF complementary β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood is constructed. Hence, using PF β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood and PF complementary β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood, three novel kinds of CPFRS are investigated and the related characteristics are analyzed. The interrelationships between Zhan et al.’s approach and our approaches are also discussed. Besides, the concept of q-ROF complementary β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood is examined. Three new Cq-ROFRS models are differentiated using the principles of q-ROF β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood and q-ROF complementary β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-neighborhood. As a result, the related properties and relationships between these various models and Hussain et al.’s model are established. Because of these correlations, we may consider our approach to be a generalization of Zhan et al.’s and Hussain et al’s approaches. Finally, we developed applications to solve MADM problems using CPFRS and Cq-ROFRS, as well as variances of the two methods using numerical examples are presented.

The concept of fuzzy rough set (FRS) and rough fuzzy set (RFS) was constructed by Dubois et al. [28]. Deng et al. [29] proposed new model of fuzzy covering according to FRS. Atef et al. and Li et al. studied additional kinds of fuzzy rough covering (FRC) [30][31][32][33][34]. Also, Ma [35] discovered kinds of fuzzy covering rough set (FCRS) using the fuzzy β-neighborhood. Moreover, the notions of a fuzzy complementary β-neighborhood and fuzzy β minimal and maximal description were found by Yang et al. [36,37].
Fuzzy set theory (FS) was initiated by Zadeh [38]. There are some problems in FS for dealing with uncertain data, so the definition of IFSs was appeared by Atanassov [39] which contain two parts membership degree and non-membership degree. In IFSs, the sum of membership and non-membership classes is between [0, 1]. Atanassov et al. [40] used IFS to make a decision multi-person problem. Huang et al. [41,42] introduced the notion of intuitionistic fuzzy multigranulation rough sets and intuitionistic fuzzy via CRS. Alcantud et al. [43] discussed the decomposition theorems.
In realistic problems, much difficult application can not be solved via IFSs. Therefore, Yager [44] introduced the concept of PFSs. The main adding in PFSs is the sum of squares of membership class and non-membership class is in [0,1]. Yager [45,46] suggested the usage of PFSs to solve real problem and make a decision. Garg [47,48] studied the generalized Pythagorean fuzzy information aggregation using Einstein operations and Pythagorean geometric aggregation operations using Einstein t-norm with their applications. Zhang et al. [49] extend the PFSs to solve MCDM issues. Hussain et al. [50] defined the concept of Pythagorean fuzzy soft rough sets. Wang and Garg [51] introduced an algorithm for MADM by Pythagorean fuzzy archimedean norm operations. Recently, Zhan et al. [52] established the definition of CPFRS according to a PF β-neighborhood. They used these notions to solve problems in multi-attribute Pythagorean fuzzy decision making (MAPFDM).
From profounding in the real application, there were some problems not solved by IFSs and PFSs. So, In 2017, Yager [53] demonstrate a new notion to solve these issues in IFSs and PFSs. This notion called the q-rung orthopair fuzzy sets (q-ROFSs) are considered a generalization of PFSs and IFSs. The sum of qth power of membership class and qth power of nonmembership class is in the interval [0, 1] in q-ROFSs. In recent years, q-ROFSs studied and applied more widely in many distinct areas. Yager and Alajlan [54] discussed the relevant characteristics of q-ROFSs. In 2019, there was a new think of q-ROFSs via orbits by Ali [55]. The notions of connection number based q-ROFS is developed by Garg [56]. Especially, Hussain et al. [57] introduced the definition of Cq-ROFRS through the notion of q-ROF β-neighborhood and applied it in multi-attribute q-ROF decision making (MAq-ROFDM). These differences illustrate that 3-PFβCRS is the best approximations among 1-PFβCRS (Zhan et al.'s model), 2-PFβCRS and 4-PFβCRS.
The main aim of this study is to improve Zhan et al.'s [52] and Hussain et al.'s [57] studies, by overgrowing the lower approximation and diminish the upper approximation of the proposed methods. Thus, we set the meaning of PF complementary β-neighborhood and hence we present a new type of CPFRS model . To generalize this study, we obtain two new PF β-neighborhoods by joining PF β-neighborhood and PF complementary β-neighborhood and then two new CPFRS models are built (3-PFβCRS and 4-PFβCRS). The properties of these models are also discussed. Further, the relationships through the Zhan et al.'s model and our models (i.e., 1-PFβCRS, 2-PFβCRS, 3-PFβCRS and 4-PFβCRS) are investigated. Moreover, we put forward the definition of q-ROF complementary βneighborhood and using it to introduce a novel model of Cq-ROFRS (2-q-ROFβCRS). Hence, we merge the definitions of q-ROF β-neighborhood and q-ROF complementary β-neighborhood to generate two new kinds of q-ROF βneighborhood (3-q-ROFβCRS and 4-q-ROFβCRS). We use these kinds to give two other paradigms of Cq-ROFRS and also study relevant properties. Relationships between Hussain et al.'s model and our's (i.e., 1-q-ROFβCRS, 2q-ROFβCRS, 3-q-ROFβCRS and 4-q-ROFβCRS) are also given. We put forward some examples to explain the differences between these two approaches which conclude that 3-PFβCRS is the best among others (i.e., 1-PFβCRS, 2-PFβCRS and 4-PFβCRS) and 3-q-ROFβCRS is more accurate than others (i.e., 1-q-ROFβCRS, 2-q-ROFβCRS and 4-q-ROFβCRS). Finally, we apply the presented work to solve MAPFDM and MAq-ROFDM problems.
The rest of the article is as follows. The basic notions about PFSs and q-ROFSs are set in "Preliminaries". "PF complementary β-neighborhood and three novel kinds of CPFRS" constructs three new models of CPFRS by means of PF β-neighborhood and PF complementary β-neighborhood. We determine the definition of q-ROF complementary βneighborhood and use it to build three models of Cq-ROFRS with the help of q-ROF β-neighborhood in "q-ROF complementary β-neighborhood and three novel kinds of Cq-ROFRS". In "Decision-making approach using PFβCAS", we give numerical examples via our methods to explain the theoretical studies. We put forward the main goals of this study in "Conclusion".

Preliminaries
In the following, we supply a short scanning of some concepts consumed over the paper.
Definition 2 [49] Consider a PFS P ∈ Λ, define the grade of indeterminacy of u ∈ Λ to P as follows.
Zhan et al. [52] introduced the notion of CPFRS and put forward the definition of PF β-neighborhood as indicated below.
Then the pair L P F 1 (X ), U P F 1 (X ) is called the 1-PFβCRS. Definition 8 Consider Θ be a universe. For every u ∈ Θ, if we have a membership function μ X : Θ → [0, 1] and a non-membership function ν X : Θ → [0, 1]. Define the q-ROFS X as indicated below.
Definition 9 [53] Consider a q-ROFS X ∈ Θ, define the grade of indeterminacy of u ∈ Θ to X as follows.
Then the pair L q-ROF 1 (X ), U q-ROF 1 (X ) is called the 1-q-

PF complementaryˇ-neighborhood and three novel kinds of CPFRS
This section's objective is to investigate the definition of PF complementary β-neighborhood and then construct three new types of a CPFRS model. Further, we discuss the relationships via these models.

Example 1 Consider a PFβCAS
The rating corresponding to them are computed and listed in Table 2. However, the values of their complementˆ P (0.7,0.4) u is given in Table 3.
Then we get the following result. and Definition 17 Consider a PFβCAS (Λ, Υ ) and PFN β = (ϑ β , ζ β ). Thus the rough and precision degrees of X ∈ PF(Λ) are respectively seen as follows.
Example 3 Consider Examples 1 and 2. Then the following results hold.
Example 4 Consider Example 1. We compute 1 ¶ β u as set in Table 4.
Then the pair L PF 3 (X ), U PF 3 (X ) is called the 3-PFβCRS.

Example 5
Consider Examples 1 and 2. Then we have the following outcomes. Now, we obtain the following theorem which has the properties of the 3-PFβCRS model. The proof of this theorem is straightforward from Definition 19 and Theorem 1, so, we omit this proof.
Example 6 Consider Examples 1 and 2. Then the following results hold.
Example 7 Consider Example 1. We compute 2 ¶ β u as follows in Table 5.
Then the pair L PF 4 (X ), U PF 4 (X ) is called the 4-PFβCRS.
Theorem 3 Consider a PFβCAS (Λ, Υ ). Then, we have the following properties Proof The proof is similar to Theorem 1 using Definition 22.
Example 9 Consider Examples 1 and 2. Then the following results hold.

Relationships between the proposed methods
Below, we proceed to explain some relationships among these kinds.
Proof It is clear from Definitions 7, 16, 19 and 22.
Based on the above remark and Example 2, the two model 1-PFβCRS and 2-PFβCRS are distinct from some of them.

q-ROF complementaryˇ-neighborhood and three novel kinds of Cq-ROFRS
To treat the insufficiency in PF, Yager's set the notion of q-ROF. In this section, we define the q-ROF complementary β-neighborhood and then we present three models of Cq-ROFRS. In addition, we study the relationships between us and the last method by Hussian et al. [56].
Thus, we can obtain their complement valuesˆ Q (0.8,0.7) u as in Table 8.
Example 12 Consider Example 11, then we have the following outcomes.
Example 13 Consider Example 10 and we can obtain the following results for 1 Q β u as listed in Table 9. Definition 28 Consider a q-ROFβCAS (Θ, ) and β = (μ β , ν β ). For all u ∈ Θ and X ∈ q − RO F(Θ). Define Then, the pair L

Example 14
Consider Example 10 and 11. Then we have the following results.
Proof It is obvious.
Example 15 Consider Example 14, then we have the following outcomes.
Example 16 Consider Example 10 and we can obtain the following results for 2 Q β u as listed in Table 10. . .
Example 18 Consider Example 17, then we have the following outcomes.

Relationships between the proposed methods
Next, we explain some relationships among these kinds.
Proof The proof is clear from Definitions 28 and 31.
(1) L Based on the above remark and Example 11, the two model 1-q-ROFβCAS and 2-q-ROFβCAS are distinct from some of them.

Decision-making approach using PFˇCAS
Now, we illustrate the proposed theoretical study with a real example to clarify how this study is beneficial for the real problems.

Description and process
Assume that Λ = {u r : r = 1, ..., k} is the set of alternatives, the m main attributes Υ = { Δ i : i = 1, 2, ..., m}. Then Υ (u r ) = (x r j , y r j ) indicates the experts assessment outcome relevant to the alternatives u r and the attribute Δ i . Also, we suppose that PFN β = (ϑ β , ζ β ). Thus (Λ, Υ ) is a PFβCAS. Therefore, by using the proposed covering method, we set up the following steps to solve problems in MAPFDM.
Step 2: Counting the adequate distances E ↑ and E ↓ as follows: and where T j = (T 1 , T 2 , ..., T m ) is the weight vector such that m j=1 T j = 1. And if we have two PFNs P 1 = (ϑ P 1 , ζ P 1 ) and P 2 = (ϑ P 2 , ζ P 2 ), then E(P 1 , Step 3: Based on the presented knowledge, calculate the lower and upper approximation of X using 3-PFβCRSs as the following equations.
Step 4: . Calculate the sorting function of the MAPFDM problem as follows.
and hence sorting the alternatives.
The following algorithm is established from the above data and it put forward in Algorithm 1.
Step 1: In the set of attributes, an expert analyses each alternative and provides its conclusions with relevant values that are concise in Table 11.
Step 2: Expert gives the following results, according to the significance of these five attributes.  Step 4: Using 3-PFβCRSs, compute the lower and upper approximation as the following results.
First, we calculate the PF β-neighborhoods as follows.
Moreover, the 1 ¶ β u neighborhood as follow in Table 14. Then using such information, we can obtain the results as follows.

Description and process
Assume that Θ = {u r : r = 1, ..., k} is the set of alternatives, the m main attributes = { δ i : i = 1, 2, .., m}. Then E in andÊ out are the experts assessment outcomes relevant to the alternatives u r and the attribute δ i by μ ri and ν ri . Thus δ i (u r ) = (μ ri , ν ri ) is q-ROFN and represents by the following matrix, whereÊ in is the membership grade μ ri andÊ out is the membership grade ν ri .
So, (Θ, ) is a q-ROFβCAS. Therefore, we give the following steps to solve MADM problems through presented method on a Cq-ROFRS.
Step 1: Construct q-ROF plus ideal and q-ROF minus ideal as the following formulas. (42) and where and denotes to "max" and "min", respectively, and S is the score function. If we have X = (μ X , ν X ), then S(X ) = 1 2 1 + μ q X − ν q X and q ≥ 1.
Step 2: Counting the adequate distances D ⊕ and D as follows: where Step 3: Compute the lower and upper approximation of X using 4-q-ROFβCRSs as the following equations.
and hence sorting the alternatives.
The following algorithm is established from the above data and it put forward in Algorithm 2.
Step 4: The lower and upper approximation of D using 3-q-ROFβCRSs are calculated as the following.
First, we investigate the q-ROF β-neighborhood and q-ROF complementary β-neighborhood as established, respectively, in Tables 16 and 17. Now, we can calculate 2 Q β u as the following Table 18. Then the order of these candidates is u 2 ≈ u 3 > u 4 > u 1 > u 5 , thus the second candidate is proper for this job.

Comparative analysis
The goal of this part is to explain the differences between our proposed study and the previous work. We split our vision into two parts, that is, CPFRS and Cq-ROFRS, respectively.
(1) The prime objective of the given method of CPFRS is capable of promoting the lower approximation and minimizing the upper approximation of the former investigation by Zhan's in [52] as apparent in Examples 2, 5 and 8. To state the rapprochements through different processes, that is Yager's process [45], Zhang's process [49], Zhan's process [52] and our process, the classification score of these decision-making samples are recorded in Table 19 and 20 . Also, we demonstrate Fig. 1, to show the values of ordering variables between Zhan's model and our model. From this figure, you can see that our outcomes are greater than Zhan's outcomes, and also the first candidate is the suitable one among all in the two presented models. Tables 19 and 20 interpreted that the optimal decision is the same alternative u 1 for the four processes i.e., Yager's process [45] Zhang's process [49] Zhan's process [52] Our process and our's), that is make our approach is feasible and effective. Figures 2 and 3 states the another way to show that the variances through Zhan's method [52] and our method. Figure 2 contains two parts. The left part illustrates that our membership ϑ of the lower approximation is higher than in Zhan et al. [52]. On the other hand, the right part shows that our non-membership ζ of the lower approximation is lower than in Zhan et al. [52]. This means that our lower is better than Zhan's lower from the view of raising the lower approximation which makes our approach is suitable than others. Figure 3 also have two figures. The left one explained that the our membership ϑ of the upper approximation is lower than in Zhan et al. [52]. In contrast, the other figure clarifies that the our non-membership ζ of the upper approximation is higher than in Zhan et al. [52]. This shows that our upper is better than Zhan's upper from the view of lowering the upper approximation that makes our model is more appropriate than others.
To sum, these two images mean that our lower approximation is better than Zhan-lower and our upper approximation is lower than Zhan-upper which makes our proposed study is more appropriate than others. Therefore, the presented method is reliable, and effective and is considered as a generalization of the Zhan's method. (2) q-ROF is considered as the generalization of PFS (where q = 2) and IFS (where q = 1). The presented model of Cq-ROFRS is the natural extension to Hussain et al. [57] which investigates the novel covering method under the  Our model u 2 ≈ u 3 > u 4 > u 1 > u 5 notion of q-ROF β-neighborhoods. Here, we present the definition of q-ROF complementary β-neighborhoods and combine these two types of neighborhoods to investigate two other kinds of q-ROF β-neighborhoods. Hence, we used these types to construct three novel kinds of Cq-ROFRS model. Now, we build the Tables 21 and 22 to demonstrate the outcomes between Hussain et al. [57] and our's.
From Tables 21 and 22, we can say that the best decree is the second candidate u 2 among two different approaches (i.e., Hussain et al. [57] and our's). This means that the decision is the same alternative. This proofs that the proposed model is effective and reliable. Figures 4 and 5 states the another way to show that the variances through Hussain's method [52] and our method. Figure 4 splits into two parts. The left split illustrates that the μ of the lower approximation is higher than in Hussain et al. [57]. On the other hand, the right split shows that the ν of the lower approximation is lower than in Hussain et al. [57]. Figure 5 also have two images. These figures explained the differences between μ and ν in Hussain et al. [57] and ours.
In particular, the above two images mean that our lower approximation is better than Hussain-lower and our upper approximation is lower than Hussain-upper. Therefore, the presented method is reliable, and effective and is considered as a generalization of the Hussain method. To simplify our studies and the relations between our presented models in CPFRS and Cq-ROFRS, we give the following Fig. 6 that explained in briefly our vision in this study. This Figure clarifies that the Cq-ROFRS is a generalization of CPFRS.

Conclusion
The main purpose of the proposed article is to improve Zhan et al's model in [52] and Hussain et al.'s model in [57]. The chief investigation of the study is summarized as follows.
(1) We extend the study of CPFRS through PF complementary β-neighborhood. By joining the concept of PF βneighborhood and PF complementary β-neighborhood, we obtain three novel methods on a CPFRS. Also, we investigate the comparisons between the Zhan et al.'s process and our process. These differences illustrate that 3-PFβCRS is the best approximations among 1-PFβCRS CPFRS is more accurate than CIFRS because it deals with membership and nonmembership degrees that their square sum is less than or equal to 1. To treat the limitation on CPFRS, Cq-ROFRS is considered as a generalization of CPFRS, CIFRS, and CFRS because of the values of q. Further studies will