Some New Covering-Based Multigranulation Fuzzy Rough Sets and Corresponding Application in Multicriteria Decision Making

Multigranulation rough set theory is an important tool to deal with the problem ofmulticriteria information system.*e notion of fuzzy β-neighborhood has been used to construct some covering-based multigranulation fuzzy rough set (CMFRS) models through multigranulation fuzzy measure. But the β-neighborhood has not been used in these models, which can be seen as the bridge of fuzzy covering-based rough sets and covering-based rough sets. In this paper, the new concept of multigranulation fuzzy neighborhood measure and some types of covering-based multigranulation fuzzy rough set (CMFRS) models based on it are proposed. *ey can be seen as the further combination of fuzzy sets: covering-based rough sets and multigranulation rough sets. Moreover, they are used to solve the problem of multicriteria decision making. Firstly, the definition of multigranulation fuzzy neighborhoodmeasure is given based on the concept of β-neighborhood. Moreover, four types of CMFRSmodels are constructed, as well as their characteristics and relationships.*en, novel matrix representations of them are investigated, which can satisfy the need of knowledge discovery from large-scale covering information systems. *e matrix representations can be more easily implemented than set representations by computers. Finally, we apply them to manage the problem of multicriteria group decision making (MCGDM) and compare them with other methods.


Introduction
Covering-based rough sets [1,2] enriched classical rough sets [3,4] to deal with the type of covering data. In fact, this data is the collection of subsets of a universe (the set of all objects). Based on it, many covering-based rough set models [5,6] were constructed to manage some practical problems, such as attribute reduction [7,8] and rule synthesis [9,10].
As the same with rough set theory, Zadeh's fuzzy set theory [11] can also deal with the problem of imperfect knowledge.
ere are many researchers who investigated their connection and distinction [12,13]. To extend fuzzy set theory and improve its application, several generalizations of fuzzy sets have been proposed, such as intuitionistic fuzzy sets [14], soft set [15], bipolar-valued fuzzy sets [16,17], hesitant bipolar-valued fuzzy soft sets [18], a historical account of types of fuzzy sets [19], and single-valued neutrosophic sets [20].
Recently, the collection of subsets of a universe has been generalized to a collection of fuzzy subsets of a universe. at is to say, coverings have been generalized to fuzzy covering under a special condition. Hence, covering-based rough sets have been extended to covering-based fuzzy rough sets under fuzzy coverings, such as fuzzy covering tough sets [21] and generalized fuzzy covering tough sets [22,23]. But in the condition of fuzzy covering, there is a constraint "1". en, Ma [24] proposed the concept of fuzzy β-covering approximation space (β-FcaS) by extending "1" to a parameter β. Motivated by Ma's work, many significative researches have been done. For example, Zhang and Wang [25] studied the β-FcaS further, D'eer et al. [26,27] investigated relationships among fuzzy neighborhood operators. Yang and Hu [28,29] constructed other covering-based fuzzy rough set models. Zhang et al. [30] used fuzzy measures and Choquet integrals in fuzzy covering rough sets for solving the decision making.
However, all these above works have been done under a β-FcaS. Hence, Zhan et al. [31] extended a fuzzy β-covering to n-fuzzy β-coverings. ey also proposed two important kinds of neighborhoods, which are the fuzzy β-neighborhood N β C(x) and the β-neighborhood N β C(x) of x. Zhan et al. [31] used the concept of fuzzy β-neighborhood to establish two types of covering-based multigranulation fuzzy rough sets (CMFRSs). Moreover, based on the concept of fuzzy β-neighborhood, they give the notion of multigranulation fuzzy measure as follows: . (1) en, it was used to construct another four types of CMFRSs. But it is time-consuming to compute them when a large-scale covering is given. On the other hand, the concept of β-neighborhood is not used in [31]. Since the concept of β-neighborhood can be used for any fuzzy β-covering to induce a covering, it can be seen as the bridge of fuzzy covering-based rough sets and covering-based rough sets.
at is to say, some methods in covering-based rough sets can be used in fuzzy covering-based rough sets from the viewpoint of the concept of β-neighborhood. In this paper, we use the concept of β-neighborhood to present the notion of multigranulation fuzzy neighborhood measure in this paper. Moreover, we construct some types of CMFRSs by it. e motivation and content of this paper are shown as follows: (i) To solve the problem of time-consuming for computing the multigranulation fuzzy measure when a large-scale covering is given, the notion of multigranulation fuzzy neighborhood measure is given by the concept of β-neighborhood as follows: It can be seen as the complement of the existing work and the first motivation of this paper. Based on this new notion, four types of CMFRS models are constructed. ey are covering-based optimistic (or pessimistic) multigranulation neighborhood fuzzy rough set models and α-covering-based optimistic (or pessimistic) multigranulation neighborhood fuzzy rough set models, where α ∈ [0, 1]. (ii) en, we investigate them from the following three ways: characteristics, relationships, and matrix representations. Moreover, we define the matrix representation of all multigranulation fuzzy neighborhood measures, which is symmetric, to study the problem of their matrix representations. e matrix methods can satisfy the need of knowledge discovery. It can well solve the time-consuming problem when a large-scale covering is given, which is the second motivation of this paper. (iii) Finally, as an application, we use them to deal with the problem of multicriteria decision making (MCGDM) and compare with other methods. e rest of this paper is organized as follows: Section 2 shows some definitions about fuzzy covering-based rough sets. In Section 3, we give covering-based (and α-coveringbased) optimistic multigranulation neighborhood fuzzy rough set models and their characteristics in Section 3.1 and covering-based (and α-covering-based) pessimistic multigranulation neighborhood fuzzy rough set models and their characteristics in Section 3.2. Section 3.3 gives their relationships. In Section 4, we use matrix approaches to compute these CMFRS models. Section 5 gives an application to show that these CMFRS models can be used to solve the problem of MCGDM. Section 6 is the conclusions and further work.

Basic Definitions
is section shows several fundamental definitions related to fuzzy covering-based rough sets. It is supposed that U is a nonempty and finite set called universe.
To combine covering-based rough sets and fuzzy sets, Ma [24] presented the notion of β-FcaS.
In a β-FcaS, there are two types of neighborhoods, which are shown in Definition 2.
Definition 2 (see [24]). Let (U, C) be a β-FcaS. For any x ∈ U, the fuzzy β-neighborhood N of x are defined as follows: en, Zhan et al. [31] generalized a fuzzy β-covering to n-fuzzy β-coverings. We call Γ � C 1 , . . . , C n a n-fuzzy β-coverings of U, where C i is a fuzzy β-covering of U for any i ∈ 1, 2, . . . , n { }. en, two types of covering-based multigranulation rough sets are proposed.

Some Types of CMFRSs
In this section, we propose the notion of multigranulation fuzzy neighborhood measure under β-neighborhoods. en, four types of CMFRSs are constructed through the notion of multigranulation fuzzy neighborhood. Finally, some relationships among them are investigated.

Covering-Based Optimistic Multigranulation Fuzzy
Rough Sets. Two types of covering-based optimistic multigranulation fuzzy rough sets are proposed in this section. Firstly, the notion of multigranulation fuzzy neighborhood measure is presented in Definition 4.

By Definition 4, for any
Hence, N 0.6 Tables 5 and 6,  respectively. erefore, D 0.6 Tables 7 and 8, respectively. en, the first type of CMFRSs is presented in Definition 5.
In Definition 5, we call A a covering-based optimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it a covering-based optimistic multigranulation neighborhood definable set. Proposition 1. Let Γ � C 1 , . . . , C n be an n-fuzzy β-coverings of U. en, for any A, B ∈ F(U), the following statements hold: For any x ∈ U and i � 1, 2, . . . , n, D

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.
On the other hand, that is, SR en, we only need to prove that . According to (1L), we can that is, SR en, we only need to prove that SR Example 3 (continued from Example 1). By Example 2, Hence, the following hold: , and the same applies for B and C.
Finally, the second type of covering-based optimistic multigranulation fuzzy rough sets is presented in Definition 6.
In Definition 6, we call A an α-covering-based optimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it an α-covering-based optimistic multigranulation neighborhood definable set.
(5LH) It is immediate by Definition 6.
Example 5 (continued from Example 1). In Example 4, we Hence, the following hold: , and the same ap-

Covering-Based Pessimistic Multigranulation Fuzzy
Rough Sets. Based on the covering-based optimistic multigranulation fuzzy rough sets in Section 3.1, we present two types of covering-based pessimistic multigranulation fuzzy rough sets in this section. Firstly, the first type is presented in Definition 7.
In Definition 7, we call A a covering-based pessimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it a covering-based pessimistic multigranulation neighborhood definable set.
Some characteristics of the PMNFLAO SR P

Proposition 3.
Let Γ � C 1 , . . . , C n be an n-fuzzy β-coverings of U. en, for any A, B ∈ F(U), we have the following statements: On the other hand, that is, SR Journal of Mathematics 9 en, we only need to prove that SR P that is, SR en, we only need to prove that SR Example 7 (continued from Example 1). By Example 6, Journal of Mathematics Hence, the following hold: , and the same Finally, the second type of covering-based pessimistic multigranulation fuzzy rough set model is presented in Definition 8.
In Definition 8, we call A an α-covering-based pessimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it an α-covering-based pessimistic multigranulation neighborhood definable set.
Hence, the following hold: , and the same ap-

Relationships between CMFRSs.
In this section, we give some relationships between CMFRSs. Firstly, the relationship between covering-based optimistic and pessimistic multigranulation fuzzy rough sets is proposed.
. . , C n be an n-fuzzy β-coverings of U. en, for any A ∈ F(U), Proof. By Definitions 5 and 7 and Proposition 1, it is immediate.
Example 10 (continued from Example 1). In Examples 2 and 6, we know that A � (0.8/x 1 ) Hence, en, the relationship between α-covering-based optimistic and pessimistic multigranulation fuzzy rough sets is given.

Theorem 2.
Let Γ � C 1 , . . . , C n be an n-fuzzy β-coverings of U. en, for any A ∈ F(U), Proof. By Definitions 6 and 8 and Proposition 3, it is immediate.

Matrix Approaches for CMFRSs
In this section, we mainly use matrix approaches to calculate approximation operators of CMFRS models presented in In [24], several matrices and matrix operations are presented in fuzzy covering-based rough sets. Hence, we consider the statement that n � 1 firstly; that is, C 1 � · · · � C n � C.
Definition 9 (see [24]). Let C � C 1 , C 2 , . . . , C m be a fuzzy β-covering of U � x 1 , . . . , x n . We call M C � (m ij ) n×m � (C j (x i )) n×m a matrix representation of C and call the Journal of Mathematics 13 en, Let A � (a ij ) m×l and B � (b jk ) l×n be two matrices. en, Lemma 1 (see [24]). Let C � C 1 , C 2 , . . . , C m be a fuzzy Example 13 (continued from Example 1). Based on Example 12, we have   is presented as follows.

Proposition 5.
Let C � C 1 , C 2 , . . . , C m be a fuzzy β-cov- In what follows, the definition of the matrix representation of all multigranulation fuzzy neighborhood measures is presented.
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Proof. By Definition 5 and Proposition 8, it is immediate.
Example 17 (continued from Example 1). Based on Example 16, we have
Proof. By Definition 6 and Proposition 9, it is immediate.
where ⊳ and ⊲ are first than ⊙ and ∘ , respectively. . en, To evaluate the efficiency of our matrix method for computing covering-based multigranulation fuzzy rough approximation operators, we choose three datasets from UCI machine learning repository. eir computational time is shown in Table 9.
ey are carried out on a personal computer with 64-bit Windows 7, Intel® Core ™ i5-4460 CPU @3.20 GHz, and 8 GB memory. e programming language is MATLAB r2016a. Since the method of attribute reduction is always presented by rough approximation operators, our matrix method for computing covering-based multigranulation fuzzy rough approximation operators can be used to solve this problem in the future.

An Application in MCGDM
In this section, these CMFRS models are applied to manage the problem of MCGDM that is stated in [31]. Zhan et al. [31] presented an algorithm for the problem of MCGDM based on the covering-based optimistic multigranulation fuzzy rough set model. We show it in Algorithm 1, where Γ is an n-fuzzy β-coverings of U and μ ∈ F(U). On the other hand, Algorithm 3 is proposed based on Definition 6 to solve the problem of MCGDM.
To show the validity of our method, we compare Algorithms 1-3 in the following example.
Example 21. Let U � x 1 , x 2 , . . . , x 5 be a set of patients, β � 0.6 be the critical value, and V � y 1 , y 2 , y 3 , y 4 be a set of five main symptoms for the disease A. Suppose that the doctor X invites 2 experts R i (i � 1, 2) to evaluate every patient x k (k � 1, 2, . . . , 5). en, the expert R i believes that the patient x k has a value of the symptom y j (j � 1, . . . , 4), denoted as C ij (x k ). For the patient x k , there exists at least one symptom y j such that the symptom value ey are shown in Tables 1 and 9, respectively. Suppose that the doctor X gives the possible value μ of the disease A of every patient as μ � (0.8/x 1 ) + (0.4/ x 2 ) + (0.6/x 3 ) + (0.2/x 4 ) + (0.5/x 5 ). e doctor X wants to decide whether or not the patients x k ∈ U have the disease A.
Step 2: by Definition 3, computing R O n i�1 C i (μ) and R O n i�1 C i (μ), respectively.
Step 4: Step 5: obtaining the ordering by the decision principle.
ALGORITHM 1: An algorithm for the problem of MCGDM in [31]. 22 Journal of Mathematics Step 5: we get x 4 ≺x 2 ≺x 5 ≺x 3 ≺x 1 . at is to say, x 1 is more likely to be sicken for the disease A en, for Algorithm 2: Step 1: D 0.6 C 1 and D 0.6 C 2 are shown in Tables 7 and 8,   respectively Step 2: SR O Step 5: we get x 4 ≺x 2 ≺x 5 ≺x 3 ≺x 1 . at is to say, x 1 is more likely to be sicken for the disease A For Algorithm 3: Step 1: D 0.6 C 1 and D 0.6 C 2 are shown in Tables 7 and 8,   respectively Step 2: ζ � 0. Step 5: we get x 4 ≺x 2 ≈ x 5 ≺x 1 ≺x 3 . at is to say, x 3 is more likely to be sicken for the disease A Finally, all results about are shown in Table 10 and Figure 1. From Table 10 and Figure 1, we find that the results Input: a multigranulation fuzzy decision information system (U, Γ, β, μ). Output: the rank of all alternatives.
Step 4: Step 5: obtaining the ordering by the decision principle.
ALGORITHM 2: An algorithm for the problem of MCGDM.
Step 3: by Definition 3, computing Step 4: Step 5: obtaining the ordering by the decision principle.  x 4 ≺x 2 ≺x 5 ≺x 3 ≺x 1 x 1 Algorithm 2 x 4 ≺x 2 ≺x 5 ≺x 3 ≺x 1 x 1 Algorithm 3 x 4 ≺x 2 ≈ x 5 ≺x 1 ≺x 3 x 3 of Algorithms 1 and 2 are same. at is to say, the patient x 1 is more likely to be sicken for the disease A. Hence, our methods are effective. On the other hand, x 3 is more likely to be sicken for the disease A by Algorithm 3. It is different with other methods. So, it will be a new choice for the decision maker.

Conclusions
is paper constructs some types of CMFRS models through fuzzy neighborhood measures. Corresponding fast calculation methods based on matrices are also proposed. Moreover, as an application, they are used to solve the problem of MCGDM. e main contributions of this paper are concluded as follows: (1) It is interesting to give the notion of multigranulation fuzzy neighborhood measure based on the concept of β-neighborhood. It can be seen as the complement of the existing work and also the dual of multigranulation fuzzy measure which is presented in [31]. ese new types of CMFRS models can be seen as the combination of covering-based rough sets, multigranulation rough sets and fuzzy sets. ese models are the complement of the existing rough set models. ey can help us to deal with multigranulation fuzzy information well.
(2) It is complicated and time-consuming to use set approaches to compute these models in a large cardinal fuzzy β-covering approximation space. e matrix methods can satisfy the needs of knowledge discovery. By using matrix approaches in CMFRS models, calculations can be easily implemented by computers. It can well solve the time-consuming problem when a large-scale covering is given. (3) According to Algorithm 3, we find that the patient x 3 is another patient that is more likely to be sicken for the disease A in Example 21, which is different from Algorithms 1 and 2. at is to say, it is a new choice for decision maker. erefore, our method is a complement of the existing methods.
In future research, the reduction in covering-based multigranulation fuzzy information systems should be investigated, as well as the matrix calculation approach. e generalizations of fuzzy sets (such as soft multisets [32], picture fuzzy sets [33], and neutrosophic sets [34]) will be combined with the content of this paper. Moreover, it can be used in the problem of MCGDM.
Data Availability e data are available at UCI machine learning repository (http://archive.ics.uci.edu/ml/index.php).

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this study.

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