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

Authors: Zaibin Chang; Lingling Mao
Journal: Journal of Mathematics
 (2021)

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## Abstract

Multigranulation rough set theory is an important tool to deal with the problem of multicriteria information system. The notion of fuzzyβ-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. They 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 neighborhood measure is given based on the concept of β-neighborhood. Moreover, four types of CMFRS models are constructed, as well as their characteristics and relationships. Then, novel matrix representations of them are investigated, which can satisfy the need of knowledge discovery from large-scale covering information systems. The 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.


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## Body

## 1. 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. There 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. That 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”. Then, 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 aβ-FcaS. Hence, Zhan et al. [31] extended a fuzzy β-covering to n-fuzzy β-coverings. They also proposed two important kinds of neighborhoods, which are the fuzzy β-neighborhood N˜C^xβ and the β-neighborhood N¯C^xβ 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)D˜C^iβx,y=∑j=1mN˜C^ixjβx∨N˜C^ixjβy∑j=1mN˜C^ixjβx∧N˜C^ixjβy.Then, 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 ofβ-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. That 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.The 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 ofβ-neighborhood as follows:(2)DC^iβx,y=N¯C^ixβ∩N¯C^iyβN¯C^ixβ∪N¯C^iyβ.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. They are covering-based optimistic (or pessimistic) multigranulation neighborhood fuzzy rough set models andα-covering-based optimistic (or pessimistic) multigranulation neighborhood fuzzy rough set models, where α∈0,1.(ii)
Then, 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. The 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.The rest of this paper is organized as follows: Section2 shows some definitions about fuzzy covering-based rough sets. In Section 3, we give covering-based (and α-covering-based) 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.


## 2. Basic Definitions

This section shows several fundamental definitions related to fuzzy covering-based rough sets. It is supposed thatU 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.Definition 1 (see [24]).
LetFU be the fuzzy power set of U. For each β∈0,1, we call C^=C1,C2,…,Cm, with Ci∈FUi=1,2,…,m, a fuzzy β-covering of U, if ∪i=1mCix≥β for any x∈U. We also call U,C^ a β-FcaS.In aβ-FcaS, there are two types of neighborhoods, which are shown in Definition 2.Definition 2 (see [24]).
LetU,C^ be a β-FcaS. For any x∈U, the fuzzy β-neighborhood N˜C^xβ and the β-neighborhood N¯C^xβ of x are defined as follows:(3)N˜C^xβ=∩C∈C^:Cx≥β,N¯C^xβ=y∈U:N˜C^xβy≥β.Then, 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. Then, two types of covering-based multigranulation rough sets are proposed.Definition 3 (see [31]).
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU, the OMGFRLAO R¯∑i=1nC^iOA and the OMGFRUAO R¯∑i=1nC^iOA are defined as follows: for any x∈U,(4)R¯∑i=1nC^iOAx=∨i=1n∧y∈U1−N˜C^ixβy∨Ay,R¯∑i=1nC^iOAx=∨i=1n∨y∈UN˜C^ixβy∧Ay.
The PMGFRLAOR¯∑i=1nC^iPA and the PMGFRUAO R¯∑i=1nC^iPA are defined as follows: for any x∈U,(5)R¯∑i=1nC^iPAx=∧i=1n∧y∈U1−N˜C^ixβy∨Ay,R¯∑i=1nC^iPAx=∨i=1n∨y∈UN˜C^ixβy∧Ay.


## 3. Some Types of CMFRSs

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


### 3.1. 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 Definition4.Definition 4.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. For any i∈1,2,…,n and x,y∈U, we call(6)DC^iβx,y=N¯C^ixβ∩N¯C^iyβN¯C^ixβ∪N¯C^iyβa multigranulation fuzzy neighborhood measure (MFNM) of x and y under C^i.By Definition4, for any i∈1,2,…,n and x,y∈U, we have DC^iβx,x=1, DC^iβx,y=DC^iβy,x and 0≤DC^iβx,y≤1.Example 1.
Γ=C^1,C^2 is a family of fuzzy β-coverings of U, where β=0.6 and C^i=Ci1,Ci2,Ci3,Ci4 (i=1,2). They are shown in Tables 1 and 2, respectively.
Then,N˜C^1xj0.6 and N˜C^2xj0.6 are shown in Tables 3 and 4, respectively.
Hence,N¯C^1xj0.6 and N¯C^2xj0.6 are shown in Tables 5 and 6, respectively.
Therefore,DC^10.6xj,xk and DC^20.6xj,xk are shown in Tables 7 and 8, respectively.
Then, the first type of CMFRSs is presented in Definition5.Table 1
C^1.
UC11C12C13C14x10.80.40.30.7x20.30.60.50.4x30.70.20.60.8x40.80.70.70.5x50.80.90.40.6Table 2
C^2.
UC21C22C23C24x10.40.70.70.3x20.30.80.60.5x30.80.60.70.4x40.60.30.30.7x50.70.40.60.8Table 3
N˜C^1xj0.6j=1,2,…,5.
Ux1x2x3x4x5N˜C^1x10.60.70.30.70.50.6N˜C^1x20.60.40.60.20.70.9N˜C^1x30.60.30.30.60.50.4N˜C^1x40.60.30.30.20.70.4N˜C^1x50.60.40.30.20.50.6Table 4
N˜C^2xj0.6j=1,2,…,5.
Ux1x2x3x4x5N˜C^2x10.60.70.60.60.30.4N˜C^2x20.60.70.60.60.30.4N˜C^2x30.60.40.30.60.30.4N˜C^2x40.60.30.30.40.60.7N˜C^2x50.60.30.30.40.30.6Table 5
N¯C^1xj0.6j=1,2,…,5.
N¯C^1x10.6N¯C^1x20.6N¯C^1x30.6N¯C^1x40.6N¯C^1x50.6x1,x3,x5x2,x4,x5x3x4x5Table 6
N¯C^2xj0.6j=1,2,…,5.
N¯C^2x10.6N¯C^2x20.6N¯C^2x30.6N¯C^2x40.6N¯C^2x50.6x1,x2,x3x1,x2,x3x3x4,x5x5Table 7
DC^10.6xj,xkj,k=1,2,…,5.
Ux1x2x3x4x5DC^10.6x1,xk11/51/301/3DC^10.6x2,xk1/5101/31/3DC^10.6x3,xk1/30100DC^10.6x4,xk01/3010DC^10.6x5,xk1/31/3001Table 8
DC^20.6xj,xkj,k=1,2,…,5.
Ux1x2x3x4x5DC^20.6x1,xk111/300DC^20.6x2,xk111/300DC^20.6x3,xk1/31/3100DC^20.6x4,xk00011/2DC^20.6x5,xk0001/21Definition 5.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. For any A∈FU, the optimistic multigranulation neighborhood fuzzy lower approximation operator (OMNFLAO) SR¯∑i=1nC^iOA and the optimistic multigranulation neighborhood fuzzy upper approximation operator (OMNFUAO) SR¯∑i=1nC^iOA are defined as follows: for any x∈U,(7)SR¯∑i=1nC^iOAx=∨i=1n∧y∈U1−DC^iβx,y∨Ay,SR¯∑i=1nC^iOAx=∧i=1n∨y∈UDC^iβx,y∧Ay.In Definition5, we call A a covering-based optimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it a covering-based optimistic multigranulation neighborhood definable set.Example 2 (continued from Example1).
LetA=0.8/x1+0.4/x1+0.6/x1+0.2/x1+0.5/x5. Then, SR¯∑i=12C^iOA=0.6667/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5 and SR¯∑i=12C^iOA=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5.Some characteristics of the OMNFLAOSR¯∑i=1nC^iOA and the OMNFUAO SR¯∑i=1nC^iOA are presented as follows.Proposition 1.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU, the following statements hold:(1LH)SR¯∑i=1nC^iOA⊆A⊆SR¯∑i=1nC^iOA(2L) IfA⊆B, then SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB(2H) IfA⊆B, then SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB(3L)SR¯∑i=1nC^iOA∩B=SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB, SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB(3H)SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB, SR¯∑i=1nC^iOA∪B=SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB(4LH)SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′, SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′Proof.
(1LH). For anyx∈U and i=1,2,…,n, DC^iβx,x=1 by Definition 4. Then, Ax=Ax∧DC^iβx,x≤∨y∈UDC^iβx,y∧Ay. Hence, Ax≤∧i=1n∨y∈UDC^iβxy∧Ay=SR¯∑i=1nC^iOAx. On the other hand, ∧y∈U1−DC^iβx,y∨Ay≤1−DC^iβx,x∨Ax=Ax. Hence, ∨i=1n∧y∈U1−DC^iβx,y∨Ay≤Ax. So, SR¯∑i=1nC^iOAx≤Ax. Therefore, SR¯∑i=1nC^iOA⊆A⊆SR¯∑i=1nC^iOA.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(8)SR¯∑i=1nC^iOAx=∨i=1n∧y∈U1−DC^iβx,y∨Ay≤∨i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(9)SR¯∑i=1nC^iOAx=∧i=1n∨y∈UDC^iβx,y∧Ay≤∧i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB.(3L) For eachx∈U,(10)SR¯∑i=1nC^iOA∩Bx=∨i=1n∧y∈U1−DC^iβx,y∨A∩By=∨i=1n∧y∈U1−DC^iβx,y∨Ay∧1−DC^iβx,y∨By=∨i=1n∧y∈U−DC^iβx,y∨Ay∧∨i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iOAx∧SR¯∑i=1nC^iOBx=SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA∩B=SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB. Then, we only need to prove that SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB. According to (1L), we can get that SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA and SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOB. Therefore, we have SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB.(3H) For eachx∈U,(11)SR¯∑i=1nC^iOA∪Bx=∧i=1n∨y∈UDC^iβx,y∧A∪By=∧i=1n∨y∈UDC^iβx,y∧Ay∨DC^iβx,y∧By=∧i=1n∨y∈UDC^iβx,y∧Ay∨∧i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iOAx∨SR¯∑i=1nC^iOBx=SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA∪B=SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB. Then, we only need to prove that SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB. According to (1H), we have SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA and SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOB. Therefore, SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB.(4LH) For eachx∈U,(12)SR¯∑i=1nC^iOA′x=∨i=1n∧y∈U1−DC^iβx,y∨1−Ay=1−∧i=1n∨y∈UDC^iβx,y∧Ay=1−R¯∑i=1nC^iOAx=R¯∑i=1nC^iOA′x,that is,SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′. Then, it can be written as SR¯∑i=1nC^iOA=SR¯∑i=1nC^iOA′′. Therefore, SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′.Example 3 (continued from Example1).
By Example2, SR¯∑i=12C^iOA=0.6667/x1+0.4/x1+0.6/x1+0.2/x1+0.8/x5, and SR¯∑i=12C^iOA=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5 for A=0.8/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5. Let B=0.8/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5 and C=0.9/x1+0.4/x2+0.3/x3+0.7/x4+0.8/x5. Then, SR¯∑i=12C^iOB=0.6667/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5, SR¯∑i=12C^iOB=0.8/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5, SR¯∑i=12C^iOC=0.6667/x1+0.4/x2+0.3/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iOC=0.9/x1+0.4/x2+0.333/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iOA∪C=0.6667/x1+0.4/x2+0.6/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iOA∪C=0.9/x1+0.4/x2+0.6/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iOA∩C=0.6667/x1+0.4/x2+0.3/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iOA∩C=0.8/x1+0.4/x2+0.333/x3+0.333/x4+0.5/x5. Hence, the following hold:(1LH)SR¯∑i=12C^iOA⊆A⊆SR¯∑i=12C^iOA, and the same applies for B and C.(2L)SR¯∑i=12C^iOA⊆SR¯∑i=12C^iOB.(2H)SR¯∑i=12C^iOA⊆SR¯∑i=12C^iOB.(3L)SR¯∑i=12C^iOA∩C=SR¯∑i=12C^iOA∩SR¯∑i=12C^iOC, SR¯∑i=12C^iOA∪C⊇SR¯∑i=12C^iOA∪SR¯∑i=12C^iOC.(3H)SR¯∑i=12C^iOA∩C⊆SR¯∑i=12C^iOA∩SR¯∑i=12C^iOC, SR¯∑i=12C^iOA∪C=SR¯∑i=12C^iOA∪SR¯∑i=12C^iOC.(4LH) SinceA=0.8/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, A′=0.2/x1+0.6/x2+0.4/x3+0.4/x4+0.5/x5. Hence, SR¯∑i=12C^iOA′=0.2/x1+0.6/x2+0.4/x3+0.6667/x4+0.5/x5, and SR¯∑i=12C^iOA′=0.333/x1+0.6/x2+0.4/x3+0.8/x4+0.5/x5. Therefore, SR¯∑i=12C^iOA′=SR¯∑i=1nC^iOA′ and SR¯∑i=12C^iOA′=SR¯∑i=1nC^iOA′.Finally, the second type of covering-based optimistic multigranulation fuzzy rough sets is presented in Definition6.Definition 6.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU and α∈0,1, the optimistic multigranulation neighborhood fuzzy α-lower approximation operator (α-OMNFLAO) R¯∑i=1nC^iO,αA and optimistic multigranulation neighborhood fuzzy α-upper approximation operator (α-OMNFUAO) R¯∑i=1nC^iO,αA are defined as follows: for any x∈U,(13)R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:DC^iβx,y≥α,R¯∑i=1nC^iO,αAx=∧i=1n∨y∈UAy:DC^iβx,y≥α.In Definition6, 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.Example 4 (continued from Example1).
Letα=0.3. Then, R¯∑i=12C^iO,0.3A=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5 and R¯∑i=12C^iO,0.3A=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5.Some characteristics of theα-OMNFLAO R¯∑i=1nC^iO,αA and the α-OMNFUAO R¯∑i=1nC^iO,αA are proposed in the following proposition.Proposition 2.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU and α,α1∈0,1, the following statements hold:(1LH)R¯∑i=1nC^iO,αA⊆A⊆R¯∑i=1nC^iO,αA(2L) IfA⊆B, then R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB(2H) IfA⊆B, then R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB(3L)R¯∑i=1nC^iO,αA∩B=R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αB, R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB(3H)R¯∑i=1nC^iO,αA∩B⊆R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αB, R¯∑i=1nC^iO,αA∪B=R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB(4LH)R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′ and R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′(5LH) Ifα≤α1, then R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,α1A and R¯∑i=1nC^iO,αA⊇R¯∑i=1nC^iO,α1AProof.
(1LH) For anyx∈U,(14)R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:DC^iβx,y≥α≤∨i=1nAx=Ax≤∧i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iO,αAx.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(15)R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:DC^iβx,y≥α≤∨i=1n∨y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iO,αBx,that is,R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(16)R¯∑i=1nC^iO,αAx=∧i=1n∨y∈UAy:DC^iβx,y≥α≤∧i=1n∨y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iO,αBx,that is,R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB.(3L) For eachx∈U,(17)R¯∑i=1nC^iO,αA∩Bx=∨i=1n∧y∈UA∩By:DC^iβx,y≥α=∨i=1n∧y∈UAy∧By:DC^iβx,y≥α=∨i=1n∧y∈UAy:DC^iβx,y≥α∧∨i=1n∧y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iO,αAx∧R¯∑i=1nC^iO,αBx=R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αBx,that is,R¯∑i=1nC^iO,αA∩B=R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αB. Then, we only need to prove that R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB. According to (2L), we have R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA and R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αB. Therefore, R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB.(3H) The proof is similar to (3L). Hence, we omit it.(4LH) For eachx∈U,(18)R¯∑i=1nC^iO,αA′x=∨i=1n∧y∈UA′y:DC^iβx,y≥α=∨i=1n∧y∈U1−Ay:DC^iβx,y≥α=1−∧i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iO,αA′x,that is,R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′. Then, it can be written as R¯∑i=1nC^iO,αA=R¯∑i=1nC^iO,αA′′. Therefore, R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′.(5LH) It is immediate by Definition6.Example 5 (continued from Example1).
In Example4, we have R¯∑i=12C^iO,0.3A=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5 and R¯∑i=12C^iO,0.3A=0.8/x1+0.5/x2+0.8/x3+0.2/x4+0.5/x5. Let B=0.8/x1+0.5/x1+0.7/x1+0.4/x1+0.6/x5 and C=0.9/x1+0.4/x1+0.3/x1+0.7/x1+0.8/x5. Then, SR¯∑i=12C^iO,0.3B=0.6/x1+0.5/x2+0.7/x3+0.4/x4+0.5/x5, SR¯∑i=12C^iO,0.3B=0.8/x1+0.6/x2+0.8/x3+0.5/x4+0.6/x5, SR¯∑i=12C^iO,0.3C=0.3/x1+0.4/x2+0.3/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iO,0.3C=0.9/x1+0.8/x2+0.9/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iO,0.3A∪C=0.6/x1+0.4/x2+0.6/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iO,0.3A∪C=0.9/x1+0.8/x2+0.9/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iO,0.3A∩C=0.3/x1+0.3/x2+0.3/x3+0.2/x4+0.4/x5, and SR¯∑i=12C^iO,0.3A∩C=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5. Hence, the following hold:(1LH)R¯∑i=12C^iO,0.3A⊆A⊆R¯∑i=12C^iO,0.3A, and the same applies for B and C(2L)R¯∑i=12C^iO,0.3A⊆R¯∑i=12C^iO,0.3B(2H)R¯∑i=12C^iO,0.3A⊆R¯∑i=12C^iO,0.3B(3L)R¯∑i=12C^iO,0.3A∩C=R¯∑i=12C^iO,0.3A∩R¯∑i=12C^iO,0.3C, R¯∑i=12C^iO,0.3A∪C⊇R¯∑i=12C^iO,0.3A∪R¯∑i=12C^iO,0.3C(3H)R¯∑i=12C^iO,0.3A∩C=R¯∑i=12C^iO,0.3A∩R¯∑i=12C^iO,0.3C, R¯∑i=12C^iO,0.3A∪C⊇R¯∑i=12C^iO,0.3A∪R¯∑i=12C^iO,0.3C(4LH) SinceR¯∑i=12C^iO,0.3A′=0.2/x1+0.5/x2+0.2/x3+0.6/x4+0.5/x5 and R¯∑i=12C^iO,0.3A′=0.5/x1+0.6/x2+0.4/x3+0.8/x4+0.6/x5, then R¯∑i=12C^iO,0.3A′=R¯∑i=1nC^iO,0.3A′ and R¯∑i=12C^iO,0.3A′=R¯∑i=1nC^iO,0.3A′(5LH) SinceR¯∑i=12C^iO,0.3A′=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5 and R¯∑i=12C^iO,0.3A′=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5, then R¯∑i=12C^iO,0.2A⊆R¯∑i=12C^iO,0.3A and R¯∑i=12C^iO,0.2A⊇R¯∑i=12C^iO,0.3A


### 3.2. Covering-Based Pessimistic Multigranulation Fuzzy Rough Sets

Based on the covering-based optimistic multigranulation fuzzy rough sets in Section3.1, we present two types of covering-based pessimistic multigranulation fuzzy rough sets in this section. Firstly, the first type is presented in Definition 7.Definition 7.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU, the pessimistic multigranulation neighborhood fuzzy lower approximation operator (PMNFLAO) SR¯∑i=1nC^iPA and the pessimistic multigranulation neighborhood fuzzy upper approximation operator (PMNFUAO) SR¯∑i=1nC^iPA are defined as follows: for any x∈U,(19)SR¯∑i=1nC^iPAx=∧i=1n∧y∈U1−DC^iβx,y∨Ay,SR¯∑i=1nC^iPAx=∨i=1n∨y∈UDC^iβx,y∧Ay.In Definition7, we call A a covering-based pessimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it a covering-based pessimistic multigranulation neighborhood definable set.Example 6 (continued from Example1).
SR¯∑i=12C^iPA=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iPA=0.8/x1+0.8/x2+0.6/x3+0.5/x4+0.5/x5.Some characteristics of the PMNFLAOSR¯∑i=1nC^iPA and the PMNFUAO SR¯∑i=1nC^iPA are presented in Proposition 3.Proposition 3.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU, we have the following statements:(1LH)SR¯∑i=1nC^iPA⊆A⊆SR¯∑i=1nC^iPA(2L) IfA⊆B, then SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB(2H) IfA⊆B, then SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB(3L)SR¯∑i=1nC^iPA∩B=SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB, SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB(3H)SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB, SR¯∑i=1nC^iPA∪B=SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB(4LH)SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′, SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′Proof.
(1LH) For anyx∈U and i=1,2,…,n, DC^iβx,x=1 by Definition 4. Then, Ax=Ax∧DC^iβx,x≤∨y∈UDC^iβx,y∧Ay. Hence, Ax≤∨i=1n∨y∈UDC^iβx,y∧Ay=SR¯∑i=1nC^iPAx. On the other hand, ∧y∈U1−DC^iβx,y∨Ay≤1−DC^iβx,x∨Ax=Ax. Hence, ∧i=1n∧y∈U1−DC^iβx,y∨Ay≤Ax. So, we have SR¯∑i=1nC^iPAx≤Ax. Therefore, SR¯∑i=1nC^iPA⊆A⊆SR¯∑i=1nC^iPA.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(20)SR¯∑i=1nC^iPAx=∧i=1n∧y∈U1−DC^iβx,y∨Ay≤∧i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(21)SR¯∑i=1nC^iPAx=∨i=1n∨y∈UDC^iβx,y∧Ay≤∨i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB.(3L) For eachx∈U,(22)SR¯∑i=1nC^iPA∩Bx=∧i=1n∧y∈U1−DC^iβx,y∨A∩By=∧i=1n∧y∈U1−DC^iβx,y∨Ay∧1−DC^iβx,y∨By=∧i=1n∧y∈U1−DC^iβx,y∨Ay∧∧i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iPAx∧SR¯∑i=1nC^iPBx=SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA∩B=SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB. Then, we only need to prove that SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB. According to (1L), we have SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA and also have SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPB. Therefore, there is SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB.(3H) For eachx∈U,(23)SR¯∑i=1nC^iPA∪Bx=∨i=1n∨y∈UDC^iβx,y∧A∪By=∨i=1n∨y∈UDC^iβx,y∧Ay∨DC^iβx,y∧By=∨i=1n∨y∈UDC^iβx,y∧Ay∨∨i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iPAx∨SR¯∑i=1nC^iPBx=SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA∪B=SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB. Then, we only need to prove that SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB. According to (1H), we have SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA and SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPB. Therefore, SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB.(4LH) For eachx∈U,(24)SR¯∑i=1nC^iPA′x=∧i=1n∧y∈U1−DC^iβx,y∨1−Ay=1−∨i=1n∨y∈UDC^iβx,y∧Ay=1−R¯∑i=1nC^iPAx=R¯∑i=1nC^iPA′x,that is,SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′. Then, it can be written as SR¯∑i=1nC^iPA=SR¯∑i=1nC^iPA′′. Therefore, SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′.Example 7 (continued from Example1).
By Example6, SR¯∑i=12C^iPA=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iPA=0.8/x1+0.8/x2+0.6/x3+0.5/x4+0.5/x5. Let 0.8/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5 and C=0.9/x1+0.4/x2+0.3/x3+0.7/x4+0.8/x5. Then, SR¯∑i=12C^iPB=0.5/x1+0.5/x2+0.6667/x3+0.4/x4+0.5/x5, SR¯∑i=12C^iPB=0.8/x1+0.8/x2+0.7/x3+0.5/x4+0.6/x5, SR¯∑i=12C^iPC=0.6667/x1+0.4/x2+0.3/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iPC=0.9/x1+0.9/x2+0.333/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iPA∪C=0.4/x1+0.4/x2+0.6/x3+0.6667/x4+0.6667/x5, SR¯∑i=12C^iPA∪C=0.9/x1+0.9/x2+0.6/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iPA∩C=0.4/x1+0.4/x2+0.3/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iPA∩C=0.8/x1+0.8/x2+0.333/x3+0.5/x4+0.5/x5. Hence, the following hold:(1LH)SR¯∑i=12C^iPA⊆A⊆SR¯∑i=12C^iPA, and the same applies for B and C(2L)SR¯∑i=12C^iPA⊆SR¯∑i=1nC^iPB(2H)SR¯∑i=12C^iPA⊆SR¯∑i=1nC^iPB(3L)SR¯∑i=12C^iPA∩C=SR¯∑i=12C^iPA∩SR¯∑i=12C^iPC, SR¯∑i=12C^iPA∪C⊇SR¯∑i=12C^iPA∪SR¯∑i=12C^iPC(3H)SR¯∑i=12C^iPA∩C⊆SR¯∑i=12C^iPA∩SR¯∑i=12C^iPC, SR¯∑i=12C^iPA∪C=SR¯∑i=12C^iPA∪SR¯∑i=12C^iPC(4LH) SinceSR¯∑i=12C^iPA′=0.2/x1+0.2/x2+0.4/x3+0.5/x4+0.5/x5, SR¯∑i=12C^iPA′=0.6/x1+0.6/x2+0.4/x3+0.8/x4+0.5/x5, then SR¯∑i=12C^iPA′=SR¯∑i=1nC^iPA′ and SR¯∑i=12C^iPA′=SR¯∑i=1nC^iPA′Finally, the second type of covering-based pessimistic multigranulation fuzzy rough set model is presented in Definition8.Definition 8.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU and α∈0,1, the pessimistic multigranulation neighborhood fuzzy α-lower approximation operator (α-PMNFLAO) R¯∑i=1nC^iP,αA and the pessimistic multigranulation neighborhood fuzzy α-upper approximation operator (α-PMNFUAO) R¯∑i=1nC^iP,αA are defined as follows: for any x∈U,(25)R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:DC^iβx,y≥α,R¯∑i=1nC^iP,αAx=∨i=1n∨y∈UAy:DC^iβx,y≥α.In Definition8, 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.Example 8 (continued from Example1).
Letα=0.3. Then, R¯∑i=12C^iP,0.3A=0.4/x1+0.2/x2+0.4/x3+0.2/x4+0.2/x5 and R¯∑i=12C^iP,0.3A=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5.Some characteristics of theα-PMNFLAO R¯∑i=1nC^iP,αA and the α-PMNFUAO R¯∑i=1nC^iP,αA are proposed in the following proposition.Proposition 4.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU and α,α1∈0,1, the following statements hold:(1LH)R¯∑i=1nC^iP,αA⊆A⊆R¯∑i=1nC^iP,αA(2L) IfA⊆B, then R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB(2H) IfA⊆B, then R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB(3L)R¯∑i=1nC^iP,αA∩B=R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αB, R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB(3H)R¯∑i=1nC^iP,αA∩B⊆R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αB, R¯∑i=1nC^iP,αA∪B=R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB(4LH)R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′, R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′(5LH) Ifα≤α1, then R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,α1A and R¯∑i=1nC^iP,αA⊇R¯∑i=1nC^iP,α1AProof.
(1LH) For anyx∈U,(26)R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:DC^iβx,y≥α≤∧i=1nAx=Ax≤∨i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iP,αAx.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(27)R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:DC^iβx,y≥α≤∧i=1n∧y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iP,αBx,that is,R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(28)R¯∑i=1nC^iP,αAx=∨i=1n∨y∈UAy:DC^iβx,y≥α≤∨i=1n∨y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iP,αBx,that is,R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB.(3L) For eachx∈U,(29)R¯∑i=1nC^iP,αA∩Bx=∧i=1n∧y∈UA∩By:DC^iβx,y≥α=∧i=1n∧y∈UAy∧By:DC^iβx,y≥α=∧i=1n∧y∈UAy:DC^iβx,y≥α∧∧i=1n∧y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iP,αAx∧R¯∑i=1nC^iP,αBx=R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αBx,that is,R¯∑i=1nC^iP,αA∩B=R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αB. Then, we only need to prove that R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB. According to (2L), we have R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA and R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αB. Therefore, we get that R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB.(3H) The proof is similar to (3L). Hence, we omit it.(4LH) For eachx∈U,(30)R¯∑i=1nC^iP,αA′x=∧i=1n∧y∈UA′y:DC^iβx,y≥α=∧i=1n∧y∈U1−Ay:DC^iβx,y≥α=1−∨i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iP,αA′x,that is,R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′. Then, it can be written as R¯∑i=1nC^iP,αA=R¯∑i=1nC^iP,αA′′. Therefore, R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′.(5LH) It is immediate by Definition8.Example 9 (continued from Example1).
In Example8, we have R¯∑i=12C^iP,0.3A=0.4/x1+0.2/x2+0.4/x3+0.2/x4+0.2/x5 and R¯∑i=12C^iP,0.3A=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5. Let B=0.8/x1+0.5/x1+0.7/x1+0.4/x1+0.6/x5 and C=0.9/x1+0.4/x1+0.3/x1+0.7/x1+0.8/x5. Then, SR¯∑i=12C^iP,0.3B=0.5/x1+0.4/x2+0.5/x3+0.4/x4+0.4/x5, SR¯∑i=12C^iP,0.3B=0.8/x1+0.8/x2+0.8/x3+0.6/x4+0.8/x5, SR¯∑i=12C^iP,0.3C=0.5/x1+0.4/x2+0.5/x3+0.4/x4+0.4/x5, SR¯∑i=12C^iP,0.3C=0.9/x1+0.9/x2+0.9/x3+0.8/x4+0.9/x5, SR¯∑i=12C^iP,0.3A∪C=0.4/x1+0.4/x2+0.4/x3+0.4/x4+0.4/x5, SR¯∑i=12C^iP,0.3A∪C=0.9/x1+0.9/x2+0.9/x3+0.8/x4+0.9/x5, SR¯∑i=12C^iP,0.3A∩C=0.3/x1+0.2/x2+0.3/x3+0.2/x4+0.2/x5, and SR¯∑i=12C^iP,0.3A∩C=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5. Hence, the following hold:(1LH)R¯∑i=12C^iP,0.3A⊆A⊆R¯∑i=12C^iP,0.3A, and the same applies for B and C(2L)R¯∑i=12C^iP,0.3A⊆R¯∑i=12C^iP,0.3B(2H)R¯∑i=12C^iP,0.3A⊆R¯∑i=12C^iP,0.3B(3L)R¯∑i=12C^iP,0.3A∩C=R¯∑i=12C^iP,0.3A∩R¯∑i=12C^iP,αC, R¯∑i=12C^iP,0.3A∪C⊇R¯∑i=12C^iP,0.3A∪R¯∑i=12C^iP,αC(3H)R¯∑i=12C^iP,0.3A∩C⊆R¯∑i=12C^iP,0.3A∩R¯∑i=12C^iP,αC, R¯∑i=12C^iP,0.3A∪C=R¯∑i=12C^iP,0.3A∪R¯∑i=12C^iP,αC(4LH) SinceR¯∑i=1nC^iP,0.3A′=0.2/x1+0.2/x2+0.2/x3+0.5/x4+0.2/x5 and R¯∑i=1nC^iP,0.3A′=0.6/x1+0.8/x2+0.6/x3+0.8/x4+0.8/x5, then R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′ and R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′.(5LH) SinceR¯∑i=12C^iP,0.5A′=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5 and R¯∑i=12C^iP,0.5A=0.8/x1+0.8/x2+0.6/x3+0.2/x4+0.5/x5, then R¯∑i=1nC^iP,0.3A⊆R¯∑i=1nC^iP,0.5A and R¯∑i=1nC^iP,0.3A⊇R¯∑i=1nC^iP,0.5A


### 3.3. 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.Theorem 1.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A∈FU,(31)SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iOA⊆A⊆SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iPA.Proof.
By Definitions5 and 7 and Proposition 1, it is immediate.Example 10 (continued from Example1).
In Examples2 and 6, we know that A=0.8/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iOA=0.6667/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iOA=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5, SR¯∑i=12C^iPA=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iPA=0.8/x1+0.8/x2+0.6/x3+0.5/x4+0.5/x5. Hence, SR¯∑i=12C^iPA⊆SR¯∑i=12C^iOA⊆A⊆SR¯∑i=12C^iOA⊆SR¯∑i=12C^iPA.Then, the relationship betweenα-covering-based optimistic and pessimistic multigranulation fuzzy rough sets is given.Theorem 2.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A∈FU,(32)SR¯∑i=1nC^iP,αA⊆SR¯∑i=1nC^iO,αA⊆A⊆SR¯∑i=1nC^iO,αA⊆SR¯∑i=1nC^iP,αA.Proof.
By Definitions6 and 8 and Proposition 3, it is immediate.Example 11 (continued from Example1).
In Examples4 and 8, we know that R¯∑i=12C^iO,0.3A=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5, R¯∑i=12C^iO,0.3A=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5, R¯∑i=12C^iP,0.3A=0.4/x1+0.2/x2+0.4/x3+0.2/x4+0.2/x5, and R¯∑i=12C^iP,0.3A=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5. Hence, SR¯∑i=1nC^iP,0.3A⊆SR¯∑i=1nC^iO,0.3A⊆A⊆SR¯∑i=1nC^iO,0.3A⊆SR¯∑i=1nC^iP,0.3A.


## 3.1. 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 Definition4.Definition 4.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. For any i∈1,2,…,n and x,y∈U, we call(6)DC^iβx,y=N¯C^ixβ∩N¯C^iyβN¯C^ixβ∪N¯C^iyβa multigranulation fuzzy neighborhood measure (MFNM) of x and y under C^i.By Definition4, for any i∈1,2,…,n and x,y∈U, we have DC^iβx,x=1, DC^iβx,y=DC^iβy,x and 0≤DC^iβx,y≤1.Example 1.
Γ=C^1,C^2 is a family of fuzzy β-coverings of U, where β=0.6 and C^i=Ci1,Ci2,Ci3,Ci4 (i=1,2). They are shown in Tables 1 and 2, respectively.
Then,N˜C^1xj0.6 and N˜C^2xj0.6 are shown in Tables 3 and 4, respectively.
Hence,N¯C^1xj0.6 and N¯C^2xj0.6 are shown in Tables 5 and 6, respectively.
Therefore,DC^10.6xj,xk and DC^20.6xj,xk are shown in Tables 7 and 8, respectively.
Then, the first type of CMFRSs is presented in Definition5.Table 1
C^1.
UC11C12C13C14x10.80.40.30.7x20.30.60.50.4x30.70.20.60.8x40.80.70.70.5x50.80.90.40.6Table 2
C^2.
UC21C22C23C24x10.40.70.70.3x20.30.80.60.5x30.80.60.70.4x40.60.30.30.7x50.70.40.60.8Table 3
N˜C^1xj0.6j=1,2,…,5.
Ux1x2x3x4x5N˜C^1x10.60.70.30.70.50.6N˜C^1x20.60.40.60.20.70.9N˜C^1x30.60.30.30.60.50.4N˜C^1x40.60.30.30.20.70.4N˜C^1x50.60.40.30.20.50.6Table 4
N˜C^2xj0.6j=1,2,…,5.
Ux1x2x3x4x5N˜C^2x10.60.70.60.60.30.4N˜C^2x20.60.70.60.60.30.4N˜C^2x30.60.40.30.60.30.4N˜C^2x40.60.30.30.40.60.7N˜C^2x50.60.30.30.40.30.6Table 5
N¯C^1xj0.6j=1,2,…,5.
N¯C^1x10.6N¯C^1x20.6N¯C^1x30.6N¯C^1x40.6N¯C^1x50.6x1,x3,x5x2,x4,x5x3x4x5Table 6
N¯C^2xj0.6j=1,2,…,5.
N¯C^2x10.6N¯C^2x20.6N¯C^2x30.6N¯C^2x40.6N¯C^2x50.6x1,x2,x3x1,x2,x3x3x4,x5x5Table 7
DC^10.6xj,xkj,k=1,2,…,5.
Ux1x2x3x4x5DC^10.6x1,xk11/51/301/3DC^10.6x2,xk1/5101/31/3DC^10.6x3,xk1/30100DC^10.6x4,xk01/3010DC^10.6x5,xk1/31/3001Table 8
DC^20.6xj,xkj,k=1,2,…,5.
Ux1x2x3x4x5DC^20.6x1,xk111/300DC^20.6x2,xk111/300DC^20.6x3,xk1/31/3100DC^20.6x4,xk00011/2DC^20.6x5,xk0001/21Definition 5.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. For any A∈FU, the optimistic multigranulation neighborhood fuzzy lower approximation operator (OMNFLAO) SR¯∑i=1nC^iOA and the optimistic multigranulation neighborhood fuzzy upper approximation operator (OMNFUAO) SR¯∑i=1nC^iOA are defined as follows: for any x∈U,(7)SR¯∑i=1nC^iOAx=∨i=1n∧y∈U1−DC^iβx,y∨Ay,SR¯∑i=1nC^iOAx=∧i=1n∨y∈UDC^iβx,y∧Ay.In Definition5, we call A a covering-based optimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it a covering-based optimistic multigranulation neighborhood definable set.Example 2 (continued from Example1).
LetA=0.8/x1+0.4/x1+0.6/x1+0.2/x1+0.5/x5. Then, SR¯∑i=12C^iOA=0.6667/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5 and SR¯∑i=12C^iOA=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5.Some characteristics of the OMNFLAOSR¯∑i=1nC^iOA and the OMNFUAO SR¯∑i=1nC^iOA are presented as follows.Proposition 1.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU, the following statements hold:(1LH)SR¯∑i=1nC^iOA⊆A⊆SR¯∑i=1nC^iOA(2L) IfA⊆B, then SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB(2H) IfA⊆B, then SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB(3L)SR¯∑i=1nC^iOA∩B=SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB, SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB(3H)SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB, SR¯∑i=1nC^iOA∪B=SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB(4LH)SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′, SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′Proof.
(1LH). For anyx∈U and i=1,2,…,n, DC^iβx,x=1 by Definition 4. Then, Ax=Ax∧DC^iβx,x≤∨y∈UDC^iβx,y∧Ay. Hence, Ax≤∧i=1n∨y∈UDC^iβxy∧Ay=SR¯∑i=1nC^iOAx. On the other hand, ∧y∈U1−DC^iβx,y∨Ay≤1−DC^iβx,x∨Ax=Ax. Hence, ∨i=1n∧y∈U1−DC^iβx,y∨Ay≤Ax. So, SR¯∑i=1nC^iOAx≤Ax. Therefore, SR¯∑i=1nC^iOA⊆A⊆SR¯∑i=1nC^iOA.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(8)SR¯∑i=1nC^iOAx=∨i=1n∧y∈U1−DC^iβx,y∨Ay≤∨i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(9)SR¯∑i=1nC^iOAx=∧i=1n∨y∈UDC^iβx,y∧Ay≤∧i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iOB.(3L) For eachx∈U,(10)SR¯∑i=1nC^iOA∩Bx=∨i=1n∧y∈U1−DC^iβx,y∨A∩By=∨i=1n∧y∈U1−DC^iβx,y∨Ay∧1−DC^iβx,y∨By=∨i=1n∧y∈U−DC^iβx,y∨Ay∧∨i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iOAx∧SR¯∑i=1nC^iOBx=SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA∩B=SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB. Then, we only need to prove that SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB. According to (1L), we can get that SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA and SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOB. Therefore, we have SR¯∑i=1nC^iOA∪B⊇SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB.(3H) For eachx∈U,(11)SR¯∑i=1nC^iOA∪Bx=∧i=1n∨y∈UDC^iβx,y∧A∪By=∧i=1n∨y∈UDC^iβx,y∧Ay∨DC^iβx,y∧By=∧i=1n∨y∈UDC^iβx,y∧Ay∨∧i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iOAx∨SR¯∑i=1nC^iOBx=SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOBx,that is,SR¯∑i=1nC^iOA∪B=SR¯∑i=1nC^iOA∪SR¯∑i=1nC^iOB. Then, we only need to prove that SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB. According to (1H), we have SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA and SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOB. Therefore, SR¯∑i=1nC^iOA∩B⊆SR¯∑i=1nC^iOA∩SR¯∑i=1nC^iOB.(4LH) For eachx∈U,(12)SR¯∑i=1nC^iOA′x=∨i=1n∧y∈U1−DC^iβx,y∨1−Ay=1−∧i=1n∨y∈UDC^iβx,y∧Ay=1−R¯∑i=1nC^iOAx=R¯∑i=1nC^iOA′x,that is,SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′. Then, it can be written as SR¯∑i=1nC^iOA=SR¯∑i=1nC^iOA′′. Therefore, SR¯∑i=1nC^iOA′=SR¯∑i=1nC^iOA′.Example 3 (continued from Example1).
By Example2, SR¯∑i=12C^iOA=0.6667/x1+0.4/x1+0.6/x1+0.2/x1+0.8/x5, and SR¯∑i=12C^iOA=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5 for A=0.8/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5. Let B=0.8/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5 and C=0.9/x1+0.4/x2+0.3/x3+0.7/x4+0.8/x5. Then, SR¯∑i=12C^iOB=0.6667/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5, SR¯∑i=12C^iOB=0.8/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5, SR¯∑i=12C^iOC=0.6667/x1+0.4/x2+0.3/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iOC=0.9/x1+0.4/x2+0.333/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iOA∪C=0.6667/x1+0.4/x2+0.6/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iOA∪C=0.9/x1+0.4/x2+0.6/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iOA∩C=0.6667/x1+0.4/x2+0.3/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iOA∩C=0.8/x1+0.4/x2+0.333/x3+0.333/x4+0.5/x5. Hence, the following hold:(1LH)SR¯∑i=12C^iOA⊆A⊆SR¯∑i=12C^iOA, and the same applies for B and C.(2L)SR¯∑i=12C^iOA⊆SR¯∑i=12C^iOB.(2H)SR¯∑i=12C^iOA⊆SR¯∑i=12C^iOB.(3L)SR¯∑i=12C^iOA∩C=SR¯∑i=12C^iOA∩SR¯∑i=12C^iOC, SR¯∑i=12C^iOA∪C⊇SR¯∑i=12C^iOA∪SR¯∑i=12C^iOC.(3H)SR¯∑i=12C^iOA∩C⊆SR¯∑i=12C^iOA∩SR¯∑i=12C^iOC, SR¯∑i=12C^iOA∪C=SR¯∑i=12C^iOA∪SR¯∑i=12C^iOC.(4LH) SinceA=0.8/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, A′=0.2/x1+0.6/x2+0.4/x3+0.4/x4+0.5/x5. Hence, SR¯∑i=12C^iOA′=0.2/x1+0.6/x2+0.4/x3+0.6667/x4+0.5/x5, and SR¯∑i=12C^iOA′=0.333/x1+0.6/x2+0.4/x3+0.8/x4+0.5/x5. Therefore, SR¯∑i=12C^iOA′=SR¯∑i=1nC^iOA′ and SR¯∑i=12C^iOA′=SR¯∑i=1nC^iOA′.Finally, the second type of covering-based optimistic multigranulation fuzzy rough sets is presented in Definition6.Definition 6.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU and α∈0,1, the optimistic multigranulation neighborhood fuzzy α-lower approximation operator (α-OMNFLAO) R¯∑i=1nC^iO,αA and optimistic multigranulation neighborhood fuzzy α-upper approximation operator (α-OMNFUAO) R¯∑i=1nC^iO,αA are defined as follows: for any x∈U,(13)R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:DC^iβx,y≥α,R¯∑i=1nC^iO,αAx=∧i=1n∨y∈UAy:DC^iβx,y≥α.In Definition6, 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.Example 4 (continued from Example1).
Letα=0.3. Then, R¯∑i=12C^iO,0.3A=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5 and R¯∑i=12C^iO,0.3A=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5.Some characteristics of theα-OMNFLAO R¯∑i=1nC^iO,αA and the α-OMNFUAO R¯∑i=1nC^iO,αA are proposed in the following proposition.Proposition 2.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU and α,α1∈0,1, the following statements hold:(1LH)R¯∑i=1nC^iO,αA⊆A⊆R¯∑i=1nC^iO,αA(2L) IfA⊆B, then R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB(2H) IfA⊆B, then R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB(3L)R¯∑i=1nC^iO,αA∩B=R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αB, R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB(3H)R¯∑i=1nC^iO,αA∩B⊆R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αB, R¯∑i=1nC^iO,αA∪B=R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB(4LH)R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′ and R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′(5LH) Ifα≤α1, then R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,α1A and R¯∑i=1nC^iO,αA⊇R¯∑i=1nC^iO,α1AProof.
(1LH) For anyx∈U,(14)R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:DC^iβx,y≥α≤∨i=1nAx=Ax≤∧i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iO,αAx.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(15)R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:DC^iβx,y≥α≤∨i=1n∨y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iO,αBx,that is,R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(16)R¯∑i=1nC^iO,αAx=∧i=1n∨y∈UAy:DC^iβx,y≥α≤∧i=1n∨y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iO,αBx,that is,R¯∑i=1nC^iO,αA⊆R¯∑i=1nC^iO,αB.(3L) For eachx∈U,(17)R¯∑i=1nC^iO,αA∩Bx=∨i=1n∧y∈UA∩By:DC^iβx,y≥α=∨i=1n∧y∈UAy∧By:DC^iβx,y≥α=∨i=1n∧y∈UAy:DC^iβx,y≥α∧∨i=1n∧y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iO,αAx∧R¯∑i=1nC^iO,αBx=R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αBx,that is,R¯∑i=1nC^iO,αA∩B=R¯∑i=1nC^iO,αA∩R¯∑i=1nC^iO,αB. Then, we only need to prove that R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB. According to (2L), we have R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA and R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αB. Therefore, R¯∑i=1nC^iO,αA∪B⊇R¯∑i=1nC^iO,αA∪R¯∑i=1nC^iO,αB.(3H) The proof is similar to (3L). Hence, we omit it.(4LH) For eachx∈U,(18)R¯∑i=1nC^iO,αA′x=∨i=1n∧y∈UA′y:DC^iβx,y≥α=∨i=1n∧y∈U1−Ay:DC^iβx,y≥α=1−∧i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iO,αA′x,that is,R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′. Then, it can be written as R¯∑i=1nC^iO,αA=R¯∑i=1nC^iO,αA′′. Therefore, R¯∑i=1nC^iO,αA′=R¯∑i=1nC^iO,αA′.(5LH) It is immediate by Definition6.Example 5 (continued from Example1).
In Example4, we have R¯∑i=12C^iO,0.3A=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5 and R¯∑i=12C^iO,0.3A=0.8/x1+0.5/x2+0.8/x3+0.2/x4+0.5/x5. Let B=0.8/x1+0.5/x1+0.7/x1+0.4/x1+0.6/x5 and C=0.9/x1+0.4/x1+0.3/x1+0.7/x1+0.8/x5. Then, SR¯∑i=12C^iO,0.3B=0.6/x1+0.5/x2+0.7/x3+0.4/x4+0.5/x5, SR¯∑i=12C^iO,0.3B=0.8/x1+0.6/x2+0.8/x3+0.5/x4+0.6/x5, SR¯∑i=12C^iO,0.3C=0.3/x1+0.4/x2+0.3/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iO,0.3C=0.9/x1+0.8/x2+0.9/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iO,0.3A∪C=0.6/x1+0.4/x2+0.6/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iO,0.3A∪C=0.9/x1+0.8/x2+0.9/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iO,0.3A∩C=0.3/x1+0.3/x2+0.3/x3+0.2/x4+0.4/x5, and SR¯∑i=12C^iO,0.3A∩C=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5. Hence, the following hold:(1LH)R¯∑i=12C^iO,0.3A⊆A⊆R¯∑i=12C^iO,0.3A, and the same applies for B and C(2L)R¯∑i=12C^iO,0.3A⊆R¯∑i=12C^iO,0.3B(2H)R¯∑i=12C^iO,0.3A⊆R¯∑i=12C^iO,0.3B(3L)R¯∑i=12C^iO,0.3A∩C=R¯∑i=12C^iO,0.3A∩R¯∑i=12C^iO,0.3C, R¯∑i=12C^iO,0.3A∪C⊇R¯∑i=12C^iO,0.3A∪R¯∑i=12C^iO,0.3C(3H)R¯∑i=12C^iO,0.3A∩C=R¯∑i=12C^iO,0.3A∩R¯∑i=12C^iO,0.3C, R¯∑i=12C^iO,0.3A∪C⊇R¯∑i=12C^iO,0.3A∪R¯∑i=12C^iO,0.3C(4LH) SinceR¯∑i=12C^iO,0.3A′=0.2/x1+0.5/x2+0.2/x3+0.6/x4+0.5/x5 and R¯∑i=12C^iO,0.3A′=0.5/x1+0.6/x2+0.4/x3+0.8/x4+0.6/x5, then R¯∑i=12C^iO,0.3A′=R¯∑i=1nC^iO,0.3A′ and R¯∑i=12C^iO,0.3A′=R¯∑i=1nC^iO,0.3A′(5LH) SinceR¯∑i=12C^iO,0.3A′=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5 and R¯∑i=12C^iO,0.3A′=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5, then R¯∑i=12C^iO,0.2A⊆R¯∑i=12C^iO,0.3A and R¯∑i=12C^iO,0.2A⊇R¯∑i=12C^iO,0.3A


## 3.2. Covering-Based Pessimistic Multigranulation Fuzzy Rough Sets

Based on the covering-based optimistic multigranulation fuzzy rough sets in Section3.1, we present two types of covering-based pessimistic multigranulation fuzzy rough sets in this section. Firstly, the first type is presented in Definition 7.Definition 7.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU, the pessimistic multigranulation neighborhood fuzzy lower approximation operator (PMNFLAO) SR¯∑i=1nC^iPA and the pessimistic multigranulation neighborhood fuzzy upper approximation operator (PMNFUAO) SR¯∑i=1nC^iPA are defined as follows: for any x∈U,(19)SR¯∑i=1nC^iPAx=∧i=1n∧y∈U1−DC^iβx,y∨Ay,SR¯∑i=1nC^iPAx=∨i=1n∨y∈UDC^iβx,y∧Ay.In Definition7, we call A a covering-based pessimistic multigranulation neighborhood fuzzy rough set; otherwise, we call it a covering-based pessimistic multigranulation neighborhood definable set.Example 6 (continued from Example1).
SR¯∑i=12C^iPA=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iPA=0.8/x1+0.8/x2+0.6/x3+0.5/x4+0.5/x5.Some characteristics of the PMNFLAOSR¯∑i=1nC^iPA and the PMNFUAO SR¯∑i=1nC^iPA are presented in Proposition 3.Proposition 3.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU, we have the following statements:(1LH)SR¯∑i=1nC^iPA⊆A⊆SR¯∑i=1nC^iPA(2L) IfA⊆B, then SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB(2H) IfA⊆B, then SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB(3L)SR¯∑i=1nC^iPA∩B=SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB, SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB(3H)SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB, SR¯∑i=1nC^iPA∪B=SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB(4LH)SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′, SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′Proof.
(1LH) For anyx∈U and i=1,2,…,n, DC^iβx,x=1 by Definition 4. Then, Ax=Ax∧DC^iβx,x≤∨y∈UDC^iβx,y∧Ay. Hence, Ax≤∨i=1n∨y∈UDC^iβx,y∧Ay=SR¯∑i=1nC^iPAx. On the other hand, ∧y∈U1−DC^iβx,y∨Ay≤1−DC^iβx,x∨Ax=Ax. Hence, ∧i=1n∧y∈U1−DC^iβx,y∨Ay≤Ax. So, we have SR¯∑i=1nC^iPAx≤Ax. Therefore, SR¯∑i=1nC^iPA⊆A⊆SR¯∑i=1nC^iPA.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(20)SR¯∑i=1nC^iPAx=∧i=1n∧y∈U1−DC^iβx,y∨Ay≤∧i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(21)SR¯∑i=1nC^iPAx=∨i=1n∨y∈UDC^iβx,y∧Ay≤∨i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iPB.(3L) For eachx∈U,(22)SR¯∑i=1nC^iPA∩Bx=∧i=1n∧y∈U1−DC^iβx,y∨A∩By=∧i=1n∧y∈U1−DC^iβx,y∨Ay∧1−DC^iβx,y∨By=∧i=1n∧y∈U1−DC^iβx,y∨Ay∧∧i=1n∧y∈U1−DC^iβx,y∨By=SR¯∑i=1nC^iPAx∧SR¯∑i=1nC^iPBx=SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA∩B=SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB. Then, we only need to prove that SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB. According to (1L), we have SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA and also have SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPB. Therefore, there is SR¯∑i=1nC^iPA∪B⊇SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB.(3H) For eachx∈U,(23)SR¯∑i=1nC^iPA∪Bx=∨i=1n∨y∈UDC^iβx,y∧A∪By=∨i=1n∨y∈UDC^iβx,y∧Ay∨DC^iβx,y∧By=∨i=1n∨y∈UDC^iβx,y∧Ay∨∨i=1n∨y∈UDC^iβx,y∧By=SR¯∑i=1nC^iPAx∨SR¯∑i=1nC^iPBx=SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPBx,that is,SR¯∑i=1nC^iPA∪B=SR¯∑i=1nC^iPA∪SR¯∑i=1nC^iPB. Then, we only need to prove that SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB. According to (1H), we have SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA and SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPB. Therefore, SR¯∑i=1nC^iPA∩B⊆SR¯∑i=1nC^iPA∩SR¯∑i=1nC^iPB.(4LH) For eachx∈U,(24)SR¯∑i=1nC^iPA′x=∧i=1n∧y∈U1−DC^iβx,y∨1−Ay=1−∨i=1n∨y∈UDC^iβx,y∧Ay=1−R¯∑i=1nC^iPAx=R¯∑i=1nC^iPA′x,that is,SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′. Then, it can be written as SR¯∑i=1nC^iPA=SR¯∑i=1nC^iPA′′. Therefore, SR¯∑i=1nC^iPA′=SR¯∑i=1nC^iPA′.Example 7 (continued from Example1).
By Example6, SR¯∑i=12C^iPA=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iPA=0.8/x1+0.8/x2+0.6/x3+0.5/x4+0.5/x5. Let 0.8/x1+0.5/x2+0.7/x3+0.4/x4+0.6/x5 and C=0.9/x1+0.4/x2+0.3/x3+0.7/x4+0.8/x5. Then, SR¯∑i=12C^iPB=0.5/x1+0.5/x2+0.6667/x3+0.4/x4+0.5/x5, SR¯∑i=12C^iPB=0.8/x1+0.8/x2+0.7/x3+0.5/x4+0.6/x5, SR¯∑i=12C^iPC=0.6667/x1+0.4/x2+0.3/x3+0.7/x4+0.7/x5, SR¯∑i=12C^iPC=0.9/x1+0.9/x2+0.333/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iPA∪C=0.4/x1+0.4/x2+0.6/x3+0.6667/x4+0.6667/x5, SR¯∑i=12C^iPA∪C=0.9/x1+0.9/x2+0.6/x3+0.7/x4+0.8/x5, SR¯∑i=12C^iPA∩C=0.4/x1+0.4/x2+0.3/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iPA∩C=0.8/x1+0.8/x2+0.333/x3+0.5/x4+0.5/x5. Hence, the following hold:(1LH)SR¯∑i=12C^iPA⊆A⊆SR¯∑i=12C^iPA, and the same applies for B and C(2L)SR¯∑i=12C^iPA⊆SR¯∑i=1nC^iPB(2H)SR¯∑i=12C^iPA⊆SR¯∑i=1nC^iPB(3L)SR¯∑i=12C^iPA∩C=SR¯∑i=12C^iPA∩SR¯∑i=12C^iPC, SR¯∑i=12C^iPA∪C⊇SR¯∑i=12C^iPA∪SR¯∑i=12C^iPC(3H)SR¯∑i=12C^iPA∩C⊆SR¯∑i=12C^iPA∩SR¯∑i=12C^iPC, SR¯∑i=12C^iPA∪C=SR¯∑i=12C^iPA∪SR¯∑i=12C^iPC(4LH) SinceSR¯∑i=12C^iPA′=0.2/x1+0.2/x2+0.4/x3+0.5/x4+0.5/x5, SR¯∑i=12C^iPA′=0.6/x1+0.6/x2+0.4/x3+0.8/x4+0.5/x5, then SR¯∑i=12C^iPA′=SR¯∑i=1nC^iPA′ and SR¯∑i=12C^iPA′=SR¯∑i=1nC^iPA′Finally, the second type of covering-based pessimistic multigranulation fuzzy rough set model is presented in Definition8.Definition 8.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. For any A∈FU and α∈0,1, the pessimistic multigranulation neighborhood fuzzy α-lower approximation operator (α-PMNFLAO) R¯∑i=1nC^iP,αA and the pessimistic multigranulation neighborhood fuzzy α-upper approximation operator (α-PMNFUAO) R¯∑i=1nC^iP,αA are defined as follows: for any x∈U,(25)R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:DC^iβx,y≥α,R¯∑i=1nC^iP,αAx=∨i=1n∨y∈UAy:DC^iβx,y≥α.In Definition8, 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.Example 8 (continued from Example1).
Letα=0.3. Then, R¯∑i=12C^iP,0.3A=0.4/x1+0.2/x2+0.4/x3+0.2/x4+0.2/x5 and R¯∑i=12C^iP,0.3A=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5.Some characteristics of theα-PMNFLAO R¯∑i=1nC^iP,αA and the α-PMNFUAO R¯∑i=1nC^iP,αA are proposed in the following proposition.Proposition 4.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A,B∈FU and α,α1∈0,1, the following statements hold:(1LH)R¯∑i=1nC^iP,αA⊆A⊆R¯∑i=1nC^iP,αA(2L) IfA⊆B, then R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB(2H) IfA⊆B, then R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB(3L)R¯∑i=1nC^iP,αA∩B=R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αB, R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB(3H)R¯∑i=1nC^iP,αA∩B⊆R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αB, R¯∑i=1nC^iP,αA∪B=R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB(4LH)R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′, R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′(5LH) Ifα≤α1, then R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,α1A and R¯∑i=1nC^iP,αA⊇R¯∑i=1nC^iP,α1AProof.
(1LH) For anyx∈U,(26)R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:DC^iβx,y≥α≤∧i=1nAx=Ax≤∨i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iP,αAx.(2L) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(27)R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:DC^iβx,y≥α≤∧i=1n∧y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iP,αBx,that is,R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB.(2H) For anyx∈U, if A⊆B, then Ax≤Bx. Hence,(28)R¯∑i=1nC^iP,αAx=∨i=1n∨y∈UAy:DC^iβx,y≥α≤∨i=1n∨y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iP,αBx,that is,R¯∑i=1nC^iP,αA⊆R¯∑i=1nC^iP,αB.(3L) For eachx∈U,(29)R¯∑i=1nC^iP,αA∩Bx=∧i=1n∧y∈UA∩By:DC^iβx,y≥α=∧i=1n∧y∈UAy∧By:DC^iβx,y≥α=∧i=1n∧y∈UAy:DC^iβx,y≥α∧∧i=1n∧y∈UBy:DC^iβx,y≥α=R¯∑i=1nC^iP,αAx∧R¯∑i=1nC^iP,αBx=R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αBx,that is,R¯∑i=1nC^iP,αA∩B=R¯∑i=1nC^iP,αA∩R¯∑i=1nC^iP,αB. Then, we only need to prove that R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB. According to (2L), we have R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA and R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αB. Therefore, we get that R¯∑i=1nC^iP,αA∪B⊇R¯∑i=1nC^iP,αA∪R¯∑i=1nC^iP,αB.(3H) The proof is similar to (3L). Hence, we omit it.(4LH) For eachx∈U,(30)R¯∑i=1nC^iP,αA′x=∧i=1n∧y∈UA′y:DC^iβx,y≥α=∧i=1n∧y∈U1−Ay:DC^iβx,y≥α=1−∨i=1n∨y∈UAy:DC^iβx,y≥α=R¯∑i=1nC^iP,αA′x,that is,R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′. Then, it can be written as R¯∑i=1nC^iP,αA=R¯∑i=1nC^iP,αA′′. Therefore, R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′.(5LH) It is immediate by Definition8.Example 9 (continued from Example1).
In Example8, we have R¯∑i=12C^iP,0.3A=0.4/x1+0.2/x2+0.4/x3+0.2/x4+0.2/x5 and R¯∑i=12C^iP,0.3A=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5. Let B=0.8/x1+0.5/x1+0.7/x1+0.4/x1+0.6/x5 and C=0.9/x1+0.4/x1+0.3/x1+0.7/x1+0.8/x5. Then, SR¯∑i=12C^iP,0.3B=0.5/x1+0.4/x2+0.5/x3+0.4/x4+0.4/x5, SR¯∑i=12C^iP,0.3B=0.8/x1+0.8/x2+0.8/x3+0.6/x4+0.8/x5, SR¯∑i=12C^iP,0.3C=0.5/x1+0.4/x2+0.5/x3+0.4/x4+0.4/x5, SR¯∑i=12C^iP,0.3C=0.9/x1+0.9/x2+0.9/x3+0.8/x4+0.9/x5, SR¯∑i=12C^iP,0.3A∪C=0.4/x1+0.4/x2+0.4/x3+0.4/x4+0.4/x5, SR¯∑i=12C^iP,0.3A∪C=0.9/x1+0.9/x2+0.9/x3+0.8/x4+0.9/x5, SR¯∑i=12C^iP,0.3A∩C=0.3/x1+0.2/x2+0.3/x3+0.2/x4+0.2/x5, and SR¯∑i=12C^iP,0.3A∩C=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5. Hence, the following hold:(1LH)R¯∑i=12C^iP,0.3A⊆A⊆R¯∑i=12C^iP,0.3A, and the same applies for B and C(2L)R¯∑i=12C^iP,0.3A⊆R¯∑i=12C^iP,0.3B(2H)R¯∑i=12C^iP,0.3A⊆R¯∑i=12C^iP,0.3B(3L)R¯∑i=12C^iP,0.3A∩C=R¯∑i=12C^iP,0.3A∩R¯∑i=12C^iP,αC, R¯∑i=12C^iP,0.3A∪C⊇R¯∑i=12C^iP,0.3A∪R¯∑i=12C^iP,αC(3H)R¯∑i=12C^iP,0.3A∩C⊆R¯∑i=12C^iP,0.3A∩R¯∑i=12C^iP,αC, R¯∑i=12C^iP,0.3A∪C=R¯∑i=12C^iP,0.3A∪R¯∑i=12C^iP,αC(4LH) SinceR¯∑i=1nC^iP,0.3A′=0.2/x1+0.2/x2+0.2/x3+0.5/x4+0.2/x5 and R¯∑i=1nC^iP,0.3A′=0.6/x1+0.8/x2+0.6/x3+0.8/x4+0.8/x5, then R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′ and R¯∑i=1nC^iP,αA′=R¯∑i=1nC^iP,αA′.(5LH) SinceR¯∑i=12C^iP,0.5A′=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5 and R¯∑i=12C^iP,0.5A=0.8/x1+0.8/x2+0.6/x3+0.2/x4+0.5/x5, then R¯∑i=1nC^iP,0.3A⊆R¯∑i=1nC^iP,0.5A and R¯∑i=1nC^iP,0.3A⊇R¯∑i=1nC^iP,0.5A


## 3.3. 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.Theorem 1.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A∈FU,(31)SR¯∑i=1nC^iPA⊆SR¯∑i=1nC^iOA⊆A⊆SR¯∑i=1nC^iOA⊆SR¯∑i=1nC^iPA.Proof.
By Definitions5 and 7 and Proposition 1, it is immediate.Example 10 (continued from Example1).
In Examples2 and 6, we know that A=0.8/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iOA=0.6667/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, SR¯∑i=12C^iOA=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5, SR¯∑i=12C^iPA=0.4/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iPA=0.8/x1+0.8/x2+0.6/x3+0.5/x4+0.5/x5. Hence, SR¯∑i=12C^iPA⊆SR¯∑i=12C^iOA⊆A⊆SR¯∑i=12C^iOA⊆SR¯∑i=12C^iPA.Then, the relationship betweenα-covering-based optimistic and pessimistic multigranulation fuzzy rough sets is given.Theorem 2.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A∈FU,(32)SR¯∑i=1nC^iP,αA⊆SR¯∑i=1nC^iO,αA⊆A⊆SR¯∑i=1nC^iO,αA⊆SR¯∑i=1nC^iP,αA.Proof.
By Definitions6 and 8 and Proposition 3, it is immediate.Example 11 (continued from Example1).
In Examples4 and 8, we know that R¯∑i=12C^iO,0.3A=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5, R¯∑i=12C^iO,0.3A=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5, R¯∑i=12C^iP,0.3A=0.4/x1+0.2/x2+0.4/x3+0.2/x4+0.2/x5, and R¯∑i=12C^iP,0.3A=0.8/x1+0.8/x2+0.8/x3+0.5/x4+0.8/x5. Hence, SR¯∑i=1nC^iP,0.3A⊆SR¯∑i=1nC^iO,0.3A⊆A⊆SR¯∑i=1nC^iO,0.3A⊆SR¯∑i=1nC^iP,0.3A.


## 4. Matrix Approaches for CMFRSs

In this section, we mainly use matrix approaches to calculate approximation operators of CMFRS models presented in Section3. They are the OMNFLAO SR¯∑i=1nC^iOA and the OMNFUAO SR¯∑i=1nC^iOA; α-OMNFLAO R¯∑i=1nC^iO,αA and the α-OMNFUAO R¯∑i=1nC^iO,αA; the PMNFLAO SR¯∑i=1nC^iPA and PMNFUAO SR¯∑i=1nC^iPA; and the α-PMNFLAO R¯∑i=1nC^iP,αA and the α-PMNFUAO R¯∑i=1nC^iP,αA.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]).
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. We call MC^=mijn×m=Cjxin×m a matrix representation of C^ and call the Boolean matrix MC^β=tijn×m a β-matrix representation of MC^, where(33)tij=1,mij≥β,0,mij<β,i.e.,tij=1,Cjxi≥β,0,Cjxi<β.Example 12 (continued from Example1).
First, we have(34)MC^1=0.80.40.30.70.30.60.50.40.70.20.60.80.80.70.70.50.80.90.40.6,MC^2=0.40.70.70.30.30.80.60.50.80.60.70.40.60.30.30.70.70.40.60.8.
Then,(35)MC^10.6=10010100101111101101,MC^20.6=01100110111010011011.LetA=aijm×l and B=bjkl×n be two matrices. Then, C=A∗B=cikm×n and D=A⋅B=dikm×n are defined as follows: cik=∧j=1l1−aij∨bjk and dik=∨j=1laij∧bjk, where i=1,2,…,m and k=1,2,…,n. AT denotes the transpose of A, and 1m×n is a matrix with all entries being 1. Then, the matrix representation of N˜C^xβ (x∈U) is presented in Lemma 1.Lemma 1 (see [24]).
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then, MC^β∗MC^T=N˜C^xiβxjn×n.Example 13 (continued from Example1).
Based on Example12, we have(36)MC^10.6∗MC^1T=10010100101111101101∗0.80.40.30.70.30.60.50.40.70.20.60.80.80.70.70.50.80.90.40.6T=0.70.30.70.50.60.40.60.20.70.90.30.30.60.50.40.30.30.20.70.40.40.30.20.50.6=N˜C^1xi0.6xj5×5,MC^20.6∗MC^2T=01100110111010011011∗0.40.70.70.30.30.80.60.50.80.60.70.40.60.30.30.70.70.40.60.8T=0.70.60.60.30.40.70.60.60.30.40.40.30.60.30.40.30.30.40.60.70.30.30.40.30.6=N˜C^2xi0.6xj5×5.Inspired by Definition2 and Lemma 1, the matrix representation of N¯C^xβ is presented as follows.Proposition 5.
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then, MC^β∗MC^Tβ=N¯C^xiβxjn×n, where N¯C^xiβxj=1xj∈N¯C^xiβ0otherwise.Proof.
By Lemma1, MC^β∗MC^T=N¯C^xiβxjn×n. Suppose that MC^β∗MC^Tβ=aijn×n. Then,(37)aij=1N˜C^xiβxj≥β;0otherwise=1xj∈N¯C^xiβ0otherwise=N˜C^xiβxj.Example 14 (continued from Example1).
Based on Example13, we have(38)MC^10.6∗MC^1T0.6=0.70.30.70.50.60.40.60.20.70.90.30.30.60.50.40.30.30.20.70.40.40.30.20.50.60.6=1010101011001000001000001=N¯C^1xi0.6xj5×5,MC^20.6∗MC^2T0.6=0.70.60.60.30.40.70.60.60.30.40.40.30.60.30.40.30.30.40.60.70.30.30.40.30.60.6=1110011100001000001100001=N¯C^2xi0.6xj5×5.In what follows, the definition of the matrix representation of all multigranulation fuzzy neighborhood measures is presented.Definition 10.
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. We call DC^β=dijn×n the matrix representation of all multigranulation fuzzy neighborhood measures DC^βxi,xj (i,j=1,2,…,n), where dij=DC^βxi,xj.Proposition6 shows that the matrix representation of all multigranulation fuzzy neighborhood measures is symmetric.Proposition 6.
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then, DC^β is a symmetric matrix.Proof.
By Definitions4 and 10, it is immediate.LetA=aijm×l, B=bjkl×n, and E=eijm×l be three matrices. Then, C=A▽B=cikm×n, D=AΔB=dikm×n, and G=A⊘E=gijm×l are defined as follows: cik=∑j=1laij∧bjk, dik=∑j=1laij∨bjk and gij=aij/eij, where i=1,2,…,m and k=1,2,…,n. Hence, all DC^β can be calculated by matrices.Proposition 7.
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then,(39)DC^β=MC^β∗MC^Tβ▽MC^β∗MC^Tβ,T⊘MC^β∗MC^Tβ△MC^β∗MC^Tβ,T,where MC^β∗MC^Tβ,T means MC^β∗MC^TβT.Proof.
Suppose thatDC^β=dijn×n, MC^β∗MC^Tβ▽MC^β∗MC^Tβ,T=cijn×n and MC^β∗MC^Tβ△MC^β∗MC^Tβ,T=bijn×n. By Proposition 5, MC^β∗MC^Tβ=N¯C^xiβxjn×n. Hence, cij=N¯C^xiβ∩N¯C^xjβ and bij=N¯C^xiβ∪N¯C^ixjβ. Therefore, dij=cij/bij=DC^βxi,xj.Example 15 (continued from Example1).
Based on Example14, we have(40)DC^10.6=1010101011001000001000001▽1010101011001000001000001T⊘1010101011001000001000001△1010101011001000001000001T=3110113011101000101011001⊘3534353433341224321233221=11513013151013131301000130101313001,DC^20.6=3310033100111000002100011⊘3335433354331325532244221=1113001113001313100000112000121.Then, we also investigate the matrix representations of the OMNFLAOSR¯∑i=1nC^iOA and the OMNFUAO SR¯∑i=1nC^iOA. Suppose that n=1; that is, SR¯C^OA and the OMNFUAO SR¯C^OA should be considered firstly.Proposition 8.
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then, for any A∈FU,(41)SR¯C^OA=DC^β∗A,SR¯C^OA=DC^β⋅A.Proof.
By Definition5, it is immediate.Example 16 (continued from Example1).
Based on Example15, we have(42)SR¯C^1OA=DC^10.6∗A=11513013151013131301000130101313001∗0.80.40.60.20.5=0.66670.40.60.20.5,SR¯C^1OA=DC^10.6⋅A=11513013151013131301000130101313001⋅0.80.40.60.20.5=0.80.40.60.3330.5,SR¯C^2OA=DC^20.6∗A=1113001113001313100000112000121∗0.80.40.60.20.5=0.40.40.60.20.5,SR¯C^2OA=DC^20.6⋅A=1113001113001313100000112000121⋅0.80.40.60.20.5=0.80.80.60.50.5.Inspired by Definition5 and Proposition 8, we give the matrix representations of the OMNFLAO SR¯∑i=1nC^iOA and the OMNFUAO SR¯∑i=1nC^iOA.Theorem 3.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A∈FU,(43)SR¯∑i=1nC^iOA=∨i=1nDC^iβ∗A,SR¯∑i=1nC^iOA=∧i=1nDC^iβ⋅A.Proof.
By Definition5 and Proposition 8, it is immediate.Example 17 (continued from Example1).
Based on Example16, we have(44)SR¯∑i=12C^iOA=∨i=12DC^i0.6∗A=0.66670.40.60.20.5∨0.40.40.60.20.5=0.66670.40.60.20.5,SR¯∑i=12C^iOA=∧i=12DC^i0.6⋅A=0.80.40.60.3330.5∧0.80.80.60.50.5=0.80.40.60.3330.5.Inspired by Definition7 and Proposition 8, we give the matrix representations of the OMNFLAO SR¯∑i=1nC^iPA and the OMNFUAO SR¯∑i=1nC^iPA.Theorem 4.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U. Then, for any A∈FU,(45)SR¯∑i=1nC^iPA=∧i=1nDC^iβ∗A,SR¯∑i=1nC^iPA=∨i=1nDC^iβ⋅A.Proof.
By Definition7 and Proposition 8, it is immediate.Example 18 (continued from Example1).
Based on Example16, we have(46)SR¯∑i=12C^iPA=∧i=12DC^i0.6∗A=0.66670.40.60.20.5∧0.40.40.60.20.5=0.40.40.60.20.5,SR¯∑i=12C^iPA=∨i=12DC^i0.6⋅A=0.80.40.60.3330.5∨0.80.80.60.50.5=0.80.80.60.50.5.LetA=aijm×l and B=bjkl×n be two matrices. Then, H=A⊙B=hikm×n and T=A∘B=tikm×n are defined as follows: hik=∧j=1laij∧bjk and tik=∨j=1laij∨bjk, where i=1,2,…,m and k=1,2,…,n. If A is a Boolean matrix and B=bjl×1, then we define C=A⊳B=cijm×l and D=A⊲B=dijm×l as follows:(47)cij=bj,aij=1,1,aij=0,dij=bj,aij=1,0,aij=0.Finally, we investigate the matrix representations of theα-OMNFLAO R¯∑i=1nC^iO,αA and the α-OMNFUAO R¯∑i=1nC^iO,αA; the α-PMNFLAO R¯∑i=1nC^iP,αA and the α-PMNFUAO R¯∑i=1nC^iP,αA. Suppose that n=1; that is, the α-OMNFLAO R¯C^O,αA and the α-OMNFUAO R¯C^O,αA (the α-PMNFLAO R¯C^P,αA and the α-PMNFLAO R¯C^P,αA) should be considered firstly. Hence, for any x∈U,(48)R¯C^O,αAx=R¯C^P,αAx=∧y∈UAy:DC^βx,y≥α,R¯C^O,αAx=R¯C^P,αAx=∨y∈UAy:DC^βx,y≥α.Proposition 9.
LetC^ be a fuzzy β-covering of U=x1,…,xm and α∈0,1. Then, for any A∈FU,(49)R¯C^O,αA=DC^βα⊳A⊙1m×1,R¯C^O,αA=DC^βα⊲A∘0m×1,where ⊳ and ⊲ have priority over ⊙ and ∘, respectively.Proof.
Suppose thatR¯C^O,αA=aim×1, DC^βα⊳A=cijm×m and DC^βα=bijm×m, where bij=1,DC^βxi,xj=10,otherwise. Hence, cij=Axj,bij=11,bij=0=Axj,DC^βxi,xj≥α1,otherwise. Therefore, R¯C^O,αAxi=ai=∧j=1mcij∧1=∧j=1mcij=∧j=1mAxj:DC^βxi,xj≥α.
On the other hand, supposeR¯C^O,αA=ai′m×1, DC^βα⊲A=cij′m×m. Hence, cij′=Axj,bij=10,bij=0=Axj,DC^βxi,xj≥α0,otherwise. Therefore, R¯C^O,αAxi=ai′=∨j=1mcij∨0=∨j=1mcij=∨j=1mAxj:DC^βxi,xj≥α.Inspired by Definition6 and Proposition 9, we present the matrix representations of the α-OMNFLAO R¯∑i=1nC^iO,αA and the α-OMNFUAO R¯∑i=1nC^iO,αA as follows.Theorem 5.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U=x1,…,xm and α∈0,1. Then, for any A∈FU,(50)R¯∑i=1nC^iO,αA=∨i=1nDC^iβα⊳A⊙1m×1,R¯∑i=1nC^iO,αA=∧i=1nDC^iβα⊲A∘0m×1,where ⊳ and ⊲ are first than ⊙ and ∘, respectively.Proof.
By Definition6 and Proposition 9, it is immediate.Example 19 (continued from Example1).
By Example15, we get DC^10.6 and DC^20.6. Then,(51)SR¯∑i=12C^iO,0.3A=∨i=12DC^i0.60.3⊳A⊙15×1=1010101011101000101011001⊳0.80.40.60.20.5⊙11111∨1110011100111000001100011⊳0.80.40.60.20.5⊙11111=0.810.610.510.410.20.50.810.61110.410.210.80.4110.5⊙11111∨0.80.40.6110.80.40.6110.80.40.6111110.20.51110.20.5⊙11111=0.50.20.60.20.4∨0.40.40.40.20.2=0.50.40.60.20.4,SR¯∑i=12C^iO,0.3A=∧i=12DC^i0.60.3⊲A∘05×1=1010101011101000101011001⊲0.80.40.60.20.5∘00000∧1110011100111000001100011⊲0.80.40.60.20.5∘00000=0.80.50.80.40.5.Inspired by Definition8 and Proposition 9, we present the matrix representations of the α-PMNFLAO R¯∑i=1nC^iP,αA and the α-PMNFUAO R¯∑i=1nC^iP,αA as follows.Theorem 6.
LetΓ˜=C^1,…,C^n be an n-fuzzy β-coverings of U=x1,…,xm and α∈0,1. Then, for any A∈FU,(52)R¯∑i=1nC^iP,αA=∧i=1nDC^iβα⊙1m×1,R¯∑i=1nC^iP,αA=∨i=1nDC^iβα∘0m×1,where ⊳ and ⊲ are first than ⊙ and ∘, respectively.Proof.
By Definition8 and Proposition 9, it is immediate.Example 20 (continued from Example1).
By Example15, we get DC^10.6 and DC^20.6. Then,(53)SR¯∑i=12C^iP,0.3A=∧i=12DC^i0.60.3⊳A⊙15×1=1010101011101000101011001⊳0.80.40.60.20.5⊙11111∧1110011100111000001100011⊳0.80.40.60.20.5⊙11111=0.810.610.510.410.20.50.810.61110.410.210.80.4110.5⊙11111∧0.80.40.6110.80.40.6110.80.40.6111110.20.51110.20.5⊙11111=0.50.20.60.20.4∧0.40.40.40.20.2=0.40.20.40.20.2,SR¯∑i=12C^iP,0.3A=∨i=12DC^i0.60.3⊲A∘05×1=1010101011101000101011001⊲0.80.40.60.20.5∘00000∨1110011100111000001100011⊲0.80.40.60.20.5∘00000=0.80.80.80.50.8.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. Their computational time is shown in Table9. They are carried out on a personal computer with 64-bit Windows 7, Intel® Core™ i5-4460 CPU @3.20 GHz, and 8 GB memory. The 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.Table 9
Comparison of different approximation operators for some datasets in UCI machine learning repository.
DatasetsSamplesAttributesOperatorsComputational time of 30%U (s)Computational time of 60%U (s)Computational time of 100%U (s)Iris1504SR¯∑i=1nC^iO0.03610.33750.3631SR¯∑i=1nC^iO0.03550.30950.3195R¯∑i=1nC^iO,α0.04540.30000.3478R¯∑i=1nC^iO,α0.03750.43730.4138Forest fires5177SR¯∑i=1nC^iO0.99382.75219.3000SR¯∑i=1nC^iO0.64492.727510.5721R¯∑i=1nC^iO,α0.90392.558710.6041R¯∑i=1nC^iO,α0.73352.484810.6961Blood7484SR¯∑i=1nC^iO0.72484.097622.2954SR¯∑i=1nC^iO0.66644.027722.5706R¯∑i=1nC^iO,α0.74924.125721.9894R¯∑i=1nC^iO,α0.72853.830222.5545


## 5. 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∈FU.Algorithm 1: An algorithm for the problem of MCGDM in [31].
Input: a multigranulation fuzzy decision information system U,Γ˜,β,μ.Output: the rank of all alternatives.Step 1: by Definition 2, calculating all N˜C^ixβ for any x∈U and i=1,2,…,n.Step 2: by Definition 3, computing R¯∑i=1nC^iOμ and R¯∑i=1nC^iOμ, respectively.Step 3: getting the right weight value of ζ, where ζ∈0,1.Step 4: calculating ∑i=1nR˜iOμ=ζR¯∑i=1nC^iOμ+1−ζR¯∑i=1nC^iOμ.Step 5: obtaining the ordering by the decision principle.Inspired by Algorithm1, we use the OMNFLAO SR¯∑i=1nC^iOμ and OMNFUAO SR¯∑i=1nC^iOμ that are presented in this paper to replace R¯∑i=1nC^iOμ and R¯∑i=1nC^iOμ, respectively. Hence, Algorithm 2 is proposed as follows:Algorithm 2: An algorithm for the problem of MCGDM.
Input: a multigranulation fuzzy decision information system U,Γ˜,β,μ.Output: the rank of all alternatives.Step 1: by Definition 4, calculating all DC^iβx,y for any x∈U and i=1,2,…,n.Step 2: by Definition 3, computingSR¯∑i=1nC^iOμx=∨i=1n∧y∈U1−DC^iβx,y∨μy,SR¯∑i=1nC^iOμx=∧i=1n∨y∈UDC^iβx,y∧μy.Step 3: getting the right weight value of ζ, where ζ∈0,1.Step 4: calculating ∑i=1nSR˜iOμ=ζSR¯∑i=1nC^iOμ+1−ζSR∑i=1nC^iOμ.Step 5: obtaining the ordering by the decision principle.On the other hand, Algorithm3 is proposed based on Definition 6 to solve the problem of MCGDM.Algorithm 3: An algorithm for the problem of MCGDM.
Input: a multigranulation fuzzy decision information system U,Γ˜,β,μ.Output: the rank of all alternatives.Step 1: by Definition 4, calculating all DC^iβx,y for any x∈U and i=1,2,…,n.Step 2: getting the right weight value of ζ, where ζ∈0,1.Step 3: by Definition 3, computingR¯∑i=1nC^iO,ζμx=∨i=1n∧y∈Uμy:DC^iβx,y≥ζ,R¯∑i=1nC^iO,ζAx=∧i=1n∨y∈Uμy:DC^iβx,y≥ζ.Step 4: calculating ∑i=1nR˜iOμ=ζSR¯∑i=1nC^iOμ+1−ζR¯∑i=1nC^iOμ.Step 5: obtaining the ordering by the decision principle.To show the validity of our method, we compare Algorithms1–3 in the following example.Example 21.
LetU=x1,x2,…,x5 be a set of patients, β=0.6 be the critical value, and V=y1,y2,y3,y4be a set of five main symptoms for the disease A. Suppose that the doctor X invites 2 experts Ri (i=1,2) to evaluate every patient xk (k=1,2,…,5). Then, the expert Ri believes that the patient xk has a value of the symptom yj (j=1,…,4), denoted as Cijxk. For the patient xk, there exists at least one symptom yj such that the symptom value Cijxk≥β. Then, Γ˜=C^1,C^2 is a family of fuzzy β-coverings of U, where C^i=Ci1,Ci2,Ci3,Ci4 (i=1,2). They 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/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5. The doctor X wants to decide whether or not the patients xk∈U have the disease A.
Firstly, we use Algorithm1 to solve this problem as follows:Step 1: allN˜C^ixk0.6 are shown in Tables 3 and 4, for any i=1,2 and k=1,2,…,5Step 2:R¯∑i=12C^iOμ=0.5/x1+0.4/x2+0.6/x3+0.4/x4+0.5/x5, and R¯∑i=12C^iOμ=0.7/x1+0.5/x2+0.6/x3+0.4/x4+0.5/x5Step 3:ζ=0.3Step 4:∑i=12R˜iOμ=0.64/x1+0.47/x2+0.6/x3+0.4/x4+0.5/x5Step 5: we getx4≺x2≺x5≺x3≺x1. That is to say, x1 is more likely to be sicken for the disease A
Then, for Algorithm2:Step 1:DC^10.6 and DC^20.6 are shown in Tables 7 and 8, respectivelyStep 2:SR¯∑i=12C^iOμ=0.6667/x1+0.4/x2+0.6/x3+0.2/x4+0.5/x5, and SR¯∑i=12C^iOμ=0.8/x1+0.4/x2+0.6/x3+0.333/x4+0.5/x5Step 3:ζ=0.3Step 4:∑i=12SR˜iOμ=0.76/x1+0.4/x2+0.6/x3+0.2933/x4+0.5/x5Step 5: we getx4≺x2≺x5≺x3≺x1. That is to say, x1 is more likely to be sicken for the disease A
For Algorithm3:Step 1:DC^10.6 and DC^20.6 are shown in Tables 7 and 8, respectivelyStep 2:ζ=0.3Step 3:R¯∑i=12C^iO,0.3μ=0.5/x1+0.4/x2+0.6/x3+0.2/x4+0.4/x5, and R¯∑i=12C^iO,0.3μ=0.8/x1+0.5/x2+0.8/x3+0.4/x4+0.5/x5Step 4:∑i=12R˜iO,0.3μ=0.71/x1+0.47/x2+0.74/x3+0.34/x4+0.47/x5Step 5: we getx4≺x2≈x5≺x1≺x3. That is to say, x3 is more likely to be sicken for the disease A
Finally, all results about are shown in Table10 and Figure 1. From Table 10 and Figure 1, we find that the results of Algorithms 1 and 2 are same. That is to say, the patient x1 is more likely to be sicken for the disease A. Hence, our methods are effective. On the other hand, x3 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.Table 10
The results utilizing the different methods of Example21.
MethodsThe final rankingThe patients are more likely to be sicken for the diseaseAAlgorithm1 (Zhan et al. [31])x4≺x2≺x5≺x3≺x1x1Algorithm2x4≺x2≺x5≺x3≺x1x1Algorithm3x4≺x2≈x5≺x1≺x3x3Figure 1
Comparisons of the ranking results in Example21 based on Algorithms 1–3.


## 6. Conclusions

This 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. The 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 ofβ-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]. These new types of CMFRS models can be seen as the combination of covering-based rough sets, multigranulation rough sets and fuzzy sets. These models are the complement of the existing rough set models. They 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 fuzzyβ-covering approximation space. The 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 Algorithm3, we find that the patient x3 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. That is to say, it is a new choice for decision maker. Therefore, 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. The 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.



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