# Matrix Approaches for Covering-Based Multigranulation Fuzzy Rough Set Models

Authors: Zaibin Chang; Junchao Wei
Journal: Journal of Mathematics
 (2023)

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## Abstract

Multigranulation rough set theory is one of the most effective tools for data analysis and mining in multicriteria information systems. Six types of covering-based multigranulation fuzzy rough set (CMFRS) models have been constructed through fuzzyβ-neighborhoods or multigranulation fuzzy measures. However, it is often time-consuming to compute these CMFRS models with a large fuzzy covering using set representation approaches. Hence, presenting novel methods to compute them quickly is our motivation for this paper. In this article, we study the matrix representations of CMFRS models to save time in data processing. Firstly, some new matrices and matrix operations are proposed. Then, matrix representations of optimistic CMFRSs are presented. Moreover, matrix approaches for computing pessimistic CMFRSs are also proposed. Finally, some experiments are proposed to illustrate the effectiveness of our approaches.


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## Body

## 1. Introduction

As a kind of extension of classical rough sets [1, 2], covering-based rough sets [3, 4] were proposed to manage this kind of covering data. In theory, this data is a family of some subsets of a universe (the set of all objects). Inspired by it, several covering-based rough set models [5, 6] were established to deal with practical problems, such as feature selection [7–9] and rule synthesis [10–13]. As the same with rough set theory, Zadeh’s fuzzy set theory [14] can also manage issues of imperfect information. Therefore, many researchers investigated their connection and distinction [15, 16].Fuzzy covering rough sets [17–19] were proposed with a special membership of “1.” Then, Ma [20] generalized it to a parameter ββ∈0,1 and constructed fuzzy β-covering rough set models. Inspired by Ma’s work, many researches were done. For example, several new fuzzy β-covering rough sets models were presented [21–23]; fuzzy β-neighborhoods were investigated [24, 25]; fuzzy β-covering approximation spaces were used in decision-making [26, 27]. Moreover, many researchers applied matrix methods to deal with fuzzy β-covering data, which can simplify calculations in the process of data analysis. For example, Ma [20] gave two types of fuzzy covering rough set models’ matrix representations; Yang and Hu [28] investigated some of the other models by matrices; Huang et al. [29] and Mao et al. [30] proposed matrix methods for computing the reduction of a fuzzy β-covering; Wang et al. [31] used matrices to deal with the issues of fuzzy description reduction and group decision-making.But, all these works above were done in a fuzzyβ-covering approximation space. Therefore, Zhan et al. [32] generalized a fuzzy β-covering to n-fuzzy β-covering and presented six types of covering-based multigranulation fuzzy rough set (CMFRS) models. Based on Zhan’s work, Zhan et al. [33] presented multigranulation I,T-fuzzy covering rough set models under fuzzy β-neighborhoods; Ma et al. [34] constructed three types of multigranulation I,T-fuzzy covering rough set models under complementary fuzzy β-neighborhoods; Atef et al. [35] presented soft fuzzy covering rough set models under complementary soft neighborhood; Atef and El Atik [36] proposed some extensions of multigranulation fuzzy covering rough sets from new perspectives. It is important to find a quick way to calculate these models, which will save time for rapid data mining and analysis and is also the motivation of this paper. In this article, we use matrix methods to investigate Zhan’s et al. CMFRS models [32], since Zhan’s CMFRS models are the basis of all these models above.Inspired by [37], we present matrix representations for computing CMFRS models [32] in this paper. Moreover, the effectiveness of our matrix approaches is given by some experiments. Firstly, several new matrices and corresponding operations are proposed. Then, matrix representations of optimistic CMFRSs are presented. Furthermore, matrix approaches are also used for computing pessimistic CMFRSs. Finally, several experiments are proposed to show the effectiveness of our matrix approaches.The rest of this article is listed as follows. Section2 recalls several basic notions about CMFRSs. In Section 3, some novel matrices and corresponding operations are proposed. Furthermore, matrix representations of optimistic CMFRSs are proposed. In Section 4, we present matrix representations of pessimistic CMFRSs. In Section 5, we illustrate the efficiency of our matrix method for computing CMFRSs through several experiments. This article is concluded and further work is indicated in Section 6.


## 2. Fundamental Definitions

This section shows several fundamental concepts related to CMFRSs. SupposingU is a nonempty and finite set called a universe.To connect covering rough sets with fuzzy sets, Ma [20] proposed the concept of fuzzy β-covering approximation space.Definition 1 (see [20]).
LetU be a universe and FU be the fuzzy power set of U. For any 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 all x∈U. We also call U,C^ a fuzzy β-covering approximation space.Several covering-based fuzzy rough set models were established under a fuzzyβ-covering approximation space. Furthermore, Zhan et al. [32] extended 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 each i∈1,2,…,n. Then, six types of CMFRSs were presented in [32].Definition 2 (see [32]).
LetΓ˜=C^1,…,C^nbe an-fuzzyβ-coverings ofU=x1,x2⋯,xmFor eachA∈FUandx∈U, we define: (1) the OMGFRLAOR¯∑i=1nC^iOAand the OMGFRUAOR¯∑i=1nC^iOA; (2) the PMGFRLAOR¯∑i=1nC^iPAand the PMGFRUAOR¯∑i=1nC^iPA; (3) theα-OMGFLAOR¯∑i=1nC^iO,αAand theα-OMGFUAOR¯∑i=1nC^iO,αA; (4) theα-PMGFLAOR¯∑i=1nC^iP,αAand theα-PMGFUAOR¯∑i=1nC^iP,αA; (5) the OMGFLAOD˜R¯∑i=1nC^iOAand the OMGFUAOD˜R¯∑i=1nC^iOA; (6) the PMGFLAOD˜R¯∑i=1nC^iPAand the PMGFUAOD˜R¯∑i=1nC^iPAas follows:(1)
R¯∑i=1nC^iOAx=∨i=1n∧y∈U1−N˜C^ixβy∨AyR¯∑i=1nC^iOAx=∧i=1n∨y∈UN˜C^ixβy∧Ay(2)
R¯∑i=1nC^iPAx=∧i=1n∧y∈U1−N˜C^ixβy∨AyR¯∑i=1nC^iPAx=∨i=1n∨y∈UN˜C^ixβy∧Ay(3)
R¯∑i=1nC^iO,αAx=∨i=1n∧y∈UAy:D˜C^iβx,y>αR¯∑i=1nC^iO,αAx=∧i=1n∨y∈UAy:D˜C^iβx,y>α(4)
R¯∑i=1nC^iP,αAx=∧i=1n∧y∈UAy:D˜C^iβx,y>αR¯∑i=1nC^iP,αAx=∨i=1n∨y∈UAy:D˜C^iβx,y>α(5)
D˜R¯∑i=1nC^iOAx=∨i=1n∧y∈U1−D˜C^iβx,y∨AyD˜R¯∑i=1nC^iOAx=∧i=1n∨y∈UD˜C^iβx,y∧Ay(6)
D˜R¯∑i=1nC^iPAx=∧i=1n∧y∈U1−D˜C^iβx,y∨AyD˜R¯∑i=1nC^iPAx=∨i=1n∨y∈UD˜C^iβx,y∧AywhereN˜C^xβ=∩C∈C^:Cx≥β is defined as the fuzzy β-neighborhood of x; α∈0,1and(1)D˜C^iβx,y=∑j=1mN˜C^ixjβx∩N˜C^ixjβy∑j=1mN˜C^ixjβx∪N˜C^ixjβyIs defined as a multigranulation fuzzy measure degree ofx and y under C^i.Example 1.
Γ=C^1,C^2 is a 2-fuzzy β-coverings of U, where β=0.5 and C^i=Ci1,Ci2,Ci3,Ci4i=1,2. They are listed in Tables 1 and 2, respectively.
Then,N˜C^1xj0.5 and N˜C^2xj0.5j=1,2,…,5 are shown in Tables 3 and 4.
Hence, we have allD˜C^10.5xk,xj andD˜C^20.5xk,xjk,j=1,2,…,5 which are shown in Tables 5 and 6.
ForA=0.9/x1+0.6/x2+0.3/x3+0.7/x4+0.4/x5 and α=0.6, we have(1)
R¯∑i=12C^iOA=0.4/x1+0.6/x2+0.4/x3+0.4/x4+0.4/x5, R¯∑i=12C^iOA=0.5/x1+0.5/x2+0.5/x3+0.6/x4+0.4/x5(2)
R¯∑i=12C^iPA=0.4/x1+0.6/x2+0.3/x3+0.3/x4+0.4/x5, R¯∑i=12C^iPA=0.6/x1+0.5/x2+0.6/x3+0.7/x4+0.6/x5(3)
R¯∑i=12C^iO,0.6A=0.4/x1+0.4/x2+0.3/x3+0.4/x4+0.4/x5, R¯∑i=12C^iO,0.6A=0.9/x1+0.9/x2+0.7/x3+0.9/x4+0.7/x5(4)
R¯∑i=12C^iP,0.6A=0.3/x1+0.4/x2+0.3/x3+0.3/x4+0.4/x5, R¯∑i=12C^iP,0.6A=0.9/x1+0.9/x2+0.9/x3+0.9/x4+0.9/x5(5)
D˜R¯∑i=12C^iOA=0.4/x1+0.4/x2+0.3/x3+0.4/x4+0.4/x5, D˜R¯∑i=12C^iOA=0.9/x1+0.6522/x2+0.7/x3+0.7/x4+0.7/x5(6)
D˜R¯∑i=12C^iPA=0.3/x1+0.4/x2+0.3/x3+0.3/x4+0.4/x5, D˜R¯∑i=12C^iPA=0.9/x1+0.7143/x2+0.7619/x3+0.7/x4+0.8750/x5Table 1
C^1.
UC11C12C13C14x10.80.60.40.7x20.40.70.50.3x30.80.30.70.8x40.80.30.60.7x50.90.70.30.6Table 2
C^2.
UC21C22C23C24x10.50.70.60.3x20.30.90.80.5x30.60.60.60.2x40.80.40.30.7x50.70.30.70.6Table 3
N˜C^1xj0.5j=1,2,…,5.
Ux1x2x3x4x5N˜C^1x10.50.60.30.30.30.6N˜C^1x20.50.40.50.30.30.3N˜C^1x30.50.40.30.70.60.3N˜C^1x40.50.40.30.70.60.3N˜C^1x50.50.60.30.30.30.6Table 4
N˜C^2xj0.5j=1,2,…,5.
Ux1x2x3x4x5N˜C^2x10.50.50.30.60.30.3N˜C^2x20.50.30.50.20.30.3N˜C^2x30.50.50.30.60.30.3N˜C^2x40.50.30.30.20.70.6N˜C^2x50.50.30.30.20.30.6Table 5
D˜C^10.5xk,xjk,j=1,2,…,5.
Ux1x2x3x4x5D˜C^10.5x1,xj10.640.56670.60710.875D˜C^10.5x2,xj0.6410.60.65220.6522D˜C^10.5x3,xj0.56670.610.9130.5172D˜C^10.5x4,xj0.60710.65220.91310.5556D˜C^10.5x5,xj0.8750.65220.51720.55561Table 6
D˜C^20.5xk,xjk,j=1,2,…,5.
Ux1x2x3x4x5D˜C^20.5x1,xj10.71430.76190.65220.6D˜C^20.5x2,xj0.714310.52170.71430.6522D˜C^20.5x3,xj0.76190.521710.480.444D˜C^20.5x4,xj0.65220.71430.4810.8182D˜C^20.5x5,xj0.60.65220.4440.81821


## 3. Matrix Approaches for Optimistic CMFRSs

In this section, we present some new matrices and use them to compute approximation operators of optimistic CMFRS models presented in Definition2. They are OMGFRLAO R¯∑i=1nC^iOA and OMGFRUAO R¯∑i=1nC^iOA; α-OMGFLAO R¯∑i=1nC^iO,αA and α-OMGFUAO R¯∑i=1nC^iO,αA; OMGFLAO D˜R¯∑i=1nC^iOA and OMGFUAO D˜R¯∑i=1nC^iOA. In [37], we have given the matrix representations of OMGFRLAO R¯∑i=1nC^iOA and OMGFRUAO R¯∑i=1nC^iOA, OMGFLAO D˜R¯∑i=1nC^iOA, and OMGFUAO D˜R¯∑i=1nC^iOA. In this section, we investigate them further and give the matrix representations of α-OMGFLAO R¯∑i=1nC^iO,αA and α-OMGFUAO R¯∑i=1nC^iO,αA.In [20], Ma has presented some matrices and corresponding operations to compute a fuzzy covering rough set model. They considered the n-fuzzy β-coverings of U when n=1.Definition 3 (see [20]).
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×maβ-matrix representation of MC^, where(2)tij=1,Cjxi≥β;0,Cjxi<β.Example 2.
(Continued from Example 1).(3)MC^1=0.80.60.40.70.40.70.50.30.80.30.70.80.80.30.60.70.90.70.30.6,MC^2=0.50.70.60.30.30.90.80.50.60.60.60.20.80.40.30.70.70.30.70.6.
Then,(4)MC^10.5=11010110101110111101,MC^20.5=11100111111010011011.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. Based on Definition 3, the matrix representations of OMGFRLAO R¯∑i=1nC^iOA and OMGFRUAO R¯∑i=1nC^iOA can be obtained as follows when n=1:Proposition 1 (see [20]).
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn:(5)R¯C^OA=MC^β∗MC^T∗A,R¯C^OA=MC^β∗MC^T⋅A.Example 3.
(Continued from Example1).(6)R¯C^1OA=0.60.30.30.30.60.40.50.30.30.30.40.30.70.60.30.40.30.70.60.30.60.30.30.30.6∗0.90.60.30.70.4=0.40.60.30.30.4,R¯C^1OA=0.60.30.30.30.60.40.50.30.30.30.40.30.70.60.30.40.30.70.60.30.60.30.30.30.6⋅0.90.60.30.70.4=0.60.50.60.60.6.Inspired by Proposition1, we give the matrix representations of OMGFRLAO R¯∑i=1nC^iOA and OMGFRUAO R¯∑i=1nC^iOA when n≥1.Proposition 2 (see [37]).
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. Then, for any A∈FU(7)R¯∑i=1nC^iOA=∨i=1nMC^iβ∗MC^iT∗A,R¯∑i=1nC^iOA=∧i=1nMC^iβ∗MC^iT⋅A.Proof.
By Definition2 and Proposition 1, it is immediate.Example 4.
(Continued fromExample1).(8)R¯∑i=12C^iOA=0.40.60.30.30.4∨0.40.60.40.40.4=0.40.60.40.40.4,R¯∑i=12C^iOA=0.60.50.60.60.6∧0.50.50.50.70.4=0.50.50.50.60.4.At the same time, the matrix representation ofN˜C^xβx∈U has been proposed in [20] as follows:Lemma 1 (see [20]).
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then, MC^β∗MC^T=N˜C^xiβxjn×n.Example 5.
(Continued from Example1). Based on Example 2, we have(9)MC^10.5∗MC^1T=0.60.30.30.30.60.40.50.30.30.30.40.30.70.60.30.40.30.70.60.30.60.30.30.30.6=N˜C^1xi0.5xj5×5,MC^20.5∗MC^2T=0.50.30.60.30.30.30.50.20.30.30.50.30.60.30.30.30.30.20.70.60.30.30.20.30.6=N˜C^2xi0.5xj5×5.On the contrary, we present the matrix representations ofα-OMGFLAO R¯∑i=1nC^iO,αA and α-OMGFUAO R¯∑i=1nC^iO,αA; OMGFLAO D˜R¯∑i=1nC^iOA and OMGFUAO D˜R¯∑i=1nC^iOA. First, we give the definition of the matrix representation of all multigranulation fuzzy measure degrees D˜C^βxi,xji,j=1,2,…,n.Definition 4 (see [37]).
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn and α∈0,1. We call D˜C^β=dijn×n the matrix representation of all multigranulation fuzzy measure degrees D˜C^βxi,xji,j=1,2,…,n, where dij=D˜C^βxi,xj. We also denote D˜C^βα=tijn×n as the α-matrix representation of D˜C^β, where(10)tij=1,dij>α;0,dij≤α.Proposition3 illustrates that the matrix representation of all multigranulation fuzzy measures degrees is symmetric.Proposition 3.
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then,D˜C^β is a symmetric matrix.Proof.
It is immediate, by Definition4.Given three matricesA=aijm×l, B=bjkl×n, and E=eijm×l. Then, C=A∇B=cikm×n, D=AΔB=dikm×n, and G=A⊘E=gijm×l are denoted by: cik=∑j=1laij∧bjk, dik=∑j=1laij∨bjk, andgij=aij/eij, where i=1,2,…,m and k=1,2,…,n. Hence, all DC^βxi,xj can be computed by matrices.Proposition 4 (see [37]).
LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn. Then,(11)D˜C^β=MC^β∗MC^TT∇MC^β∗MC^T⊘MC^β∗MC^TTΔMC^β∗MC^T.Proof.
AssumeD˜C^β=dijn×n, MC^β∗MC^TT∇MC^β∗MC^T=cijn×n, and MC^β∗MC^TTΔMC^β∗MC^T=bijn×n. By Lemma 1, MC^β∗MC^T=N˜C^xiβxjn×n. Therefore, dij=cij/bij=DC^βxi,xj.Example 6.
(Continued from Example1). Let α=0.6. Based on Example 5, we have(12)D˜C^10.5=2.41.61.71.72.11.61.71.51.51.51.71.52.32.11.51.71.52.12.11.52.11.51.51.52.1⊘2.42.532.82.42.51.72.52.32.332.52.32.32.92.82.32.32.12.72.42.32.92.72.1=10.640.56670.60710.8750.6410.60.65220.65220.56670.610.9130.51720.60710.65220.91310.55560.8750.65220.51720.55561D˜C^20.5=1.91.51.61.51.51.51.71.21.51.51.61.21.81.21.21.51.51.21.91.81.51.51.21.82.1⊘1.92.12.12.32.52.11.72.32.12.32.12.31.82.52.72.32.12.51.92.22.52.32.72.22.1=10.71430.76190.65220.60.714310.52170.71430.65220.76190.521710.480.44440.65220.71430.4810.81820.60.65220.44440.81821.
Hence, we have(13)D˜C^10.50.6=1101111011001101111011001,D˜C^20.50.6=1111011011101001101101011.LetA=aijm×l, 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:(14)cij=bj,aij=1;1,aij=0.,anddij=bj,aij=1;0,aij=0.After that, we investigate the matrix representations ofα-OMGFLAO R¯∑i=1nC^iO,αA and α-OMGFUAO R¯∑i=1nC^iO,αA. We suppose n=1, i.e., α-OMGFLAO R¯C^O,αA and α-OMGFUAO R¯C^O,αA should be considered first. Hence, for any x∈U(15)R¯C^O,αAx=∧y∈UAy:D˜C^βx,y>α,R¯C^O,αAx=∨y∈UAy:D˜C^βx,y>α.Proposition 5.
LetC^ be a fuzzy β-covering of U=x1,…,xm and α∈0,1. Then, for any A∈FU(16)R¯C^O,αA=D˜C^βα⊳A⊙1m×1,R¯C^O,αA=D˜C^βα⊲A∘0m×1,where ⊳ and ⊲ are first than ⊙ and °, respectively.Proof.
Suppose,R¯C^O,αA=aim×1, D˜C^βα⊳A=cijm×m, and D˜C^βα=bijm×m,where(17)bij=1,D˜C^βxi,xj>α;0,Otherwise.
Hence,(18)cij=Axj,bij=1;1,bij=0,=Axj,D˜C^βxi,xj>α;1,Otherwise.
Therefore,(19)R¯C^O,αAxi=ai=∧j=1mcij∧1=∧j=1mcij=∧j=1mAxj:DC^βxi,xj>α.
On the contrary, suppose(20)R¯C^O,αA=ai′m×1,D˜C^βα⊲A=cij′m×m.
Hence,(21)cij′=Axj,bij=1;0,bij=0,=Axj,D˜C^βxi,xj>α;0,Otherwise.
Therefore,(22)R¯C^O,αAxi=ai′=∨j=1mcij∨0=∨j=1mcij=∨j=1mAxj:D˜C^βxi,xj>α.Inspired by Proposition5, we propose the matrix representations of α-OMGFLAO R¯∑i=1nC^iO,αA and α-OMGFUAO R¯∑i=1nC^iO,αA as follows.Theorem 1.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U=x1,…,xm and α∈0,1. Then, for any A∈FU(23)R¯∑i=1nC^iO,αA=∨i=1nD˜C^iβα⊳A⊙1m×1,R¯∑i=1nC^iO,αA=∧i=1nD˜C^iβα⊲A∘0m×1,where ⊳ and ⊲ are first than ⊙ and °, respectively.Proof.
It is immediate, by Proposition5.Example 7.
(Continued from Example1). By Example 6, we get DC^10.50.6 and DC^20.50.6. Then,(24)SR¯∑i=12C^iO,0.6A=∨i=12D˜C^i0.50.6⊳A⊙15×1=1101111011001101111011001⊳0.90.60.30.70.4⊙11111∨1111011011101001101101011⊳0.90.60.30.70.4⊙11111=0.90.610.70.40.90.610.70.4110.30.710.90.60.30.710.90.6110.4⊙11111∨0.90.60.30.710.90.610.70.40.910.3110.90.610.70.410.610.70.4⊙11111=0.40.40.30.30.4∨0.30.40.30.40.4=0.40.40.30.40.4,SR¯∑i=12C^iO,0.6A=∧i=12D˜C^i0.50.6⊲A∘05×1=1101111011001101111011001⊲0.90.60.30.70.4∘00000∧1111011011101001101101011⊲0.90.60.30.70.4∘00000=0.90.600.70.40.90.600.70.4000.30.700.90.60.30.700.90.6000.4∘00000∧0.90.60.30.700.90.600.70.40.900.3000.90.600.70.400.600.70.4∘00000=0.90.90.70.90.9∧0.90.90.90.90.7=0.90.90.70.90.7.Finally, we propose the matrix representations of OMGFLAOD˜R¯∑i=1nC^iOA and OMGFUAO D˜R¯∑i=1nC^iOA.Theorem 2 (see [37]).
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. Then, for any A∈FU(25)D˜R¯∑i=1nC^iOA=∨i=1nD˜C^iβ∗A,D˜R¯∑i=1nC^iOA=∧i=1nD˜C^iβ⋅A.Proof.
By Definition2 and Proposition 2, it is immediate.Example 8.
(Continued from Example1). We have(26)D˜R¯∑i=12C^iOA=∨i=12DC^i0.6∗A=0.40.40.30.30.4∨0.30.40.30.40.4=0.40.40.30.40.4,D˜R¯∑i=12C^iOA=∧i=12DC^i0.6⋅A=0.90.65220.70.70.875∧0.90.71430.76190.70.7=0.90.65220.70.70.7.


## 4. Matrix Approaches for Pessimistic MCFRS

Based on Section3, we mainly use matrix methods to calculate approximation operators of pessimistic MCFRS models presented in Definition 2. They are PMGFRLAO R¯∑i=1nC^iPA and PMGFRUAO R¯∑i=1nC^iPA; α-PMGFLAO R¯∑i=1nC^iP,αA and α-PMGFUAO R¯∑i=1nC^iP,αA; PMGFLAO D˜R¯∑i=1nC^iPA and PMGFUAO D˜R¯∑i=1nC^iPA.Firstly, we consider the matrix representations of PMGFRLAOR¯∑i=1nC^iPA and PMGFRUAO R¯∑i=1nC^iPA when n=1.Proposition 6.
.LetC^=C1,C2,…,Cm be a fuzzy β-covering of U=x1,…,xn.(27)R¯C^PA=MC^β∗MC^T∗A,R¯C^PA=MC^β∗MC^T⋅A.Proof.
By Proposition1, it is immediate.Example 9.
(Continued from Example1). In Example 5, we have(28)MC^20.5∗MC^2T=0.50.30.60.30.30.30.50.20.30.30.50.30.60.30.30.30.30.20.70.60.30.30.20.30.6.
Then,(29)R¯C^2PA=MC^20.5∗MC^2T∗A=0.50.30.60.30.30.30.50.20.30.30.50.30.60.30.30.30.30.20.70.60.30.30.20.30.6∗0.90.60.30.70.4=0.40.60.40.40.4,R¯C^2PA=MC^20.5∗MC^2T⋅A=0.50.30.60.30.30.30.50.20.30.30.50.30.60.30.30.30.30.20.70.60.30.30.20.30.6⋅0.90.60.30.70.4=0.50.50.50.70.4.Then, we consider the statement ofn≥1.Proposition 7.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. Then, for any A∈FU(30)R¯∑i=1nC^iPA=∧i=1nMC^iβ∗MC^iT∗A,R¯∑i=1nC^iPA=∨i=1nMC^iβ∗MC^iT⋅A.Proof.
By Definition2 and Proposition 6, it is immediate.Example 10.
(Continued from Example1)(31)R¯∑i=12C^iPA=0.40.60.30.30.4∧0.40.60.40.40.4=0.40.60.30.30.4,R¯∑i=12C^iPA=0.60.50.60.60.6∨0.50.50.50.70.4=0.60.50.60.70.6.Inspired by Proposition5, we present the matrix representations of α-PMGFLAO R¯∑i=1nC^iP,αA and α-PMGFUAO R¯∑i=1nC^iP,αA as follows:Theorem 3.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U=x1,…,xm and α∈0,1. Then, for any A∈FU(32)R¯∑i=1nC^iP,αA=∧i=1nD˜C^iβα⊙1m×1,R¯∑i=1nC^iP,αA=∨i=1nD˜C^iβα°0m×1,where ⊳ and ⊲ are first than ⊙ and °, respectively.Proof.
By Proposition5, it is immediate.Example 11.
(Continued from Example1). In Example 6, we get(33)D˜C^10.50.6=1101111011001101111011001andD˜C^20.50.6=1111011011101001101101011.
Then,(34)SR¯∑i=12C^iP,0.6A=∧i=12D˜C^i0.50.6⊳A⊙15×1=1101111011001101111011001⊳0.90.60.30.70.4⊙11111∧1111011011101001101101011⊳0.90.60.30.70.4⊙11111=0.90.610.70.40.90.610.70.4110.30.710.90.60.30.710.90.6110.4⊙11111∧0.90.60.30.710.90.610.70.40.910.3110.90.610.70.410.610.70.4⊙11111=0.40.40.30.30.4∧0.30.40.30.40.4=0.30.40.30.30.4,SR¯∑i=12C^iP,0.6A=∨i=12D˜C^i0.50.6⊲A∘05×1=1101111011001101111011001⊲0.90.60.30.70.4∘00000∨1111011011101001101101011⊲0.90.60.30.70.4∘00000=0.90.600.70.40.90.600.70.4000.30.700.90.60.30.700.90.6000.4∘00000∨0.90.60.30.700.90.600.70.40.900.3000.90.600.70.400.600.70.4∘00000=0.90.90.70.90.9∨0.90.90.90.90.7=0.90.90.90.90.9.Finally, we investigate the matrix representations of OMGFLAOD˜R¯∑i=1nC^iPA and OMGFUAO D˜R¯∑i=1nC^iPA.Theorem 4.
LetΓ˜=C^1,…,C^n be a n-fuzzy β-coverings of U. Then, for any A∈FU(35)D˜R¯∑i=1nC^iPA=∧i=1nD˜C^iβ∗A,D˜R¯∑i=1nC^iPA=∨i=1nD˜C^iβ⋅A.Proof.
By Definition2 and Theorem 2, it is immediate.Example 12.
(Continued from Example1). We have(36)D˜R¯∑i=12C^iPA=∧i=12DC^i0.6∗A=0.40.40.30.30.4∧0.30.40.30.40.4=0.30.40.30.30.4,D˜R¯∑i=12C^iPA=∧i=12DC^i0.6⋅A=0.90.65220.70.70.875∨0.90.71430.76190.70.7=0.90.71430.76190.70.875.


## 5. Applied Examples

To evaluate the efficiency of our matrix method for computing covering-based multigranulation fuzzy rough approximation operators, we construct some fuzzyβ-covering approximation spaces to compare the matrix approaches with set approaches. In Definition 3, we know that any fuzzy β-covering approximation space can be seen as a matrix. The procedure of constructing the fuzzy β-covering matrix is as follows: (1) The elements of the matrix are randomly chosen from 0,0.1,0.2,…,0.9,1. (2) For any row of the matrix, if the maximal number of the row is less than β, then we denote β as its maximal number. Hence, the matrix is the fuzzy β-covering matrix, i.e., a fuzzy β-covering approximation space. Hence, we consider a fuzzy β-covering approximation space only in this section. That is to say, we consider n=1 for optimistic CMFRSs and pessimistic CMFRSs. Since optimistic CMFRSs and pessimistic CMFRSs are the same when n=1, we consider optimistic CMFRSs only in this section.Finally, we compare the computational time of matrix methods and set methods (i.e., traditional methods) with different values ofβ, sizes of a universe and sizes of a fuzzy β-covering. All of the experiments were carried out on a personal computer with 64-bit Windows 10, ADM Ryzen 7 3700X 8-Core Processor 3.59 GHz, and 16 GB memory. The programming language was MATLAB r2016a.To compare the computational time of the matrix method and set method with different values ofβ in Figure 1, we set the size of U to 500 and the size of C^ to 500. The value of β ranges from 0.1 to 1, gradually increasing by a step of 0.1. In Figure 1, we can see that the computational time of the matrix method and set method decreases with the gradual increase in the value of β. The matrix method is more efficient than the set method with different values of β, especially with a small value of β.Figure 1
Computational time of matrix methods and set methods with differentβU=500,C^=500. (a) Comparison of matrix methods and set methods of lower approximation operators with different β. (b) Comparison of matrix methods and set methods of upper approximation operators with different β.
(a)(b)To compare the computational time of the matrix method and set method with different sizes ofU in Figure 2, we set the value of β to 0.5 and the size of C^ to 500. The size of U ranges from 100 to 1000, gradually increasing by a step of 100. In Figure 2, we can see that the computational time of the matrix method and set method increases with the gradual increase in the size of U. The matrix method is more efficient than the set method with different sizes of U.Figure 2
Computational time of matrix methods and set methods with different sizes ofUβ=0.5,C^=500. (a) Comparison of matrix methods and set methods of lower approximation operators with different sizes of U. (b) Comparison of matrix methods and set methods of upper approximation operators with different sizes of U.
(a)(b)To compare the computational time of the matrix method and set method with different sizes ofC^ in Figure 3, we set the value of β to 0.5 and the size of U to 500. The size of C^ ranges from 200 to 1100, gradually increasing by a step of 100. In Figure 3, we can see that the computational time of the matrix method and set method increases with the gradual increase in the size of U. The matrix method is more efficient than the set method with different sizes of C^, especially with large size of C^.Figure 3
Computational time of matrix methods and set methods with different sizes ofC^β=0.5,U=500. (a) Comparison of matrix methods and set methods of lower approximation operators with different sizes of C^. (b) Comparison of matrix methods and set methods of upper approximation operators with different sizes of C^.
(a)(b)


## 6. Conclusions

In this article, we present the matrix representations of MCFRS models. Hence, through some experiments, calculations can be easily implemented by computers. It gives a way to study the reduction of multigranulation fuzzy covering information systems by matrices in the future. Other multigranulation fuzzy covering rough set models (such as [34–36]) can also be represented by matrices in the future. On the contrary, the recent generalized types of fuzzy sets: (i) (2, 1)-Fuzzy sets: properties, weighted aggregated operators, and their applications to multicriteria decision-making methods [38]; (ii) new generalization of fuzzy soft sets: (a, b)-fuzzy soft sets [39]; (iii) three-way fuzzy sets [40] will be connected with the research content of this paper in further research.



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