# Certain Types of Covering-Based Multigranulation (ℐ,T)-Fuzzy Rough Sets with Application to Decision-Making

Authors: Jue Ma; Mohammed Atef; Shokry Nada; Ashraf Nawar
Journal: Complexity
 (2020)

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## Abstract

As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to define the family of complementary fuzzyβ-neighborhoods and thus three kinds of covering-based multigranulation (ℐ,T)-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation (ℐ,T)-fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.


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## Body

## 1. Introduction

Group decision-making aims at aggregating individual judgments to construct a composite group decision, which must be a true representative of individual preferences. The MAGDM methods choose among a discrete set of alternatives evaluated on multiple attributes and overall utility of the decision makers. MAGDM have some of the popular methods such as the weighted sum and the weighted product method (see, e.g., [1–7]). The theory of rough set was founded by Pawlak [8, 9] for dealing with the vagueness and granularity in information systems and data analysis. This theory has been applied to many different fields (see, e.g., [10–20]). Furthermore, we have noticed a wide range of generalized rough set models (see, e.g., [21–23]). Covering-based rough sets (CRSs) are considered to be one of the most studied generalized models. Pomykala [24, 25] obtained two pairs of dual approximation operators. Yao [26] studied these approximation operators by the concepts of neighborhood and granularity. Couso and Dubois [27] examined the two pairs within the context of incomplete information. Bonikowski et al. [28] established a CRS model based on the notion of minimal description. Zhu and Wang [29–32] presented several CRS models and discussed their relationships. Tsang et al. [33] and Xu and Zhang [34] proposed additional CRS models. Liu and Sai [35] compared Zhu′s CRS models and Xu and Zhang′s CRS models. Ma [36] constructed some types of neighborhood-related covering rough sets by using the definitions of the neighborhood and complementary neighborhood. In 2016, Ma [37] introduced the definition of fuzzy β-neighborhood. In 2017, Yang and Hu [38] constructed the definition of the fuzzy β-complementary neighborhood to establish some types of fuzzy covering-based rough sets. Zhang et al. [39], in 2019, established the fuzzy covering-based ℐ,T-fuzzy rough set models and applications to multiattribute decision-making. Also, in 2019, Zhan et al. [40] proposed covering-based multigranulation ℐ,T-fuzzy rough set models and applications in multiattribute group decision-making.The concept of a family fuzzyβ-neighborhoods was defined and their properties were studied by Zhan et al. [40]. Hence, to increase the lower approximation and decrease the upper approximation of Zhan’s model, this article’s contribution is to introduce three kinds of covering-based multigranulation ℐ,T-fuzzy rough sets models and explore the properties of these models with their relationships. Also, six kinds of covering-based variable precision multigranulation ℐ,T-fuzzy rough sets are demonstrated. An application to a practical problem illustrates their ability to help practitioners to make decisions. The outline of this paper is as follows. Section 2 gives technical preliminaries. Section 3 describes our three new types of covering-based multigranulation ℐ,T-fuzzy rough sets and also we introduce variable precision in order to produce the six types of covering-based variable precision multigranulation ℐ,T-fuzzy rough sets. Section 4 establishes relationships among our models. Section 5 puts forward a decision-making procedure that takes advantage of our theoretical framework. The conclusion is written in Section 6.


## 2. Preliminaries

In this section, we provide a brief survey of some notions used throughout the paper.Definition 1 (see [1]).
Suppose that the mappingT:0,1×0,1⟶0,1 is commutative, associative, and satisfies the increasing laws plus the boundary condition T1,x=x for each x∈0,1. We say that such a T is a t-norm (for triangular norm) of 0,1.
Among the most important continuoust-norms, we can cite(1)
The minimum operatorTℳx,y=x∧y(2)
The algebraic productTPx,y=x∗y(3)
The Lukasiewiczt-norm Tℒx,y=0∨x+y−1Definition 2 (see [1]).
Suppose that the mappingS:0,1×0,1⟶0,1 is commutative, associative, and satisfies the increasing laws plus the boundary condition S0,x=x for each x∈0,1. We say that such an S is an s-norm or a t-conorm of 0,1.
Continuouss-norms are(1)
The maximum operatorSℳx,y=x∨y(2)
The algebraic summationSSx,y=x+y(3)
The bounded summationSℒx,y=1∧x+y(4)
The probabilistic summationℬpx,y=x+y−x∗yDefinition 3 (see [1]).
A negator operator isN:0,1⟶0,1, an order-reversing mapping with the properties N0=1 and N1=0.
We say thatN is involutive when NNx=x for every x∈0,1.
The standard negator operator is defined asNx=1−x, for any x∈0,1.
Involutive negators are continuous.
Negators produce fuzzy complements. Involutive negators assure that whenX^∈ℱΩ and x∈Ω, we always obtain NNX^x=X^x.
We say thatT, a t-norm, and S, a t-conorm, are dual with respect to negator N, when for each x,y∈0,1, it must be the case that SNx,Ny=NTx,y and TNx,Ny=NSx,y.Definition 4 (see [1]).
A fuzzy implicator operator isℐ:0,1×0,1⟶0,1, and a mapping with the properties ℐ0,0=ℐ0,1=ℐ1,1=1 and ℐ1,0=0.
If, in addition,ℐ is such that x≤y⇒ℐx,z≥ℐy,z, respectively, y≤z⇒ℐx,y≤ℐx,z, for every x,y,z∈0,1, then ℐ is left monotonic decreasing, respectively, increasing.
We say thatℐ is hybrid monotonic when it is both left and right monotonic.
An implicatorℐ is a border implicator when ℐ1,x=x for each x∈0,1.
Next, we recall three relevant classes of implicator operators [1].Definition 5.
(1)
TheS-implicator defined by S and N is given: for each x,y∈0,1, ℐSx,y=SNx,y(2)
Theℛ-implicator defined by a continuous t-norm T is given: for each x,y∈0,1, ℐℛx,y=∨u∈0,1:Tx,u≤y(3)
IfT and S are dual with respect to N, the Qℒ-implicator defined from T, S, and N is given: for all x,y∈0,1, ℐQℒu,v=SNu,Tu,vWell-knownS-implicators are(1)
ℐℒx,y=1∧1−x+y, according to Sℒ and NS(2)
ℐKDx,y=1−x∨y, according to Sℳ and NS(3)
ℐPx,y=1−x+x∗y, according to Sℳ and NSDefinition 6 (see [42, 43]).
LetΩ be an arbitrary universal set, and ℱΩ be the fuzzy power set of Ω. We call Γ˜=C1˜,C2˜,…,Cm˜, with Ci˜∈ℱΩi=1,2,…,m, a fuzzy covering of Ω, if ∪i=1mCi˜x=1 for each x∈Ω.
As a generalization of fuzzy covering, Ma [37] defined a fuzzy β-covering by replacing 1 with a parameter β0<β≤1, that is, we call Γ˜=C1˜,C2˜,…,Cm˜, with Ci˜∈ℱΩi=1,2,…,m, a fuzzy β-covering of Ω, if ∪i=1mCi˜x≥β for each x∈Ω. Moreover, Ω,Γ˜ is called a fuzzy β-covering approximation space (briefly, FβCAS).Definition 7 (see [37]).
LetΩ,Γ˜ be a FβCAS with Γ˜=C1˜,C2˜,…,Cm˜ for some β∈0,1. Then, for each x∈Ω, define the fuzzy β-neighborhood of x as follows:(1)Nxβ˜=∩C˜i∈Γ˜:C˜i≥β.Definition 8 (see [38]).
LetΩ,Γ˜ be a FβCAS for some β∈0,1. Then, for each x,y∈Ω, define the fuzzy complementary β-neighborhood of x as follows:(2)Mxβ˜y=Nyβ˜x.Definition 9.
(see [39]). Let Ω,Γ˜ be a FβCAS for some β∈0,1. For each x∈Ω and X^∈ℱΩ, define the fuzzy set C˜1−X^ (resp., C˜1+X^, C˜2−X^, C˜2+X^, C˜3−X^, C˜3+X^, C˜4−X^, and C˜4+X^) which is called the first type fuzzy lower approximation (resp., the first type of the fuzzy upper approximation, the second type of the fuzzy lower approximation, the second type of the fuzzy upper approximation, the third type of the fuzzy lower approximation, the third type of the fuzzy upper approximation, the fourth type of the fuzzy lower approximation, and the fourth type of the fuzzy upper approximation), briefly, 1-FCITFLA (resp., 1-FCITFUA, 2-FCITFLA, 2-FCITFUA, 3-FCITFLA, 3-FCITFUA, 4-FCITFLA, and 4-FCITFUA), where(3)C˜1−X^x=∧y∈ΩℐNxβ˜y,X^y,C˜1+X^x=∨y∈ΩTNxβ˜y,X^y∀x∈Ω,C˜2−X^x=∧y∈ΩℐMxβ˜y,X^y,C˜2+X^x=∨y∈ΩTMxβ˜y,X^y∀x∈Ω,C˜3−X^x=∧y∈ΩℐNxβ˜y∧Mxβ˜y,X^y,C˜3+X^x=∨y∈ΩTNxβ˜y∧Mxβ˜y,X^y∀x∈Ω,C˜4−X^x=∧y∈ΩℐNxβ˜y∨Mxβ˜y,X^y,C˜4+X^x=∨y∈ΩTNxβ˜y∨Mxβ˜y,X^y∀x∈Ω.
IfC˜1−X^ (resp., C˜2−X^, C˜3−X^, C˜4−X^)≠C˜1+X^ (resp., C˜2+X^, C˜3+X^, C˜4+X^), then X^ is called the first type of a fuzzy β-covering-based ℐ,T-fuzzy rough set (resp., the second type of a fuzzy β-covering-based ℐ,T-fuzzy rough set, the third type of a fuzzy β-covering-based ℐ,T-fuzzy rough set, and the fourth type of a fuzzy β-covering-based ℐ,T-fuzzy rough set), briefly, 1-FCITFRS (resp., 2-FCITFRS, 3-FCITFRS, and 4-FCITFRS).
Zhan et al. [40] defined the covering-based multigranulation ℐ,T-fuzzy rough set models (briefly, CMGITFRSs). So, in the following, some basic notions related to CMGITFRSs are given.
Suppose thatΓ˜=C1˜,C2˜,…,Cm˜ be m fuzzy β-coverings of Ω for some β∈0,1, where Ci˜=Ci1˜,Ci2˜,…,Cimi˜, for all i=1,2,…,n. Then, for each x∈Ω, define the family of fuzzy β-neighborhoods as follows:(4)N˜Ci˜xβ=Cij∈Ci˜:x∈Cij,j=1,…,m.Definition 10 (see [40]).
LetΩ,Γ˜ be an FβCAS and Γ˜=C1˜,C2˜,…,Cm˜ be m fuzzy β-coverings of Ω for some β∈0,1, where Ci˜=Ci1˜,Ci2˜,…,Cimi˜, for all i=1,2,…,n. For each X^∈ℱΩ, the set ℒ∑i=1nCi˜ℐo0X^ (resp., U∑i=1nCi˜To0X^, ℒ∑i=1nCi˜ℐp0X^, and U∑i=1nCi˜Tp0X^) is called the optimistic multigranulation ℐ,T-fuzzy rough lower approximation (resp., the optimistic multigranulation ℐ,T-fuzzy rough upper approximation, the pessimistic multigranulation ℐ,T-fuzzy rough lower approximation, and the pessimistic multigranulation ℐ,T-fuzzy rough upper approximation), briefly, 0-OMGITFRLA (resp., 0-OMGITFRUA, 0-PMGITFRLA, and 0-PMGITFRUA), where(5)ℒ∑i=1nCi˜ℐo0X^x=∨i=1n∧y∈ΩℐN˜Ci˜xβy,X^y,U∑i=1nCi˜To0X^x=∧i=1n∨y∈ΩTN˜Ci˜xβy,X^y,∀x∈Ω,ℒ∑i=1nCi˜ℐp0X^x=∧i=1n∧y∈ΩℐN˜Ci˜xβy,X^y,U∑i=1nCi˜Tp0X^x=∨i=1n∨y∈ΩTN˜Ci˜xβy,X^y∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo0X^ (resp., ℒ∑i=1nCi˜ℐp0X^)≠U∑i=1nCi˜To0X^ (resp., U∑i=1nCi˜Tp0X^), then X^ is called a covering-based optimistic multigranulation ℐ,T-fuzzy rough set (resp., a covering-based pessimistic multigranulation ℐ,T-fuzzy rough set), briefly, 0-COMGITFRS (resp., 0-CPMGITFRS).Definition 11 (see [40]).
LetΩ,Γ˜ be an FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1, where Ci˜=Ci1˜,Ci2˜,…,Cimi˜, for all i=1,2,…,n. For any X^∈ℱΩ and a variable precision parameter γ∈0,1. Then, the set Iℒ∑i=1nCi˜ℐvpX^ (resp., IU∑i=1nCi˜TvpX^, IIℒ∑i=1nCi˜ℐvpX^, and IIU∑i=1nCi˜TvpX^) is called the first type of the variable precision multigranulation ℐ,T-fuzzy lower approximation (resp., the first type of the variable precision multigranulation ℐ,T-fuzzy upper approximation, the second type of the variable precision multigranulation ℐ,T-fuzzy lower approximation, and the second type of the variable precision multigranulation ℐ,T-fuzzy upper approximation), briefly, I-VPMGITFLA (resp., I-VPMGITFUA, II-VPMGITFLA, and II-VPMGITFUA), where(6)Iℒ∑i=1nCi˜ℐvpX^x=∨i=1n∧X^y≤γℐN˜Ci˜xβy,γ∧∧X^y>γℐN˜Ci˜xβy,X^y,∀x,y∈Ω,IU∑i=1nCi˜TvpX^x=∧i=1n∨X^y≥1−γTN˜Ci˜xβy,1−γ∨∨X^y<1−γTN˜Ci˜xβy,X^y,∀x,y∈Ω,IIℒ∑i=1nCi˜ℐvpX^x=∧i=1n∧X^y≤γℐN˜Ci˜xβy,γ∧∧X^y>γℐN˜Ci˜xβy,X^y,∀x,y∈Ω,IIU∑i=1nCi˜TvpX^x=∨i=1n∨X^y≥1−γTN˜Ci˜xβy,1−γ∨∨X^y<1−γTN˜Ci˜xβy,X^y,∀x,y∈Ω.
IfIℒ∑i=1nCi˜ℐvpX^ (resp., IIℒ∑i=1nCi˜ℐvpX^)≠IU∑i=1nCi˜TvpX^ (resp., IIU∑i=1nCi˜TvpX^), then X^ is called a covering-based I-variable precision multigranulation ℐ,T-fuzzy rough set (resp., a covering-based II-variable precision multigranulation ℐ,T-fuzzy rough set), briefly, I-VPMGITFRS (resp., II-VPMGITFRS); otherwise, it is I-variable precision multigranulation fuzzy definable (resp., II-variable precision multigranulation fuzzy definable).


## 3. Three Types of Covering-Based Multigranulationℐ,T-Fuzzy Rough Sets

Here, three new kinds of covering-based multigranulationℐ,T-fuzzy rough sets models (briefly, CMGITFRS) are introduced. Also, some of their properties are investigated.Assume thatΓ˜=C1˜,C2˜,…,Cm˜ be m fuzzy β-coverings of Ω for some β∈0,1, where Ci˜=Ci1˜,Ci2˜,…,Cimi˜, for all i=1,2,…,n. Then, for each x∈Ω, define the family of fuzzy β-neighborhoods as follows:(7)M˜Ci˜xβy=N˜Ci˜yβx.Example 1.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜ be 2 fuzzy β-coverings of Ω, where Ω=x1,x2,x3,x4,x5,x6, β=0.6, and C1˜=C11˜,C12˜,…,C14˜ and C2˜=C21˜,C22˜,…,C25˜ as in Tables 1 and 2.
Then, we introduce the family of a fuzzyβ-neighborhood and fuzzy complementary β-neighborhood for Γ˜=C1˜,C2˜ as follows in Tables 3–6.Table 1
Table forC˜1.
C˜11C˜12C˜13C˜14x10.70.80.30.4x20.30.60.50.2x30.40.10.90.7x40.80.40.60.3x50.20.70.40.8x60.50.90.60.1Table 2
Table forC˜2.
C˜21C˜22C˜23C˜24C˜25x10.60.30.50.40.7x20.90.70.10.30.4x30.30.60.40.80.3x40.40.20.90.70.6x50.20.80.40.60.5x60.70.40.30.90.2Table 3
Table forN˜xβ of C1˜.
x1x2x3x4x5x6N˜x10.50.70.30.10.40.20.5N˜x20.50.30.50.10.40.40.6N˜x30.50.30.20.70.30.40.1N˜x40.50.30.30.40.60.20.5N˜x50.50.40.20.10.30.70.1N˜x60.50.30.30.10.40.20.5Table 4
Table forN˜xβ of C2˜.
x1x2x3x4x5x6N˜x10.50.50.10.30.40.20.2N˜x20.50.30.70.30.20.20.4N˜x30.50.30.30.60.20.20.4N˜x40.50.40.10.30.60.40.2N˜x50.50.40.30.30.20.50.2N˜x60.50.40.30.30.40.20.7Table 5
Table forM˜xβ of C1˜.
x1x2x3x4x5x6M˜x10.50.70.30.30.30.40.3M˜x20.50.30.50.20.30.20.3M˜x30.50.10.10.70.40.10.1M˜x40.50.40.40.30.60.30.4M˜x50.50.20.40.40.20.70.2M˜x60.50.50.60.10.50.10.5Table 6
Table forM˜xβ of C2˜.
x1x2x3x4x5x6M˜x10.50.50.30.30.40.40.4M˜x20.50.10.70.30.10.30.3M˜x30.50.30.30.60.30.30.3M˜x40.50.40.20.20.60.20.4M˜x50.50.20.20.20.40.50.2M˜x60.50.20.40.40.20.20.7


### 3.1. Three Types of the Optimistic Multigranulationℐ,T-Fuzzy Rough Sets

In the following, three kinds of COMGITFRS models are given and some of their properties are presented.Definition 12.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1, for each X^∈ℱΩ. Then, the first type of the optimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 1-OMGITFLA) ℒ∑i=1nCi˜ℐo1X^ and the first type of the optimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 1-OMGITFUA) U∑i=1nCi˜To1X^ are, respectively, defined as follows:(8)ℒ∑i=1nCi˜ℐo1X^x=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y,U∑i=1nCi˜To1X^x=∧i=1n∨y∈ΩTM˜Ci˜xβy,X^y,∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo1X^≠U∑i=1nCi˜To1X^, then X^ is called a covering-based optimistic multigranulation (ℐ,T)-fuzzy rough set (briefly, 1-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.Example 2.
Let us consider Example1. If X^=0.6/x1+0.1/x2+0.7/x3+0.2/x4+0.5/x5+0.4/x6, then we have the following.Case 1
Let us fixℐ=ℐ∗ based on SP and NS, and T=TP:(9)ℒ∑i=1nCi˜ℐo1X^=0.72x1+0.55x2+0.73x3+0.52x4+0.68x5+0.58x6,U∑i=1nCi˜To1X^=0.3x1+0.18x2+0.42x3+0.24x4+0.25x5+0.28x6.Case 2
Let us fixℐ=ℐKD based on Sℳ and NS, and T=Tℳ. Then, we have the following results:(10)ℒ∑i=1nCi˜ℐo1X^=0.6x1+0.5x2+0.7x3+0.4x4+0.5x5+0.4x6,U∑i=1nCi˜To1X^=0.5x1+0.3x2+0.6x3+0.4x4+0.5x5+0.4x6.Theorem 1.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1, for each X^∈ℱΩ. Then, we have the following properties:(1)
Ifℐ is an S-implicator based on S, a continuous t-conorm, and N, an involutive negator,(i)
ℒ∑i=1nCi˜ℐo1X^=U∑i=1nCi˜To1X^cNcN(ii)
U∑i=1nCi˜To1X^=ℒ∑i=1nCi˜ℐo1X^cNcN(2)
If eitherℐ is an ℛ-implicator based on T, a continuous t-norm, and N, a negator induced by ℐ, or ℐ is a Qℒ-implicator based on T, a continuous t-norm, and N, an involutive negator,(i)
ℒ∑i=1nCi˜ℐo1X^⊆U∑i=1nCi˜To1X^cNcN(ii)
U∑i=1nCi˜To1X^⊆ℒ∑i=1nCi˜ℐo1X^cNcN(3)
Ifℐ satisfies left monotonicity, then(i)
ℒ∑i=1nCi˜ℐo1Ω=Ω(ii)
U∑i=1nCi˜To1∅=∅(4)
Ifℐ satisfies right monotonicity and X^⊆Y^, then(i)
ℒ∑i=1nCi˜ℐo1X^⊆ℒ∑i=1nCi˜ℐo1Y^(ii)
U∑i=1nCi˜To1X^⊆U∑i=1nCi˜To1Y^(5)
Ifℐ satisfies right monotonicity, then ℒ∑i=1nCi˜ℐo1X^∩Y^=ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Furthermore, if ℐ and T satisfy ℐu,Tv,w≥Tℐu,v,ℐu,w, for all u,v,w∈0,1, then ℒ∑i=1nCi˜ℐo1X^∩Y^⊇ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Also, if T,T∗ satisfies Tu,T∗v,w≤T∗Tu,v,Tu,w for all u,v,w∈0,1, then U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1X^∩U∑i=1nCi˜To1Y^.(6)
Ifℐ and S satisfy ℐu,Sv,w≥Sℐu,v,ℐu,w for every u,v,w∈0,1, then ℒ∑i=1nCi˜ℐo1X^∪Y^⊇ℒ∑i=1nCi˜ℐo1X^∪ℒ∑i=1nCi˜ℐo1Y^. Also, if T and S satisfy the weakened distributivity laws, then U∑i=1nCi˜To1X^∪Y^⊆U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1Y^. Specially, U∑i=1nCi˜To1X^∪Y^=U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1Y^.Proof.
We only need to prove (1)–(4) (i), (5), and (6), since the other proofs are similar:(1)
U∑i=1nCi˜To1X^cNcN=N∧i=1n∨y∈ΩTM˜Ci˜xβy,NX^y=N∧i=1n∨y∈ΩNℐM˜Ci˜xβy,X^y=NN∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y=ℒ∑i=1nCi˜ℐo1X^.(2)
1U∑i=1nCi˜ToX^cNcN=N∧i=1n∨y∈ΩTM˜Ci˜xβy,NX^y=N∧i=1n∨y∈ΩNℐM˜Ci˜xβy,X^y=NN∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y≥∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y=ℒ∑i=1nCi˜ℐo1X^.(3)
Sinceℐ is left monotonic, we have ℒ∑i=1nCi˜ℐo1Ω=∨i=1n∧y∈ΩℐM˜Ci˜xβy,Ω=∨i=1n∧y∈ΩℐM˜Ci˜xβy,1=1=Ω. Also, we have U∑i=1nCi˜To1∅=∧i=1n∨y∈ΩTM˜Ci˜xβy,∅=∧i=1n∨y∈ΩTM˜Ci˜xβy,0=0=∅.(4)
Sinceℐ is right monotonic, we have ℒ∑i=1nCi˜ℐo1X^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^⊆∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1Y^. Thus, ℒ∑i=1nCi˜ℐo1X^⊆ℒ∑i=1nCi˜ℐo1Y^ holds.(5)
Sinceℐ is right monotonic, we have ℒ∑i=1nCi˜ℐo1X^∩Y^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧Y^∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Since ℐ and T satisfy ℐu,Tv,w≥Tℐu,v,ℐu,w, for all u,v,w∈0,1. Then, we have ℒ∑i=1nCi˜ℐo1X^∩Y^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧Y^≥∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Thus, ℒ∑i=1nCi˜ℐo1X^∩Y^⊇ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Also, since X^∩Y^⊆X^ and X^∩Y^⊆Y^, from (3) above, we have U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1X^ and U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1Y^. Therefore, U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1X^∩U∑i=1nCi˜To1Y^.(6)
Sinceℐ and S satisfy ℐu,Tv,w≥Tℐu,v,ℐu,w, for all u,v,w∈0,1. We conclude that ℒ∑i=1nCi˜ℐo1X^∪Y^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∨Y^≥∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∨∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1X^∪ℒ∑i=1nCi˜ℐo1Y^.Hence,ℒ∑i=1nCi˜ℐo1X^∪Y^⊇ℒ∑i=1nCi˜ℐo1X^∪ℒ∑i=1nCi˜ℐo1Y^. Also, ∀x∈Ω, we have U∑i=1nCi˜To1X^∪Y^x=∧i=1n∨y∈ΩTM˜Ci˜xβy,SX^∪Y^y≤∧i=1n∨y∈ΩSTM˜Ci˜xβy,X^y,TM˜Ci˜xβy,Y^y.Since the weakened distributivity laws are satisfied,(11)≤S∧i=1n∨y∈ΩTM˜Ci˜xβy,X^y,∧i=1n∨y∈ΩTM˜Ci˜xβy,Y^y=SU∑i=1nCi˜To1X^x,U∑i=1nCi˜To1Y^x=U∑i=1nCi˜To1X^x∨U∑i=1nCi˜To1Y^x=U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1X^x.In particular, ifx∈Ω, we have(12)U∑i=1nCi˜To1X^∪Y^x=∧i=1n∨y∈ΩTM˜Ci˜xβy,X^∪Y^y=∧i=1n∨y∈ΩTM˜Ci˜xβy,X^y∨∧y∈ΩTM˜Ci˜xβy,Y^y=U∑i=1nCi˜To1X^x∨1UTY^x=U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1Y^x.Definition 13.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, the second type of the optimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 2-OMGITFLA) ℒ∑i=1nCi˜ℐo2X^ and the second type of the optimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 2-OMGITFUA) U∑i=1nCi˜To2X^ are, respectively, defined as follows:(13)ℒ∑i=1nCi˜ℐo2X^x=∨i=1n∧y∈ΩℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y,U∑i=1nCi˜To2X^x=∧i=1n∨y∈ΩTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo2X^≠U∑i=1nCi˜To2X^, then X^ is called a covering-based optimistic multigranulation ℐ,T-fuzzy rough set (briefly, 2-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.Example 3.
(continued from Example2). We compute N˜xiβ∧M˜xiβ for all xi∈Ω, where i=1,2,…,6 for some β=0.5, as shown in Tables 7 and 8 as follows.
Now, we calculate the 2-OMGITFLA and 2-OMGITFUA as explored in the following two cases.Case 1
Let us fixℐ=ℐ∗ based on SP and NS and T=TP. So,(14)ℒ∑i=1nCi˜ℐo2X^=0.72x1+0.55x2+0.76x3+0.52x4+0.73x5+0.68x6,U∑i=1nCi˜To2X^=0.3x1+0.18x2+0.42x3+0.21x4+0.25x5+0.2x6.Case 2
Let us fixℐ=ℐKD based on Sℳ and NS and T=Tℳ. Thus,(15)ℒ∑i=1nCi˜ℐo2X^=0.6x1+0.5x2+0.7x3+0.4x4+0.5x5+0.5x6,U∑i=1nCi˜To2X^=0.5x1+0.3x2+0.6x3+0.4x4+0.5x5+0.4x6.Table 7
Table forN˜xi0.5∧M˜xi0.5 of C˜1.
x1x2x3x4x5x6N˜x10.5∧M˜x10.50.70.30.10.30.20.3N˜x20.5∧M˜x20.50.30.50.10.30.20.3N˜x30.5∧M˜x30.50.10.10.70.30.10.1N˜x40.5∧M˜x40.50.30.30.30.60.20.4N˜x50.5∧M˜x50.50.20.20.10.20.70.1N˜x60.5∧M˜x60.50.30.30.10.40.10.5Table 8
Table forN˜xi0.5∧M˜xi0.5 of C˜2.
x1x2x3x4x5x6N˜x10.5∧M˜x10.50.50.10.30.40.20.2N˜x20.5∧M˜x20.50.10.70.30.10.20.3N˜x30.5∧M˜x30.50.30.30.60.20.20.3N˜x40.5∧M˜x40.50.40.10.20.60.20.2N˜x50.5∧M˜x50.50.20.20.20.20.50.2N˜x60.5∧M˜x60.50.20.30.30.20.20.7Remark 1.
Definition13 satisfies Theorem 1.Definition 14.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. For each X^∈ℱΩ, the third type of the optimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 3-OMGITFLA) ℒ∑i=1nCi˜ℐo3X^ and the third type of the optimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 3-OMGITFUA) U∑i=1nCi˜To3X^ are, respectively, defined as follows:(16)ℒ∑i=1nCi˜ℐo3X^x=∨i=1n∧y∈ΩℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y,U∑i=1nCi˜To3X^x=∧i=1n∨y∈ΩTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo3X^≠U∑i=1nCi˜To3X^, then X^ is called a covering-based optimistic multigranulation (ℐ,T)-fuzzy rough set (briefly, 3-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.Example 4.
(continued from Example2). We compute N˜xiβ∨M˜xiβ for all xi∈Ω, where i=1,2,…,6 for some β=0.5, as shown in Tables 9 and 10.
The following two cases calculate the 3-OMGITFLA and 3-OMGITFUA, respectively.Case 1
Let us fixℐ=ℐ∗ based on SP and NS and T=TP. So,(17)ℒ∑i=1nCi˜ℐo3X^=0.68x1+0.64x2+0.73x3+0.52x4+0.68x5+0.58x6,U∑i=1nCi˜To3X^=0.3x1+0.21x2+0.42x3+0.24x4+0.25x5+0.28x6.Case 2
Let us fixℐ=ℐKD based on Sℳ and NS and T=Tℳ. Thus,(18)ℒ∑i=1nCi˜ℐo3X^=0.6x1+0.4x2+0.6x3+0.4x4+0.5x5+0.4x6,U∑i=1nCi˜To3X^=0.5x1+0.4x2+0.6x3+0.4x4+0.5x5+0.4x6.Table 9
Table forN˜xiβ∨M˜xiβ of C˜1.
x1x2x3x4x5x6N˜x10.5∨M˜x10.50.70.30.30.40.40.5N˜x20.5∨M˜x20.50.30.50.20.40.40.6N˜x30.5∨M˜x30.50.30.20.70.40.40.1N˜x40.5∨M˜x40.50.40.40.40.60.30.5N˜x50.5∨M˜x50.50.40.40.40.30.70.2N˜x60.5∨M˜x60.50.50.60.10.50.20.5Table 10
Table forN˜xiβ∨M˜xiβ of C˜2.
x1x2x3x4x5x6N˜x10.5∨M˜x10.50.50.30.30.40.40.4N˜x20.5∨M˜x20.50.30.70.30.20.30.4N˜x30.5∨M˜x30.50.30.30.60.30.30.4N˜x40.5∨M˜x40.50.40.20.30.60.40.4N˜x50.5∨M˜x50.50.40.30.30.40.50.2N˜x60.5∨M˜x60.50.40.40.40.40.20.7Remark 2.
Definition14 satisfies Theorem 1.


### 3.2. Three Types of the Pessimistic Multigranulationℐ,T-Fuzzy Rough Sets

In the following, we introduce three kinds of CPMGITFRS models and study some of their properties.LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. For each X^∈ℱΩ. We have three models of the pessimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 1-PMGITFLA, 2-PMGITFLA, and 3-PMGITFLA) and three model of the pessimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 1-PMGITFUA, 2-PMGITFUA, and 3-PMGITFUA) are, respectively, defined as follows.Model 1:(19)ℒ∑i=1nCi˜ℐp1X^x=∧i=1n∧y∈ΩℐM˜Ci˜xβy,X^y,U∑i=1nCi˜Tp1X^x=∨i=1n∨y∈ΩTM˜Ci˜xβy,X^y,∀x∈Ω.Model 2:(20)ℒ∑i=1nCi˜ℐp2X^x=∧i=1n∧y∈ΩℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y,U∑i=1nCi˜Tp2X^x=∨i=1n∨y∈ΩTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω.Model 3:(21)ℒ∑i=1nCi˜ℐp3X^x=∧i=1n∧y∈ΩℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y,U∑i=1nCi˜Tp3X^x=∨i=1n∨y∈ΩTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω.Ifℒ∑i=1nCi˜ℐp1X^ (resp., ℒ∑i=1nCi˜ℐp2X^, ℒ∑i=1nCi˜ℐp3X^) ≠U∑i=1nCi˜Tp1X^ (resp., U∑i=1nCi˜Tp2X^, U∑i=1nCi˜Tp3X^), then X^ is called a covering-based pessimistic multigranulation (ℐ,T)-fuzzy rough set (briefly, 1-CPMGITFRS, 2-CPMGITFRS, and 3-CPMGITFRS); otherwise, it is pessimistic multigranulation fuzzy definable.It is obvious that the properties of these mentioned models satisfy Theorem1.Example 5.
(continued from Examples2 and 3 and Remark 2). We have the following results.Case 1
Let us fixℐ=ℐ∗ based on SP and NS and T=TP. So,(i)
ℒ∑i=1nCi˜ℐp1X^=0.68/x1+0.37/x2+0.68/x3+0.52/x4+0.64/x5+0.46/x6 and U∑i=1nCi˜Tp1X^=0.42/x1+0.21/x2+0.49/x3+0.24/x4+0.28/x5+0.3/x6(ii)
ℒ∑i=1nCi˜ℐp2X^=0.68/x1+0.37/x2+0.73/x3+0.52/x4+0.58/x5+0.58/x6 and U∑i=1nCi˜Tp1X^=0.42/x1+0.21/x2+0.49/x3+0.24/x4+0.35/x5+0.28/x6(iii)
ℒ∑i=1nCi˜ℐp3X^=0.68/x1+0.37/x2+0.68/x3+0.52/x4+0.64/x5+0.46/x6 and U∑i=1nCi˜To3X^=0.42/x1+0.24/x2+0.49/x3+0.28/x4+0.35/x5+0.3/x6Case 2
Let us fixℐ=ℐKD based on Sℳ and NS and T=Tℳ. Thus,(i)
ℒ∑i=1nCi˜ℐp1X^=0.6/x1+0.3/x2+0.6/x3+0.4/x4+0.5/x5+0.4/x6 and U∑i=1nCi˜Tp1X^=0.6/x1+0.3/x2+0.7/x3+0.4/x4+0.5/x5+0.5/x6(ii)
ℒ∑i=1nCi˜ℐp2X^=0.6/x1+0.3/x2+0.7/x3+0.4/x4+0.5/x5+0.4/x6 and U∑i=1nCi˜Tp2X^=0.6/x1+0.3/x2+0.7/x3+0.4/x4+0.5/x5+0.4/x6(iii)
ℒ∑i=1nCi˜ℐp3X^=0.5/x1+0.3/x2+0.6/x3+0.4/x4+0.5/x5+0.4/x6 and U∑i=1nCi˜To3X^=0.6/x1+0.4/x2+0.7/x3+0.4/x4+0.5/x5+0.5/x6


### 3.3. Some Types of Covering-Based Variable Precision Multigranulationℐ,T-Fuzzy Rough Sets

In the following, six new kinds of CVPMITFRS are defined and their properties are investigated. It is clear that the properties of these mentioned models satisfy Theorem1. So, we only present the concepts and omit the properties.Definition 15.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. For each X^∈ℱΩ and a variable precision parameter γ∈0,1. Define the i-variable precision multigranulation ℐ,T-fuzzy lower approximation (briefly, i-VPMGITFLA) and the i-variable precision multigranulation ℐ,T-fuzzy upper approximation (briefly, i-VPMGITFUA), ∀i∈I,II,III,IV,V,VI are, respectively, defined as follows:(1)
ℒ∑i=1nCi˜ℐvpIX^x=∨i=1n∧X^y≤γℐM˜Ci˜xβy,γ∧∧X^y≤γℐM˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIX^x=∧i=1n∨X^y≥1−γTM˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy,X^y,∀x∈Ω(2)
ℒ∑i=1nCi˜ℐvpIIX^x=∧i=1n∧X^y≤γℐM˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIIX^x=∨i=1n∨X^y≥1−γTM˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy,X^y,∀x∈Ω(3)
ℒ∑i=1nCi˜ℐvpIIIX^x=∨i=1n∧X^y≤γℐM˜Ci˜xβy∨N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIIIX^x=∧i=1n∨X^y≥1−γTM˜Ci˜xβy∨N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω(4)
ℒ∑i=1nCi˜ℐvpIVX^x=∧i=1n∧X^y≤γℐM˜Ci˜xβy∨N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIVX^x=∨i=1n∨X^y≥1−γTM˜Ci˜xβy∨N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω(5)
ℒ∑i=1nCi˜ℐvpVX^x=∨i=1n∧X^y≤γℐM˜Ci˜xβy∧N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpVX^x=∧i=1n∨X^y≥1−γTM˜Ci˜xβy∧N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω(6)
ℒ∑i=1nCi˜ℐvpVIX^x=∧i=1n∧X^y≤γℐM˜Ci˜xβy∧N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpVIX^x=∨i=1n∨X^y≥1−γTM˜Ci˜xβy∧N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω
Ifℒ∑i=1nCi˜ℐvpiX^≠U∑i=1nCi˜TvpiX^, then X^ is called a covering-based i-variable precision multigranulation ℐ,T-fuzzy rough set (briefly, i-VPMGITFRS); otherwise, it is i-variable precision multigranulation fuzzy definable.
The best way to explain the above definition is to give the following example.Example 6.
(continued from Example2). Assume that γ=0.5. Then, we have the following results:(1)
ℒ∑i=1nCi˜ℐvpIX^=0.8/x1+0.75/x2+0.82/x3+0.7/x4+0.75/x5+0.7/x6andU∑i=1nCi˜TvpIX^=0.25/x1+0.15/x2+0.3/x3+0.2/x4+0.25/x5+0.25/x6(2)
ℒ∑i=1nCi˜ℐvpIIX^=0.72/x1+0.65/x2+0.79/x3+0.7/x4+0.65/x5+0.65/x6andU∑i=1nCi˜TvpIIX^=0.35/x1+0.15/x2+0.35/x3+0.2/x4+0.35/x5+0.28/x6(3)
ℒ∑i=1nCi˜ℐvpIIIX^=0.8/x1+0.75/x2+0.82/x3+0.7/x4+0.75/x5+0.7/x6andU∑i=1nCi˜TvpIIIX^=0.25/x1+0.15/x2+0.3/x3+0.2/x4+0.25/x5+0.25/x6(4)
ℒ∑i=1nCi˜ℐvpIVX^=0.72/x1+0.65/x2+0.79/x3+0.7/x4+0.65/x5+0.65/x6andU∑i=1nCi˜TvpIVX^=0.35/x1+0.15/x2+0.35/x3+0.16/x4+0.35/x5+0.28/x6(5)
ℒ∑i=1nCi˜ℐvpVX^=0.8/x1+0.7/x2+0.8/x3+0.7/x4+0.75/x5+0.7/x6andU∑i=1nCi˜TvpVX^=0.25/x1+0.16/x2+0.3/x3+0.2/x4+0.25/x5+0.25/x6(6)
ℒ∑i=1nCi˜ℐvpVIX^=0.72/x1+0.65/x2+0.79/x3+0.7/x4+0.65/x5+0.65/x6andU∑i=1nCi˜TvpVIX^=0.35/x1+0.24/x2+0.35/x3+0.2/x4+0.35/x5+0.28/x6


## 3.1. Three Types of the Optimistic Multigranulationℐ,T-Fuzzy Rough Sets

In the following, three kinds of COMGITFRS models are given and some of their properties are presented.Definition 12.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1, for each X^∈ℱΩ. Then, the first type of the optimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 1-OMGITFLA) ℒ∑i=1nCi˜ℐo1X^ and the first type of the optimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 1-OMGITFUA) U∑i=1nCi˜To1X^ are, respectively, defined as follows:(8)ℒ∑i=1nCi˜ℐo1X^x=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y,U∑i=1nCi˜To1X^x=∧i=1n∨y∈ΩTM˜Ci˜xβy,X^y,∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo1X^≠U∑i=1nCi˜To1X^, then X^ is called a covering-based optimistic multigranulation (ℐ,T)-fuzzy rough set (briefly, 1-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.Example 2.
Let us consider Example1. If X^=0.6/x1+0.1/x2+0.7/x3+0.2/x4+0.5/x5+0.4/x6, then we have the following.Case 1
Let us fixℐ=ℐ∗ based on SP and NS, and T=TP:(9)ℒ∑i=1nCi˜ℐo1X^=0.72x1+0.55x2+0.73x3+0.52x4+0.68x5+0.58x6,U∑i=1nCi˜To1X^=0.3x1+0.18x2+0.42x3+0.24x4+0.25x5+0.28x6.Case 2
Let us fixℐ=ℐKD based on Sℳ and NS, and T=Tℳ. Then, we have the following results:(10)ℒ∑i=1nCi˜ℐo1X^=0.6x1+0.5x2+0.7x3+0.4x4+0.5x5+0.4x6,U∑i=1nCi˜To1X^=0.5x1+0.3x2+0.6x3+0.4x4+0.5x5+0.4x6.Theorem 1.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1, for each X^∈ℱΩ. Then, we have the following properties:(1)
Ifℐ is an S-implicator based on S, a continuous t-conorm, and N, an involutive negator,(i)
ℒ∑i=1nCi˜ℐo1X^=U∑i=1nCi˜To1X^cNcN(ii)
U∑i=1nCi˜To1X^=ℒ∑i=1nCi˜ℐo1X^cNcN(2)
If eitherℐ is an ℛ-implicator based on T, a continuous t-norm, and N, a negator induced by ℐ, or ℐ is a Qℒ-implicator based on T, a continuous t-norm, and N, an involutive negator,(i)
ℒ∑i=1nCi˜ℐo1X^⊆U∑i=1nCi˜To1X^cNcN(ii)
U∑i=1nCi˜To1X^⊆ℒ∑i=1nCi˜ℐo1X^cNcN(3)
Ifℐ satisfies left monotonicity, then(i)
ℒ∑i=1nCi˜ℐo1Ω=Ω(ii)
U∑i=1nCi˜To1∅=∅(4)
Ifℐ satisfies right monotonicity and X^⊆Y^, then(i)
ℒ∑i=1nCi˜ℐo1X^⊆ℒ∑i=1nCi˜ℐo1Y^(ii)
U∑i=1nCi˜To1X^⊆U∑i=1nCi˜To1Y^(5)
Ifℐ satisfies right monotonicity, then ℒ∑i=1nCi˜ℐo1X^∩Y^=ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Furthermore, if ℐ and T satisfy ℐu,Tv,w≥Tℐu,v,ℐu,w, for all u,v,w∈0,1, then ℒ∑i=1nCi˜ℐo1X^∩Y^⊇ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Also, if T,T∗ satisfies Tu,T∗v,w≤T∗Tu,v,Tu,w for all u,v,w∈0,1, then U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1X^∩U∑i=1nCi˜To1Y^.(6)
Ifℐ and S satisfy ℐu,Sv,w≥Sℐu,v,ℐu,w for every u,v,w∈0,1, then ℒ∑i=1nCi˜ℐo1X^∪Y^⊇ℒ∑i=1nCi˜ℐo1X^∪ℒ∑i=1nCi˜ℐo1Y^. Also, if T and S satisfy the weakened distributivity laws, then U∑i=1nCi˜To1X^∪Y^⊆U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1Y^. Specially, U∑i=1nCi˜To1X^∪Y^=U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1Y^.Proof.
We only need to prove (1)–(4) (i), (5), and (6), since the other proofs are similar:(1)
U∑i=1nCi˜To1X^cNcN=N∧i=1n∨y∈ΩTM˜Ci˜xβy,NX^y=N∧i=1n∨y∈ΩNℐM˜Ci˜xβy,X^y=NN∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y=ℒ∑i=1nCi˜ℐo1X^.(2)
1U∑i=1nCi˜ToX^cNcN=N∧i=1n∨y∈ΩTM˜Ci˜xβy,NX^y=N∧i=1n∨y∈ΩNℐM˜Ci˜xβy,X^y=NN∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y≥∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^y=ℒ∑i=1nCi˜ℐo1X^.(3)
Sinceℐ is left monotonic, we have ℒ∑i=1nCi˜ℐo1Ω=∨i=1n∧y∈ΩℐM˜Ci˜xβy,Ω=∨i=1n∧y∈ΩℐM˜Ci˜xβy,1=1=Ω. Also, we have U∑i=1nCi˜To1∅=∧i=1n∨y∈ΩTM˜Ci˜xβy,∅=∧i=1n∨y∈ΩTM˜Ci˜xβy,0=0=∅.(4)
Sinceℐ is right monotonic, we have ℒ∑i=1nCi˜ℐo1X^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^⊆∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1Y^. Thus, ℒ∑i=1nCi˜ℐo1X^⊆ℒ∑i=1nCi˜ℐo1Y^ holds.(5)
Sinceℐ is right monotonic, we have ℒ∑i=1nCi˜ℐo1X^∩Y^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧Y^∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Since ℐ and T satisfy ℐu,Tv,w≥Tℐu,v,ℐu,w, for all u,v,w∈0,1. Then, we have ℒ∑i=1nCi˜ℐo1X^∩Y^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧Y^≥∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∧∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Thus, ℒ∑i=1nCi˜ℐo1X^∩Y^⊇ℒ∑i=1nCi˜ℐo1X^∩ℒ∑i=1nCi˜ℐo1Y^. Also, since X^∩Y^⊆X^ and X^∩Y^⊆Y^, from (3) above, we have U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1X^ and U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1Y^. Therefore, U∑i=1nCi˜To1X^∩Y^⊆U∑i=1nCi˜To1X^∩U∑i=1nCi˜To1Y^.(6)
Sinceℐ and S satisfy ℐu,Tv,w≥Tℐu,v,ℐu,w, for all u,v,w∈0,1. We conclude that ℒ∑i=1nCi˜ℐo1X^∪Y^=∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∨Y^≥∨i=1n∧y∈ΩℐM˜Ci˜xβy,X^∨∨i=1n∧y∈ΩℐM˜Ci˜xβy,Y^=ℒ∑i=1nCi˜ℐo1X^∪ℒ∑i=1nCi˜ℐo1Y^.Hence,ℒ∑i=1nCi˜ℐo1X^∪Y^⊇ℒ∑i=1nCi˜ℐo1X^∪ℒ∑i=1nCi˜ℐo1Y^. Also, ∀x∈Ω, we have U∑i=1nCi˜To1X^∪Y^x=∧i=1n∨y∈ΩTM˜Ci˜xβy,SX^∪Y^y≤∧i=1n∨y∈ΩSTM˜Ci˜xβy,X^y,TM˜Ci˜xβy,Y^y.Since the weakened distributivity laws are satisfied,(11)≤S∧i=1n∨y∈ΩTM˜Ci˜xβy,X^y,∧i=1n∨y∈ΩTM˜Ci˜xβy,Y^y=SU∑i=1nCi˜To1X^x,U∑i=1nCi˜To1Y^x=U∑i=1nCi˜To1X^x∨U∑i=1nCi˜To1Y^x=U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1X^x.In particular, ifx∈Ω, we have(12)U∑i=1nCi˜To1X^∪Y^x=∧i=1n∨y∈ΩTM˜Ci˜xβy,X^∪Y^y=∧i=1n∨y∈ΩTM˜Ci˜xβy,X^y∨∧y∈ΩTM˜Ci˜xβy,Y^y=U∑i=1nCi˜To1X^x∨1UTY^x=U∑i=1nCi˜To1X^∪U∑i=1nCi˜To1Y^x.Definition 13.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, the second type of the optimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 2-OMGITFLA) ℒ∑i=1nCi˜ℐo2X^ and the second type of the optimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 2-OMGITFUA) U∑i=1nCi˜To2X^ are, respectively, defined as follows:(13)ℒ∑i=1nCi˜ℐo2X^x=∨i=1n∧y∈ΩℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y,U∑i=1nCi˜To2X^x=∧i=1n∨y∈ΩTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo2X^≠U∑i=1nCi˜To2X^, then X^ is called a covering-based optimistic multigranulation ℐ,T-fuzzy rough set (briefly, 2-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.Example 3.
(continued from Example2). We compute N˜xiβ∧M˜xiβ for all xi∈Ω, where i=1,2,…,6 for some β=0.5, as shown in Tables 7 and 8 as follows.
Now, we calculate the 2-OMGITFLA and 2-OMGITFUA as explored in the following two cases.Case 1
Let us fixℐ=ℐ∗ based on SP and NS and T=TP. So,(14)ℒ∑i=1nCi˜ℐo2X^=0.72x1+0.55x2+0.76x3+0.52x4+0.73x5+0.68x6,U∑i=1nCi˜To2X^=0.3x1+0.18x2+0.42x3+0.21x4+0.25x5+0.2x6.Case 2
Let us fixℐ=ℐKD based on Sℳ and NS and T=Tℳ. Thus,(15)ℒ∑i=1nCi˜ℐo2X^=0.6x1+0.5x2+0.7x3+0.4x4+0.5x5+0.5x6,U∑i=1nCi˜To2X^=0.5x1+0.3x2+0.6x3+0.4x4+0.5x5+0.4x6.Table 7
Table forN˜xi0.5∧M˜xi0.5 of C˜1.
x1x2x3x4x5x6N˜x10.5∧M˜x10.50.70.30.10.30.20.3N˜x20.5∧M˜x20.50.30.50.10.30.20.3N˜x30.5∧M˜x30.50.10.10.70.30.10.1N˜x40.5∧M˜x40.50.30.30.30.60.20.4N˜x50.5∧M˜x50.50.20.20.10.20.70.1N˜x60.5∧M˜x60.50.30.30.10.40.10.5Table 8
Table forN˜xi0.5∧M˜xi0.5 of C˜2.
x1x2x3x4x5x6N˜x10.5∧M˜x10.50.50.10.30.40.20.2N˜x20.5∧M˜x20.50.10.70.30.10.20.3N˜x30.5∧M˜x30.50.30.30.60.20.20.3N˜x40.5∧M˜x40.50.40.10.20.60.20.2N˜x50.5∧M˜x50.50.20.20.20.20.50.2N˜x60.5∧M˜x60.50.20.30.30.20.20.7Remark 1.
Definition13 satisfies Theorem 1.Definition 14.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. For each X^∈ℱΩ, the third type of the optimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 3-OMGITFLA) ℒ∑i=1nCi˜ℐo3X^ and the third type of the optimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 3-OMGITFUA) U∑i=1nCi˜To3X^ are, respectively, defined as follows:(16)ℒ∑i=1nCi˜ℐo3X^x=∨i=1n∧y∈ΩℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y,U∑i=1nCi˜To3X^x=∧i=1n∨y∈ΩTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω.
Ifℒ∑i=1nCi˜ℐo3X^≠U∑i=1nCi˜To3X^, then X^ is called a covering-based optimistic multigranulation (ℐ,T)-fuzzy rough set (briefly, 3-COMGITFRS); otherwise, it is optimistic multigranulation fuzzy definable.Example 4.
(continued from Example2). We compute N˜xiβ∨M˜xiβ for all xi∈Ω, where i=1,2,…,6 for some β=0.5, as shown in Tables 9 and 10.
The following two cases calculate the 3-OMGITFLA and 3-OMGITFUA, respectively.Case 1
Let us fixℐ=ℐ∗ based on SP and NS and T=TP. So,(17)ℒ∑i=1nCi˜ℐo3X^=0.68x1+0.64x2+0.73x3+0.52x4+0.68x5+0.58x6,U∑i=1nCi˜To3X^=0.3x1+0.21x2+0.42x3+0.24x4+0.25x5+0.28x6.Case 2
Let us fixℐ=ℐKD based on Sℳ and NS and T=Tℳ. Thus,(18)ℒ∑i=1nCi˜ℐo3X^=0.6x1+0.4x2+0.6x3+0.4x4+0.5x5+0.4x6,U∑i=1nCi˜To3X^=0.5x1+0.4x2+0.6x3+0.4x4+0.5x5+0.4x6.Table 9
Table forN˜xiβ∨M˜xiβ of C˜1.
x1x2x3x4x5x6N˜x10.5∨M˜x10.50.70.30.30.40.40.5N˜x20.5∨M˜x20.50.30.50.20.40.40.6N˜x30.5∨M˜x30.50.30.20.70.40.40.1N˜x40.5∨M˜x40.50.40.40.40.60.30.5N˜x50.5∨M˜x50.50.40.40.40.30.70.2N˜x60.5∨M˜x60.50.50.60.10.50.20.5Table 10
Table forN˜xiβ∨M˜xiβ of C˜2.
x1x2x3x4x5x6N˜x10.5∨M˜x10.50.50.30.30.40.40.4N˜x20.5∨M˜x20.50.30.70.30.20.30.4N˜x30.5∨M˜x30.50.30.30.60.30.30.4N˜x40.5∨M˜x40.50.40.20.30.60.40.4N˜x50.5∨M˜x50.50.40.30.30.40.50.2N˜x60.5∨M˜x60.50.40.40.40.40.20.7Remark 2.
Definition14 satisfies Theorem 1.


## 3.2. Three Types of the Pessimistic Multigranulationℐ,T-Fuzzy Rough Sets

In the following, we introduce three kinds of CPMGITFRS models and study some of their properties.LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. For each X^∈ℱΩ. We have three models of the pessimistic multigranulation ℐ,T-fuzzy lower approximation (briefly, 1-PMGITFLA, 2-PMGITFLA, and 3-PMGITFLA) and three model of the pessimistic multigranulation ℐ,T-fuzzy upper approximation (briefly, 1-PMGITFUA, 2-PMGITFUA, and 3-PMGITFUA) are, respectively, defined as follows.Model 1:(19)ℒ∑i=1nCi˜ℐp1X^x=∧i=1n∧y∈ΩℐM˜Ci˜xβy,X^y,U∑i=1nCi˜Tp1X^x=∨i=1n∨y∈ΩTM˜Ci˜xβy,X^y,∀x∈Ω.Model 2:(20)ℒ∑i=1nCi˜ℐp2X^x=∧i=1n∧y∈ΩℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y,U∑i=1nCi˜Tp2X^x=∨i=1n∨y∈ΩTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω.Model 3:(21)ℒ∑i=1nCi˜ℐp3X^x=∧i=1n∧y∈ΩℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y,U∑i=1nCi˜Tp3X^x=∨i=1n∨y∈ΩTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω.Ifℒ∑i=1nCi˜ℐp1X^ (resp., ℒ∑i=1nCi˜ℐp2X^, ℒ∑i=1nCi˜ℐp3X^) ≠U∑i=1nCi˜Tp1X^ (resp., U∑i=1nCi˜Tp2X^, U∑i=1nCi˜Tp3X^), then X^ is called a covering-based pessimistic multigranulation (ℐ,T)-fuzzy rough set (briefly, 1-CPMGITFRS, 2-CPMGITFRS, and 3-CPMGITFRS); otherwise, it is pessimistic multigranulation fuzzy definable.It is obvious that the properties of these mentioned models satisfy Theorem1.Example 5.
(continued from Examples2 and 3 and Remark 2). We have the following results.Case 1
Let us fixℐ=ℐ∗ based on SP and NS and T=TP. So,(i)
ℒ∑i=1nCi˜ℐp1X^=0.68/x1+0.37/x2+0.68/x3+0.52/x4+0.64/x5+0.46/x6 and U∑i=1nCi˜Tp1X^=0.42/x1+0.21/x2+0.49/x3+0.24/x4+0.28/x5+0.3/x6(ii)
ℒ∑i=1nCi˜ℐp2X^=0.68/x1+0.37/x2+0.73/x3+0.52/x4+0.58/x5+0.58/x6 and U∑i=1nCi˜Tp1X^=0.42/x1+0.21/x2+0.49/x3+0.24/x4+0.35/x5+0.28/x6(iii)
ℒ∑i=1nCi˜ℐp3X^=0.68/x1+0.37/x2+0.68/x3+0.52/x4+0.64/x5+0.46/x6 and U∑i=1nCi˜To3X^=0.42/x1+0.24/x2+0.49/x3+0.28/x4+0.35/x5+0.3/x6Case 2
Let us fixℐ=ℐKD based on Sℳ and NS and T=Tℳ. Thus,(i)
ℒ∑i=1nCi˜ℐp1X^=0.6/x1+0.3/x2+0.6/x3+0.4/x4+0.5/x5+0.4/x6 and U∑i=1nCi˜Tp1X^=0.6/x1+0.3/x2+0.7/x3+0.4/x4+0.5/x5+0.5/x6(ii)
ℒ∑i=1nCi˜ℐp2X^=0.6/x1+0.3/x2+0.7/x3+0.4/x4+0.5/x5+0.4/x6 and U∑i=1nCi˜Tp2X^=0.6/x1+0.3/x2+0.7/x3+0.4/x4+0.5/x5+0.4/x6(iii)
ℒ∑i=1nCi˜ℐp3X^=0.5/x1+0.3/x2+0.6/x3+0.4/x4+0.5/x5+0.4/x6 and U∑i=1nCi˜To3X^=0.6/x1+0.4/x2+0.7/x3+0.4/x4+0.5/x5+0.5/x6


## 3.3. Some Types of Covering-Based Variable Precision Multigranulationℐ,T-Fuzzy Rough Sets

In the following, six new kinds of CVPMITFRS are defined and their properties are investigated. It is clear that the properties of these mentioned models satisfy Theorem1. So, we only present the concepts and omit the properties.Definition 15.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. For each X^∈ℱΩ and a variable precision parameter γ∈0,1. Define the i-variable precision multigranulation ℐ,T-fuzzy lower approximation (briefly, i-VPMGITFLA) and the i-variable precision multigranulation ℐ,T-fuzzy upper approximation (briefly, i-VPMGITFUA), ∀i∈I,II,III,IV,V,VI are, respectively, defined as follows:(1)
ℒ∑i=1nCi˜ℐvpIX^x=∨i=1n∧X^y≤γℐM˜Ci˜xβy,γ∧∧X^y≤γℐM˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIX^x=∧i=1n∨X^y≥1−γTM˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy,X^y,∀x∈Ω(2)
ℒ∑i=1nCi˜ℐvpIIX^x=∧i=1n∧X^y≤γℐM˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIIX^x=∨i=1n∨X^y≥1−γTM˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy,X^y,∀x∈Ω(3)
ℒ∑i=1nCi˜ℐvpIIIX^x=∨i=1n∧X^y≤γℐM˜Ci˜xβy∨N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIIIX^x=∧i=1n∨X^y≥1−γTM˜Ci˜xβy∨N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω(4)
ℒ∑i=1nCi˜ℐvpIVX^x=∧i=1n∧X^y≤γℐM˜Ci˜xβy∨N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∨N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpIVX^x=∨i=1n∨X^y≥1−γTM˜Ci˜xβy∨N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∨N˜Ci˜xβy,X^y,∀x∈Ω(5)
ℒ∑i=1nCi˜ℐvpVX^x=∨i=1n∧X^y≤γℐM˜Ci˜xβy∧N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpVX^x=∧i=1n∨X^y≥1−γTM˜Ci˜xβy∧N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω(6)
ℒ∑i=1nCi˜ℐvpVIX^x=∧i=1n∧X^y≤γℐM˜Ci˜xβy∧N˜Ci˜xβy,γ∧∧X^y>γℐM˜Ci˜xβy∧N˜Ci˜xβy,X^y and U∑i=1nCi˜TvpVIX^x=∨i=1n∨X^y≥1−γTM˜Ci˜xβy∧N˜Ci˜xβy,1−γ∨∨X^y<1−γTM˜Ci˜xβy∧N˜Ci˜xβy,X^y,∀x∈Ω
Ifℒ∑i=1nCi˜ℐvpiX^≠U∑i=1nCi˜TvpiX^, then X^ is called a covering-based i-variable precision multigranulation ℐ,T-fuzzy rough set (briefly, i-VPMGITFRS); otherwise, it is i-variable precision multigranulation fuzzy definable.
The best way to explain the above definition is to give the following example.Example 6.
(continued from Example2). Assume that γ=0.5. Then, we have the following results:(1)
ℒ∑i=1nCi˜ℐvpIX^=0.8/x1+0.75/x2+0.82/x3+0.7/x4+0.75/x5+0.7/x6andU∑i=1nCi˜TvpIX^=0.25/x1+0.15/x2+0.3/x3+0.2/x4+0.25/x5+0.25/x6(2)
ℒ∑i=1nCi˜ℐvpIIX^=0.72/x1+0.65/x2+0.79/x3+0.7/x4+0.65/x5+0.65/x6andU∑i=1nCi˜TvpIIX^=0.35/x1+0.15/x2+0.35/x3+0.2/x4+0.35/x5+0.28/x6(3)
ℒ∑i=1nCi˜ℐvpIIIX^=0.8/x1+0.75/x2+0.82/x3+0.7/x4+0.75/x5+0.7/x6andU∑i=1nCi˜TvpIIIX^=0.25/x1+0.15/x2+0.3/x3+0.2/x4+0.25/x5+0.25/x6(4)
ℒ∑i=1nCi˜ℐvpIVX^=0.72/x1+0.65/x2+0.79/x3+0.7/x4+0.65/x5+0.65/x6andU∑i=1nCi˜TvpIVX^=0.35/x1+0.15/x2+0.35/x3+0.16/x4+0.35/x5+0.28/x6(5)
ℒ∑i=1nCi˜ℐvpVX^=0.8/x1+0.7/x2+0.8/x3+0.7/x4+0.75/x5+0.7/x6andU∑i=1nCi˜TvpVX^=0.25/x1+0.16/x2+0.3/x3+0.2/x4+0.25/x5+0.25/x6(6)
ℒ∑i=1nCi˜ℐvpVIX^=0.72/x1+0.65/x2+0.79/x3+0.7/x4+0.65/x5+0.65/x6andU∑i=1nCi˜TvpVIX^=0.35/x1+0.24/x2+0.35/x3+0.2/x4+0.35/x5+0.28/x6


## 4. The Relationships between COMGITFRS Models and CPMGITFRS Models

In this section, we explain relationships among our models. Through the proposed study, we have the following results.From Definitions10 and 12, we conclude the following results.Proposition 1.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, we have the following properties: (i) ℒ∑i=1nCi˜ℐo1X^=∪i=1nC˜2−X^ and ℒ∑i=1nCi˜ℐp1X^=∩i=1nC˜2−X^. (ii) U∑i=1nCi˜To1X^=∩i=1nC˜2+X^ and U∑i=1nCi˜Tp1X^=∪i=1nC˜2+X^.
By Definitions11 and 13, we have the following results.Proposition 2.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, we have the following properties:(i)
ℒ∑i=1nCi˜ℐo2X^=∪i=1nC˜3−X^ and ℒ∑i=1nCi˜ℐp2X^=∩i=1nC˜3−X^(ii)
U∑i=1nCi˜To2X^=∩i=1nC˜3+X^ and U∑i=1nCi˜Tp2X^=∪i=1nC˜3+X^
From Definitions11 and 14, we have the following results.Proposition 3.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, we have the following properties.(i)
ℒ∑i=1nCi˜ℐo3X^=∪i=1nC˜4−X^ and ℒ∑i=1nCi˜ℐp3X^=∩i=1nC˜4−X^(ii)
U∑i=1nCi˜To3X^=∩i=1nC˜4+X^ and U∑i=1nCi˜Tp3X^=∪i=1nC˜4+X^Proposition 4.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, we have the following properties:(i)
ℒ∑i=1nCi˜ℐo3X^⊆ℒ∑i=1nCi˜ℐo0X^⊆ℒ∑i=1nCi˜ℐo2X^(ii)
ℒ∑i=1nCi˜ℐo3X^⊆ℒ∑i=1nCi˜ℐo1X^⊆ℒ∑i=1nCi˜ℐo2X^(iii)
U∑i=1nCi˜To2X^⊆U∑i=1nCi˜To0X^⊆U∑i=1nCi˜To3X^(iv)
U∑i=1nCi˜To2X^⊆U∑i=1nCi˜To1X^⊆U∑i=1nCi˜To3X^Proposition 5.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, we have the following properties:(i)
ℒ∑i=1nCi˜ℐo2X^=ℒ∑i=1nCi˜ℐo0X^∨ℒ∑i=1nCi˜ℐo1X^(ii)
U∑i=1nCi˜To2X^=U∑i=1nCi˜To0X^∧U∑i=1nCi˜To1X^(iii)
ℒ∑i=1nCi˜ℐo3X^=ℒ∑i=1nCi˜ℐo0X^∧ℒ∑i=1nCi˜ℐo1X^(iv)
U∑i=1nCi˜To3X^=U∑i=1nCi˜To0X^∨U∑i=1nCi˜To1X^Remark 3.
LetΩ,Γ˜ be a FβCAS and Γ˜=C1˜,C2˜,…,Cn˜ be n fuzzy β-coverings of Ω for some β∈0,1. Then, for each X^∈ℱΩ, we have the following properties:(1)
ℒ∑i=1nCi˜ℐp1X^⊆ℒ∑i=1nCi˜ℐo1X^ and U∑i=1nCi˜To1X^⊆U∑i=1nCi˜Tp1X^(2)
ℒ∑i=1nCi˜ℐp2X^⊆ℒ∑i=1nCi˜ℐo2X^ and U∑i=1nCi˜To2X^⊆U∑i=1nCi˜Tp2X^(3)
ℒ∑i=1nCi˜ℐp3X^⊆ℒ∑i=1nCi˜ℐo3X^ and U∑i=1nCi˜To3X^⊆U∑i=1nCi˜Tp3X^(4)
ℒ∑i=1nCi˜ℐvpIIX^⊆ℒ∑i=1nCi˜ℐvpIX^ and U∑i=1nCi˜TvpIX^⊆U∑i=1nCi˜TvpIIX^(5)
ℒ∑i=1nCi˜ℐvpIVX^⊆ℒ∑i=1nCi˜ℐvpIIIX^ and U∑i=1nCiTvpIIIX^⊆U∑i=1nCiTvpIVX^(6)
ℒ∑i=1nℐvpVIX^⊆ℒi=1ℐvpVX^ and U∑i=1nCi˜TvpVX^⊆U∑i=1nCi˜TvpVIX^


## 5. An Application to Decision-Making

In this section, we apply the proposed method to make a decision on a real-life problem.


### 5.1. Description and Process

LetΩ=u1,u2,…,un be n alternatives and ℰ=e1,e2,…,el be l decision makers. Suppose for each νi is a weighted vector correspondingly to ei, where νi≥0 for i=1,…,l and ∑i=1lνi=1. Hence, Ci˜=Ci1˜,Ci2˜,…,Cimi˜, for all i=1,2,…,n is a set of attributes. A family of mappings G=gl, where gl:Ω×Ci˜⟶0,1. So, we construct the MAGDM with fuzzy information system Ω,C˜,ℰ,G. Based on the proposed covering methods, we present a decision-making algorithm to find the best alternative through the following steps:Step 1:
Construct the decision-making object with fuzzy information of the universe of discourse. Through the rule of fuzzy TOPSIS method, we have(22)P¯l=C˜lj,∨1≤i≤nglui,C˜lj:j=1,…,ml=1,…,t=C˜l1,∨gl1,…,C˜lm,∨glm,P¯l=C˜lj,∧1≤i≤nglui,C˜lj:j=1,…,ml=1,…,t=C˜l1,∧gl1,…,C˜lm,∧glmwhere∧ and ∧ denote “max” and “min,” respectively.Step 2:
Compute the respective distancesD¯ and D¯ as follows:(23)D¯l=SC˜ljui,C˜ljP¯=1m∑j=1mC˜ljui−∨glj2,D¯l=SC˜ljui,C˜ljP¯=1m∑j=1mC˜ljui−∧glj2,whereSY^,Z^=1/m∑j=1mY^ui−Z^ui2 and m is the cardinality of Ω.Step 3:
Calculate the lower and upper approximations of the best and worst decision-making objects with fuzzy information by Definition13 (2-OMGITFLA and 2-OMGITFUA).Step 4:
Calculate the closeness coefficient degree byℛlui=Wl¯ui/Wl¯ui+Wl¯ui, where(24)Wl¯ui=ℬpℒ∑r=1kCr˜ℐo3D¯lui,U∑r=1kCr˜To3D¯lui,Wl¯ui=ℬpℒ∑r=1kCr˜ℐo3D¯lui,U∑r=1kCr˜To3D¯lui,be the worst and the best decision-making objects for individual ranking function of expertl for the candidates ui, and 0≤Wl¯ui,Wl¯ui≤1.Step 5:
Calculate the group ranking function by the following equationℛui=∑l=1tνlℛlui, and hence rank the alternatives.According to these steps, we give an algorithm to solve the decision-making problems based on the 2-COMGITFRS model. The steps corresponding to it are summarized in Algorithm1.Algorithm 1: Algorithm for MAGDM with the TOPSIS method.
Input: Fuzzy information systems Ω,C˜,ℰ,G.Output: Decision-Making.(1)
EnterX^, β, Γ=Ci˜:i=1,…,n and Ω=xj:j=1,…,m.(2)
From Definition7, calculate fuzzy β-neighborhood.(3)
FromStep 2 and by Definition 8, calculate complementary fuzzy β-neighborhood(4)
FromSteps 2 and 3, calculate N˜C1˜u1β∧M˜C1˜uiβ.(5)
EnterP¯l and P¯l.(6)
Calculate the distancesD¯l and D¯l.(7)
From Definition13, calculate the lower approximation ℒ∑r=1kCr˜ℐo3D¯l and the upper approximation U∑r=1kCr˜ℐo3D¯l.(8)
Calculate the worst and the best decision-making objectsWl¯ and Wl¯ for each individual decision-maker.(9)
Calculate the individual ranking functionℛl.(10)
Calculate the group ranking functionℛ.(11)
Obtain the decision.


### 5.2. Applied Example

The abovementioned steps have been illustrated with a numerical example as shown next.Example 7 (see [40]).
LetΩ=u1,u2,…,u6 be six system analysis engineers and Γ˜=emotional steadiness C1, oral communication skill C2, personality C3, past experience C4, self-confidence C5 be the attribute set of the basic description of the candidates. Suppose that three experts e1,e2, and e3 are invited to evaluate the system analysis engineers according to their specialized knowledge. The weights of every expert are ν1=0.4,ν2=0.1, and ν3=0.5. The following steps of the stated algorithm are implemented here.Step 1:
Experts evaluate each candidate under the set of the attribute and present their judgments with the real values. These values are summarized in Tables11–13.Step 2:
According to the importance of these five attributes, we give the following results for each expert:(25)P¯1=C11˜,0.82,C12˜,0.76,C13˜,0.74,C14˜,0.78,C15˜,0.91,P¯1=C11˜,0.28,C12˜,0.32,C13˜,0.36,C14˜,0.45,C15˜,0.43,P¯2=C21˜,0.85,C22˜,0.77,C23˜,0.79,C24˜,0.81,C25˜,0.71,P¯2=C21˜,0.35,C22˜,0.34,C23˜,0.46,C24˜,0.26,C25˜,0.43,P¯3=C31˜,0.84,C32˜,0.75,C33˜,0.74,C34˜,0.69,C35˜,0.78,P¯3=C31˜,0.37,C32˜,0.36,C33˜,0.35,C34˜,0.42,C35˜,0.48.Step 3:
If the thresholdβ=0.6, it produces N˜Ci˜β,M˜Ci˜β and N˜Ci˜0.6∧M˜Ci˜0.6 as displayed in Tables 14–22.Step 4:
Calculate the distancesD¯l and D¯l as follows:(26)D¯1=0.248u1+0.269u2+0.315u3+0.189u4+0.261u5+0.306u6,D¯1=0.307u1+0.307u2+0.241u3+0.317u4+0.247u5+0.259u6,D¯2=0.276u1+0.276u2+0.293u3+0.146u4+0.261u5+0.228u6,D¯2=0.246u1+0.278u2+0.209u3+0.352u4+0.293u5+0.278u6,D¯3=0.243u1+0.233u2+0.21u3+0.219u4+0.184u5+0.259u6,D¯3=0.225u1+0.231u2+0.25u3+0.237u4+0.232u5+0.212u6.Step 5:
Calculate the lower and upper approximations of the best and worst decision-making objects as follows.Takee=e1, and we have(27)ℒ∑r=1kCr˜ℐo3D¯1=0.49616u1+0.52485u2+0.55475u3+0.48907u4+0.51965u5+0.54196u6,U∑r=1kCr˜ℐo3D¯1=0.16616u1+0.17485u2+0.20475u3+0.11907u4+0.16965u5+0.20196u6,ℒ∑r=1kCr˜ℐo3D¯1=0.53569u1+0.54955u2+0.50665u3+0.56971u4+0.51055u5+0.51094u6,U∑r=1kCr˜ℐo3D¯1=0.20569u1+0.19955u2+0.15665u3+0.19971u4+0.16055u5+0.17094u6.Takee=e2, and we have(28)ℒ∑r=1kCr˜ℐo3D¯2=0.51492u1+0.5294u2+0.54045u3+0.46198u4+0.51965u5+0.49048u6,U∑r=1kCr˜ℐo3D¯2=0.18492u1+0.1794u2+0.19045u3+0.09198u4+0.16965u5+0.15048u6,ℒ∑r=1kCr˜ℐo3D¯2=0.49482u1+0.5307u2+0.48585u3+0.59176u4+0.54045u5+0.52348u6,U∑r=1kCr˜ℐo3D¯2=0.16482u1+0.1807u2+0.13585u3+0.22176u4+0.19045u5+0.18348u6.Takee=e3, and we have(29)ℒ∑r=1kCr˜ℐo3D¯3=0.49281u1+0.50145u2+0.4865u3+0.50797u4+0.4696u5+0.51094u6,U∑r=1kCr˜ℐo3D¯3=0.16281u1+0.15145u2+0.1365u3+0.13797u4+0.1196u5+0.17094u6,ℒ∑r=1kCr˜ℐo3D¯3=0.48075u1+0.50015u2+0.5125u3+0.51931u4+0.5008u5+0.47992u6,U∑r=1kCr˜ℐo3D¯3=0.15085u1+0.15015u2+0.1625u3+0.14931u4+0.1508u5+0.13992u6.Step 6
Based on the importance of these five attributes, we give the worst and the best decision-making objects as follows:(30)W1¯=0.57988u1+0.60793u2+0.645915u3+0.549904u4+0.60114u5+0.63447u6,W1¯=0.631194u1+0.63944u2+0.58393u3+0.65564u4+0.58913u5+0.59848u6,W2¯=0.604621u1+0.61383u2+0.62797u3+0.52247u4+0.60114u5+0.57115u6,W2¯=0.578084u1+0.615503u2+0.5557u3+0.68229u4+0.62797u5+0.61091u6,W3¯=0.57539u1+0.57696u2+0.55659u3+0.57586u4+0.53304u5+0.59454u6,W3¯=0.55908u1+0.57520u2+0.58667u3+0.591082u4+0.57608u5+0.55269u6.Thus, we evaluate a closeness coefficient as follows:(31)ℛ1=0.521185u1+0.51263u2+0.45625u3+0.54385u4+0.49495u5+0.48540u6,ℛ2=0.48878u1+0.50068u2+0.46947u3+0.57155u4+0.51091u5+0.51682u6,ℛ3=0.49281u1+0.49924u2+0.51316u3+0.50652u4+0.51940u5+0.48176u6.Step 7
Based on these results, we calculate the group optimal index as follows.ℛ=0.503757/u1+0.50474/u2+0.486027/u3+0.527955/u4+0.508771/u5+0.486722/u6, and hence get the ranking order as u4≥u5≥u2≥u1≥u6≥u3.From the calculations, we conclude that the 4th system analysis engineer is the best alternative among the others.Furthermore, we get the solution forCase 2 by the same analysis in Case 1. Therefore, we have the group optimal index as follows:(32)ℛ=0.50055u1+0.500528u2+0.500528u3+0.500636u4+0.500528u5+0.500539u6,and hence get the ranking order asu4≥u1≥u6≥u2≥u3≥u5. Through the previous computation, we obtain the 4th system analysis engineer is the best alternative among the others.Table 11
Table forC˜1.
ΩC˜11C˜12C˜13C˜14C˜15u10.820.710.460.550.52u20.730.320.650.580.84u30.560.680.360.780.44u40.530.480.740.650.91u50.660.530.570.720.43u60.280.760.520.450.77Table 12
Table forC˜2.
ΩC˜21C˜22C˜23C˜24C˜25u10.780.560.670.260.59u20.350.770.490.690.55u30.510.370.790.420.67u40.850.680.570.750.48u50.580.340.730.810.43u60.530.750.460.590.71Table 13
Table forC˜3.
ΩC˜31C˜32C˜33C˜34C˜35u10.560.750.390.670.48u20.760.360.680.450.55u30.840.550.350.580.65u40.430.530.740.690.63u50.590.710.650.480.55u60.370.660.560.420.78Table 14
Table forN˜C1˜ui0.6.
u1u2u3u4u5u6N˜C1˜u10.60.710.320.560.480.530.28N˜C1˜u20.60.460.650.360.530.430.28N˜C1˜u30.60.550.320.680.480.530.45N˜C1˜u40.60.460.580.360.650.430.45N˜C1˜u50.60.550.580.560.530.660.28N˜C1˜u60.60.520.320.440.480.430.76Table 15
Table forN˜C2˜ui0.6.
u1u2u3u4u5u6N˜C2˜u10.60.670.350.510.570.580.46N˜C2˜u20.60.260.690.370.680.340.59N˜C2˜u30.60.590.490.670.480.430.46N˜C2˜u40.60.260.350.370.680.340.53N˜C2˜u50.60.260.490.420.570.730.46N˜C2˜u60.60.560.550.370.480.340.71Table 16
Table forN˜C3˜ui0.6.
u1u2u3u4u5u6N˜C3˜u10.60.670.360.550.530.480.42N˜C3˜u20.60.390.680.350.430.590.37N˜C3˜u30.60.480.550.650.430.550.37N˜C3˜u40.60.390.450.350.630.480.42N˜C3˜u50.60.390.360.350.530.650.56N˜C3˜u60.60.480.360.550.530.550.66Table 17
Table forM˜C1˜ui0.6.
u1u2u3u4u5u6M˜C1˜u10.60.710.460.550.460.550.52M˜C1˜u20.60.320.650.320.580.580.32M˜C1˜u30.60.560.360.680.360.560.44M˜C1˜u40.60.480.530.480.650.530.48M˜C1˜u50.60.530.430.530.430.660.43M˜C1˜u60.60.280.280.450.450.280.76Table 18
Table forM˜C2˜ui0.6.
u1u2u3u4u5u6M˜C2˜u10.60.670.260.590.260.260.56M˜C2˜u20.60.350.690.490.350.490.55M˜C2˜u30.60.510.370.670.370.420.37M˜C2˜u40.60.570.680.480.680.570.48M˜C2˜u50.60.580.340.430.340.730.34M˜C2˜u60.60.460.590.460.530.460.71Table 19
Table forM˜C3˜ui0.6.
u1u2u3u4u5u6M˜C3˜u10.60.670.390.480.390.390.48M˜C3˜u20.60.360.680.550.450.360.36M˜C3˜u30.60.550.350.650.350.350.55M˜C3˜u40.60.530.430.430.630.530.53M˜C3˜u50.60.480.590.550.480.650.55M˜C3˜u60.60.420.370.370.420.560.66Table 20
Table forN˜C1˜ui0.6∧M˜C1˜ui0.6.
u1u2u3u4u5u6N˜C1˜u10.6∧M˜C1˜u10.60.710.320.550.460.530.28N˜C1˜u20.6∧M˜C1˜u20.60.320.650.320.530.430.28N˜C1˜u30.6∧M˜C1˜u30.60.550.320.680.360.530.44N˜C1˜u40.6∧M˜C1˜u40.60.460.530.360.650.430.45N˜C1˜u50.6∧M˜C1˜u50.60.530.430.530.430.660.28N˜C1˜u60.6∧M˜C1˜u60.60.280.280.440.450.280.76Table 21
Table forN˜C2˜ui0.6∧M˜C2˜ui0.6
u1u2u3u4u5u6N˜C2˜u10.6∧M˜C2˜u10.60.670.260.510.260.260.46N˜C2˜u20.6∧M˜C2˜u20.60.260.690.370.350.340.55N˜C2˜u30.6∧M˜C2˜u30.60.510.370.670.370.420.37N˜C2˜u40.6∧M˜C2˜u40.60.260.350.370.680.340.48N˜C2˜u50.6∧M˜C2˜u50.60.260.340.420.340.730.34N˜C2˜u60.6∧M˜C2˜u60.60.460.550.370.480.340.71Table 22
Table forN˜C3˜ui0.6∧M˜C3˜ui0.6.
u1u2u3u4u5u6N˜C3˜u10.6∧M˜C3˜u10.60.670.360.480.390.390.42N˜C3˜u20.6∧M˜C3˜u20.60.360.680.350.430.360.36N˜C3˜u30.6∧M˜C3˜u30.60.480.350.650.350.350.37N˜C3˜u40.6∧M˜C3˜u40.60.390.430.350.630.480.42N˜C3˜u50.6∧M˜C3˜u50.60.390.360.350.480.650.55N˜C3˜u60.6∧M˜C3˜u60.60.420.360.370.420.550.66


### 5.3. Comparative Analysis

The main aim of the current work is to present a method that increases the lower approximation and decreases the upper approximation of Zhan’s methods in [40]. This can be seen easily from Examples 2– 4. Moreover and by looking at Tables 23 and 24, we can see that the ranking results of the two decision-making models. It is obvious that the optimal selected alternative is the same, although there exist some differences in the ranking results because we choose different decision-making methods.Table 23
Table for the ranking results for Case 1.
Two modelsObtain a decisionZhan modelx4≥x1≥x6≥x2=x3=x5Our modelx4≥x1≥x6≥x2=x3=x5Table 24
Table for the ranking results for Case 2.
Two modelsObtain a decisionZhan modelx4≥x1≥x5≥x2≥x3≥x6Our modelx4≥x5≥x2≥x1≥x6≥x3An easy way to see the effectiveness of our method and the differences between the four models (i.e., our three proposed models and Zhan’s model) are shown in Figures1 and 2.Figure 1
The representations of the lower approximations by using our model and Zhan model in two cases.
(a)(b)Figure 2
The representations of the upper approximations by using our model and Zhan model in two cases.
(a)(b)Figure1 explained the comparisons between the lower approximations for the four models (i.e., 0-OMGITFLA, 1-OMGITFLA, 2-OMGITFLA, and 3-OMGITFLA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). This figure justifies that the 2-OMGITFLA is better than the others.Figure2 clarified the differences between the upper approximations for the four models (i.e., 0-OMGITFUA, 1-OMGITFUA, 2-OMGITFUA, and 3-OMGITFUA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). This figure illustrates that the 2-OMGITFUA is lower than the others.


## 5.1. Description and Process

LetΩ=u1,u2,…,un be n alternatives and ℰ=e1,e2,…,el be l decision makers. Suppose for each νi is a weighted vector correspondingly to ei, where νi≥0 for i=1,…,l and ∑i=1lνi=1. Hence, Ci˜=Ci1˜,Ci2˜,…,Cimi˜, for all i=1,2,…,n is a set of attributes. A family of mappings G=gl, where gl:Ω×Ci˜⟶0,1. So, we construct the MAGDM with fuzzy information system Ω,C˜,ℰ,G. Based on the proposed covering methods, we present a decision-making algorithm to find the best alternative through the following steps:Step 1:
Construct the decision-making object with fuzzy information of the universe of discourse. Through the rule of fuzzy TOPSIS method, we have(22)P¯l=C˜lj,∨1≤i≤nglui,C˜lj:j=1,…,ml=1,…,t=C˜l1,∨gl1,…,C˜lm,∨glm,P¯l=C˜lj,∧1≤i≤nglui,C˜lj:j=1,…,ml=1,…,t=C˜l1,∧gl1,…,C˜lm,∧glmwhere∧ and ∧ denote “max” and “min,” respectively.Step 2:
Compute the respective distancesD¯ and D¯ as follows:(23)D¯l=SC˜ljui,C˜ljP¯=1m∑j=1mC˜ljui−∨glj2,D¯l=SC˜ljui,C˜ljP¯=1m∑j=1mC˜ljui−∧glj2,whereSY^,Z^=1/m∑j=1mY^ui−Z^ui2 and m is the cardinality of Ω.Step 3:
Calculate the lower and upper approximations of the best and worst decision-making objects with fuzzy information by Definition13 (2-OMGITFLA and 2-OMGITFUA).Step 4:
Calculate the closeness coefficient degree byℛlui=Wl¯ui/Wl¯ui+Wl¯ui, where(24)Wl¯ui=ℬpℒ∑r=1kCr˜ℐo3D¯lui,U∑r=1kCr˜To3D¯lui,Wl¯ui=ℬpℒ∑r=1kCr˜ℐo3D¯lui,U∑r=1kCr˜To3D¯lui,be the worst and the best decision-making objects for individual ranking function of expertl for the candidates ui, and 0≤Wl¯ui,Wl¯ui≤1.Step 5:
Calculate the group ranking function by the following equationℛui=∑l=1tνlℛlui, and hence rank the alternatives.According to these steps, we give an algorithm to solve the decision-making problems based on the 2-COMGITFRS model. The steps corresponding to it are summarized in Algorithm1.Algorithm 1: Algorithm for MAGDM with the TOPSIS method.
Input: Fuzzy information systems Ω,C˜,ℰ,G.Output: Decision-Making.(1)
EnterX^, β, Γ=Ci˜:i=1,…,n and Ω=xj:j=1,…,m.(2)
From Definition7, calculate fuzzy β-neighborhood.(3)
FromStep 2 and by Definition 8, calculate complementary fuzzy β-neighborhood(4)
FromSteps 2 and 3, calculate N˜C1˜u1β∧M˜C1˜uiβ.(5)
EnterP¯l and P¯l.(6)
Calculate the distancesD¯l and D¯l.(7)
From Definition13, calculate the lower approximation ℒ∑r=1kCr˜ℐo3D¯l and the upper approximation U∑r=1kCr˜ℐo3D¯l.(8)
Calculate the worst and the best decision-making objectsWl¯ and Wl¯ for each individual decision-maker.(9)
Calculate the individual ranking functionℛl.(10)
Calculate the group ranking functionℛ.(11)
Obtain the decision.


## 5.2. Applied Example

The abovementioned steps have been illustrated with a numerical example as shown next.Example 7 (see [40]).
LetΩ=u1,u2,…,u6 be six system analysis engineers and Γ˜=emotional steadiness C1, oral communication skill C2, personality C3, past experience C4, self-confidence C5 be the attribute set of the basic description of the candidates. Suppose that three experts e1,e2, and e3 are invited to evaluate the system analysis engineers according to their specialized knowledge. The weights of every expert are ν1=0.4,ν2=0.1, and ν3=0.5. The following steps of the stated algorithm are implemented here.Step 1:
Experts evaluate each candidate under the set of the attribute and present their judgments with the real values. These values are summarized in Tables11–13.Step 2:
According to the importance of these five attributes, we give the following results for each expert:(25)P¯1=C11˜,0.82,C12˜,0.76,C13˜,0.74,C14˜,0.78,C15˜,0.91,P¯1=C11˜,0.28,C12˜,0.32,C13˜,0.36,C14˜,0.45,C15˜,0.43,P¯2=C21˜,0.85,C22˜,0.77,C23˜,0.79,C24˜,0.81,C25˜,0.71,P¯2=C21˜,0.35,C22˜,0.34,C23˜,0.46,C24˜,0.26,C25˜,0.43,P¯3=C31˜,0.84,C32˜,0.75,C33˜,0.74,C34˜,0.69,C35˜,0.78,P¯3=C31˜,0.37,C32˜,0.36,C33˜,0.35,C34˜,0.42,C35˜,0.48.Step 3:
If the thresholdβ=0.6, it produces N˜Ci˜β,M˜Ci˜β and N˜Ci˜0.6∧M˜Ci˜0.6 as displayed in Tables 14–22.Step 4:
Calculate the distancesD¯l and D¯l as follows:(26)D¯1=0.248u1+0.269u2+0.315u3+0.189u4+0.261u5+0.306u6,D¯1=0.307u1+0.307u2+0.241u3+0.317u4+0.247u5+0.259u6,D¯2=0.276u1+0.276u2+0.293u3+0.146u4+0.261u5+0.228u6,D¯2=0.246u1+0.278u2+0.209u3+0.352u4+0.293u5+0.278u6,D¯3=0.243u1+0.233u2+0.21u3+0.219u4+0.184u5+0.259u6,D¯3=0.225u1+0.231u2+0.25u3+0.237u4+0.232u5+0.212u6.Step 5:
Calculate the lower and upper approximations of the best and worst decision-making objects as follows.Takee=e1, and we have(27)ℒ∑r=1kCr˜ℐo3D¯1=0.49616u1+0.52485u2+0.55475u3+0.48907u4+0.51965u5+0.54196u6,U∑r=1kCr˜ℐo3D¯1=0.16616u1+0.17485u2+0.20475u3+0.11907u4+0.16965u5+0.20196u6,ℒ∑r=1kCr˜ℐo3D¯1=0.53569u1+0.54955u2+0.50665u3+0.56971u4+0.51055u5+0.51094u6,U∑r=1kCr˜ℐo3D¯1=0.20569u1+0.19955u2+0.15665u3+0.19971u4+0.16055u5+0.17094u6.Takee=e2, and we have(28)ℒ∑r=1kCr˜ℐo3D¯2=0.51492u1+0.5294u2+0.54045u3+0.46198u4+0.51965u5+0.49048u6,U∑r=1kCr˜ℐo3D¯2=0.18492u1+0.1794u2+0.19045u3+0.09198u4+0.16965u5+0.15048u6,ℒ∑r=1kCr˜ℐo3D¯2=0.49482u1+0.5307u2+0.48585u3+0.59176u4+0.54045u5+0.52348u6,U∑r=1kCr˜ℐo3D¯2=0.16482u1+0.1807u2+0.13585u3+0.22176u4+0.19045u5+0.18348u6.Takee=e3, and we have(29)ℒ∑r=1kCr˜ℐo3D¯3=0.49281u1+0.50145u2+0.4865u3+0.50797u4+0.4696u5+0.51094u6,U∑r=1kCr˜ℐo3D¯3=0.16281u1+0.15145u2+0.1365u3+0.13797u4+0.1196u5+0.17094u6,ℒ∑r=1kCr˜ℐo3D¯3=0.48075u1+0.50015u2+0.5125u3+0.51931u4+0.5008u5+0.47992u6,U∑r=1kCr˜ℐo3D¯3=0.15085u1+0.15015u2+0.1625u3+0.14931u4+0.1508u5+0.13992u6.Step 6
Based on the importance of these five attributes, we give the worst and the best decision-making objects as follows:(30)W1¯=0.57988u1+0.60793u2+0.645915u3+0.549904u4+0.60114u5+0.63447u6,W1¯=0.631194u1+0.63944u2+0.58393u3+0.65564u4+0.58913u5+0.59848u6,W2¯=0.604621u1+0.61383u2+0.62797u3+0.52247u4+0.60114u5+0.57115u6,W2¯=0.578084u1+0.615503u2+0.5557u3+0.68229u4+0.62797u5+0.61091u6,W3¯=0.57539u1+0.57696u2+0.55659u3+0.57586u4+0.53304u5+0.59454u6,W3¯=0.55908u1+0.57520u2+0.58667u3+0.591082u4+0.57608u5+0.55269u6.Thus, we evaluate a closeness coefficient as follows:(31)ℛ1=0.521185u1+0.51263u2+0.45625u3+0.54385u4+0.49495u5+0.48540u6,ℛ2=0.48878u1+0.50068u2+0.46947u3+0.57155u4+0.51091u5+0.51682u6,ℛ3=0.49281u1+0.49924u2+0.51316u3+0.50652u4+0.51940u5+0.48176u6.Step 7
Based on these results, we calculate the group optimal index as follows.ℛ=0.503757/u1+0.50474/u2+0.486027/u3+0.527955/u4+0.508771/u5+0.486722/u6, and hence get the ranking order as u4≥u5≥u2≥u1≥u6≥u3.From the calculations, we conclude that the 4th system analysis engineer is the best alternative among the others.Furthermore, we get the solution forCase 2 by the same analysis in Case 1. Therefore, we have the group optimal index as follows:(32)ℛ=0.50055u1+0.500528u2+0.500528u3+0.500636u4+0.500528u5+0.500539u6,and hence get the ranking order asu4≥u1≥u6≥u2≥u3≥u5. Through the previous computation, we obtain the 4th system analysis engineer is the best alternative among the others.Table 11
Table forC˜1.
ΩC˜11C˜12C˜13C˜14C˜15u10.820.710.460.550.52u20.730.320.650.580.84u30.560.680.360.780.44u40.530.480.740.650.91u50.660.530.570.720.43u60.280.760.520.450.77Table 12
Table forC˜2.
ΩC˜21C˜22C˜23C˜24C˜25u10.780.560.670.260.59u20.350.770.490.690.55u30.510.370.790.420.67u40.850.680.570.750.48u50.580.340.730.810.43u60.530.750.460.590.71Table 13
Table forC˜3.
ΩC˜31C˜32C˜33C˜34C˜35u10.560.750.390.670.48u20.760.360.680.450.55u30.840.550.350.580.65u40.430.530.740.690.63u50.590.710.650.480.55u60.370.660.560.420.78Table 14
Table forN˜C1˜ui0.6.
u1u2u3u4u5u6N˜C1˜u10.60.710.320.560.480.530.28N˜C1˜u20.60.460.650.360.530.430.28N˜C1˜u30.60.550.320.680.480.530.45N˜C1˜u40.60.460.580.360.650.430.45N˜C1˜u50.60.550.580.560.530.660.28N˜C1˜u60.60.520.320.440.480.430.76Table 15
Table forN˜C2˜ui0.6.
u1u2u3u4u5u6N˜C2˜u10.60.670.350.510.570.580.46N˜C2˜u20.60.260.690.370.680.340.59N˜C2˜u30.60.590.490.670.480.430.46N˜C2˜u40.60.260.350.370.680.340.53N˜C2˜u50.60.260.490.420.570.730.46N˜C2˜u60.60.560.550.370.480.340.71Table 16
Table forN˜C3˜ui0.6.
u1u2u3u4u5u6N˜C3˜u10.60.670.360.550.530.480.42N˜C3˜u20.60.390.680.350.430.590.37N˜C3˜u30.60.480.550.650.430.550.37N˜C3˜u40.60.390.450.350.630.480.42N˜C3˜u50.60.390.360.350.530.650.56N˜C3˜u60.60.480.360.550.530.550.66Table 17
Table forM˜C1˜ui0.6.
u1u2u3u4u5u6M˜C1˜u10.60.710.460.550.460.550.52M˜C1˜u20.60.320.650.320.580.580.32M˜C1˜u30.60.560.360.680.360.560.44M˜C1˜u40.60.480.530.480.650.530.48M˜C1˜u50.60.530.430.530.430.660.43M˜C1˜u60.60.280.280.450.450.280.76Table 18
Table forM˜C2˜ui0.6.
u1u2u3u4u5u6M˜C2˜u10.60.670.260.590.260.260.56M˜C2˜u20.60.350.690.490.350.490.55M˜C2˜u30.60.510.370.670.370.420.37M˜C2˜u40.60.570.680.480.680.570.48M˜C2˜u50.60.580.340.430.340.730.34M˜C2˜u60.60.460.590.460.530.460.71Table 19
Table forM˜C3˜ui0.6.
u1u2u3u4u5u6M˜C3˜u10.60.670.390.480.390.390.48M˜C3˜u20.60.360.680.550.450.360.36M˜C3˜u30.60.550.350.650.350.350.55M˜C3˜u40.60.530.430.430.630.530.53M˜C3˜u50.60.480.590.550.480.650.55M˜C3˜u60.60.420.370.370.420.560.66Table 20
Table forN˜C1˜ui0.6∧M˜C1˜ui0.6.
u1u2u3u4u5u6N˜C1˜u10.6∧M˜C1˜u10.60.710.320.550.460.530.28N˜C1˜u20.6∧M˜C1˜u20.60.320.650.320.530.430.28N˜C1˜u30.6∧M˜C1˜u30.60.550.320.680.360.530.44N˜C1˜u40.6∧M˜C1˜u40.60.460.530.360.650.430.45N˜C1˜u50.6∧M˜C1˜u50.60.530.430.530.430.660.28N˜C1˜u60.6∧M˜C1˜u60.60.280.280.440.450.280.76Table 21
Table forN˜C2˜ui0.6∧M˜C2˜ui0.6
u1u2u3u4u5u6N˜C2˜u10.6∧M˜C2˜u10.60.670.260.510.260.260.46N˜C2˜u20.6∧M˜C2˜u20.60.260.690.370.350.340.55N˜C2˜u30.6∧M˜C2˜u30.60.510.370.670.370.420.37N˜C2˜u40.6∧M˜C2˜u40.60.260.350.370.680.340.48N˜C2˜u50.6∧M˜C2˜u50.60.260.340.420.340.730.34N˜C2˜u60.6∧M˜C2˜u60.60.460.550.370.480.340.71Table 22
Table forN˜C3˜ui0.6∧M˜C3˜ui0.6.
u1u2u3u4u5u6N˜C3˜u10.6∧M˜C3˜u10.60.670.360.480.390.390.42N˜C3˜u20.6∧M˜C3˜u20.60.360.680.350.430.360.36N˜C3˜u30.6∧M˜C3˜u30.60.480.350.650.350.350.37N˜C3˜u40.6∧M˜C3˜u40.60.390.430.350.630.480.42N˜C3˜u50.6∧M˜C3˜u50.60.390.360.350.480.650.55N˜C3˜u60.6∧M˜C3˜u60.60.420.360.370.420.550.66


## 5.3. Comparative Analysis

The main aim of the current work is to present a method that increases the lower approximation and decreases the upper approximation of Zhan’s methods in [40]. This can be seen easily from Examples 2– 4. Moreover and by looking at Tables 23 and 24, we can see that the ranking results of the two decision-making models. It is obvious that the optimal selected alternative is the same, although there exist some differences in the ranking results because we choose different decision-making methods.Table 23
Table for the ranking results for Case 1.
Two modelsObtain a decisionZhan modelx4≥x1≥x6≥x2=x3=x5Our modelx4≥x1≥x6≥x2=x3=x5Table 24
Table for the ranking results for Case 2.
Two modelsObtain a decisionZhan modelx4≥x1≥x5≥x2≥x3≥x6Our modelx4≥x5≥x2≥x1≥x6≥x3An easy way to see the effectiveness of our method and the differences between the four models (i.e., our three proposed models and Zhan’s model) are shown in Figures1 and 2.Figure 1
The representations of the lower approximations by using our model and Zhan model in two cases.
(a)(b)Figure 2
The representations of the upper approximations by using our model and Zhan model in two cases.
(a)(b)Figure1 explained the comparisons between the lower approximations for the four models (i.e., 0-OMGITFLA, 1-OMGITFLA, 2-OMGITFLA, and 3-OMGITFLA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). This figure justifies that the 2-OMGITFLA is better than the others.Figure2 clarified the differences between the upper approximations for the four models (i.e., 0-OMGITFUA, 1-OMGITFUA, 2-OMGITFUA, and 3-OMGITFUA) for the two cases (i.e., Case 1 (resp., Case 2) is in the left (resp., right) figure). This figure illustrates that the 2-OMGITFUA is lower than the others.


## 6. Conclusion and Future Work

The main aim of the present work is to increase the effectiveness of Zhan’s method by increasing the lower approximation and decreasing the upper approximation. So, based on the concepts of a family of fuzzyβ-neighborhood (and a family of fuzzy complementary ℐ,T-neighborhood), we introduced new three types of covering-based multigranulation ℐ,T-fuzzy rough sets models and their properties. Furthermore, we give six kinds of covering-based variable precision multigranulation ([[1008]])-fuzzy rough sets. The relationships among these models are investigated. Also, an illustrative example with algorithm is given. Therefore, it is clear to see that 2-COMGITFRS is better than the other models (i.e., 0-COMGITFRS, 1-COMGITFRS, and 3-COMGITFRS).In future research, we plan to further investigate along with the following: (1) topological properties of the presented methods [44, 45], (2) combination with the soft set and the proposed methods [46, 47], and (3) combination with the neutrosophic set and the current methods [48].



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*Source: 6661782-2020-11-29.xml*