# Uncertainty Measure for Multisource Intuitionistic Fuzzy Information System

Authors: Hong Wang; Hong Li
Journal: Complexity
 (2022)

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## Abstract

Multisource information systems and multigranulation intuitionistic fuzzy rough sets are important extended types of Pawlak’s classical rough set model. Multigranulation intuitionistic fuzzy rough sets have been investigated in depth in recent years. However, few studies have considered this combination of multisource information systems and intuitionistic fuzzy rough sets. In this paper, we give the uncertainty measure for multisource intuitionistic fuzzy information system. Against the background of multisource intuitionistic fuzzy information system, each information source is regarded as a granularity level. Considering the different importance of information sources, we assign different weights to them. Firstly, the paper proposes an optimal source selection method. Secondly, we study the weighted generalized, weighted optimistic, and weighted pessimistic multigranularity intuitionistic fuzzy rough set models and uncertainty measurement methods in the multisource intuitionistic fuzzy information system, and we further study the relationship between the three models and related properties. Finally, an example is given to verify the validity of the models and methods.


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## Body

## 1. Introduction

Rough set theory [1] was proposed by Polish mathematician Pawlak in 1982. It is an effective mathematical tool to analyze and process inaccurate data and uncertain information. Rough set theory has received more and more attention in recent years. It is widely used in many fields such as natural science, social science, and engineering technology [2–11]. Uncertainty measurement [12, 13] is one of the important research contents in rough set theory. It can measure the dependency and similarity between attributes and provide an effective measurement tool for attribute reduction and cluster analysis [14]. Traditional uncertainty measurement considers single-source information system. With the advent of the era of big data, it is necessary to study uncertainty measurement methods in multisource intuitionistic fuzzy information system [15].In current study, by extending the equivalence relationship to a general binary relationship, Qian et al. extended the single-granularity rough set model to multigranularity structure [16, 17]. By combining fuzzy set theory and rough set theory, a fuzzy rough set model [18] and rough fuzzy set model [19, 20] are obtained. Literature [21] extends the fuzzy set theory to the intuitionistic fuzzy set theory and extends the relationship between elements and sets from membership degree to nonmembership degree and hesitation degree. Therefore, the intuitionistic fuzzy rough set theory is a very effective mathematical tool when analyzing and processing inaccurate, incomplete, and other rough information, and the result accuracy is significantly improved. Literature [22, 23] combined rough set theory with intuitionistic fuzzy set theory and established an intuitionistic fuzzy rough set model. Literature [24] investigated the upper approximation reduction problem of intuitionistic fuzzy information system based on dominant relationship. Literature [25, 26] extended the single-granularity intuitionistic fuzzy rough set model to the multigranularity intuitionistic fuzzy rough set model and presented the optimistic multigranularity and pessimistic multigranularity intuitionistic fuzzy rough set models under the dominant relationship. Literature [27] proposes the intuitionistic fuzzy soft set, which is an effective tool for solving multiple attribute decision-making with intuitionistic fuzzy information. Literature [28] proposes the hesitant fuzzy sets and studies their relationship with intuitionistic fuzzy sets.In the research of multigranularity intuitionistic fuzzy rough set model, if all granularity levels are considered equally important, optimistic multigranularity only requires one granularity knowledge and target concept to meet inclusion relation, while pessimistic multigranularity requires all granularity knowledge and target cocept to meet inclusion relation, ultimately leading to inaccurate decision-making results. The importance of granulation is usually different for decision-making in real life. Taking various factors into account, some granularity is very important, so we give it a larger weight, but some granularity is not important, so we give it a smaller weight. Considering the importance of different granularity levels, literature [29] proposed a weighted multigranularity intuitionistic fuzzy rough set model.Now, in our real life, we no longer face a single-source information system, but a multisource information system. Multisource information systems [30] are used to represent information that comes from multiple sources. Literature [31] proposed a fuzzy multigranulation decision-theoretic rough set model in multisource fuzzy information systems. Literature [32] investigated the attribute reduction in multisource decision systems. Literature [33] combined rough set model and multisource decision systems and established a decision-theoretic rough set model of multisource decision systems. Literature [34] built the information source selection criteria and proposed some principles of information fusion. Literature [35–42] proposed information fusion methods of multisource information system. In this paper, we treat each single information system as a granular structure. As the number of information sources increases, a large amount of data is unreliable. Therefore, we propose a corresponding algorithm to select reliable information sources and one that can greatly improve the efficiency of information processing. Up to now, few scholars have combined the rough set model with the multisource intuitionistic fuzzy information system. Different weights are given to the granularity; the weighted multigranularity intuitionistic fuzzy rough set models and the uncertainty measurement methods for multisource intuitionistic fuzzy information system are proposed. This is the purpose of this article.The rest of this paper is organized as follows. In Section2, we mainly review the relevant concepts and properties of intuitionistic fuzzy rough sets, multigranulation intuitionistic fuzzy rough sets, and multisource intuitionistic fuzzy information system. In Section 3, we first study the optimal source selection of multisource intuitionistic fuzzy information system. Further, the weighted multigranulation intuitionistic fuzzy rough set model and its related properties of multisource intuitionistic fuzzy information system are researched. In Section 4, we give the uncertainty measurement methods for the weighted multigranulation intuitionistic fuzzy rough set model of multisource intuitionistic fuzzy information system. At the same time, we verify the effectiveness of the proposed models and methods through a specific example. Section 5 uses a numerical experiment to verify the effectiveness of the proposed methods. In Section 6, we present the conclusion and the future work.


## 2. Preliminaries

This section mainly reviews related concepts and properties of intuitionistic fuzzy rough set models, multigranulation intuitionistic fuzzy rough set models, and multisource intuitionistic fuzzy information system.


### 2.1. Intuitionistic Fuzzy Rough Set Model

Definition 1.
(see [21]): Given the universe of discourse U, an intuitionistic fuzzy rough set A on U is defined by A=x,μAx,νAx|x∈U, where the functions μAx:μ⟶0,1 and vAx:v⟶0,1 satisfy 0≤μAx+νAx≤1 for all x∈U. μAx and vAx are called the degrees of membership and nonmembership of element x∈U to A, respectively. The family of all intuitionistic fuzzy sets in U is denoted by IFU. When μAx+νAx=1, A degenerates into a fuzzy set.Definition 2.
(see [21]): Let A=x,μAx,νAx|x∈U∈IFU, B=x,μBx,νBx|x∈U∈IFU; then:(1)
The supplementary set ofA∼A=x,vAx,μAx|x∈U.(2)
A=B⇔μAx=μBx,νAx=νBx,∀x∈U.(3)
A⊆B⇔μAx=μBx,νAx=νBx,∀x∈U.(4)
A∩B=x,μAx∧μBx,νAx∨νBx|x∈U.(5)
A∪B=x,μAx∧μBx,νAx∨νBx|x∈U.(6)
A−B=A∩∼B.(7)
A△B=A∩Bc∪B∩Ac.Definition 3.
(see [21]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, where U=x1,x2…xn is a nonempty finite set of objects called the universe of discourse; AT=a1,a2…an is a nonempty finite attribute set; V=∪a∈ATVa is the attribute value set; f:U×AT⟶V is the information function, A⊆AT; a binary dominant relation RA is defined as follows:(1)RA=xi,xj∈U×U|fxi,a≤fxj,a,∀a∈A=xi,xj∈U×U|μaxi≤μaxj∧νaxi≥νaxj,∀a∈A.
Obviously, this binary dominant relation satisfies reflexivity, antisymmetry, and transitivity, andRA=∩Ra∈A. Based on the above dominant relation, the dominant class RAxi definition of object xi can be obtained as follows:(2)RAxi=xj∈U:fxi,a≤fxj,a,∀a∈A.Definition 4.
(see [22]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system; for any X⊆U,A⊆AT, the lower approximation and upper approximation of X for RA are defined as follows:(3)RAX=x∈U|RAxi⊆X;RA¯X=x∈U|RAxi∩X≠∅.
By definition,RA¯X consists of all objects that are definitely contained in the set X. RA¯X consists of all objects that are possibly contained in the set X. If RA¯X=RA¯X, then X is the exact set of RA; otherwise, it is the rough set about RA.The positive, negative, and boundary regions of X can be defined as follows:(4)POSRAX=RA¯X,NEGRAX=u−RA¯X,BNDRAX=RA¯X−RA¯X.Definition 5.
(see [22]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system; for any X⊆U,A⊆AT, the approximate accuracy and the roughness of the target set X are defined as follows:(5)αRA,X=RA¯XRA¯X;ρRA,X=1−=RA¯XRA¯X,when RA¯X=0, αRA,X=0,ρRA,X=1.Definition 6.
(see [23]): Let A=x,μAx|x∈U be a fuzzy sets on universe U; the nonmembership degree of A is defined as follows:(6)νAx=0,μAx>0.5,0.5,μAx≤0.5.
Then,x,μAx,νAx|x∈U is the correspondence intuitionistic fuzzy sets of the fuzzy set A.


### 2.2. Multigranulation Intuitionistic Fuzzy Rough Set Model

Multigranulation rough set model was first proposed by Qian et al. It is an extension of the classical rough set theory. In this subsection, the single-granulation intuitionistic fuzzy rough set is extended to the multigranulation intuitionistic fuzzy rough set model under multiple dominant relations. Generalized multigranulation intuitionistic fuzzy rough set is the generalization of optimistic multigranulation intuitionistic fuzzy rough set and pessimistic multigranulation intuitionistic fuzzy rough set.Given the definition of the support characteristic function, we use this function to complete the object selection.Definition 7.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; for any X⊆U, the support characteristic function of x for X is denoted as(7)SXAix=1,RAix⊆X,0,else.
The support characteristic function is used to describe the inclusion relation between dominance classRAix and concept X, Which indicates whether object x accurately supports X by Ai, or whether object x has a positive description of X by Ai.Definition 8.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; SXAix is the characteristic function of x for X. Given information level β∈0.5,1, for any X⊆U, the generalized lower and upper approximation of X are defined as follows:(8)GM¯∑i=1mAiXβ=x∈U|∑i=1mSXAixm≥βGM¯∑i=1mAiXβ=x∈U|∑i=1m1−SXcAixm>1−β.
IfGM¯∑I=1mAiXβ=GM¯∑I=1mAiXβ, then the target set X is generalized and definable; otherwise, it is generalized and rough. GM¯∑I=1mAiXβ=GM¯∑I=1mAiXβ is called the generalized multigranulation intuitionistic fuzzy rough set model GMIFRS.Definition 9.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; SXAix is the support characteristic function of x for X; for any X⊆U, the pessimistic lower and upper approximation of X are defined as follows:(9)PM¯∑i=1mAiX=x∈U|∧i=1mRAix⊆X=x∈U|∑i=1mSXAixm≥1,PM¯∑i=1mAiX==x∈U|∨i=1mRAix∩X≠∅=x∈U|∑i=1m1−SXcAixm>0,where “∨” denotes “or” and “∧” denotes “and.”
IfPM¯∑I=1MAiX=PM¯∑I=1MAiX, then the target set X is pessimistic and definable; otherwise, X is pessimistic and rough. PM¯∑I=1MAiX=PM¯∑I=1MAiX is called the pessimistic multigranulation intuitionistic fuzzy rough set model PMIFRS.Definition 10.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; SXAix is the support characteristic function of x for X; for any X⊆U, the optimistic lower and upper approximation of X are defined as follows:(10)OM¯∑i=1mAiX=x∈U|∧i=1mRAix⊆X=x∈U|∑i=1mSXAixm>0,OM¯∑i=1mAiX==x∈U|∨i=1mRAix∩X≠∅=x∈U|∑i=1m1−SXcAixm>1,where “∨” denotes “or” and “∧” denotes “and.”
IfOM¯∑I=1MAiX=OM¯∑I=1MAiX, then the target set X is optimistic and definable; otherwise, X is optimistic and rough. OM¯∑I=1MAiX=OM¯∑I=1MAiX is called the optimistic multigranulation intuitionistic fuzzy rough set model OMIFRS.


### 2.3. Attribute Reduction of Intuitionistic Fuzzy Information System Based on Dominant Relation

In this subsection, we define the attribute importance of intuitionistic fuzzy information system based on the dominant relation. We also give the attribute reduction algorithm of intuitionistic fuzzy information system based on knowledge rough entropy.Definition 11.
(see [9]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, A⊆AT, U/RA=RAxi|xi∈U; RA is a binary dominant relation; the rough entropy of A is defined as(11)ErA=∑i=1URAxiUlog2RAxi.
It is obvious that there are minimum and maximum values of the rough entropy. WhenRA is the finest classification, the rough entropy of A has a minimum value of 0; if RA is the coarsest classification, the rough entropy of A has a maximum value of Ulog2U.
Each information system has many attributes, but some of these are redundant. To measure the significance of a single attribute, [3, 4] give the concepts of relative importance and absolute importance of attribute.Definition 12.
LetIFIS=U,AT,V,f be an intuitionistic fuzzy information system, A⊆AT; for any a∈A, for any b∈AT−A, the relative importance and absolute importance of attribute a in attribute set A is defined as(12)sigina,A=ErA−a−ErA;siginb,A=ErA−ErA∪b.
According to this definition, the following properties are established:(1)
0≤sigina,A≤log2U.(2)
Attributea is necessary ⇔sigina,A>0.(3)
The attribute core ofA is coreA=a∈A|sigina,A>0.Definition 13.
LetIFIS=U,AT,V,f be an intuitionistic fuzzy information system, A⊆AT; if ErA=ErAT, for any a∈A and ErA−a≠ErAT, A is the attribute reduction of intuitionistic fuzzy information system based on dominance relation.
Since the attribute core is a subset of attribute reduction, in the heuristic reduction process starting from the attribute core, we often add attributes to attribute core through a measurement method to obtain attribute reduction. Next, an attribute reduction algorithm for intuitionistic fuzzy information system based on knowledge rough entropy is given.Algorithm 1: The minimum attribute reduction of intuitionistic fuzzy information system.
Input:IFIS=U,AT,V,f, AT=a1,a2,…,an.Output: The minimum attribute reductionRedAT.(1)
begin(2)
setcoreAT←∅; RedAT←∅; A0←∅;(3)
fora∈AT do(4)
sigina,A=ErAT−a−ErAT;(5)
ifsigina,A>0 then(6)
coreAT=coreAT∪a;(7)
end(8)
end(9)
setA0←coreAT(10)
whileErA0≠ErAT do(11)
fora∗∈AT/A0(12)
sigouta∗,A0=ErA0−ErA0∪a∗;(13)
a∗∗=maxsigouta∗,A0;(14)
setA1←a∗∗(15)
end(16)
setA0←A0∪A1(17)
endreturn:RedAT=A0(18)
endExample 1.
Table1 shows an intuitionistic fuzzy information system IFIS=U,AT,V,f, where the universe U=x1,x2,x3,x4,x5,x6 and attribute set AT=a1,a2,a3,a4. We use Algorithm 1 to calculate the minimum attribute reduction of the intuitionistic fuzzy information system given in Table 1.
From this table, it is easy to calculate the granular structure:
U/AT=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6,x5,x5,x6.
According to Definition11, the rough entropy of AT is as follows:
ErAT=2/6log22+5/6log25+4/6log24+6/6log26+1/6log21+2/6log22≈6.520.
After calculating the core of attribute setAT,
U/AT−a1=U/AT−a2=U/AT−a3=U/AT, U/AT−a4=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6,x5,x5,x6.
Hence,ErAT−a1=ErAT−a2=ErAT−a3=ErAT≈6.520, ErAT−a4=2/6log22+5/6log25+4/6log24+6/6log26+1/6log21+2/6log22≈7.122.
sigina1,AT=sigina2,AT=sigina3,AT=0, sigina4,AT=0.602>0.
Accordingly,coreAT=a4. Let A0=a4.
U/a4=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6, x5,x1,x3,x5,x6.
Therefore,Era4≈9.170, Era4≈ErAT.
A/A0=a1,a2,a3; for any ai∈A/A0i=1,2,3, we calculate ErA0∪ai as follows:
ErA0∪a1=Era1,a4≈6.520,
ErA0∪a2=Era2,a4≈7.629,
ErA0∪a3=Era3,a4≈6.520.
Thus,ErA0∪a1=ErA0∪a3<ErA0∪a2.
Then, letA0=a1,a4 or A0=a3,a4.
Similarly, by calculation, we know thatEra1,a4=ErAT, Era3,a4=ErAT.
Therefore, the minimal attribute reduction ofAT is RedAT=a1,a4 or a3,a4.Table 1
Intuitionistic fuzzy information system.
Ua1a2a3a4x1(0.3, 0.5)(0.6, 0.4)(0.5, 0.2)(0.7, 0.1)x2(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.1, 0.8)x3(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.7, 0.1)x4(0.1, 0.8)(0.1, 0.8)(0.2, 0.7)(0.1, 0.8)x5(0.9, 0.1)(0.8, 0.1)(0.8, 0.1)(0.9, 0.0)x6(0.4, 0.6)(0.7, 0.3)(0.6, 0.3)(0.7, 0.1)


### 2.4. Multisource Intuitionistic Fuzzy Information System (MSIFIS)

In this subsection, we introduce multisource intuitionistic fuzzy information systems. When a person obtains information about a group of objects from different sources, each source can be regarded as a classical information system, which is an attribute with some intuitionistic fuzzy attribute values. The information system is called multisource intuitionistic fuzzy information system (MSIFIS). AMSIFIS is shown in Figure 1.Figure 1
A multisource intuitionistic fuzzy information system.This paper mainly discusses the case which shares the same structure. That is, the same object, attributes, and the value of the object’s attributes have the same digital characteristics under different information sources.Definition 14.
(see [30]): Let MSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system, where:(1)
U is a set of nonempty finite objects called the universe.(2)
Ai is a finite nonempty set of the attributes of each subsystem.(3)
Vi=Vaa∈Ai, Va is the value range of attribute a.(4)
fi represents the corresponding relation between the object and the feature under the information source i.
LetMSIFDIS=IFDIS1,IFDIS2,…,IFDISn be a multisource intuitionistic fuzzy decision information system, where IFDISi=U,Ai∪D,Vi,fii≤n and D is the decision attribute.


## 2.1. Intuitionistic Fuzzy Rough Set Model

Definition 1.
(see [21]): Given the universe of discourse U, an intuitionistic fuzzy rough set A on U is defined by A=x,μAx,νAx|x∈U, where the functions μAx:μ⟶0,1 and vAx:v⟶0,1 satisfy 0≤μAx+νAx≤1 for all x∈U. μAx and vAx are called the degrees of membership and nonmembership of element x∈U to A, respectively. The family of all intuitionistic fuzzy sets in U is denoted by IFU. When μAx+νAx=1, A degenerates into a fuzzy set.Definition 2.
(see [21]): Let A=x,μAx,νAx|x∈U∈IFU, B=x,μBx,νBx|x∈U∈IFU; then:(1)
The supplementary set ofA∼A=x,vAx,μAx|x∈U.(2)
A=B⇔μAx=μBx,νAx=νBx,∀x∈U.(3)
A⊆B⇔μAx=μBx,νAx=νBx,∀x∈U.(4)
A∩B=x,μAx∧μBx,νAx∨νBx|x∈U.(5)
A∪B=x,μAx∧μBx,νAx∨νBx|x∈U.(6)
A−B=A∩∼B.(7)
A△B=A∩Bc∪B∩Ac.Definition 3.
(see [21]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, where U=x1,x2…xn is a nonempty finite set of objects called the universe of discourse; AT=a1,a2…an is a nonempty finite attribute set; V=∪a∈ATVa is the attribute value set; f:U×AT⟶V is the information function, A⊆AT; a binary dominant relation RA is defined as follows:(1)RA=xi,xj∈U×U|fxi,a≤fxj,a,∀a∈A=xi,xj∈U×U|μaxi≤μaxj∧νaxi≥νaxj,∀a∈A.
Obviously, this binary dominant relation satisfies reflexivity, antisymmetry, and transitivity, andRA=∩Ra∈A. Based on the above dominant relation, the dominant class RAxi definition of object xi can be obtained as follows:(2)RAxi=xj∈U:fxi,a≤fxj,a,∀a∈A.Definition 4.
(see [22]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system; for any X⊆U,A⊆AT, the lower approximation and upper approximation of X for RA are defined as follows:(3)RAX=x∈U|RAxi⊆X;RA¯X=x∈U|RAxi∩X≠∅.
By definition,RA¯X consists of all objects that are definitely contained in the set X. RA¯X consists of all objects that are possibly contained in the set X. If RA¯X=RA¯X, then X is the exact set of RA; otherwise, it is the rough set about RA.The positive, negative, and boundary regions of X can be defined as follows:(4)POSRAX=RA¯X,NEGRAX=u−RA¯X,BNDRAX=RA¯X−RA¯X.Definition 5.
(see [22]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system; for any X⊆U,A⊆AT, the approximate accuracy and the roughness of the target set X are defined as follows:(5)αRA,X=RA¯XRA¯X;ρRA,X=1−=RA¯XRA¯X,when RA¯X=0, αRA,X=0,ρRA,X=1.Definition 6.
(see [23]): Let A=x,μAx|x∈U be a fuzzy sets on universe U; the nonmembership degree of A is defined as follows:(6)νAx=0,μAx>0.5,0.5,μAx≤0.5.
Then,x,μAx,νAx|x∈U is the correspondence intuitionistic fuzzy sets of the fuzzy set A.


## 2.2. Multigranulation Intuitionistic Fuzzy Rough Set Model

Multigranulation rough set model was first proposed by Qian et al. It is an extension of the classical rough set theory. In this subsection, the single-granulation intuitionistic fuzzy rough set is extended to the multigranulation intuitionistic fuzzy rough set model under multiple dominant relations. Generalized multigranulation intuitionistic fuzzy rough set is the generalization of optimistic multigranulation intuitionistic fuzzy rough set and pessimistic multigranulation intuitionistic fuzzy rough set.Given the definition of the support characteristic function, we use this function to complete the object selection.Definition 7.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; for any X⊆U, the support characteristic function of x for X is denoted as(7)SXAix=1,RAix⊆X,0,else.
The support characteristic function is used to describe the inclusion relation between dominance classRAix and concept X, Which indicates whether object x accurately supports X by Ai, or whether object x has a positive description of X by Ai.Definition 8.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; SXAix is the characteristic function of x for X. Given information level β∈0.5,1, for any X⊆U, the generalized lower and upper approximation of X are defined as follows:(8)GM¯∑i=1mAiXβ=x∈U|∑i=1mSXAixm≥βGM¯∑i=1mAiXβ=x∈U|∑i=1m1−SXcAixm>1−β.
IfGM¯∑I=1mAiXβ=GM¯∑I=1mAiXβ, then the target set X is generalized and definable; otherwise, it is generalized and rough. GM¯∑I=1mAiXβ=GM¯∑I=1mAiXβ is called the generalized multigranulation intuitionistic fuzzy rough set model GMIFRS.Definition 9.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; SXAix is the support characteristic function of x for X; for any X⊆U, the pessimistic lower and upper approximation of X are defined as follows:(9)PM¯∑i=1mAiX=x∈U|∧i=1mRAix⊆X=x∈U|∑i=1mSXAixm≥1,PM¯∑i=1mAiX==x∈U|∨i=1mRAix∩X≠∅=x∈U|∑i=1m1−SXcAixm>0,where “∨” denotes “or” and “∧” denotes “and.”
IfPM¯∑I=1MAiX=PM¯∑I=1MAiX, then the target set X is pessimistic and definable; otherwise, X is pessimistic and rough. PM¯∑I=1MAiX=PM¯∑I=1MAiX is called the pessimistic multigranulation intuitionistic fuzzy rough set model PMIFRS.Definition 10.
(see [26]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, Ai⊆AT, i=1,2,…mm≤2AT; SXAix is the support characteristic function of x for X; for any X⊆U, the optimistic lower and upper approximation of X are defined as follows:(10)OM¯∑i=1mAiX=x∈U|∧i=1mRAix⊆X=x∈U|∑i=1mSXAixm>0,OM¯∑i=1mAiX==x∈U|∨i=1mRAix∩X≠∅=x∈U|∑i=1m1−SXcAixm>1,where “∨” denotes “or” and “∧” denotes “and.”
IfOM¯∑I=1MAiX=OM¯∑I=1MAiX, then the target set X is optimistic and definable; otherwise, X is optimistic and rough. OM¯∑I=1MAiX=OM¯∑I=1MAiX is called the optimistic multigranulation intuitionistic fuzzy rough set model OMIFRS.


## 2.3. Attribute Reduction of Intuitionistic Fuzzy Information System Based on Dominant Relation

In this subsection, we define the attribute importance of intuitionistic fuzzy information system based on the dominant relation. We also give the attribute reduction algorithm of intuitionistic fuzzy information system based on knowledge rough entropy.Definition 11.
(see [9]): Let IFIS=U,AT,V,f be an intuitionistic fuzzy information system, A⊆AT, U/RA=RAxi|xi∈U; RA is a binary dominant relation; the rough entropy of A is defined as(11)ErA=∑i=1URAxiUlog2RAxi.
It is obvious that there are minimum and maximum values of the rough entropy. WhenRA is the finest classification, the rough entropy of A has a minimum value of 0; if RA is the coarsest classification, the rough entropy of A has a maximum value of Ulog2U.
Each information system has many attributes, but some of these are redundant. To measure the significance of a single attribute, [3, 4] give the concepts of relative importance and absolute importance of attribute.Definition 12.
LetIFIS=U,AT,V,f be an intuitionistic fuzzy information system, A⊆AT; for any a∈A, for any b∈AT−A, the relative importance and absolute importance of attribute a in attribute set A is defined as(12)sigina,A=ErA−a−ErA;siginb,A=ErA−ErA∪b.
According to this definition, the following properties are established:(1)
0≤sigina,A≤log2U.(2)
Attributea is necessary ⇔sigina,A>0.(3)
The attribute core ofA is coreA=a∈A|sigina,A>0.Definition 13.
LetIFIS=U,AT,V,f be an intuitionistic fuzzy information system, A⊆AT; if ErA=ErAT, for any a∈A and ErA−a≠ErAT, A is the attribute reduction of intuitionistic fuzzy information system based on dominance relation.
Since the attribute core is a subset of attribute reduction, in the heuristic reduction process starting from the attribute core, we often add attributes to attribute core through a measurement method to obtain attribute reduction. Next, an attribute reduction algorithm for intuitionistic fuzzy information system based on knowledge rough entropy is given.Algorithm 1: The minimum attribute reduction of intuitionistic fuzzy information system.
Input:IFIS=U,AT,V,f, AT=a1,a2,…,an.Output: The minimum attribute reductionRedAT.(1)
begin(2)
setcoreAT←∅; RedAT←∅; A0←∅;(3)
fora∈AT do(4)
sigina,A=ErAT−a−ErAT;(5)
ifsigina,A>0 then(6)
coreAT=coreAT∪a;(7)
end(8)
end(9)
setA0←coreAT(10)
whileErA0≠ErAT do(11)
fora∗∈AT/A0(12)
sigouta∗,A0=ErA0−ErA0∪a∗;(13)
a∗∗=maxsigouta∗,A0;(14)
setA1←a∗∗(15)
end(16)
setA0←A0∪A1(17)
endreturn:RedAT=A0(18)
endExample 1.
Table1 shows an intuitionistic fuzzy information system IFIS=U,AT,V,f, where the universe U=x1,x2,x3,x4,x5,x6 and attribute set AT=a1,a2,a3,a4. We use Algorithm 1 to calculate the minimum attribute reduction of the intuitionistic fuzzy information system given in Table 1.
From this table, it is easy to calculate the granular structure:
U/AT=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6,x5,x5,x6.
According to Definition11, the rough entropy of AT is as follows:
ErAT=2/6log22+5/6log25+4/6log24+6/6log26+1/6log21+2/6log22≈6.520.
After calculating the core of attribute setAT,
U/AT−a1=U/AT−a2=U/AT−a3=U/AT, U/AT−a4=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6,x5,x5,x6.
Hence,ErAT−a1=ErAT−a2=ErAT−a3=ErAT≈6.520, ErAT−a4=2/6log22+5/6log25+4/6log24+6/6log26+1/6log21+2/6log22≈7.122.
sigina1,AT=sigina2,AT=sigina3,AT=0, sigina4,AT=0.602>0.
Accordingly,coreAT=a4. Let A0=a4.
U/a4=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6, x5,x1,x3,x5,x6.
Therefore,Era4≈9.170, Era4≈ErAT.
A/A0=a1,a2,a3; for any ai∈A/A0i=1,2,3, we calculate ErA0∪ai as follows:
ErA0∪a1=Era1,a4≈6.520,
ErA0∪a2=Era2,a4≈7.629,
ErA0∪a3=Era3,a4≈6.520.
Thus,ErA0∪a1=ErA0∪a3<ErA0∪a2.
Then, letA0=a1,a4 or A0=a3,a4.
Similarly, by calculation, we know thatEra1,a4=ErAT, Era3,a4=ErAT.
Therefore, the minimal attribute reduction ofAT is RedAT=a1,a4 or a3,a4.Table 1
Intuitionistic fuzzy information system.
Ua1a2a3a4x1(0.3, 0.5)(0.6, 0.4)(0.5, 0.2)(0.7, 0.1)x2(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.1, 0.8)x3(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.7, 0.1)x4(0.1, 0.8)(0.1, 0.8)(0.2, 0.7)(0.1, 0.8)x5(0.9, 0.1)(0.8, 0.1)(0.8, 0.1)(0.9, 0.0)x6(0.4, 0.6)(0.7, 0.3)(0.6, 0.3)(0.7, 0.1)


## 2.4. Multisource Intuitionistic Fuzzy Information System (MSIFIS)

In this subsection, we introduce multisource intuitionistic fuzzy information systems. When a person obtains information about a group of objects from different sources, each source can be regarded as a classical information system, which is an attribute with some intuitionistic fuzzy attribute values. The information system is called multisource intuitionistic fuzzy information system (MSIFIS). AMSIFIS is shown in Figure 1.Figure 1
A multisource intuitionistic fuzzy information system.This paper mainly discusses the case which shares the same structure. That is, the same object, attributes, and the value of the object’s attributes have the same digital characteristics under different information sources.Definition 14.
(see [30]): Let MSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system, where:(1)
U is a set of nonempty finite objects called the universe.(2)
Ai is a finite nonempty set of the attributes of each subsystem.(3)
Vi=Vaa∈Ai, Va is the value range of attribute a.(4)
fi represents the corresponding relation between the object and the feature under the information source i.
LetMSIFDIS=IFDIS1,IFDIS2,…,IFDISn be a multisource intuitionistic fuzzy decision information system, where IFDISi=U,Ai∪D,Vi,fii≤n and D is the decision attribute.


## 3. Weighted Multigranulation Intuitionistic Fuzzy Rough Set Model for MSIFIS

In the theory of multigranulation intuitionistic fuzzy rough set, optimistic multigranulation only requires the existence of the granular knowledge and the target concept to satisfy the inclusion relation; the requirements are too loose for approximate characterization. However, the pessimistic multigranularity requires that the knowledge granule and the target concept satisfy the inclusion relation at all granularity spaces; the requirements are too strict for approximate characterization. These multigranularity models treat each granularity space equally.However, in practical applications, considering the application background and user preferences, the granularity spaces are not equally important. Therefore, in this section, we study multigranularity intuitionistic fuzzy rough set with weights in multisource intuitionistic fuzzy information system.


### 3.1. Optimal Source Selection forMSIFIS

MSIFISs are used to express information from multiple sources. As the number of information sources increases, the selection of reliable information sources is a key issue in the field of information technology research. To characterize the effectiveness of an information source, we define the two source quality metrics of the internal-confidence degree and external-confidence degree.Definition 15.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system. For each single information source IFISi∈MSIFIS, let RedATi be the reduction of IFISi. The internal-confidence degree of IFISi can be defined as follows:(13)ICIFISi=RedAiAi.
From the above definition, the following is obvious:(1)
ICIFISi is the ratio of the cardinalities of RedATi and ATi, and 0≤ICIFISi≤1.(2)
IfICIFISi>0.5, the majority of the attributes are useful and the source is reliable. In practical applications, different thresholds ICIFISi>α can be defined in different fields according to the specific requirements.Definition 16.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi,IFISj∈MSIFIS, the difference between them can be defined as follows:(14)DIFISi,IFISj=∑k=1URAixk∪RAjxk−RAixk∩RAjxk,where RATixk is the dominant class of xk with respect to RAi.
According to this definition, the following is obvious:(1)
∀IFISi,IFISj∈MSIFIS,DIFISi,IFISj≥0.(2)
∀IFISi,IFISj∈MSIFIS,DIFISi,IFISj=DIFISj,IFISi.(3)
When the single information sourcesIFISi and IFISj have the same granular structure, the difference between IFISi and IFISj has a minimum of 0. If the single information source IFISi has the finest granular structure and IFISj has the coarsest structure, then the difference between IFISi and IFISj reaches a maximum of UU−1=U2−U.Definition 17.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi,IFISj∈MSIFIS, the external correlation between IFISi and IFISj can be defined as(15)ecIFISi,IFISj=1−DIFISi,IFISjU2−U,where DIFISi,IFISj is the difference between IFISi and IFISj.
From this definition, the following is obvious:
To clarify the relationship between the external-confidence degrees ofIFISi and IFISj, we can construct an external-confidence degree matrix MEC as follows:(16)MEC=e11e12⋯e1ne21e22⋯e2n⋯⋯⋯⋯en1en2⋯enn,where eij=ecIFISi,IFISj.Definition 18.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi,IFISj∈MSIFIS, the external-confidence degree between IFISi and IFISj can be defined as(17)ECIFISi=1n∑j=1n1−DIFISi,IFISjU2−U=1n∑j=1necIFISi,IFISj.
From this definition, the following is obvious:(1)
For anyIFISi∈MS, 0≤ECIFISi≤1.(2)
Similar to the internal-confidence degree, different thresholdsECIFISi>β may be applicable in different fields according to specific requirements.Definition 19.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi∈MSIFIS, we define a total score for information source IFISi as follows:(18)TCIFISi=ICIFISi+ECIFISi.where ICIFISi is the internal-confidence degree of a source IFISi, and ECIFISi is the external-confidence degree of a source IFISi.Example 2.
Table2 shows a multisource intuitionistic fuzzy information system, which consists of three intuitionistic fuzzy information tables; xi∈Ui=1,2,3,4,5,6 represents six evaluated objects, and ai∈Ai=1,2,3,4| represents conditional attribute set.
From this table, it is easy to calculate the granular structure for each information source:
U/AT1=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6,x5,x5,x6, U/AT2=x1,x2,x2,x3,x4,x2,x3,x4,x5,x2,x3,x4,x6, U/AT3=x1,x3,x2,x3,x2,x3,x4,x3,x5,x2,x3,x6.
From the result of example 1,RedAT1=a1,a4ora3,a4.
Similarly,RedAT2=a2,a3,a4, and RedAT3=a1,a2,a4ora1,a3,a4.The calculation results of all sources IFISii=1,2,3 are presented in Table 3.
Then, the internal-confidence degree of sourceIFISii=1,2,3 is as follows:
ICIFIS1=RedAT1/AT1=2/4=0.5,
ICIFIS2=RedAT2/AT2=3/4=0.75,
ICIFIS3=RedAT3/AT3=3/4=0.75.
Next, the external-confidence degree betweenIFISi and IFISj can be calculated as follows:
DIFIS1,IFIS1=DIFIS2,IFIS2=DIFIS3,IFIS3=0,
DIFIS1,IFIS2=21, DIFIS1,IFIS3=16, DIFIS2,IFIS3=7.
By Definition18, the external correlation between IFISi and IFISj can be computed. The result is as follows:
ecIFIS1,IFIS1=ecIFIS2,IFIS2=ecIFIS3,IFIS3=1,
ecIFIS1,IFIS2=1−21/36−1=0.3, ecIFIS1,IFIS3=1−16/36−1≈0.467, ecIFIS2,IFIS3=1−7/36−1=0.767.Thus, we can get the external-confidence degree matrix MEC as follows:(19)MEC=10.30.4670.310.7670.4670.7671.
Therefore, the external-confidence degrees for each information sourceIFISii=1,2,3 is ECIFISi=0.589,0.689,0.745.
Furthermore, the total score for each information sourceIFISii=1,2,3 can be calculated using Definition 19:TCIFIS1=0.5+0.589=1.089,TCIFIS2=0.5+0.589=1.439,TCIFIS3=0.5+0.589=1.495.
Therefore, the quality ranking of information sources isIFIS3>IFIS2>IFIS1, and IFIS3 is the optimal source of the multisource intuitionistic fuzzy information system.Table 2
Multisource intuitionistic fuzzy information system.
IFIS1IFIS2IFIS3a1a2a3a4a1a2a3a4a1a2a3a4x1(0.3, 0.5)(0.6, 0.4)(0.5, 0.2)(0.7, 0.1)(0.3, 0.5)(0.7, 0.3)(0.5, 0.1)(0.7, 0.1)(0.2, 0.5)(0.6, 0.2)(0.4, 0.6)(0.3, 0.5)x2(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.1, 0.8)(0.8, 0.2)(0.8, 0.1)(0.7, 0.1)(1.0, 0.0)(0.4, 0.5)(0.5, 0.4)(0.3, 0.5)(0.6, 0.4)x3(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.7, 0.1)(0.8, 0.2)(0.9, 0.0)(0.7, 0.1)(0.8, 0.2)(0.6, 0.2)(0.7, 0.1)(0.4, 0.3)(0.5, 0.2)x4(0.1, 0.8)(0.1, 0.8)(0.2, 0.7)(0.1, 0.8)(0.9, 0.1)(0.9, 0.0)(0.8, 0.1)(0.6, 0.3)(0.3, 0.7)(0.2, 0.8)(0.2, 0.7)(0.4, 0.6)x5(0.9, 0.1)(0.8, 0.1)(0.8, 0.1)(0.9, 0.0)(0.7, 0.2)(0.6, 0.4)(0.6, 0.2)(0.5, 0.3)(0.6, 0.2)(0.2, 0.7)(0.1, 0.7)(0.1, 0.8)x6(0.4, 0.6)(0.7, 0.3)(0.6, 0.3)(0.7, 0.1)(0.6, 0.3)(0.2, 0.5)(0.7, 0.2)(0.4, 0.5)(0.2, 0.7)(0.1, 0.8)(0.1, 0.6)(0.5, 0.4)Table 3
The calculation results for eachIFISi.
ErAisigina1,Aisigina2,Aisigina3,Aisigina4,AiRedAiIFIS16.520000.602a1,a4ora3,a4IFIS2300.3330.6022.126a2,a3,a4IFIS32.2521000.874a1,a2,a4ora1,a3,a4


### 3.2. Weighted Generalized Multigranularity Intuitionistic Fuzzy Rough Set Model forMSIFIS

Definition 20.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, parameter minωi≤β≤1i=1,2,…,n denotes the information level with respect to ∑i=1nAi; for any X⊆U, the lower approximation and upper approximation of X weighted generalized multigranularity intuitionistic fuzzy rough set are defined as follows:(20)MSIF¯∑i=1nAiωXβ=x∈U|∑i=1nωiSXAix≥β;MSIF¯∑i=1nAiωXβ=∼MSIF¯∑i=1nAiω∼Xβ=x∈U|∑i=1nωi1−SXcAix>1−β.
IfMSIF¯∑i=1nAiωXβ=MSIF¯∑i=1nAiωXβ, then the target set X is definable for ∑i=1nAi; otherwise, X is rough for ∑i=1nAi. MSIF¯∑i=1nAiωXβ,MSIF¯∑i=1nAiωXβ is called the weighted generalized multigranularity intuitionistic fuzzy rough set model for MSIFIS.
In the weighted generalized multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions ofX can be defined as follows:(21)POS∑i=1nAiωXβ=MSIF¯∑i=1nAiωXβ;NEG∑i=1nAiωXβ=U−MSIF¯∑i=1nAiωXβ;BND∑i=1nAiωXβ=MSIF¯∑i=1nAiωXβ−MSIF¯∑i=1nAiωXβ.
The basic properties of the weighted generalized multigranularity intuitionistic fuzzy rough set forMSIFIS are given by the following theorem.Theorem 1.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, parameter minωi≤β≤1i=1,2,…,n denotes the information level with respect to ∑i=1nAi; ∀X,Y⊆U, the following conclusions hold:(1)
MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ,MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ.(2)
MSIF¯∑i=1nAiωXβ⊆X⊆MSIF¯∑i=1nAiωXβ.(3)
MSIF¯∑i=1nAiω∅β=MSIF¯∑i=1nAiω∅β=∅.(4)
MSIF¯∑i=1nAiωUβ=MSIF¯∑i=1nAiωUβ=U.(5)
MSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MISF¯∑i=1nAiωYβ,MSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ.(6)
MSIF¯∑i=1nAiωX∩Yβ⊆MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ,MSIF¯∑i=1nAiωX∩Yβ⊆MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ.(7)
X⊆Y⇒MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ,X⊆Y⇒MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ.(8)
β1≤β2⇒MSIF¯∑i=1nAiωXβ2⊆MSIF¯∑i=1nAiωXβ1,β1≤β2⇒MSIF¯∑i=1nAiωXβ2⊇MSIF¯∑i=1nAiωXβ1.(9)
MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ,MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ.Proof.
(1)
Because(22)∼MSIF¯∑i=1nAiωXβ=x∈U|∑i=1nωi1−SXcAix≤1−β=x∈U|∑i=1nωi−∑i=1nωiSXcAix≤1−β=x∈U|1−∑i=1nωiSXcAix≤1−β=x∈U|∑i=1nωiSXcAix≥β=MSIF¯∑i=1nAiω∼Xβ,MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ.Similarly,MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ.Therefore, the property is clearly established.(2)
For anyx∈MSIF¯∑i=1nAiωXβ, we can know ∑i=1nωiSXAix≥β. In addition, minωi≤β≤1, so ∃Ai, s.t. SXAix=1. Namely, xAi⊆X. We have x∈X; thus, MSIF¯∑i=1nAiωXβ⊆X.MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ=∼⊆X. Hence, X⊆MSIF¯∑i=1nAiωXβ.Therefore, property (2) has been proved.(3)
For anyx∈U, we can easily get S∅Aix=0,SUAix=1. Therefore, ∑i=1nωiS∅Aix=0,∑i=1nωiSUAix=1.Thus,MSIF¯∑i=1nAiω∅β=x∈U|∑i=1nωiS∅Aix≥β=x∈U|0≥β=∅,MSIF¯∑i=1nAiω∅β=x∈U|∑i=1nωi1−SUAix>1−β=x∈U|0>1−β=∅.Therefore, property (3) has been proved.(4)
According to (3), MSIF¯∑i=1nAiωUβ=x∈U|∑i=1nωiSUAix≥β=x∈U|1≥β=U,MSIF¯∑i=1nAiωUβ=x∈U|∑i=1nωi1−S∅Aix>1−β=x∈U|1>1−β=U.Therefore, property (4) has been proved.(5)
For anyx∈MSIF¯∑i=1nAiωX∪Yβ⇔x∈MSIF¯∑i=1nAiωXβorx∈MSIF¯∑i=1nAiωYβ⇔∑i=1nωiSXAix≥βor∑i=1nωiSYAix≥β⇒∑i=1nωiSX∪YAix≥∑i=1nωiSXAix∨SYAix≥∑i=1nωiSXAix∧∑i=1nωiSYAix≥β.Therefore,x∈MSIF¯∑i=1nAiωX∪Yβ.Then, we can get thatMSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ.Meanwhile, for anyx∈MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ,⇔x∈MSIF¯∑i=1nAiωXβorx∈MSIF¯∑i=1nAiωYβ,⇔∑i=1nωi1−SXcAix>1−βor∑i=1nωi1−SYcAix>1−β⇔∑i=1nωiSXcAix<βor∑i=1nωiSYcAix<β⇒∑i=1nωiSX∪YcAix=∑i=1nωiSXc∩YcAix=∑i=1nωiSXcAix∧SYcAix≤∑i=1nωiSXcAix∧∑i=1nωiSYcAix<β⇒∑i=1nωi1−SX∪YcAix>1−β⇒x∈MSIF¯∑i=1nAiωX∪Yβ.Then, we obtain thatMSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ.Therefore, the property is clearly established.(6)
∀x∈MSIF¯∑i=1nAiωX∩Yβ⇒∑i=1nωiSX∩YAix=∑i=1nωiSXAix∧SYAix≥β⇒∑i=1nωiSXAix∧∑i=1nωiSYAix≥∑i=1nωiSXAix∧SYAix≥β⇒∑i=1nωiSXAix≥βand∑i=1nωiSYAix≥β⇒x∈MSIF¯∑i=1nAiωXβandx∈MSIF¯∑i=1nAiωYβ⇒x∈MSIF¯∑i=1nAiωXβ∩MSIF∑i=1nAiωYβ.Thus,MSIF¯∑i=1nAiωX∩Yβ⊆MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ.Meanwhile, for anyx∈MSIF¯∑i=1nAiωX∩Yβ⇒∑i=1nωi1−SX∩YcAix>1−β⇒∑i=1nωiSX∩YcAix<β⇒∑i=1nωiSXc∪YcAix<β⇒∑i=1nωiSXcAix∨∑i=1nωiSYcAix≤∑i=1nωiSXcAix∨SYcAix≤∑i=1nωiSXc∪YcAix<β⇒∑i=1nωiSXcAix<βand∑i=1nωiSYcAix<β⇒1−∑i=1nωiSXcAix>1−βand1−∑i=1nωiSYcAix>1−β⇒x∈MSIF¯∑i=1nAiωXβandx∈MSIF¯∑i=1nAiωYβ⇒x∈MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ.Therefore, the property is clearly established.(7)
BecauseX⊆Y, we can know X∪Y=Y.According to property (5), MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωX∪Yβ.Therefore,MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωYβ.Thus,MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ.Meanwhile, according to property (5), MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωX∪Yβ.Therefore,MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωYβ.Thus,MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ.Therefore, the property is clearly established.(8)
For anyx∈MSIF¯∑i=1nAiωXβ2, we have ∑i=1nωiSXAix≥β2≥β1.Therefore,x∈MSIF¯∑i=1nAiωXβ1.Then,β1≤β2 implies that MSIF¯∑i=1nAiωXβ2⊆MSIF¯∑i=1nAiωXβ1.Similarly, we can prove thatβ1≤β2⇒MSIF¯∑i=1nAiωXβ2⊇MSIF¯∑i=1nAiωXβ1.Therefore, property (8) has been proved.(9)
According to properties (2) and (7),MSIF¯∑i=1nAiωXβ⊆X⇒MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ⊆MSIF¯∑i=1nAiωXβ.
The following is the proof:MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ.
For anyx∈MSIF¯∑i=1nAiωXβ, we can know ∑i=1nωiSXAix≥β.
minωi≤β≤1; thus, ∃Ai, s.t.SXAix=1; namely, xAi⊆X.
Then, according to property (7), we have MSIF¯∑i=1nAiωxAiβ⊆MSIF¯∑i=1nAiωXβ.
Therefore,xAi⊆MSFI¯∑i=1nAiωXβ; we can get ∑i=1nωiSMSIF¯∑i=1nAiωXβAix≥β, and x∈MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ.
Therefore,MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ, from which one can get that MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ.
Meanwhile, we can prove thatMSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ.
Property (1) shows that in multisource intuitionistic fuzzy information system, the lower approximation operator, and the upper approximation operator of weighted generalized multigranularity intuitionistic fuzzy rough sets satisfy duality. Property (2) illustrates the inclusion relationship between the lower approximation, the upper approximation, and the target concept. Properties (3) and (4) show the approximation of two special sets. The lower and upper approximation of the empty set ∅ and the universe U are themselves. Properties 5−8 state the monotonicity of MSIF¯∑i=1nAiω and MSIF¯∑i=1nAiω. Moreover, property (9) expresses the idempotency between the lower approximation operator and the upper approximation operator.


### 3.3. Weighted Optimistic Multigranularity Intuitionistic Fuzzy Rough Set Model forMSIFIS

Definition 21.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, ∀X⊆U, the lower approximation and upper approximation of X weighted optimistic multigranularity intuitionistic fuzzy rough set are defined as follows:(23)OMSIF¯∑i=1nAiωX=x∈U|∑i=1nωiSXAix>0;OMSIF¯∑i=1nAiωX=∼OMSIF¯∑i=1nAiω∼X=x∈U|∑i=1nωi1−SXcAix≥1.
IfOMSIF¯∑i=1nAiωX=OMSIF¯∑i=1nAiωX, then the target set X is optimistic and definable for ∑i=1nAi; otherwise, X is optimistic and rough for ∑i=1nAi. OMSIF¯∑i=1nAiωX,OMSIF¯∑i=1nAiωX is called the weighted optimistic multigranularity intuitionistic fuzzy rough set model for MSIFIS.
In the weighted optimistic multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions ofX can be defined as follows:(24)POS∑i=1nAiωXO=OMSIF¯∑i=1nAiωX;NEG∑i=1nAiωXO=U−OMSIF¯∑i=1nAiωX;BND∑i=1nAiωXO=OMSIF¯∑i=1nAiωX−OMSIF¯∑i=1nAiωX.
The basic properties of the weighted optimistic multigranularity intuitionistic fuzzy rough set forMSIFIS are given by the following theorem.Theorem 2.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, ∀X,Y⊆U, the following conclusions hold:(1)
OMSIF¯∑i=1nAiω∼X=∼OMSIF¯∑i=1nAiωX,OMSIF¯∑i=1nAiω∼X=∼OMSIF¯∑i=1nAiωX.(2)
OMSIF¯∑i=1nAiωX⊆X⊆OMSIF¯∑i=1nAiωX.(3)
OMSIF¯∑i=1nAiω∅=OMSIF¯∑i=1nAiω∅=∅.(4)
OMSIF¯∑i=1nAiωU=OMSIF¯∑i=1nAiωU=U.(5)
OMSF¯∑i=1nAiωX∪Y⊇OMSIF¯∑i=1nAiωX∪OMSIF¯∑i=1nAiωY,OMSIF¯∑i=1nAiωX∪Y⊇OMSIF¯∑i=1nAiωX∪OMSIF¯∑i=1nAiωY.(6)
OMSIF¯∑i=1nAiωX∩Y⊆OMSIF¯∑i=1nAiωX∩OMSIF¯∑i=1nAiωY,OMSIF¯∑i=1nAiωX∩Y⊆OMSIF¯∑i=1nAiωX∪OMSIF¯∑i=1nAiωY.(7)
X⊆Y⇒OMSIF¯∑i=1nAiωX⊆OMSIF¯∑i=1nAiωY,X⊆Y⇒OMSIF¯∑i=1nAiωX⊆OMSIF¯∑i=1nAiωY.(8)
OMSIF¯∑i=1nAiωOMSIF¯∑i=1nAiωX=OMSIF¯∑i=1nAiωX,OMSIF¯∑i=1nAiωOMSIF¯∑i=1nAiωX=OMSIF¯∑i=1nAiωX.Proof.
It is similar to that of Theorem1.


### 3.4. Weighted Pessimistic Multigranularity Intuitionistic Fuzzy Rough Set Model forMSIFIS

Definition 22.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X⊆U, the lower approximation and upper approximation of X weighted pessimistic multigranularity intuitionistic fuzzy rough set are defined as follows:(25)PMSIF¯∑i=1nAiωX=x∈U∣∑i=1nωiSXAix≥1;PMSIF¯∑i=1nAiωX=∼PMSIF¯∑i=1nAiω∼X=x∈U∣∑i=1nωi1−SXcAix>0.
IfPMSIF¯∑i=1nAiωX=PMSIF¯∑i=1nAiωX, then the target set X is pessimistic and definable for ∑i=1nAi; otherwise, X is pessimistic and rough for ∑i=1nAi. PMSIF¯∑i=1nAiωX,PMSIF¯∑i=1nAiωX is called the weighted pessimistic multigranularity intuitionistic fuzzy rough set model for MSIFIS.
In the weighted pessimistic multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions ofX can be defined as follows:(26)POS∑i=1nAiωXP=PMSIF¯∑i=1nAiωX;NEG∑i=1nAiωXP=U−PMSIF¯∑i=1nAiωX;BND∑i=1nAiωXP=PMSIF¯∑i=1nAiωX−PMSIF¯∑i=1nAiωX.The basic properties of the weighted pessimistic multigranularity intuitionistic fuzzy rough set for MSIFIS are given by the following theorem.Theorem 3.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X,Y⊆U, the following conclusions hold:(1)
PMSIF¯∑i=1nAiω∼X=∼PMSIF¯∑i=1nAiωX,PMSIF¯∑i=1nAiω∼X=∼PMSIF¯∑i=1nAiωX.(2)
PMSIF¯∑i=1nAiωX⊆X⊆PMSIF¯∑i=1nAiωX.(3)
PMSIF¯∑i=1nAiω∅=PMSIF¯∑i=1nAiω∅=∅.(4)
PMSIF¯∑i=1nAiωU=PMSIF¯∑i=1nAiωU=U.(5)
PMSIF¯∑i=1nAiωX∪Y⊇PMSIF¯∑i=1nAiωX∪PMSIF¯∑i=1nAiωY,PMSIF¯∑i=1nAiωX∪Y=PMSIF¯∑i=1nAiωX∪PMSIF¯∑i=1nAiωY.(6)
PMSIF¯∑i=1nAiωX∩Y=PMSIF¯∑i=1nAiωX∩PMSIF¯∑i=1nAiωY,PMSIF¯∑i=1nAiωX∩Y⊆PMSIF¯∑i=1nAiωX∪PMSIF¯∑i=1nAiωY.(7)
X⊆Y⇒PMSIF¯∑i=1nAiωX⊆PMSIF¯∑i=1nAiωY,X⊆Y⇒PMSIF¯∑i=1nAiωX⊆PMSIF¯∑i=1nAiωY.(8)
PMSIF¯∑i=1nAiωPMSIF¯∑i=1nAiωX=PMSIF¯∑i=1nAiωX,PMSIF¯∑i=1nAiωPMSIF¯∑i=1nAiωX=PMSIF¯∑i=1nAiωX.Proof.
It is similar to that of Theorem1.


### 3.5. The Relationship between the Three Models ofMSIFIS

In this subsection, we investigate the relationship between the three models ofMSIFIS.Theorem 4.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X⊆U, the following equations can be obtained:(1)
Ifβ=1, then(27)MSIF¯∑i=1nAiωXβ=PMSIF¯∑i=1nAiωX,MISF¯∑i=1nAiωXβ=PMSIF¯∑i=1nAiωX.(2)
Ifβ=minωii=1,2,…,n, then(28)MSIF¯∑i=1nAiωXβ=OMSIF¯∑i=1nAiωX,MSIF¯∑i=1nAiωXβ=OMSIF¯∑i=1nAiωX.Proof.
It is easily obtained from Definition15, Definition 20, and Definition 21.
From the above theorems, ifβ=1, weighted generalized multigranularity intuitionistic fuzzy rough set degenerates into weighted pessimistic multigranularity intuitionistic fuzzy rough set. If β=minωAi, weighted generalized multigranularity intuitionistic fuzzy rough set degenerates into weighted optimistic multigranularity intuitionistic fuzzy rough set. Hence, weighted generalized multigranularity intuitionistic fuzzy rough set is a generalization of weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set. On the other hand, weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set are the special cases of weighted generalized multigranularity intuitionistic fuzzy rough set.Theorem 5.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X⊆U, the following properties are established:(1)
PMSIF¯∑i=1nAiωX⊆MSIF¯∑i=1nAiωXβ⊆OMSIF¯∑i=1nAiωX.(2)
OMSIF¯∑i=1nAiωX⊆MSIF¯∑i=1nAiωXβ⊆PMSIF¯∑i=1nAiωX.Proof.
From Definition20–Definition 22, the theorem clearly holds.
The theorem shows that the lower and upper approximation of the weighted generalized multigranularity intuitionistic fuzzy rough set are between weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set.


## 3.1. Optimal Source Selection forMSIFIS

MSIFISs are used to express information from multiple sources. As the number of information sources increases, the selection of reliable information sources is a key issue in the field of information technology research. To characterize the effectiveness of an information source, we define the two source quality metrics of the internal-confidence degree and external-confidence degree.Definition 15.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system. For each single information source IFISi∈MSIFIS, let RedATi be the reduction of IFISi. The internal-confidence degree of IFISi can be defined as follows:(13)ICIFISi=RedAiAi.
From the above definition, the following is obvious:(1)
ICIFISi is the ratio of the cardinalities of RedATi and ATi, and 0≤ICIFISi≤1.(2)
IfICIFISi>0.5, the majority of the attributes are useful and the source is reliable. In practical applications, different thresholds ICIFISi>α can be defined in different fields according to the specific requirements.Definition 16.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi,IFISj∈MSIFIS, the difference between them can be defined as follows:(14)DIFISi,IFISj=∑k=1URAixk∪RAjxk−RAixk∩RAjxk,where RATixk is the dominant class of xk with respect to RAi.
According to this definition, the following is obvious:(1)
∀IFISi,IFISj∈MSIFIS,DIFISi,IFISj≥0.(2)
∀IFISi,IFISj∈MSIFIS,DIFISi,IFISj=DIFISj,IFISi.(3)
When the single information sourcesIFISi and IFISj have the same granular structure, the difference between IFISi and IFISj has a minimum of 0. If the single information source IFISi has the finest granular structure and IFISj has the coarsest structure, then the difference between IFISi and IFISj reaches a maximum of UU−1=U2−U.Definition 17.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi,IFISj∈MSIFIS, the external correlation between IFISi and IFISj can be defined as(15)ecIFISi,IFISj=1−DIFISi,IFISjU2−U,where DIFISi,IFISj is the difference between IFISi and IFISj.
From this definition, the following is obvious:
To clarify the relationship between the external-confidence degrees ofIFISi and IFISj, we can construct an external-confidence degree matrix MEC as follows:(16)MEC=e11e12⋯e1ne21e22⋯e2n⋯⋯⋯⋯en1en2⋯enn,where eij=ecIFISi,IFISj.Definition 18.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi,IFISj∈MSIFIS, the external-confidence degree between IFISi and IFISj can be defined as(17)ECIFISi=1n∑j=1n1−DIFISi,IFISjU2−U=1n∑j=1necIFISi,IFISj.
From this definition, the following is obvious:(1)
For anyIFISi∈MS, 0≤ECIFISi≤1.(2)
Similar to the internal-confidence degree, different thresholdsECIFISi>β may be applicable in different fields according to specific requirements.Definition 19.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; ∀IFISi∈MSIFIS, we define a total score for information source IFISi as follows:(18)TCIFISi=ICIFISi+ECIFISi.where ICIFISi is the internal-confidence degree of a source IFISi, and ECIFISi is the external-confidence degree of a source IFISi.Example 2.
Table2 shows a multisource intuitionistic fuzzy information system, which consists of three intuitionistic fuzzy information tables; xi∈Ui=1,2,3,4,5,6 represents six evaluated objects, and ai∈Ai=1,2,3,4| represents conditional attribute set.
From this table, it is easy to calculate the granular structure for each information source:
U/AT1=x1,x5,x1,x2,x3,x5,x6,x1,x3,x5,x6,x1,x2,x3,x4,x5,x6,x5,x5,x6, U/AT2=x1,x2,x2,x3,x4,x2,x3,x4,x5,x2,x3,x4,x6, U/AT3=x1,x3,x2,x3,x2,x3,x4,x3,x5,x2,x3,x6.
From the result of example 1,RedAT1=a1,a4ora3,a4.
Similarly,RedAT2=a2,a3,a4, and RedAT3=a1,a2,a4ora1,a3,a4.The calculation results of all sources IFISii=1,2,3 are presented in Table 3.
Then, the internal-confidence degree of sourceIFISii=1,2,3 is as follows:
ICIFIS1=RedAT1/AT1=2/4=0.5,
ICIFIS2=RedAT2/AT2=3/4=0.75,
ICIFIS3=RedAT3/AT3=3/4=0.75.
Next, the external-confidence degree betweenIFISi and IFISj can be calculated as follows:
DIFIS1,IFIS1=DIFIS2,IFIS2=DIFIS3,IFIS3=0,
DIFIS1,IFIS2=21, DIFIS1,IFIS3=16, DIFIS2,IFIS3=7.
By Definition18, the external correlation between IFISi and IFISj can be computed. The result is as follows:
ecIFIS1,IFIS1=ecIFIS2,IFIS2=ecIFIS3,IFIS3=1,
ecIFIS1,IFIS2=1−21/36−1=0.3, ecIFIS1,IFIS3=1−16/36−1≈0.467, ecIFIS2,IFIS3=1−7/36−1=0.767.Thus, we can get the external-confidence degree matrix MEC as follows:(19)MEC=10.30.4670.310.7670.4670.7671.
Therefore, the external-confidence degrees for each information sourceIFISii=1,2,3 is ECIFISi=0.589,0.689,0.745.
Furthermore, the total score for each information sourceIFISii=1,2,3 can be calculated using Definition 19:TCIFIS1=0.5+0.589=1.089,TCIFIS2=0.5+0.589=1.439,TCIFIS3=0.5+0.589=1.495.
Therefore, the quality ranking of information sources isIFIS3>IFIS2>IFIS1, and IFIS3 is the optimal source of the multisource intuitionistic fuzzy information system.Table 2
Multisource intuitionistic fuzzy information system.
IFIS1IFIS2IFIS3a1a2a3a4a1a2a3a4a1a2a3a4x1(0.3, 0.5)(0.6, 0.4)(0.5, 0.2)(0.7, 0.1)(0.3, 0.5)(0.7, 0.3)(0.5, 0.1)(0.7, 0.1)(0.2, 0.5)(0.6, 0.2)(0.4, 0.6)(0.3, 0.5)x2(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.1, 0.8)(0.8, 0.2)(0.8, 0.1)(0.7, 0.1)(1.0, 0.0)(0.4, 0.5)(0.5, 0.4)(0.3, 0.5)(0.6, 0.4)x3(0.2, 0.7)(0.1, 0.8)(0.4, 0.5)(0.7, 0.1)(0.8, 0.2)(0.9, 0.0)(0.7, 0.1)(0.8, 0.2)(0.6, 0.2)(0.7, 0.1)(0.4, 0.3)(0.5, 0.2)x4(0.1, 0.8)(0.1, 0.8)(0.2, 0.7)(0.1, 0.8)(0.9, 0.1)(0.9, 0.0)(0.8, 0.1)(0.6, 0.3)(0.3, 0.7)(0.2, 0.8)(0.2, 0.7)(0.4, 0.6)x5(0.9, 0.1)(0.8, 0.1)(0.8, 0.1)(0.9, 0.0)(0.7, 0.2)(0.6, 0.4)(0.6, 0.2)(0.5, 0.3)(0.6, 0.2)(0.2, 0.7)(0.1, 0.7)(0.1, 0.8)x6(0.4, 0.6)(0.7, 0.3)(0.6, 0.3)(0.7, 0.1)(0.6, 0.3)(0.2, 0.5)(0.7, 0.2)(0.4, 0.5)(0.2, 0.7)(0.1, 0.8)(0.1, 0.6)(0.5, 0.4)Table 3
The calculation results for eachIFISi.
ErAisigina1,Aisigina2,Aisigina3,Aisigina4,AiRedAiIFIS16.520000.602a1,a4ora3,a4IFIS2300.3330.6022.126a2,a3,a4IFIS32.2521000.874a1,a2,a4ora1,a3,a4


## 3.2. Weighted Generalized Multigranularity Intuitionistic Fuzzy Rough Set Model forMSIFIS

Definition 20.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, parameter minωi≤β≤1i=1,2,…,n denotes the information level with respect to ∑i=1nAi; for any X⊆U, the lower approximation and upper approximation of X weighted generalized multigranularity intuitionistic fuzzy rough set are defined as follows:(20)MSIF¯∑i=1nAiωXβ=x∈U|∑i=1nωiSXAix≥β;MSIF¯∑i=1nAiωXβ=∼MSIF¯∑i=1nAiω∼Xβ=x∈U|∑i=1nωi1−SXcAix>1−β.
IfMSIF¯∑i=1nAiωXβ=MSIF¯∑i=1nAiωXβ, then the target set X is definable for ∑i=1nAi; otherwise, X is rough for ∑i=1nAi. MSIF¯∑i=1nAiωXβ,MSIF¯∑i=1nAiωXβ is called the weighted generalized multigranularity intuitionistic fuzzy rough set model for MSIFIS.
In the weighted generalized multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions ofX can be defined as follows:(21)POS∑i=1nAiωXβ=MSIF¯∑i=1nAiωXβ;NEG∑i=1nAiωXβ=U−MSIF¯∑i=1nAiωXβ;BND∑i=1nAiωXβ=MSIF¯∑i=1nAiωXβ−MSIF¯∑i=1nAiωXβ.
The basic properties of the weighted generalized multigranularity intuitionistic fuzzy rough set forMSIFIS are given by the following theorem.Theorem 1.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, parameter minωi≤β≤1i=1,2,…,n denotes the information level with respect to ∑i=1nAi; ∀X,Y⊆U, the following conclusions hold:(1)
MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ,MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ.(2)
MSIF¯∑i=1nAiωXβ⊆X⊆MSIF¯∑i=1nAiωXβ.(3)
MSIF¯∑i=1nAiω∅β=MSIF¯∑i=1nAiω∅β=∅.(4)
MSIF¯∑i=1nAiωUβ=MSIF¯∑i=1nAiωUβ=U.(5)
MSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MISF¯∑i=1nAiωYβ,MSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ.(6)
MSIF¯∑i=1nAiωX∩Yβ⊆MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ,MSIF¯∑i=1nAiωX∩Yβ⊆MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ.(7)
X⊆Y⇒MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ,X⊆Y⇒MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ.(8)
β1≤β2⇒MSIF¯∑i=1nAiωXβ2⊆MSIF¯∑i=1nAiωXβ1,β1≤β2⇒MSIF¯∑i=1nAiωXβ2⊇MSIF¯∑i=1nAiωXβ1.(9)
MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ,MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ.Proof.
(1)
Because(22)∼MSIF¯∑i=1nAiωXβ=x∈U|∑i=1nωi1−SXcAix≤1−β=x∈U|∑i=1nωi−∑i=1nωiSXcAix≤1−β=x∈U|1−∑i=1nωiSXcAix≤1−β=x∈U|∑i=1nωiSXcAix≥β=MSIF¯∑i=1nAiω∼Xβ,MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ.Similarly,MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ.Therefore, the property is clearly established.(2)
For anyx∈MSIF¯∑i=1nAiωXβ, we can know ∑i=1nωiSXAix≥β. In addition, minωi≤β≤1, so ∃Ai, s.t. SXAix=1. Namely, xAi⊆X. We have x∈X; thus, MSIF¯∑i=1nAiωXβ⊆X.MSIF¯∑i=1nAiω∼Xβ=∼MSIF¯∑i=1nAiωXβ=∼⊆X. Hence, X⊆MSIF¯∑i=1nAiωXβ.Therefore, property (2) has been proved.(3)
For anyx∈U, we can easily get S∅Aix=0,SUAix=1. Therefore, ∑i=1nωiS∅Aix=0,∑i=1nωiSUAix=1.Thus,MSIF¯∑i=1nAiω∅β=x∈U|∑i=1nωiS∅Aix≥β=x∈U|0≥β=∅,MSIF¯∑i=1nAiω∅β=x∈U|∑i=1nωi1−SUAix>1−β=x∈U|0>1−β=∅.Therefore, property (3) has been proved.(4)
According to (3), MSIF¯∑i=1nAiωUβ=x∈U|∑i=1nωiSUAix≥β=x∈U|1≥β=U,MSIF¯∑i=1nAiωUβ=x∈U|∑i=1nωi1−S∅Aix>1−β=x∈U|1>1−β=U.Therefore, property (4) has been proved.(5)
For anyx∈MSIF¯∑i=1nAiωX∪Yβ⇔x∈MSIF¯∑i=1nAiωXβorx∈MSIF¯∑i=1nAiωYβ⇔∑i=1nωiSXAix≥βor∑i=1nωiSYAix≥β⇒∑i=1nωiSX∪YAix≥∑i=1nωiSXAix∨SYAix≥∑i=1nωiSXAix∧∑i=1nωiSYAix≥β.Therefore,x∈MSIF¯∑i=1nAiωX∪Yβ.Then, we can get thatMSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ.Meanwhile, for anyx∈MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ,⇔x∈MSIF¯∑i=1nAiωXβorx∈MSIF¯∑i=1nAiωYβ,⇔∑i=1nωi1−SXcAix>1−βor∑i=1nωi1−SYcAix>1−β⇔∑i=1nωiSXcAix<βor∑i=1nωiSYcAix<β⇒∑i=1nωiSX∪YcAix=∑i=1nωiSXc∩YcAix=∑i=1nωiSXcAix∧SYcAix≤∑i=1nωiSXcAix∧∑i=1nωiSYcAix<β⇒∑i=1nωi1−SX∪YcAix>1−β⇒x∈MSIF¯∑i=1nAiωX∪Yβ.Then, we obtain thatMSIF¯∑i=1nAiωX∪Yβ⊇MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ.Therefore, the property is clearly established.(6)
∀x∈MSIF¯∑i=1nAiωX∩Yβ⇒∑i=1nωiSX∩YAix=∑i=1nωiSXAix∧SYAix≥β⇒∑i=1nωiSXAix∧∑i=1nωiSYAix≥∑i=1nωiSXAix∧SYAix≥β⇒∑i=1nωiSXAix≥βand∑i=1nωiSYAix≥β⇒x∈MSIF¯∑i=1nAiωXβandx∈MSIF¯∑i=1nAiωYβ⇒x∈MSIF¯∑i=1nAiωXβ∩MSIF∑i=1nAiωYβ.Thus,MSIF¯∑i=1nAiωX∩Yβ⊆MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ.Meanwhile, for anyx∈MSIF¯∑i=1nAiωX∩Yβ⇒∑i=1nωi1−SX∩YcAix>1−β⇒∑i=1nωiSX∩YcAix<β⇒∑i=1nωiSXc∪YcAix<β⇒∑i=1nωiSXcAix∨∑i=1nωiSYcAix≤∑i=1nωiSXcAix∨SYcAix≤∑i=1nωiSXc∪YcAix<β⇒∑i=1nωiSXcAix<βand∑i=1nωiSYcAix<β⇒1−∑i=1nωiSXcAix>1−βand1−∑i=1nωiSYcAix>1−β⇒x∈MSIF¯∑i=1nAiωXβandx∈MSIF¯∑i=1nAiωYβ⇒x∈MSIF¯∑i=1nAiωXβ∩MSIF¯∑i=1nAiωYβ.Therefore, the property is clearly established.(7)
BecauseX⊆Y, we can know X∪Y=Y.According to property (5), MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωX∪Yβ.Therefore,MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωYβ.Thus,MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ.Meanwhile, according to property (5), MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωX∪Yβ.Therefore,MSIF¯∑i=1nAiωXβ∪MSIF¯∑i=1nAiωYβ⊆MSIF¯∑i=1nAiωYβ.Thus,MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωYβ.Therefore, the property is clearly established.(8)
For anyx∈MSIF¯∑i=1nAiωXβ2, we have ∑i=1nωiSXAix≥β2≥β1.Therefore,x∈MSIF¯∑i=1nAiωXβ1.Then,β1≤β2 implies that MSIF¯∑i=1nAiωXβ2⊆MSIF¯∑i=1nAiωXβ1.Similarly, we can prove thatβ1≤β2⇒MSIF¯∑i=1nAiωXβ2⊇MSIF¯∑i=1nAiωXβ1.Therefore, property (8) has been proved.(9)
According to properties (2) and (7),MSIF¯∑i=1nAiωXβ⊆X⇒MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ⊆MSIF¯∑i=1nAiωXβ.
The following is the proof:MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ.
For anyx∈MSIF¯∑i=1nAiωXβ, we can know ∑i=1nωiSXAix≥β.
minωi≤β≤1; thus, ∃Ai, s.t.SXAix=1; namely, xAi⊆X.
Then, according to property (7), we have MSIF¯∑i=1nAiωxAiβ⊆MSIF¯∑i=1nAiωXβ.
Therefore,xAi⊆MSFI¯∑i=1nAiωXβ; we can get ∑i=1nωiSMSIF¯∑i=1nAiωXβAix≥β, and x∈MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ.
Therefore,MSIF¯∑i=1nAiωXβ⊆MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ, from which one can get that MSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ.
Meanwhile, we can prove thatMSIF¯∑i=1nAiωMSIF¯∑i=1nAiωXββ=MSIF¯∑i=1nAiωXβ.
Property (1) shows that in multisource intuitionistic fuzzy information system, the lower approximation operator, and the upper approximation operator of weighted generalized multigranularity intuitionistic fuzzy rough sets satisfy duality. Property (2) illustrates the inclusion relationship between the lower approximation, the upper approximation, and the target concept. Properties (3) and (4) show the approximation of two special sets. The lower and upper approximation of the empty set ∅ and the universe U are themselves. Properties 5−8 state the monotonicity of MSIF¯∑i=1nAiω and MSIF¯∑i=1nAiω. Moreover, property (9) expresses the idempotency between the lower approximation operator and the upper approximation operator.


## 3.3. Weighted Optimistic Multigranularity Intuitionistic Fuzzy Rough Set Model forMSIFIS

Definition 21.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, ∀X⊆U, the lower approximation and upper approximation of X weighted optimistic multigranularity intuitionistic fuzzy rough set are defined as follows:(23)OMSIF¯∑i=1nAiωX=x∈U|∑i=1nωiSXAix>0;OMSIF¯∑i=1nAiωX=∼OMSIF¯∑i=1nAiω∼X=x∈U|∑i=1nωi1−SXcAix≥1.
IfOMSIF¯∑i=1nAiωX=OMSIF¯∑i=1nAiωX, then the target set X is optimistic and definable for ∑i=1nAi; otherwise, X is optimistic and rough for ∑i=1nAi. OMSIF¯∑i=1nAiωX,OMSIF¯∑i=1nAiωX is called the weighted optimistic multigranularity intuitionistic fuzzy rough set model for MSIFIS.
In the weighted optimistic multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions ofX can be defined as follows:(24)POS∑i=1nAiωXO=OMSIF¯∑i=1nAiωX;NEG∑i=1nAiωXO=U−OMSIF¯∑i=1nAiωX;BND∑i=1nAiωXO=OMSIF¯∑i=1nAiωX−OMSIF¯∑i=1nAiωX.
The basic properties of the weighted optimistic multigranularity intuitionistic fuzzy rough set forMSIFIS are given by the following theorem.Theorem 2.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, ∀X,Y⊆U, the following conclusions hold:(1)
OMSIF¯∑i=1nAiω∼X=∼OMSIF¯∑i=1nAiωX,OMSIF¯∑i=1nAiω∼X=∼OMSIF¯∑i=1nAiωX.(2)
OMSIF¯∑i=1nAiωX⊆X⊆OMSIF¯∑i=1nAiωX.(3)
OMSIF¯∑i=1nAiω∅=OMSIF¯∑i=1nAiω∅=∅.(4)
OMSIF¯∑i=1nAiωU=OMSIF¯∑i=1nAiωU=U.(5)
OMSF¯∑i=1nAiωX∪Y⊇OMSIF¯∑i=1nAiωX∪OMSIF¯∑i=1nAiωY,OMSIF¯∑i=1nAiωX∪Y⊇OMSIF¯∑i=1nAiωX∪OMSIF¯∑i=1nAiωY.(6)
OMSIF¯∑i=1nAiωX∩Y⊆OMSIF¯∑i=1nAiωX∩OMSIF¯∑i=1nAiωY,OMSIF¯∑i=1nAiωX∩Y⊆OMSIF¯∑i=1nAiωX∪OMSIF¯∑i=1nAiωY.(7)
X⊆Y⇒OMSIF¯∑i=1nAiωX⊆OMSIF¯∑i=1nAiωY,X⊆Y⇒OMSIF¯∑i=1nAiωX⊆OMSIF¯∑i=1nAiωY.(8)
OMSIF¯∑i=1nAiωOMSIF¯∑i=1nAiωX=OMSIF¯∑i=1nAiωX,OMSIF¯∑i=1nAiωOMSIF¯∑i=1nAiωX=OMSIF¯∑i=1nAiωX.Proof.
It is similar to that of Theorem1.


## 3.4. Weighted Pessimistic Multigranularity Intuitionistic Fuzzy Rough Set Model forMSIFIS

Definition 22.
LetMSIFIS=IFISi|IFISi=U,Ai,Vi,fii=1,2,…,n be a multisource intuitionistic fuzzy information system; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X⊆U, the lower approximation and upper approximation of X weighted pessimistic multigranularity intuitionistic fuzzy rough set are defined as follows:(25)PMSIF¯∑i=1nAiωX=x∈U∣∑i=1nωiSXAix≥1;PMSIF¯∑i=1nAiωX=∼PMSIF¯∑i=1nAiω∼X=x∈U∣∑i=1nωi1−SXcAix>0.
IfPMSIF¯∑i=1nAiωX=PMSIF¯∑i=1nAiωX, then the target set X is pessimistic and definable for ∑i=1nAi; otherwise, X is pessimistic and rough for ∑i=1nAi. PMSIF¯∑i=1nAiωX,PMSIF¯∑i=1nAiωX is called the weighted pessimistic multigranularity intuitionistic fuzzy rough set model for MSIFIS.
In the weighted pessimistic multigranularity intuitionistic fuzzy rough set model, the positive, negative, and boundary regions ofX can be defined as follows:(26)POS∑i=1nAiωXP=PMSIF¯∑i=1nAiωX;NEG∑i=1nAiωXP=U−PMSIF¯∑i=1nAiωX;BND∑i=1nAiωXP=PMSIF¯∑i=1nAiωX−PMSIF¯∑i=1nAiωX.The basic properties of the weighted pessimistic multigranularity intuitionistic fuzzy rough set for MSIFIS are given by the following theorem.Theorem 3.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X,Y⊆U, the following conclusions hold:(1)
PMSIF¯∑i=1nAiω∼X=∼PMSIF¯∑i=1nAiωX,PMSIF¯∑i=1nAiω∼X=∼PMSIF¯∑i=1nAiωX.(2)
PMSIF¯∑i=1nAiωX⊆X⊆PMSIF¯∑i=1nAiωX.(3)
PMSIF¯∑i=1nAiω∅=PMSIF¯∑i=1nAiω∅=∅.(4)
PMSIF¯∑i=1nAiωU=PMSIF¯∑i=1nAiωU=U.(5)
PMSIF¯∑i=1nAiωX∪Y⊇PMSIF¯∑i=1nAiωX∪PMSIF¯∑i=1nAiωY,PMSIF¯∑i=1nAiωX∪Y=PMSIF¯∑i=1nAiωX∪PMSIF¯∑i=1nAiωY.(6)
PMSIF¯∑i=1nAiωX∩Y=PMSIF¯∑i=1nAiωX∩PMSIF¯∑i=1nAiωY,PMSIF¯∑i=1nAiωX∩Y⊆PMSIF¯∑i=1nAiωX∪PMSIF¯∑i=1nAiωY.(7)
X⊆Y⇒PMSIF¯∑i=1nAiωX⊆PMSIF¯∑i=1nAiωY,X⊆Y⇒PMSIF¯∑i=1nAiωX⊆PMSIF¯∑i=1nAiωY.(8)
PMSIF¯∑i=1nAiωPMSIF¯∑i=1nAiωX=PMSIF¯∑i=1nAiωX,PMSIF¯∑i=1nAiωPMSIF¯∑i=1nAiωX=PMSIF¯∑i=1nAiωX.Proof.
It is similar to that of Theorem1.


## 3.5. The Relationship between the Three Models ofMSIFIS

In this subsection, we investigate the relationship between the three models ofMSIFIS.Theorem 4.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X⊆U, the following equations can be obtained:(1)
Ifβ=1, then(27)MSIF¯∑i=1nAiωXβ=PMSIF¯∑i=1nAiωX,MISF¯∑i=1nAiωXβ=PMSIF¯∑i=1nAiωX.(2)
Ifβ=minωii=1,2,…,n, then(28)MSIF¯∑i=1nAiωXβ=OMSIF¯∑i=1nAiωX,MSIF¯∑i=1nAiωXβ=OMSIF¯∑i=1nAiωX.Proof.
It is easily obtained from Definition15, Definition 20, and Definition 21.
From the above theorems, ifβ=1, weighted generalized multigranularity intuitionistic fuzzy rough set degenerates into weighted pessimistic multigranularity intuitionistic fuzzy rough set. If β=minωAi, weighted generalized multigranularity intuitionistic fuzzy rough set degenerates into weighted optimistic multigranularity intuitionistic fuzzy rough set. Hence, weighted generalized multigranularity intuitionistic fuzzy rough set is a generalization of weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set. On the other hand, weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set are the special cases of weighted generalized multigranularity intuitionistic fuzzy rough set.Theorem 5.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1,0≤ωi≤1, ∀X⊆U, the following properties are established:(1)
PMSIF¯∑i=1nAiωX⊆MSIF¯∑i=1nAiωXβ⊆OMSIF¯∑i=1nAiωX.(2)
OMSIF¯∑i=1nAiωX⊆MSIF¯∑i=1nAiωXβ⊆PMSIF¯∑i=1nAiωX.Proof.
From Definition20–Definition 22, the theorem clearly holds.
The theorem shows that the lower and upper approximation of the weighted generalized multigranularity intuitionistic fuzzy rough set are between weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set.


## 4. Uncertainty Measurement ofMSIFIS

Like Pawlak rough sets, the uncertainty of knowledge is caused by the boundary region. The larger the boundary area, the lower the accuracy and the higher roughness. For the weighted multigranularity intuitionistic fuzzy rough set model in the MSIFIS, this section gives definition of rough accuracy, roughness, and attribute dependence.Definition 23.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, parameter minωi≤β≤1i=1,2,…,n denotes the information level with respect to ∑i=1nAi; ∀X⊆U, the approximation accuracy and roughness of the weighted generalized multigranularity intuitionistic fuzzy rough set model of the set X with respect to ∑i=1nAi are defined as follows:(29)αMSIFωXβ=|MSIF¯∑i=1nAiωXβ|MSIF¯∑i=1nAiωXβ;ρMSIFωXβ=1−αMSIFωXβ.
Similarly, the approximation accuracy and roughness of the weighted optimistic multigranularity intuitionistic fuzzy rough set model of the setX with respect to ∑i=1nAi are defined as follows:(30)αOMSIFωX=OMSIF¯∑i=1nAiωXOMSIF¯∑i=1nAiωX;ρOMSIFωX=1−αOMSIFωX.
The approximation accuracy and roughness of the weighted pessimistic multigranularity intuitionistic fuzzy rough set model of the setX with respect to ∑i=1nAi are defined as follows:(31)αPMSIFωX=PMSIF¯∑i=1nAiωXPMSIF¯∑i=1nAiωX;ρPMSIFωX=1−αPMSIFωX.Definition 24.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=10≤ωi≤1, parameter minωi≤β≤1i=1,2,…,n denotes the information level with respect to ∑i=1nAi; ∀X⊆U, the approximation accuracy and roughness of the weighted generalized multigranularity intuitionistic fuzzy rough set model of the set X with respect to ∑i=1nAi are defined as follows:(32)γMSFωXβ=MSIF¯∑i=1nAiωXβU=POS∑i=1nAiωXβU.
Similarly, the dependence of the weighted optimistic multigranularity intuitionistic fuzzy rough set model of the setX with respect to ∑i=1nAi is defined as follows:(33)γOMSFωX=OMSIF¯∑i=1nAiωXU=POS∑i=1nAiωXOU.
The dependence of the weighted pessimistic multigranularity intuitionistic fuzzy rough set model of the setX with respect to ∑i=1nAi is defined as follows:(34)γPMSFωX=PMSIF¯∑i=1nAiωXU=POS∑i=1nAiωXPU.Theorem 6.
LetMSIFIS=IFIS1,IFIS2,…,IFISn be a multisource intuitionistic fuzzy information system, where IFISi=U,Ai,Vi,IFii≤n; SXAix is the support characteristic function of x for X. If the weight corresponding to each granularity space is derived from each dominance relation as ω=ω1,ω2,…,ωn,∑i=1nωi=1, for any X⊆U, under the three rough set models, the relationships between the approximate accuracy, roughness, and dependence of the set X with respect to ∑i=1nAi are as follows:(1)
αPMSIFωX≤αMSIFωXβ≤αOMSIFωXβ.(2)
ρOMSIFωX≤ρMSIFωXβ≤ρPMSIFωXβ.(3)
γPMSIFωX≤γMSIFωXβ≤γOMSIFωX.Proof.
It is easy to obtain by Definition23 and Definition 24.
The theorem shows that the accuracy, roughness, and dependence of the weighted generalized multigranularity intuitionistic fuzzy rough set model are between weighted pessimistic multigranularity intuitionistic fuzzy rough set and weighted optimistic multigranularity intuitionistic fuzzy rough set model.Example 3.
In example 1, the quality ranking of the three information sources in the multisource intuitionistic fuzzy information system is calculated. Let us suppose that the granularity weights corresponding to the three information systems are assigned asω0.2,0.3,0.5; the distribution of granularity weights in specific applications can be given subjectively based on the experience of domain experts, with threshold β=0.6.
According to Definition2, we calculate the dominant classes of object under each source.
Dominant classes under source 1 are as follows:
RA1x1=x1,x5, RA1x2=x1,x2,x3,x5,x6,
RA1x3=x1,x3,x5,x6, RA1x4=x1,x2,x3,x4,x5,x6,
RA1x5=x5, RA1x6=x5,x6.
Dominant classes under source 2 are as follows:
RA2x1=x1,x2, RA2x2=x2,
RA2x3=x3, RA2x4=x4,
RA2x5=x2,x3,x4,x5, RA2x6=x2,x3,x4,x6.
Dominant classes under source 3 are as follows:
RA3x1=x1,x3, RA3x2=x2,
RA3x3=x3, RA3x4=x2,x3,x4,
RA3x5=x3,x5, RA3x6=x2,x3,x6.
Given a concept setX=x1,x2,x3,x5, the support characteristic function of X and support characteristic function of Xc under each source are computed.
SXA1x1=1, SXA2x1=1, SXA3x1=1,
SXA1x2=0, SXA2x2=1, SXA3x2=1,
SXA1x3=0, SXA2x3=1, SXA3x3=1,
SXA1x4=0, SXA2x4=0, SXA3x4=0,
SXA1x5=1, SXA2x5=0, SXA3x5=1,
SXA1x6=0, SXA2x6=0, SXA3x6=0.
Xc=x4,x6,
SXcA1x1=0, SXcA2x1=0, SXcA3x1=0,
SXcA1x2=0, SXcA2x2=0, SXcA3x2=0,
SXcA1x3=0, SXcA2x3=0, SXcA3x3=0,
SXcA1x4=0, SXcA2sx4=1, SXcA3x4=0,
SXcA1x5=0, SXcA2x5=0, SXcA3sx5=0,
SXcA1sx6=0, SXcA2x6=0, SXcA3x6=0.
According to Definition20, the lower approximation and upper approximation of the weighted generalized multigranularity intuitionistic fuzzy rough set X are obtained as follows:
MSIF¯∑i=13AiωXβ=x1,x2,x3,x5,
MSIF¯∑i=13AiωXβ=x1,x2,x3,x4,x5,x6.
The positive region, boundary region, and negative region of the weighted generalized multigranularity intuitionistic fuzzy rough setX are as follows:
POS∑i=13AiωXβ=x1,x2,x3,x5,
BND∑i=13AiωXβ=x4,x6,
NEG∑i=13AiωXβ=∅.
The approximation accuracy, roughness, and dependence of the weighted generalized multigranularity intuitionistic fuzzy rough setX are as follows:
αMSIFωXβ=4/6=2/3,ρMSIFωXβ=1−2/3=1/3,γMSIFωXβ=4/6=2/3.
According to Definition21, the lower approximation and upper approximation of the weighted optimistic multigranularity intuitionistic fuzzy rough set X are obtained as follows:
OMSIF¯∑i=1nAiωX=x1,x2,x3,x5,OMSIF¯∑i=1nAiωX=x1,x2,x3,x5,x6.
The positive region, boundary region, and negative region of the weighted optimistic multigranularity intuitionistic fuzzy rough setX are as follows:
POS∑i=1nAiωXO=x1,x2,x3,x5,BND∑i=1nAiωXO=x6,NEG∑i=1nAiωXO=x4,x6.
The approximation accuracy, roughness, and dependence of the weighted optimistic multigranularity intuitionistic fuzzy rough setX are as follows:
αOMSIFωX=4/5,ρOMSIFωX=1−4/5=1/5,γOMSIFωX=4/6=2/3,
According to Definition22, the lower approximation and upper approximation of the weighted pessimistic multigranularity intuitionistic fuzzy rough set X are obtained as follows:
PMSIF¯∑i=13AiωX=x1,PMSIF¯∑i=13AiωX=x1,x2,x3,x4,x5,x6.
The positive region, boundary region, and negative region of the weighted pessimistic multigranularity intuitionistic fuzzy rough setX are as follows:
POS∑i=13AiωXP=x1,
BND∑i=13AiωXP=x1,x2,x3,x4,x5,x6,
NEG∑i=13AiωXP=∅.
The approximation accuracy, roughness, and dependence of the weighted pessimistic multigranularity intuitionistic fuzzy rough setX are as follows:
αPMSIFωX=1/6,
ρPMSIFωX=1−1/6=5/6,
γPMSIFωX=1/6.
Hence,
PMSIF¯∑i=13AiωX⊆MSIF¯∑i=13AiωXβ⊆OMSIF¯∑i=13AiωX,
OMSIF¯∑i=13AiωX⊆MSIF¯∑i=13AiωXβ⊆PMSIF¯∑i=13AiωX.
αPMSIFωX≤αMSIFωXβ≤αOMSIFωXβ,
ρOMSIFωX≤ρMSIFωXβ≤ρPMSIFωXβ,
γPMSIFωsX≤γMSIFωXβ≤γOMSIFωX.


## 5. Experimental Evaluation and Analysis

In this paper, three weighted multigranulation intuitionistic fuzzy rough set models forMSIFIS are studied, and the uncertainty measurement methods of different models are discussed. In this section, we use an experiment to show the effectiveness of the models and methods proposed in this paper. We propose Algorithm 2 to calculate the uncertainty measure of the weighted generalized multigranulation intuitionistic fuzzy rough set in MSIFIS. Similarly, by changing the threshold β, a calculated weighted optimistic and pessimistic uncertainty measurement algorithm can be obtained.The time complexity analysis of Algorithm 2 is as folloxws. From Steps2−20, we calculate the dominant class of each object under each information source, and its time complexity is OU2∗A∗q (q is the number of sources). From Steps 21−35, we calculate the support feature function of X and XC under each information source, and the time complexity is OU∗q. From Steps 36−45, we compute the upper and lower approximations of X in MSIFIS, and the time complexity is OU.As we all know, we cannot getMSIFIS directly from UCI (http://archive.ics.uci.edu/ml/datasets.html), so we need to process the data. First, we need to download the “winequality-red” and “winequality-white” data sets from UCI and divide them by the maximum number of each column in the data table to make them fuzzy data tables. Then, we use the MATLAB software to randomly generate two fuzzy data sets. Finally, the method given in literature [23] is used to make them intuitionistic fuzzy data sets.The details of the four data tables are shown in Table4. We use the above four data sets as the original intuitionistic fuzzy information system. Finally, we randomly select 40% of the original data to add white noise and then randomly select 20% of the remaining data to add random noise, and the rest of the data remains unchanged to generate four MSIFISs with ten sources. The entire experiment was run on a private computer. The specific operating environment, including hardware and software, is shown in Table 5.Table 4
Specific information about the operating environment.
NameModelParameterCPUIntel® core™ i3-2370M2.40 GHzPlatformMATLABR2016bSystemWindows 764 bitsMemoryDDR34 GB; 1600 MHzHard diskMQ01ABD050500 GBTable 5
Uncertainty measurement on “own-data 1.”
αβγGOPGOPGOP10.842 110.082 40.157 900.917 60.752 90.80.082 420.722 610.047 10.277 400.952 90.658 80.80.047 130.727 810.070 60.272 200.929 40.676 50.80.070 640.784 010.111 80.21600.888 20.747 10.80.111 850.726 110.052 90.273 900.947 10.670 60.80.052 960.516 10.992 70.052 90.483 90.007 30.947 10.470 60.80.052 970.783 80.797 30.052 90.216 20.202 70.947 10.682 40.694 10.052 980.682 40.802 70.052 90.317 60.197 30.947 10.594 10.694 10.052 990.478 00.992 70.023 50.5220.007 30.976 50.447 10.80.023 5100.337 60.992 7≤ 0.0010.662 40.007 310.311 80.80We add white noise as follows: theq real numbers with a normal distribution n1,n2,…,nq were first generated by MATLAB.(35)IFISiμx,νx,a=IFISiμx+ni,νx+ni,a,0≤μx+ni+νx+ni≤1,IFISiμx,νx,a,else.We add random noise as follows:(36)IFISiμx,νx,a=IFISiμx+ri,νx+ri,a,0≤μx+ri+νx+ri≤1,IFISiμx,νx,a,else.We conducted ten experiments on each data set, assuming that the granularity weights corresponding to the three information systems are assigned asω=0.08,0.04,0.1,0.03,0.07,0.02,0.06,0.2,0.3,0.1, with threshold β=0.6, and the concept set X is randomly generated from the target set. The uncertainty measurement of the three rough set models corresponding to each data set is shown in Tables 6–8, and the results are illustrated inFigure 2–5.Table 6
Uncertainty measurement on “winequality-red.”
αβγGOPGOPGOP10.746 10.985 30.531 60.253 90.014 70.468 40.657 90.794 90.500 320.742 20.936 90.636 70.257 80.063 10.363 30.653 50.770 50.585 430.745 20.970 00.638 30.254 80.030.361 70.656 70.789 90.586 040.730 70.969 40.638 30.269 30.030 60.361 70.649 80.793 00.586 050.735 60.953 50.637 20.264 40.046 50.362 80.656 00.781 70.585 460.731 10.962 50.502 30.268 90.037 50.497 70.652 90.785 50.471 570.730 90.940 90.637 20.269 10.059 10.362 80.652 30.776 70.585 480.743 50.996 10.629 80.256 50.003 90.370 20.657 90.803 00.579 790.728 40.939 30.585 40.271 60.060 70.414 60.647 30.774 20.585 4100.739 80.931 10.586 60.260 20.068 90.413 40.656 00.769 20.586 6Table 7
Uncertainty measurement on “own-data 2.”
αβγGOPGOPGOP10.251 90.804 70.000 30.748 10.195 30.999 70.2470.804 70.000 320.277 00.994 20.000 70.7230.005 80.999 30.273 70.8040.000 730.344 20.996 70.000 70.655 80.003 30.999 30.337 70.805 30.000 740.297 70.992 20.001 30.702 30.007 80.998 70.2930.804 70.001 350.322 50.995 90.002 70.677 50.004 10.997 30.316 70.8050.002 760.311 90.990 20.002 70.688 10.009 80.997 30.307 70.804 30.002 770.306 50.993 40.0010.693 50.006 60.9990.302 30.804 30.00180.277 50.994 2≤ 0.0010.722 50.005 810.272 30.805≤ 0.00190.334 00.995 50.0020.6660.004 50.9980.3280.805 30.002100.291 30.993 80.000 70.708 70.006 20.999 30.2870.804 30.000 7Table 8
Uncertainty measurement on “winequality-white.”
αβγGOPGOPGOP10.638 30.997 90.035 10.361 70.002 10.964 90.603 10.793 80.035 120.647 60.999 50.031 20.352 40.000 50.968 80.611 50.793 80.031 230.662 10.999 70.040 80.337 90.000 30.959 20.626 20.793 80.040 840.652 20.999 50.035 70.347 80.000 50.964 30.614 90.793 80.035 750.657 810.034 70.342 200.965 30.618 20.793 80.034 760.657 20.998 70.039 20.342 80.001 30.960 80.617 60.793 60.039 270.624 10.999 20.034 70.375 90.000 80.965 30.590 90.793 80.034 780.627 20.999 20.033 50.372 80.000 80.966 50.592 90.793 80.033 590.629 50.999 20.032 30.370 50.000 80.967 70.596 00.793 80.032 3100.635 90.999 50.029 40.364 10.000 50.970 60.600 90.793 80.029 4Figure 2
Uncertainty measurement of weighted multigranulation model for “own-data 1.”Figure 3
Uncertainty measurement of weighted multigranulation model for “winequality-red.”Figure 4
Uncertainty measurement of weighted multigranulation model for “own-data 2.”Figure 5
Uncertainty measurement of weighted multigranulation model for “winequality-white.”From Figures2–5, we can find that the weighted pessimistic approximation accuracy is less than or equal to the weighted generalized approximation accuracy, which is also less than or equal to the weighted optimistic approximation accuracy; the weighted optimistic approximation roughness is less than or equal to the weighted generalized approximation roughness, which is also less than or equal to the weighted pessimistic approximation roughness. Similarly, the weighted pessimistic approximation dependence is less than or equal to the weighted generalized approximation dependence, which is also less than or equal to the weighted optimistic approximation dependence.Algorithm 2: Uncertainty measurement of weighted generalized multigranulation intuitionistic fuzzy rough sets for MSIFIS.
Input:ω, β, MSIFIS=IFIS1,IFIS2,…,IFISq, IFIS=U,Ai,Vi,IFi, X⊆UOutput:αMSIFωXβ, ρMSIFωXβ and γMSIFωXβ(1)
begin(2)
forl = 1:q do(3)
fori = 1:n do(4)
xiAl=∅;(5)
forj = 1:n do(6)
flag1=1;(7)
fork = 1:m do(8)
ifμAlxi,k≤μAlxj,k∧νAlxi,k≥∧νAlxj,k then(9)
flag1=1;(10)
else(11)
flag1=0;(12)
break;(13)
end(14)
end(15)
fflag1==1;(16)
xiAl=xiAl∪xj;(17)
end(18)
end(19)
end(20)
end(21)
fori = 1:n do(22)
SXAli=0,SXCAli=0(23)
forl = 1:q do(24)
ifxiAl⊆X then(25)
SXAli=SXAli+1;(26)
else(27)
SXAli=SXAli;(28)
end(29)
ifxiAl⊆XC then(30)
SXCAli=SXCAli+1;(31)
else(32)
SXCAli=SXCAli;(33)
end(34)
end(35)
end(36)
MSIF¯∑i=1qAlXβ←∅;MSIF¯∑i=1qAlXβ←∅;(37)
fori=1:n do(38)
if∑ωiSXAix≥β then(39)
MSIF¯∑i=1qAlXβ=MSIF¯∑i=1qAlXβ∪xi;(40)
end(41)
if∑ωi1−SXAix>1−β then(42)
MSIF¯∑i=1qAlXβ=MSIF¯∑i=1qAlXβ∪xi;(43)
end(44)
endreturn:αMSIFωXβ, ρMSIFωXβ and γMSIFωXβ(45)
endMSIFIS′s optimistic multigranularity intuitionistic fuzzy rough set model requirements are too loose in selecting objects, but the pessimistic multigranularity intuitionistic fuzzy rough set model requirements are too strict in selecting objects. Therefore, in practical applications, the object selection can be completed by the weighted generalized multigranular intuitionistic fuzzy rough set model.From Figures2–5, in the same data set, MSIFIS has different calculation results in different multigranularity intuitionistic fuzzy rough set models. Therefore, in practical applications, we can complete the object selection by changing the threshold β.


## 6. Conclusions

In this paper, in order to solve the problem of knowledge discovery in theMSIFIS, the weighted multigranulation intuitionistic fuzzy rough set models, combined with the idea of multigranulation, are studied. We also studied the relationship between them. In order to further study the multigranulation intuitionistic fuzzy rough set model in the MSIFIS, the uncertainty measurement methods of different models are discussed. Finally, the effectiveness of the proposed models and methods is verified through an example. In the future, we need to continue to study the granularity weight distribution of multisource intuitionistic fuzzy information system and its application to decision-making.



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