Intuitionistic fuzzy dominance–based rough set approach ... · lation, intuitionistic fuzzy dominance–based rough set, ap-proximate distribution reducts I. INTRODUCTION Rough

Intuitionistic fuzzy dominance–based rough setapproach: model and attribute reductions

Yanqin ZhangSchool of Economics, Xuzhou Institute of Technology, Xuzhou, 221000, P.R. China

Email: [email protected] Yang

School of Computer Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu,212003, P.R. China

School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing, Jiangsu,210094, P.R. China

Email: [email protected]

Abstract— The dominance–based rough set approach playsan important role in the development of the rough settheory. It can be used to express the inconsistencies comingfrom consideration of the preference–ordered domains ofthe attributes. The purpose of this paper is to furthergeneralize the dominance–based rough set model to fuzzyenvironment. The constructive approach is used to definethe intuitionistic fuzzy dominance–based lower and upperapproximations respectively. Basic properties of the intu-itionistic fuzzy dominance–based rough approximations arethen examined. By introducing the concept of approximatedistribution reducts into intuitionistic fuzzy dominance–based rough approximations, four different forms of reductsare defined. The judgment theorems and discernibilitymatrixes associated with these reducts are also obtained.Such results are all intuitionistic fuzzy generalizations ofthe classical dominance–based rough set approach. Somenumerical examples are employed to substantiate the con-ceptual arguments.

Index Terms— dominance–based rough set, dominance–based fuzzy rough set, intuitionistic fuzzy dominance re-lation, intuitionistic fuzzy dominance–based rough set, ap-proximate distribution reducts

I. INTRODUCTION

Rough set theory [44]–[47], proposed by Pawlak, is anew mathematical tool which can be used to deal withvague and uncertain information. The lower and upperapproximate operators are key notions in rough set theory,they were constructed on the basis of an indiscernibilityrelation (equivalence relation, i.e. reflexive, symmetricand transitive). By using such two approximations, knowl-edge hidden in the information tables may be unravelledand expressed in the form of decision rules.

It is well known that the indiscernibility relation istoo restrictive for classification analysis in practical ap-plications. Therefore, many authors have generalized thenotions of rough approximations by using some more

This work is supported by the Natural Science Foundation of China(Nos. 61100116, 61103133), Natural Science Foundation of JiangsuProvince of China (No. BK2011492), Natural Science Foundation ofJiangsu Higher Education Institutions of China (No. 11KJB520004),Postdoctoral Science Foundation of China (No. 20100481149), Postdoc-toral Science Foundation of Jiangsu Province of China (No. 1101137C).

general binary relations, e.g., tolerance relation [35], [37],similarity relation [52]–[54], characteristic relation [27],[28], [39], etc. These extensions of the rough approxima-tions may be used on reasoning and acquisition of knowl-edge in incomplete systems [38], [48], [58], [61], [63],[66]–[68], continues–valued systems [29], [56] and somemore complex forms of information systems. Moreover, itshould be noticed that the generalization of rough approx-imations to fuzzy environments also plays an importantrole in the development of rough set theory. For example,in Ref. [17], the model of rough fuzzy set was proposedby using the indiscernibility relation to approximate afuzzy concept. Alternatively, the fuzzy rough model isthe approximation of a crisp set or a fuzzy set in a fuzzyapproximation space. In Ref. [55], Sun et al. presentedthe interval–valued fuzzy rough set by combining theinterval–valued fuzzy set and rough set. By employingan approximation space constituted from an intuitionisticfuzzy triangular norm, an intuitionistic fuzzy implicator,and an intuitionistic fuzzy 𝑇–equivalence relation, Cor-nelis et al. [12] defined the concept of intuitionistic fuzzyrough sets in which the lower and upper approximationsare both intuitionistic fuzzy sets on the universe of dis-course. Zhou et al. proposed the intuitionistic fuzzy roughapproximation from the viewpoint of constructive andaxiomatic approaches respectively in Ref. [75]. Bhatt [4]presented the fuzzy–rough sets on compact computationaldomain. Zhao et al. [74] investigated the fuzzy variableprecision rough sets by combining fuzzy rough set andvariable precision rough set with the goal of makingfuzzy rough set a special case. Hu et al. [31] proposedthe Gaussian Kernel based fuzzy rough set, it uses theGaussian Kernel to compute fuzzy 𝑇–equivalence relationfor objective approximation. The same authors also pro-posed the fuzzy preference–based rough sets in Ref. [32].Ouyang et al. [43] presented a fuzzy rough model, whichis based on the fuzzy tolerance relation. More detailsabout recent advancements of fuzzy rough set can befound in the literatures [9], [10], [16], [40], [41], [57],[62], [69], [70], [72].

On the other hand, though the rough set has been

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demonstrated to be useful in the fields of knowledgediscovery, decision analysis, pattern recognition and soon, it is not able, however, to discover inconsistenciescoming from consideration of criteria, that is, attributeswith preference–ordered domains, such as product quality,market share and debt ratio. To solve this problem, Grecoet al. have proposed an extension of Pawlak’s rough setapproach, which is called the Dominance–based RoughSet Approach (DRSA) [5]–[8], [13]–[15], [18]–[26], [30],[34], [71]. This innovation is mainly based on substitutionof the indiscernibility relation by a dominance relation.Presently, work on dominance–based rough set modelalso progressing rapidly. For example, by consideringtwo different types of semantic explanations of unknownvalues, Shao et al. and Yang et al. generalized the DRSAto incomplete environments in Ref. [50] and Ref. [65]respectively. Wei et al. presented the concept of valueddominance–based rough approximations in Ref. [60].With introduction of the concept of variable precisionrough set [76] into DRSA, Błszczynki et al. proposedthe variable consistency dominance–based rough set ap-proach [6], [8], Inuiguchi et al. proposed the variableprecision dominance–based rough set [33]. Kotłowski etal. [34] introduced a new approximation of DRSA whichis based on the probabilistic model for the ordinal clas-sification problems. Greco et al. generalized the DRSAto fuzzy environment and then presented the model ofdominance–based rough fuzzy set in Ref. [20]. By using afuzzy dominance relation, the same authors also presenteddominance–based fuzzy rough set [26] in their literatures.

As a generalization of the Zadeh fuzzy set, the notionof intuitionistic fuzzy set was suggested for the firsttime by Atanassov [1], [2]. An intuitionistic fuzzy setallocates to each element both a degree of membershipand one of non–membership, and it was applied to thefields of approximate inference, signal transmission andcontroller, etc. In this paper, the intuitionistic fuzzy setwill be combined with the DRSA and then the model ofIntuitionistic Fuzzy Dominance–based Rough Set(IFDRS)is presented. The IFDRS is a new generalization of theclassical DRSA because we use an intuitionistic fuzzydominance relation instead of the crisp or fuzzy domi-nance relation to approximate the upward and downwardunions of the decision classes. It should be noticed herethat we use the constructive approach to define the IFDRSin this paper.

In traditional DRSA, the dominance relation can onlybe used to judge whether an object is dominating anotherone. Furthermore, to express the credibility that an objectis dominating another one, the fuzzy dominance relationis then presented (eg. Ref. [26] and Ref. [60]). In fuzzydominance relation, if an object 𝑥 dominates anotherobject 𝑦 with a credibility 𝛼, then it naturally followsthat 𝑥 does not dominate 𝑦 to the extent 1 − 𝛼. Tofurther generalize such idea, it is naturally to introducethe intuitionistic fuzzy approach into DRSA, i.e. IFDRS.In our IFDRS, the intuitionistic fuzzy dominance relationcan express not only the credibility that 𝑥 dominates 𝑦,

but also the non–credibility that 𝑥 dominates 𝑦.Once a new rough set model is presented, the immedi-

ate problem is attribute reduction. It involves the searchfor particular subsets of attributes, which provide the sameinformation for some purpose as the full set of availableattributes. Such subsets are called reducts. In traditionalrough set theory, Pawlak proposed the positive–regionbased reduct, which can be used to preserve the unionof all lower approximations. Following Pawlak’s work,Kryszkiewicz [36] investigated and compared five notionsof knowledge reductions in inconsistent systems, Zhanget al. [73] proposed the concepts of distribution reduct andmaximal distribution reduct. Moreover, Wang et al. [59]presented a systematic approach to knowledge reductionwhich is based on the general binary relation and thecorresponding rough approximation, Chen et al. [11] in-vestigated the problem of knowledge reduction in decisionsystem with covering based rough approximation, Yang etal. [64] constructed a new reduction theory by redefiningthe approximation space and the reducts of coveringgeneralized rough set. By using the variable precisionrough set model [76], Beynon [3] proposed the conceptof 𝛽–reduct, Mi et al. proposed the lower and upperapproximate distribution reducts in Ref. [42].

In this paper, we will further introduce Mi’s approx-imate distribution reducts into our IFDRS. Four notionsof approximate distribution reducts are then presented be-cause there are two pairs of approximations in DRSA. Thejudgment theorems and discernibility matrices associatedwith these reducts are also established, from which weobtain the practical approaches to compute approximatedistribution reducts in IFDRS.

To facilitate our discussion, we first present basicnotions of classical DRSA and dominance–based fuzzyrough set in Section 2. The constructive approach todefine IFDRS is presented in Section 3. We also employan illustrative example to show how the IFDRS can beused in decision system with probabilistic interpretation.In Section 4, the approximate distribution reducts in termsof our IFDRS are investigated. Results are summarized inSection 5.

II. DOMINANCE–BASED ROUGH SET APPROACH

A. Greco’s DRSA

A decision system is a pair I =< 𝑈,𝐴𝑇 ∪ {𝑑} >,where

∙ 𝑈 is a non–empty finite set of objects, it is calledthe universe;

∙ 𝐴𝑇 is a non–empty finite set of conditional at-tributes;

∙ 𝑑 is the decision attribute where 𝐴𝑇 ∩ {𝑑} = ∅.∀𝑎 ∈ 𝐴𝑇 , 𝑉𝑎 is used to represent the domain of

attribute 𝑎 and then 𝑉 = 𝑉𝐴𝑇 =∪

𝑎∈𝐴𝑇 𝑉𝑎 is the domainof all attributes. Moreover, for each 𝑥 ∈ 𝑈 , let us denoteby 𝑎(𝑥) the value that 𝑥 holds on 𝑎 (𝑎 ∈ 𝐴𝑇 ).

By considering the preference–ordered domains of at-tributes (criteria), Greco et al. have proposed an extension

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of the classical rough set that is able to deal with incon-sistencies typical to exemplary decisions in Multi–CriteriaDecision Making (MCDM) problems, which is calledthe Dominance–based Rough Set Approach (DRSA). letર𝑎 be a weak preference relation on 𝑈 (often calledoutranking) representing a preference on the set of objectswith respect to criterion 𝑎 (𝑎 ∈ 𝐴𝑇 ); 𝑥 ર𝑎 𝑦 means “𝑥 isat least as good as 𝑦 with respect to criterion 𝑎”. We saythat 𝑥 dominates 𝑦 with respect to 𝐴 ⊆ 𝐴𝑇 , iff 𝑥 ર𝑎 𝑦for each 𝑎 ∈ 𝐴.

By the above discussion, we can define the followingtwo sets for each object 𝑥 in I such that:

∙ the set of objects dominate 𝑥, i.e. [𝑥]≥𝐴 = {𝑦 ∈ 𝑈 :∀𝑎 ∈ 𝐴, 𝑦 ર𝑎 𝑥};

∙ the set of objects dominated by 𝑥, i.e. [𝑥]≤𝐴 = {𝑦 ∈𝑈 : ∀𝑎 ∈ 𝐴, 𝑥 ર𝑎 𝑦}.

In the traditional DRSA, we assume here that thedecision attribute 𝑑 determines a partition of 𝑈 into afinite number of classes; let CL = {𝐶𝐿𝑛, 𝑛 ∈ 𝑁}, 𝑁 ={1, 2, ⋅ ⋅ ⋅ ,𝑚}, be a set of these classes that are ordered.Different from Pawlak’s rough approximation, in DRSA,the sets to be approximated are an upward union and adownward union of decision classes, which are definedrespectively as 𝐶𝐿≥

𝑛 =∪

𝑛′≥𝑛

𝐶𝐿𝑛′ , 𝐶𝐿≤𝑛 =

∪𝑛′≤𝑛

𝐶𝐿𝑛′ ,

𝑛, 𝑛′ ∈ 𝑁 .

In Greco’s DRSA, the A-lower approximation and A-upper approximation of 𝐶𝐿≥

𝑛 are:

𝐴(𝐶𝐿≥𝑛 ) = {𝑥 ∈ 𝑈 : [𝑥]≥𝐴 ⊆ 𝐶𝐿≥

𝑛 },𝐴(𝐶𝐿≥

𝑛 ) = {𝑥 ∈ 𝑈 : [𝑥]≤𝐴 ∩ 𝐶𝐿≥𝑛 ∕= ∅}.

the A-lower approximation and A-upper approximationof 𝐶𝐿≤

𝑛 are:

𝐴(𝐶𝐿≤𝑛 ) = {𝑥 ∈ 𝑈 : [𝑥]≤𝐴 ⊆ 𝐶𝐿≤

𝑛 },𝐴(𝐶𝐿≤

𝑛 ) = {𝑥 ∈ 𝑈 : [𝑥]≥𝐴 ∩ 𝐶𝐿≤𝑛 ∕= ∅};

the A-boundaries of 𝐶𝐿≥𝑛 and 𝐶𝐿≤

𝑛 are:

𝐵𝑁𝐴(𝐶𝐿≥𝑛 ) = 𝐴(𝐶𝐿≥

𝑛 )−𝐴(𝐶𝐿≥𝑛 ),

𝐵𝑁𝐴(𝐶𝐿≤𝑛 ) = 𝐴(𝐶𝐿≤

𝑛 )−𝐴(𝐶𝐿≤𝑛 ).

B. Dominance–based fuzzy rough set

Dominance–based fuzzy rough set is a fuzzy general-ization of DRSA. In dominance–based fuzzy rough setmodel, the dominance relation is replaced by a fuzzydominance relation.

Definition 1: Let 𝑅𝑎 be a fuzzy dominance relation on𝑈 with respect to attribute 𝑎, i.e. 𝑅𝑎 : 𝑈 × 𝑈 → [0, 1],∀𝑥, 𝑦 ∈ 𝑈 , 𝑅𝑎(𝑥, 𝑦) represents the credibility of theproposition “𝑥 is at least as good as 𝑦 with respect toattribute 𝑎”. A fuzzy dominance relation on 𝑈 (denotation𝑅𝐴(𝑥, 𝑦)) can be defined for each 𝐴 ⊆ 𝐴𝑇 as:

𝑅𝐴(𝑥, 𝑦) = ∧{𝑅𝑎(𝑥, 𝑦) : 𝑎 ∈ 𝐴}.Definition 2: Let I be a decision system in which

𝐴 ⊆ 𝐴𝑇 , ∀𝑛 ∈ 𝑁 , the 𝐴–lower approximationand 𝐴–upper approximation of 𝐶𝐿≥

𝑛 with respect to

fuzzy dominance relation are denoted by 𝐴𝑅(𝐶𝐿≥𝑛 ) and

𝐴𝑅(𝐶𝐿≥𝑛 )respectively, whose memberships for each 𝑥 ∈

𝑈 , are defined as:

𝜇𝐴𝑅(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝜇𝐶𝐿

≥𝑛(𝑦) ∨ (1−𝑅𝐴(𝑦, 𝑥))

)𝜇𝐴𝑅(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝜇𝐶𝐿

≥𝑛(𝑦) ∧𝑅𝐴(𝑥, 𝑦)

)∀𝑛 ∈ 𝑁 , the 𝐴–lower approximation and 𝐴–upperapproximation of 𝐶𝐿≤

𝑛 with respect to fuzzy dominancerelation are denoted by 𝐴𝑅(𝐶𝐿≤

𝑛 ) and 𝐴𝑅(𝐶𝐿≤𝑛 ) respec-

tively, whose memberships for each 𝑥 ∈ 𝑈 , are definedas:

𝜇𝐴𝑅(𝐶𝐿

≤𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝜇𝐶𝐿

≤𝑛(𝑦) ∨ (1−𝑅𝐴(𝑥, 𝑦))

)𝜇𝐴𝑅(𝐶𝐿

≤𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝜇𝐶𝐿

≤𝑛(𝑦) ∧𝑅𝐴(𝑦, 𝑥)

)More details about the dominance–based fuzzy rough

set can be found in Ref. [26].

III. INTUITIONISTIC FUZZY DOMINANCE–BASEDROUGH SET

A. Construction of intuitionistic fuzzy dominance–basedrough sets

An intuitionistic fuzzy set F in 𝑈 is given by

F = {< 𝑥, 𝑢F (𝑥), 𝑣F (𝑥) >: 𝑥 ∈ 𝑈}

where 𝑢F : 𝑈 → [0, 1] and 𝑣F : 𝑈 → [0, 1] withthe condition such that 0 ≤ 𝑢F (𝑥) + 𝑣F (𝑥) ≤ 1. Thenumbers 𝑢F (𝑥), 𝑣F (𝑥) ∈ [0, 1] denote the degree ofmembership and non–membership of 𝑥 to F , respec-tively. Obviously, when 𝑢F (𝑥) + 𝑣F (𝑥) = 1, for allelements in the universe, the traditional fuzzy set conceptis recovered. The family of all intuitionistic fuzzy subsetson 𝑈 is denoted by I F (𝑈). Let us review some basicoperations on I F (𝑈) as follows:∀F1,F2 ∈ I F (𝑈)

1) 𝑈 − F1 = {< 𝑥, 𝑣F1(𝑥), 𝑢F1(𝑥) >: 𝑥 ∈ 𝑈};2) F1 ∧ F2 = {< 𝑥, 𝑢F1(𝑥) ∧ 𝑢F2(𝑥), 𝑣F1(𝑥) ∨

𝑣F2(𝑥) >: 𝑥 ∈ 𝑈};3) F1 ∨ F2 = {< 𝑥, 𝑢F1(𝑥) ∨ 𝑢F2(𝑥), 𝑣F1(𝑥) ∧

𝑣F2(𝑥) >: 𝑥 ∈ 𝑈};4) F1 ⊆ F2 ⇔ 𝑢F1(𝑥) ≤ 𝑢F2(𝑥), 𝑣F1(𝑥) ≥

𝑣F2(𝑥),∀𝑥 ∈ 𝑈 ;5) F1 ⊇ F2 ⇔ F2 ⊆ F1;6) F1 = F2 ⇔ F1 ⊆ F2,F1 ⊇ F2.By the definition of intuitionistic fuzzy set, we know

that an intuitionistic fuzzy relation R on 𝑈 is an intu-itionistic fuzzy subset of 𝑈 × 𝑈 , namely, R is given by

R = {< (𝑥, 𝑦), 𝑢R(𝑥, 𝑦), 𝑣R(𝑥, 𝑦) >: (𝑥, 𝑦) ∈ 𝑈×𝑈 >},

where

𝑢R : 𝑈 × 𝑈 → [0, 1] and 𝑣R : 𝑈 × 𝑈 → [0, 1]

satisfy with the condition 0 ≤ 𝑢R(𝑥, 𝑦) + 𝑣R(𝑥, 𝑦) ≤ 1for each (𝑥, 𝑦) ∈ 𝑈×𝑈 . The set of all intuitionistic fuzzyrelation on 𝑈 is denoted by I FR(𝑈 × 𝑈).

Definition 3: Let 𝑈 be the universe of discourse, ∀R ∈I FR(𝑈 × 𝑈), if



1) 𝑢R(𝑥, 𝑦) represents the credibility of the proposi-tion “𝑥 is at least as good as 𝑦 in R”;

2) 𝑣R(𝑥, 𝑦) represents the non–credibility of theproposition “𝑥 is at least as good as 𝑦 in R”;

then R is referred to as an intuitionistic fuzzy dominancerelation.

By the above definition, we can see that different fromthe fuzzy dominance relation we used in Section 2.2,intuitionistic fuzzy dominance relation can express notonly the credibility of dominance principle between dif-ferent objects, but also the non–credibility of dominanceprinciple between these objects. In a decision system,suppose that for each 𝑎 ∈ 𝐴𝑇 , we have an intuitionisticfuzzy dominance relation R𝑎, then the intuitionistic fuzzydominance relation in terms of 𝐴𝑇 is denoted by R𝐴𝑇

where

R𝐴𝑇 (𝑥, 𝑦) = < 𝑢R𝐴(𝑥, 𝑦), 𝑣R𝐴

(𝑥, 𝑦) >

=⟨∧ {𝑢R𝑎(𝑥, 𝑦) : 𝑎 ∈ 𝐴𝑇},

∨{𝑣R𝑎(𝑥, 𝑦) : 𝑎 ∈ 𝐴𝑇}⟩

(1)

for each (𝑥, 𝑦) ∈ 𝑈 × 𝑈 . To simplify our discussion, theintuitionistic fuzzy dominance relation we used in thispaper is always reflexive, i.e. R𝑎(𝑥, 𝑥) = 1 (𝑢R𝑎(𝑥, 𝑥) =1, 𝑣R𝑎(𝑥, 𝑥) = 0) for each 𝑥 ∈ 𝑈 and each 𝑎 ∈ 𝐴𝑇 .

Definition 4: Let I be a decision system in which𝐴 ⊆ 𝐴𝑇 , R𝐴 is an intuitionistic fuzzy dominance relationwith respect to 𝐴, ∀𝑛 ∈ 𝑁 , the 𝐴–lower approximationand 𝐴–upper approximation of 𝐶𝐿≥

𝑛 with respect tointuitionistic fuzzy dominance relation R𝐴 are denotedby 𝐴R(𝐶𝐿≥

𝑛 ) and 𝐴R(𝐶𝐿≥𝑛 ), respectively and

𝐴R(𝐶𝐿≥𝑛 ) = {< 𝑥, 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥), 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) >: 𝑥 ∈ 𝑈},

𝐴R(𝐶𝐿≥𝑛 ) = {< 𝑥, 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥), 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) >: 𝑥 ∈ 𝑈},

where

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿

≥𝑛(𝑦) ∨ 𝑣R𝐴

(𝑦, 𝑥));

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑣𝐶𝐿

≥𝑛(𝑦) ∧ 𝑢R𝐴

(𝑦, 𝑥));

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑢𝐶𝐿


(𝑥, 𝑦));

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑣𝐶𝐿


(𝑥, 𝑦));

the 𝐴–lower approximation and 𝐴–upper approximationof 𝐶𝐿≤

𝑛 with respect to intuitionistic fuzzy dominancerelation R𝐴 are denoted by 𝐴R(𝐶𝐿≤

𝑛 ) and 𝐴R(𝐶𝐿≤𝑛 ),

respectively and

𝐴R(𝐶𝐿≤𝑛 ) = {< 𝑥, 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥), 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) >: 𝑥 ∈ 𝑈},

𝐴R(𝐶𝐿≤𝑛 ) = {< 𝑥, 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥), 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) >: 𝑥 ∈ 𝑈},

where

𝑢𝐴R(𝐶𝐿

≤𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿

≤𝑛(𝑦) ∨ 𝑣R𝐴

(𝑥, 𝑦));

𝑣𝐴R(𝐶𝐿

≤𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑣𝐶𝐿

≤𝑛(𝑦) ∧ 𝑢R𝐴

(𝑥, 𝑦));

𝑢𝐴R(𝐶𝐿

≤𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑢𝐶𝐿

≤𝑛(𝑦) ∧ 𝑢R𝐴

(𝑦, 𝑥));

𝑣𝐴R(𝐶𝐿

≤𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑣𝐶𝐿

≤𝑛(𝑦) ∨ 𝑣R𝐴

(𝑦, 𝑥)).

By the above definition, we know that

1) 𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥)(𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥)) is the

membership(non–membership) of 𝑥 belongsto the lower approximation 𝐴R(𝐶𝐿≥

𝑛 );2) 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥)(𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥)) is the

membership(non–membership) of 𝑥 belongsto the upper approximation 𝐴R(𝐶𝐿≥

𝑛 );3) 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥)(𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥)) is the

membership(non–membership) of 𝑥 belongsto the lower approximation 𝐴R(𝐶𝐿≤

𝑛 );4) 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥)(𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥)) is the

membership(non–membership) of 𝑥 belongsto the upper approximation 𝐴R(𝐶𝐿≤

𝑛 ).Since mathematically, an intuitionistic fuzzy set may be

equivalently characterized by an interval–valued fuzzy set,then our intuitionistic fuzzy dominance–based rough setmodels can also be considered as a type of interval–valuedfuzzy rough set model. However, it should be noticed thatour IFDRS is different from the common interval–valuedfuzzy rough set because the intuitionistic fuzzy dominancerelation we used here has its own semantic explanation,i.e. it represents both the credibility and non–credibilityof the dominance principle.

Theorem 1: Let I be a decision system in which 𝐴 ⊆𝐴𝑇 , R𝐴 is the intuitionistic fuzzy dominance relationrelated to 𝐴, if 𝑢R𝐴

(𝑥, 𝑦) + 𝑣R𝐴(𝑥, 𝑦) = 1 for each

(𝑥, 𝑦) ∈ 𝑈 × 𝑈 , then for each 𝑥 ∈ 𝑈 , we have1) 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) + 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = 1;


≥𝑛 )(𝑥) + 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = 1;


≤𝑛 )(𝑥) + 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) = 1;


≤𝑛 )(𝑥) + 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) = 1.

Proof: We only prove (1), the proofs of (2), (3) and(4) are similar to the proof of (1).∀𝑥 ∈ 𝑈 , by Definition 4, we have 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) +

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿

≥𝑛(𝑦) ∨ 𝑣R𝐴(𝑦, 𝑥)

)+

∨𝑦∈𝑈

(𝑣𝐶𝐿


(𝑦, 𝑥)).

∀𝑦 ∈ 𝑈 ,∙ If 𝑦 ∈ 𝐶𝐿≥

𝑛 , i.e. 𝑢𝐶𝐿

≥𝑛(𝑦) = 1 and 𝑣

𝐶𝐿≥𝑛(𝑦) = 0,

then𝑢𝐶𝐿

≥𝑛(𝑦) ∨ 𝑣R𝐴(𝑦, 𝑥) = 1,

𝑣𝐶𝐿


(𝑦, 𝑥) = 0.∙ If 𝑦 /∈ 𝐶𝐿≥


≥𝑛(𝑦) = 0 and 𝑣

𝐶𝐿≥𝑛(𝑦) = 1,

then𝑢𝐶𝐿


(𝑦, 𝑥) = 𝑣R𝐴(𝑦, 𝑥),

𝑣𝐶𝐿

≥𝑛(𝑦) ∧ 𝑢R𝐴(𝑦, 𝑥) = 𝑢R𝐴(𝑦, 𝑥).

From discussion above, if 𝑛 = 1, then∧𝑦∈𝑈

(𝑢𝐶𝐿

≥𝑛(𝑦) ∨ 𝑣R𝐴(𝑦, 𝑥)

)+ ∨𝑦∈𝑈

(𝑣𝐶𝐿

≥𝑛(𝑦) ∧

𝑢R𝐴(𝑦, 𝑥))

= 1 holds obviously. On the otherhand, if 𝑛 ∕= 1, then there must be 𝑦 /∈ 𝐶𝐿≥

𝑛 suchthat ∧𝑦∈𝑈

(𝑢𝐶𝐿


(𝑦, 𝑥))

= 𝑣R𝐴(𝑦, 𝑥) and

∨𝑦∈𝑈

(𝑣𝐶𝐿


(𝑦, 𝑥))= 𝑢R𝐴

(𝑦, 𝑥).Since 𝑢R𝐴

(𝑥, 𝑦) + 𝑣R𝐴(𝑥, 𝑦) = 1 for each (𝑥, 𝑦) ∈

𝑈×𝑈 , then 𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥)+𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = 𝑣R𝐴(𝑦, 𝑥)+

𝑢R𝐴(𝑦, 𝑥)

= 1.By the above theorem, we can see that if the intu-

itionistic fuzzy dominance relation degenerated to be the



fuzzy dominance relation, then the intuitionistic fuzzydominance–based rough set we showed in Definition 4will degenerate to be the fuzzy dominance–based roughset. From this point of view, the intuitionistic fuzzydominance–based rough set is a generalization of thetraditional fuzzy dominance–based rough set.

Theorem 2: Let I be a decision system in which 𝐴 ⊆𝐴𝑇 , we have


≥𝑛 )(𝑥) = ∧{𝑣R𝐴

(𝑦, 𝑥) : 𝑦 /∈ 𝐶𝐿≥𝑛 } (𝑛 =

2, ⋅ ⋅ ⋅ ,𝑚);2) 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = ∨{𝑢R𝐴(𝑦, 𝑥) : 𝑦 /∈ 𝐶𝐿≥

𝑛 } (𝑛 =

2, ⋅ ⋅ ⋅ ,𝑚);3) 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = ∨{𝑢R𝐴

(𝑥, 𝑦) : 𝑦 ∈ 𝐶𝐿≥𝑛 } (𝑛 =

1, ⋅ ⋅ ⋅ ,𝑚);4) 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = ∧{𝑣R𝐴

(𝑥, 𝑦) : 𝑦 ∈ 𝐶𝐿≥𝑛 } (𝑛 =

1, ⋅ ⋅ ⋅ ,𝑚);5) 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥) = ∧{𝑣R𝐴

(𝑥, 𝑦) : 𝑦 /∈ 𝐶𝐿≤𝑛 } (𝑛 =

1, ⋅ ⋅ ⋅ ,𝑚− 1);6) 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) = ∨{𝑢R𝐴

(𝑥, 𝑦) : 𝑦 /∈ 𝐶𝐿≤𝑛 } (𝑛 =

1, ⋅ ⋅ ⋅ ,𝑚− 1);7) 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥) = ∨{𝑢R𝐴

(𝑦, 𝑥) : 𝑦 ∈ 𝐶𝐿≤𝑛 } (𝑛 =

1, ⋅ ⋅ ⋅ ,𝑚);8) 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) = ∧{𝑣R𝐴

(𝑦, 𝑥) : 𝑦 ∈ 𝐶𝐿≤𝑛 } (𝑛 =

1, ⋅ ⋅ ⋅ ,𝑚).Proof: We only prove 1), others can be proved

analogously.∀𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚, by Definition 4, we have

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿


(𝑦, 𝑥)).

∙ If 𝑦 ∈ 𝐶𝐿≥𝑛 , then 𝑢

𝐶𝐿≥𝑛(𝑦) = 1, it follows that

𝑢𝐶𝐿


(𝑦, 𝑥) = 1;∙ If 𝑦 /∈ 𝐶𝐿≥

𝑛 , then 𝑢𝐶𝐿

≥𝑛(𝑦) = 0, it follows that

𝑢𝐶𝐿


(𝑦, 𝑥) = 𝑣R𝐴(𝑦, 𝑥).

Since 𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚, then there must be 𝑦 /∈ 𝐶𝐿≥𝑛 such

that 𝑢𝐶𝐿

≥𝑛(𝑦)∨ 𝑣R𝐴

(𝑦, 𝑥) = 𝑣R𝐴(𝑦, 𝑥). From discussion

above, it is not difficult to conclude that 𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) =

∧{𝑣R𝐴(𝑦, 𝑥) : 𝑦 /∈ 𝐶𝐿≥𝑛 }.

Theorem 3: Let I be a decision system in which𝐴 ⊆ 𝐴𝑇 , the intuitionistic fuzzy dominance–based roughapproximations have the following properties:

1) Contraction and extension:

𝐴R(𝐶𝐿≥𝑛 ) ⊆ 𝐶𝐿≥

𝑛 ⊆ 𝐴R(𝐶𝐿≥𝑛 );

𝐴R(𝐶𝐿≤𝑛 ) ⊆ 𝐶𝐿≤

𝑛 ⊆ 𝐴R(𝐶𝐿≤𝑛 );

2) Complements:

𝐴R(𝐶𝐿≥𝑛 ) = 𝑈 −𝐴R(𝐶𝐿≤

𝑛−1), 𝑛 = 2 ⋅ ⋅ ⋅𝑚𝐴R(𝐶𝐿≤

𝑛 ) = 𝑈 −𝐴R(𝐶𝐿≥𝑛+1), 𝑛 = 1 ⋅ ⋅ ⋅𝑚− 1

𝐴R(𝐶𝐿≥𝑛 ) = 𝑈 −𝐴R(𝐶𝐿≤

𝑛−1), 𝑛 = 2 ⋅ ⋅ ⋅𝑚𝐴R(𝐶𝐿≤

𝑛 ) = 𝑈 −𝐴R(𝐶𝐿≤𝑛+1), 𝑛 = 1 ⋅ ⋅ ⋅𝑚− 1

3) Monotones with attributes:

𝐴R(𝐶𝐿≥𝑛 ) ⊆ 𝐴𝑇R(𝐶𝐿≥

𝑛 );𝐴R(𝐶𝐿≥𝑛 ) ⊇ 𝐴𝑇R(𝐶𝐿≥

𝑛 );

𝐴R(𝐶𝐿≤𝑛 ) ⊆ 𝐴𝑇R(𝐶𝐿≤

𝑛 );𝐴R(𝐶𝐿≤𝑛 ) ⊇ 𝐴𝑇R(𝐶𝐿≤

𝑛 );

4) Monotones with decision classes:

𝑛1, 𝑛2 ∈ 𝑁 such that 𝑛1 ≤ 𝑛2

𝐴R(𝐶𝐿≥𝑛1) ⊇ 𝐴R(𝐶𝐿≥

𝑛2);𝐴R(𝐶𝐿≥

𝑛1) ⊇ 𝐴R(𝐶𝐿≥

𝑛2);

𝐴R(𝐶𝐿≤𝑛1) ⊆ 𝐴R(𝐶𝐿≤

𝑛2);𝐴R(𝐶𝐿≤

𝑛1) ⊆ 𝐴R(𝐶𝐿≤

𝑛2).

Proof:1) ∀𝑥 /∈ 𝐶𝐿≥


≥𝑛(𝑥) = 0 and 𝑣

𝐶𝐿≥𝑛(𝑥) = 1,

we have

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿


(𝑦, 𝑥))

≤ 𝑢𝐶𝐿

≥𝑛(𝑥) ∨ 𝑣R𝐴

(𝑥, 𝑥)

= 0 = 𝑢𝐶𝐿

≥𝑛(𝑥)

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑣𝐶𝐿


(𝑦, 𝑥))

≥ 𝑣𝐶𝐿

≥𝑛(𝑥) ∧ 𝑢R𝐴

(𝑥, 𝑥)

= 1 = 𝑣𝐶𝐿

≥𝑛(𝑥)

From discussion above, we can conclude that𝐴R(𝐶𝐿≥

𝑛 ) ⊆ 𝐶𝐿≥𝑛 .

On the other hand, ∀𝑥 ∈ 𝑈 , if 𝑥 ∈ 𝐶𝐿≥𝑛 , i.e.

𝑢𝐶𝐿

≥𝑛(𝑥) = 1 and 𝑣

𝐶𝐿≥𝑛(𝑥) = 0, we have

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑢𝐶𝐿


(𝑥, 𝑦))

≥ 𝑢𝐶𝐿

≥𝑛(𝑥) ∧ 𝑢R𝐴

(𝑥, 𝑥)

= 𝑢𝐶𝐿

≥𝑛(𝑥)

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑣𝐶𝐿


(𝑥, 𝑦))

≤ 𝑣𝐶𝐿

≥𝑛(𝑥) ∨ 𝑣R𝐴

(𝑥, 𝑥)

= 𝑣𝐶𝐿

≥𝑛(𝑥)

From discussion above, we can conclude that𝐶𝐿≥

𝑛 ⊆ 𝐴R(𝐶𝐿≥𝑛 ).

Similarity, it is not difficult to prove 𝐴R(𝐶𝐿≤𝑛 ) ⊆

𝐶𝐿≤𝑛 ⊆ 𝐴R(𝐶𝐿≤

𝑛 ).2) ∀𝑥 ∈ 𝑈 , since 𝑢

𝐶𝐿≥𝑛(𝑥) = 𝑣

𝐶𝐿≤𝑛−1

(𝑥) and𝑣𝐶𝐿

≥𝑛(𝑥) = 𝑢

𝐶𝐿≤𝑛−1

(𝑥) where 𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚, wehave

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿


(𝑦, 𝑥))

= ∧𝑦∈𝑈

(𝑣𝐶𝐿

≤𝑛−1

(𝑦) ∨ 𝑣R𝐴(𝑦, 𝑥)

)= 𝑣

𝐴R(𝐶𝐿≤𝑛−1)

(𝑥)

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑣𝐶𝐿


(𝑦, 𝑥))

= ∨𝑦∈𝑈

(𝑢𝐶𝐿

≤𝑛−1

(𝑦) ∧ 𝑢R𝐴(𝑦, 𝑥)

)= 𝑢

𝐴R(𝐶𝐿≤𝑛−1)

(𝑥)


𝑛 ) = 𝑈 − 𝐴R(𝐶𝐿≤𝑛−1), 𝑛 = 2 ⋅ ⋅ ⋅𝑚.

Others can be proved analogously.3) By Eq. 1), ∀(𝑥, 𝑦) ∈ 𝑈 ×𝑈 , we have 𝑢R𝐴(𝑥, 𝑦) ≥

𝑢R𝐴𝑇(𝑥, 𝑦) and 𝑣R𝐴

(𝑥, 𝑦) ≤ 𝑣R𝐴𝑇(𝑥, 𝑦) because

𝐴 ⊆ 𝐴𝑇 , thus

𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿


(𝑦, 𝑥))

≤ ∧𝑦∈𝑈

(𝑢𝐶𝐿

≥𝑛(𝑦) ∨ 𝑣R𝐴𝑇 (𝑦, 𝑥)

)= 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥)

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∨𝑦∈𝑈

(𝑣𝐶𝐿


(𝑦, 𝑥))

≥ ∨𝑦∈𝑈

(𝑣𝐶𝐿

≥𝑛(𝑦) ∧ 𝑢R𝐴𝑇

(𝑦, 𝑥))



= 𝑣𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥)


𝑛 ) ⊆ 𝐴𝑇R(𝐶𝐿≥𝑛 ). Others can be proved

analogously.4) Since 𝑛1 ≤ 𝑛2, we obtain that 𝐶𝐿≥

𝑛1⊇ 𝐶𝐿≥

𝑛2, i.e.

𝑢𝐶𝐿

≥𝑛1

(𝑥) ≥ 𝑢𝐶𝐿

≥𝑛2

(𝑥) and 𝑣𝐶𝐿

≥𝑛1

(𝑥) ≤ 𝑣𝐶𝐿

≥𝑛2

(𝑥)

for each 𝑥 ∈ 𝑈 , thus

𝑢𝐴R(𝐶𝐿

≥𝑛1

)(𝑥) = ∧𝑦∈𝑈

(𝑢𝐶𝐿

≥𝑛1


);

≥ ∧𝑦∈𝑈

(𝑢𝐶𝐿

≥𝑛2


);

= 𝑢𝐴R(𝐶𝐿

≥𝑛2

)(𝑥)

𝑣𝐴R(𝐶𝐿

≥𝑛1

)(𝑥) = ∨𝑦∈𝑈

(𝑣𝐶𝐿

≥𝑛1


)≤ ∨𝑦∈𝑈

(𝑣𝐶𝐿

≥𝑛2


)= 𝑣

𝐴R(𝐶𝐿≥𝑛2

)(𝑥)


𝑛1) ⊇ 𝐴R(𝐶𝐿≥

𝑛2). Others can be proved

analogously.

Results 1), 2), 3) and 4) of Theorem 3 can be regardedas intuitionistic fuzzy counterparts of results well knownwithin the classical DRSA. More precisely, 1) says thatthe upward (downward) union of decision classes includeits intuitionistic fuzzy rough lower approximation and isincluded in its intuitionistic fuzzy rough upper approxi-mation; 2) represents complementarity properties of theproposed intuitionistic fuzzy dominance–based rough ap-proximations; 3) expresses monotonicity of the proposedintuitionistic fuzzy dominance–based rough set in termsof the monotonous varieties of condition attributes; 4) ex-presses monotonicity of the proposed intuitionistic fuzzydominance–based rough set in terms of the monotonousvarieties of unions of decision classes.

B. Intuitionistic fuzzy dominance–based rough set in de-cision system with probabilistic interpretation

It is well known that Greco’s traditional DRSA wasfirstly proposed for dealing with complete system withpreference–ordered domains of the attributes. In this sec-tion, we will illustrate how the proposed intuitionisticfuzzy dominance–based rough set can be used in thedecision system with probabilistic interpretation.

For a decision system I , if ∀𝑥 ∈ 𝑈 and ∀𝑎 ∈ 𝐴𝑇 ,𝑎(𝑥) ⊆ 𝑉𝑎 instead of 𝑎(𝑥) ∈ 𝑉𝑎, i.e.

𝑎 : 𝑈 → 𝑃 (𝑉𝑎)

where 𝑃 (𝑉𝑎) is the collection of all nonempty subsetsof 𝑉𝑎, then such system is referred to as a set–valueddecision system. Obviously, in a set–valued decisionsystem I , 𝑥 holds a set of values instead of a singlevalue on each attribute.

Furthermore, in a set–valued decision system withprobabilistic interpretation, ∀𝑣 ∈ 𝑉𝑎, 𝑎(𝑥)(𝑣) ∈ [0, 1]

represents the possibility of state 𝑣. ∀𝑥 ∈ 𝑈 , ∀𝑎 ∈ 𝐴𝑇 ,we assume here that∑

𝑣∈𝑉𝑎

𝑎(𝑥)(𝑣) = 1

It is clear that every set value is expressed in aprobability distribution over the elements contained insuch set. This leads to that the set value can be expressedin terms of a probability distribution such that

𝑎(𝑥) = {𝑣1/𝑎(𝑥)(𝑣1), 𝑣2/𝑎(𝑥)(𝑣2), ⋅ ⋅ ⋅ , 𝑣𝑘/𝑎(𝑥)(𝑣𝑘)}

where 𝑣1, 𝑣2, ⋅ ⋅ ⋅ , 𝑣𝑘 ∈ 𝑉𝑎.Actually, the set–valued decision system with prob-

abilistic interpretation has been analyzed by rough settechnique. For example, in valued tolerance relation [53],[54] and the valued dominance relation [60] based roughsets for dealing with incomplete information systems,each unknown value is expressed in a uniform prob-ability distribution over the elements contained in thedomain of the corresponding attribute. Suppose that 𝑉𝑎 ={𝑎1, 𝑎2, 𝑎3, 𝑎4}, if 𝑎(𝑥) = ∗ where ∗ denotes the “do notcare” unknown value, then the probability distribution canbe written such that

𝑎(𝑥) = {𝑎1/0.25, 𝑎2/0.25, 𝑎3/0.25, 𝑎4/0.25}.

This tells us that if the value that 𝑥 holds on 𝑎 is unknown,then 𝑥 may hold any one of the values in 𝑉𝑎. Moreover,the probabilistic degrees that 𝑥 holds each value are equal.However, valued tolerance and dominance relations onlyconsider the memberships of tolerance degree and domi-nance degree, they do not take the non–memberships intoaccount. To overcome this limitation, the intuitionisticfuzzy rough technique has become a necessity.

Let us consider Table 1, it is a set–valued decisionsystem with probabilistic interpretation. In Table 1,

∙ 𝑈 = {𝑥1, 𝑥2, ⋅ ⋅ ⋅ , 𝑥10} is the universe of discourse;∙ 𝐴𝑇 = {𝑎, 𝑏, 𝑐, 𝑑, 𝑒} denotes the set of condition

attributes;∙ 𝑉𝑎 = {𝑎0, 𝑎1, 𝑎2}, 𝑉𝑏 = {𝑏0, 𝑏1, 𝑏2}, 𝑉𝑐 =

{𝑐0, 𝑐1, 𝑐2}, 𝑉𝑑 = {𝑑0, 𝑑1, 𝑑2}, 𝑉𝑒 = {𝑒0, 𝑒1, 𝑒2},𝑎0 < 𝑎1 < 𝑎2, 𝑏0 < 𝑏1 < 𝑏2, 𝑐0 < 𝑐1 < 𝑐2,𝑑0 < 𝑑1 < 𝑑2, 𝑒0 < 𝑒1 < 𝑒2;

∙ 𝑓 is the decision attribute where 𝑉𝑓 = {1, 2}∀(𝑥, 𝑦) ∈ 𝑈 × 𝑈 , let us denote the intuitionistic fuzzy

dominance relation as following:

R𝐴𝑇 (𝑥, 𝑦) =

{[1, 0]:𝑥 = 𝑦

< 𝑢R𝐴𝑇(𝑥, 𝑦), 𝑣R𝐴𝑇

(𝑥, 𝑦) >:otherwise

where ∀𝑎 ∈ 𝐴𝑇 ,

𝑢R𝑎(𝑥, 𝑦) =∑

𝑣1>𝑣2,𝑣1,𝑣2∈𝑉𝑎

𝑎(𝑥)(𝑣1) ⋅ 𝑎(𝑥)(𝑣2)

𝑣R𝑎(𝑥, 𝑦) =∑

𝑣1<𝑣2,𝑣1,𝑣2∈𝑉𝑎

𝑎(𝑥)(𝑣1) ⋅ 𝑎(𝑥)(𝑣2)

In the above definition, 𝑢R𝐴𝑇 (𝑥, 𝑦) denotes the degreeof dominance principle in terms of the set of attributes 𝐴𝑇



TABLE I.A DECISION SYSTEM WITH PROBABILISTIC INTERPRETATION

𝑈 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓𝑥1 {𝑎1/1} {𝑏0/0.7, 𝑏1/0.3} {𝑐0/1} {𝑑1/0/4, 𝑑2/0.6} {𝑒2/1} 2𝑥2 {𝑎0/0.3, 𝑎1/0.7} {𝑏2/1} {𝑐1/0.5, 𝑐2/0.5} {𝑑0/1} {𝑒0/1} 1𝑥3 {𝑎0/1} {𝑏1/0.4, 𝑏2/0.6} {𝑐1/1} {𝑑0/0.3, 𝑑1/0.7} {𝑒0/1} 1𝑥4 {𝑎0/0.9, 𝑎1/0.1} {𝑏1/1} {𝑐1/1} {𝑑1/1} {𝑒0/0.2, 𝑒2/0.8} 1𝑥5 {𝑎1/0.8, 𝑎2/0.2} {𝑏1/1} {𝑐0/0.6, 𝑐1/0.4} {𝑑0/1} {𝑒1/1} 2𝑥6 {𝑎0/0.5, 𝑎2/0.5} {𝑏1/1} {𝑐0/0.3, 𝑐1/0.7} {𝑑0/1} {𝑒1/1} 1𝑥7 {𝑎1/1} {𝑏0/0.2, 𝑏2/0.8} {𝑐0/0.1, 𝑐1/0.9} {𝑑1/1} {𝑒2/1} 2𝑥8 {𝑎0/1} {𝑏2/1} {𝑐1/1} {𝑑0/1} {𝑒0/0.9, 𝑒1/0.1} 1𝑥9 {𝑎1/1} {𝑏0/0.8, 𝑏1/0.2} {𝑐0/0.5, 𝑐2/0.5} {𝑑1/1} {𝑒2/1} 2𝑥10 {𝑎1/1} {𝑏1/1} {𝑐2/1} {𝑑0/0.8, 𝑑1/0.2} {𝑒2/1} 2

TABLE II.INTUITIONISTIC FUZZY DOMINANCE RELATION IN TABLE 1

𝑥∖𝑦 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9 𝑥10

𝑥1 [1.0,0.0] [0.0,1.0] [0.0,1.0] [0.0,1.0] [0.0,0.7] [0.0,0.7] [0.0,0.9] [0.0,1.0] [0.0,0.5] [0.0,1.0]𝑥2 [0.0,1.0] [1.0,0.0] [0.0,0.7] [0.0,1.0] [0.0,1.0] [0.0,1.0] [0.0,1.0] [0.0,0.1] [0.0,1.0] [0.0,1.0]𝑥3 [0.0,1.0] [0.0,0.7] [1.0,0.0] [0.0,0.8] [0.0,1.0] [0.0,1.0] [0.0,1.0] [0.0,0.4] [0.0,1.0] [0.0,1.0]𝑥4 [0.0,0.9] [0.0,1.0] [0.0,0.6] [1.0,0.0] [0.0,0.92] [0.0,0.5] [0.0,0.9] [0.0,1.0] [0.0,0.9] [0.0,1.0]𝑥5 [0.0,1.0] [0.0,1.0] [0.0,0.7] [0.0,1.0] [1.0,0.0] [0.0,0.42] [0.0,1.0] [0.0,1.0] [0.0,1.0] [0.0,1.0]𝑥6 [0.0,1.0] [0.0,1.0] [0.0,0.7] [0.0,1.0] [0.0,0.5] [1.0,0.0] [0.0,1.0] [0.0,1.0] [0.0,1.0] [0.0,1.0]𝑥7 [0.0,0.6] [0.0,0.55] [0.0,0.2] [0.0,0.2] [0.0,0.2] [0.27,0.5] [1.0,0.0] [0.0,0.2] [0.0,0.5] [0.0,1.0]𝑥8 [0.0,1.0] [0.0,0.7] [0.0,0.7] [0.0,1.0] [0.0,1.0] [0.0,0.9] [0.0,1.0] [1.0,0.0] [0.0,1.0] [0.0,1.0]𝑥9 [0.0,0.6] [0.0,1.0] [0.0,0.92] [0.0,0.8] [0.0,0.8] [0.0,0.8] [0.0,0.8] [0.0,1.0] [1.0,0.0] [0.0,0.8]𝑥10 [0.0,0.92] [0.0,1.0] [0.0,0.6] [0.0,0.8] [0.0,0.2] [0.0,0.5] [0.0,0.8] [0.0,1.0] [0.0,0.8] [1.0,0.0]

while 𝑣R𝐴𝑇(𝑥, 𝑦) denotes the degree of non-dominance

principle in terms of the set of attributes 𝐴𝑇 . For instance,

𝑢R𝑏(𝑥2, 𝑥1) =

∑𝑣1≥𝑣2,𝑣1,𝑣2∈𝑉𝑏

𝑏(𝑥2)(𝑣1) ⋅ 𝑏(𝑥1)(𝑣2)

= 𝑏(𝑥2)(𝑏2) ⋅ 𝑏(𝑥1)(𝑏0)

+ 𝑏(𝑥2)(𝑏2) ⋅ 𝑏(𝑥1)(𝑏1)

= 1

𝑣R𝑏(𝑥2, 𝑥1) =

∑𝑣1<𝑣2,𝑣1,𝑣2∈𝑉𝑏

𝑏(𝑥2)(𝑣1) ⋅ 𝑏(𝑥1)(𝑣2)

= 0

Similarity, the result of intuitionistic fuzzy dominancerelation in Table 1 is showed in Table 2.

By the above intuitionistic fuzzy dominance relation,we can obtain the corresponding rough approximatememberships and non–memberships by Definition 4. Bythe decision attribute 𝑓 , the universe can be partitionedinto decision classes such that CL = {𝐶𝐿1, 𝐶𝐿2} ={{𝑥2, 𝑥3, 𝑥4, 𝑥6, 𝑥8}, {𝑥1, 𝑥5, 𝑥7, 𝑥9, 𝑥10}}. The resultsof intuitionistic fuzzy dominance–based rough approxi-mations in Table 1 are showed in Table 3.

IV. APPROXIMATE DISTRIBUTION REDUCTS OFINTUITIONISTIC FUZZY DOMINANCE–BASED ROUGH

SET

The concept of approximate distribution reduct wasfirstly proposed by Mi in Ref. [42]. Following Mi’swork, we have introduced such reduct into DRSA fordealing with the incomplete system with lost unknownvalues [65]. Moreover, Qian et al. [49] also introducedsuch reduct into maximal consistent block based roughset approach for dealing with the incomplete system with

TABLE III.INTUITIONISTIC FUZZY DOMINANCE–BASED ROUGH

APPROXIMATIONS IN TABLE 1

𝑥∖𝑦 𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9 𝑥10

𝑢𝐴𝑇R(𝐶𝐿

≥1 )

(𝑥) 1 1 1 1 1 1 1 1 1 1


≥2 )

(𝑥) 0.9 0 0 0 0.5 0 0.9 0 0.9 1

𝑣𝐴𝑇R(𝐶𝐿

≥1 )

(𝑥) 0 0 0 0 0 0 0 0 0 0


≥2 )

(𝑥) 0 1 1 1 0 1 0 1 0 0


≥1 )

(𝑥) 1 1 1 1 1 1 1 1 1 1


≥2 )

(𝑥) 1 0 0 0 1 0 1 0 1 1


≥1 )

(𝑥) 0 0 0 0 0 0 0 0 0 0


≥2 )

(𝑥) 0 1 1 0.9 0 0.5 0 1 0 0


≤1 )

(𝑥) 0 1 1 0.9 0 0.5 0 1 0 0


≤2 )

(𝑥) 1 1 1 1 1 1 1 1 1 1


≤1 )

(𝑥) 1 0 0 0 1 0 1 0 1 1


≤2 )

(𝑥) 0 0 0 0 0 0 0 0 0 0


≤1 )

(𝑥) 0 1 1 1 0 1 0 1 0 0


≤2 )

(𝑥) 1 1 1 1 1 1 1 1 1 1


≤1 )

(𝑥) 0.9 0 0 0 0.5 0 0.9 0 0.9 1


≤2 )

(𝑥) 0 0 0 0 0 0 0 0 0 0

“do not care” unknown values. In the following, we willfurther generalize the concept of approximate distribu-tion reduct into our intuitionistic fuzzy dominance–basedrough set model.



A. Concept and approach to approximate distributionreducts

Definition 5: Let I be a decision system, 𝐴 ⊆ 𝐴𝑇 ,let us denote by

𝐿≥𝐴𝑇 = {𝐴𝑇R(𝐶𝐿≥

1 ), 𝐴𝑇R(𝐶𝐿≥2 ), ⋅ ⋅ ⋅ , 𝐴𝑇R(𝐶𝐿≥

𝑛 )};𝐿≤𝐴𝑇 = {𝐴𝑇R(𝐶𝐿≤

1 ), 𝐴𝑇R(𝐶𝐿≤2 ), ⋅ ⋅ ⋅ , 𝐴𝑇R(𝐶𝐿≤

𝑛 )};𝐻≥

𝐴𝑇 = {𝐴𝑇R(𝐶𝐿≥1 ), 𝐴𝑇R(𝐶𝐿≥

2 ), ⋅ ⋅ ⋅ , 𝐴𝑇R(𝐶𝐿≥𝑛 )};

𝐻≤𝐴𝑇 = {𝐴𝑇R(𝐶𝐿≤

1 ), 𝐴𝑇R(𝐶𝐿≤2 ), ⋅ ⋅ ⋅ , 𝐴𝑇R(𝐶𝐿≤

𝑛 )};

1) If 𝐿≥𝐴 = 𝐿≥

𝐴𝑇 , then 𝐴 is referred to as the ≥-lowerapproximate distribution consistent set; if 𝐿≥

𝐴 =𝐿≥𝐴𝑇 and 𝐿≥

𝐵 ∕= 𝐿≥𝐴 for ∀𝐵 ⊂ 𝐴, then 𝐴 is referred

to as a ≥-lower approximate distribution reduct ofI ;

2) If 𝐿≤𝐴 = 𝐿≤

𝐴𝑇 , then 𝐴 is referred to as the ≤-lowerapproximate distribution consistent set; if 𝐿≤

𝐴 =𝐿≤𝐴𝑇 and 𝐿≤

𝐵 ∕= 𝐿≤𝐴 for ∀𝐵 ⊂ 𝐴, then 𝐴 is referred

to as a ≤-lower approximate distribution reduct ofI ;

3) If 𝐻≥𝐴 = 𝐻≥

𝐴𝑇 , then 𝐴 is referred to as the ≥-upper approximate distribution consistent set; if𝐻≥

𝐴 = 𝐻≥𝐴𝑇 and 𝐻≥

𝐵 ∕= 𝐻≥𝐴 for ∀𝐵 ⊂ 𝐴, then 𝐴

is referred to as a ≥-upper approximate distributionreduct of I ;

4) If 𝐻≤𝐴 = 𝐻≤

𝐴𝑇 , then 𝐴 is referred to as the ≤-upper approximate distribution consistent set; if𝐻≤

𝐴 = 𝐻≤𝐴𝑇 and 𝐻≤

𝐵 ∕= 𝐻≤𝐴 for ∀𝐵 ⊂ 𝐴, then 𝐴

is referred to as a ≤-upper approximate distributionreduct of I .

A ≥–lower(upper) approximate distribution consistentset is a subset of attributes that preserves the intuitionisticfuzzy dominance–based lower(upper) approximations ofall the upward unions of the decision classes; a ≤–lower(upper) approximate distribution consistent set isa subset of attributes that preserves the intuitionisticfuzzy dominance–based lower(upper) approximations ofall the downward unions of the decision classes; a ≥–lower(upper) approximate distribution reduct is a minimalsubset of attributes that preserves the intuitionistic fuzzydominance–based lower(upper) approximations of all theupward unions of the decision classes; a ≤–lower(upper)approximate distribution reduct is a minimal subset of at-tributes that preserves the intuitionistic fuzzy dominance–based lower(upper) approximations of all the downwardunions of the decision classes.


1) 𝐴 is ≥-lower approximate distribution consistent set⇔ 𝐴 is ≤-upper approximate distribution consistentset;

2) 𝐴 is ≤-lower approximate distribution consistent set⇔ 𝐴 is ≥-upper approximate distribution consistentset.

Proof: It can be derived directly from (2) of Theo-rem 3 and Definition 5.


1) 𝐴 is ≥-lower approximate distribution reduct ⇔ 𝐴is ≤-upper approximate distribution reduct;

2) 𝐴 is ≤-lower approximate distribution reduct ⇔ 𝐴is ≥-upper approximate distribution reduct.

Proof: It can be derived directly from Theorem 4and Definition 5.

Theorem 6: Let I be a decision system in which 𝐴 ⊆𝐴𝑇 , for ∀𝑥 ∈ 𝑈 , we denote

𝑃≥𝐴𝑇 (𝑥) = {< 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥), 𝑣

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) >: 𝑛 ∈ 𝑁},

𝑃≤𝐴𝑇 (𝑥) = {< 𝑢

𝐴𝑇R(𝐶𝐿≤𝑛 )(𝑥), 𝑣

𝐴𝑇R(𝐶𝐿≤𝑛 )(𝑥) >: 𝑛 ∈ 𝑁},

𝑄≥𝐴𝑇 (𝑥) = {< 𝑢


𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) >: 𝑛 ∈ 𝑁},

𝑄≤𝐴𝑇 (𝑥) = {< 𝑢

𝐴𝑇R(𝐶𝐿≤𝑛 )(𝑥), 𝑣

𝐴𝑇R(𝐶𝐿≤𝑛 )(𝑥) >: 𝑛 ∈ 𝑁},

then we have the following:1) 𝐴 is ≥-lower approximate distribution consistent set

⇔ for ∀𝑥 ∈ 𝑈 , 𝑃≥𝐴 (𝑥) = 𝑃≥

𝐴𝑇 (𝑥);2) 𝐴 is ≤-lower approximate distribution consistent set

⇔ for ∀𝑥 ∈ 𝑈 , 𝑃≤𝐴 (𝑥) = 𝑃≤

𝐴𝑇 (𝑥);3) 𝐴 is ≥-upper approximate distribution consistent set

⇔ for ∀𝑥 ∈ 𝑈 , 𝑄≥𝐴(𝑥) = 𝑄≥

𝐴𝑇 (𝑥);4) 𝐴 is ≤-upper approximate distribution consistent set

⇔ for ∀𝑥 ∈ 𝑈 , 𝑄≤𝐴(𝑥) = 𝑄≤

𝐴𝑇 (𝑥).Proof: We only prove (1), others can be proved

analogously.𝐿≥𝐴 = 𝐿≥

𝐴𝑇 ⇔ 𝐴R(𝐶𝐿≥𝑛 ) = 𝐴𝑇R(𝐶𝐿≥

𝑛 )(𝑛 ∈𝑁) ⇔ 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) = 𝑢


𝐴R(𝐶𝐿≥𝑛 )(𝑥) =


≥𝑛 )(𝑥)(∀𝑥 ∈ 𝑈) ⇔ 𝑃≥

𝐴 (𝑥) = 𝑃≥𝐴𝑇 (𝑥).

Definition 6: Let I be a decision system in which𝐴 ⊆ 𝐴𝑇 , define

𝐷≥𝐿 = {(𝑥, 𝑦) ∈ 𝑈2 : 𝑥 ∈ 𝑈, 𝑦 /∈ 𝐶𝐿≥

𝑛 , 𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚}𝐷≤

𝐿 = {(𝑥, 𝑦) ∈ 𝑈2 : 𝑥 ∈ 𝑈, 𝑦 /∈ 𝐶𝐿≤𝑛 , 𝑛 = 1, ⋅ ⋅ ⋅ ,𝑚− 1}

𝐷≥𝐻 = {(𝑥, 𝑦) ∈ 𝑈2 : 𝑥 ∈ 𝑈, 𝑦 ∈ 𝐶𝐿≥

𝑛 , 𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚}𝐷≤

𝐻 = {(𝑥, 𝑦) ∈ 𝑈2 : 𝑥 ∈ 𝑈, 𝑦 ∈ 𝐶𝐿≤𝑛 , 𝑛 = 1, ⋅ ⋅ ⋅ ,𝑚− 1}

where1) if (𝑥, 𝑦) ∈ 𝐷≥

𝐿 , then 𝐷≥𝑢𝐿 (𝑥, 𝑦) = {𝑎 ∈

𝐴𝑇 : 𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) ≤ 𝑣R𝑎(𝑦, 𝑥)}, otherwise,

𝐷≥𝑢𝐿 (𝑥, 𝑦) = ∅;

2) if (𝑥, 𝑦) ∈ 𝐷≥𝐿 , then 𝐷≥𝑣

𝐿 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑣

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) ≥ 𝑢R𝑎(𝑦, 𝑥)}, otherwise,

𝐷≥𝑣𝐿 (𝑥, 𝑦) = ∅;

3) if (𝑥, 𝑦) ∈ 𝐷≤𝐿 , then 𝐷≤𝑢

𝐿 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑢

𝐴𝑇R(𝐶𝐿≤𝑛 )(𝑥) ≤ 𝑣R𝑎(𝑥, 𝑦)}, otherwise,

𝐷≤𝑢𝐿 (𝑥, 𝑦) = ∅;

4) if (𝑥, 𝑦) ∈ 𝐷≤𝐿 , then 𝐷≤𝑣

𝐿 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑣

𝐴𝑇R(𝐶𝐿≤𝑛 )(𝑥) ≥ 𝑢R𝑎(𝑥, 𝑦)}, otherwise,

𝐷≤𝑣𝐿 (𝑥, 𝑦) = ∅;

5) if (𝑥, 𝑦) ∈ 𝐷≥𝐻 , then 𝐷≥𝑢

𝐻 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) ≥ 𝑢R𝑎(𝑥, 𝑦)}, otherwise,

𝐷≥𝑢𝐻 (𝑥, 𝑦) = ∅;

6) if (𝑥, 𝑦) ∈ 𝐷≥𝐻 , then 𝐷≥𝑣

𝐻 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑣

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) ≤ 𝑣R𝑎(𝑥, 𝑦)}, otherwise,

𝐷≥𝑣𝐻 (𝑥, 𝑦) = ∅;



7) if (𝑥, 𝑦) ∈ 𝐷≥𝐻 , then 𝐷≤𝑢

𝐻 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) ≥ 𝑢R𝑎(𝑦, 𝑥)}, otherwise,

𝐷≤𝑢𝐻 (𝑥, 𝑦) = ∅;

8) if (𝑥, 𝑦) ∈ 𝐷≥𝐻 , then 𝐷≤𝑣

𝐻 (𝑥, 𝑦) = {𝑎 ∈𝐴𝑇 : 𝑣

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) ≤ 𝑣R𝑎(𝑥, 𝑦)}, otherwise,

𝐷≤𝑣𝐻 (𝑥, 𝑦) = ∅;

𝐷≥𝑢𝐿 (𝑥, 𝑦), 𝐷≥𝑣

𝐿 (𝑥, 𝑦), 𝐷≤𝑢𝐿 (𝑥, 𝑦), 𝐷≤𝑣

𝐿 (𝑥, 𝑦),𝐷≥𝑢

𝐻 (𝑥, 𝑦), 𝐷≥𝑣𝐻 (𝑥, 𝑦), 𝐷≤𝑢

𝐻 (𝑥, 𝑦), 𝐷≤𝑣𝐻 (𝑥, 𝑦) are

referred to as the ≥𝑢–lower, ≥𝑣–lower, ≤𝑢–lower, ≤𝑣–lower, ≥𝑢–upper, ≥𝑣–upper, ≤𝑢–upper and ≤𝑣–upperapproximate discernibility sets for pair of the objects(𝑥, 𝑦) respectively, the matrixes

M≥𝑢𝐿 = {𝐷≥𝑢

𝐿 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≥𝐿 },

M≥𝑣𝐿 = {𝐷≥𝑣

𝐿 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≥𝐿 },

M≤𝑢𝐿 = {𝐷≤𝑢

𝐿 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≤𝐿 },

M≤𝑣𝐿 = {𝐷≤𝑣

𝐿 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≤𝐿 },

M≥𝑢𝐻 = {𝐷≥𝑢

𝐻 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≥𝐻},

M≥𝑣𝐻 = {𝐷≥𝑣

𝐻 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≥𝐻},

M≤𝑢𝐻 = {𝐷≤𝑢

𝐻 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≤𝐻},

M≤𝑣𝐻 = {𝐷≤𝑣

𝐻 (𝑥, 𝑦) : (𝑥, 𝑦) ∈ 𝐷≤𝐻},

are referred to as ≥𝑢–lower, ≥𝑣–lower, ≤𝑢–lower, ≤𝑣–lower, ≥𝑢–upper, ≥𝑣–upper, ≤𝑢–upper and ≤𝑣–upper ap-proximate distribution discernibility matrixes respectively.


1) 𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) for each 𝑥 ∈ 𝑈

and 𝑛 ∈ 𝑁 if and only if 𝐴 ∩ 𝐷≥𝑢𝐿 (𝑥, 𝑦) ∕= ∅ for

each (𝑥, 𝑦) ∈ 𝐷≥𝐿 ;

2) 𝑣𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) = 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) for each 𝑥 ∈ 𝑈 and

𝑛 ∈ 𝑁 if and only if 𝐴 ∩𝐷≥𝑣𝐿 (𝑥, 𝑦) ∕= ∅ for each

(𝑥, 𝑦) ∈ 𝐷≥𝐿 ;


≤𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥) for each 𝑥 ∈ 𝑈

and 𝑛 ∈ 𝑁 if and only if 𝐴 ∩ 𝐷≤𝑢𝐿 (𝑥, 𝑦) ∕= ∅ for

each (𝑥, 𝑦) ∈ 𝐷≤𝐿 ;


≤𝑛 )(𝑥) = 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) for each 𝑥 ∈ 𝑈 and

𝑛 ∈ 𝑁 if and only if 𝐴 ∩𝐷≤𝑣𝐿 (𝑥, 𝑦) ∕= ∅ for each

(𝑥, 𝑦) ∈ 𝐷≤𝐿 ;


≥𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) for each 𝑥 ∈ 𝑈

and 𝑛 ∈ 𝑁 if and only if 𝐴 ∩ 𝐷≥𝑢𝐻 (𝑥, 𝑦) ∕= ∅ for

each (𝑥, 𝑦) ∈ 𝐷≥𝐻 ;


≥𝑛 )(𝑥) = 𝑣

𝐴R(𝐶𝐿≥𝑛 )(𝑥) for each 𝑥 ∈ 𝑈 and

𝑛 ∈ 𝑁 if and only if 𝐴 ∩𝐷≥𝑣𝐻 (𝑥, 𝑦) ∕= ∅ for each

(𝑥, 𝑦) ∈ 𝐷≥𝐻 ;


≤𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≤𝑛 )(𝑥) for each 𝑥 ∈ 𝑈

and 𝑛 ∈ 𝑁 if and only if 𝐴 ∩ 𝐷≤𝑢𝐻 (𝑥, 𝑦) ∕= ∅ for

each (𝑥, 𝑦) ∈ 𝐷≤𝐻 ;


≤𝑛 )(𝑥) = 𝑣

𝐴R(𝐶𝐿≤𝑛 )(𝑥) for each 𝑥 ∈ 𝑈 and

𝑛 ∈ 𝑁 if and only if 𝐴 ∩𝐷≤𝑣𝐻 (𝑥, 𝑦) ∕= ∅ for each

(𝑥, 𝑦) ∈ 𝐷≤𝐻 .

Proof: We only prove (1), others can be provedanalogously.

If 𝑛 = 1, then 𝑢𝐴𝑇R(𝐶𝐿

≥1 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥1 )(𝑥) = 1

because 𝐶𝐿≥1 = 𝑈 . What should be considered in the

following are 𝑛 > 1.⇒: Suppose ∃(𝑥, 𝑦) ∈ 𝐷≥

𝐿 such that 𝐴∩𝐷≥𝑢𝐿 (𝑥, 𝑦) =

∅, then for each 𝑎 ∈ 𝐴, we have 𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) >

𝑣R𝑎(𝑦, 𝑥) by Definition 6. By formula (1) we have𝑣R𝐴(𝑦, 𝑥) = ∨{𝑣R𝑎(𝑦, 𝑥) : 𝑎 ∈ 𝐴}, from which wecan conclude that 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) > 𝑣R𝐴

(𝑦, 𝑥). Sinceby assumption we have 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥),

thus 𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) > 𝑣R𝐴

(𝑦, 𝑥) holds, which contradic-tive to the condition 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) ≤ 𝑣R𝐴

(𝑦, 𝑥) because𝑢𝐴R(𝐶𝐿

≥𝑛 )(𝑥) = ∧{𝑣R𝐴

(𝑦, 𝑥) : 𝑦 /∈ 𝐶𝐿≥𝑛 } (𝑛 =

2, ⋅ ⋅ ⋅ ,𝑚).⇐: Suppose that ∃𝑥 ∈ 𝑈 and 𝑛 ∈ 𝑁 where


≥𝑛 )(𝑥) ∕= 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥), then 𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚

and 𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) > 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) (by Theorem 3).

Therefore, there must be 𝑦 /∈ 𝐶𝐿≥𝑛 such that 𝑣R𝐴

(𝑦, 𝑥) <𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥), it follows that for each 𝑎 ∈ 𝐴,

𝑣R𝑎(𝑦, 𝑥) < 𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) holds, i.e. 𝐴 ∩𝐷≥𝑢

𝐿 (𝑥, 𝑦) =

∅, here (𝑥, 𝑦) ∈ 𝐷≥𝐿 . From discussion above, we can draw

the following conclusion: if for each (𝑥, 𝑦) ∈ 𝐷≥𝐿 where

𝐴∩𝐷≥𝑢𝐿 (𝑥, 𝑦) ∕= ∅, then 𝑢

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥)

for each 𝑥 ∈ 𝑈 and 𝑛 = 2, ⋅ ⋅ ⋅ ,𝑚.Theorem 8: Let I be a decision system in which 𝐴 ⊆

𝐴𝑇 , we have

1) 𝐿≥𝐴 = 𝐿≥

𝐴𝑇 ⇔ ∀(𝑥, 𝑦) ∈ 𝐷≥𝐿 such that 𝐴 ∩

(𝐷≥𝑢𝐿 (𝑥, 𝑦) ∩𝐷≥𝑣

𝐿 (𝑥, 𝑦)) ∕= ∅;2) 𝐿≤

𝐴 = 𝐿≤𝐴𝑇 ⇔ ∀(𝑥, 𝑦) ∈ 𝐷≤

𝐿 such that 𝐴 ∩(𝐷≤𝑢

𝐿 (𝑥, 𝑦) ∩𝐷≤𝑣𝐿 (𝑥, 𝑦)) ∕= ∅;

3) 𝐻≥𝐴 = 𝐻≥

𝐴𝑇 ⇔ ∀(𝑥, 𝑦) ∈ 𝐷≥𝐻 such that 𝐴 ∩

(𝐷≥𝑢𝐻 (𝑥, 𝑦) ∩𝐷≥𝑣

𝐻 (𝑥, 𝑦)) ∕= ∅;4) 𝐻≤

𝐴 = 𝐻≤𝐴𝑇 ⇔ ∀(𝑥, 𝑦) ∈ 𝐷≤

𝐻 such that 𝐴 ∩(𝐷≤𝑢

𝐻 (𝑥, 𝑦) ∩𝐷≤𝑣𝐻 (𝑥, 𝑦)) ∕= ∅.

Proof: We only prove (1), others can be provedanalogously.⇒: If 𝐿≥

𝐴 = 𝐿≥𝐴𝑇 , then ∀𝑛 ∈ 𝑁 and ∀𝑥 ∈ 𝑈 , we have


≥𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) and 𝑣

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) =

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥). By Theorem 7, we have 𝐴∩𝐷≥𝑢

𝐿 (𝑥, 𝑦) ∕= ∅and 𝐴 ∩𝐷≥𝑣

𝐿 (𝑥, 𝑦) ∕= ∅ for each (𝑥, 𝑦) ∈ 𝐷≥𝐿 , it follows

that 𝐴 ∩ (𝐷≥𝑢𝐿 (𝑥, 𝑦) ∩𝐷≥𝑣

𝐿 (𝑥, 𝑦)) ∕= ∅.⇐: If 𝐴 ∩ (𝐷≥𝑢

𝐿 (𝑥, 𝑦) ∩ 𝐷≥𝑣𝐿 (𝑥, 𝑦)) ∕= ∅ for each

(𝑥, 𝑦) ∈ 𝐷≥𝐿 , then we have 𝐴 ∩ 𝐷≥𝑢

𝐿 (𝑥, 𝑦) ∕= ∅and 𝐴 ∩ 𝐷≥𝑣

𝐿 (𝑥, 𝑦) ∕= ∅. By Theorem 7, we have𝑢𝐴𝑇R(𝐶𝐿

≥𝑛 )(𝑥) = 𝑢

𝐴R(𝐶𝐿≥𝑛 )(𝑥) and 𝑣

𝐴𝑇R(𝐶𝐿≥𝑛 )(𝑥) =

𝑣𝐴R(𝐶𝐿

≥𝑛 )(𝑥) for each 𝑛 ∈ 𝑁 and 𝑥 ∈ 𝑈 , it follows

that 𝐿≥𝐴 = 𝐿≥

𝐴𝑇 .Definition 7: Let I be a decision system, define

Δ≥𝐿 =

⋀(𝑥,𝑦)∈𝐷

≥𝐿

((⋁

𝐷≥𝑢𝐿 (𝑥, 𝑦))

⋀(⋁

𝐷≥𝑣𝐿 (𝑥, 𝑦))); (2)

Δ≤𝐿 =

⋀(𝑥,𝑦)∈𝐷

≤𝐿

((⋁

𝐷≤𝑢𝐿 (𝑥, 𝑦))

⋀(⋁

𝐷≤𝑣𝐿 (𝑥, 𝑦))); (3)



Δ≥𝐻 =

⋀(𝑥,𝑦)∈𝐷

≥𝐻

((⋁

𝐷≥𝑢𝐻 (𝑥, 𝑦))

⋀(⋁

𝐷≥𝑣𝐻 (𝑥, 𝑦))); (4)

Δ≤𝐻 =

⋀(𝑥,𝑦)∈𝐷

≤𝐻

((⋁

𝐷≤𝑢𝐻 (𝑥, 𝑦))

⋀(⋁

𝐷≤𝑣𝐻 (𝑥, 𝑦))); (5)

Δ≥𝐿 , Δ≤

𝐿 , Δ≥𝐻 and Δ≤

𝐻 are referred to as the ≥–lower, ≤–lower, ≥–upper and ≤–upper approximate discernibilityfunctions respectively.

By using Boolean reasoning techniques, we can obtainthe following Theorem 9 from Theorem 8.

Theorem 9: Let I be a decision system in which 𝐴 ⊆𝐴𝑇 , then we have

1) 𝐴 is ≥–lower approximate distribution reduct ⇔⋀𝐴 is a prime implicant of Δ≥

𝐿 ;2) 𝐴 is ≤–lower approximate distribution reduct ⇔⋀

𝐴 is a prime implicant of Δ≤𝐿 ;

3) 𝐴 is ≥–upper approximate distribution reduct ⇔⋀𝐴 is a prime implicant of Δ≥

𝐻 ;4) 𝐴 is ≤–upper approximate distribution reduct ⇔⋀

𝐴 is a prime implicant of Δ≤𝐻 .

Proof: We only prove (1), others can be provedanalogously.

“⇒”: Since 𝐴 is ≥–lower approximate distributionreduct, then 𝐴 is also a ≥–lower approximate distributionconsistent set. By Theorem 8, we have 𝐴∩ (𝐷≥𝑢

𝐿 (𝑥, 𝑦)∩𝐷≥𝑣

𝐿 (𝑥, 𝑦)) ∕= ∅,∀(𝑥, 𝑦) ∈ 𝐷≥𝐿 . We claim that for

each 𝑎 ∈ 𝐴, there must be (𝑥, 𝑦) ∈ 𝐷≥𝐿 such that

𝐴 ∩ (𝐷≥𝑢𝐿 (𝑥, 𝑦) ∩ 𝐷≥𝑣

𝐿 (𝑥, 𝑦)) = {𝑎}. In fact, if foreach pair (𝑥, 𝑦) ∈ 𝐷≥

𝐿 , there exists 𝑎 ∈ 𝐷≥𝐿 (𝑥, 𝑦) such

that 𝐶𝑎𝑟𝑑(𝐴 ∩ (𝐷≥𝑢𝐿 (𝑥, 𝑦) ∩ 𝐷≥𝑣

𝐿 (𝑥, 𝑦))) > 2 where𝑎 ∈ 𝐴 ∩ (𝐷≥𝑢

𝐿 (𝑥, 𝑦) ∩ 𝐷≥𝑣𝐿 (𝑥, 𝑦)), let 𝐴′ = 𝐴 − {𝑎},

then by Theorem 8 we can see that 𝐴′ is a ≥–lowerapproximate distribution consistent set, which contradictsthat 𝐴 is a ≥–lower approximate distribution reduct . Itfollows that

⋀𝐴 is a prime implicant of Δ≥

𝐿 .“⇐”: If

⋀𝐴 is a prime implicant of Δ≥

𝐿 , then byTheorem 8 there must be 𝐴∩(𝐷≥𝑢

𝐿 (𝑥, 𝑦)∩𝐷≥𝑣𝐿 (𝑥, 𝑦)) ∕=

∅, (∀(𝑥, 𝑦) ∈ 𝐷≥𝐿 ). Moreover, for each 𝑎 ∈ 𝐴, there exists

(𝑥, 𝑦) ∈ 𝐷≥𝐿 such that 𝐴 ∩ (𝐷≥𝑢

𝐿 (𝑥, 𝑦) ∩ 𝐷≥𝑣𝐿 (𝑥, 𝑦)) =

{𝑎}. Consequently, ∀𝐴′ where 𝐴′ ⊆ 𝐴 and 𝐴′ = 𝐴−{𝑎},𝐴′ is not the ≥–lower approximate distribution consistentset. We conclude that 𝐴 is a ≥–lower approximate distri-bution reduct.

B. Illustrative example

Following Section 3.2, compute the ≥–lower approx-imate distribution reduct, ≤–lower approximate distribu-tion reduct, ≥–upper approximate distribution reduct and≤–upper approximate distribution reduct of Table 1.

By Definition 6, we can obtain eight different types ofdistribution discernibility matrixes. Here, we only present≥𝑢–lower, ≥𝑣–lower, ≥𝑢–upper, ≥𝑣–upper approximatedistribution discernibility matrixes matrixes as Table 4,Table 5, Table 6 and Table 7 show respectively.

Therefore, by Definition 7, we obtain the following ≥–

TABLE IV.≥𝑢–LOWER APPROXIMATE DISTRIBUTION DISCERNIBILITY MATRIX

𝐷≥𝐿 (𝑥, 𝑦)𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9𝑥10

𝑥1 {𝑑, 𝑒} {𝑎, 𝑒} {𝑎} {𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥2 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥3 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥4 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥5 {𝑒} {𝑎, 𝑒} {𝑎} {𝑎} {𝑎, 𝑒}𝑥6 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥7 {𝑑, 𝑒} {𝑎, 𝑒} {𝑎} {𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥8 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥9 {𝑑, 𝑒} {𝑎, 𝑒} {𝑎} {𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥10 {𝑒} {𝑎, 𝑐, 𝑒}{𝑐} {𝑐, 𝑒} {𝑎, 𝑐, 𝑒}

TABLE V.≥𝑣–LOWER APPROXIMATE DISTRIBUTION DISCERNIBILITY MATRIX

𝐷≥𝐿 (𝑥, 𝑦)𝑥1 𝑥2 𝑥3 𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9𝑥10

𝑥1 {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥2 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥3 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥4 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥5 {𝑎, 𝑑, 𝑒} {𝑎, 𝑒} {𝑎, 𝑏} {𝑏, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥6 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥7 {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥8 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥9 {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑑, 𝑒} {𝑎, 𝑑, 𝑒}𝑥10 {𝑎, 𝑐, 𝑑, 𝑒}{𝑎, 𝑐, 𝑒}{𝑎, 𝑏, 𝑐, 𝑒} {𝑏, 𝑐, 𝑑, 𝑒} {𝑎, 𝑐, 𝑑, 𝑒}

TABLE VI.≥𝑢–UPPER APPROXIMATE DISTRIBUTION DISCERNIBILITY MATRIX

𝐷≥𝐻(𝑥, 𝑦) 𝑥1 𝑥2𝑥3𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9 𝑥10

𝑥1 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥2 {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒}{𝑎, 𝑐, 𝑑, 𝑒}𝑥3 {𝑎, 𝑑, 𝑒} {𝑎, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑐, 𝑒}𝑥4 {𝑎, 𝑑, 𝑒} {𝑎, 𝑏} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒}{𝑎, 𝑏, 𝑐, 𝑒}𝑥5 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥6 {𝑑, 𝑒} {𝑏, 𝑑, 𝑒} {𝑑, 𝑒} {𝑑, 𝑒} {𝑏, 𝑐, 𝑑, 𝑒}𝑥7 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥8 {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒}{𝑎, 𝑐, 𝑑, 𝑒}𝑥9 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥10 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇

TABLE VII.≥𝑣–UPPER APPROXIMATE DISTRIBUTION DISCERNIBILITY MATRIX

𝐷≥𝐻(𝑥, 𝑦) 𝑥1 𝑥2𝑥3𝑥4 𝑥5 𝑥6 𝑥7 𝑥8 𝑥9 𝑥10

𝑥1 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥2 {𝑑, 𝑒} {𝑒} {𝑑, 𝑒} {𝑑, 𝑒} {𝑒}𝑥3 {𝑎, 𝑒} {𝑎, 𝑒} {𝑎, 𝑒} {𝑎, 𝑒} {𝑎, 𝑐, 𝑒}𝑥4 {𝑎} {𝑎} {𝑎} {𝑎} {𝑎, 𝑐}𝑥5 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥6 {𝑎, 𝑑, 𝑒} {𝑎} {𝑎, 𝑏, 𝑑, 𝑒} {𝑎, 𝑐, 𝑑, 𝑒}{𝑎, 𝑐, 𝑒}𝑥7 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥8 {𝑎, 𝑑, 𝑒} {𝑎} {𝑎, 𝑑, 𝑒} {𝑎, 𝑑, 𝑒} {𝑎, 𝑐, 𝑒}𝑥9 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇𝑥10 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇 𝐴𝑇



lower, ≥–upper approximate discernibility functions:

Δ≥𝐿 =

⋀(𝑥,𝑦)∈𝐷

≥𝐿

((⋁

𝐷≥𝑢𝐿 (𝑥, 𝑦))

⋀(⋁

𝐷≥𝑣𝐿 (𝑥, 𝑦)));

= 𝑎⋀

𝑐⋀

𝑒

Δ≥𝐻 =

⋀(𝑥,𝑦)∈𝐷

≥𝐻

((⋁

𝐷≥𝑢𝐻 (𝑥, 𝑦))

⋀(⋁

𝐷≥𝑣𝐻 (𝑥, 𝑦)))

= 𝑎⋀

𝑒

By the above results and Theorem 9, we know that{𝑎, 𝑐, 𝑒} is the ≥–lower approximate distribution reductof Table 1, {𝑎, 𝑒} is the ≥–upper approximate distributionreduct of Table 1. In other words, to preserve the intu-itionistic fuzzy dominance–based lower approximationsof all the upward unions of the decision classes, attributes𝑏 and 𝑑 can be deleted; to preserve the intuitionisticfuzzy dominance–based upper approximations of all theupward unions of the decision classes, attributes 𝑏, 𝑐, 𝑑are redundant.

Similar to the above progress, it is not difficult toobtain that {𝑎, 𝑒} is the ≤–lower approximate distributionreduct of Table 1, {𝑎, 𝑐, 𝑒} is the ≤–upper approximatedistribution reduct of Table 1. Such results demonstratethe correctness of Theorem 5.

V. CONCLUSIONS

In this paper, we have developed a general frameworkfor the generalization of dominance–based rough set. Inour approach, the concept of intuitionistic fuzzy set iscombined with the DRSA and then the intuitionistic fuzzydominance–based rough set is defined. We also introducedthe concept of approximate distribution reducts into intu-itionistic fuzzy dominance–based rough set model, fourtypes of approximate distribution reducts are presented,the practical approaches to compute these reducts are alsodiscussed. Different from the previous DRSA, we use anintuitionistic fuzzy dominance relation instead of the crispor fuzzy dominance relation to defined dominance–basedrough set model.

Furthermore, a lot of experiment analysis are alsoneeded to conduct in the future for practical applicationsof our intuitionistic fuzzy dominance–based rough setapproach.

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Yanqin Zhang received her MS degrees in computer scienceand education from the Xuzhou Normal University, Xuzhou,in 2002, she received her BS degree in computer applicationsfrom the China University of Mining and Technology, Xuzhou,in 2007.

She is a lecturer at the Xuzhou Institute of Technology. Herresearch interests include rough set theory.

Xibei Yang received his MS degree in computer applicationsfrom the Jiangsu University of Science and Technology, Zhen-jiang, in 2006, he received his Ph.D. degree in computer appli-cations from the Nanjing University of Science and Technology,Nanjing, in 2010. In 2009, he was a visiting scholar at the SanJose State University, San Jose, USA.

He is currently lecturer at the Jiangsu University of Scienceand Technology. Presently, he is also a postdoctoral researcherat the Nanjing University of Science and Technology. He haspublished more than thirty papers in international journals andinternational conferences. His research interests include roughset theory and granular computing.



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