Variable precision multigranulation decision-theoretic fuzzy rough sets

Knowledge-Based Systems 91 (2016) 93–101

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Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys

Variable precision multigranulation decision-theoretic fuzzy rough sets Tao Feng a,∗, Ju-Sheng Mi b a b

School of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, China College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China

a r t i c l e

i n f o

Article history: Received 2 February 2015 Revised 4 October 2015 Accepted 5 October 2015 Available online 23 October 2015 Keywords: Multigranulation fuzzy approximation space Variable precision rough sets Lower and upper approximations Three-way decisions

a b s t r a c t This paper studies variable precision multigranulation fuzzy decision-theoretic rough sets in an information system. We firstly review definitions and properties of multigranulation fuzzy rough sets. A novel membership degree based on single granulation rough sets is proposed. Then two operators based on this membership degree are defined. By employing these operators, two types of variable precision multigranulation fuzzy rough sets in an information system are proposed. Finally, inspired by three-way decisions, we propose Type-1 variable precision multigranulation decision-theoretic fuzzy rough sets.

1. Introduction Human brain is a very great machine to capture useful information quickly and accurately, so it is an interesting subject to simulate the thinking way of human brain. There are many attempts. One of the methods is Pawlak’s rough set theory [21], which deals with insufficient and incomplete data. It is one of the characteristics of rough set theory that uncertain concepts and phenomena are approximated by the existing knowledge [22]. From both theoretical and practical viewpoints, Pawlak’s rough approximation is very stringent, and may limit application scopes. With more than thirty years’ development, many authors have generalized Pawlak’s rough set theory by using nonequivalence binary relations [1,13,32,34,43,45,49], and developed these rough set models based on reasoning and knowledge acquisition in incomplete information tables. Moreover, many authors generalized rough approximations to fuzzy environments, for example, rough fuzzy sets and fuzzy rough sets [3,7,23,27,36,52,53,55]. These models have been employed to handle fuzzy and quantitative data. Previously, many scholars used granular computing to analyze information sources. The method of granular computing is proposed by Zadeh [54], which is based on a single granulation structure. Recently, rough set theory becomes a popular mathematical framework for granular computing. In this theory, concepts are expressed by upper and lower approximations induced by a single granulation structure [8,9,24,44]. Thus, it is called single granulation rough set model. Since we can catch an element from different aspects [26] or different levels [18,37], and we always meet different useful information

∗

Corresponding author. Tel.: +8613785119408. E-mail addresses: [email protected] (T. Feng), [email protected] (J.-S. Mi).

http://dx.doi.org/10.1016/j.knosys.2015.10.007 0950-7051/© 2015 Elsevier B.V. All rights reserved.

© 2015 Elsevier B.V. All rights reserved.

sources for the same element, so we need to give an overall consideration for these information sources. Thus, the theory of granular computing should be generalized to suit multiple information sources. In order to meet actual needs, Qian et al. [27] first proposed multigranulation rough sets (MGRS, in short). It has a more widely application scope, for example, decision making, feature selection, and so on [12,28–30,42]. Since for different requirements, a concept can be described by different multiple binary relations, many extensions of MGRSs have been proposed. For example, Qian and Liang et al. generalized classical multigranulation rough sets to neighborhood-based ones [11] and covering ones [7]. The neighborhood multigranulation rough sets are useful for hybrid data sets. Dou et al. [2] investigated variable precision multigranulation rough sets. Yao et al. discussed rough set models in multigranulation spaces [50]. Qian et al. [31] discussed multigranulation decision-theoretic rough sets. The topological structures of multigranulation rough sets were discussed by She et al. [35]. Li et al. [6] made a detailed comparison between multigranulation rough sets and concept lattices via rule acquisition. Furthermore, multigranulation rough sets based on fuzzy binary relations [41] and multigranulation fuzzy rough sets based on classical tolerant relations [38] were defined. Liu et al. [14,15] proposed fuzzy covering multigranulation rough sets. Liang et al. [10] proposed an efficient algorithm for feature selection in large-scale and multiple granulation data sets. Since there always exist some noises, uncertainty and fuzziness in an information system, it is difficult for us to deal with all the problems only using the above MGRS theories. It is necessary for us to study the fuzzy MGRSs. There always exist a few errors in approximations, but some uncertainty in classification process is admitted in making decision. To handle uncertain and imprecise information, in 1993, Ziarko proposed variable precision rough set model [58], which is directly

94

T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101

derived from Pawlak’s rough set model without any additional assumptions. It may make a better utilization of data being analyzed, and a lower likelihood of incorrect decision. Moreover, this model is also useful for eliminating noise attributes. In order to weaken the errors generated by incompleteness, uncertainty and noises, we try to construct variable precision rough sets based on multigranulation fuzzy rough sets. The essential ideas of three-way decisions are commonly used in different fields and disciplines by different names and notations. It is more suitable for decision making of human cognition [19]. Yao first proposed a unified framework of the theory of three-way decisions in 2010 [46]. The theory of three-way decisions is constructed based on the notions of acceptance, rejection and noncommitment, adding a noncommitment notion with respect to two-way decisions. It is also used to interpret three regions of Pawlak’s rough sets [47,48]. Corresponding to the three regions, one may construct rules for acceptance from the positive region, construct rules for rejection from the negative region and construct rules for noncommitment from the boundary region. In recent years, three-way decision theory develops in a wide range [49] and is more banausic in many aspects, for example, email spam filtering [57] and social networks [25]. Three-way decision rough sets in interval-value and fuzzy circumstances are studied [16,17,56]. Three-way decision model based on the evidence theory is also studied by Xue [40]. Yu et al. studied a tree-based incremental overlapping clustering method using the three-way decision theory [51]. In this paper, we would like to combine the method of threeway decisions and variable precision rough sets based on multigranulation fuzzy approximation spaces, which can help us to make a reasonable and suitable decision for every element. The rest of this paper is organized as follows: Section 2 reviews definitions and properties of optimistic multigranulation fuzzy approximations and pessimistic multigranulation fuzzy approximations. Section 3 proposes a novel membership degree, and two operators based on this membership degree in multigranulation fuzzy approximation space are also defined. Then, we define two types of variable precision multigranulation fuzzy rough sets in an information system. Section 4 studies decision-theory of the Type-1 variable precision fuzzy rough sets. We then conclude the paper with a summary and give an outlook for further researches in Section 5. 2. Basic concepts In this section, we first review definitions and propositions of fuzzy rough sets [20] and multigranulation fuzzy rough sets [39]. Let S = (U, AT ) be an information system, in which U is a nonempty and finite universe of discourse, and AT is a non-empty finite set of attributes. The set of all fuzzy sets defined on U is denoted by F(U). Every attribute is a fuzzy set, that is, ∀x ∈ U, a ∈ AT, the value of x on attribute a is a(x) ∈ [0, 1]. R: U × U → [0, 1] is a fuzzy tolerance relation [20] satisfying (1) reflexivity: R(x, x) = 1, ∀x ∈ U, (2) symmetry: R(x, y) = R(y, x), ∀x, y ∈ U. Given A⊆AT, RA is a fuzzy tolerance relation, ∀x ∈ U, RA (x) is a fuzzy set such that RA (x)(y) = RA (x, y), ∀y ∈ U. Using fuzzy tolerance relation RA , ∀X ∈ F(U), the lower and upper approximations of X can be computed by approximation operators RA

Multiple granulation structures can be obtained by different fuzzy binary relations. Combining the granulation structures, multigranulation fuzzy rough sets can be defined.

2.1. Optimistic multigranulation fuzzy rough sets Associating fuzzy tolerance rough sets with the theory of granular computing, Xu et al. defined optimistic and pessimistic multigranulation fuzzy rough set models on fuzzy tolerance relations, and discussed their properties in 2011 [39]. These models are reasonable generalizations of crisp multigranulation rough set models. In this subsection, we review the optimistic multigranulation fuzzy rough set model. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai , then m O  O ∀X ∈ F(U), m i=1 RA (X ) and i=1 RA (X ) are the optimistic multigrani

m 



((1 − RA (x, y)) ∨ X (y)),

(1)

(RA (x, y) ∧ X (y)).

(2)

y∈U

RA (X )(x) =



ROAi (X )(x) =

i=1

m 

m 



ROA i

(X )(x) = 1 −



m 

The partial relation of two fuzzy tolerance relations is defined as: for two fuzzy tolerance relations R1 and R2 , R1 R2 if and only if R1 (x)⊆R2 (x) for each x ∈ U. In this case, R2 is coarser than R1 .

((1 − RAi (x, y)) ∨ X (y)) ,

ROAi

(∼X ))(x) =

i=1

(3)

m 





 (RAi (x, y) ∧ X (y)) ,

y∈U

i=1

(4) where ∼ X (x) = 1 − X (x), ∀x ∈ U. If

m i=1

RO (X ) = A i

m i=1

RO (X ), then A i

X is an optimistic multigranulation fuzzy definable set. Otherwise, X is an optimistic multigranulation fuzzy rough set. The optimistic multigranulation fuzzy boundary region of X is O BN m i=1

R Ai

(X ) =

m 



ROA (X ) ∩ i



m  ∼ ROAi (X ) .

i=1

(5)

i=1

By the definitions of optimistic multigranulation fuzzy lower and upper approximations, we have the following properties of optimistic multigranulation fuzzy rough sets based on a fuzzy tolerance approximation space: ∀X, Y ∈ F(U), 1. 2. 3. 4. 5. 6.

7.

8.

m

 O RO (X ) ⊆ X ⊆ m i=1 RAi (X ); Ai m O m O m O m O i=1 RAi (∅) = i=1 RAi (∅) = ∅, i=1 RAi (U ) = i=1 RAi (U ) = U; m O  m O m O O X ⊆ Y ⇒ i=1 RA (X ) ⊆ i=1 RA (Y ), i=1 RA (X ) ⊆ m i=1 RAi (Y ); i i i m m m O m O i=1 RAi (X ) = i=1 RAi (X ), i=1 RAi (X ) = i=1 RAi (X ); m O m O m O m O i=1 RAi (∼X ) = ∼ ( i=1 RAi (X )), i=1 RAi (∼X ) = ∼ ( i=1 RAi (X )); m O m i=1 RAi (X ∩ Y ) = i=1 (RAi (X ) ∩ RAi (Y )), m O m i=1 RA (X ∪ Y ) = i=1 (RA (X ) ∪ RAi (Y )); m O i m O m Oi R ( X ∩ Y ) ⊆ i=1 Ai i=1 RAi (X ) ∩ i=1 RAi (Y ), m O m O m O i=1 RA (X ∪ Y ) ⊇ i=1 RA (X ) ∪ i=1 RA (Y ); m Oi m Oi m Oi i=1 RAi (X ∪ Y ) ⊇ i=1 RAi (X ) ∪ i=1 RAi (Y ), m O m O m O i=1 RA (X ∩ Y ) ⊆ i=1 RA (X ) ∩ i=1 RA (Y ). i=1

i

y∈U



y∈U

i=1

i=1

(X) and RA (X ). They are defined as follows: ∀x ∈ U,

RA (X )(x) =

i

ulation fuzzy lower and upper approximations, respectively. They are defined as follows: ∀x ∈ U,

i

i

m O If RA1 RA2 · · · RAm , then we have (1) i=1 RAi (X ))(x) = m O RA1 (X )(x); (2) i=1 RA (X ))(x) = RA1 (X )(x), ∀X ∈ F(U). Thus the opi

timistic multigranulation fuzzy lower and upper approximations are dependent on the thinnest fuzzy relation.

T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101

2.2. Pessimistic multigranulation fuzzy rough sets In this subsection, we review the pessimistic multigranulation fuzzy lower and upper approximations. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai , then m P  P ∀X ∈ F(U), m i=1 RA (X ) and i=1 RA (X ) are the pessimistic multigrani

i

ulation fuzzy lower and upper approximations of X, respectively. They are defined as follows: ∀x ∈ U, m 

RPAi (X )(x) =

i=1

m 

m 



m 

i

i=1

((1 − RAi (x, y)) ∨ X (y)) ,

(6)

y∈U

i=1

RPA (X )(x) = 1 −





RPAi (∼X )(x) =

i=1

m 



i=1



 (RAi (x, y) ∧ X (y)) .

y∈U

(7) If

m

P i=1 RAi (X )

=

m

P i=1 RAi (X ),

then X is a pessimistic multigran-

ulation fuzzy definable set. The pessimistic multigranulation fuzzy boundary region of X is P BN m i=1

R Ai

(X ) =

m 

RPA (X )



i

m 

(∼

i=1

RPAi (X )).

(8)

i=1

The properties of pessimistic multigranulation fuzzy lower and upper approximations are similar to properties 1–5 of optimistic multigranulation fuzzy lower and upper approximations listed in Section 2.1. Proposition 2.1 ([39]). Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . ∀X, Y ∈ F(U), we have the following properties 1. 2. 3. 4. 5.

m

m

RPA (X ) ⊆ X ⊆ i=1 RPA (X ); i i m P m P m P m P R i=1 Ai (∅) = i=1 RAi (∅) = ∅, i=1 RAi (U ) = i=1 RAi (U ) = U; m P m P  m P P X ⊆ Y ⇒ i=1 RA (X ) ⊆ i=1 RA (Y ), i=1 RA (X ) ⊆ m i=1 RAi (Y ); i i i m P m P m m i=1 RAi (X ) = i=1 RAi (X ), i=1 RAi (X ) = i=1 RAi (X ); m P m P m P m P i=1 RA (∼X ) = ∼ ( i=1 RA (X )), i=1 RA (∼X ) = ∼ ( i=1 RA (X )). i=1

i

i

i

i

3. Variable precision multigranulation fuzzy rough sets

define a membership degree of an element in a fuzzy subset responding to the inclusion measure. Suppose S = (U, AT ) is a fuzzy information system, in which U is a non-empty and finite universe of discourse, A⊆AT. ∀X ∈ F(U), x ∈ U, the membership degree of x in X based on tolerance relation RA is R denoted by X A (x). It is defined by

XRA (x) = 1 − 

where |X | =

|RA (x) ∩ (∼X )| , |RA (x)|

x∈U

XRA (x)

(9)

X (x) is the cardinality of a fuzzy set X.

∈ [0, 1] describes the inclusion measure between fuzzy neighborhood of x and fuzzy set X, and can be regarded as the rough membership degree of x with respect to X. When R is a classical equiv|[x]R ∩X |

R

alence relation and X is a crisp set, X A (x) = |[x]A | is the memberRA ship degree of x with respect to X, which is Pawlak rough set’s case. In single granulation rough set models, variable precision rough sets can be defined by one membership degree. It should be noticed that more than one granulation structures are used in multigranulation fuzzy rough sets. Thus, we would like to combine these membership degrees to one novel membership degree in order to define variable precision multigranulation fuzzy rough sets, which is a generalization of the multigranulation fuzzy rough set model. Therefore, we give the following definition. Definition 3.1. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect minimal membership degrees to Ai . ∀X ∈ F(U), x ∈ U, the maximal and  

of x with respect to X are denoted by ωX spectively, and defined by m 

ωX

i=1

m 

R Ai

m i=1

RA

i

(x) and ψX

m i=1 RAi

R

Ai (x) = maxm i=1 X (x);

R Ai

(x) re-

(10)

R

Ai ψXi=1 (x) = minm i=1 X (x).

(11)

Using the above definition, we can obtain some useful properties of the maximal and minimal membership degrees as follows. Proposition 3.1. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . Then, we have m

1. ωX

i=1

RA

i

(x) = 1 ⇔

m

2. 0 < ψX

i=1 RAi

m

In decision making, some uncertainty in classification process is admitted. To handle uncertain and imprecise information, Ziarko in 1993 proposed the variable precision rough set model [58]. This method is directly derived from Pawlak’s rough set model without any additional assumptions. It may make a better utilization of data being analyzed, and a lower likelihood of incorrect decision. Moreover, this model is also useful for eliminating noise attributes. Therefore, we put variable precision rough sets and multigranulation fuzzy rough sets together to obtain a novel rough set model. In practice, the hypotheses derived from the standard set inclusion criterion may be too restrictive in some situations characterized by strong trends but the absence of total inclusion. In decision making, any partially incorrect classification rule provides valued trend information if the majority of available data to which such a rule applied can be correctly classified. By replacing the inclusion relation with a majority inclusion relation in Pawlak’s rough sets, variable precision rough sets allow for a controlled degree of misclassification in its formalism which leads to more general notions of set approximations [58]. There is a close relationship between the degree of the partially incorrect classification and the inclusion measure. Thus, we

95

3. ψX

i=1 RAi

m

4. 0 < ωX

i=1

(x) ≤ 1 ⇔

(x) = 1 ⇔

i=1 RAi

m

RO (X )(x) = 1; A i

m i=1

m

RO (X )(x) > 0; A i

P i=1 RA (X )(x) = 1; i

(x) ≤ 1 ⇔

m i=1

RPA (X )(x) > 0. i

RA

Proof: 1. By the definition of X i (x), we have m 

ωX

i=1

R Ai

(x) = 1

R Ai

⇔ X

(x) = 1, ∃RAi

|RA (x)∩(∼X )| ⇔ 1 − i|R (x)| = 1, ∃RAi Ai

⇔ RAi (x) ∩ (∼X ) = ∅, ∃RAi  ⇔ y∈U ((1 − RAi (x, y)) ∨ X (y)) = 1, ∃RAi  O ⇔ m i=1 RAi (X )(x) = 1. Similarly, we can prove 2, 3 and 4. By the above properties, we know that the value of x belonging to the optimistic multigranulation fuzzy lower approximation of X is 1  m i=1 RA

i if and only if ωX (x) = 1. The maximal membership degree and the rate of classification errors are in close relations, that is, 1 subtracting the maximal membership degree is the rate of classification

96

T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101

R A3 = R { a

Table 1 A fuzzy information system. a1 x1 x2 x3 x4 x5 x6 x7 x8

a2

0.7 1 0.8 1 0.7 0.9 1 0.2

1 ,a2 ,a3 }

⎛

a3

1 0.9 0.7 0.7 1 1 0.7 0.1

⎜ 0.44 ⎜ ⎜ 0.11 ⎜ ⎜ ⎜0.625 =⎜ ⎜ 0.75 ⎜ ⎜ ⎜ 0.75 ⎜ ⎝ 0.25

a4

0.8 1 0.6 1 1 0.6 0.2 1

1

0.4 0.2 1 0.3 0.2 1 1 1

0.44

0.11

0.625

0.75

0.75

0.25

1

0.5

0.78

0.44

0.44

0

0.5

1

0.44

0.11

0.11

0.44

0 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0.25⎟

0.78

0.44

1

0.625

0.5

0

0

0.44

0.11

0.625

1

0.5

0

0

0.44

0.11

0.5

0.5

1

0.5

0

0

0.44

0

0

0.5

1

0

0

0.25

0

0

0

0

1

0.44

0.11

0.625

0.75

0.25

0.25

0

⎞

RA4 = RAT

errors. Thus, we can use the maximal membership degree to define variable precision rough sets in multigranulation fuzzy cases. Similarly, the minimal membership degree is also used to construct another type of variable precision rough sets in multigranulation fuzzy cases. Since the threshold value α denotes the membership degree in variable precision rough sets, the range of α must satisfy 0.5 < α ≤ 1. Definition 3.2. Suppose S = (U, AT ) is a fuzzy information system, in which U is a non-empty and finite universe of discourse, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . ∀X ∈ F(U), α ∈ (0.5, 1], the Type-1 variable precision multigranulation fuzzy lower and upper approximations are denoted by ωα  m ωα (X ) and m (X ), respectively, and defined by i=1 RAi i=1 RAi m 

RAi

i=1 m 



ωα

m 

(X ) = x : ωX

RAi

⎜ 0.44 ⎜ ⎜ 0.11 ⎜ ⎜ ⎜0.625 =⎜ ⎜ 0.75 ⎜ ⎜ ⎜ 0.25 ⎜ ⎝ 0.25

0

0.78

0.44

0

0

1

0.125

0

0.11

0.44

0.125

1

0.625

0.125

0

0

0

0.625

1

0

0

0

0

0.11

0.125

0

1

0.5

0

0

0.44

0

0

0.5

1

0

0

0.25

0

0

0

0

1

0.7 x1

+

R

1 x2

+

0.8 x3

0.9 x4

+

1 x5

+

0 x6

+

1 x7

+

0 x8 ,

then

0.3 + 0 + 0.2 + 0.1 + 0 + 0.75 + 0 + 0.375 1 + 0.625 + 0.875 + 0.625 + 1 + 0.75 + 0.625 + 0.375

= 0.7. RA

(13)

+

⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

|RA1 (x1 ) ∩ (∼X )| |RA1 (x1 )|

Similarly, we have X

(x) > 1 − α .

0

⎞

0.25⎟

0.78

 X A1 ( x 1 ) = 1 −

(12)

i=1

0

0.44

0

If X =

1 0

= 1−

(X ) = x : ωXi=1 (x) ≥ α ,

m ωα  i=1

1

R Ai

R Ai

⎛

1.26 3.925

RA

= 0.68, X

4

2

(x1 ) = 1 −

(x1 ) = 1 −

1.26 3.425

1.26 4.52

RA

= 0.72, X

3

(x1 ) = 1 −

= 0.63. If {RA1 , RA2 , RA3 , RA4 } is m

i=1 RA

i (x1 ) = 0.72, the set of fuzzy tolerance relations, then, ωX m R  ωα i=1 Ai m ψX (x1 ) = 0.63. If α = 0.75, then x1 ∈/ i=1 RAi (X ).

The Type-1 boundary region of X is defined as

BNωα m 

i=1

R Ai

(X ) =

m 

ωα

(X ) −

RAi

i=1

m 

ωα

(X ).

RAi

Therefore, we have

(14)

m

i=1

ωX i=1

1

0.375

0.625

0.875

0.625

1

0.75

0.625

1

0.75

1

0.625

0.875

1

0.75

1

0.75

0.875

0.875

0.75

1

0.75

1

0.625

0.875

1

0.625

0.875

0.625

1

0.75

0.625

0.875

0.875

0.875

0.75

1

0.875

1

0.75

1

0.625

0.875

1

0

0.25

0

0.375

0.125

0

0.375

1

RA2 = R{a

⎛

1 ,a2 }

1

⎜ 0.44 ⎜ ⎜ 0.11 ⎜ ⎜ ⎜0.625 =⎜ ⎜ 1 ⎜ ⎜ ⎜ 0.75 ⎜ ⎝0.625 0

0.44

0.11

0.625

1

0.75

0.625

1

0.67

0.78

0.44

0.44

0.78

0.67

1

0.44

0.11

0.11

0.44

0.78

0.44

1

0.625

0.67

1

0.44

0.11

0.625

1

0.75

0.625

0.44

0.11

0.67

0.75

1

0.67

0.78

0.44

1

0.625

0.67

1

0

0.25

0

0

0

0.33

0

⎞

⎟ ⎟ ⎟ 0.25⎟ ⎟ 0 ⎟ ⎟, 0 ⎟ ⎟ ⎟ 0 ⎟ ⎟ 0.33⎠ 0

1

ωX

i=1 RAi

ωX

i=1 RAi

ωX

i=1 RAi

ωX

i=1 RAi

m m m

⎞

⎟ ⎟ ⎟ 0.75 ⎟ ⎟ 0 ⎟ ⎟, 0.375⎟ ⎟ ⎟ 0.125⎟ ⎟ 0 ⎠ 0

ωX

i=1 RAi

m

|at (x )−at (x )|

⎜0.625 ⎜ ⎜0.875 ⎜ ⎜ ⎜0.625 =⎜ ⎜ 1 ⎜ ⎜ ⎜ 0.75 ⎜ ⎝0.625

ωX

i=1 RAi

m

i j binary relation Rat (xi , x j ) = 1 − max{|a (x )−a , ∀xi , xj ∈ U, t l t (xk )|:xl ,xk ∈U } RA (xi , x j ) = at ∈A Rat (xi , x j ), A⊆AT, then we have

⎛

m

i

m

Example 3.1. Let U = {x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 }, AT = {a1 , a2 , a3 , a4 }, and S = (U, AT ) is a fuzzy information system, described as Table 1. Suppose ∀at ∈ AT, max{|at (xi ) − at (x j )| : xi , x j ∈ U } = 0, and fuzzy

RA1 = Ra1

RA

(x2 ) = 0.85, ψX

i=1 RAi

(x3 ) = 0.75, ψX

i=1 RAi

m

m

(x4 ) = 0.8, ψX

i=1 RAi

m

(x3 ) = 0.62;

(x4 ) = 0.71;

(x5 ) = 0.86, ψX

i=1 RAi

m

(x2 ) = 0.71;

RA

(x5 ) = 0.72;

(x6 ) = 0.72, ψX

i=1

i

(x6 ) = 0.26;

(x7 ) = 0.75, ψX

i=1 RAi

(x7 ) = 0.57;

(x8 ) = 0.23, ψX

i=1 RAi

m m

(x8 ) = 0.04.

 ω0.75 By Definition 3.2, when α = 0.75, m (X ) = {x2 , x3 , x4 , i=1 RAi ω0.75 m ω0.75 (X ) = {x1 , x2 , x3 , x4 , x5 , x6 , x7 }, and BN x5 , x7 }, m i=1 RAi R i=1

(X ) = {x1 , x6 }.

Ai

Proposition 3.2. Suppose S = (U, AT ) is a fuzzy information system, RAT is the single fuzzy tolerance relation. α ∈ (0.5, 1], ∀X ∈ F(U), we have m 

ωα

RAi

i=1

    |R (x) ∩ (∼X )| (X ) = x : ωXRAT (x) ≥ α = x : 1 − AT ≥α , |RAT (x)| (15)

m 

ωα

RAi

  (X ) = x : ωXRAT (x) > 1 − α

i=1

 =

x:1−

 |RAT (x) ∩ (∼X )| >1−α . |RAT (x)|

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T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101

If X⊆U, then m 

|RAT (x) ∩ X | ≥α , |RAT (x)|

(X ) = x :

RAi

i=1 m 

ωα

RAi

i=1

m 





ωα

(17)

Theorem 3.3. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . α ∈ (0.5, 1], the Type-1 variable precision multigranulation fuzzy lower and upper approximations satisfy the following properties: m i=1

RAi

ωα

m

(X ) = ∼

m

RAi

i=1

ωα

m

(∼X )

ωX i=1

iff

RA

i

(x) ≥ α ⇔

i=1 RA

2.

3. 4.

5. 6.

i 1 − ω∼X (x) ≥ α , ∀x ∈ U, ∀X ∈ F(U); ωα m  m ωα ωα R ( ∅ )= m (∅) = ∅; (U ) = i=1 Ai i=1 RAi i=1 RAi ωα m (U ) = U; i=1 RAi ωα m  ωα (X ) ⊆ m (X ), ∀X ∈ F(U); i=1 RAi i=1 RAi ωα m m m ωα ωα (X ) ⊆ X ⊆ Y ⇒ i=1 RAi (X ) ⊆ i=1 RAi (Y ), i=1 RAi ωα m (Y ), i=1 RAi ∀ X, Y ∈ F(U);   m ωα ωα ωα (X ∩ Y ) ⊆ m (X ) ∩ m (Y ), ∀X, Y ∈ i=1 RAi i=1 RAi i=1 RAi

F(U); m i=1

F(U).

RAi

ωα

(X ∪ Y ) ⊇

m i=1

RAi

ωα

(X ) ∪

m i=1

RAi

ωα

(Y ), ∀X, Y ∈

R Ai

x : maxm i=1 X

(x) ≥ α





x : maxm i=1 {1 −

=

m 

ωα

(Y ).

RAi

i=1

Similarly, it is not difficult to prove that

m

m i=1

ωα

RAi

ωα

(X ) ⊆

(Y ). i=1 RAi The essential differences in comparison to the properties listed in [39] is due to the absence of the maximal membership degree in multigranulation fuzzy rough sets. Similarly, we can propose another kind of variable precision multigranulation fuzzy rough sets using the minimal membership degree and give the properties of variable precision multigranulation fuzzy rough sets in this case.

Definition 3.3. Suppose S = (U, AT ) is a fuzzy information system, in which U is a non-empty and finite universe of discourse, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . ∀X ∈ F(U), α ∈ (0.5, 1], the Type-2 variable precision multigranulation fuzzy lower and upper approximations are denoted by ψα  m ψα (X ) and m (X ), respectively, and defined by i=1 RAi i=1 RAi m 



ψα

m 

(X ) = x : ψX

i=1

RAi

R Ai

(x) ≥ α ,

(19)

i=1

m 



ψα

m 

R Ai

(X ) = x : ψXi=1 (x) > 1 − α .

RAi

(20)

The Type-2 boundary region of X is defined as



m 

R Ai

i=1

= =



R Ai

x : maxm i=1 ∅



x:



ωα

maxm i=1 m 

(x) > 1 − α



=

= ωα



R Ai

x : maxm i=1 U



x:

maxm i=1 m

RAi (X ) = {x : ωX  ωα α} = m (X ). i=1 RAi i=1

4. ∀X, Y ∈ F(U),

i=1

(x) ≥ α

m 

ψα

RAi

(X ).

(21)

i=1



RA

(X ) = {x1 , x3 , x7 }. i

Theorem 3.4. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . α ∈ (0.5, 1], the Type-2 variable precision multigranulation fuzzy lower and upper approximations satisfy the following properties: m i=1

RAi

ψα

m

(X ) =∼

m i=1

RAi

ψα

(∼X )

m

iff

ψX

i=1

RA

i

(x) ≥ α ⇔

i=1 RA



m

i

(X ) −

i=1

i=1

1.

|R (x) ∩ ∅| } ≥ α = U. { 1 − Ai |RAi (x)|

RA

ψα

RAi

ψ 0.7

(U ) = x : ωUi=1 (x) ≥ α

i=1

m 

x4 , x5 , x7 }, and BNm

 |RAi (x) ∩ U | {1 − } > 1 − α = ∅. |RAi (x)|

R Ai

R Ai

(X ) =

Example 3.2. (Following Example 3.1) By Definition 3.3, take ψ 0.7   ψ 0.7 α = 0.7, m (X ) = {x2 , x4 , x5 }, m (X ) = {x1 , x2 , x3 , i=1 RAi i=1 RAi

(∅) = x : ω∅i=1 (x) > 1 − α

RAi

m

(x) ≥ α

 |RAi (x) ∩ (∼X )| }≥α |RAi (x)|   |RAi (x) ∩ (∼Y )| } ≤ x : maxm { 1 − ≥ α i=1 |RAi (x)|

ψ

i=1

3.



=

BN m

ωα

RAi

R Ai

i=1

2. We have

m 

(X ) = x : ωX

RAi

i=1

Proof: 1, 5 and 6 are obvious, now we prove the others.

m 

m  i=1

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By Proposition 3.2, we know that the Type-1 variable precision multigranulation fuzzy lower and upper approximations are generalizations of Ziarko’s variable precision lower and upper approximations, respectively. In the following, we will demonstrate some fundamental properties of variable precision multigranulation fuzzy approximations. With a few exceptions, these properties are identical to the properties of multigranulation fuzzy approximations given in [39].

1.



ωα

=

  |R (x) ∩ X | (X ) = x : AT >1−α . |RAT (x)|

97

(x) > 1 − α} ⊇ {x : ωX i=1

RA

i

(x) ≥

i 1 − ψ∼X (x) ≥ α , ∀x ∈ U, ∀X ∈ F(U); ψα m  ψα 2. (∅) = m (∅) = ∅; i=1 RAi i=1 RAi ψα m (U ) = U; i=1 RAi ψα  m ψα (X ) ⊆ m (X ), ∀X ∈ F(U); 3. i=1 RAi i=1 RAi m m ψα ψα 4. X ⊆ Y ⇒ i=1 RAi (X ) ⊆ i=1 RAi (Y ), ψα m (Y ), i=1 RAi ∀X, Y ∈ F(U);

m i=1

m i=1

RAi

RAi

ψα

ψα

(U ) =

(X ) ⊆

98

5. 6.

T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101

m i=1

F(U); m i=1

F(U).

RAi RAi

ψα ψα

(X ∩ Y ) ⊆ (X ∪ Y ) ⊇

m i=1

m i=1

RAi RAi

ψα ψα

(X ) ∩ (X ) ∪

m i=1

m i=1

RAi RAi

ψα ψα

(Y ), ∀X, Y ∈ (Y ), ∀X, Y ∈

Proof: Similarly to Theorem 3.3, it is not difficult to prove these properties. In the following, we give some relationships of the two kinds of variable precision multigranulation fuzzy rough sets. Theorem 3.5. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . ∀X ∈ F(U), α , α 1 , α 2 ∈ (0.5, 1], then we have the following properties: ωα  m ψα (X ) =∼ m (∼X ), ∀X⊆U; 1. i=1 RAi i=1 RAi ψα  m ωα 2. (X ) =∼ m (∼X ), ∀X⊆U; i=1 RAi i=1 RAi ωα ψα   m m ψα ωα (X ) ⊆ i=1 RAi (X ), m (X ) ⊆ m (X ), 3. i=1 RAi i=1 RAi i=1 RAi

∀X ∈ F(U).

Proof: We only prove 1. 1. ∀X⊆U,

∼

m 

ωα

RAi

(∼X )

i=1



m 

R Ai

x : ω∼X

=U−

i=1

(x) > 1 − α



RA

i = U − x : maxm i=1 ∼X (x) > 1 − α





|R (x) ∩ (∼(∼X )) =U− x: { 1 − Ai }>1−α |RAi (x)|   |RAi (x) ∩ (∼(∼X )) m = x : maxi=1 {1 − }≤1−α |RAi (x)|   |RAi (x) ∩ ((∼X )) m }≥α = x : mini=1 {1 − |RAi (x)|



maxm i=1

=

m 

ψα

RAi

(X ).

i=1

1 − α denotes the specified majority requirement, that is, the rate of admissible classification errors. The above properties are sensitive to the changes of α . Proposition 3.6. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . ∀X ∈ F(U), α 1 , α 2 ∈ (0.5, 1], if α 1 ≥ α 2 , then m    ωα1 ωα2 ψα1 ψα2 (X ) ⊆ m (X ), m (X ) ⊆ m (X ); i=1 RAi i=1 RAi i=1 RAi i=1 RAi ωα1 ψα m  m  1 ψα2 (X ) ⊇ i=1 RAi ωα2 (X ), m (X ) ⊇ m (X ). i=1 RAi i=1 RAi i=1 RAi Proof: It is obvious. Proposition 3.7. Suppose S = (U, AT ) is a fuzzy information system, A1 , A2 , . . . , Am ⊆ AT, and RAi is a fuzzy tolerance relation with respect to Ai . α ∈ (0.5, 1], if A ⊆ {A1 , . . . , An }, then ∀X ∈ F(U), we have



RAi

ωα

(X ) ⊆

Ai ∈A

 Ai ∈A

m 

ωα

RAi

(X ),

ψα

(X ) ⊇

m  i=1

RAi

ωα

(X ) ⊆

Ai ∈A

i=1

RAi



m 

ψα

RAi

(X ),

 Ai ∈A

ωα

RAi

(X );

i=1

RAi

ψα

(X ) ⊇

m  i=1

ψα

RAi

(X ).

The Type-1 and Type-2 variable precision multigranulation fuzzy rough sets are more suitable for handling uncertain and noise data. They will make a better utilization of data being analyzed, and a lower likelihood of incorrect decision. 4. Decision-theory of Type-1 variable precision multigranulation fuzzy rough set model One of the most important tasks in information systems is to give appropriate guidance or decision for every element. The theory of three-way decisions is constructed based on the notions of acceptance, rejection and noncommitment. Meanwhile it is closer to the theory of rough sets. The most significant difference between threeway decisions and two-way decisions is representing a concept by three regions instead of two ones [22]. Three-way decisions play an important role in decision-making and have been widely used in many fields and disciplines. Due to the advantages of three-way decisions, we would like to put three-way decisions and the Type-1 variable precision multigranulation fuzzy rough sets together, in order to make decision in the Type-1 variable precision multigranulation fuzzy approximation space. Suppose S = (U, AT ) is a fuzzy information system, in which U is a non-empty and finite universe of discourse, A1 , A2 , . . . , Am ⊆ AT, RAi is a fuzzy tolerance relation with respect to Ai and T⊆U. Based on variable precision multigranulation fuzzy approximations of T, the universe U can be divided into three disjoint regions, i.e., the positive region POSω (T), the boundary region BNω (T), and the negative region NEGω (T): for α ∈ (0.5, 1],  ωα POSω (T ) = m (T ); i=1 RAi ωα  m ωα ω (T ); BN (T ) = i=1 RAi (T ) − m i=1 RAi ωα  ω ω ω NEG (T ) = U − POS (T ) − BN (T ) = U − m (T ). i=1 RAi Suppose we have a set of 2 states given by  = {T, ¬T }( T is a crisp set), indicating that an element is in T or not in T. In two-way decisions, for every element, we can take two actions, which represent classifying an element in the positive region or negative region. With the two regions, one makes a committed decision, that is, the positive region for acceptance and the negative region for rejection. With respect to three-way decisions based on variable precision multigranulation fuzzy information systems, since some elements may belong to the boundary region or we cannot immediately give the best answer, the expert may give three types of rules, namely, rules for acceptance P, rejection N and deferment B corresponding to the three regions, respectively. Let Des(x) denote the logic formula defining fuzzy neighborhood of x, which is typically a conjunction of attribute-value pairs in an information system [22]. Thus we have Des(x) →ω Des(T ), for x ∈ POSω (T); P ω Des(x) →B Des(T ), for x ∈ BNω (T); Des(T ), for x ∈ NEGω (T). Des(x) →ω N The general term formula of all decision rules can be denoted by Des(x) →ω  Des(T ), where  ∈ {P, B, N}. And because of the complexity and incompleteness of information systems, it is certain that there are some uncertainty and errors of every decision rule. Thus, we can give the accuracy or confidence of the rule of x belonging to T as follows: m 

ω c(Des(x) →ω  Des(T )) = Pr (T |x) = m

i=1 RA

ωT

i=1

R Ai

(x).

i Where ωT (x) is an uncertainty measure of x belonging to T, which can be regarded as the conditional probability measure [5], but perhaps it is not a very practical way to estimate probability. Since the probability estimated by the Bayesian formula is more practical, and the Bayesian decision theory deals with decision making with

T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101

minimum risk based on the known information, for approximating a subset of a universe, one may apply the Bayesian decision procedure to derive a three-way decision model [4,33]. For describing the risk loss degree, let λPP , λBP and λNP denote the loss functions by taking actions P, B and N respectively, when an element belongs to T. And λPN , λBN and λNN denote the loss functions by taking the same actions P, B and N respectively, when an element does not belong to T. To minimize the losses of the whole universe U, one can minimize the overall risk. Let R be the overall risk, which is the expected loss associated with a given decision rule, and is called losses or risks of the whole universe U. For every x, τ (x) is a function that specifies which rule to take. R(τ (x)|x) is the conditional risk associated with action τ (x). Let Pr(x) be the probability of x, ∀x ∈ U, and Pr(x) = |U1 | . Then the overall risk R can be denoted by the following formula:

R=



R(τ (x)|x)Pr(x).

(22)

x∈U

If τ (x) is chosen such that R(τ (x)|x) is as small as possible for every x, the overall risk R is minimized. Let Prω (T|x) (Prω (¬T|x), respectively) be the conditional probability of an object belonging to state T ( ¬T, respectively) given that the object is described by x. The expected loss of x can be computed as follows: P, B and N are three individual actions,

R(P |x) = λPP Prω (T |x) + λPN Prω (¬T |x);

R(P |x) = λPP ωT

m 

R(B|x) = λBP ωT

i=1

R Ai

R Ai

m 

R(N|x) = λNP ωTi=1

m 

m 

R Ai

−λ

λ −λ

PP

BP

BP

NP

PP

NP

BP

NP

BNω (T); λ −λ λ −λ (N1 ) If f (x, T ) < λNN−λ PN and f (x, T ) ≤ λNN−λ BN , decide x ∈ NEGω (T).

For a given information system, the threshold parameters can be gotten by using the loss function. λ −λ λ −λ For the rule (B1 ), if λBN −λPN ≥ λNN−λ BN , the loss function satisfies PP

the condition:

BP

BP

NP

λBN − λPN λBN − λPN + λNN − λBN λNN − λPN λNN − λBN ≥ ≥ ≥ . λPP − λBP λPP − λBP + λBP − λNP λPP − λNP λBP − λNP λ −λ

λ

−λ

λ

−λ

Thus, 1 ≥ λBN −λPN ≥ λNN−λ PN ≥ λNN−λ BN > 0, then we have: PP BP PP NP BP NP λ −λ

(P2) If f (x, T ) ≥ λBN −λPN , decide x ∈ POSω (T); PP BP λ −λ λ −λ (B2) If λNN−λ BN < f (x, T ) < λBN −λPN , decide x ∈ BNω (T); NP

λ

−λ

PP

BP

m

RA

RA

i i In particular, if ωT (x) = 1 − ω¬T (x), ∀x ∈ U, the rules and the loss function can be simplified. Under condition (c0 ), the decision rules can be rewritten as:

(P3) If ωT

i=1 RAi

(B3) If ωT

i=1 RA

(N3) If ωT

i=1 RA

m

R Ai

m 

λ

(P1 ) If f (x, T ) ≥ λNN−λ PN and f (x, T ) ≥ λBN −λPN , decide x ∈ PP NP PP BP POSω (T); λ − λ λ − λ (B1 ) If f (x, T ) < λBN −λPN and f (x, T ) > λNN−λ BN , decide x ∈

m

R Ai

i=1

m

RA

(x) ≥ ξ and ωT

i=1

(x) < ξ and ωT

i=1 RA

i

m

i

m

(x) + λBN ω¬T (x); i=1

Under condition (c1 ), then the decision rules can also be rewritten as:

i=1

(x) + λPN ω¬T (x); i=1

λNN < λBN < λPN ,

m

Therefore, for every individual action, the above expected loss can also be rewritten as: m 

λPP < λBP < λNP ,

(N2) If f (x, T ) ≤ λNN−λ BN , decide x ∈ NEGω (T). BP NP

R(N|x) = λNP Prω (T |x) + λNN Prω (¬T |x).

i=1

Consider a special kind of loss function satisfying condition (c1 ):

BP

R(B|x) = λBP Prω (T |x) + λBN Prω (¬T |x);

99

m

i

(x) ≤ β and ωT

i

i=1 RA

(x) ≥ γ , decide x ∈ POSω (T); (x) > β , decide x ∈ BNω (T); i

(x) < γ , decide x ∈ NEGω (T).

where the parameters ξ , β and γ are defined as:

R Ai

i=1 (x) + λNN ω¬T (x).

ξ =

(λPN − λBN ) , (λPN − λBN ) + (λBP − λPP )

(23)

(P) If R(P |x) ≤ R(N|x) and R(P |x) ≤ R(B|x); then x ∈ POS(T); (B) If R(B|x) < R(P |x) and R(B|x) < R(N|x); then x ∈ BN(T); (N) If R(N|x) < R(P |x) and R(N|x) ≤ R(B|x); then x ∈ NEG(T).

β=

λBN − λNN , (λBN − λNN ) + (λNP − λBP )

(24)

For Type-1 variable precision multigranulation fuzzy rough sets, the rules based on the loss function λ can be simplified. Consider a special kind of loss function satisfying condition (c0 ):

γ =

λPN − λNN . (λPN − λNN ) + (λNP − λPP )

(25)

λPP ≤ λBP < λNP ,

(λPN − λBN ) λBN − λNN ≥ , (λPN − λBN ) + (λBP − λPP ) (λBN − λNN ) + (λNP − λBP )

The minimum-risk of decision rules according to the procedure of Bayesian decision can be denoted as [45]:

λNN ≤ λBN < λPN , m

Under condition (c0 ), if ωT m

If ω¬T

i=1

m

ω¬Ti=1

RA

i

RA

i

i=1

then RA

i

(x) = 0, then x ∈ NEGω (T); m

(x) = 0, then x ∈ POSω (T); If ωT

(x) = 0, let f (x, T ) =

be rewritten as: λ

−λ

m RA ωT i=1 i m RA ω¬Ti=1 i

(x) (x)

i=1

RA

i

(x) = 0 and

, then the decision rules can

λ −λ

1 ω PP BP (P1) If f (x, T ) ≥ λNN−λ PN and f (x,T ) ≤ λBN −λPN , decide x ∈ POS (T); PP NP λPP −λBP λNN −λBN 1 ω (B1) If f (x,T ) > λ −λ and f (x, T ) > λ −λ , decide x ∈ BN (T); BN

λ

PN

−λ

For the rule (B3), if ξ ≥ β , the loss function satisfies the condition:

BP

NP

λ

−λ

(N1) If f (x, T ) < λNN−λ PN and f (x, T ) ≤ λNN−λ BN , decide x ∈ PP NP BP NP NEGω (T).

(λPN − λBN ) (λPN − λBN ) + (λBP − λPP ) (λPN − λBN ) + (λBN − λNN ) ≥ (λPN − λBN ) + (λBP − λPP ) + (λBN − λNN ) + (λNP − λBP ) λPN − λNN λBN − λNN = ≥ . (λPN − λNN ) + (λNP − λPP ) (λBN − λNN ) + (λNP − λBP ) Thus, β ≤ γ ≤ ξ . If 0 ≤ β ≤ γ ≤ ξ ≤ 1, then we have: m

(P4) If ωT

i=1 RAi

(x) ≥ ξ , decide x ∈ POSξ , β (T);

100

T. Feng, J.-S. Mi / Knowledge-Based Systems 91 (2016) 93–101 m

(B4) If β < ωT m

(N4) If ωT

i=1 RAi

i=1 RAi

References

(x) < ξ , decide x ∈ BNξ , β (T);

(x) ≤ β , decide x ∈ NEGξ , β (T).

In this case, if α = ξ and β = 1 − α , where α ∈ (0.5, 1], ωα   ωα (T ), NEGωξ,β (T ) = U − m (T ) then POSξω,β (T ) = m i=1 RAi i=1 RAi ωα m m ωα ω and BNξ ,β (T ) = i=1 RAi (T ) − i=1 RAi (T ). When S is a fuzzy information system in which the only granular is generated by AT, and T⊆U, we have

ωT (x) = TAT (x) = 1 −

|RAT (x) ∩ (¬T )| . |RAT (x)|

Thus

R(P |x) = λPP ωT (x) + λPN ω¬T (x), R(B|x) = λBP ωT (x) + λBN ω¬T (x), R(N|x) = λNP ωT (x) + λNN ω¬T (x). Since ωT (x) + ω¬T (x) = 1, the rules and the loss function can be simplified. Under condition (c0 ), the decision rules can be rewritten as: (P5) If ωT (x) ≥ ξ and ωT (x) ≥ γ , decide x ∈ POSω (T); (B5) If ωT (x) < ξ and ωT (x) > β , decide x ∈ BNω (T); (N5) If ωT (x) ≤ β and ωT (x) < γ , decide x ∈ NEGω (T). If 0 ≤ β ≤ γ ≤ ξ ≤ 1, then we have: (P6) If ωT (x) ≥ ξ , decide x ∈ POSξ , β (T); (B6) If β < ωT (x) < ξ , decide x ∈ BNξ , β (T); (N6) If ωT (x) ≤ β , decide x ∈ NEGξ , β (T). If α = ξ and β = 1 − α , where α ∈ (0.5, 1], then POSξ ,β (T ) =

RAT ωα (T ) = {x : ωT (x) ≥ α},

ωT (x) > 1 − α} and ωT (x) > 1 − α}.

NEGξ ,β (T ) = U − RAT

BNξ ,β (T ) = RAT

ωα

ωα

(T ) = U − {x : α>

(T ) − RAT ωα (T ) = {x :

5. Conclusion In this paper, we proposed a membership degree of an object with respect to a fuzzy set in a single granulation fuzzy rough set model. By using the maximal and minimal membership degrees of an object with respect to a fuzzy set based on multi-fuzzy tolerance relations, we gave two types of variable precision multigranulation fuzzy rough sets of an information system. Then, we discussed the decisiontheory of Type-1 variable precision multigranulation fuzzy rough set model, using the method of three-way decisions. The decisiontheory of Type-2 variable precision multigranulation fuzzy rough set model is similar to the one of Type-1 variable precision multigranulation fuzzy rough set model. Reductions and uncertainty measures of variable precision multigranulation fuzzy rough sets are also worth studying. In the future, we will discuss these topics. Acknowledgments This paper is supported by the National Natural Science Foundation of China (Nos. 61300121, 61170107, 61573127, 61502144), by Natural Science Foundation of Hebei Province (Nos. A2013208175, A2014205157), by Training Program for Leading Talents of Innovation Teams in the Universities of Hebei Province (LJRC022), and by the Doctoral Starting up Foundation of Hebei University of Science and Technology (QD201228).

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