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Uncertainty measurement of the rough interval-valued intuitionistic fuzzy sets
WANG Yanping, WANG Jinying
School of Science, Liaoning University of Technology, Jinzhou 121001, China
Abstract: In order to give an uncertainty measurement of the interval-valued intuitionistic fuzzy rough set, initially based on a trigonometric function, an interval-valued intuitionistic fuzzy entropy formula is proposed by using a concise axiom definition of the interval-valued intuitionistic fuzzy entropy. The problems concerning the interval-valued intuitionistic fuzzy entropy formula discussed in the previous literature are also pointed out. Then the rough membership function of the interval-valued intuitionistic fuzzy set is defined and when using the interval-valued intuitionistic fuzzy entropy of rough membership functions, uncertainty measurement of the interval-valued intuitionistic fuzzy rough set is given, and some related properties of the measurement are discussed for the purpose of proving the rationality of the definition.
Key words: interval-valued intuitionistic fuzzy sets     fuzzy entropy     rough interval-valued intuitionistic fuzzy sets     rough membership function; uncertainty measurement

1 粗糙区间直觉模糊集的基本理论

U上所有区间直觉模糊集构成的集合为IVIF(U)．

1) AB iff μA(x)≤μB(x)且vA(x)≥vB(x)；

2) A=B iff μA(x)=μB(x)且vA(x)=vB(x)；

3)~A={〈x, vA(x), μA(x)〉|xU}．

 (1)

2 区间直觉模糊集的粗糙隶属函数

 (2)

1) ∀A, B∈IVIF(U)，若AB，则R(A)⊆R(B)；

2)若A∈IVIF(U)，则R(~A)=~R(A)。

3 区间直觉模糊熵

1) E(A)=0当且仅当A是一个分明集；

2) E(A)=1当且仅当对∀xU都有μA(x)=vA(x)；

3) E(A)=E(~A)；

4)对∀xU及∀A, B∈IVIF(U)，如果当μB(x)≥vB(x)时，有AB，或当μB(x)≤vB(x)时，有AB，则E(A)≤E(B)．

 (3)

 (4)

μB(xi)≤vB(xi)时，有AB时，同理可证，所以有

4 粗糙区间直觉模糊集的不确定性度量

 (5)

1) ∀xA，有[1, 1]=μA(x)=μ R(A)(x)=inf{μA(y)|y∈[x]R}，因此∀y∈[x]R都有μA(y)=[μAL(y), μAU(y)]=[1, 1]，即

2)∀xA，有[1, 1]=vA(x)=v R(A)(x)=inf{vA(y)|y∈[x]R}，因此y∈[x]R都有vA(y)=[vAL(y), vAU(y)]=[1, 1]，即

5 结束语

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DOI: 10.3969/j.issn.1673-4785.201303013

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#### 文章信息

WANG Yanping, WANG Jinying

Uncertainty measurement of the rough interval-valued intuitionistic fuzzy sets

CAAI Transactions on Intelligent Systems, 2014, 9(4): 449-453
http://dx.doi.org/10.3969/j.issn.1673-4785.201303013