In the rough fuzzy set theory, the rough degree is used to characterize the uncertainty of a fuzzy set, and the rough entropy of a knowledge is used to depict the roughness of a rough classification. Both of them are effective, but they are not accurate enough. In this paper, we propose a new rough entropy of a rough fuzzy set combining the rough degree with the rough entropy of a knowledge. Theoretical studies and examples show that the new rough entropy of a rough fuzzy set is suitable. As an application, we introduce it into a fuzzy-target decision-making table and establish a new method for evaluating the entropy weight of attributes.
The rough set theory, introduced by Pawlak [
However, there exist some deficiencies about these uncertainty measures of rough fuzzy sets mentioned above. On the one hand, the rough degree is not strictly monotonic with finer knowledge. That implies that the rough degree is not accurate enough. On the other hand, the rough entropy of knowledge does not reflect the uncertainty produced by the boundary region of approximation space. In order to overcome these limitations, we propose a new rough entropy about rough fuzzy sets, which considers not only the rough degree but also the rough entropy of a knowledge. This is the first aim in this paper (see Section
In multiattribute decision-making problems, the research of evaluating attribute weights is always a hot area. There are some traditional methods to determine attribute weights, such as expert grading, binary comparing, fuzzy statistics, and grey relational analysis. Nevertheless these methods depend largely on decision maker’s experience. Now, the rough set theory has been introduced into research of determining weights. Usually, there are two kinds of methods. From the view of algebraic theory [
In order to deal with complicated fuzzy information systems integrating the rough set theory, Dubois and Prade in [
Let
Let
In Definition
Let
Let
If
Two kinds of methods to measure uncertainty of rough fuzzy sets are often used. One is the rough degree, which is defined from an algebra point of view. Another is the rough entropy, which is defined from an information theory point of view.
Since a fuzzy target is described by use of the lower and upper approximation sets, the noncoincidence of the lower and upper approximation sets results in roughness. The larger the boundary region formed by the lower and upper approximation sets is, the more the roughness is. In order to quantify the roughness, rough degree was introduced.
Let
In Definition
Let
If If
Let
Theorem
Let
Let
Fuzzy-target decision-making table.
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|
|
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1 | 1 | 1 | 0.6 |
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1 | 1 | 2 | 0.7 |
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1 | 2 | 1 | 0.7 |
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1 | 2 | 2 | 0.9 |
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2 | 3 | 3 | 0.5 |
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2 | 3 | 3 | 0.4 |
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2 | 3 | 3 | 0.7 |
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3 | 4 | 4 | 0.8 |
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3 | 4 | 4 | 0.7 |
It is possible that the values of rough degree are invariant under the finer knowledge from Example
Entropy is an important notion measuring uncertainty of information systems in the information theory; see [
Let
Let
From Theorem
The uncertainty of a rough fuzzy set not only depends on the roughness of the fuzzy target itself, but also depends on the uncertainty of a knowledge. From the discussion in Section
Let
Rough entropy of For any given
(i) Let
In addition, by Definition
(ii) If
Let
Assuming that
Theorem
For the fuzzy set
This example validates that the rough entropy in Definition
Given a fuzzy-target decision-making table
The significance of an attribute is obtained by the change value of rough degree when the attribute is removed from the attribute set. The larger the change is, the more the significance of the attribute is. The significance
We can see that
Let
Let
Since
Theorem
Given a fuzzy-target decision-making table for people’s trip in an area influenced by weather conditions, see Table
Let
Let
Then, by computation using rough degree, the significance
On the other hand, we compute the modified significance
By contrasting two kinds of weights above, we find that the latter method involving the rough entropy of a fuzzy set given in Definition
Fuzzy-target decision-making table.
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Outlook |
Temperature |
Humidity |
Windy |
Decision |
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Sunny | Hot | High | False | 0.4 |
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Sunny | Hot | High | True | 0.5 |
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Overcast | Hot | High | False | 0.6 |
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Rain | Mild | High | False | 0.8 |
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Rain | Cool | Normal | False | 0.7 |
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Rain | Cool | Normal | True | 0.7 |
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Overcast | Cool | Normal | True | 0.5 |
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Sunny | Mild | High | False | 0.6 |
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Sunny | Cool | Normal | False | 0.6 |
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Rain | Mild | High | False | 0.9 |
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Sunny | Mild | Normal | True | 0.5 |
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Overcast | Mild | High | True | 0.7 |
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Overcast | Hot | Normal | False | 0.6 |
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Rain | Mild | High | True | 0.8 |
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Sunny | Hot | High | True | 0.6 |
A new rough entropy for measuring uncertainty in the rough fuzzy set theory has been proposed. Using it, we can measure not only the rough degree induced by the boundary region of a fuzzy target in an approximation space, but also the roughness from the classification of a knowledge. In a fuzzy-target decision-making table, we have established a new method based on the new rough entropy to evaluate attribute weights.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referee for his/her valuable comments and suggestions. This work is supported by the Guangdong Province’s Nature and Science Project (S201101006103) of China, the Jiangmen City’s Science and Technology Project of China, and the Wuyi University Youth’s Nature and Science Project of China.