Study On The Fuzzy Set Extensions And Applications

Since the theory of fuzzy sets was proposed by professor L.Zadeh, it has been successfully applied in many fields. Fuzzy set is the extension of classical crisp set. In recent years, many new fuzzy sets extensions have been proposed along with the development of fuzzy set theory, such as Intuitionistic Fuzzy Sets, Grey Sets, L-Fuzzy Sets, Interval-valued Fuzzy Sets and Vague Sets, etc., and the relationships among these fuzzy set extensions are analyzed in detail.Vague set (Intuitionistic Fuzzy Sets) is the most representative theory among those fuzzy set extensions. A kind of three dimension expression of Vague Sets was given; the operation formula and properties were discussed. Based on the consideration of the tendency of neutrals in the vote model, three transformations from Vague Sets to Fuzzy Sets were proposed. Fuzzy entropy is an important method to measure the fuzzy degree of Vague Sets, but the traditional fuzzy entropy of Vague Sets are not reasonable because they only considered the fuzziness of elements and ignore the fuzziness of subject degree. We consider the fuzziness of elements and subject degree together, the concepts of hesitate degree and conflict degree are introduced, then, a new kind of fuzzy entropy of vague sets is presented, examples show that this kind of fuzzy entropy definition is very reasonable. Finally, the definition of continuous vague sets was proposed.Based on the characteristic of the values of the subjecting functions, after investigating the development process of fuzzy theory, we reclassify the reclassical crisp set and Fuzzy Set as point value set; put Vague Set and other fuzzy set extensions into interval valued sets. Even the interval valued set can describe the fuzzy information very well, but is can not show the distribution of the subject degree. In order to solve this problem, we propose the concept of Normal Distribution Fuzzy Set. The Normal Distribution Fuzzy Set establish a map from a discussed domain X to a set of normal distribution functions, the every normal distribution function can directly describe the distribution and subject degree of the corresponding element. The properties of union, intersection and complementation about the Normal Distribution Fuzzy Set are discussed. The fuzziness degree of Normal Distribution Fuzzy Set is proposed. Normal Distribution Fuzzy Set is the extension of Vague Set, then, the relationships and mutual transformation among Fuzzy Set,Vague Set and Normal Distribution Fuzzy Set are discussed. The similarity measure between Vague Sets based on normal distribution is given. Examples show that this method is better than any existed similarity measures; furthermore, this method is suit to be used in linguistic variables.The fuzzy theory provides a powerful tool for soft classification. It has been used to deal with cluster problems since it was presented. Fuzzy cluster analysis includes two main contents: the similarity measure between two patterns and the cluster algorithm. Base on the spontaneous relationship between Normal Distribution Fuzzy Sets and Vague Set, Normal Dstribution Fzzy Sets provide a new similarity measure method for patterns and a new kind of cluster algorithm of interval values. It has been proved can gain satisfied result easily.