Fuzzy Entropy Of Vague Set Theory Based On Vague Set In Decision-making

In the real world, there exists quite a number of information that is vague, uncertain, incomplete and fuzzy. How to exactly express this information is a very important issue in science research. Currently, most of methods to process fuzzy information are based on fuzzy sets theory proposed by Zadeh. However, there are many fuzzy information can not be expressed and processed by fuzzy sets theory in real world.Vague sets proposed by W.L. Gau and D.J. Buehrer, Taiwan scholar, which is a generalization of the concept of fuzzy sets, have stronger capability to describe uncertainty than fuzzy sets. Appearance of vague sets enhances the research process on the fuzzy phenomenons and provides a new and powerful tool for information process and decision-making. Since it has been proposed just for ten years more, its basic theories are still incomplete, and need us to develop its various fields of applications. Therefore, we choose to research vague sets in this paper, mainly focused on theory research, and discussed its application in decision-making.We investigated some corresponding theories on similarity measures between vague sets, analyzed the formulas of similarity measure that have been proposed, and found that they have some problems. For overcoming these problems, we proposed more reasonable axiom definition of similarity measure between vague sets. A new method of similarity measure between vague sets with continuous universe of discourse is introduced, and proved that it satisfies the new axiom definition. Finally, an example is given to illustrate its application. How to make vague set and classical set convert into each other, have important value in practical applications. Decomposition theorems of fuzzy sets play important roles in fuzzy mathematics, since they can convert fuzzy set into classical set. Representation theorems of fuzzy sets show that any set embedding of fuzzy sets can consist of a fuzzy set. Inspired and motivated by decomposition theorems and representation theorem of fuzzy sets, in this paper we propose decomposition theorems and representation theorem on vague sets based on binary cut-set. The concept of entropy is used to measure of fuzziness, has important use-value in informationism. Inspired by Shannon's probability entropy, the axiom definition of vague set's fuzzy entropy is introduced, and a new method of calculating vague set's fuzzy entropy is given. We consider that"support","opposition"and"neutrality"three aspects information, made the calculation of vague set's entropy more agreement with the actual situation. At last in this paper, a new multicriteria fuzzy decision-making method based on the weighted correlation coefficient using entropy weights is proposed under vague sets for some situations where the information about criteria weights for alternatives is completely unknown. To determine the entropy weights with respect to a set of criteria represented by vague sets, we establish an entropy weight model, which can be used to get the criteria weights, and then propose an evaluation formula of weighted correlation coefficient between an alternative and the ideal alternative. The alternatives can be ranked and the most desirable one(s) can be selected according to the weighted correlation coefficients. Finally, an illustrative example demonstrates the practicality and effectiveness of the proposed method.