Characteristics And Application Of Axiomatic Fuzzy Sets

Fuzzy set is an important mathematical tool to deal with fuzzy problems and simulate the fuzzy logical thinking of human brain.It has been widely used in natural science,management science,industrial technology and many other fields.Based on the essence and characteristics of fuzzy phenomenon,axiomatic fuzzy sets is established,which is a new mathematical model.It puts forward a new understanding of fuzzy phenomenon from the overall viewpoint,and makes Zadeh’s fuzzy set strict and definite.Starting from the definition of axiomatic fuzzy sets,this paper studies the mathematical properties of the membership functions of axiomatic fuzzy sets and the application of axiomatic fuzzy sets in fuzzy multi-attribute decision-making,two main aspects are as follows:Firstly,we define the normal form of axiomatic fuzzy sets based on the constructive def-inition of axiomatic fuzzy sets,and give the mathematical characteristics of the membership functions of axiomatic fuzzy sets.We define two normal forms of axiomatic fuzzy sets,they are modular intersection fuzzy set normal form and modular union fuzzy set normal form.Then,we prove the normal form existence theorem in the axiomatic fuzzy sets(N,(?)_M,⊕_M)-F((?))with Zadeh operator and strong negative operator.On the basis of the normal form existence theorem of axiomatic fuzzy sets,we prove the characteristic theorem of membership functions in the axiomatic fuzzy sets(N,(?)_M,⊕_M)-F((?))with Zadeh operator and strong negative operator.Futhermore,we give a classification of axiomatic fuzzy sets and analyze the relation between the classes under this classification.Secondly,we study the VIKOR method based on axiomatic fuzzy sets.A new projection similarity is defined,then,we define new group benefit value by simultaneously consider-ing the projection similarity between scheme and positive ideal scheme and the projection similarity between scheme and negative ideal scheme.On this basis,a fuzzy multi-attribute decision-making method based on axiomatic fuzzy sets is proposed.Finally,an example is given to analyze this effectiveness of the decision-making method.