The Fundamental Theory And Applications Of Fuzzy Sets Based On Cut Set With Parameters

This dissertation studies systematically the fundamental theories about the cut set,cut product,decomposition theorem and representation theorem of fuzzy sets.It discusses the properties of the different cut sets,the relationships among the different cut sets,the properties of the different cut products,the relationships among the different cut products on fuzzy sets,intuitionistic fuzzy sets and interval-valued fuzzy sets.The equivalence of the existing decomposition theorems based on different cut sets and different cut products is researched,as well as the representation theorems.Based on the above results,a cut set with parameters was introduced as the unification of the existing four cut sets of fuzzy sets,and the corresponding fundamental theory of fuzzy sets were established,such as the decomposition theorems,the representation theorems,and so on.Furthermore,the cut set with parameters was applied into the research of convex fuzzy sets theory and fuzzy linear programming theory.All of the conclusions we obtained are extending and unifying the existing classical fuzzy sets theory,they are of great theoretical significances and practical values.The main results obtained in this dissertation can be summarized as follows:In chapter 3,the properties and the interrelation of the different cut sets,and that of the different cut products on fuzzy sets are discussed.The equivalence of the existing decomposition theorems and the equivalence of the existing representation theorems based on different cut sets and different cut products are researched.Furthermore,the corresponding conclusions about the intuitionistic fuzzy sets and interval-valued fuzzy sets are obtained similarly.In chapter 4,considering the interrelation of the four kind of cut sets given in the above chapter,the cut set with parameters on fuzzy sets is defined as the unification of the four kind of cut sets,and the properties of the cut set with parameters are given out.Basing on the cut set with parameters,the appropriate decomposition theorems and the representation theorems on fuzzy sets are established.Chapter 5 is devoted to apply the cut set with parameters into the reasearch of the convex fuzzy sets,several kinds of convex fuzzy sets based on the cut set with parameters are given out,the basic properties and the operational properties of them are researched,and the interrelation of them is discussed.And,in chapter 6,the definition and properties of the cut set with parameters on fuzzy sets are applied into the reasearch of the fuzzy linear programming,a class of fuzzy linear programming problems based on the cut set with parameters is researched,the corresponding model is established,and the general algorithm is discussed.At last,in the context of practical problems,a practical example about investment problems is given out and solved to illustrate the feasibility and practicability of the model we established.