The Binary Relation Of Fuzzy Numbers And Its Application

Since the fuzzy set theory had been proposed as a new subject, it quickly developed into three branches of fuzzy mathematics: fuzzy analytic, fuzzy topology and fuzzy algebra. The fuzzy number theory which is one of the most basic and most important part of the fuzzy analytic become a hot research topic, in particular the problem of ranking fuzzy numbers catch widespread attention by scholars. From the definition and properties of fuzzy set, we know that the order for fuzzy numbers is not the total order but the partial order under the lattice structure, so ranking fuzzy numbers becomes one of both important and difficult tasks in fuzzy decision-making. In recent years, many scholars have focused on research in this area and achieved encouraging results. In this dissertation, we propose two new methods for ranking fuzzy numbers based on the results of the predecessors: endpoint method and the mean by quasi-mean and fuzzy degree; then define a new fuzzy 2-cell number—pyramid fuzzy number, discuss its features and give the ranking method about it; finally we apply fuzzy numbers to game theory. The main works of this paper are as follows:(1) The first method in this paper for ranking trapezoidal fuzzy numbers is introduced—Endpoint method, this method is based on the satisfaction degree about interval number by paper [40]. When using this method, the only condition need to know is the special points (the points which membership degrees are 0 or 1) about triangular fuzzy numbers or trapezoidal fuzzy numbers, then we can easily get their membership function and ranking them. The advantage of this method is that for two fuzzy numbers, we can get in which degree A is prior to B and B is prior to A at the same time, then the order for multiple fuzzy numbers can be obtained between pair wise comparison.(2) Propose the second method in this paper for ranking fuzzy numbers which using both quasi-mean and fuzzy degree. In this method, firstly we introduce a new index—quasi-mean, it is proposed by the mean of fuzzy number, and this make us have a more comprehensive understanding about fuzzy number. Then we take into account the influence of fuzzy degree, and give the new method by comprehensive considering these two indexes. Finally many examples are given and compared with other existing methods, and from the results we can see the new method is indeed preferable.(3) We define a special fuzzy 2-cell number—four-pyramid fuzzy number, then analyze the relationship between it and a 2-dimensional fuzzy vector whose components are both triangular fuzzy numbers, and prove that they can represent each other uniquely. After that we give the ranking algorithm about it, this provides a convenient for the use of fuzzy 2-cell numbers in the engineering field.(4) Finally we introduce the dominant strategy and dominant-strategy equilibrium in the game theory, combine them with fuzzy number theories and give the application of raking fuzzy numbers in this field.