Special Type Of Fuzzy Ellipsoid Number And A Weak Order In Its Space

Professor L.A.Zadeh put forward the concept of fuzzy set[1] for the first time in1965. And from now on, a new subject had been set up. Afterwards, Zadeh and Chang[2] introduced the concept of fuzzy number in1972, which contributed to the development in a new field of the researchh of fuzzy mathematics theory. With an increasing number of mathematicians studying the theory of fuzzy numbers, more and more results of the study have been showed. Nowadays, the theory of fuzzy numbers has become one important research direction of the theory of fuzzy set.Fuzzy n-cell numbers is a special kind of fuzzy subset on R" and can be used to cope with some application problems with the feature of uncertainty or imprecise information. It plays an important role in practical application. However, fuzzy n-cell numbers will miss some important information while is used in representing some information with uncertainty or imprecision(see details in the Appendix [56,47]). We need to study the fuzzy n-ellipsoid numbers for further exploration fuzzy n-ellipsoid numbers are able to be more objective and more rational in expressing uncertain multi-channel digital information than fuzzy n-cell numbers. In document [47], Wang put forward the concept of fuzzy ellipsoid number for the first time. What’s more, he gave the definition of number of n dimensional symmetric ellipsoid. In order to be more objective and more reasonable in repressing some inaccurate or uncertain multi-channel digital information, this paper discusses some special types of dimension fuzzy n-ellipsoid numbers on the basis of the Wang’s research from the perspective of practical application. What’s more, we have given out their construction method, discussed their quality and relationship between each other and set up some weak orders "<p " in the fuzzy n-ellipsoid numbers space, What’s more, we apply these to the evaluation of city environment. We will show the main arrangement of works of this paper as follows:1. In chapter1, we primarily introduce the research background of fuzzy number, some basic concepts of fuzzy n-ellipsoid numbers, its research status and the arrangement of the main content.2. In chapter2, we introduce the basic concept, algorithm of fuzzy set and fuzzy n-ellipsoid numbers and their properties.3. In chapter3, we give the definition of six special kinds of fuzzy n-ellipsoid numbers C-fuzzy ellipsoid number, P-fuzzy ellipsoid number, CT-fuzzy ellipsoid number, PT-fuzzy ellipsoid number, TCT-fuzzy ellipsoid number, TPT-fuzzy ellipsoid number, TC-fuzzy ellipsoid number and TP-fuzzy ellipsoid number. Then we give the method of construction of them and study their properties. Then we discuss the relationship of these special fuzzy n-ellipsoid numbers. At last, we illustrate the rationality that in expressing uncertain or imprecise multi-channel digital information with these six fuzzy n-ellipsoid numbers.4. In chapter4, on the basis of studying the properties of the six kind of fuzzy n-ellipsoid numbers in chapter3, we set up partially order "≤" on the space of some special fuzzy n-ellipsoid number. Despite the theory of partial order has good properties, but with strict conditions that could satisfy the requirement of partial order. Therefore we put forward another weak order“(?)p”to facilitate the practical application value. Then we define a method of ranking based on the weak order and study their properties. At last we give a practical example to show the proposed weak orders in the fuzzy n-ellipsoid number space are used to rank some objects.5. In chapter5,we mainly summarize this paper, at the same time, looking forward to the future of the research work.