The Approximation Of The Fuzzy Numbers

Fuzzy number is used in various engineering fields,and shows its great function when it is produced.But the concept of the fuzzy number has a flaw which is misty.The researchers discuss different fuzzy numbers by their membership functions.But sometimes they are difficult in studying because of the complex membership function of some fuzzy numbers.At this time some scholars investigated fuzzy number by the substitute of the given fuzzy numbers.So,there are some approximation problems of the fuzzy numbers.Its known that approximation theory is an important research field in mathematics,and has strong application background.In this research field,many important results have been made.C.T.Yeh and A.I.Ban studied the problems of trapezoidal,triangular,weighted trapezoidal,weighted semi-trapezoidal approximations of fuzzy numbers,and get some important results in recent years;P.Grzegorzewski proposed a new interval approximation operator,which the best one with respect to a certain measure of distance between fuzzy numbers in 2002;P.Grzegorzewski and Edyta Mr ?owka gave a new nearest trapezoidal approximation operator preserving expected interval in 2005;L.Coroianu,M.Gagolewski and P.Grzegorzewski obtained approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study by using 1-knot fuzzy numbers in 2013.But it has a flaw that piecewise 1-knot fuzzy number is no longer content to sealing of addition: it will appear two knots in add usual add operation.So we define a new fuzzy number–piecewise 2-knots fuzzy number and it can be used to approximate fuzzy number.And this approximation is more accurate than the former.In fact,whether interval approximations of fuzzy numbers.Such approximations are indeed simple and practical enough as the used piecewise linear functions have no knot or have only one knot.However,with the increase of the number of the knots,such approximations will become more and more complex.We will try to find a new fuzzy number approximation of given fuzzy number with simple structure and simple membership function and more knots in this paper.The main work and results of this paper lie in the following:1.In the chapter 1,we mainly introduce the research background,purpose and meaning.2.In the chapter 2,we introduced some basic concepts of fuzzy set theory and fuzzy numbers.3.In the chapter 3,the concept of piecewise linear 2-knot is given by using piecewise linear 1-knot fuzzy number,and some theorems are given in this chapter.Then,we obtained the algorithm by using the theorem.At last,we consider some properties of the nearest trapezoidal approximation operator.4.In the chapter 4,we gave the definition of simple fuzzy numbers and the expressions of their membership functions,and obtained that a fuzzy set of real number field R is a simple fuzzy number if and only if it is a step type fuzzy set.Then we showed that simple fuzzy number space is dense in the fuzzy numbers space with respect to the metric d,and proposed the nearest approximations of fuzzy numbers as the left threshold value set and right threshold value set are given.Such nearest approximations of fuzzy numbers,as number of the knots get large,are still simple enough and flexible.And then,we investigated the properties of such approximation operatorr?s: E ? S(E),and obtained theorem 4.8 and proposition 4.1.5.In the chapter 5,we summarize the article and talk about the future research work.6.In the chapter 5,thanks.