The Method Of Approximating General Fuzzy Number By Piecewise Linear Fuzzy Numbers

In real life,fuzzy phenomenon exists everywhere.In order to accurately describe the fuzzy phenomenon with uncertain information by mathematical methods,membership functions of fuzzy numbers are often complicated.Therefore,we often use some simple fuzzy numbers to approximate complex fuzzy numbers,such as interval approximations of fuzzy numbers or triangular approximations of fuzzy numbers or trapezoidal approximations of fuzzy numbers.In fact,these approximation methods all belong to the use of special simple(few nodes)piecewise linear membership function to approximate the general fuzzy number.However,for some practical applications,some problems that need more accurate approximation may be encountered,so it is meaningful work to study using multi-knots piecewise linear fuzzy number to approximate general fuzzy number.In this thesis,based on Single-knot piecewise linear fuzzy number(piecewise linear fuzzy number with one left node and one right node),a new method of approximating general fuzzy number by Multi-knots piecewise linear fuzzy numbers is proposed.The kernel set and support set of fuzzy numbers are no longer maintained unchanged,thus the method retains more information about the fuzzy number being approximated.In addition,a calculation method based on Thomas algorithm is presented.The main work of this thesis is divided into two parts: In the first part,based on unweighted distance,under the condition that the level value set of Multi-knots piecewise linear fuzzy numbers is known,the approximation problems of ?-?-node piecewise linear fuzzy number and ? ? {0,1}-? ? {0,1}-node piecewise linear fuzzy number are discussed respectively.In the second part,based on the weighted distance,under the condition that the level value set of Single-knots piecewise linear fuzzy numbers is known,the fuzzy number of ?-?-node piecewise linear and the fuzzy number of {0,?,1}-{0,?,1}-node piecewise linear are discussed respectively.The structure of this thesis is as follows:1.In the first chapter,we summarize the fuzzy set theory,the development history of fuzzy numbers,and the present situation,purpose and significance of fuzzy number approximation research.2.In the second chapter,we summarize some concepts in the theory about approx-imation of fuzzy numbers,such as basic concepts and properties of fuzzy set and fuzzy number,and two definitions of distance used in this thesis are given.3.In the third chapter,we use unweighted distance as metric to give two methods of best approximation of general fuzzy numbers by Multi-knots piecewise linear fuzzy numbers under the conditions of given level of given Multi-knots piecewise linear fuzzy numbers,and give the expression of its best approximation.Then,the two approximation methods are compared by example under unweighted distance.At last,we also investigate some properties of the approximation operators introduced by us.4.In the fourth chapter,we use weighted distance as metric to give two methods of best approximation of general fuzzy numbers by Single-knots piecewise linear fuzzy numbers under the conditions of given level set of Single-knots piecewise linear fuzzy numbers,and give the expression of its best approximation,and some properties of the best approximation.At last,the two approximation methods are compared by example under weighted distance,and combine the examples in chapter 3,two kinds of approximation methods at different distances are compared.5.In the fifth chapter,we make a summary and outlook of the thesis.