Approximation Of One Dimensional Fuzzy Numbers And High Dimensional Fuzzy Cell Numbers By Using Piecewise Linear Fuzzy Numbers

Fuzzy events are ubiquitous in real life,and fuzzy set theory is widely used to describe these events.Fuzzy numbers are a special kind of fuzzy sets.In order to accurately describe uncertain or imprecise quantitative attribute information,the expression forms of some fuzzy numbers are often complex,leading to their ineffective application in many fields.In general,in order to overcome the limitations of complex structured fuzzy numbers,in the application research of fuzzy number theory,it is often used to approximate complex structured fuzzy numbers using simple structured fuzzy numbers,such as triangular approximation,trapezoidal approximation,interval number approximation,and other methods.In current research,in order to facilitate calculation,approximation based on general metric has been well applied,but there are some shortcomings in terms of accuracy for some highly demanding problems.Weighted metric is more reasonable and objective in describing the degree of difference between two fuzzy numbers.Therefore,the piecewise linear approximation results based on weighted metric will be more objective,reasonable,and accurate than the piecewise linear approximation results based on unweighted metric.In addition,when describing complex fuzzy events,high dimensional fuzzy numbers are more widely used.High dimensional fuzzy cell numbers are a class of high dimensional fuzzy numbers that are often used in practical applications.Therefore,it is also a meaningful work to study the approximation of high dimensional fuzzy cell numbers to polygonal fuzzy numbers.Therefore,this paper is mainly divided into two parts:In the first part,for one dimensional fuzzy numbers,based on the weighted metric d*,under the condition that the level value set is known,definitions of Ⅰ-and Ⅱ-of nearest r-s-knots piecewise linear approximation are proposed,and the specific recursive formulas for solving their nearest approximation are given.Finally,through specific examples,the rationality and effectiveness of the two kinds of approximation methods given are illustrated.In the second part,the the definition of α-β knots piecewise linear fuzzy n-cell number is proposed.Under the condition that the level value set is known,the concepts of approximation of the Ⅰ-and the Ⅱ-nearest α-β knots piecewise linear fuzzy n-cell numbers for general fuzzy cell numbers are proposed,and the specific recursive formulas for solving the approximation of the Ⅰ-and the Ⅱ-nearest α-β knots piecewise linear fuzzy n-cell number are given.Finally,through an example,the advantages and disadvantages of these two kinds of approximation are compared.