Approximation of Fuzzy Numbers by Using $$\alpha $$ - $$\beta $$ -knots Piecewise Linear Fuzzy Numbers

Abstract

In this paper, the problems of approximating general fuzzy number by using \(\alpha \)–\(\beta \)–knots piecewise linear fuzzy number are studied. Firstly, \(\alpha \)–\(\beta \)–knots piecewise linear fuzzy number are defined, and the conceptions of nearest \(\alpha -\beta -\)knots piecewise linear approximation and nearest \(\{0,\alpha ,1\}\)–\(\{0,\beta ,1\}\)–knots piecewise linear approximation are introduced for a general fuzzy number. Then, it is also the most important work of this paper that for a general fuzzy number, we obtain a formula to get the nearest \(\alpha -\beta \)–knots piecewise linear approximation and the nearest \(\{0,\alpha ,1\}-\{0,\beta ,1\}\)–knots piecewise linear approximation using weighted metric as a criterion. And then, we give specific example to show more reasonable and effective of the methods proposed by us.