Weighted approximation of fuzzy numbers by using m-n-step type fuzzy number

In this paper, based on the weighted metric, the problem of approximating a general continuous fuzzy number $u$ with derivable $\underline{u}(r)$ and $\overline{u}(r)$ (left and right cut functions of level value $r$) for $r \in (0,1]$ by using the $m-n-$step type fuzzy number is studied. Two kinds of approximations ($I-$nearest approximation and $II-$nearest approximation which satisfy different conditions) of using $m-n-$step type fuzzy numbers to approximate general continuous fuzzy numbers are defined, and the relationship between the $I-$nearest approximation and $II-$nearest approximation is obtained. Then the methods of the two kinds of approximations are respectively presented by the theorems more specifically. At last, an example is given to show the effectiveness and usability of the methods set up by us.