Approximation of Parabolic Fuzzy Numbers

Most of the real world problems are complex that is raised from uncertainity in the form of ambiguity. To overcome the ambiguity and impreciseness, Zadeh [20] introduced fuzzy set theory in the year 1965. The underlying power of fuzzy sets is that linguistic variables can be used rather than quantitative variables to represent imprecise concepts which are continuous transitions. The continuous imprecise concepts need to be modeled by continuous fuzzy numbers instead of intervals and real numbers. Many researchers applied triangular and trapezoidal fuzzy numbers which are piecewise linear and continuous for imprecise concepts because of its convenience and ease. But imprecise variables in the fields of decision making, data analysis, fuzzy control, image processing, fuzzy clustering etc., need not be linear always. So parabolic fuzzy numbers can be used for modeling nonlinear fuzzy concepts to have a better accuracy in real life problems. Since parabolic fuzzy numbers are not completely studied till now and the triangular fuzzy numbers have been thoroughly studied, triangular approximations of parabolic fuzzy numbers may be used as supplementary tool along with parabolic fuzzy numbers. Approximation is a kind of de-fuzzification that has to change non-linear functions into linear functions which reduces the fuzziness for the undefined information and produces a better accuracy. So the study of approximation of parabolic fuzzy numbers is necessary and hence, in this paper, the problem of triangular approximations of parabolic fuzzy numbers is attempted using distance function in terms of α-cuts. Some properties of the parabolic fuzzy numbers which are useful in multi criteria decision making (MCDM) have also been discussed. Finally pertinent illustrations and applications of the approximation of parabolic fuzzy number have also been given.