New definitions (measures) of skewness, mean and dispersion of fuzzy numbers -- by way of a new representation as parameterized curves
Abstract
We give a geometrically motivated measure of skewness, define a mean value triangle number, and dispersion (in that order) of a fuzzy number without reference or seeking analogy to the namesake but parallel concepts in probability theory. These measures come about by way of a new representation of fuzzy numbers as parameterized curves respectively their associated tangent bundle. Importantly skewness and dispersion are given as functions of α (the degree of membership) and such may be given separately and pointwise at each α-level, as well as overall. This allows for e.g., when a mathematical model is formulated in fuzzy numbers, to run optimization programs level-wise thereby encapsuling with deliberate accuracy the involved membership functions' characteristics while increasing the computational complexity by only a multiplicative factor compared to the same program formulated in real variables and parameters. As an example the work offers a contribution to the recently very popular fuzzy mean-variance-skewness portfolio optimization.