New definitions (measures) of skewness, mean and dispersion of fuzzy numbers -- by way of a new representation as parameterized curves

TL;DR

This paper introduces geometrically motivated measures of skewness, mean, and dispersion for fuzzy numbers using a new parameterized curve representation, enabling level-wise optimization in fuzzy modeling.

Contribution

It presents novel definitions of fuzzy number measures based on a curve representation, allowing pointwise and overall analysis of skewness and dispersion.

Findings

New geometric measures of fuzzy skewness, mean, and dispersion.

Measures can be computed at each membership level for detailed analysis.

Application demonstrated in fuzzy portfolio optimization.

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 $\alpha$ (the degree of membership) and such may be given separately and pointwise at each $\alpha$-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.