Closed form formulas for computing the centroid of a general type-2 fuzzy set

Centroid of a general type-2 (GT2) fuzzy set (FS) has been a critical concept. With the introduction of the α plane representation / the z slice representation for a GT2 FS, the centroid of a GT2 FS can be computed by calculating the centroid endpoints of its α planes using the Karnik-Mendel (KM) iterative algorithm. However, there lack closed form formulas for the centroid endpoints of the α planes for a GT2 FS. To fill this gap, this paper considers GT2 FSs with triangular, trapezoid, gaussian or bell-shaped secondary MFs. Above all, it is shown that for a GT2 FS, the upper MF (UMF) and lower MF (LMF) of any α plane can respectively be expressed as a generalized linear combination of those of its α = 0 and α = 1 planes, regardless of the shape of its secondary MFs. Using these expressions, closed form formulas for the centroid endpoints of the α planes of a GT2 FS are established. These formulas provide a theoretical tool for studying the centroid of a GT2 FS. Next, properties are presented to reveal how the footprint of uncertainty (FOU) of a GT2 FS and the shape of its secondary MFs affect its centroid, respectively.