Grouping Indices: Definition, Properties and Construction Methods

Zadeh has defined its consistency index
$O_{Z}$
in order to compare two fuzzy subsets
$A$
and
$B$
of a referential set
$U$
, in the sense that “The lower
$O_{Z} (A, B)$
is, the closer to
$\emptyset A\cap B$
is.” Several axiomatizations of this concept can be found in the literature, in terms of overlap indices, allowing its application in different contexts, as in classification and clustering. The present work, in an opposite direction, introduces the grouping index, a comparison index
$\mathcal{G}$
of two fuzzy subsets
$A$
and
$B$
that provides a measure of the proximity of the fuzzy union of
$A$
and
$B$
to the referential set
$U$
, that is, 'the higher
$\mathcal{G}(A, B)$
is, the closer to
$U A\cup B$
is.” Many important properties are analyzed, and construction methods are presented, whereas establishing the formal relation between overlap and grouping indices via duality. The theory is illustrated by several detailed examples.