The area of aggregation operations is one of the most promising off-springs of
fuzzy sets theory. Before Zadeh introduced fuzzy set connectives in 1965, there
was a wide gap between logic and decision sciences. On the one hand, multiplevalued
logics proposed many-valued extensions of the conjunctions and
disjunctions for which the triangular norms and conorms today provide a natural
setting. On the other hand, in decision sciences, the prototypical operation turned
out to be the weighted average, whether for decision under uncertainty (the
expected utility) or for multi-criteria decision-making, or yet in voting theory (the
majority voting).
However, the setting of fuzzy sets theory, and soon after, the emergence of socalled
fuzzy measures and integrals by Sugeno in 1974 led to the breaking of those
borders. It was clear that while the triangular norms and co-norms, respectively,
stood below the minimum and above the maximum, and the arithmetic means
stood in-between. This led to the study of the family of averaging operations. All
logical connectives could be viewed as aggregation operations of some sort, and
the arithmetic mean could be viewed as an alternative set-theoretic operation. The
Sugeno integral appeared as a natural ordinal family of weighted averaging
operations based on the minimum and maximum. Another crucial step was the
importance given to the notion of generalized quantifiers by Zadeh at the turn of
the eighties, followed by several researchers including Ronald Yager. In
multicriteria decision-making, it is natural to try and construct aggregation
schemes computing to what extent most criteria are satisfied. This was achieved
for additive aggregations by the introduction of the ordered weighted average
(OWA) by Ronald Yager in 1988: instead of weighting criteria, the basic idea was
to put weights on components of rating vectors after a preliminary ranking of the
individual ratings. In this way it was possible to give importance weights to the
fact of having prescribed numbers of criteria to be fulfilled. The other crucial
contribution of Yager was to connect these weights (that sum up to 1 like for the
usual averages) to fuzzy quantifiers like most, few, etc. This connective is a
symmetric generalization of the arithmetic mean, the minimum and the maximum
in the fuzzy set theory.