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Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice

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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.

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