Extremal symmetrization of aggregation functions

For aggregating observed unordered n values, based on an n-ary aggregation function A, two extremal symmetric aggregation functions $$A^*$$ A ∗ and $$A_*$$ A ∗ are introduced and discussed. In the case of weighted arithmetic means, the representation of $$A^*$$ A ∗ and $$A_*$$ A ∗ as particular $${{\mathrm{OWA}}}$$ OWA operators is shown. Considering weighted aggregation function $${{{A}}_{{\mathbf w} }}$$ A w with unordered weights and input values to be aggregated, another two symmetric aggregation functions $$({{{A}}_{{\mathbf w} }})^\Diamond $$ ( A w ) ◊ and $$({{{A}}_{{\mathbf w} }})_\Diamond $$ ( A w ) ◊ are introduced and discussed. A relation between our approach and the Hungarian algorithm known from the linear optimization domain is also shown.