A method of introducing weights into OWA operators and other symmetric functions

This paper proposes a new way of introducing weights into OWA functions which are popular in fuzzy systems modelling. The proposed method is based on replicating the inputs of OWA the desired number of times (which reflect the importances of the inputs), and then using a pruned n-ary tree construction to calculate the weighted OWA. It is shown that this tree-based construction preserves many useful properties of the OWA, and in fact produces the discrete Choquet integral. A computationally efficient algorithm is provided. The tree-based construction is universal in its applicability to arbitrary symmetric idempotent n-ary functions such as OWA, and transparent in its handling the weighting vectors. It will be a valuable tool for decision making systems in the presence of uncertainty and for weighted compensative logic.