Ordered Weighted Averaging

Ordered weighted averaging (OWA) is a multiattribute fuzzy method that assigns to each alternative a weighted sum with ordered attributes; in addition to criterion weights, order weights are used; the method provides continuous fuzzy aggregation operations between the fuzzy intersection and union, with a weighted average combination falling midway in between. OWA is a weighted sum with ordered evaluation criteria. Thus in addition to the criterion weights, order weights are used. The order weights allow for direct control over the levels of trade-off among criteria. The degree of overall trade-off is thte degree to which criterion/trade-off weights are applied in the aggregation procedure (Eastman and Jiang 1996; Eastman 1997).A fundamental aspect of this aggregation rule is the reordering step. The weighting coefficients are not associated directly with a particular criterion but rather, are assigned to an ordered position of criterion values for a given alternative. The evaluation criterion with the highest value, after criterion weights are applied, is given the first order weight. The criterion with the next-highest value is given the second order weight, and so on.

The OWA operators include the fuzzy MIN and MAX operators. The Min or intersection operator corresponds to the local AND. It produces the largest fuzzy set from among those produced by all possible fuzzy intersections. This interpretation implies no positive compensation (trade-off) between degree of membership of the fuzzy sets under consideration. From the MADM perspective, this means that an alternative is rejected on the basis of poor performance with respect to at least one attribute, even if it performs well above average on other attributes. The MAX operator corresponds to the logical OR and generates the smallest fuzzy set among the fuzzy sets produced by all possible fuzzy unions. It is the maximum degree of membership achieved by any of the fuzzy sets representing evaluation criteria (or/and constraints). This amounts to a full compensation of lower degrees of membership by the maximum degree of membership. It implies that an alternatives recognized as the best one on the basis of an exceptionally high value of one attribute, irrespective of poor performance with respect to other attributes.

Besides MIN and MAX, which can be considered as limited cases, there exist a fairly large number of averaging operators. In general, by varying the assignment of weights for the MIN and MAX operators, one can cover the entire range between the MIN and MAX operators. The degree to which the order weights are evenly distributed across all positions controls the level of overall trade-off. In a decision –making process the concept of trade-offs corresponds to the global evaluation of alternative decisions as lying between the worst and best local ratings. This occurs in the presence of conflicting evaluation criteria, when compensation between the corresponding compatibilities is allowed. Averaging operators realize trade-offs between evaluation criteria by allowing a positive compensation between ratings.