ON QUALITATIVE WEIGHTED MEANS Lluís Godo Institut d'Investigació en Intel.ligència Artificial Consejo Superior de Investigaciones Científicas Campus UAB s/n, 08193 Bellaterra, Spain godo@iiia.csic.es Vicenç Torra Departament Enginyeria Informàtica (ETSE) Universitat Rovira i Virgili Carretera de Salou s/n, 43006 Tarragona, Spain vtorra@etse.urv.es Abstract In many applications values to be aggregated are qualitative. In that case, if one wants to compute an average value in a pure qualitative setting, one is basically restricted to weighted versions of max-min combinations. In this paper we explore the feasibility of defining a qualitative counterpart of the weighted mean operator without having to necessarily use some kind of numerical interpretation of the values. We propose a method to average qualitative values belonging to an ordinal scale weighted with natural numbers based on the use of qualitative (finite) t-norms defined on the set of values. Keywords: aggregation operators, qualitative aggregation, weighted mean, finite t-norms 1 INTRODUCTION Aggregation operators have been traditionally defined in the numerical setting (notorious exceptions are [3, 7]). Most notorious numerical aggregation operations are those based on the weighted mean (WM) [1] and those based on the ordered weighted average (OWA) [10, 11]. However, in many applications, values to be aggregated are qualitative, or ordinal, rather than quantitative. In such a cases, the usual technique is to map the qualitative values and weights into a numerical scale, and then perform the aggregation of those numerical values, and optionally get back to a value in an ordinal scale, if needed. If one wants to strictly remain in a pure qualitative setting, where the ordering of the values and weights is what only matters, then one is basically led to weighted versions of max or min combinations (Sugeno integrals), provided that the domain of values U and the domain of weights W are commensurate scales. For instance, if we take U = W, the weighted-max and weighted-min of a set of values {x 1 , ..., x n } according to a set of weights {w 1 , ..., w n } are respectively defined as x + = max(min(x 1 , w 1 ), ..., min(x n , w n ))) x - = min(max(x 1 , n(w 1 )), ..., max(x n , n(w n ))) where n is the order reversing involution in W and the normalisation condition max(w 1 , ..., w n ) = 1 W is assumed to hold. Then, it can be shown that the inequalities min(x 1 , ..., x n ) < x - < x + < max(x 1 , ..., x n ) always hold. The above qualitative aggregations are usual in possibility and fuzzy sets theory. The expressions x + and x - respectively correspond to the possibility and the necessity measures of a fuzzy event A over a (finite) domain D = {d 1 , ..., d n }, with membership function µ A : D ∅ [0, 1], given some possibility distribution π : D ∅ [0, 1]. Namely Pos π (A) and Nec π (A) are nothing but the weighted-max and weighted-min of the membership values x i = µ A (d i ) according to the weights w i = π (d i ), and the involution n(w i ) = 1 – w i . But even in a qualitative setting, very often a notion of average is also needed. In this paper we explore the possibility of defining a qualitative counterpart of the weighted mean operator without having to necessarily use some kind of numerical interpretation of the values. Actually, we revise and extend a previous report [6] of the authors. Namely, in [6] there are some claims (Theorem 3 and Corollary 1) that have turned out to be false. We try here to overcome those difficulties and propose alternative conditions to get proper qualitative weighted mean-like operators.