Determining OWA Weights by Maximizing Consensus

Abstract — InthispaperweproposeamethodforgeneratingOWAweightingvectorsfromtheindividualassessmentsonasetofalterna-tivesinsuchawaythattheseweightsmaximizetheconsensusamongindividual assessments with respect to the outcome provided by theOWA operator.Keywords — OWA operators; consensus; distances; mathematicalprogramming. 1 Introduction In 1988 Yager [9] introduced OWA operators as a tool foraggregating numerical values in multi-criteria decision mak-ing. An OWA operator is similar to a weighted mean, but withthe values of the variables previously ordered in a decreasingway. Thus, contrary to the weighted means, the weights arenot associated with specific variables and, therefore, they areanonymous. Moreover, they satisfy other interesting proper-ties, such as monotonicity, unanimity, continuity and compen-sativeness.Initially, the weights of an OWA operator may be fixed tak-ing into account the importance we want to give to the assess-ments. So, the outcome of an OWA operator may be the maxi-mum, the minimum, the average or a median of the individualassessments, among a large number of possibilities.It is important to note that the determination of the weightsof OWA operators is a relevant issue since the origins of thetheory of OWA operators. In this way, Yager [9] proposesto use linguistic quantifiers for generating the OWA weights;O’Hagan [6] generates the OWA weights by maximizing theirentropy whenever a degree of orness has been fixed; Filevand Yager [3] consider an exponential smoothing approach forgenerating the OWA weights. After these seminal papers, alarge variety of techniques have been proposed in the litera-ture (see, for instance, Wang and Parkan [7] and Xu [8]).When a group of individuals provides assessments on analternative and these values are aggregated, it is relevant toknow the degree of agreement or consensus among the indi-vidual assessments with respect to the aggregated value. Infact, it is desirable that the aggregation function used to obtainthe collective value reflects the opinions of as many agentsas possible. Using a specific OWA operator for aggregatingindividual assessments does not necessarily ensure such con-sensus for every opinion situation.In our proposal, we do not fix the OWA weighting vector,but we generate an OWA operator for each profile of individ-ual assessments, just one that maximizes the consensus (orequivalently, minimizes the disagreement) in the group withrespect to the outcome provided by the OWA operator. Moreconcretely, once the agents opinions are known, we first cal-culate the distances among individual assessments on the al-ternatives and the collective assessments generated by an ar-bitrary OWA operator. Secondly, we use an aggregation op-erator for obtaining a representative measure of disagreementfrom the individual assessments to the collective one. By solv-ing a mathematical program, we obtain the weighting vec-tor(s) that maximize(s) the consensus among individual andcollective opinions.The paper is organized as follows. Section 2 is devoted tointroduce notation, basic notions and our proposal for gener-ating an OWA operator for each profile of individual assess-ments. Section 3 contains some illustrative examples. Finally,Section 4 shows some open problems and further research.