Mathematical Approach for Consensus Formation

Kayoko Tanabe ； Hiroaki Ishii

《Asia Pacific Management Review》 12卷3期 (2007/06) Pp. 145-149

並列摘要

In this paper, we extend the distance measure of Cook and Kress and examine a procedure to rank order from various perspectives. When we choose the most favorable object among a number of them, we often use 'ranking'. In general, objects are placed in order of decreasing votes they had. This method is not sufficient or appropriate for rank ordering with respect to fairness since the average and variance of each object are different in viewpoints and standings next to each other do not always have the same gap. Approaches with Data Envelopment Analysis (DEA) have been studied to extend this problem. Cook and Kress, using DEA as their starting point, have proposed rank ordering of the objects in a preferential election. Each object is permitted to choose the most favorable weights to his/her standings (first place, second place, etc. votes) in the usual DEA manner with the additional 'assurance region' restriction that the weight for a j place vote should be more than that for a j+1 place vote by some amount. Green has presented an alternative procedure which retains Cook and Kress's central idea but except each object's rating, he has made use of each object's ratings of all the objects. Noguchi and Ishii have extended Green's rank ordering to adopt practical cases. The distance measure is based on a different idea that individual preferences for a set of alternatives should be aggregated. In attempt to rationalize various existing consensus procedures, a number of comparative studies have been carried out, say, Blin, Cook and Seiford, Fishburn, Kemeny and Snell and Cook and Kress. While some studies have concentrated on the commonality of properties exhibited by consensus procedures, others have examined the likelihood of different procedures producing the same first-ranked object. In this paper, we note the consensus measure of distance function. The distance function indicates degree of disagreement of voters and sum of distances is minimized so that consensus will be formed. The distance measure has two kinds of positioning; absolute and relative one. The absolute distance has been proposed by Blin and it evaluates voter disagreement merely in terms of whether the rank assigned to an object by two voters is same or different. This distance dose not reflect individual preferences clearly since data spread dose not have much effect on rank ordering and objects are placed in the order where each of them has the most votes. In contrast, Cook and Kress have constructed relative positioning to express the degree of differences. We extend Cook and Kress measure and add viewpoints organized by principle component analysis and weights of rankings and viewpoints. We adopt a strong ordering in which every object is not placed in the same standings. Further, we compare our distance measure to rank ordering with DEA proposed by Noguchi and Ishii. Cook and Kress distance is expressed with inner product. Analytical results are dependent on how to define distance. So, we extend the distance measure to construct joint ballot where the most favorable pairs are chosen.

並列關鍵字

Data envelopment analysis DEA ； Distance function ； Linear programming

參考文獻

Noguchi, H,Ishii, H.(2000).The application of rank ordinal data using DEA model to conjoint analysis.Math Japonica.51(1),21-34.

Noguchi H, Ishii H.(2000).The appropriate total ranking method using DEA for multiple categorized purposes.Journal of The University of Marketing and Distribution Sciences, Information/Economics & Management Scienc.9(2),21-34.

Cook, W. D.(2006).Distance-based and ad hoc consensus models in ordinal preference ranking.European Journal of Operational Research.172(2),369-385.

Cook, W. D.,Kress M(1984).Relationships between 1 metrics on linear ranking spaces.SIAM Journal on Applied Mathematics.44(1),209-220.

Green R. H.,Doyle J. R.,Cook W. D.(1996).Preference voting and project ranking using DEA and cross-evaluation.European Journal of Operational Research.90(6),461-472.