A New Definition of Consistency of Pairwise Comparisons

A new definition of consistency is introduced. It allows us to locate the roots of inconsistency and is easy to interpret. It also forms a better basis than the old eigenvalue consistency for selecting a threshold based on commnn sense. The new definition of consistency is applicable to expert systems and to knowledge acquisition. It is instrumental in applications of fuzzy sets and the theory of evidence where the definitions of the membership and belief functions are fundamental issues. A method of pairwise comparison introduced by Thurstone [l] in 1927 was a milestone in decision making science. It is comparable to the introduction of derivatives in calculus or eigenvalues in linear algebra. The decision making process nearly always involves some kind of constituency in modern democratic societies. We have various boards of governors or directors, committees, task groups, city councils, panels of experts, etc. Stormy discussion and various ways of dispute, reasoning, and argumentation take place to arrive at certain decisions. Most constituencies have worked out precise and practical policies for running meetings in an orderly and effective way. What we lack, however, is a device for drawing solid conclusions, and very often the loudest individual wins! Unfortunately, loudness does not necessarily go along with wisdom. Casual thinking does not work well in predicting complex outcomes. Casual thinking is partial, fragmentary, and has no effective way to measure intangibles. In the decision making process, many factors must be considered simultaneously and with about the same degree of importance, therefore an approach with more finesse is necessary to obtain a clear and unambiguous conclusion. It has been shown by numerous examples [2,3] that the pairwise comparison method can always be used to draw the final conclusions in a comparatively easy and elegant way. The brilliance of the pairwise comparison could be reduced to a common sense rule: take two at a time if you are unable to handle more than that. It is intriguing why such a natural and powerful tool has never become widely accepted by decision makers despite its extreme practicality. The author of this paper truly believes that failure of the pair-wise comparison method to become more popular has its roots in the consistency definition. The consistency definition is given by the following formula: AA order(A) cf = (order(A) 1) &n&m ’ 0) for a reciprocal matrix: A = [wij], where wij is a positive element which expresses the relative importance of two stimuli: i and j. This project wss partially supported by the Natural Science and Engineering Council of Canada under Grant OGP 003838.