Deriving weights from pairwise comparison ratio matrices: An axiomatic approach. (English) Zbl 0652.90002

Cook, Wade D.; Kress, Moshe

Eur. J. Oper. Res. 37, No. 3, 355-362 (1988).

This paper examines the problem of extracting object or attribute weights from a pairwise comparision ratio matrix. This problem is approached from the point of view of a distance measure on the space of all such matrices. A set of axioms is presented which such a distance measure should satisfy, and the uniqueness of the measure is proven. The problem of weight derivation is then shown to be equivalent to that of finding a totally transitive matrix which is a minimum distance from the given matrix. This problem reduces to a goal programming model. Finally, it is shown that the problem of weight derivation is related to that of ranking players in a round robin tournament. The space of all binary tournament matrices is proven to be isometric to a subset of the space of ratio matrices.

MSC:

91B06

Decision theory

Keywords:

pairwise comparision ratio matrix; set of axioms; distance measure; weight derivation; goal programming; round robin tournament

References:

[1]	Ali, I.; Cook, W.D.; Kress, M., On the minimum violations ranking of a tournament, Management science, 32, 660-672, (1986) · Zbl 0601.90003
[2]	Barzilai, J.; Cook, W.D.; Golany, B., Consistent weights for judgement matrices of the relative importance of alternatives, Operations research letters, 6, 131-134, (1987) · Zbl 0622.90004
[3]	Bradley, R.A., Science, statistics and paired comparisons, Biometrics, 32, 213-232, (1976) · Zbl 0365.62072
[4]	Brunk, H.D., Mathematical models for ranking from paired comparisons, Journal of the American statistical association, 55, 503-520, (1960) · Zbl 0101.11902
[5]	Chu, A.T.W.; Kalaba, R.E.; Spingarn, K., A comparison of two methods for determining the weights of belonging to fuzzy sets, Journal of optimization theory and applications, 27, 531-538, (1979) · Zbl 0377.94002
[6]	Cogger, K.O.; Yu, P.L., Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios, Journal of optimization theory and applications, 46, 483-491, (1985) · Zbl 0552.90050
[7]	Cook, Wade D.; Kress, Moshe, Ordinal ranking with intensity of preference, Management science, 31, 26-32, (1985) · Zbl 0608.90003
[8]	Cook, Wade D.; Seiford, L.M., Priority ranking and consensus formation, Management science, 24, 1721-1732, (1978) · Zbl 0491.90006
[9]	Crawford, G.; Williams, C., A note on the analysis of subjective judgement matrices, Journal of mathematical psychology, 29, 387-405, (1985) · Zbl 0585.62183
[10]	Daniels, H.E., Round-Robin tournament scores, Biometrika, 56, 295-299, (1969)
[11]	David, H.A., Ranking the players in a round Robin tournament, Review of the international statistical institute, 39, 137-147, (1971) · Zbl 0224.90080
[12]	deCani, J.S., A branch and bound algorithm for maximum likelihood paired comparison ranking, Biometrika, 59, 131-135, (1972) · Zbl 0245.62037
[13]	Jeck, T., The ranking of incomplete tournaments: A Mathematician’s guide to popular sports, American mathematical monthly, 90, 246-266, (1983) · Zbl 0519.05034
[14]	Kemeny, J.G.; Snell, L.J., Preference ranking: an axiomatic approach, (), 9-23
[15]	Saaty, T.L., The analytic hierarchy process, (1980), McGraw-Hill New York · Zbl 1176.90315
[16]	Stob, M., A supplement to “A Mathematician’s guide to popular sports”, American mathematical monthly, 91, 277-282, (1984) · Zbl 0538.62064