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Decision-making with the AHP: Why is the principal eigenvector necessary. (English) Zbl 1012.90015

Summary: In this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix \(A=(a_{ij})\) when \(A\) is a small perturbation of a consistent matrix. When providing numerical judgments, an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios. Near consistent matrices are essential because when dealing with intangibles, human judgment is of necessity inconsistent, and if with new information one is able to improve inconsistency to near consistency, then that could improve the validity of the priorities of a decision. In addition, judgment is much more sensitive and responsive to large rather than to small perturbations, and hence once near consistency is attained, it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one. If such perturbations were forced, they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision.

MSC:

90B50 Management decision making, including multiple objectives
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[1] Harker, P.T., Derivatives of the Perron root of a positive reciprocal matrix: with applications to the analytic hierarchy process, Applied mathematics and computation, 22, 217-232, (1987) · Zbl 0619.15017
[2] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press New York · Zbl 0576.15001
[3] Lancaster, P.; Tismenetsky, M., The theory of matrices, (1985), Academic Press New York · Zbl 0516.15018
[4] T.L. Saaty, Decision making with the AHP: Why is the principal eigenvector necessary? Proceedings of the Sixth International Symposium on the Analytic Hierarchy Process, Berne-Switzerland, August 2-4, 2001 · Zbl 1012.90015
[5] Saaty, T.L., Decision making with dependence and feedback: the analytic network process, (2001), RWS Publications Pittsburgh, PA · Zbl 1176.90315
[6] Saaty, T.L.; Vargas, L., Inconsistency and rank preservation, Journal of mathematical psychology, 28, 2, (1984) · Zbl 0557.62093
[7] Saaty, T.L.; Hu, G., Ranking by the eigenvector versus other methods in the analytic hierarchy process, Applied mathematical letters, 11, 4, 121-125, (1998) · Zbl 0942.91020
[8] Vargas, L.G., Analysis of sensitivity of reciprocal matrices, Applied mathematics and computation, 12, 301-320, (1983) · Zbl 0547.15006
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