## Hierarchies, reciprocal matrices and ratio scales |

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14 elements 3k nh-l Amos Tversky analysis backward process Cairo Tokyo Chicago Montreal Cluster analysis column column vector complete hierarchy components composite vector CORNELL UNIVERSITY LIBRARY corresponding decomposition differences distance from Philadelphia Eating Places efficiency Eigen eigenvalue problem error estimate example favor one activity given hierarchical structures hierarchy levels inconsistency individual influence interactions Interval Scale judgments largest element Learning Friends School Location Reputation Mathematical matrix measure method multiplicative nm-l number of elements number of levels objects obtain ordered set outcome overall pairwise comparison matrices Preparation Music Classes priority functions priority vector psychological rank ratio scale RECIPROCAL MATRICES reciprocal value relation relative represented Research Growth Benefits result Saaty scale values Science seven clusters seven elements small perturbations subset Sudan systems theory theorem tion University of Pennsylvania upper bounds variables vector S Q weighting function weights Xmax York zero