  The Analytic Hierarchy Process (AHP) is a much discussed method in ranking
business alternatives based on empirical and judgemental information. We focus
here upon the key component of deducing efficient vectors for a reciprocal
matrix of pair-wise comparisons. It is not yet known how to produce all
efficient vectors. It has been shown that the entry-wise geometric mean of all
columns is efficient for any reciprocal matrix. Here, by combining some new
basic observations with some known theory, we 1) give a method for inductively
generating large collections of efficient vectors, and 2) show that the
entry-wise geometric mean of any collection of distinct columns of a reciprocal
matrix is efficient. We study numerically, using different measures, the
performance of these geometric means in approximating the reciprocal matrix by
a consistent matrix.
