Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 9

The eigenvectors are those vectors v such that, for some eigenvalue ,

λ

Mv ¼ k v :

The trace of a matrix is the sum of its eigenvalues. It is also equal to the sum of
the diagonal elements. For reciprocal matrices, since all diagonal elements are equal
to 1, the trace is equal to the number of rows or columns. Because the rank of any
consistent reciprocal matrix is 1, its non-null eigenvectors are all in the same
direction and, consequently, the non-null eigenvalue is that number of rows or
columns. In particular, it is a real number.
In contrast, if the matrix is inconsistent, it has negative eigenvalues; thus, its
highest eigenvalue is larger than the number of rows and columns. A detailed proof
of this result may be found in Saaty ( 1990 ).
If the matrix of pairwise preferences between criteria is consistent, the vector of
weights is the normalized eigenvector of the matrix. Saaty proposes then, to deal
with inconsistency, to take as the vector of weights the unitary eigenvector asso-
ciated with the highest eigenvalue, employing the value of this highest eigenvalue
to decide if the level of inconsistency in the matrix is suf ciently small.

For high levels of inconsistency, the pairwise comparison of the criteria must be
revised.
Saaty employs a measure of consistency called the Consistency Index, which is
based on the deviation of the highest eigenvalue to m, the number of criteria:

CI ¼ k ð max m Þ= ð m 1 Þ:

This index may also be seen as the negative average of the other eigenvalues of
the inconsistent matrix.
After knowing the Consistency Index, the next question is how to use it. Saaty
proposed to use this index by comparing it with an appropriate threshold. To
determine such an appropriate threshold, one can employ the Random Consistency
Index, RI, an index obtained by examining reciprocal matrices randomly generated
by Vargas ( 1982 ) using the scale 1/9, 1/8 8, 9. The random consistency

index, computed as the average of a sample of 500 matrices, for a number of criteria
varying from 3 to 10, is shown in Table 2.1 .
Then, Saaty employs what is called the Consistency Ratio, which is a com-
parison between the observed Consistency Index and the Random Consistency
Index, or, formally,

… 1, …

CR ¼ CI = RI :

Table 2.1 Random consistency index
n 3 4 5 6 7 8 9 10

RI

0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49