Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 8

2.2.2 AHP Tools

The most noticeable feature of AHP is the form employed to address the incon-
sistencies arising from the pairwise comparisons. The relative preferences are
registered in a square matrix M, where the ij-th entry, m ij , measures the ratio
between the preference for the j-th and the i-th criterion. Thus, the m ij are positive
numbers with

m ij ¼ 1 = m ji :

A square matrix with these properties is called a positive reciprocal matrix.
This matrix of preference ratios is consistent if and only if, not only

m ij *m ji ¼ m ii ¼ 1

for every pair (i, j) but also, for every triple (j 1 ,j 2 ,j 3 ),

m j1j2 *m j2j3 ¼ m j1j3 :

Obviously, given a row or a column of a reciprocal matrix, consistency deter-
mines the rest of the matrix. However, when informing the preference ratios for
each pair of criteria on the scale from 1 to 9, the dif

culty in evaluating abstract
criteria leads to inaccuracies in such a way that it is not expected that these reci-
procal matrices will be consistent in practice.
The Analytic Hierarchy Process (AHP) is designed to allow for inconsistencies
due to the fact that, in making judgments, people are more likely to be cardinally
inconsistent than cardinally consistent (Saaty 2003 ). The decision makers are not
able to estimate precisely measurement values even with a tangible scale, and the
dif culty is worse when they address abstract concepts.

If a reciprocal matrix is consistent, all its rows and columns are proportional to
each other. This means that they span a linear space of dimension 1. In other words,
the rank of consistent reciprocal matrices is equal to 1.
By the rank of a matrix we mean the dimension of the space generated by their
columns (or their rows), i.e., the maximal number of linear independent columns (or
rows).
In addition to the concept of rank, the concepts of the trace of a square matrix
and their eigenvalues and eigenvectors play an important role in the weighting of
the criteria in AHP.
The eigenvalues of a matrix M are the real or complex numbers λ such that, for

some vector v,

Mv ¼ k v :