Analytic hierarchy process (AHP), what is the significance of eigenvalues/eigenvectors? I was to a new years party today. Mathematics interested as I am I tried to discuss
eigenvalues with my friend there. He did not know what eigenvalues are but said
that he has heard about them in his research in something called
Analytic Hierarchy Process (AHP).
Analytic hierarchy process, wikipedia link
I found the following passage in wikipedia that mentions the word eigenvector, and the reference to Oskar Perron:
"Non-Monotony of some weight extraction methods
Within a comparison matrix one may replace a judgement with a less favourable judgement and then check to see if the indication of the new priority becomes less favourable then the original priority. In the context of tournament matrices, it has been proven by Oskar Perron in,[32] that the principle right eigenvector method is not monotonic. This behaviour can also be demonstrated for reciprocal n x n matrices, where n > 3. Alternative approaches are discussed in.[33][34][35]"
Reading about Oskar Perron I found that he worked in differential equations. Then reading about differential equations and eigenvalues in my book Advanced Engineering Mathematics, in the chapter Eigenvectors from the Start, page 299, I found:
"The general solution of system of differential equations $x' = Ax$ is known as soon as the eigenvalues and eigenvectors are known. In fact, if the eigenvalues are $\lambda_1$, $\lambda_2$
and the corresponding eigenvectors are $v_1, v_2$ then the general solution would be
$$x = a \cdot \exp(\lambda_1 \cdot t) \cdot v_1 + b \cdot \exp(\lambda_2 \cdot t) \cdot v_2$$
So my question is: What is the significance of eigenvalues in Analytic Hierarchy Process (AHP)?
 A: what is priority or
more generally what meaning should we attach to a
priority vector of a set of alternatives?
We can think of two meanings. The first is a numerical ranking of
the alternatives that indicates an order of preference
among them. The other is that the ordering
should also reflect intensity or cardinal preference
as indicated by the ratios of the numerical values
and is thus unique to within a positive multiplicative
constant (a similarity transformation). It is the
latter that interests us here as it relates to the
principle of hierarchic composition under a single
criterion. Given the priorities of the alternatives
and given the matrix of preferences for each alternative
over every other alternative, what meaning
do we attach to the vector obtained by weighting
the preferences by the corresponding priorities of
the alternatives and adding? It is another priority
vector for the alternatives. We can use it again to
derive another priority vector ad infinitum. Even
then what is the limit priority and what is the real
priority vector to be associated with the alternatives?
It all comes down to this: What condition
must a priority vector satisfy to remain invariant
under the hierarchic composition principle? A priority
vector must reproduce itself on a ratio scale
because it is ratios that preserve the strength of
preferences. Thus a necessary condition that the
priority vector should satisfy is not only that it
should belong to a ratio scale, which means that it
should remain invariant under multiplication by a
positive constant c, but also that it should be invariant
under hierarchic composition for its own
judgment matrix so that one does not keep getting
new priority vectors from that matrix. In sum, a
priority vector x must satisfy the relation Ax = cx,
c > 0. Hence, it is clear that principal eigenvector is the only plausible candidate for representing priorities derived from a positive reciprocal near
consistent pairwise comparison matrix.
Source: Saaty T L (2003) "Decision making with the AHP: Why is the principal eigenvector necessary", European Journal of Operational Research, 145, 85-91
A: The second link by Amzoti in the comment above is very clear about how the algorithm goes.
Illustration of Analytic hierarchy process
I have begun to study this by looking at the more arithmetic table $n/k$ as the pairwise comparison:
$$\begin{array}{llllll}
 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\
 2 & 1 & \frac{2}{3} & \frac{1}{2} & \frac{2}{5} & \frac{1}{3} \\
 3 & \frac{3}{2} & 1 & \frac{3}{4} & \frac{3}{5} & \frac{1}{2} \\
 4 & 2 & \frac{4}{3} & 1 & \frac{4}{5} & \frac{2}{3} \\
 5 & \frac{5}{2} & \frac{5}{3} & \frac{5}{4} & 1 & \frac{5}{6} \\
 6 & 3 & 2 & \frac{3}{2} & \frac{6}{5} & 1
\end{array}$$
Mathematica:
TableForm[Table[
  Rationalize[
   Total[Transpose[
      MatrixPower[N[Table[Table[n/k, {k, 1, i}], {n, 1, i}]], i]]]/
    Total[Total[
      Transpose[
       MatrixPower[N[Table[Table[n/k, {k, 1, i}], {n, 1, i}]], 
        i]]]]], {i, 1, 6}]]

The output is very orderly:
$$\begin{array}{llllll}
 1 & \text{} & \text{} & \text{} & \text{} & \text{} \\
 \frac{1}{3} & \frac{2}{3} & \text{} & \text{} & \text{} & \text{} \\
 \frac{1}{6} & \frac{1}{3} & \frac{1}{2} & \text{} & \text{} & \text{} \\
 \frac{1}{10} & \frac{1}{5} & \frac{3}{10} & \frac{2}{5} & \text{} & \text{} \\
 \frac{1}{15} & \frac{2}{15} & \frac{1}{5} & \frac{4}{15} & \frac{1}{3} & \text{} \\
 \frac{1}{21} & \frac{2}{21} & \frac{1}{7} & \frac{4}{21} & \frac{5}{21} & \frac{2}{7}
\end{array}$$
which is $$\frac{k}{n(n+1)/2}$$
I will continue editing this later when I have the time.
nn = 5
Do[
 A = N[Table[Table[n/k, {k, 1, nn}], {n, 1, nn}]];
 A = MatrixPower[A, n];
 MatrixForm[N[A]];
 MatrixForm[Total[Transpose[N[A]]]];
 Total[Total[Transpose[N[A]]]];
 Print[Rationalize[
   Total[Transpose[N[A]]]/Total[Total[Transpose[N[A]]]]]], {n, 1, 6}]

