Analytical Hierarchy Process (AHP)

The importances of the criteria could be approximated by the AHP using pairwise comparisons:

. T. L. Saaty, The Analytic Hierarchy Process:Planning, Priority Setting, Resource Allocation, McGraw-Hill, New York, 1980.

Suppose that the value function has the form

If wi = 0, the corresponding outcome y_i can be deleted from consideration. Thus, we shall assume that w_i > 0, i = 1, 2, . . . , q.

Define the weight ratio by

Note that, for any i, j, k indexes

Define the matrix of weight ratios as W = [wij]q×q:

A matrixW is called consistent if its components satisfy the equalities w_ij = w_ikw_kj for any i, j and k.

Observe that:

• . Since each row of W is a multiple of the first row, the rank of W is one, and thus there is onlu one nonzero eigenvalue which is q.

• This due to the fact that wii = 1 and that the sum of all eigenvalues is equal to the trace of W

• . We can easily chack that

Ww = qw

therefore w must be the eigenvector of W corresponding to the maximum eigenvalue q.

As a living system, human perception and judgment are subject to change when the information inputs or psychological states of the decsion maker change.

A fixed weight vector is difficult to find. Saaty proposed the following to overcome this difficulty:

Estimate or elicit the weight ratio wij by aij and let A = [a_ij]_q×q be the matrix of components {a_ij}.

Note that as each wij > 0, we expect and shall assume that all aij > 0.

Furthermore, as , Saaty suggested that in practice, only aij, j > I need to be assessed.

Since A is found as an approximate for W, when the consistency conditions are almost satisfied for A, one would expect that the normalized eigenvector corresponding to the maximum eigenvector of A, denoted by λ_max, will also be close to w.

Theorem 1. The maximum eigenvalue, λ_max,of A is a positive real number.

Let ˆ w be the normalized eigenvector corresponding to λ_max of A. Then ˆ w_i > 0 for all 1 ≤ i ≤ q.

Theorem 2. The maximum eigenvalue of A satisfies the inequality

λ_max ≥ q.

Assume we have q objectives and we want to construct a scale, rating these objectives as to their importance with respect to the decision, as seen by the analyst.

We ask the decision maker to compare the objectives in paired comparisons.

If we are comparing objective i with objective j, we assign the values a_ij and a_ji as follows:

.

. If objective i is more important than objective j then aij gets assigned a number as follows:

Note that the above observation is valid for any matrix which is consistent.

Intensity of relative importance	Definition
1	equal importance
3	weak importance (of one over the other)
5	strong importance
7	demonstrated importance over the other
9	absolute importance
2, 4, 6, 8	intermediate values between

Saaty’s scale of relative importances.

Example 1. Let us consider the following matrix

To find λ_max we solve

det[A . λI] = 0

that is,

= (1 . λ)3 . 3(1 . λ) + 9/35 + 35/9 = 0

The maximum solution is

λ_max = 3.21.

After normalization we get

ˆ w₁ = 0.77, ˆ w₂ = 0.05, ˆ w₃ = 0.17.

We illustrate Saaty’s method on a job selection problem (3 alternatives compared on 6 criteria)

Choice of job.

The question asked was, which of a given pair of criteria is seen as contributing more to overall satisfaction with a job and what is the intensity or strength of the difference?

Pairwise comparison matrix of criteria.

The relative weights of criteria (priorities) can be computed as normalized geometric means of the rows (which are very close to the eigenvector corresponding to the largest eigenvalue of the matrix)

The geometric means are computed as

m1 = (1 × 1 × 1 × 4 × 1 × ½)^1/6

m2 = (1 × 1 × 2 × 4 × 1 × ½)^1/6

m3 = (1 × 1/2 × 1 × 5 × 3 × ½)^1/6

m4 = (1/4 × 1/4 × 1/5 × 1 × 1/3 × 1/3) ^1/6

m5 = (1 × 1 × 1/3 × 3 × 1 × 1) ^1/6

m6 = (2 × 2 × 2 × 3 × 1 × 1) ^1/6

So, the relative weight (priority) of the criterion research is obtained as

p1 = m1 / (m1 + m2 + m3 + m4 + m5 + m6)

Then we compare the alternatives on each of the

We obtain

So job A should be selected as the best alternative.