AHP - Analytic Hierarchy Process

Overview

The priority vector entries sum to 1 and represent relative importance. Always inspect CR; if CR > 0.10, revisit the pairwise judgments before using the weights downstream.

Output

Weight, higher is better

Data

Crisp, expert input required

Weights

Derived internally, no weight source needed

Size

2+ alternatives, 3-12 criteria works best

Used for

Site selection, Supplier selection, policy ranking, project portfolio

How it works

1. 1

   Construct the pairwise comparison matrix A on Saaty's 1-9 fundamental scale; check reciprocity.

   $$A=[a_{ij}{]}_{n\times n},a_{ij}>0,a_{ji}=1/a_{ij},a_{ii}=1$$

   Saaty 1980, Ch.1-3 (pairwise comparison construction; pending PDF page verification)

2. 2

   Compute the principal eigenvalue λ_max and its right eigenvector; normalise so its entries sum to 1.

   $$Aw={\lambda }_{\max }w,\sum _{i=1}^n{w}_i=1,w_i>0$$

   Saaty 1980, Ch.7 (principal-eigenvector priority extraction; pending PDF page verification)

3. 3

   Compute the Consistency Index CI = (λ_max − n)/(n − 1).

   $$CI=\frac{{\lambda }_{\max }-n}{n-1}$$

   Saaty 1980, Ch.11 (Consistency Index definition; pending PDF page verification)

4. 4

   Compute the Consistency Ratio CR = CI / RI(n), where RI(n) is Saaty's Random Index for matrix order n.

   $$CR=\frac{CI}{RI(n)},\text{accept if }CR\le 0.10$$

   Saaty 1980, Ch.11 (Random Index table; RI_3=0.58 pending PDF table verification)

Fits when / Look elsewhere when

Assumptions to verify

• Domain experts available
• Experts can express consistent comparisons

Edge cases and pitfalls

• Filling the lower triangle independently of the upper triangle breaks reciprocity a_ij · a_ji = 1.

• Using the geometric-mean approximation in place of the principal eigenvector for inconsistent matrices yields different weights - choose one method and document it.

Works with

How to cite