AHP: the analytic hierarchy process

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

AHP in one sentence

AHP (Analytic Hierarchy Process) was built by Thomas Saaty in the 1970s. The idea is to turn a complex problem into a hierarchy (goal, criteria, sub-criteria, alternatives) and compare only two things at a time, because the human mind is more accurate at pairwise comparison than at weighting ten things at once.

Four principles

1. Reciprocity: if the preference of $A$ over $B$ is $a$, the preference of $B$ over $A$ is $1/a$.
2. Homogeneity: the compared elements must be of the same order of importance (“beauty” cannot be compared with “the price of a screw”).
3. Hierarchical dependence: the elements of each level depend on the level above.
4. Expectations: any change in the hierarchy’s structure must restart the evaluation.

Steps

1. Build the hierarchy.
2. Pairwise comparison of the elements of each level with Saaty’s scale: 1 (equal importance), 3 (slightly more important), 5 (more important), 7 (much more important), 9 (extremely more important) and the even numbers between.
3. Build the pairwise comparison matrix $A$.
4. Compute the weights from $A$: the eigenvector method (most accurate), or approximations such as the arithmetic mean of normalised columns, the geometric mean, row and column sums; or least squares.
5. Check consistency and merge the weights to reach the final weights of the alternatives.

Consistency and the inconsistency ratio

If $a_{ij}\cdot a_{jk}=a_{ik}$ holds for all triples, the matrix is perfectly consistent; then the largest eigenvalue equals $n$. In practice judgements are slightly inconsistent. Two indicators:

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

where $RI$ is Saaty’s random index ($n=3$: 0.58; 4: 0.90; 5: 1.12; 6: 1.24; 7: 1.32; 8: 1.41; 9: 1.45; 10: 1.51). If $CR<0.1$, the inconsistency is acceptable; otherwise the judgements must be revised.

Numerical example

Comparing three criteria:

$A=\left(\begin{array}{ccc}1& 2& 8\\ 1/2& 1& 6\\ 1/8& 1/6& 1\end{array}\right)$

Normalising the columns and averaging the rows: $w\approx (0.593,0.341,0.066)$. With ${\lambda }_{max}\approx 3.018$ we get $CI\approx 0.009$ and $CR\approx 0.016<0.1$: the judgements are acceptable. (The eigenvector method gives nearly the same weights.)

Group AHP

With several experts, we combine the judgements with the geometric mean ($a_{ij}^G=({\prod }_k{a}_{ij}^{(k)}{)}^{1/K}$), because the geometric mean preserves reciprocity and the arithmetic mean does not. (This is the AIJ method; the alternative is AIP, which takes each expert’s weights separately and then combines them.)

Beyond AHP

• ANP (Analytic Network Process): when the criteria depend on each other and feed back, it uses a supermatrix instead of a hierarchy. It is more complex and therefore used less.
• DEMATEL: measures cause-and-effect relations among elements with a direct-influence matrix (row sum $R$ and column sum $J$; $R+J$ is importance and $R-J$ the causal/effect type).
• BWM: a cheaper version of pairwise comparison. (Weighting criteria)

Critique

The well-known critique of AHP is “rank reversal” (adding a new alternative sometimes changes the ranking of the old ones) and that the 1-to-9 scale is subjective. I think AHP is excellent for group decisions and for making a discussion transparent, but its result should not be felt as “precise”.

Related: Weighting criteria and Scoring methods: SAW, WPM and WASPAS.

AHPpairwise_comparisonmulti_criteria_decision_making