AHP (Analytic Hierarchy Process)

AHP is a weighting method for use when you already have criteria in hand and want to turn their relative importance into numbers through expert or decision-maker judgement. Its output is two things: a weight vector summing to 1, and a ratio showing how internally consistent the judgements behind those weights are. It does not rank alternatives and does not evaluate options. Within DecisionMind, AHP works only at the criterion-weighting layer; the weights it produces feed as inputs into ranking methods such as TOPSIS or VIKOR. Saaty proposed the method in 1980. An application survey found it used in 128 of 393 papers reviewed (32.57 per cent), reportedly the most frequently applied method among MCDM techniques (Mardani et al., 2015).

AHP's underlying idea is this: rather than putting a criterion's importance into a single number directly, set criteria against one another two at a time. Direct scoring says "this criterion rates 70 out of 100 in importance"; AHP instead asks "how many times more important is this criterion than that one?" The human mind performs relative comparison more reliably than absolute scoring. For n criteria, n(n-1)/2 comparisons are called for. That is more than strictly necessary to fix the weights. This very redundancy is what sets AHP apart from the other two subjective weighting methods: the same judgements suffice both to compute the weights and to measure whether those judgements contradict one another, in other words their consistency.

This idea carries a philosophical consequence. Among subjective weighting methods, AHP is the only one able to audit itself. BWM asks only for comparisons against the most and least important criterion. SWARA follows an ordered chain without building any comparison matrix at all. Both demand fewer judgements, but neither carries AHP's dense internal cross-checking. AHP's cost shows up in exactly the same place: as the number of criteria grows, the comparison burden grows quadratically.

The method proceeds through four steps.

In the first step, a pairwise comparison matrix is built. Criteria are compared two at a time on Saaty's scale, which runs from 1 to 9. On this scale, 1 means "equally important" and 9 means "definitely more important." The matrix is reciprocal: if one criterion is 3 times as important as another, the other is worth exactly the inverse, that is 1/3, as important. The diagonal is always 1, because a criterion is being compared with itself.

In the second step, the priority vector is extracted. The correct way to obtain a single weight vector from the matrix is to find its principal eigenvector. Dividing each column by its own sum and then averaging the rows is a common shortcut. But this is only an approximation, and it drifts from the true eigenvector whenever the matrix is not fully consistent. DecisionMind uses the full eigenvector solution, not the shortcut.

In the third step, the Consistency Index (CI) is calculated. The closer the matrix's largest eigenvalue (λmax) sits to the number of criteria (n), the more consistent the matrix is. For a fully consistent matrix, λmax equals n exactly. CI is this gap, normalised.

In the fourth step, the Consistency Ratio (CR) is calculated. CI is divided by the average inconsistency of randomly filled matrices of the same size, an average called the Random Index (RI). If the result falls below 0.10, the judgements are considered acceptable; above it, the weights are not used until the comparisons have been reviewed.

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

A weight shows a criterion's relative importance against the others within this particular comparison set. Weights always sum to 1, and a criterion's weight is meaningful only relative to the other criteria in the same analysis. CR measures something different: whether the judgements contradict one another. The two should not be confused. A low CR does not show that the decision-maker has correctly captured the "true" importance; it shows only that the judgements given are mathematically consistent with one another. A completely wrong but internally consistent set of judgements can still yield CR=0.

Thus instead of:

"CR came out close to zero, so these weights are correct"

write:

"These judgements are internally consistent (CR≈0); whether the weights are accurate still depends on whether the decision-maker has assessed the criteria correctly"

AHP works with crisp data: every cell of the pairwise comparison matrix takes a single value from Saaty's scale (1–9 and its reciprocals). Where expert judgement is uncertain, or experts disagree with one another, AHP has fuzzy, hesitant fuzzy and other extensions; DecisionMind holds five AHP family members alongside the base method.

You need a list of the criteria to be compared and, for every pair, a judgement of "which one is more important, and by how much." The matrix must be filled reciprocally; the lower triangle should never be entered independently of the upper. AHP produces weights and requires none from outside. It needs no alternative data at all, only the criterion set to be weighted. A minimum of two criteria is required; three to twelve work comfortably, but as the criterion count grows, the n(n-1)/2 comparisons grow rapidly and the expert struggles to stay consistent.

AHP is a sound choice when you must set criteria's relative importance through expert judgement and want to audit numerically whether those judgements are internally consistent. It works best with a small number of criteria (three to twelve) and an expert or board able to bear the comparison workload.

Avoid it when no expert opinion is available, when repeated comparisons keep producing high inconsistency (CR above 0.10) that cannot be fixed, or when the criterion count is so large that the comparison burden becomes impractical. Where no expert exists, let the weights come from the data itself.

Expert judgement plus a consistency audit wanted, criteria few to moderate → AHP

Same need, but too many criteria, comparison burden must shrink → BWM

Criteria easily ranked, speed is the priority → SWARA

No expert, weights derived from data → Entropy, CRITIC (objective)

Not weights but a ranking is needed → TOPSIS, VIKOR, PROMETHEE, the ELECTRE family (any of which can take AHP's weights as input)

AHP's clearest advantage is that, among subjective weighting methods, it is the only one with a built-in consistency audit (CR). The decision-maker does not receive weights alone but also sees whether the judgements underlying them contradict one another. The pairwise-comparison format produces human judgement that is both easier to give and more reliable than direct scoring. Application surveys report AHP as the most frequently used MCDM method (Mardani et al., 2015), which brings it broad literature support and comparability.

Its limitations stem from the same structure. First, the number of comparisons grows quadratically with the number of criteria: ten criteria require 45 comparisons, and the expert struggles to keep that many judgements consistent. Second, if the column-normalise/row-average shortcut replaces the full eigenvector method, it yields a different weight vector, particularly on inconsistent matrices; whichever method is used must be stated in the report. Third, in AHP's full hierarchical version, where alternatives are also compared pairwise, rank reversal is a known problem: adding a new alternative can change the existing ranking (Belton and Gear, 1983). In DecisionMind's weight-generating use, this risk shifts instead to criteria added or removed. Fourth, a low CR shows the internal consistency of the judgements, not their correctness. Different normalisation and comparison variants can assign different weights to the same set of judgements (Triantaphyllou, 2000).

The most common mistake is breaking reciprocity. If, once the upper triangle is filled, the lower triangle is entered independently so that a_ij times a_ji does not equal 1, the matrix becomes undefined and the result unreliable.

A second mistake is using the column-normalise/row-average shortcut instead of the full eigenvector method while reporting it as though it were the same thing; the two give different weights on inconsistent matrices. A third is ignoring CR and proceeding with judgements that come out above 0.10. A fourth is claiming, when CR is low, that the weights have "captured the true importance"; CR measures only internal consistency. A fifth is pushing the criterion count well beyond the comfortable range (3–12) and still expecting the expert to stay consistent.

The governing principle is this:

AHP's weights are an internally consistent summary of the pairwise comparisons supplied. CR audits that consistency, not whether the judgements correctly reflect the criteria; that responsibility remains the decision-maker's.