Fuzzy AHP (Van Laarhoven & Pedrycz, 1983)

Four things change; the pairwise-comparison logic does not.

Cells. In crisp AHP every cell is a single number from Saaty's 1-9 scale. Here every cell is three numbers: the lowest, most likely and highest ratio. A judgement such as "this criterion is definitely more important than that one" now corresponds to a triangle such as (7; 9; 9); the reciprocal cell is its inverse (1/9; 1/9; 1/7). The form this card covers supports group decision-making: multiple experts' triangles can be reduced to a single group matrix by the geometric mean.

Scale equalisation. Crisp AHP has no counterpart to this step; scale equalisation is not seen in a decision matrix, as in crisp TOPSIS, but in the final rescaling of the weights so that they sum to 1. The difference is that this rescaling happens not at the fuzzy stage but at the very last step, once the numbers have already been reduced to a single figure.

Synthesis (in place of the eigenvector). Crisp AHP solves the matrix's principal eigenvector. At the fuzzy stage no eigenvalue problem is set up; instead, the fuzzy geometric mean of every row is taken, and this row mean is multiplied by the fuzzy inverse of the sum of all row means to assign each criterion a fuzzy weight (three numbers). This is Buckley's (1985) geometric mean method, and the synthesis route DecisionMind fixes for this classical Fuzzy AHP; other synthesis routes exist in the literature, such as Chang's (1996) extent analysis, but they are not used here.

Result and defuzzification. What comes out of synthesis is still three numbers for every criterion. Defuzzification is carried out at the very last step by the centre-of-area method: the three components are averaged, and these single numbers are then rescaled so that they sum to 1. Only after this final step does a single crisp weight vector emerge; the ranking is read from this crisp vector.

How consistency is handled here. This is the point of greatest departure from crisp AHP, and it is plainly visible in the manifest's list of steps (F1-F5): Buckley's geometric mean method contains no calculation corresponding to crisp AHP's Consistency Ratio (CR). None of the five steps (building the fuzzy matrix, group combination, row geometric mean, fuzzy weight, defuzzification plus normalisation) enters a λmax/CI/CR chain. Although this method in DecisionMind carries a "consistency threshold" parameter (default 0.10), this parameter is not tied to any output computed in any of the five steps. As a result, Fuzzy AHP does not automatically provide the assurance crisp AHP gives, that "these judgements are internally consistent"; the consistency of the judgements remains largely the decision-maker's or the analyst's own responsibility to check. This is addressed further in the "Mistakes Specific to This Extension" section below.

The output is a weight vector summing to 1, as in crisp AHP, and it is read the same way: a criterion's weight is meaningful only relative to the other criteria in the same analysis.

The difference is this: in crisp AHP, a low CR at least gives an assurance that the judgements do not contradict one another. In the form of Fuzzy AHP described on this card, no such assurance enters the calculation at all; the weights are synthesised directly from the fuzzy comparisons and then defuzzified. Fuzzy AHP itself therefore gives no answer to the question of whether a small score gap might mean "the judgements could be contradictory"; this question must be asked separately, for instance by computing a CR on a crisp version of the comparisons.

Thus instead of writing:

"Because Fuzzy AHP takes fuzziness into account, these weights are more reliable than crisp AHP's"

the report should read:

"These weights carry the approximation in the expert judgements through to the synthesis step; whether the judgements contradict one another has not been separately checked in this method"

Use this extension where the judgement about criteria's relative importance is verbal or approximate, that is, where a statement such as "definitely more important" is too flexible to reduce to a single 7 or 9, and this flexibility should be carried through into the weights. Where a measured ratio exists (the ratio of two criteria's past budget shares, say), turning it into a triangle is not modelling uncertainty but manufacturing it; DecisionMind requires a single data type, so a measured comparison is embedded in the same matrix by writing the triangle's three components as equal (l=m=u).

Crisp AHP's own exit condition applies here too: the method works comfortably with two to twelve criteria, and the comparison burden grows quadratically as the criterion count rises. Furthermore, if automatically auditing the consistency of the judgements is part of the decision process, that is, if a CR is a requirement of the report, this classical form of Fuzzy AHP produces no CR, so either crisp AHP should be used, or a separate consistency check should be carried out.

Accepting a zero weight without questioning the method. In synthesis routes of the Chang extent-analysis type, one row's synthetic extent can dominate the others completely, and an important criterion can come out with a zero weight (Wang, Luo and Hua, 2008); cross-checking with the Buckley method fixed here is recommended.

Normalising the weights before defuzzifying them. The normalisation step is carried out on the defuzzified (crisp) weight values; normalising the fuzzy components (l, m, u) separately and defuzzifying afterwards can produce a different, mistaken ranking.

Mistaking the "consistency threshold" parameter for a CR calculation. The consistency_threshold the engine carries is a default value; none of this method's five steps produces a number to compare against this threshold. A sentence in the report such as "CR below threshold" presents something that was never calculated here as though it had been.

Defuzzifying first and skipping the fuzzy synthesis. Reducing the triangles to a single number from the outset and running crisp AHP's eigenvector method on them is not Fuzzy AHP; moreover, this route produces crisp AHP's own CR, and the reader may mistakenly think it came from the fuzzy method. The two methods become conflated.

The governing principle is this:

Fuzzy AHP exists to carry the approximation in pairwise comparisons through to the synthesis step; in doing so, it does not produce the consistency assurance crisp AHP gives, and this assurance must be sought separately if it is wanted.