Z-Number AHP (Nuriyev, 2020)

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 has two parts: the constraint (A), the "how many times more important" judgement; and the reliability (B), how much this judgement is trusted. The constraint is given as a triangular fuzzy number, exactly as in Fuzzy AHP. Reliability is a separate triangular number, or a pre-declared verbal term (in Nuriyev's scale, seven steps from "very low" to "very high"). This extension supports group decisions: several experts' comparison matrices can be reduced to a single group matrix, each carrying its own expert's reliability judgement.

Scale equalisation. As in crisp AHP, there is no separate normalisation step here either; weights are rescaled at the very last step so that they sum to 1. Up to this point it is identical to Fuzzy AHP.

Embedding reliability into the comparison. This is the real step that departs from the base method and from Fuzzy AHP. The reliability triangle is first reduced to a single number (α); this number is the triangle's centroid. The constraint triangle's three components are then multiplied by the square root of this number.

If reliability is high (α close to one), the constraint stays almost as it was. If reliability is low, the constraint shrinks, that is, the magnitude of the "how many times more important" judgement weakens. The more solid the source a given comparison rests on, the more strongly that comparison enters the synthesis step.

Synthesis and defuzzification. The constraint triangles, scaled by reliability, are combined by Buckley's (1985) fuzzy geometric-mean method, exactly as in Fuzzy AHP, and reduced to a single number by the centroid. Where there is more than one expert, DecisionMind keeps a fixed order. First each expert's own comparison is scaled by reliability; then the experts' scaled comparisons are combined by a plain average; the geometric mean and defuzzification are carried out last. Reversing this order, that is, averaging the raw Z-numbers first, gives a different, incorrect result.

DecisionMind fixes this order and the centroid defuzzification (not the graded mean) for classical Z-AHP. Crisp AHP's Consistency Ratio (CR) is not calculated here; unlike in Fuzzy AHP, this safeguard is not automatic.

The weight shows a criterion's relative importance against the others within this comparison set, exactly as in crisp AHP; the weights always sum to 1.

The difference is here. A criterion's weight now depends not only on "how important it was seen to be" but also on "how reliable a judgement that importance rests on." The same "twice as important" judgement, if it comes from a solid source, affects the weight substantially; if it comes from a weak source, its effect shrinks and the other comparisons come to the fore. So if two criteria's weights are close to one another, it should be asked separately which comparison's reliability this closeness is sensitive to.

Thus instead of writing:

"Since the environmental criterion received the highest weight, this is the most important criterion for the decision-makers"

the report should read:

"The environmental criterion received the highest weight with these comparisons; this advantage is sensitive to the reliability of the comparison between the environmental and economic criteria, and the rank may change if this reliability is considered low"

Use this extension when comparisons come from different sources and the reliability of those sources differs from one another. If one comparison comes from a long-experienced expert and another from a newcomer, treating the two as equally weighted hides this difference. If all comparisons come from a source of the same reliability, the reliability component carries the same value in every cell and adds no discriminating information; in that case Fuzzy AHP is sufficient.

Crisp AHP's exit condition applies here too: it works comfortably with two to twelve criteria. If a measured ratio exists (for instance, if two criteria's past budget shares are known precisely), expanding this into a triangle and a reliability term is producing uncertainty, not modelling it. In that case the comparison is written as crisp, and reliability is also written as "certain" (α=1).

Averaging the raw Z-numbers first and applying reliability scaling afterwards. The correct order is to scale each expert's own comparison by reliability first, then average the scaled numbers. If the order is reversed, that is, if the raw comparisons are averaged first and reliability applied afterwards, a different weight results.

Confusing the constraint (A) triangle with the reliability (B) triangle. Both can be triangular numbers, but they answer separate questions: one is "how many times more important", the other "how much is this judgement trusted." Writing the same triangle into both erases this distinction.

Enforcing reciprocity for the constraint but not for reliability. When the upper triangle is filled in, the lower triangle's reciprocal cell must be filled in automatically for both the constraint and reliability. Reversing only the constraint and entering reliability separately by hand produces an inconsistent matrix.

Confusing the centroid defuzzification with the graded mean (dividing l+4m+u by 6). DecisionMind uses the centroid, that is, the plain average of the three components, in classical Z-AHP. These two defuzzification methods give different numbers; the report should state which one was used.

The governing principle is this:

Z-AHP weights carry both the magnitude and the reliability of the comparisons. If reliability is low, the comparison enters the synthesis step more weakly; this requires the report to show which comparison is driving how much of the weight.