Fermatean Fuzzy MOORA

Two things change; the vector normalisation and ratio-system logic do not.

Cells. In crisp MOORA every cell is a single number. Here every cell is a pair of support (μ) and rejection (ν) degrees given to a judgement. The constraint is μ³+ν³≤1. Criterion weights are taken from outside as crisp numbers.

Defuzzification comes right at the start, and the vector norm is taken over the scores. Every (μ, ν) pair collapses to a single number in the very first step of the calculation, via the Fermatean score function (s = μ³ − ν³). Since the score can range from −1 to 1, 1 is added to shift it to the positive side. Crisp MOORA's vector normalisation, dividing every value by the square root of the sum of squares of the values in its column, is applied to these shifted scores; there is no separate fuzzy normalisation applied to a triangular number or a support-rejection pair.

Ratio system. The normalised scores are multiplied by the weights; for every alternative, the sum of weighted shares on the cost criteria is subtracted from the sum of weighted shares on the benefit criteria. This is exactly the same operation as crisp MOORA's ratio system, except that the input now comes from the Fermatean score.

Reference point. Although it appears as a separate step in the manifest, DecisionMind's engine for this family does not compute the reference-point approach independently of the ratio system; this step simply copies the ratio-system score as it stands. This means the only real number in this extension is the ratio-system score; "cross-checking with the reference point" is not a second, independent calculation here.

DecisionMind holds the Fermatean score function (μ³ − ν³), the +1 shift of the score, and the vector norm over these shifted scores fixed for this extension. Weights are taken from outside; the method does not produce weights.

The MOORA score is read as in crisp MOORA: it is an alternative's net advantage on beneficial criteria relative to its share on harmful criteria, and it cannot be compared with a different analysis. The difference is this: this share is not a reversible support-rejection pair, only the shifted and normalised form of the Fermatean score. If, on a given criterion, all alternatives' scores sit close together (even if their μ,ν pairs differ considerably but their scores are equal), that criterion's discriminating power weakens here too, just as in crisp MOORA.

Thus instead of writing:

"According to Fermatean fuzzy MOORA, this is the best alternative, because both the ratio system and the reference point confirmed the same result"

the report should read:

"With these weights and this alternative set, the alternative with the highest net score according to the ratio system is this one; in this family the reference point is not an independent calculation and does not count as a second confirmation"

This extension is considered when experts' support-rejection pairs also exceed Pythagorean fuzzy data's squared constraint (where the sum of squares exceeds 1 but the sum of cubes does not); details are on the Pythagorean fuzzy data-type card. If the two degrees already fit within the Pythagorean constraint, moving to Fermatean is not necessary, and only discriminating power is lost. Converting a measured value directly into a support-rejection pair is not modelling uncertainty but manufacturing it. The exit condition of crisp MOORA applies here too: where a criterion admits no compromise, this extension is also compensatory.

Violating the value domain. Every cell must satisfy μ³+ν³ ≤ 1; this is checked from the sum of cubes, not squares.

Writing ν as 1 − μ. In that case the sum is always exactly 1, and the extra room the cube constraint provides is never used.

Treating the reference-point row as an independent second result. In this family the reference point is a copy of the ratio-system score. Cross-checking the ranking by claiming "both methods gave the same result" is misleading; crisp MOORA's own warning against relying on a single ranking without testing it against the reference point applies differently in this extension: the test simply is not possible here.

Marking criterion direction wrongly. If a cost criterion is marked as a benefit criterion, that criterion's worst score is pulled towards the ideal and the ranking becomes meaningless.

The governing principle is this:

Fermatean fuzzy MOORA's cube constraint adds something only when experts' pairs genuinely exceed the Pythagorean constraint; in this engine the only real number is the ratio-system score, and the reference point is not an independent verification.