Fermatean Fuzzy WPM (based on Senapati and Yager's 2020 foundation)

Two things change; the multiplicative logic does not.

Cells. In crisp WPM every cell is a single, strictly positive number. Here every cell is a support (μ) and rejection (ν) degree pair given to a judgement. The difference from intuitionistic fuzzy lies in the constraint: intuitionistic fuzzy requires μ+ν≤1, whereas here μ³+ν³≤1; the sum of cubes permits some high (μ,ν) pairs that intuitionistic fuzzy does not allow. Criterion weights are taken from outside as crisp numbers.

Direction handling and aggregation. Crisp WPM ratios every cell against its own column's best value: it divides by the largest for a benefit criterion, and divides the smallest into the cell for a cost criterion. This is WPM's single-step direction handling; it does not group columns and sum them separately. Here too, direction is handled the same way, in a single step, at the cell level: on the cost criterion, the support and rejection degrees swap places, (μ, ν) becomes (ν, μ). This is the Fermatean fuzzy counterpart of crisp WPM's "ratio every cell directly" logic. After this direction correction, every criterion is raised to a power equal to the criterion's weight and combined with the weighted geometric aggregation operator (FFWG): the combined support is computed as Πμ_j^{w_j}, and the combined rejection as the cube root of 1 − Π(1−ν_j³)^{w_j}. Support combines multiplicatively, rejection combines cube-additively; this means WPM's character of not forgiving a weak criterion is preserved in the support component, while the rejection share accumulates.

Result and defuzzification. The combined (μ, ν) pair is reduced to a single number at the final step by the Fermatean score function (s = μ³ − ν³). The score can take a value between −1 and 1; unlike crisp WPM's always-positive score, a negative score is possible here, and, as shown below, this makes it particularly fragile against a single weak criterion.

DecisionMind fixes the Fermatean score function (μ³ − ν³) and the FFWG aggregation operator in this extension. Weights are taken from outside; the method does not generate weights.

The score shows a ranking within this set of alternatives; unlike crisp WPM's score, it is not a percentage or a probability, and it can be negative. A negative score does not mean "the alternative has failed," only that the combined rejection share exceeds the combined support share; nor can it be compared with a score from a different Fermatean fuzzy WPM analysis.

The real difference lies here: because the combined support (μ) combines multiplicatively, an alternative whose support degree on a SINGLE criterion is very close to zero has its combined support driven almost to zero. The combined rejection (ν), by contrast, rises only by that criterion's rejection degree; the rejection degrees of the other criteria also enter the share. In the score formula (μ³ − ν³) there is no compensation on the support side for this "zeroing penalty," because neither crisp WPM's normalisation step nor the +1 constant found in the neutrosophic family's score function is present here. A single weak criterion can therefore turn negative the score of an alternative that is otherwise very strong on the remaining criteria.

Thus instead of writing:

"Because A1's score comes out negative, A1 is a weak alternative on every criterion"

the report should read:

"A1's score stems from a single criterion's very low support degree, which, because of multiplicative aggregation, drives the combined support share almost to zero; A1's performance on the other criteria is not visible in this score, and the T components must be examined criterion by criterion"

Fermatean fuzzy is used where the criteria come from a judgement rather than a measurement, and where the expert wants both the support and the rejection degree expressed at levels higher than intuitionistic fuzzy allows. That is, this extension applies once μ+ν>1 and intuitionistic fuzzy becomes invalid; where the μ+ν≤1 bound is already sufficient, intuitionistic fuzzy WPM is enough.

Crisp WPM's own exit condition applies here too: the criteria must be on a genuine ratio scale, and the matrix must be of a single type. In addition, no cell's support degree (μ) should be very close to zero; otherwise, whatever that criterion's weight, the combined score is pulled down severely.

Feeding an invalid pair with μ³+ν³>1 into the calculation without noticing it. This breaches the Fermatean fuzzy set's definition, and the score becomes meaningless.

Failing to notice that a single low support degree can turn the score negative. The kernel has been tested directly in Python; in the illustrative example, when one criterion's support degree was lowered from 0.7 to 0.02 (holding the rejection degree fixed), that alternative's score fell from +0.062 to −0.123. This drop occurs even if that criterion's weight is only 0.40; the reason is that FFWG combines the support component multiplicatively, and the score function (μ³−ν³) carries no normalisation term as crisp WPM's does.

Silently changing the score function. μ³−ν³ is the fixed rule here. Using μ alone, or adding an indeterminacy term, can produce a different ranking; such a change must not be made without stating it in the report.

Presenting Fermatean fuzzy as a "more advanced version" of intuitionistic fuzzy. The cube constraint only permits a wider (μ,ν) region; which constraint to use depends on the nature of the data, and is not a "better" or "worse" choice.

The governing principle is this:

In Fermatean fuzzy WPM, combined support aggregates multiplicatively; a single criterion's very low support degree can therefore inflict a penalty severe enough to turn the score negative, even where that criterion's weight is small. This is a sharper version, within Fermatean fuzzy, of crisp WPM's own trait of not forgiving a weak criterion.