Fermatean Fuzzy WASPAS (Senapati and Yager, 2020)

Three things change; the WASPAS logic of blending the sum and the product with λ does not.

Cells. In crisp WASPAS every cell is a single number. Here every cell is a support (μ) and rejection (ν) degree pair given to a judgement. The constraint is μ³+ν³≤1. The sum of cubes grows more slowly than the sum of squares, so this constraint permits some pairs (both above 0.8, say) that Pythagorean fuzzy does not allow. Criterion weights are taken from outside as crisp numbers.

Complementing on the cost criterion. Crisp WASPAS equalises every column by dividing by the best value; this division is not defined on (μ, ν) pairs. Instead, on cost criteria, every cell's support and rejection degree swap places: on a "less is better" criterion, high support is now read as low rejection. Benefit criteria stay as they are. This correction corresponds to the base method's column division, but it is a swap, not a division.

FFWA and FFWG in place of the sum and the product. Crisp WASPAS's weighted sum (WSM) is matched here by the Fermatean Fuzzy Weighted Average (FFWA): support degrees are combined optimistically across the criteria. Its weighted product (WPM) is matched by the Fermatean Fuzzy Weighted Geometric (FFWG): here rejection degrees dominate, and a strong rejection on a single criterion drags the overall result down. Each combination produces its own (μ, ν) pair; these are reduced to a single number by the Fermatean score function (support cubed minus rejection cubed).

Result and combination. The two scores, one from FFWA and one from FFWG, are blended with λ. In crisp WASPAS, λ is an input that DecisionMind assumes to be 0.5 if unspecified, but which can be changed where needed. When this card's author read the DM3 manifest and kernel code, FF-WASPAS's D.parameters list contained only the weights and the criterion directions; λ is defined nowhere as a parameter. The kernel code falls back directly to 0.5 whenever no extras.lambda is supplied externally; this matches crisp WASPAS's own default, but the user can never change it from the interface.

DecisionMind fixes the complementing rule, the FFWA/FFWG combination, the Fermatean score function and λ=0.5 in this extension. Weights are taken from outside as crisp numbers.

The output is a combined score and a rank, as in crisp WASPAS; it is read the same way: it is not a percentage or a probability, and the ranking can shift when the alternative set changes.

The difference is this. The score is no longer a single measurement but a blend of two distinct views: one optimistic (FFWA), the other cautious (FFWG). Each answers the question "how much is this supported, how much is it rejected" from its own angle. If the two components give the same ranking, the result is robust; if they suggest a different ranking, the combined score depends on λ. But because λ is fixed here, this dependency can remain hidden at a point the user cannot see.

Thus instead of writing:

"According to FF-WASPAS, the alternative with the highest score is definitively the best"

the report should read:

"The FFWA and FFWG components have been blended at λ=0.5; the result should not be presented as certain without first checking whether these two components, computed separately, give the same ranking"

This extension is suitable where the criterion evaluation comes from a judgement and the rejection degree carries as much information as the support degree. It is also preferable where this pair's sum can fall outside the region Pythagorean fuzzy allows (sum of squares ≤1). It suits assessments where experts' views can both be high, for example, both strong support and a genuine, if slight, degree of rejection. Where the rejection degree is computed as 1 minus the support, this extension is unnecessary; the crisp or the fuzzy structure carries the same information. Where the value is measured, the base method is used instead.

WASPAS's own exit condition applies just the same: the FFWA component is compensatory, and is unsuitable where no compromise is acceptable on one criterion; where weights rather than a ranking are needed, methods such as AHP, BWM, CRITIC or Entropy should be considered instead.

Assuming λ is adjustable. In crisp WASPAS, λ is an input parameter, and in some of DecisionMind's other extensions (the Hesitant Fuzzy WASPAS, for instance) it can be set from the interface. Here, λ is neither defined as a field in the D.parameters list nor changeable from the interface; the kernel code always uses 0.5. A sentence in the report such as "we tried it with λ=0.3" is invalid for this extension.

Letting the sum of the cubes of support and rejection exceed 1. A pair with μ³+ν³>1 does not fit this structure; once this constraint is broken, the FFWA/FFWG combination gives an undefined or misleading result.

Writing rejection as 1 minus support. This erases the information that the pair has a separate source for rejection, and removes what the Fermatean structure contributes.

Skipping the complementing step on the cost criterion. The direction correction is a (μ, ν) swap, not a division; skipping this step wrongly rewards high support on a "less is better" criterion.

Reporting only FFWA or only FFWG. The two have different characters; showing only one and omitting the other conceals how robust the combined score really is.

The governing principle is this:

Support and rejection degrees must come from separate sources, and the sum of their cubes must not exceed 1; λ is fixed here and must not be presented in the report as if it were a choice.

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