Spherical fuzzy WASPAS (Boltürk and Kutlu Gündoğdu, 2021)

Cells. In crisp WASPAS every cell is a single number. In the spherical fuzzy form every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π). The expert supplies all three directly; hesitancy is not computed as a residual share. The only constraint is that the sum of the three degrees' squares cannot exceed 1. Criterion weights come from outside DecisionMind as crisp numbers.

Scale equalisation. Crisp WASPAS equalises every column by scaling it against its own best value. In the spherical fuzzy form there is no separate scale-equalisation step; every cell is already on the same 0–1 scale across three degrees. Instead, on a cost criterion the support and rejection swap places, while hesitancy stays as it is: (μ, ν, π) → (ν, μ, π).

Sum and product components. What crisp WASPAS's weighted sum (WSM) does here is done by the spherical fuzzy weighted arithmetic mean (SFWA): every alternative's three degrees across the criteria are reduced to a single triple (Q̃¹). What crisp WASPAS's weighted product (WPM) does here is done by the spherical fuzzy weighted geometric mean (SFWG), which likewise comes down to a single triple (Q̃²). SFWA magnifies support in an optimistic way, while SFWG brings rejection to the fore; this corresponds to the compensatory character of summation and the punishing character of multiplication respectively.

Result and crisping. The two triples (Q̃¹, Q̃²) are blended by spherical fuzzy addition into a single triple (Q̃) in the form λ·Q̃¹ ⊕ (1−λ)·Q̃²; DecisionMind does not present λ as a field in the interface, and it comes as an additional input with a default of 0.5. This combined triple is only crisped into a single number at the very last step, with Boltürk and Kutlu Gündoğdu's (2021) score function, (2μ − π/2)² − (ν − π/2)²; a larger Q_i is better.

DecisionMind holds this SFWA/SFWG combination and this score function fixed in spherical fuzzy WASPAS; weights come from outside as crisp numbers, and the method does not generate weights. DecisionMind does not support group decisions in this family (the step that merges several evaluators' matrices); the user enters a single, already merged decision matrix.

Q_i is read like the combined score in crisp WASPAS; it only ranks alternatives within this particular set, is not a percentage or a probability, and a larger value is better.

The difference is here. The score formula squares μ together with π in one term, (2μ − π/2)², and this term is not always increasing with respect to μ while π is held fixed. In cells where small μ and large π occur together, the score can first fall and then rise as μ increases a little. This is a feature noted from the literature that DecisionMind's own internal review has flagged in this family too, including SF-WASPAS; it is not an error in the engine but a mathematical consequence of the chosen score function, and it must not be ignored in the report.

Thus instead of writing:

"Q_i has risen, so support has increased"

the report should read:

"Q_i is a score that jointly squares support, rejection and hesitancy; when low support and high hesitancy occur together, a small rise in support can lower the score, so μ, ν and π should also be reported separately"

Use this extension when experts give the degrees of support, rejection and hesitancy for a judgement separately, and these three pieces of information run too high to fit the picture fuzzy structure's constraint (a sum of at most 1). A measured criterion is not expanded into three degrees by force. Crisp WASPAS's exit condition applies here too: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is also compensatory.

Confusing the value-space constraint. The constraint is the sum of the squares of the three degrees: μ²+ν²+π²≤1. Neither the picture fuzzy structure's constraint (the sum of the three degrees themselves) nor the intuitionistic fuzzy structure's constraint (hesitancy as a derived share) applies here.

Assuming the score always rises with μ. The score (2μ − π/2)² − (ν − π/2)² is not increasing everywhere with respect to μ while ν and π are held fixed. For instance, with ν=0.20 and π=0.40 held fixed, the score is 0.0256 at μ=0.02, 0.0000 at μ=0.10, and 0.0100 again at μ=0.15; as μ rises, the score first falls and then rises. This region occurs in cells where small μ and large π occur together, and can be misread if it is not noticed in the report.

Confusing this score with the closure ratio. In SF-TOPSIS's closure ratio a small value is good; here a large Q_i is good. The reading direction of the two families must be kept separate.

Looking for λ in the interface. λ is not an interface field like the weights here; it comes as an additional input with a default of 0.5.

The governing principle is this:

In spherical fuzzy WASPAS, a larger Q_i is better, but the score formula is not increasing everywhere with respect to μ; this feature should be checked separately, and stated in the report, in cells where low support and high hesitancy occur together.