Spherical fuzzy WPM (Kutlu Gündoğdu and Yörükoğlu, 2021)

Three things change. The decision logic does not.

Cells. In crisp WPM 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. The only constraint is that the sum of the three degrees' squares cannot exceed 1. Weights come from outside DecisionMind as crisp numbers; the method does not generate weights, nor does it support the group-decision step that merges several experts' judgements.

Scale equalisation. Crisp WPM scales every column against its own best value. In the spherical fuzzy form, every cell is instead reduced to a single number with the source chapter's own score function: Score(α) = (3μ − π/2)² − (ν − π/2)². This corresponds not to crisp WPM's scaling but to SF-SAW's own crisping step; the first four steps of SF-WPM are identical to those of SF-SAW, since both come from the same chapter.

Weighted product. In crisp WPM every column is raised to the power of its weight and multiplied along the row: SV_i = Π_j x̄_ij^{w_j}. The same operation is applied here; the difference is that x̄_ij is now a score already crisped from three degrees. This is the one step that separates SF-WPM from SF-SAW: SF-SAW multiplies these same x̄_ij values by the weight and sums them, whereas SF-WPM raises them to the power of the weight and multiplies them.

Result and the negative-score issue. The Score(α) formula can come out negative for certain combinations of μ, ν and π; even in the source chapter's own example, the two worst linguistic terms ("very poor," "extremely poor") produce a negative score. When this happens, the kernel clips any cell that comes out negative or zero to a very small constant (1e-12). Otherwise, raising a negative base to a fractional exponent (the weight) is undefined in the real numbers. This clipping destroys information: every clipped cell converges on the same constant, so how poor the cell actually was (−0.66 or −0.94) never reaches the result. DecisionMind does not hide this; the kernel reports every clipping as a separate warning and states that the ranking of the affected alternatives is unreliable.

The total product (SV) is read as in crisp WPM: it only ranks alternatives within this particular set, is not a percentage, and cannot be compared with a different analysis. The difference is here. SV is a value that has been crisped per cell and then multiplied; how the hesitancy share feeds into the score is not directly visible.

More importantly, if any alternative's Score(α) value on any criterion has fallen to zero or below, the report must state this separately. In that case, that cell has been clipped to EPS, and the alternative's total SV carries only the information "it was clipped," not the true magnitude of the weakness on that criterion.

Thus instead of writing:

"A high SV in SF-WPM shows the alternative stayed strong on every criterion"

the report should read:

"A high SV shows that Score(α) did not fall to zero or below on any criterion and that the product reflects genuine magnitudes. If a clipping warning exists on a criterion, only the affected alternative's position in the ranking should be read cautiously, not the size of its SV"

Use this extension when experts give the degrees of support, rejection and hesitancy for a judgement separately. If these three pieces of information run too high to fit the picture fuzzy structure's constraint, the spherical fuzzy form is needed. A measured criterion is not expanded into three degrees by force. The boundary that must not be confused with neighbouring types (Pythagorean, picture fuzzy) is set out on the data-type card.

Crisp WPM's severity applies doubly here. If a criterion's Score(α) sits just above zero, the product already penalises that criterion harshly; if it falls below zero, clipping takes over and that criterion loses almost all its discriminating power. If no compromise at all is acceptable on one criterion, this extension is still compensatory; the SAW family's exit condition applies here too.

Ignoring the clipping warning. The kernel reports every clipping when it happens. If this warning is ignored and SV values are compared directly, the affected alternative's true weakness appears at a different magnitude than it actually is.

Confusing the value-space constraint. The constraint is the sum of the squares of the three degrees: μ²+ν²+π²≤1. It is not the sum of the three degrees themselves; that is the picture fuzzy structure's constraint.

Changing the score function. Every cell is crisped with the source chapter's own score function. Carrying over the Pythagorean family's or another family's score function gives a different score, and in some tables a different rank.

Failing to apply the cost direction. If the support and rejection degrees are not swapped on a "lower is better" criterion, the score is computed in the wrong direction. The worst alternative on that criterion then looks like the best.

The governing principle is this:

SF-WPM's multiplicative step operates on crisped per-cell scores, and these scores can be negative. A negative score is resolved by clipping, but clipping loses information; this loss must be shown in the report together with the warning, not passed over silently.