Spherical fuzzy SAW (Kutlu Gündoğdu and Yörükoğlu, 2021)

Three things change. The decision logic does not.

Cells. In crisp SAW 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 here as a residual share. 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. The source book chapter also includes a group-decision step that merges several experts' judgements; DecisionMind does not support group decisions in this family, and the user enters a single, already merged decision matrix.

Scale equalisation. In crisp SAW every column is scaled against its own best value, because criteria are in different units. In the spherical fuzzy form there is no separate "scale against the best value" step; every cell is already on the same 0–1 scale across three degrees. Instead, the method reduces every cell to a single number with the source chapter's own score function: Score(α) = (3μ − π/2)² − (ν − π/2)². This scoring is the spherical fuzzy counterpart of crisp SAW's "use the criterion column directly" logic; it is not a column-wise division but a cell-wise crisping.

Weighted sum. As in crisp SAW, every crisped cell is multiplied by its criterion's weight and summed along the row: SV_i = Σ_j x̄_ij · w_j. This step carries exactly the same summation logic as crisp SAW; the only difference is that the input (x̄_ij) is now a score already crisped from three degrees.

Result and crisping. As in crisp SAW, a larger SV value is better, and ranking runs from highest to lowest value; this direction does not change. Uncertainty is not consumed within SAW itself here, but at the crisping step, the score function; the summation already operates on numbers that have already been crisped.

DecisionMind holds the source chapter's score function ((3μ − π/2)² − (ν − π/2)²) fixed in this family. Weights are crisp numbers. The step that merges several experts' scores does not run in this implementation.

The total score is read as in crisp SAW: it is not a percentage, it cannot be compared with a different analysis, and a larger score is better. The difference is here. The score is now a weighted sum of a value already reduced from three degrees to one, and how the hesitancy share feeds into the score is not directly visible.

Thus instead of writing:

"A high score in SF-SAW also shows the experts' firm certainty"

the report should read:

"The score has been calculated after the three degrees were crisped by the score function; which criterion still carries a high hesitancy share cannot be read from the score itself, and the raw triple table must be consulted"

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, that is, their sum being at most 1, the spherical fuzzy form is needed. A criterion that is directly measured (price, time) 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.

SAW's exit condition applies here too: if no compromise at all is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold. The matrix must be of a single type; some criteria cannot be written as crisp and others as spherical fuzzy.

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. Nor is hesitancy a share derived as 1 − μ − ν.

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.

Computing hesitancy afterwards and writing it into the spherical structure. If the expert did not supply the third degree, the data belongs to the intuitionistic or Pythagorean fuzzy structure. The spherical structure should not be forced onto such data.

The governing principle is this:

In spherical fuzzy SAW, uncertainty is crisped at cell level by the score function and then combined with SAW's own direct summation logic. Any implementation that changes the constraint or the score function blurs the point where these two steps are meant to stay separate.