Spherical fuzzy TOPSIS (Kutlu Gündoğdu and Kahraman, 2019)

Four things change. The decision logic does not.

Cells. In crisp TOPSIS 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. Hesitancy here is not computed as a residual share; this is what distinguishes it from intuitionistic or Pythagorean structures. Weights come from outside DecisionMind as crisp numbers. The method's own literature also includes a group-decision step that merges several experts' three-degree judgements. DecisionMind does not support group decisions in this family. The user enters a single, already merged decision matrix, and this merging step does not run.

Scale equalisation. Crisp TOPSIS divides every column by its own magnitude, because criteria are in different units: TL, days, points. 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, and the columns are comparable from the outset. DecisionMind instead moves directly to weighting. It scales every cell by the criterion's weight through spherical fuzzy multiplication. This operation enlarges or shrinks all three degrees together; it does not multiply them one by one.

Distance. The ideal and anti-ideal points are built, as in crisp TOPSIS, from the best and worst observed value on each criterion. The difference is here: "best" is not a single number here. A score function evaluates and compares all three degrees together. The ideal point is the real triple that attains the highest of this score. The anti-ideal is the one that attains the lowest. Neither, unlike in crisp TOPSIS, is a fabricated extreme point; both are chosen from among the alternatives themselves. Every alternative's distance to these two references is a Euclidean distance that jointly sums the squares of the differences of the three degrees.

Closure ratio. Crisp TOPSIS's closeness score is better the larger it is. SF-TOPSIS's closure ratio (ξ) runs the opposite way. Distance to the ideal is divided by one reference, distance to the anti-ideal is divided by another reference, and the difference between the two is taken. The value closest to 0 shows the alternative closest to the ideal. DecisionMind holds the score function and the normalised Euclidean distance fixed in this family. Weights are crisp numbers. The step that merges several experts' scores does not run in this implementation.

The closure ratio only ranks alternatives within this particular set relative to one another. It cannot be compared with a different analysis and it changes when the alternative set changes; this is shared with crisp TOPSIS. What differs is the reading direction.

Thus instead of writing:

"In SF-TOPSIS too, the alternative with the highest score is first"

the report should read:

"SF-TOPSIS's output is the closure ratio ξ, and it runs the opposite way; the smallest ξ shows the one closest to the ideal, and ranking runs from smallest to largest"

If ξ values are ordered from largest to smallest in the report and presented as "the highest score," the rank is effectively inverted. Readers, out of habit from crisp TOPSIS, look for the large number and take the wrong alternative for the winner.

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 measured criterion (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.

The exit condition is the same as for crisp TOPSIS. The matrix must be of a single type; some criteria cannot be written as crisp and others as spherical fuzzy. If no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.

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; that is the intuitionistic fuzzy structure's constraint. Confusing these three means either the data entered is invalid or the wrong family's calculation has been run.

Changing the score function. The ideal and anti-ideal points are built with the spherical fuzzy family's own score function. Carrying over the Pythagorean family's score function here leads to a different ideal point, and the rank changes.

Ranking the closure ratio from largest to smallest. This is the most commonly confused point in the method. Smaller ξ is better. Setting up the ranking in reverse out of crisp-TOPSIS habit shows the last-place alternative as first.

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 SF-TOPSIS, uncertainty is carried through all three degrees to the very end of the calculation and gathered into a single closure ratio. A smaller value in this ratio is better, its constraint is a sum of squares, and its score function is specific to this family.