Spherical fuzzy VIKOR (Sharaf, 2021)

Four things change. The compromise logic does not.

Cells. In crisp VIKOR 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 is asked for separately; it is not derived as a residual share. Weights come from outside DecisionMind as crisp numbers. The literature's own algorithm begins with a 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 step does not run.

Scale equalisation. Crisp VIKOR divides every alternative's distance to the ideal, on every criterion, by the range between that criterion's best and worst value. This is called linear normalisation and brings values into the 0–1 range. The spherical fuzzy form keeps the same logic. The difference is here: the numerator and denominator are no longer single numbers but three-degree distances. Every cell's distance to the best reference (PIS) is divided by the best reference's distance to the worst reference (NIS). Crisp VIKOR's idea of position within a range is applied in exactly the same way, over the spherical fuzzy distance. On a cost criterion the support and rejection degrees are swapped; direction is preserved this way.

Group utility and individual regret. In crisp VIKOR, S (group utility) is the sum of weighted distances, and R (individual regret) is the largest of these distances; both start out as single numbers. In the spherical fuzzy form, every criterion's weighted distance is first held as a three-degree value. S and R are summed, or the largest taken, over these values. Only after this step does a score function bring them down to a single number. Uncertainty stays within the calculation until S and R are reached.

Compromise index and the two conditions. From the scored S and R onward, the method proceeds exactly as crisp VIKOR does. Q is built with the compromise coefficient v (default 0.5). The acceptable-advantage and acceptable-stability conditions are tested exactly as before. DecisionMind holds the score function and the spherical fuzzy distance fixed in this family. The user can change v and whether the PIS/NIS are built relatively or absolutely; the default relative construction uses every criterion's observed best and worst triple.

S, R and Q are read together; this is shared with crisp VIKOR. Writing only the alternative with the smallest Q as the winner is an incomplete report here too. What differs is that S and R have been obtained by scoring spherical fuzzy distances. This means how much uncertainty has fed into S and R can be tracked separately.

Thus instead of writing:

"According to SF-VIKOR, the alternative with the smallest Q is the winner"

the report should read:

"With these weights, the alternative with the smallest Q is a candidate; if the acceptable-advantage and acceptable-stability conditions hold, it is the single compromise solution, and if they do not, the compromise set is reported as it stands"

A compromise set built from spherical fuzzy input should also be read as a consequence of data uncertainty. Because S and R have been scored from three degrees, how close two alternatives are to one another should be judged more cautiously than in crisp VIKOR.

Use this extension when experts give the degrees of support, rejection and hesitancy for a judgement separately. Crisp VIKOR's question, "which alternative will draw the least objection," is then asked under this uncertainty. 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 VIKOR. The matrix must be of a single type. If no compromise at all is acceptable on one criterion, this extension also limits regret but does not eliminate it.

Confusing the value-space constraint. The constraint is the sum of the squares of the three degrees. Deriving hesitancy as 1−μ−ν and writing it into the spherical structure is a different data type; that is the intuitionistic fuzzy structure's constraint.

Setting up PIS/NIS the wrong way round on a cost criterion. On a cost criterion the roles of the support and rejection degrees swap. If this is not done, the ideal point is built from the worst alternative and the ranking is reversed. This is the spherical fuzzy counterpart of crisp VIKOR's "the best value on a cost criterion is the smallest" error.

Skipping S and R and showing only Q. In the spherical fuzzy form, S and R have been scored from three-degree distances. If this intermediate step is not shown and Q alone is reported, which criterion's uncertainty the ranking comes from becomes invisible. The same mistake in crisp VIKOR costs more here; what is lost is not just an intermediate number but the hesitancy information itself.

Choosing v as 0 or 1. As in crisp VIKOR, the method stops being a compromise method at these extreme values. Spherical fuzzy S and R also lose their meaning at these extremes.

The governing principle is this:

In SF-VIKOR, uncertainty is carried through all three degrees until it is scored into S and R. The compromise index and the two conditions are the same as in crisp VIKOR. The method is not considered properly applied unless the report shows that S and R come from spherical fuzzy distance.