Spherical fuzzy CODAS (Karaşan, Boltürk & Kutlu Gündoğdu, 2021)

Four things change. The decision logic stays the same.

Cells. In crisp CODAS every cell is a single number. In spherical fuzzy, every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π). The expert gives all three directly; hesitancy is not calculated as a leftover share. The only constraint is that the sum of the three degrees' squares must not exceed 1. Weights come from outside in DecisionMind as crisp numbers.

Scale equalisation. Crisp CODAS divides every column by its largest or smallest value. In spherical fuzzy there is no separate division step; every cell already consists of three degrees on a 0–1 scale. The method scales every cell directly by the criterion's weight, using the spherical fuzzy multiplication operation.

Negative-ideal and distance. In crisp CODAS the negative-ideal is built from each criterion's worst single number. In spherical fuzzy, a score function (μ−π/2)²−(ν−π/2)² evaluates the three degrees together; the negative-ideal is the real triple that receives the lowest such score on each criterion, not a fabricated extreme point. Two different distances are then calculated: normalised Euclidean distance (D_E) and an angular, that is arccos-based, spherical distance (D_S). The difference from crisp CODAS's Euclidean and taxicab distance is that the second distance here is angular rather than horizontal-vertical.

Assessment score. As in crisp CODAS, it is built through pairwise comparisons and can come out positive or negative. The threshold value (τ) is not exposed to the user here; DecisionMind fixes it at 0.02. This differs from crisp CODAS and the picture fuzzy extension, where τ is a user parameter.

DecisionMind fixes, for this family, the Ch.12 score function, normalised Euclidean distance and angular distance. Weights are crisp numbers; the group step that combines several experts' scores does not run in this implementation.

The assessment score is read as in crisp CODAS: it is a relative score, a negative value is not a failure, and the negative-ideal shifts when the alternative set changes. The difference is here: beneath the score now lies a three-degree uncertainty, and this uncertainty has been measured separately with two different distances (angular and Euclidean).

Thus instead of writing:

"In spherical fuzzy CODAS too, the alternative with the highest score is always the most reliable alternative"

the report should read:

"The assessment score is specific to this alternative set and these weights; it should not be forgotten that the score function is built with the spherical fuzzy family's own definition"

This extension is used when experts give a judgement's degree of support, rejection and hesitancy separately. Spherical fuzzy is needed when these three pieces of information are too high to fit the picture fuzzy constraint; the detail is on the Spherical fuzzy data-type card. A measured criterion is not expanded into three degrees directly. The base CODAS's exit condition applies here too: if no compromise at all is acceptable on one criterion, this compensatory family is not the right choice.

Confusing the value-space constraint. The constraint is μ²+ν²+π² ≤ 1; it is not the sum of the three degrees themselves. This must not be confused with the picture fuzzy structure's constraint of μ+η+ν ≤ 1.

Changing the score function. The negative-ideal is built with the Ch.12 score function. A different score function (for instance μ²−ν²) leads to a different negative-ideal point and a different ranking. There is no single standard in the literature for the spherical fuzzy family's score function; this is a known fragility of the family, and which score was used must be stated in the report.

Forgetting that the threshold value is fixed. τ = 0.02 is not exposed to the user. In this family it is wrong to report that "the threshold value was changed."

Calculating hesitancy afterwards and writing it into the spherical structure. If the expert did not give a third degree, the data belongs to an intuitionistic or Pythagorean fuzzy structure and should not be forced into the spherical one.

The governing principle is this:

In spherical fuzzy CODAS uncertainty is carried through three degrees and measured with two different distances. The score function is specific to the family itself, and the negative-ideal changes when it is changed.