Spherical fuzzy MARCOS (Kovač et al., 2021)

Four things change. The idea of measuring the ratio to the ideal and the anti-ideal does not.

Cells. In crisp MARCOS 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. The only constraint is that the sum of the three degrees' squares must not exceed 1. Hesitancy is not a leftover share; it is asked for separately. Weights come from outside in DecisionMind as crisp numbers. DecisionMind does not support group decisions in this family; a single, previously combined decision matrix is entered.

Extended table and direction. Crisp MARCOS adds two hypothetical rows, ideal and anti-ideal, alongside the real alternatives. In spherical fuzzy the same idea is kept; ideal and anti-ideal rows are built here too, but as three-degree triples rather than single numbers. On a cost criterion, every cell's degrees of support and rejection swap places before the table is built. This is the spherical fuzzy counterpart of crisp MARCOS's rule that "on a cost criterion the ideal is the smallest value."

Choosing the ideal and anti-ideal. In crisp MARCOS the ideal and anti-ideal are the best and worst single numbers observed on each criterion. In spherical fuzzy, "best" is compared with a score function. The ideal is the real triple that receives the highest such score; the anti-ideal the one receiving the lowest. Every cell is then scaled by the criterion's weight with the spherical fuzzy multiplication operation.

Distance. Crisp MARCOS builds two ratios by dividing the alternative's total score by the ideal's and the anti-ideal's total scores. In spherical fuzzy an angular distance takes this ratio's place. The two triples are treated as directions, and the angle between them is measured with the arccos operation. This is where the name comes from: spherical fuzzy triples behave like points on a sphere, and the most natural distance between two points is not a straight line but the arc on the sphere. Every alternative's angular distance to the ideal and the anti-ideal gives a utility degree each (K+ and K−). These two degrees combine, as in crisp MARCOS, into a single final utility degree (F). DecisionMind fixes the score function and the angular distance in this family.

The final utility degree carries the same meaning as in crisp MARCOS: it shows where the alternative stands relative to the ideal and anti-ideal references. It is not compared with a different analysis; this is shared with crisp MARCOS. What differs is that this position has been measured with an angular distance. A high utility degree does not mean "closest to the sphere" but the alternative in the most balanced position relative to the ideal and anti-ideal directions.

Thus instead of writing:

"SF-MARCOS's utility degree is a plain distance ratio, as in crisp MARCOS"

the report should read:

"The utility degree here comes from an angular distance; if two alternatives' utility degrees are close to one another, this closeness can easily change places with a small shift in the three degrees"

This extension is used when experts give a judgement's degree of support, rejection and hesitancy separately. Crisp MARCOS's own question, "how much of the ideal has been reached and how far has the alternative moved from the anti-ideal," is asked here under this uncertainty. The boundary that must not be confused with the spherical fuzzy identity and neighbouring types (Pythagorean, picture fuzzy) is on the data-type card.

The exit condition is the same as for crisp MARCOS. The matrix must be of a single type. If no compromise is acceptable on one criterion, this extension is also compensatory.

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

Skipping the swap on a cost criterion. On a cost criterion, the degrees of support and rejection must swap places before the extended table is built. If this is skipped, the ideal point is built from the most expensive or slowest alternative, and the ranking reverses.

Interpreting the angular distance as if it were a plain Euclidean distance. The K+ and K− values come from the arccos operation; small differences in these values cannot be read on the same scale as the ratio differences in crisp MARCOS. If two alternatives' K+ values are close, the robustness of this closeness must be separately tested.

Reading the final utility degree as a percentage or a probability. The same mistake as in crisp MARCOS applies here too. F only compares this alternative set against itself on its own ideal-anti-ideal axis.

The governing principle is this:

In SF-MARCOS, distance to the ideal and the anti-ideal is angular; the method is not considered properly applied unless the ideal-anti-ideal references built with the score function and this angular distance are shown.