Spherical fuzzy EDAS (Garg & Sharaf, 2022)

Four things change. The idea of positioning relative to the average does not.

Cells. In crisp EDAS every cell is a single number. In spherical fuzzy, every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π), given directly by the expert. The only constraint is that the sum of the three degrees' squares must not exceed 1; hesitancy is asked for separately, not left as a remainder. Weights come from outside as crisp numbers. The literature works with a single expert's matrix or one already combined by a board; DecisionMind likewise does not support group decisions in this family.

Average solution. In crisp EDAS every criterion's average is a single number, the arithmetic mean of the column. In spherical fuzzy the average is also a three-degree value: the column's triples are combined with the spherical weighted-average operation into a single SFN, an average triple. On a cost criterion, every cell's support and rejection swap places before averaging, just as crisp EDAS reverses which side of the average counts as favourable for a cost criterion.

Deviation. In crisp EDAS a deviation from the average is a plain subtraction, positive or negative. Three degrees cannot be subtracted this way, so the spherical fuzzy difference operation is used instead, producing a new triple that holds, in three separate degrees, how far one triple is in favour of, against, and undecided relative to another. Crisp EDAS's "above or below the average" distinction is carried here across the three components of this difference at once.

Combination. Crisp EDAS normalises the in-favour and against totals separately against their own largest values, then averages the two into the assessment score. Spherical fuzzy has no such normalise-and-average step: the weighted differences are combined across criteria into a single triple with a dedicated aggregation operation (SFAgg), which takes a separate squared mean for each component. The score function applies only to this final triple, in the very last step, giving a single assessment score (AS); alternatives are ranked from highest to lowest, the same direction as in crisp EDAS.

The assessment score only ranks this alternative set against the set's own average. It is not compared with a different analysis; this is shared with crisp EDAS. What differs is that the average itself is also three-degree. An alternative's score depends not only on the question "above or below the average" but also on the hesitancy within that deviation.

Thus instead of writing:

"The SF-EDAS score carries only the same positional information relative to the average as crisp EDAS's"

the report should read:

"The SF-EDAS score carries both the position relative to the set's average and the hesitancy within that position; if the gap between two alternatives' scores is small, this gap can close with a small change in the input's degrees of hesitancy"

This extension is used when experts give a judgement's degree of support, rejection and hesitancy separately. Crisp EDAS's own question, "position relative to the set's average," 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 EDAS. The matrix must be of a single type. If no compromise is acceptable on one criterion, this extension is also compensatory and does not screen out anything below a threshold.

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.

Not applying the conjugate on a cost criterion. On a cost criterion, the degrees of support and rejection must swap places before the average is taken. If this is skipped, the average is built in the wrong direction, and the expensive or slow alternative appears favourable.

Reducing the average to a crisp number and running crisp EDAS. Reducing the spherical fuzzy triples to a single number (for instance μ alone) from the start and applying crisp EDAS is not SF-EDAS. Hesitancy is erased in the first step, and the gaps between scores gain a certainty that does not actually exist.

Forgetting that an alternative has been added to the set. The average is built from every alternative in the set. As in crisp EDAS, an alternative added to or removed from the set changes the average, and hence every score; in spherical fuzzy this effect spreads across all three degrees at once.

The governing principle is this:

In SF-EDAS both the average and the deviation stay three-degree; they drop to a single number only in the very last step, after the three degrees accumulated across the criteria have passed through a single spherical fuzzy aggregation operation.