Spherical fuzzy MOORA (Aydın & Kutlu Gündoğdu, 2021)

Four things change; the ratio system logic does not.

Cells. In crisp MOORA every cell is a single number. Here every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π); 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; the method does not generate weights. This extension does not support group decisions; it works with a single, combined decision matrix.

Scale equalisation. Crisp MOORA divides the column directly by the square root of the sum of squares. Here every cell is first reduced to a single number with a score function: s = (μ−π)² − (ν−π)². On a "lower is better" criterion μ and ν swap places, so that low support turns into a high score. Because the score can come out between −1 and 1, 1 is added to every score to avoid dividing with negative values. Only after this is the single number equalised, as in crisp MOORA, by dividing it by the square root of the sum of squares column by column. That is, the spherical structure drops to a single number immediately before scale equalisation; unlike most MULTIMOORA extensions, the uncertainty is not carried through to the end.

Distance / score / combination. The equalised single number is multiplied directly by the criterion weight, and crisp MOORA's ratio-system operation is applied: the weighted shares on "lower is better" criteria are subtracted from the sum of the weighted shares on "higher is better" criteria. DecisionMind runs only the ratio system here; the reference-point approach and the full multiplicative form are not part of this extension. This is exactly the same choice as the DecisionMind implementation (ratio system only) described on the crisp MOORA card.

Result and defuzzification. The output is a single number; the three-degree structure is never carried through to the result. Defuzzification happens in the first step, where the score function is applied, before scale equalisation; this ordering is a design decision DecisionMind fixes in this family.

DecisionMind fixes, in this extension: the score function (μ−π)²−(ν−π)², the +1 shift on the score, column-based vector normalisation, and direct weighted addition-subtraction after normalisation. The reference-point approach and the full multiplicative form are not added; consistent with its name, this extension is only MOORA's ratio system.

The score is read as in crisp MOORA: it is not a percentage or a probability, it is not compared with a different analysis, and the normalisation denominator changes when the alternative set changes. Being positive or negative does not mean "good" or "bad"; it only shows relative position within this alternative set.

The difference is here: the three-degree uncertainty beneath the score has dropped to a single score before scale equalisation. In the illustrative example below, the gap between A2 (0.6291) and A1 (0.5622) almost closes when C2's weight is raised from 0.35 to 0.55 (A1=0.5881, A2=0.5970), and A1 moves ahead once C2's weight is raised to 0.80 and C1's weight is lowered to 0.10 (A1=0.6113, A2=0.5640).

Thus instead of writing:

"Because spherical fuzzy MOORA preserves the uncertainty, its result is more reliable"

the report should read:

"The three-degree uncertainty has been reduced to a single score before scale equalisation; the score is calculated on a crisp number that results from this reduction, and the ranking is sensitive to the weight distribution"

This extension is used when experts give a judgement's degree of support, rejection and hesitancy separately, and this triple is too high to fit the picture fuzzy constraint (sum at most 1). The constraint rule and the boundary that must not be confused with neighbouring types (Pythagorean, picture fuzzy) are on the data-type card. A measured criterion (price, time) should not be expanded directly into three degrees.

Crisp MOORA's exit condition applies here too: if no compromise can ever be given on one criterion, this extension is also compensatory and does not screen out anything below a threshold. If both the ratio system and the reference-point logic are wanted together, this extension does not provide that; for that need, the spherical fuzzy extension of the MULTIMOORA family should be consulted.

Confusing the value-space constraint. The constraint is μ²+ν²+π²≤1; it is not the sum of the three degrees themselves. This is the picture fuzzy structure's constraint, and confusing it means invalid data gets processed as if it were crisp MOORA's.

Changing the score function. The defuzzification before scale equalisation is fixed in DecisionMind at (μ−π)²−(ν−π)². More than one score proposal exists in the spherical fuzzy family's literature, and one of them may not always be monotonically increasing; a different function can produce a different ranking, and this choice must be stated in the report.

Skipping the shifting step. Because the score can come out negative, normalising directly without the +1 shift causes sign errors when negative and positive scores are divided by the same denominator.

Confusing the spherical structure with MULTIMOORA. This extension is only the ratio system; the reference-point approach and the full multiplicative form do not enter the calculation. Saying "MULTIMOORA's three viewpoints are here too" is wrong.

The governing principle is this:

In spherical fuzzy MOORA the three-degree uncertainty is reduced to a single score immediately before scale equalisation; the method is only the ratio system and does not include the reference-point approach or the full multiplicative form.