Spherical fuzzy CoCoSo

Four things change; the five-step skeleton and the decision logic do not.

Cells. In crisp CoCoSo 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; hesitancy is not a leftover share here, it is given separately by the expert. Criterion weights come from outside as crisp numbers. The method's own literature includes a group-decision step that combines several experts' judgements; DecisionMind does not support group decisions in this family, and the user enters a single, previously combined matrix.

Score and scale equalisation. In a cost criterion the triple is complemented first: (μ, ν, π) is rewritten as (ν, μ, π). Every cell is then reduced to a single number with Kutlu Gündoğdu and Kahraman's (2019) score function: s = (μ−π)² − (ν−π)². The score is shifted by 1, then pulled onto 0–1 column by column with min–max scaling.

An important limit: the score function is not monotone. Differentiating s = (μ−π)² − (ν−π)² with respect to μ gives 2(μ−π). When π (hesitancy) is high and μ is smaller than π, s DECREASES as μ grows; that is, the score worsens as the degree of support rises. This is a property of the KG-K (2019) score function itself, and it holds across DecisionMind's entire SF family (spherical fuzzy) of extensions; it was noticed during a catalogue scan (2026-09-13) and was independently verified with Python while preparing this card for SF-COCOSO (see the approval notes). A concrete example: at μ=0.3, s comes out at −0.07. With π held fixed at the same value, μ is raised to 0.7 and ν is lowered to 0.1; that is, support rises and rejection falls, an "improvement" in both raw degrees. Despite this, s falls to −0.24. Stronger support and lower rejection here result in a worse score. This is demonstrated with a full kernel run in the "Mistakes Specific to This Extension" section below.

Additive and multiplicative measures. The weighted sum (P) and weighted power-sum (R) measures are computed from the scaled scores, and the three compromise strategies then follow with the same formulas as crisp CoCoSo. If the sum of P or R is zero, if either measure's minimum is zero, or if the balanced-strategy denominator is zero, DecisionMind does not run the calculation silently; it returns a typed error (DEGENERATE_DATA_FOR_METHOD).

DecisionMind fixes, for this extension, the KG-K (2019) score function, the cost complementation and λ=0.5. Weights come from outside.

As in crisp CoCoSo, the K score is only a ranking within this alternative set. Whether the three compromise strategies agree with one another shows how robust it is.

The real difference appears here: because of the non-monotonicity limit above, a cell that looks like "more support, less rejection" on a criterion can in fact receive a lower score if its hesitancy share is large. This never arises in crisp CoCoSo, which works with plain numbers; there, a larger raw value always means a larger score.

Thus instead of writing:

"This alternative's degree of support is higher and its degree of rejection is lower, so its score on this criterion must be better than the other's"

the report should read:

"The score weighs support and rejection together against the hesitancy share; when hesitancy is large, a rise in support may not improve the score, so which cell receives which score must be checked separately"

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 (sum ≤ 1); the distinction is set out on the data-type card. A measured criterion is not expanded into three degrees directly. The base method's exit condition applies here too: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is also compensatory.

Assuming the score is monotone. This is the most critical mistake. The small example below was produced by running DecisionMind's engine directly. In a two-criterion, three-alternative table, C1's hesitancy share is high (π=0.6):

On C1, A3 has both the highest degree of support (μ=0.70) and the lowest degree of rejection (ν=0.10); both are better than A1's and A2's. Despite this, A3 receives the worst scaled score on this criterion, while A1, with the weakest-looking input (μ=0.30), receives the best score. The reason is that π=0.60 exceeds μ, and in this region the score formula decreases as μ grows. In the full two-criterion calculation this makes the final ranking A2, A1, A3; that is, "the strongest support and the lowest rejection" puts A3 last.

Mistaking the DEGENERATE_DATA_FOR_METHOD error for a data fault. If the sum of P and R is zero, or if either P's or R's minimum is zero, DecisionMind halts the calculation and returns this error. This is not a flaw in the engine; it is an honest report that the k_a/k_b/k_c ratios are mathematically undefined for that dataset.

Confusing the value-space constraint. The constraint is μ²+ν²+π²≤1; it is not the sum of the three degrees themselves.

Changing the score function and expecting the same result. The KG-K (2019) score is a fixed rule; a different score function can give a different ranking, and even different non-monotonicity behaviour.

The governing principle is this:

SF-COCOSO's score function (and that of the entire spherical fuzzy family) is not monotone; on a criterion with a high hesitancy share, higher support and lower rejection can result in a worse score. This comes from the mathematical structure of the score function, not from the data, and must not be hidden in the report.