Spherical fuzzy TODIM (Sharaf and Khalil, 2021)

Four things change; the reference-criterion logic continues, but in this extension the reference criterion is always the criterion with the highest weight, and the loss-aversion coefficient θ is fixed at 1; neither can be changed from the interface.

Cells. In crisp TODIM every cell is a single number. In the spherical fuzzy form every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π); the only constraint is that the sum of their squares cannot exceed 1. Hesitancy here, unlike in intuitionistic or Pythagorean structures, is not a residual share; the expert supplies it separately. Weights are crisp numbers. DecisionMind does not support group decisions in this family.

Scale equalisation. Crisp TODIM extracts a share by dividing every column by its own total, because the raw data is in different units. Here there is no separate scale-equalisation step; every cell is already on the same 0–1 scale across three degrees. On a cost criterion, the method swaps the support and rejection degrees; hesitancy stays as it is.

Score and distance. In crisp TODIM the difference is a direct subtraction. Here which alternative "wins" on a criterion is determined by a score function that weighs all three degrees together (support minus the square of hesitancy, rejection subtracted from the square of hesitancy). Distance is a Euclidean distance that jointly measures the difference of the three degrees' squares; the magnitude of gain or loss rests on this distance, its direction on the score difference.

Result. The spherical value is again a single number normalised between 0 and 1. The tension among the three degrees is carried through to the distance and score calculation, and only comes down to a single number at the last step.

DecisionMind holds Kutlu Gündoğdu and Kahraman's (2019) score function and the squares-difference Euclidean distance fixed in this family; θ is fixed at 1, and the reference criterion is always the one with the highest weight.

The spherical value is read as in crisp TODIM: the lowest total dominance takes 0, the highest takes 1, and this holds only for this particular set of alternatives. The difference is here. Beneath this value now lies a three-degree judgement, and the score function that determines the winning-losing direction does not always grow as support increases; when the hesitancy share is high, a rise in support can lower the score.

Thus instead of writing:

"As the support degree rises, the score rises too, and so does the spherical value"

the report should read:

"The score is a function jointly built from support, rejection and hesitancy; when hesitancy is high, a rise in support may not always raise the score, so which criterion still carries a high hesitancy share must be reported separately"

Use this extension when experts give the degrees of support, rejection and hesitancy for a judgement separately, and these three run too high to fit the picture fuzzy structure's constraint (a sum of at most 1). It also suits cases where the intuition that the decision-maker is more sensitive to losses than to gains matches the nature of the decision.

If hesitancy is not measured separately, intuitionistic or Pythagorean fuzzy TODIM carries the same information with two degrees and is sufficient. A measured criterion is not expanded into three degrees by force. TODIM's exit condition applies exactly as before: if no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.

Assuming "greater support always means a greater score." The Kutlu Gündoğdu-Kahraman score function is not monotonic when hesitancy exceeds support. For instance, (μ=0.2; ν=0.1; π=0.6) gives a score of −0.09, whereas raising only the support to 0.3, (μ=0.3; ν=0.1; π=0.6), lowers the score to −0.16; the score has fallen even though support rose. This behaviour is a known feature of the same score function in the SF-COPRAS and SF-ARAS extensions as well.

Confusing the value-space constraint. The constraint is the sum of the squares of the three degrees (μ²+ν²+π²≤1), not the sum of the three degrees themselves; that is the picture fuzzy structure's constraint.

Trying to change θ or the reference criterion. Neither can be changed from the interface in this extension; θ is fixed at 1, and the reference criterion is always the one with the highest weight.

Computing hesitancy afterwards and writing it into the spherical structure. If the expert did not supply the third degree, the data belongs to the intuitionistic or Pythagorean fuzzy structure; the spherical structure should not be forced onto such data.

The governing principle is this:

In spherical fuzzy TODIM, the score function that determines the winning-losing direction weighs all three degrees together and does not behave monotonically when hesitancy is high; predicting the result by looking only at the support degree is therefore misleading.