Spherical fuzzy MABAC

Four things change; the border approximation area logic does not.

Cells. In crisp MABAC 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; hesitancy is not calculated here as a leftover share. 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; combining group decisions does not work in this family.

Scale equalisation. Crisp MABAC places every column between 0 and 1 according to its own minimum and maximum. In spherical fuzzy there is no separate min–max step. On cost criteria the three degrees swap together: support becomes rejection, rejection becomes support, and hesitancy stays as it is. Because the cells are already on the same 0–1 scale, the columns are comparable from the outset.

Weighting and the border approximation area. Every cell is scaled by the criterion's weight with spherical fuzzy algebra's own exponential rule, an operation that enlarges or shrinks the three degrees together. Every cell's score is then reduced to a single number with a score function; the border approximation area is built as the arithmetic mean of these scores, in place of crisp MABAC's geometric mean.

Distance and total score. Every alternative's score is subtracted from the border's score; this is a plain subtraction, as in crisp MABAC, but it works on values already reduced to a single score rather than on the three-degree cell itself. These differences are summed across criteria, and alternatives are ranked from highest to lowest by the total score.

DecisionMind fixes, in this form, the spherical fuzzy scaling and the score function. Weights are taken from outside as crisp numbers.

The total score is read as in crisp MABAC: a positive score means above the border, a negative one below it, and this holds only for this alternative set. The difference is here: beneath this score lies a three-degree support-rejection-hesitancy triple, and the score carries only this triple's score, not the triple itself.

Thus instead of writing:

"According to spherical fuzzy MABAC, this alternative is the most reliable one"

the report should read:

"With the weights given, this alternative sits highest above the border approximation area; this position rests on the score that support, rejection and hesitancy together produce, not on high support alone"

This extension is used when the expert's hesitancy is not a leftover share from support and rejection but information that can be separately measured and reported. Spherical fuzzy is needed when these three pieces of information are too high to fit the picture fuzzy constraint, that is, the sum of all three being at most 1. If hesitancy is not separately measured, an intuitionistic or Pythagorean fuzzy structure is sufficient.

Crisp MABAC's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory and does not screen out anything below a threshold. A measured criterion is not expanded directly into three degrees.

Confusing the value-space constraint. The constraint is the sum of the three degrees' squares: μ²+ν²+π²≤1. It is not the sum of the three degrees themselves; that is the picture fuzzy structure's constraint. Nor is hesitancy a share derived as 1−μ−ν; that is the intuitionistic fuzzy structure's constraint.

Calculating hesitancy afterwards and writing it into the spherical structure. If the expert did not give a third degree, the data belongs to an intuitionistic or Pythagorean fuzzy structure and the spherical fuzzy structure should not be forced onto it.

Assuming the score always reads as "support up improves it, hesitancy up worsens it." This family's score function evaluates the three degrees together, over their squares. Which direction the hesitancy share pushes the score depends on which of support and rejection dominates; comparing two alternatives' scores merely by "which has the higher support" can therefore be misleading. The form of the score used in this family is referred to with more than one variant in the literature; DecisionMind documents in its manifest which form it fixes, and this is a point that will be separately reviewed at the scientific-approval stage.

The "more advanced" fallacy. Spherical fuzzy MABAC does not produce a "more correct" ranking than Pythagorean or picture fuzzy MABAC; it is only the form required when hesitancy is separately measured.

The governing principle is this:

In spherical fuzzy MABAC, support, rejection and hesitancy are bounded by the sum of their squares; the border approximation area and the distance are built on a single score derived from these three degrees, and the score alone does not show the triple itself.