Spherical fuzzy ARAS

Four things change. The benefit-degree logic remains largely the same, but there is an important difference at the final step.

Cells. In crisp ARAS every cell is a single number. Here every cell consists of three degrees: support (μ), rejection (ν) and hesitancy (π). The expert gives all three directly; hesitancy is not a leftover share. The only constraint is that the sum of the squares of the three degrees must not exceed 1. Weights come from outside in DecisionMind as crisp numbers.

Scale equalisation. Crisp ARAS reverses the cost criterion and divides every column by its own total. In spherical fuzzy, a cost criterion is reversed by swapping the support and rejection degrees. But the step of dividing by the column total is not needed here, because the support-rejection-hesitancy triple is already bounded between 0 and 1 and the columns are already comparable with one another. DecisionMind produces this step internally as well, but it does not enter the final result; the actual weighting happens at the next step.

Optimal alternative and combination. As in crisp ARAS, a hypothetical optimal alternative is built here too: a row is added consisting of the highest support, the lowest rejection and the lowest hesitancy observed on each criterion. Then, for both the real alternatives and this optimal row, a weighted spherical fuzzy aggregation (SFWA) is performed across the criteria, and the result is again a single triple.

Result and defuzzification. This is where the real difference appears. The combined triple is reduced to a single number by a score function, and each real alternative's score is divided by the optimal alternative's score to obtain the benefit degree (K). In crisp ARAS this score is always positive, and K is read as a percentage between 0 and 1. In spherical fuzzy, the score function used (Kutlu Gündoğdu-Kahraman, 2019) is built by subtracting hesitancy from support, and it can come out negative in cells where support stays below hesitancy. In that case, even though the optimal alternative's score stays positive, some real alternatives' scores can come out negative, and so can K. This is exactly what happens in the illustrative example below.

DecisionMind fixes the score function in this method; unlike SF-COPRAS, no shifting step that pulls a negative score back to positive is applied in SF-ARAS.

The benefit degree K is a ratio comparing an alternative with the optimal alternative, and as in crisp ARAS it cannot be compared with a different analysis; it must be recalculated whenever the alternative set changes.

The difference is here: K's sign does not promise a percentage between 0 and 1 here. Because the score function depends on the relationship between support and hesitancy, K can come out either positive or negative. A negative K does not mean that alternative delivers "minus per cent benefit" relative to the optimal one; it shows that the score function does not behave in the same direction as the optimal alternative in this region. K in SF-ARAS should therefore be read not as a percentage but only as a ranking score.

Thus instead of writing:

"K came out at -0.10, so this alternative gives 10 per cent less benefit than the optimal one"

the report should read:

"K here is not a percentage; it is a score used only to rank the alternatives. This alternative getting a negative K shows that the score function evaluates the support-hesitancy balance in these cells in a different direction from the optimal alternative"

Use this extension when experts give a degree of support, rejection and hesitancy for a judgement separately. But if K is expected to give a directly interpretable percentage the way crisp ARAS does, that is, if the report needs to state "per cent benefit relative to the optimal," a spherical fuzzy method that pre-shifts negative scores, such as SF-COPRAS, should be preferred over this extension; the boundary that must not be confused with the spherical fuzzy identity and neighbouring types is on the data-type card.

The exit condition is the same as for crisp ARAS. The matrix must be of a single type; if no compromise is acceptable on one criterion, this extension is also compensatory.

Reading K as a percentage benefit. This is the most important mistake specific to this extension. Because K can come out negative, a statement of "per cent benefit relative to the optimal" is wrong and misleading; K is for ranking purposes only.

Assuming a negative K is a calculation error and changing the data. A negative K is a consequence of the score function's mathematical behaviour in that region; it results from a valid triple that satisfies the input constraint (sum of squares ≤ 1), not from a calculation error.

Deriving hesitancy as 1 − μ − ν and writing it into the spherical structure. This is a different data type (intuitionistic fuzzy); in the spherical structure, hesitancy is obtained separately from the expert.

Carrying SF-COPRAS's shifting step over here. SF-ARAS does not apply this shift; it divides the score function directly by the optimal alternative's score. The two methods' results are therefore on different scales and must not be confused.

The governing principle is this:

Unlike crisp ARAS, SF-ARAS's benefit degree does not promise a percentage between 0 and 1; because the score function can come out negative for this data type, K should be read only as a ranking score.