Fermatean fuzzy ARAS

Five things change; the logic of taking a ratio to the optimal alternative does not.

Cells. In crisp ARAS every cell is a single, positive number. Here every cell is a pair: a degree of supporting a judgement (μ) and a degree of rejecting it (ν). In intuitionistic fuzzy, the sum of these two cannot exceed 1; in Fermatean fuzzy the bound is wider, the sum of their cubes cannot exceed 1 (μ³+ν³≤1). This allows pairs such as μ = 0.9 and ν = 0.5, whose sum exceeds 1 but whose cube sum stays below 1, to remain valid too; that is, a judgement can be both strongly supported and strongly rejected at once, which is not possible in intuitionistic fuzzy.

Score and cost direction. A pair's score, corresponding to a single number, is μ³−ν³, ranging between −1 and 1. For a cost criterion, direction reversal is not done with crisp ARAS's 1/x operation but by swapping support and rejection (the complement); this is the same logic as the swap rule in intuitionistic fuzzy ARAS.

The optimal alternative. The optimal-alternative row is built, for every criterion, from the highest support and lowest rejection components observed; crisp ARAS's idea of "the best value on every criterion" is carried here into the support-rejection space.

Scale equalisation (shifting and normalising). Because the score can come out negative, a constant equal to the absolute value of the smallest score is first added to every score so that all become positive. These shifted scores, the optimal alternative included, are then normalised by dividing by the column sum. This is the same as crisp ARAS's dividing-by-column-sum step; only what is being divided is now the shifted score rather than the raw value.

Weighting and the utility degree. The normalised scores are multiplied by the criterion weight and summed along the row; this sum is exactly the same as crisp ARAS's optimality function. The utility degree K is the ratio of the real alternative's sum to the optimal alternative's sum.

DecisionMind fixes, in this classical form, the score function (μ³−ν³), the shifting constant, and the reversal of the support-rejection pair by complement.

The utility degree K is read as in crisp ARAS: a percentage-style benefit relative to the optimal alternative, meaningful only for this set of options and these weights. The difference is here: beneath K, a judgement's support and rejection degree first collapse into a single score, which is then shifted and normalised into a ratio. The score function (μ³−ν³) is not the only option; other score definitions exist in the literature, and which one is used affects the result.

Thus instead of writing:

"According to Fermatean fuzzy ARAS, A2 is the best option"

the report should read:

"Relative to the optimal alternative derived from this set of options, A2 has the highest utility degree (K = 0.89); this ratio is meaningful only with these weights and the μ³−ν³ score function, and must be recomputed if the weight distribution changes"

This extension is appropriate when criterion evaluation rests on a judgement, and the support and rejection degree of that judgement need to be expressed together more strongly than intuitionistic fuzzy allows. Example: an expert supports a supplier with high confidence while also holding a serious reservation, and these two feelings do not fit within intuitionistic fuzzy's μ+ν≤1 bound. If a criterion only states "how suitable" and there is no separate rejection evidence, this wider bound contributes nothing; intuitionistic fuzzy or plain fuzzy is sufficient. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory.

Entering a cell without checking the cube-sum bound (μ³+ν³≤1). A pair outside this bound is invalid; applying intuitionistic fuzzy's μ+ν≤1 bound here is wrong, Fermatean fuzzy's own bound is wider and must be checked separately.

Applying 1/x instead of the complement for a cost criterion. This breaks the support-rejection pair's cube-sum bound and produces an invalid pair.

Assuming the score function (μ³−ν³) is the only correct choice and not stating it in the report. Different score definitions can give a different ranking; the function used must be stated in the report.

Ignoring the shifting constant and comparing scores directly. If negative scores are normalised without shifting, the column sum can come out negative and the utility degrees become meaningless.

The governing principle is this:

Fermatean fuzzy ARAS exists to carry strong support-rejection combinations that intuitionistic fuzzy cannot accommodate, through to a ratio against the optimal alternative; every application made without checking the cube-sum bound, or without stating the score function, corrupts this information.