Fermatean Fuzzy SAW

Three things change; the weighted-sum logic does not.

Cells. In crisp SAW every cell is a single number. Here every cell is a support (μ) and rejection (ν) degree pair given to a judgement; the constraint is μ³+ν³ ≤ 1. Criterion weights are taken from outside as crisp numbers; only the decision matrix becomes fuzzy, not the weight vector.

Scale equalisation. Crisp SAW divides every column by its own best value. There is no such step here, because support-rejection pairs already lie between 0 and 1. Instead, on a cost criterion, support and rejection swap places: (μ, ν) → (ν, μ). This is this family's counterpart to crisp SAW's divide-by-best step, but it reverses the direction rather than the scale.

Weighted sum and defuzzification. In crisp SAW, every column is multiplied by its weight and summed directly. Here, each alternative's support-rejection pairs across the criteria are reduced to a single combined support-rejection pair by the Fermatean fuzzy weighted average (FFWA) operator. This combination is an algebraic operator, distinct from crisp SAW's linear sum; its result is again a support-rejection pair, not yet a single number. Only at this final step does the combined pair descend to a single number through the Fermatean score function (s = μ³ − ν³), and alternatives are ranked from the highest score to the lowest.

DecisionMind fixes the FFWA combination operator and the score function (μ³−ν³) for classical FF-SAW. Weights are taken from outside; the method does not generate weights.

The score ranks only this set of alternatives, much as the total score does in crisp SAW; it is not a percentage or a probability, and it cannot be compared with a different analysis.

The difference lies here. Beneath the score, both SAW's full-compensation logic and a compression arising from combining the support-rejection pair through FFWA are stacked on top of one another. Two alternatives can reach the same score from very different support-rejection pairs: one from strong support alongside strong rejection, the other from both being weak.

Thus instead of writing:

"Because the FF-SAW score comes out highest, this alternative is definitively the best"

the report should read:

"This score is a difference (μ³−ν³) emerging from the Fermatean weighted average of the support-rejection pairs; full compensation still applies, and which alternative's support-rejection pair carries more uncertainty must be shown separately"

Consider this extension when experts' support-rejection pairs exceed even the Pythagorean fuzzy structure's squared bound (μ²+ν²>1 but μ³+ν³≤1); the detail is on the Pythagorean fuzzy data-type card. Where the pairs already fit the intuitionistic or Pythagorean bound, moving to Fermatean adds no information, only reduces discrimination. Converting a measured value directly into a support-rejection pair is not modelling uncertainty but manufacturing it. SAW's own exit condition still applies here: if no compromise is acceptable on one criterion, this extension too carries full-compensation logic and will not eliminate anything below a threshold.

Domain violation. Every cell must satisfy μ³+ν³ ≤ 1; this is checked from the sum of cubes, not squares.

Writing ν as 1 − μ. In that case the sum is always exactly 1, and the extra room Fermatean provides is never used.

Choosing Fermatean for pairs that already fit the intuitionistic or Pythagorean bound. If the pairs fit the narrower bound, the wider region adds no information, only reduces discrimination.

Changing the score function and expecting the same result. s = μ³ − ν³ is the canonical choice but not the only one; a different score definition can give a different ranking.

The governing principle is this:

Fermatean fuzzy SAW's cube constraint adds something only when the experts' pairs genuinely exceed the Pythagorean bound; the support-rejection pair descends to a single score only at the final step, once combined via FFWA.