Fermatean Fuzzy MARCOS

Three things change; the extended table and the ideal-anti-ideal ratio logic do not.

Cells. In crisp MARCOS every cell is a single number. Here every cell consists of a support (μ) and a rejection (ν) degree; the sum of their cubes (μ³+ν³) does not exceed 1. Criterion weights are taken from outside as crisp numbers; the method does not produce weights.

Ideal and anti-ideal, normalisation. In crisp MARCOS the ideal and anti-ideal are built from the observed best and worst number. Here, on a benefit column, the highest support and lowest rejection are taken as the ideal, and the reverse on a cost column. Every cell is scaled into the 0-1 range relative to its ideal by taking a cube root (the ratio norm of Fermatean algebra).

Aggregation and defuzzification. Crisp MARCOS multiplies every row by the weights and sums them. Here every row is first combined into a single support-rejection pair with a Fermatean fuzzy weighted average (FFWA), and this pair is then reduced to a single number by a score function: s = μ³ − ν³. What remains is crisp MARCOS itself: the ratio to the ideal and anti-ideal (K+, K−), the two utility functions and the final utility degree that combines them are built with the same steps.

DecisionMind holds the score function, the FFWA aggregation operator and the cube-root norm fixed for this extension.

The final utility degree is read as in crisp MARCOS: it is a position relative to this set's own ideal and anti-ideal references, not a percentage or a probability (see the MARCOS card). Because the score function (μ³−ν³) can be negative, intermediate values and final utility degrees can also fall outside the 0-1 range; this does not mean the calculation has gone wrong, it only means the degree cannot be read directly as a percentage.

Thus instead of writing:

"In Fermatean fuzzy MARCOS, the final utility degree is a percentage between 0 and 1"

the report should read:

"The final utility degree is only a ranking measure relative to the ideal and anti-ideal within this alternative set; its numerical size can fall outside the 0-1 range and should not be read as a percentage on its own"

This extension is considered when a judgement's support and rejection degree can be simultaneously high over a region wider than intuitionistic fuzzy data (μ+ν≤1) or Pythagorean fuzzy data (μ²+ν²≤1) allow. The typical case is one where experts hold both strong support and a strong reservation at once, and a pair such as (0.8; 0.6) violates the intuitionistic and Pythagorean constraints but satisfies the Fermatean constraint (0.8³+0.6³=0.728≤1). If support and rejection do not need this width, a simpler constraint (intuitionistic or Pythagorean) is sufficient and should be preferred. The exit condition of crisp MARCOS applies here too: where a criterion admits no compromise, this extension is also compensatory.

Violating the value domain. Every cell must satisfy μ³+ν³ ≤ 1; a pair that satisfies the intuitionistic or Pythagorean constraint automatically satisfies the Fermatean constraint too, but the reverse does not hold. Entering data without checking this constraint invalidates the calculation.

Forgetting that the score function is not the only option. DecisionMind uses s=μ³−ν³ as its canonical choice; a different score function can produce a different ranking. Which function was used should be stated in the report.

Treating the Fermatean constraint as a "more advanced" uncertainty model. Fermatean widens the region permitted by intuitionistic and Pythagorean fuzzy data; this wider region only carries meaning if the data genuinely falls into it. If the data already satisfies the intuitionistic constraint, moving to Fermatean adds no extra accuracy.

The governing principle is this:

The only difference in Fermatean fuzzy MARCOS is that the region permitted for the support-rejection pair widens through the cube constraint; the ideal-anti-ideal ratio logic and the reading of the final utility degree are the same as in crisp MARCOS.