Fermatean Fuzzy GRA

Four things change; the reference-deviation-coefficient skeleton does not.

Cells. In crisp GRA every cell is a single number. Here every cell is a support (μ) and a rejection (ν) degree. Where intuitionistic fuzzy data allow μ+ν up to 1 at most, and Pythagorean fuzzy data allow μ²+ν² up to 1 at most, here the sum of cubes is bounded: μ³+ν³ ≤ 1. This allows pairs such as μ=0.8 and ν=0.7, whose sum and sum of squares both exceed 1 but whose sum of cubes (0.512+0.343=0.855) stays below 1, to remain valid. Weights remain crisp numbers; GRA does not produce weights, it takes them from outside.

Complementation on a cost criterion. Crisp GRA reverses a cost criterion within the normalisation formula. Here there is no normalisation; instead, in every cell of a "lower is better" criterion, the support and rejection degrees are swapped (μ ↔ ν). This is the Fermatean counterpart of the same complementation rule found in Pythagorean and intuitionistic fuzzy GRA.

Reference and distance. In crisp GRA the reference sequence is a fixed (1; 1; …; 1) normalised value. Here the reference is a Fermatean ideal α⁰ = (max μ, min ν), built from every column's own highest support and lowest rejection degree; as in intuitionistic and Pythagorean fuzzy GRA, only a single positive ideal is considered, and no second anti-ideal is built. The distance, too, is not an absolute difference but a Euclidean-like distance computed over the cubes of the components: it is the square root of half the sum of the squared differences between the cubes of the support degrees, the cubes of the rejection degrees, and the cubes of the indeterminacy degrees. Using cubes rather than squares follows directly from the Fermatean constraint (μ³+ν³≤1).

Grey relational coefficient and degree. These distances are converted, with the same formula as crisp GRA (discriminating coefficient ρ = 0.5, fixed in DecisionMind), into a grey relational coefficient, then into a grey relational degree via a weighted sum. These last two steps work by exactly the same logic as crisp GRA; only the input is the Fermatean distance.

DecisionMind holds the single-positive-ideal reference, the cube-based distance definition, and the discriminating coefficient (ρ = 0.5) fixed for Fermatean fuzzy GRA; weights are taken from outside as crisp numbers.

The grey relational degree shows, as in crisp GRA, an alternative's relative closeness to the Fermatean reference in this analysis; it cannot be compared with a different analysis.

The difference is here. The reference is a composite point seeking both "strongest support" and "lowest rejection" at once, and these two components are compared after cubing. An apparently small μ difference may not change much once cubed; but as μ approaches 1, the difference between cubes grows rapidly. The report should therefore state not only the grey relational degree, but also which criterion, in terms of support or rejection, moves the alternative away from the reference.

Thus instead of writing:

"According to Fermatean fuzzy GRA, A2 came out as the most reliable alternative"

the report should read:

"A2's grey relational degree is highest at 0.8274; this advantage comes from A2 staying close to the reference on the two heaviest criteria, C1 and C2, despite its weakness on the third criterion"

This extension is used when the sum of squares of experts' pairs also exceeds 1, that is, when even the Pythagorean constraint proves insufficient. When a judgement needs to be given both very strong support and a marked reservation at once, and the square of these two exceeds the Pythagorean limit, Fermatean fuzzy data carry these pairs without clipping them.

If experts' pairs already fit within the Pythagorean constraint, moving to Fermatean adds nothing, and only discriminating power is lost. If the criteria are measured, the base GRA should be kept. If the table is mixed, DecisionMind requires a single data type; a measured criterion is then embedded as (t, 0), that is, with full support and zero rejection. The exit condition of base GRA applies here too: if a criterion carries a threshold that can never be compromised, GRA's additive structure does not preserve this.

Violating the value domain. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ³ + ν³ ≤ 1; this constraint is wider than Pythagorean data's μ²+ν²≤1 constraint, but it is not unlimited, and it must not enter the calculation unchecked.

Writing ν as 1 − μ. This reduces Fermatean fuzzy data to crisp data; the indeterminacy share is zeroed out and the one thing the structure adds to GRA disappears.

Forgetting complementation on a cost criterion. If support and rejection degrees are not swapped on a "lower is better" criterion, the reference sequence is built from the wrong end on that criterion.

Using Fermatean fuzzy data for a pair that already fits within Pythagorean data. If none of the experts' pairs have a sum of squares exceeding 1, moving to Fermatean adds no information, and only reduces discriminating power.

Never questioning the discriminating coefficient. ρ = 0.5 is DecisionMind's fixed value; in tables where distances sit close together, this choice affects the result to a greater or lesser degree.

The governing principle is this:

In Fermatean fuzzy GRA, the reference is built, via cubes, from that analysis's own strongest support and lowest rejection values. Writing ν as 1 − μ, skipping complementation on a cost criterion, or using this wider region for a pair that already fits within Pythagorean data wastes the method's one contribution.

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