Fermatean Fuzzy TODIM

Three things change; the loss-aversion logic and the reference-criterion approach do not.

Cells. In crisp TODIM every cell is a single number. Here every cell is a pair: a support degree μ and a rejection degree ν. In intuitionistic fuzzy, μ+ν cannot exceed 1; here the sum of cubes μ³+ν³ cannot exceed 1. For the same μ and ν, this allows a wider region than intuitionistic fuzzy does; for example, μ=0.8 and ν=0.7 are invalid in intuitionistic fuzzy but valid here. Weights are crisp numbers. DecisionMind does not support group decision-making in this extension.

Scale equalisation. In crisp TODIM, cost criteria are converted to the benefit direction by a column-wise ratio. Here the same aim is achieved through the Fermatean complement: every (μ, ν) pair on the cost criterion swaps places with (ν, μ). This is a reversal of direction, not a division.

Distance and score. In crisp TODIM, the difference between two values is a direct subtraction. Here two steps separate out. The score s=μ³−ν³ determines which alternative wins on that criterion. The distance, by contrast, is computed via the Fermatean Hamming distance (the sum of the differences in the cubes of μ, ν and the hesitancy margin π). In crisp TODIM, the difference gives both magnitude and direction in a single step; here the distance gives the magnitude, and the score comparison gives the direction.

Result. The global value is again a single number normalised between 0 and 1; the uncertainty in the support-rejection pair enters the distance calculation but is resolved by the end, so the final figure is not itself Fermatean fuzzy.

DecisionMind fixes μ³−ν³ as the score function and the Fermatean Hamming distance as the distance function in this extension. The loss-aversion coefficient θ is held fixed at 1 within the kernel; unlike in HF-TODIM and IVIF-TODIM, θ is not a parameter DecisionMind exposes to the user in this extension. The reference criterion is likewise automatic and, unlike in crisp TODIM, cannot be changed by the user.

The global value is read as in crisp TODIM: the lowest total dominance takes 0, the highest takes 1; it is not an absolute "good/bad" measure.

The difference lies here. The winner-loser direction comes from the score difference, and the magnitude comes from the Fermatean Hamming distance. Because a support-rejection pair too high to be valid in intuitionistic fuzzy can be valid here, what looks like the same judgement can correspond to a different distance under the two data types. The report should therefore state clearly which data type was used.

Thus instead of writing:

"According to Fermatean fuzzy TODIM, A2 is the best alternative"

the report should read:

"The support-rejection pairs were compared using the Fermatean Hamming distance, and the win-loss direction was determined by the μ³−ν³ score; A2 has the highest global value, and this ranking holds between θ=0.5 and θ=5"

This extension is suitable when a judgement's support and rejection degrees fall outside the region intuitionistic fuzzy allows, that is, when an expert assigns a high value to both support and rejection at once. It is equally suitable where the intuition that the decision-maker is more sensitive to losses than to gains fits the nature of the decision.

Where support and rejection stay within intuitionistic fuzzy's bounds (μ+ν≤1), this extra breadth adds nothing, and intuitionistic fuzzy TODIM (IF-TODIM) suffices. Where the criteria are measured, crisp TODIM remains the right choice. TODIM's own exit condition applies here just as it does elsewhere: if no compromise is acceptable on one criterion, screening should be applied first; if the loss-aversion assumption does not fit, a symmetrically compensatory method such as Fermatean fuzzy TOPSIS should be preferred instead.

Domain violation. Every cell must be checked to ensure μ³+ν³ does not exceed 1; this must not be confused with the μ+ν≤1 rule that applies to intuitionistic fuzzy.

Changing the score function and comparing the result. DecisionMind fixes the score μ³−ν³. A different score function (one based on μ alone, for instance) can produce a different winner-loser order.

Assuming θ is adjustable in the DecisionMind interface. In this extension the loss-aversion coefficient is held fixed at 1 within the kernel; it is not a user parameter as it is in HF-TODIM or IVIF-TODIM.

The "more advanced" fallacy. Opening data that already stays within intuitionistic fuzzy's bounds to Fermatean fuzzy, merely because it allows a wider region, adds no information.

The governing principle is this:

In Fermatean fuzzy TODIM, the winner-loser direction comes from the μ³−ν³ score, and its magnitude comes from the Hamming distance in cube space; this extension adds information only where the support-rejection pair genuinely exceeds intuitionistic fuzzy's bounds.