Extension card · Fuzzy
Fermatean fuzzy CoCoSo (built on Senapati and Yager's 2020 foundation)
Fermatean fuzzy CoCoSo is the form of CoCoSo used for situations where criterion evaluations are expressed as the support and rejection degree given to a judgement. The constraint here is not the sum of these two degrees but the sum of their CUBES, and this sum cannot exceed 1. Every cell is first reduced to a single Fermatean score; everything that follows runs exactly as in crisp CoCoSo.
Base method
CoCoSo →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
What this data type is, when to use it, how to write it in a cell: the family's full account is here.
What Changes from the Base Method?
Two things change; the five-step skeleton and the decision logic do not.
Cells. In crisp CoCoSo every cell is a single number. Here every cell is a support (μ) and rejection (ν) degree pair given to a judgement. The difference from intuitionistic fuzzy lies in the constraint: intuitionistic fuzzy requires μ+ν≤1, while here μ³+ν³≤1. A cube sum grows more slowly than a square sum. This is why this constraint permits some pairs that intuitionistic fuzzy would not (for example, a pair in which both values are as high as 0.8). Criterion weights are taken from outside as crisp numbers; they are not a μ/ν pair.
The point of defuzzification. Here too, defuzzification happens at the very start. Every (μ, ν) pair is reduced to a single number, in the first step of the calculation, by the Fermatean score function (s = μ³ − ν³); DecisionMind shifts this score into a positive range by adding 1. This score plays the same role as the "criterion value" number expected by crisp CoCoSo; it is first scaled to the 0-1 range by direction-sensitive column min-max scaling (relative to the lowest value for a benefit criterion, reversed for a cost criterion). The weighted-sum measure and the weighted power-sum measure are then computed, three compromise strategies are applied, and the final K score is found; all of these steps work exactly as in crisp CoCoSo's formulas. The Fermatean fuzzy structure determines only the INPUT, that is, how the score is arrived at; the OUTPUT mechanics are the same as the crisp method's.
DecisionMind fixes, in this extension, the Fermatean score function (μ³ − ν³) as the defuzzification rule; it also fixes the balance coefficient (λ = 0.5) in the third compromise strategy. Weights are taken from outside; the method does not generate weight itself.
How to Read the Output
As in crisp CoCoSo, the K score too is only a ranking within this set of options; it can exceed 1, and is not a percentage or a probability. How closely the three compromise strategies' (the sum, the worst-case, and the balanced) separate rankings agree with one another shows how robust the result is.
The difference is here: μ and ν carry separate information, as in intuitionistic fuzzy, but the cube constraint lets them spread over a wider region. Two alternatives can reach the same μ³−ν³ score through very different (μ, ν) pairs; one may have arrived at this same difference through strong support and strong rejection, another through moderate support and moderate rejection. This information is lost from the moment the score is computed and cannot be distinguished in the K score.
Thus instead of writing:
"Fermatean fuzzy CoCoSo is more reliable because it allows more extreme judgements than intuitionistic fuzzy does"
the report should read:
"The cube constraint only allows a wider (support, rejection) region; this does not automatically make the result more accurate. The robustness of the result must still be tested separately (do the three strategies agree, does the ranking break down if one judgement shifts by one step)"
When to Prefer This over the Base Method
Fermatean fuzzy is used when the criteria come not from a measurement but from a judgement, and the expert's support and rejection degree both need to be expressed at levels higher than intuitionistic fuzzy allows, that is, when μ+ν>1 and intuitionistic fuzzy is no longer valid, this extension is the one to move to. If the μ+ν≤1 bound is already sufficient, intuitionistic fuzzy CoCoSo is enough; unnecessary complexity should not be added.
The crisp method's exit conditions apply here too: the matrix must be of a single type, and the criterion direction (higher or lower is better) must be marked correctly.
Mistakes Specific to This Extension
Feeding an invalid pair with μ³+ν³>1 into the calculation without noticing. This violates the definition of the Fermatean fuzzy set, and the score becomes meaningless.
Silently changing the score function. μ³−ν³ is the fixed rule here. Using a different defuzzification, for instance using only μ or adding an accuracy function (μ³+ν³), can produce a different ranking; this change must not be made without stating it in the report.
Presenting Fermatean as a "more advanced version" of intuitionistic fuzzy. The cube constraint only allows a wider (μ, ν) region; which constraint to use depends on the nature of the data, it is not a "better" or "worse" choice.
All scores in one column coming out equal. This is a condition inherited from crisp CoCoSo; the min-max normalisation denominator becomes zero and the calculation becomes undefined.
The governing principle is this:
Fermatean fuzzy CoCoSo carries support and rejection degrees only as far as the very start of the calculation, up to a single score. Once the score has been computed, it is crisp CoCoSo itself that carries the ranking from that point on.
Cases
The first case is DecisionMind's validation example. For the Fermatean fuzzy set family there is no shared, page-and-table-traceable CoCoSo application example in the literature. DecisionMind has therefore built a synthetic 3×3 table resting on the F.steps formulas, and the kernel has independently recomputed and verified these figures in Python. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Three suppliers scored with Fermatean fuzzy pairs on three criteria
A firm evaluates three suppliers (A1, A2, A3) on three criteria and gives the scores as support/rejection pairs (the third criterion is cost, lower is better).
| Supplier | Criterion 1 | Criterion 2 | Criterion 3 (Cost) |
|---|---|---|---|
| A1 | (0.7; 0.4) | (0.5; 0.5) | (0.6; 0.5) |
| A2 | (0.8; 0.3) | (0.6; 0.4) | (0.4; 0.6) |
| A3 | (0.6; 0.5) | (0.7; 0.4) | (0.5; 0.5) |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method reduces every pair to a μ³−ν³ score, scales the columns with direction-sensitive min-max, computes the weighted-sum (P) and weighted power-sum (R) measures, combines them with the three compromise strategies (the sum, the worst-case, and the balanced; λ = 0.5), and builds the final k score.
| Supplier | k score | Rank |
|---|---|---|
| A2 | 4.873 | 1 |
| A3 | 2.864 | 2 |
| A1 | 1.211 | 3 |
The result reads as follows: A2 has the highest support/lowest rejection pair on the first criterion, the second-best pair on the second criterion, and also the lowest (best) pair on cost; A2 is superior to the other two suppliers on all three criteria. All three compromise strategies (the sum, the worst-case, and the balanced) give the SAME ranking (A2, A3, A1). This result has been verified with a separate Python calculation.
The firm's hesitation: A2's superiority on all three criteria at once raises the question of how robust this ranking is to changes in weight. When the second criterion's weight is raised from 0.35 to 0.65 and the other two weights are reduced proportionally (0.20/0.65/0.15), A2 still stays ahead (k≈8.41), and A3 keeps second place (k≈7.94); the gap between these two suppliers narrows but the ranking does not break down. But when the second criterion's weight is raised to 0.70 (0.05/0.70/0.25), A3 moves ahead (k≈14.45) and A2 drops to second (k≈13.33). Both of these tests have also been computed independently with Python.
In the report: "With the current weights (0.40/0.35/0.25), A2 has the highest k score (4.873), and all three compromise strategies agree on this ranking. This superiority is robust to moderate changes in weight; A2 stays ahead even if the weight on the second criterion rises to 65 per cent. However, if the great majority of the weight (70 per cent) is given to the second criterion, A3 moves ahead."
Source: DecisionMind's Fermatean fuzzy CoCoSo validation example; because there is no page/table-traceable application in the literature, it has been constructed synthetically based on the F.steps formulas (see the approval notes). The scores and sensitivity scenarios were obtained by independently recomputing the kernel logic in Python.
2. Fisheries: Choosing a model for a new fishing-boat investment
A fisheries cooperative will choose one of three boat models (A1, A2, A3) for a new fishing-boat investment. Three criteria are used: fuel efficiency, cargo capacity, and maintenance cost (lower is better). Two of the models are still new to the market. Technical experts therefore hold both a strong expectation and a strong reservation about these models' performance. In some evaluations the sum of the support and rejection degrees exceeds the bound intuitionistic fuzzy allows (μ+ν≤1), but the cube sum (μ³+ν³≤1) still remains valid.
The method reduces every model to a Fermatean score on all three criteria, scales it, computes the additive and multiplicative measures, and combines them with the three compromise strategies. Suppose the model with the highest fuel efficiency comes out ahead, because it also has the lowest maintenance cost, but it was the model with the lowest cargo capacity.
The cooperative's hesitation: the three compromise strategies do not give the same ranking here. The sum and balanced strategies put the fuel-efficient model ahead, while the worst-case strategy puts the model with the highest cargo capacity ahead. This disagreement shows that the final k score is sensitive to the choice of aggregation; the cooperative should base its decision not only on the combined score but also on which strategy sits closer to its own priority (fuel saving or carrying capacity).
In the report: "By the combined score, the fuel-efficient model is ahead; however, the three compromise strategies do not fully agree on this ranking, so the result should be reassessed depending on whether the cooperative prioritises fuel saving or carrying capacity."
3. What Not to Do
The first error is marking the direction of the third criterion (cost) in the illustrative example as "higher is better"; in that case the most expensive supplier is shown as advantageous and the ranking becomes meaningless. The second error is reporting only the final k score (something like 4.873) without noticing that A2 is superior on all three criteria at once, and never testing how robust this superiority is (it holds up to a weight of 65 per cent, breaks down at 70 per cent). The third error is reading the k score (a value that can exceed 1) as a percentage or a probability, and comparing it with another analysis's score.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ff-cocoso
Senapati, T., & Yager, R. R. (2020). Fermatean fuzzy sets. Journal of Ambient Intelligence and Humanized Computing, 11, 663–674. DOI: 10.1007/s12652-019-01377-0
Yazdani, M., Zarate, P., Zavadskas, E. K., & Turskis, Z. (2019). A combined compromise solution (CoCoSo) method for multi-criteria decision-making problems. Management Decision, 57(9), 2501–2519. DOI: 10.1108/MD-05-2017-0458
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X