Extension card · Fuzzy
Fermatean Fuzzy EDAS
The form of EDAS for situations where the sum of the cubes of a judgement's support and rejection degrees does not exceed 1. This constraint allows the two degrees to be simultaneously high over a region wider even than Pythagorean fuzzy data permit; the method reduces every cell to a score straightaway and runs the remainder exactly as in crisp EDAS.
Base method
EDAS →
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 six-step skeleton and the decision logic do not.
Cells. In crisp EDAS every cell is a single number. Here every cell is a pair of degrees given to a judgement: support (μ) and rejection (ν). The constraint is μ³+ν³≤1. The sum of cubes grows more slowly than the sum of squares, so this constraint permits some pairs (both 0.85, for example) that Pythagorean fuzzy data would not. Criterion weights are taken from outside as crisp numbers.
Defuzzification comes right at the start, and the average is also taken over the scores. Every (μ, ν) pair collapses to a single number in the very first step of the calculation, via the Fermatean score function (s = μ³ − ν³). Unlike Pythagorean fuzzy EDAS, the column average is built NOT by taking a weighted average of a support-rejection pair, but by taking the plain arithmetic mean of these scores directly. This means this extension's "average solution" is not a Fermatean pair; it is only a number, and it cannot be converted back into μ,ν. Everything else is crisp EDAS itself: every cell's positive and negative deviation from this average is measured, summed with the weights, normalised, and combined into a single appraisal score.
DecisionMind holds the Fermatean score function (μ³ − ν³) and the plain arithmetic mean of the scores fixed for this extension. Weights are taken from outside; the method does not produce weights.
How to Read the Output
The appraisal score is read as in crisp EDAS: it is a position relative to the set's own average, not a percentage or a probability. The difference is this: that average is not a reversible support-rejection pair, it is only the plain mean of the scores. If, on a given criterion, all alternatives' scores sit close together (even if their μ,ν pairs differ considerably but their scores are equal), that criterion's discriminating power weakens here too, just as in crisp EDAS.
Thus instead of writing:
"Fermatean fuzzy EDAS is more reliable because it allows more extreme judgements than Pythagorean fuzzy data do"
the report should read:
"The cube constraint only permits a wider (support, rejection) region; the result remains a position relative to this alternative set's own average, and it is not automatically more accurate"
When to Prefer This over the Base Method
This extension is considered when experts' support-rejection pairs also exceed Pythagorean fuzzy data's squared constraint (for example, μ=0.9, ν=0.7; the sum of squares is 1.30, which exceeds the Pythagorean limit, but if the sum of cubes, 0.729+0.343=1.07, still exceeds it, q-Rung is needed; if it does not, Fermatean suffices); details are on the Pythagorean fuzzy data-type card. If the two degrees already fit within the Pythagorean constraint, moving to Fermatean is not necessary, and only discriminating power is lost. Converting a measured value directly into a support-rejection pair is not modelling uncertainty but manufacturing it. The exit condition of crisp EDAS applies here too: where a criterion admits no compromise, this extension is also compensatory.
Mistakes Specific to This Extension
Violating the value domain. 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 the cube constraint provides is never used.
Trying to interpret the average solution as a support-rejection pair. In this engine the average is only the plain mean of the scores; it cannot be converted back into "the set's average μ,ν is such-and-such".
Choosing Fermatean for pairs that already fit within the intuitionistic or Pythagorean constraint. If the pairs already fit within the narrower constraint, the wider region adds no extra information; it only reduces discriminating power.
The governing principle is this:
Fermatean fuzzy EDAS's cube constraint adds something only when experts' pairs genuinely exceed the Pythagorean constraint; in this engine the average solution is a number, not a support-rejection pair, and the report must not confuse the two.
Cases
The first case is DecisionMind's validation example; the synthetic 3x3 table in the manifest has been built faithfully to the formulas and carries no literature page. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing a raw-material supplier
A manufacturer compares three raw-material suppliers on three criteria: delivery reliability (higher is better), quality consistency (higher is better), and rejected-batch rate (lower is better). Every supplier's support and rejection on every criterion is given as a Fermatean pair (μ,ν).
| Supplier | Delivery reliability | Quality consistency | Rejected-batch rate (lower is better) |
|---|---|---|---|
| T1 | (0.70; 0.40) | (0.50; 0.50) | (0.60; 0.50) |
| T2 | (0.80; 0.30) | (0.60; 0.40) | (0.40; 0.60) |
| T3 | (0.60; 0.50) | (0.70; 0.40) | (0.50; 0.50) |
| Weight | 0.40 | 0.35 | 0.25 |
The method computes every cell's Fermatean score (μ³−ν³) and finds the plain mean of each column's scores. It then measures every supplier's positive and negative deviation from this average, sums with the weights, and combines them into a single appraisal score.
| Supplier | Appraisal score | Rank |
|---|---|---|
| T2 | 1.000 | 1 |
| T3 | 0.435 | 2 |
| T1 | 0.000 | 3 |
The result reads as follows. T2 holds the highest score on delivery reliability, the most heavily weighted criterion, and is also best positioned on rejected-batch rate (lower is better); these two carry T2 well ahead. T1 sits in the opposite position and finishes last.
The manufacturer's hesitation is sensitive to weight. If delivery reliability's weight is raised to 0.80 with the other two criteria pulled to 0.10 each, T3's score falls from 0.435 to 0.054; T2 remains first, but its gap with T3 widens considerably. Conversely, if quality consistency is raised to 0.80 with the other two pulled to 0.10 each, T3's score rises to 0.931 and nearly catches T2. Even then it cannot overtake T2, because T2 is also strong on delivery reliability and rejected-batch rate. This shows that T2's first place is robust to weight trade-offs, but how close T3 can get to T2 depends on the weighting.
In the report: "With the given weights, T2 is clearly first (1.000); this ranking is robust to weight trade-offs, because T2 is strong on two criteria at once. T3's closeness to T2 only increases when the weight on quality consistency is raised very substantially."
Source: DecisionMind's Fermatean fuzzy EDAS validation example. The score function and the EDAS skeleton rest on Senapati and Yager's (2020) definition of Fermatean fuzzy sets and on Keshavarz Ghorabaee et al.'s (2015) definition of EDAS. The appraisal scores and the weight-trade-off scenarios have been independently computed by this card's author by running the kernel directly.
2. Nurseries: A company's choice of contracted nursery for employees' children
A company will enter into a corporate agreement with one of three nurseries for its employees' children. There are three criteria: the richness of the educational programme, the adequacy of the staff-to-child ratio, and the frequency of absences and closures (lower is better). The human resources unit records, for each nursery, both the degree of support and the degree of reservation obtained from the visiting team members as a Fermatean pair. At some nurseries both values are high together, because team members observed both strong points and serious concerns at once.
The method computes every nursery's score, measures its positive and negative deviation from the column average, and combines it into a single appraisal score. Suppose the nursery with the richest educational programme also came in below average (favourably) on absence frequency, and finished first.
The company's hesitation is this: at a nursery where support and reservation are both high, this two-sided information is lost once it collapses to a single score. The company should carry out a separate reference check for nurseries that score highly but also carry a high degree of reservation; looking at the score alone conceals this reservation.
In the report: "One nursery stands out owing to the richness of its educational programme and its low absence frequency; the visiting team's marked reservations about this nursery have also been recorded, and a separate reference check is recommended."
3. What Not to Do
Had the rejected-batch rate been marked "higher is better" in the illustrative example, the supplier with the most rejects would have been rewarded and the ranking would become meaningless. The second error is trying to convert T2's average solution back into a (μ,ν) pair, as in "the set's average support-rejection pair is such-and-such"; in this engine the average is only a number. The third error is reading T2's score of 1.000 as "a one hundred per cent reliable supplier"; the score is only a comparison among these three suppliers relative to the set's own average.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ff-edas
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
Yager, R. R. (2014). Pythagorean membership grades in multicriteria decision making. IEEE Transactions on Fuzzy Systems, 22(4), 958–965. DOI: 10.1109/TFUZZ.2013.2278989
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Keshavarz Ghorabaee, M., Zavadskas, E. K., Olfat, L., & Turskis, Z. (2015). Multi-criteria inventory classification using a new method of evaluation based on distance from average solution (EDAS). Informatica, 26(3), 435–451. DOI: 10.15388/Informatica.2015.57