Extension card · Fuzzy
Fermatean Fuzzy CODAS
Fermatean fuzzy CODAS is the form of CODAS used when an expert states, together, how strongly a criterion is supported and how strongly it is rejected. This pair is admitted over a wider region than intuitionistic fuzzy data allow, and the result is again ranked by a single assessment score.
Base method
CODAS →
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?
Four things change. The two-distance comparison logic does not.
Cells. In crisp CODAS every cell is a single number. Here every cell is a pair made up of a support and a rejection degree, and the sum of their cubes cannot exceed one. Criterion weights remain crisp numbers. The method does not directly support group decisions.
Scale equalisation. Crisp CODAS divides every cell by the ratio of the column's largest or smallest value. There is no such division here, because support and rejection degrees already lie between 0 and 1. For a cost criterion, support and rejection are first swapped. Each pair is then scaled directly by the criterion's weight through a power operation specific to Fermatean algebra; this operation shrinks the support and rejection degrees together while preserving the cube constraint.
Distance. Crisp CODAS computes two distances to the negative-ideal point: Euclidean and taxicab. Here a negative-ideal pair is first built, for every criterion, from the lowest support and highest rejection value among the scaled pairs. Every alternative's distance to this point is then computed from the cubes of its support and rejection degrees together with a third component, the degree of indeterminacy, which shows how close a pair sits to the cube constraint. DecisionMind applies the weight in this extension both at the scaling step and at the step where distances are summed; this differs from classical CODAS's single-layer weighting.
Result and defuzzification. The assessment score is built with the same pairwise-comparison rule as crisp CODAS (threshold τ, Euclidean first, taxicab if it does not suffice) and is already a single number. There is no separate defuzzification step, because the support and rejection degrees stay paired throughout until they enter the distance calculation, only descending to a single number at that step.
DecisionMind holds fixed, for classical Fermatean fuzzy CODAS, the score function (support cubed minus rejection cubed) and the threshold value (τ = 0.02). If a different scoring rule is chosen, this is stated in the report.
How to Read the Output
The assessment score is read exactly as in crisp CODAS. It is a relative measure of position, not a percentage, and it cannot be compared with a different analysis. The difference is this: because the score derives from the cube of the support and rejection degrees, a small change in degree can produce a larger effect than a linear change would. An alternative's score can therefore shift visibly even when the support or rejection degree on a single criterion changes only slightly.
Thus instead of writing:
"Fermatean fuzzy CODAS carries two degrees, so the result is more reliable than crisp CODAS"
the report should read:
"Because criterion scores are given as support and rejection degrees, the uncertainty has been carried through a cube power and then reduced to a single number by distance to the negative-ideal; the ranking should be reported against this weighting"
When to Prefer This over the Base Method
This extension is used when an expert states, separately, how strongly they support and how strongly they reject a criterion. These two degrees can be simultaneously high to a stronger extent than intuitionistic fuzzy data allow. Measured criteria should not be moved into this extension. If the matrix must be of a single type, a measured value is written by setting support and rejection accordingly (support 1, rejection 0, for example). The base CODAS exit condition applies here in exactly the same way: where a criterion admits no compromise, a compensatory method is already unsuitable.
Mistakes Specific to This Extension
Marking criterion direction wrongly. Treating a cost criterion as a benefit and skipping the support-rejection swap builds the negative-ideal point incorrectly. In the illustrative example below, this mistake swaps the positions of the weakest and the second-strongest alternative.
Entering an invalid support-rejection pair. A pair is invalid if the sum of the support and rejection cubes exceeds one. The engine does not raise an error in this case; it treats the degree of indeterminacy as zero. This conceals the fact that the entered value violates the constraint, so the user must check for violations themselves.
Changing the score function without question. The rule of support cubed minus rejection cubed is widely accepted, but it is not the only option. A different scoring rule can produce a different ranking from the same pairs.
Forgetting that the weight is applied twice. In this extension the weight is used both at the scaling step and when distances are summed. Verifying the result by hand while assuming the weight is applied only once gives the wrong number.
The governing principle is this:
Fermatean fuzzy CODAS exists to carry support and rejection degrees together, in keeping with the cube constraint; if the constraint is violated, or the scoring rule is changed without justification, the apparent precision is undermined.
Cases
The first case is DecisionMind's own validation example: a small, three-alternative, three-criterion table produced by the engine itself, faithful to the formulas. The manifest's own record states that no commonly shared literature example exists for Fermatean fuzzy CODAS. Case 1 is therefore an illustrative validation example, not a literature case. The second case is an illustrative construction.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
Three alternatives are assessed on three criteria with support and rejection degrees. The first two criteria are "higher is better", the third is "lower is better" (cost). The threshold value is τ = 0.02.
| Alternative | C1 (support; rejection) | C2 (support; rejection) | C3, cost (support; rejection) |
|---|---|---|---|
| A1 | 0.70; 0.40 | 0.50; 0.50 | 0.60; 0.50 |
| A2 | 0.80; 0.30 | 0.60; 0.40 | 0.40; 0.60 |
| A3 | 0.60; 0.50 | 0.70; 0.40 | 0.50; 0.50 |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method first swaps support and rejection in C3. It scales every pair by the criterion's weight, and builds the negative-ideal from the pair with the lowest support and highest rejection. It computes each alternative's Euclidean and taxicab distance to this point and sums the pairwise comparisons.
| Alternative | Assessment score | Rank |
|---|---|---|
| A2 | 0.3965 | 1 |
| A3 | -0.1672 | 2 |
| A1 | -0.2293 | 3 |
The result reads as follows. A2 holds the highest support (0.80) on the most heavily weighted criterion, C1, and the lowest, best, support on cost criterion C3. These two advantages more than offset its relative weakness on C2. The gap between A3 and A1 is small, and A3's advantage on C2 largely cancels out A1's advantage on C1.
To see how robust the decision is, the weights were changed and the calculation redone. C2's weight was raised from 0.35 to 0.63, C1 lowered to 0.12, and C3 held fixed at 0.25. Under this configuration A3 moves ahead (0.2067), and A2 drops to second (0.2054). The gap between them is only 0.0013, meaning A2's first place is moderately sensitive to the weight on C2.
In the report: "With the given weights (C1 = 0.40, C2 = 0.35, C3 = 0.25) and a threshold of τ = 0.02, A2 holds the highest assessment score (0.3965); when C2's weight is raised to approximately 0.63, A3 moves ahead (0.2067 against 0.2054), so A2's first place is sensitive to the weight on C2."
Source: DecisionMind's validation fixture for the FF-CODAS engine. The steps rest on Senapati and Yager's (2020) definition of Fermatean fuzzy sets and on the two-distance comparison logic of Keshavarz Ghorabaee et al. (2016) in CODAS. No single founding paper combining these two under a distinct name could be confirmed; details are in the verification notes. All figures have been independently recomputed by this card's author and verified against the engine's output.
2. Logistics: An e-commerce company's choice of last-mile delivery partner
An e-commerce company will contract with one of three delivery firms for last-mile fulfilment ahead of its peak season. The criteria are delivery speed, breadth of service coverage, and cost per shipment, the last being "lower is better". Because the company's operations team has previously trialled the firms on a small scale, it has scored each criterion by how strongly it found the firm to perform well and how strongly it found weaknesses, separately.
The method scales the three firms' pairs, builds the negative-ideal, and computes the two distances. Suppose the fastest firm on delivery is also the most expensive. It nonetheless comes first, because the weight on speed exceeds the weight on cost. The Euclidean gap between the second and third firm falls below the threshold, so the ranking between them is settled by taxicab distance instead.
The company's hesitation is this. Choosing the most expensive firm must be defended to senior management as a cost increase during peak season. It should also be stated in the report that the ranking between the second and third firm rests on taxicab distance, and that these two could swap places if the threshold were changed.
In the report: "With the high weight given to delivery speed, the fastest firm reaches the highest assessment score; the ranking between the second and third firm is sensitive to the threshold value, and the source of these firms' support-rejection scores should be stated separately in the report."
3. What Not to Do
Had C3 been mistakenly treated as a benefit criterion in the illustrative example, that is, had the support-rejection swap been skipped, the negative-ideal would have been built from the wrong criterion. A1 would then move to second place and A3 would finish last, giving the order A2, A1, A3, whereas the correct ranking places A3 second. The second error is entering a pair whose support and rejection cubes sum to more than one without noticing. The engine does not flag this as an error; it treats the indeterminacy as zero and conceals the violation. The third error is forgetting that the weight in this extension is used both in scaling and in summing distances, and trying to verify the result by hand with a single-layer weighting.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ff-codas
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
Keshavarz Ghorabaee, M., Zavadskas, E. K., Turskis, Z., & Antucheviciene, J. (2016). A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making. Economic Computation and Economic Cybernetics Studies and Research, 50(3), 25–44. (no DOI. This article is not registered in Crossref; see the Sources section of the base CODAS card.)
Keshavarz Ghorabaee, M., Amiri, M., Zavadskas, E. K., Hooshmand, R., & Antucheviciene, J. (2017). Fuzzy extension of the CODAS method for multi-criteria market segment evaluation. Journal of Business Economics and Management, 18(1), 1–19. DOI: 10.3846/16111699.2016.1278559
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X