Extension card · Fuzzy
Fermatean Fuzzy MABAC
Fermatean fuzzy MABAC is the form of MABAC 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 alternatives are again ranked by their distance to a hypothetical border approximation area.
Base method
MABAC →
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?
Three things change; the border-approximation-area comparison logic does not.
Cells. In crisp MABAC 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 and border approximation area. Crisp MABAC places every column into the 0-1 range with min-max scaling, multiplies by the weight, shifts by "+1", and builds the border value with a geometric mean. This "+1" shift exists because the geometric mean cannot work with a zero or negative value. No such shift is needed here. 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 preserves the cube constraint. The score of every scaled pair (support cubed minus rejection cubed) is then computed, and the border value is built as the arithmetic mean of these scores across the alternatives. Crisp MABAC's need for a geometric mean and a "+1" shift does not apply here, because the score has already collapsed to a single number and can be negative.
Result and defuzzification. In crisp MABAC every alternative's distance to the border is summed, and the total is already a single number. The same summation happens here too, but defuzzification takes place one step earlier, while the border value is being built: the support-rejection pair collapses to a single number via the score function, the border is built from these single numbers, and only then are distances to the border summed. This means the uncertainty is carried, not to the very last step, but to the step just before the border is built; crisp MABAC has no defuzzification at all, whereas in this extension defuzzification happens at an early step, but before distances are summed.
DecisionMind holds the score function (support cubed minus rejection cubed) fixed for classical Fermatean fuzzy MABAC. If a different scoring rule is chosen, this is stated in the report.
How to Read the Output
The MABAC score is read exactly as in crisp MABAC: it is an alternative's net position relative to this set's average performance border. A positive score does not mean "good" and a negative score does not mean "bad"; it only shows relative position within this alternative set.
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. The border value is also the average of these cube scores; adding or removing an alternative changes the border.
Thus instead of writing:
"Fermatean fuzzy MABAC carries two degrees, so the result is more reliable than crisp MABAC"
the report should read:
"Because criterion scores are given as support and rejection degrees, the uncertainty has been carried through a cube power, reduced to a single number at an early step by the score function, and then summed as distance to the border approximation area; 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 MABAC 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 gives that criterion's scores, and hence the border value, the wrong sign.
Entering an invalid support-rejection pair. A pair is invalid if the sum of the support and rejection cubes exceeds one; this violation shows that the entered value breaches the constraint, and the user must check for it themselves before the analysis.
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 change the border value, and with it the entire ranking.
Trying to build the border value with a geometric mean, as in crisp MABAC. In this extension the border is the arithmetic mean of the scaled pairs' scores; crisp MABAC's "+1"-shifted geometric mean does not apply here, because the scores are already single numbers that can be negative.
The governing principle is this:
Fermatean fuzzy MABAC carries the support and rejection degree in keeping with the cube constraint and reduces it to a score at an early step; 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. The manifest's own record states that no commonly shared literature example exists for Fermatean fuzzy MABAC; a web search likewise could not confirm a single founding paper specific to the method with an MCDM application. 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).
| 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 computes its score (support cubed minus rejection cubed). The average of these scores on every criterion gives the border value (C1: -0.169; C2: -0.328; C3: -0.552). Every alternative's score is subtracted from this border and summed across the criteria.
| Alternative | MABAC score | Rank |
|---|---|---|
| A2 | 0.3176 | 1 |
| A3 | -0.1013 | 2 |
| A1 | -0.2162 | 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; it sits above the border on all three criteria, most strongly on C1 (+0.183) and C3 (+0.108), and by a finer margin on C2 (+0.027). A1, by contrast, sits below the border on all three criteria, most markedly on C2 (-0.110) and C3 (-0.097), which is why it finishes last.
The robustness of the decision has been tested by changing the weights. Even when C2's weight is raised from 0.35 to 0.90, with C1 and C3's ratio held fixed and rescaled, A2 stays ahead (0.093 against A3's 0.076); only when C2's weight is raised to 0.95 does A3 move ahead (0.104 against 0.053). A2's first place is fairly robust in this example, breaking down only in the extreme case where nearly all the weight is given to C2.
In the report: "With the given weights (C1=0.40, C2=0.35, C3=0.25), A2 has the highest MABAC score (0.3176); this ranking holds even when C2's weight is raised to approximately 90 per cent, and only in the extreme scenario of giving nearly all the weight to C2 does A3 move ahead."
Source: DecisionMind's validation fixture for the FF-MABAC engine (SYNTHETIC_VERIFIED). The steps rest on Senapati and Yager's (2020) definition of Fermatean fuzzy sets and on the border-approximation-area logic of Pamučar and Ćirović (2015) in MABAC. No single founding paper combining these two under a distinct name could be confirmed either in the manifest's record or in the web search; details are in the verification notes. All figures have been independently recomputed by this card's author and verified against the engine's output to a tolerance of 1e-9.
2. Fisheries: A fishery cooperative's choice of cold-chain processing facility tender
A fishery cooperative will choose among three contractor tenders to build a new cold-chain processing facility. The criteria are the adequacy of cooling capacity, the breadth of hygiene certification coverage, and installation unit cost, the last being "lower is better". Following site visits, the cooperative's management has scored each tender by how strongly it found the tender to perform well and how strongly it found weaknesses, separately, for every criterion.
The method scales the three tenders' pairs, computes their scores, builds the border value, and sums each tender's distance to the border. Suppose the tender with the highest cooling capacity is also the most expensive. It nonetheless comes first, because the weight on capacity exceeds the weight on cost.
The cooperative's hesitation is this. Choosing the most expensive tender must be defended to members on cost grounds. It should also be stated in the report that the border value is built from the average score of these three tenders, and that adding a fourth tender would change the border and require the ranking to be recalculated.
In the report: "With the high weight given to cooling capacity, the tender with the highest capacity reaches the highest MABAC score; its cost disadvantage is offset by its advantage in capacity, and the border value is specific only to the comparison among these three tenders."
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 ranking would stay the same (A2, A3, A1), but the gap between A3 and A1 would shrink from 0.115 to 0.0016; the second and third places would nearly merge, hiding the real gap between them. The second error is entering a pair whose support and rejection cubes sum to more than one without noticing; this conceals the fact that the entered value violates the constraint. The third error is presenting the border value as "the best tender's score"; the border is the average of the three tenders' scores, not the value of the best one.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ff-mabac
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
Pamučar, D., & Ćirović, G. (2015). The selection of transport and handling resources in logistics centers using Multi-Attributive Border Approximation area Comparison (MABAC). Expert Systems with Applications, 42(6), 3016–3028. DOI: 10.1016/j.eswa.2014.11.057
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. DOI: 10.1016/S0019-9958(65)90241-X
Torkayesh, A. E., Tirkolaee, E. B., Bahrini, A., Pamucar, D., & Khakbaz, A. (2022). A Systematic Literature Review of MABAC Method and Applications: An Outlook for Sustainability and Circularity. Informatica, 34(4), 899–929. DOI: 10.15388/23-INFOR511