Extension card · Fuzzy
Fermatean Fuzzy TOPSIS
This is the form of TOPSIS for situations where it is not the sum but the sum of the cubes of the support and rejection degrees given to a judgement that must not exceed 1. It accepts stronger support-rejection pairs than intuitionistic fuzzy allows, and processes the rest exactly as crisp TOPSIS does.
Base method
TOPSIS →
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 five-step skeleton and the decision logic do not.
Cells. In crisp TOPSIS every cell is a single number. Here every cell is a pair made up of a support degree (μ) and a rejection degree (ν); the sole constraint is μ³+ν³≤1. Under intuitionistic fuzzy, μ=0.8 and ν=0.6 are invalid (their sum exceeds 1), and they are invalid under Pythagorean fuzzy too (the sum of their squares exceeds 1); under Fermatean fuzzy the pair is valid, because the sum of their cubes (0.512+0.216=0.728) stays below 1. Support and rejection swap places on the cost criterion. Criterion weights are taken from outside as crisp numbers.
Weighting. Crisp TOPSIS multiplies every column by the criterion's weight. Under Fermatean fuzzy, weighting is carried out through an operation that shrinks the support degree and raises the rejection degree to a small power; the result of this operation is still a Fermatean fuzzy pair, and it preserves the μ³+ν³≤1 constraint.
Ideal-anti-ideal point and distance. The ideal point (FFPIS) is built from the highest support and lowest rejection degree on every criterion, and the anti-ideal (FFNIS) from the lowest support and highest rejection degree; this follows the same logic as crisp TOPSIS's third step. The distance between the two pairs is a Euclidean distance based on the differences between the cubes (μ³, ν³) and the hesitancy margin derived from them; it is crisp TOPSIS's plain Euclidean distance carried over into cube space.
DecisionMind fixes the cube-based weighting and distance formulas in this extension. Weights are taken from outside; the method does not generate weights.
How to Read the Output
The closeness score is read as in crisp TOPSIS: it states how close an alternative sits to the ideal, given this set of alternatives and these weights, and it is neither a percentage nor a probability. The difference lies here: because the support and rejection degrees are processed in cube space, the difference between two pairs that look numerically alike (μ=0.8 against μ=0.7, say) can grow or shrink once cubed, even where it looks small on the raw figures; estimating the score gap from the raw μ-ν difference is therefore misleading.
Thus instead of writing:
"Because Fermatean fuzzy TOPSIS uses a wider value range, the result is more reliable"
the report should read:
"This score is computed from the distance, in cube space, between the given support and rejection degrees; support-rejection pairs too strong to be valid under intuitionistic fuzzy are valid here, but this only widens the value range and does not automatically make the result more accurate"
When to Prefer This over the Base Method
This extension is suitable where experts assign both strong support and marked reservation to a judgement, and this pair exceeds the intuitionistic or Pythagorean fuzzy constraint (μ+ν>1 or μ²+ν²>1, but μ³+ν³≤1). For example, if an expert says "I support this at 0.9, but I also reject it at 0.6," this pair is invalid under intuitionistic and Pythagorean fuzzy, yet the sum of its cubes (0.729+0.216=0.945) remains valid under Fermatean fuzzy. Where the support-rejection pair already fits the intuitionistic fuzzy constraint, Fermatean fuzzy adds nothing further, and intuitionistic fuzzy TOPSIS is sufficient. Crisp TOPSIS's own exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.
Mistakes Specific to This Extension
Domain violation. Every pair must satisfy μ³+ν³≤1; feeding this into the calculation without checking it invalidates the method.
Moving data that already fits the intuitionistic fuzzy constraint to Fermatean fuzzy unnecessarily. If the support-rejection pair already fits the μ+ν≤1 constraint, moving to Fermatean fuzzy adds no further information, and only complicates the calculation and the report.
Changing the score function. The canonical score is S=μ³-ν³; choosing a different score function can change the ranking, and this choice must be stated in the report.
Reading the cube-space difference as if it were the raw numerical difference. Two pairs' μ-ν difference may look small, yet the distance can grow once cubed; this non-linear relationship must be kept in mind when interpreting the score gap.
The governing principle is this:
Fermatean fuzzy TOPSIS exists to validate strong support-rejection pairs that exceed the intuitionistic and Pythagorean fuzzy constraint; this extra breadth adds something only where the data genuinely exceeds that constraint, and otherwise complicates the calculation needlessly.
Cases
The first case is DecisionMind's validation example; the synthetic 3x3 table in the manifest is built faithfully to the formulas and carries no literature source. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing among three training programmes
A sports club compares three training programmes on three criteria: expected performance gain (more is better), injury risk (less is better), and ease of athlete adherence (more is better). The coaches give each programme's support and rejection on every criterion as a pair; some pairs are strong enough to exceed the intuitionistic fuzzy constraint.
| Programme | Performance gain (μ; ν) | Injury risk (μ; ν, less is better) | Ease of adherence (μ; ν) |
|---|---|---|---|
| P1 | 0.7; 0.4 | 0.6; 0.5 | 0.5; 0.5 |
| P2 | 0.8; 0.3 | 0.4; 0.6 | 0.6; 0.4 |
| P3 | 0.6; 0.5 | 0.5; 0.5 | 0.7; 0.4 |
| Weight | 0.40 | 0.25 | 0.35 |
While some of these pairs (0.7 and 0.4, for instance) remain valid under intuitionistic fuzzy, the μ³+ν³ constraint is comfortably satisfied by all of them; the method uses this flexibility precisely for such strong pairs. The method swaps the support-rejection pair on injury risk, builds the ideal and anti-ideal point in cube space for every criterion, sums the weighted cube distances, and computes the closeness score.
| Programme | Closeness score | Rank |
|---|---|---|
| P2 | 0.823 | 1 |
| P3 | 0.370 | 2 |
| P1 | 0.323 | 3 |
The result reads as follows. P2 holds the highest support and lowest rejection degree on performance gain, the heaviest criterion, and also carries the lowest degree of injury risk. P1 stands out on no criterion, and comes last.
The club's hesitation: once the weight is concentrated on ease of adherence (0.10; 0.10; 0.80) and the calculation is redone, P3 (0.803) clearly overtakes P2 (0.456); this shows that P2's first place depends entirely on the weight given to performance gain and injury risk (recomputed independently with the kernel).
In the report: "With the given weights, P2 has the highest closeness score (0.823); once the weight is concentrated on ease of adherence (0.10; 0.10; 0.80), P3 moves ahead, so the ranking is sensitive to this weight."
Source: DecisionMind's FF-TOPSIS validation example. The TOPSIS skeleton rests on the formulas in the manifest; the Fermatean fuzzy set definition rests on Senapati and Yager's (2020) work. The closeness scores and the weight-trade-off scenario were independently computed by this card's author by running the kernel directly.
2. Sports facility: A municipality's choice of operator for a new swimming pool
A municipality will hand the operation of a new swimming-pool complex to one of three candidate firms. There are three criteria: water-quality management capacity, lifeguard and safety standard, and operating cost (less is better). The municipal commission gives each firm both strong support and marked reservation; for some firms, the two together are high enough to exceed the intuitionistic fuzzy constraint.
The method swaps the support-rejection pair on the cost criterion, builds the ideal and anti-ideal point in cube space for every criterion, and sums the weighted distances. Suppose the cheapest firm also has the weakest support-rejection pair on the safety standard; it still falls behind on the closeness score, because the weight on the safety standard has been kept high.
The commission's hesitation is this: if the score gap between two firms is small, that gap may stem from a small change in the support-rejection pair on the safety-standard criterion. On a sensitive criterion such as safety, the commission should show separately which criterion the score gap comes from.
In the report: "With the weight given to the safety standard, the first firm stands out; the gap between the two firms is sensitive to the support-rejection pair on this criterion and must be assessed separately before the contract is signed."
3. What Not to Do
Arbitrarily shrinking P1's support-rejection pair in the illustrative example to force it within the intuitionistic fuzzy constraint (writing 0.5; 0.3 instead of 0.7; 0.4, say) distorts the expert's actual judgement and needlessly weakens the strong pair that Fermatean fuzzy is precisely meant to allow. A second error is feeding a pair with μ³+ν³>1 into the calculation without checking it; the engine then operates on an invalid value. A third error is reading P2's score of 0.823 as "eighty-three per cent suitable"; the score is only a comparison among these three programmes, relative to the set's own ideal and anti-ideal points.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/ff-topsis
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
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications, A State-of-the-Art Survey. Lecture Notes in Economics and Mathematical Systems, Vol. 186. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9