Extension card · Pythagorean
Pythagorean fuzzy ARAS
Pythagorean fuzzy ARAS is the form of ARAS used when criterion assessments are given as a membership and non-membership degree pair, and the sum of these two degrees is permitted to exceed 1 provided the sum of their squares does not. It computes the additive utility ratio over these pairs, reducing the result to a single degree of utility.
Base method
ARAS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Pythagorean →
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 decision logic does not.
Cells. In crisp ARAS every cell is a single number. Here every cell is a pair: a membership degree (μ) and a non-membership degree (ν). Unlike the intuitionistic fuzzy pair, it is not their sum but the sum of their squares that must not exceed 1 (μ²+ν²≤1). This leaves the expert a wider region: a support degree of 0.8 and a rejection degree of 0.5 can be given, since 0.8²+0.5²=0.89≤1, whereas intuitionistic fuzzy would reject 0.8+0.5=1.3 as invalid.
Scale equalisation. Crisp ARAS inverts a cost criterion (1/x) and divides it by the column total. Here, inverting direction on a cost criterion means taking a complement: μ and ν swap places. Every cell is then reduced to a score, the membership degree's square minus the non-membership degree's square (s=μ²−ν²). Because these scores can be negative, they are shifted first, then set as a ratio against the column total, the optimal-alternative row included — the intuitionistic fuzzy extension's shifted-score logic, adapted to Pythagorean algebra.
The optimal alternative. The optimal alternative is built not from the raw membership-non-membership pairs but from these scores: on each criterion, the cell carrying the highest score is taken as optimal.
Result and defuzzification. The shifted, normalised scores are multiplied by the criterion weight, and every row, the optimal alternative included, is summed across the criteria. The utility degree K keeps its crisp-ARAS meaning: the ratio of a real alternative's total score to the optimal alternative's total score. DecisionMind fixes this score function and shifting step; the (μ, ν) pair is never ranked directly, only K is.
How to Read the Output
The utility degree K carries the same meaning here too. The best alternative is taken as 100, and the others receive a percentage relative to it, valid only for this alternative set and these weights. The difference lies here: beneath K sits an uncertainty from the Pythagorean fuzzy structure's wide region of freedom. A cell where both degrees are high (0.7 and 0.6, say) represents a judgement that is at once strongly supported and strongly qualified; the score function subtracts these two large numbers from each other, leaving a small net value.
Thus instead of writing:
"According to Pythagorean fuzzy ARAS, A2 is the best alternative"
the report should read:
"With the weights given (0.40; 0.35; 0.25), A2 has the highest utility degree (K=0.902); once the weight is concentrated on the second criterion (0.10; 0.80; 0.10), A3 moves into first place, so the weight distribution must be separately justified in the report"
When to Prefer This over the Base Method
This method suits situations where a judgement's support and rejection degrees are given so that it is not their sum but the sum of their squares that does not exceed 1. Examples: expert judgements in which strong support and strong reservation can be voiced at the same time, and situations where the intuitionistic fuzzy structure's sum-to-1 constraint would artificially narrow the expert's opinion.
If the sum of the support and rejection degrees does not already exceed 1, moving to the Pythagorean fuzzy structure adds nothing; intuitionistic fuzzy ARAS is then sufficient, and simpler. Crisp ARAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory and will not screen out anything below a threshold.
Mistakes Specific to This Extension
Domain violation. In every cell, the sum of the squares of the membership and non-membership degrees must not exceed 1 (Yager, 2014). This must not be confused with the intuitionistic fuzzy constraint (that their sum does not exceed 1); the two are different constraints, and a pair valid under one can be invalid under the other.
Applying crisp ARAS's 1/x operation to a cost criterion instead of taking the complement. This breaks the membership-non-membership pair's sum-of-squares constraint and produces an invalid pair.
Using the score without shifting it. The score (μ²−ν²) can be negative; if the shifting step is skipped, the normalised values, and hence the utility degrees, can also come out negative.
Changing the score function without saying so. μ²−ν² is the canonical choice, but not the only one; a different score function (one that also folds in the hesitancy margin, say) can change the ranking. Which score was used must be stated in the report.
The governing principle is this:
Pythagorean fuzzy ARAS exists to carry the wide freedom of a judgement that can hold both strong support and strong reservation through to the ratio against the optimal alternative; any application that confuses the constraint with the intuitionistic fuzzy one, or uses the score without shifting it, corrupts the information the pair carries.
Cases
The first case is DecisionMind's validation example. Because the manifest records contain no application paper specific to Pythagorean fuzzy ARAS that presents a shared decision table, DecisionMind's 2026-05-25 protocol required building a small, hand-traceable table instead. The second case is an illustrative construction.
1. Illustrative example: Three candidates, three criteria (DecisionMind validation example)
An assessment board scores three candidates on three criteria; the first and second criteria are "more is better", the third is "less is better". Every score is given as a membership and non-membership degree. The board has assigned weights of 0.40, 0.35 and 0.25 to the three criteria respectively.
| Candidate | C1 | C2 | C3 (less is better) |
|---|---|---|---|
| A1 | μ=0.7 ν=0.4 | μ=0.5 ν=0.5 | μ=0.6 ν=0.5 |
| A2 | μ=0.8 ν=0.3 | μ=0.6 ν=0.4 | μ=0.4 ν=0.6 |
| A3 | μ=0.6 ν=0.5 | μ=0.7 ν=0.4 | μ=0.5 ν=0.5 |
| Weight | 0.40 | 0.35 | 0.25 |
The method takes the complement on the third criterion, scores every cell (μ²−ν²), shifts the scores and sets them as a ratio against the column total (the optimal alternative included), then multiplies by the weights and sums.
| Candidate | Utility degree (K) | Rank |
|---|---|---|
| A2 | 0.902 | 1 |
| A3 | 0.560 | 2 |
| A1 | 0.331 | 3 |
The result reads as follows. A2 has the highest membership degree and the lowest non-membership degree on the first criterion; being also the best on the first criterion, the most heavily weighted one, it comes out clearly first. A1 sits in the weakest position on the third criterion (once reversed) and finishes last.
The board has a hesitation. Once the weight is concentrated on the second criterion (0.10; 0.80; 0.10) and the calculation redone, the utility degrees come out at 0.857 for A3, 0.769 for A2 and 0.259 for A1, and the ranking is COMPLETELY reversed: A3 rises to first place (computed independently by running the same algorithm in Python). This shows that A2's first place depends entirely on the weight given to the first and third criteria; the board must defend, in the report, why it chose this weight distribution.
In the report: "With the weights given (0.40; 0.35; 0.25), A2 has the highest utility degree (K=0.902); once the weight is concentrated on the second criterion (0.10; 0.80; 0.10) the ranking is completely reversed, and A3 rises to first place. The weight distribution must therefore be separately justified by the board."
Source: DecisionMind's validation example for the Pythagorean fuzzy ARAS engine; the score function and the constraint are faithful to Yager's (2014) definition of the Pythagorean fuzzy set, but no application paper specific to this method presenting a shared decision table could be found in the manifest records. The decision matrix and the results were built by hand and computed in Python by this card's author; this point is separately noted in the verification note.
2. Tourism: A travel agency's choice of tour operator
A travel agency will choose among three operators for a new overseas tour programme. Two criteria apply: service quality and the programme's flexibility (adaptation to changing demands). The agency's managers have stated, for each operator, how far they support and how far they reject the proposition "this operator will meet the expected quality"; for one operator, both the support and the rejection degree come out high at once, because its past tours have gathered both very positive and a few negative reviews.
The method compares the three operators: it scores every cell, shifts the scores and sets them as a ratio against the column total, then multiplies by the weights and sums. Suppose the operator with both high support and high rejection comes out second rather than first, because its net score stays at a middling level. An operator with a lower, but less qualified, support value comes first.
The agency has a hesitation. The second-ranked operator's high rejection share calls for investigating the cause of the past negative reviews. The agency should not decide between the two operators on the strength of the net score alone before carrying out this investigation.
In the report: "The first operator has both a higher net score and a lower rejection share; price negotiation between these two operators should not proceed until the source of the second operator's high rejection share is clarified."
3. What Not to Do
The first error is checking the illustrative example's membership-non-membership pairs against the intuitionistic fuzzy constraint (that their sum does not exceed 1); A2's pair on C1, (0.8; 0.3), appears to violate this constraint with a sum of 1.1, whereas under the Pythagorean constraint (sum of squares) it is valid, at 0.73. The second error is applying crisp ARAS's 1/x operation to a cost criterion instead of taking the complement; this produces a pair that breaks the sum-of-squares constraint. The third error is reporting A2's utility degree of 0.902 as "the certainly correct choice, at 90 per cent"; K shows only these three candidates' proportional utility relative to the optimal candidate, not a probability.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pf-aras
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
Zhang, X., & Xu, Z. (2014). Extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. International Journal of Intelligent Systems, 29(12), 1061–1078. DOI: 10.1002/int.21676
Zavadskas, E. K., & Turskis, Z. (2010). A new additive ratio assessment (ARAS) method in multicriteria decision-making. Technological and Economic Development of Economy, 16(2), 159–172. DOI: 10.3846/tede.2010.10