Extension card · Pythagorean
Pythagorean fuzzy GRA
This is the form of GRA that works for situations where criterion scores are given as the degree to which a judgement is supported and rejected, and the sum of these two degrees can exceed 1. It computes the distance to the reference from these two degrees, and ranks the result again with a grey relational grade.
Base method
GRA →
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 reference-distance-coefficient skeleton does not.
Cells. In crisp GRA every cell is a single number. Here every cell is a support (μ) and a rejection (ν) degree. In intuitionistic fuzzy, μ plus ν may be at most 1; here it is the sum of the squares that is bounded: μ² plus ν² is at most 1. This allows pairs such as μ = 0.7 and ν = 0.4, whose sum exceeds 1 (1.1), but whose sum of squares (0.49 plus 0.16 equals 0.65) stays under 1, to remain valid. Weights remain crisp numbers; GRA does not generate weights, it takes them from outside.
Complementing the cost criterion. Crisp GRA inverts a cost criterion within its normalisation formula. There is no normalisation here; instead, on every cell of a "less is better" criterion, the support and rejection degrees swap places (μ and ν exchange).
Reference and distance. The reference is a Pythagorean ideal, built from each column's own highest support and lowest rejection degree. This point is assembled from components coming from different rows of the column, and need not actually belong to any single alternative; it carries the same logic as crisp GRA's hypothetical reference sequence. The distance, too, is not an absolute difference but a Euclidean-like distance calculated over the squared components: the square root of half the sum of the squared differences between the support squares, the rejection squares and the hesitancy-margin squares.
The grey relational coefficient and grade. These distances are converted into a grey relational coefficient with the same formula as crisp GRA (distinguishing coefficient ρ = 0.5, fixed in DecisionMind), and then, through a weighted sum, into a grey relational grade.
DecisionMind fixes the single-positive-ideal reference, the squared Euclidean distance definition, and the distinguishing coefficient (ρ = 0.5) in Pythagorean fuzzy GRA. Weights are taken from outside as crisp numbers.
How to Read the Output
As in crisp GRA, the grey relational grade shows an alternative's relative closeness to the Pythagorean reference within this particular analysis; it cannot be compared with a different analysis.
The difference lies here. The reference is a composite point seeking both "the strongest support" and "the lowest rejection", and these two components are compared through their squares. A seemingly small μ difference grows rapidly in the squared difference as μ approaches 1. For this reason, the report should state not only the grey relational grade but also which criterion moved the alternative away from the reference, and whether that was in support or in rejection.
Thus instead of writing:
"According to Pythagorean fuzzy GRA, A2 came out as the most reliable alternative"
the report should read:
"A2's grey relational grade is the highest, at 0.8192; this advantage comes from A2 sitting right at the reference on the most heavily weighted criterion (C1), and from its edge on the cost criterion"
When to Prefer This over the Base Method
This extension is used where an expert pair's sum exceeds 1, that is, where a judgement of the form "I strongly support this, but I also have a serious reservation" does not fit the intuitionistic fuzzy constraint. The detail is on the Pythagorean fuzzy data-type card, which also explains the boundary that must not be confused with q-Rung orthopair.
If the expert pairs already fit the intuitionistic constraint, moving to Pythagorean adds nothing, and only reduces discriminating power. If the criteria are measured, one should stay with the base GRA. The crisp method's exit condition holds here too: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is also compensatory.
Mistakes Specific to This Extension
Domain violation. In every cell, μ must lie in [0,1], ν in [0,1], and μ² plus ν² must be at most 1; the calculation should not proceed without this check.
Writing ν as 1 minus μ. This reduces Pythagorean fuzzy data to crisp data; the hesitancy margin is zeroed, and the one thing the structure adds to GRA disappears.
Forgetting to complement the cost criterion. If the support and rejection degrees are not swapped on a "less is better" criterion, the reference sequence is built from the wrong end on that criterion.
Using a pair that already fits the intuitionistic constraint in Pythagorean form. If none of the expert pairs sums to more than 1, moving to Pythagorean adds no information, and only reduces discriminating power.
Never questioning the distinguishing coefficient. ρ = 0.5 is DecisionMind's fixed value; in tables where the distances sit close to one another, this choice affects the result.
The governing principle is this:
In Pythagorean fuzzy GRA, the reference is built, through squares, from that analysis's own strongest support and lowest rejection value. Writing ν as 1 minus μ, skipping the cost-criterion complement, or using this wide region for a pair that already fits the intuitionistic constraint wastes the method's one contribution.
Cases
The first case is DecisionMind's validation example. Because no common PF-GRA application example exists in the literature, a synthetic table with three alternatives and three criteria has been built, staying faithful to the formula chain in the manifest. The second case is an illustrative construction.
1. Illustrative example: Pythagorean fuzzy evaluation of three suppliers on three criteria
A firm compares three suppliers on three criteria: delivery reliability (C1) and quality assurance (C2) are "more is better", unit cost (C3) is "less is better". Weights are C1 = 0.40, C2 = 0.35, C3 = 0.25.
| Supplier | C1 | C2 | C3 (less is better) |
|---|---|---|---|
| A1 | (0.70; 0.40) | (0.50; 0.50) | (0.50; 0.60) |
| A2 | (0.80; 0.30) | (0.60; 0.40) | (0.60; 0.40) |
| A3 | (0.60; 0.50) | (0.70; 0.40) | (0.50; 0.50) |
The method swaps support and rejection on C3, builds the Pythagorean reference from each column's highest support and lowest rejection pair, measures the distance through squares, and converts it to a grey relational grade with ρ = 0.5.
| Supplier | Grey relational grade | Rank |
|---|---|---|
| A2 | 0.8192 | 1 |
| A3 | 0.6196 | 2 |
| A1 | 0.4248 | 3 |
The result reads as follows. A2 is itself the reference on C1, the most heavily weighted criterion (0.80; 0.30), and also sits close to the reference on the inverted C3. A3's edge on C2 (0.70; 0.40) is not enough to close this gap, because C1's weight exceeds C2's.
The firm's hesitation lies in the weights. Verified with an independent Python calculation, if C2's weight is raised from 0.35 to 0.60 and C1 is drawn down from 0.40 to 0.20, and C3 from 0.25 to 0.20, A3 (0.7757) overtakes A2 (0.6901). In other words, A2's lead depends on the high weight given to delivery reliability; if quality assurance is brought to the fore, first place moves to A3.
In the report: "With the weights given (0.40; 0.35; 0.25), A2 is the supplier closest to the reference sequence (grey relational grade 0.8192). Once the weight is markedly shifted towards quality assurance (0.20; 0.60; 0.20), first place moves to A3 (0.7757); the report should therefore justify which criterion is treated as the priority."
Source: a hand-traceable illustrative example with 3 suppliers by 3 criteria; DecisionMind's validation example for the Pythagorean fuzzy GRA engine. The grey relational grades and the weight sensitivity were independently computed in Python by this card's author; the results match exactly the expected result in DecisionMind's manifest (A2 > A3 > A1, the same decimal values).
2. Retail: A chain store's choice of location for a new branch
A retail chain will choose among three locations for a new branch. Three criteria are used: pedestrian-traffic potential and the location's fit with the target customer profile (more is better), monthly rent (less is better). For every location, how much support this judgement receives and how much reservation it carries have been scored separately; because the sum of support and reservation exceeds the intuitionistic constraint in some locations, Pythagorean fuzzy pairs have been used.
The method compares the three locations: it swaps support and rejection on the rent criterion, builds the reference, measures the distance through squares, and computes the grey relational grades. Suppose the location with the strongest support-rejection pair on pedestrian traffic comes out first, despite its relatively high rent, because the weight on pedestrian traffic exceeds that on rent.
The chain's hesitation lies here. If the weight on customer-profile fit is raised, a location with lower rent but weaker pedestrian traffic may move ahead. This means that the relative order of importance among the three criteria reflects the chain's growth strategy, and must be reported.
In the report: "With the high weight given to pedestrian traffic, the location strong on this criterion comes first; its relative disadvantage in rent does not change this advantage. If the weight on customer-profile fit is raised, the ranking may change, and this sensitivity must be separately reported."
3. What Not to Do
The first error is deriving ν as 1 minus μ when writing A2's C1 cell (0.80; 0.30), that is, (0.80; 0.20), instead; this brings the calculation closer to crisp data without ever using the Pythagorean structure's extra region of acceptance. The second error is leaving support and rejection unswapped on C3 (less is better); in that case the reference sequence is built relative to the most expensive supplier. The third error is ignoring the sensitivity by which A2 and A3 swap places under a weight change, and reporting instead that "A2 is clearly first"; the ranking depends on the choice of weights, and this must not be hidden from the report.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pf-gra
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
Deng, J. L. (1989). Introduction to grey system theory. The Journal of Grey System, 1(1), 1–24. (no DOI)
Kuo, Y., Yang, T., & Huang, G. W. (2008). The use of grey relational analysis in solving multiple attribute decision-making problems. Computers & Industrial Engineering, 55(1), 80–93. DOI: 10.1016/j.cie.2007.12.002