Extension card · Pythagorean
Pythagorean fuzzy SAW
This is the Pythagorean fuzzy form of SAW. The support and rejection degrees given to a judgement may sum to more than 1, provided the sum of their squares does not. This pair is carried through the whole calculation, and comes down to a single weighted score only at the last step.
Base method
SAW →
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?
Three things change; the weighted-sum logic does not.
Cells. In crisp SAW every cell is a single number. Here every cell is a pair: μ, the support degree, and ν, the rejection degree. These two values must satisfy μ² + ν² ≤ 1. Criterion weights are taken from outside as crisp numbers; only the cells become fuzzy, not the weights.
Scale equalisation. Crisp SAW ratios every column to its own best value. There is no such step here, because support-rejection pairs already lie between 0 and 1. In its place, μ and ν swap places on a cost criterion. This operation does not equalise scale; it reverses direction.
Weighted sum and defuzzification. In crisp SAW every column is multiplied by its weight and summed directly. Here, every alternative's support-rejection pair across the criteria is reduced to a single combined pair by the Pythagorean weighted-average (PFWA) operator. This combination is an algebraic operator, different from crisp SAW's linear sum; its result is again a support-rejection pair, not yet a single number. Only at this final step does the combined pair come down to a single number through the Pythagorean score function (s = μ² − ν²), and alternatives are ranked by this score from highest to lowest.
DecisionMind fixes, in classical PF-SAW, the PFWA combination operator and the score function (μ²−ν²). Weights are taken from outside; the method does not generate weights.
How to Read the Output
The score, like the total score in crisp SAW, ranks only this particular alternative set; it is not a percentage or a probability, and it is not compared with another analysis.
The difference is this: beneath the score, both SAW's full-compensation logic and a compression coming from combining the support-rejection pair with PFWA are stacked on top of one another. Two alternatives can reach the same score from very different support-rejection pairs: one from strong support and strong rejection, the other from both being weak.
Thus instead of writing:
"This alternative is definitively the best because its PF-SAW score is the highest"
the report should read:
"This score is a difference (μ²−ν²) that comes from the Pythagorean weighted average of the support-rejection pairs; full compensation still applies, and which alternative's support-rejection pair is more uncertain should be shown separately"
When to Prefer This over the Base Method
Use this extension where experts give a judgement both strong support and a marked reservation. If the two degrees sum to more than 1, the intuitionistic fuzzy constraint forces these pairs to shrink; Pythagorean fuzzy solves this problem. Where the sum does not already exceed 1, intuitionistic fuzzy is sufficient, and moving to Pythagorean is unnecessary. Where the sum of squares also exceeds 1, Pythagorean falls short, and Fermatean or q-Rung orthopair is required. SAW's 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 cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1.
Writing ν as 1 − μ. In that case the sum always comes to exactly 1, the hesitancy margin is zeroed, and the extra region the Pythagorean structure offers goes unused.
Using, in Pythagorean form, a pair that already fits the intuitionistic constraint. If none of the expert pairs sums to more than 1, moving to Pythagorean adds no information, and only discrimination is lost.
Changing the score function and expecting the same result. s = μ² − ν² is the canonical choice, but not the only one; a different score definition can give a different ranking.
The governing principle is this:
PF-SAW exists to carry the expert's support and rejection degrees without clipping them. Writing ν as 1−μ, and using here a pair that already fits the intuitionistic constraint, waste the method's one contribution: a wide and honest region of acceptance.
Cases
The first case is DecisionMind's validation example; the manifest's synthetic 3×3 table has been built faithfully to the formulas and carries no literature page reference. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Choosing a contract research organisation (CRO)
A pharmaceutical company will choose one of three contract research organisation (CRO) proposals. Three criteria apply: experience and expertise (higher is better), schedule adherence (higher is better), and contract cost (lower is better). The assessment committee has recorded, as a Pythagorean fuzzy pair (μ, ν), the support and reservation it holds for each organisation.
| Organisation | Experience and expertise | Schedule adherence | Contract cost (lower is better) |
|---|---|---|---|
| K1 | (0.70; 0.40) | (0.50; 0.50) | (0.60; 0.50) |
| K2 | (0.80; 0.30) | (0.60; 0.40) | (0.40; 0.60) |
| K3 | (0.60; 0.50) | (0.70; 0.40) | (0.50; 0.50) |
| Weight | 0.40 | 0.35 | 0.25 |
The method swaps μ and ν on the cost criterion, reduces every organisation's support-rejection pairs across the three criteria to a single pair with PFWA, and calculates the score (μ²−ν²).
| Organisation | Score | Rank |
|---|---|---|
| K2 | 0.3645 | 1 |
| K3 | 0.1711 | 2 |
| K1 | 0.1281 | 3 |
The result reads as follows. K2 holds the highest support (0.80) and lowest rejection (0.10) pair on the most heavily weighted criterion, experience and expertise; it also has the advantage on cost (once reversed). K3 holds the strongest pair on schedule adherence, (0.70; 0.40), but this is not enough to offset K2's advantage in experience.
The company's hesitation is this: what happens if experience and expertise's weight is lowered from 0.40 to 0.10 and schedule adherence is given 0.75? Once verified with an independent Python calculation, redistributing the weights to (0.10; 0.75; 0.15) drops K3's score to 0.2683 and K2's to 0.2447; first place passes to K3, because K3's advantage on schedule now determines the heaviest criterion.
In the report: "With the given weights (0.40; 0.35; 0.25), K2 has the highest score (0.3645). Once the weight is markedly shifted to schedule adherence (0.10; 0.75; 0.15), first place passes to K3; the priority given to each criterion in the contract should therefore be separately justified."
Source: DecisionMind's PF-SAW validation example. The score function and the PFWA combination operator rest on Yager's (2014) and Zhang and Xu's (2014) Pythagorean fuzzy set definition. Since the manifest attaches no literature source to this table, it is presented as an illustrative example; the scores and the weight-swap scenario were independently calculated by this card's author by running the kernel directly.
2. Public transport: A municipality's choice of bus-fleet renewal tender
A municipality will choose one of three bus tenders to renew its urban route fleet. Three criteria apply: passenger capacity and comfort (higher is better), fuel/energy efficiency (higher is better), and purchase price (lower is better). The transport department's technical board has recorded, for each tender, its support and reservation as a Pythagorean fuzzy pair; on some tenders both values are high together, because the tender carries both strong technical features and a serious budget risk.
The method complements the purchase price, combines the three criteria's support-rejection pairs with PFWA, and scores the result. Suppose the tender with the highest capacity also came out with low reservation on energy efficiency, and finished first.
The municipality's hesitation is this: the reservation on the purchase price, that is, the budget risk the finance unit considers serious, has been dissolved into the score. Before approving the tender, this risk (for example, exchange-rate sensitivity) needs to be examined separately; looking only at the score conceals it.
In the report: "One tender stands out because of its capacity and energy efficiency; the finance unit's stated reservation about the purchase price has been dissolved into the score and needs to be assessed separately."
3. What Not to Do
Had contract cost been marked "higher is better" in the illustrative example, the most expensive organisation would receive full marks on this criterion, and K2's advantage of being cheap would be reversed. The second mistake is writing K1's schedule-adherence cell, (0.50; 0.50), by taking ν as 1−μ, which regresses it to intuitionistic fuzzy; this means running the calculation without ever using the Pythagorean structure's extra region. The third mistake is reading K2's score of 0.3645 as "thirty-six per cent reliability"; the score only ranks these three organisations relative to one another.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pf-saw
Fishburn, P. C. (1967). Additive utilities with incomplete product sets: Application to priorities and assignments. Operations Research, 15(3), 537–542. DOI: 10.1287/opre.15.3.537
Yager, R. R. (2013). Pythagorean fuzzy subsets. 2013 Joint IFSA World Congress and NAFIPS Annual Meeting, 57–61. DOI: 10.1109/IFSA-NAFIPS.2013.6608375
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