Extension card · Pythagorean
Pythagorean fuzzy MABAC
This is the form of MABAC for situations where criterion scores are given as support and rejection degrees for a judgement, and the sum of these two degrees may exceed 1. It builds the border approximation area on a score derived from these two degrees, and ranks the result, once again, with a single figure.
Base method
MABAC →
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 border approximation area's logic does not.
Cells. In crisp MABAC every cell is a single number. Here every cell consists of two degrees: support (μ) and rejection (ν). The one constraint is that the sum of the squares of the two degrees does not exceed 1. Hesitancy is not taken from the expert here; it is derived from μ and ν. Weights come from outside as crisp numbers in DecisionMind; group-decision combination does not operate in this family, and the user enters a single, already-combined matrix.
Scale equalisation. Crisp MABAC places every column between 0 and 1 relative to its own smallest and largest value. In Pythagorean fuzzy there is no separate min-max step. On cost criteria, μ and ν simply swap places: support becomes rejection, and rejection becomes support. Because the cells already sit on the same 0-1 scale, the columns are comparable from the outset.
Weighting and the border approximation area. Every cell is scaled by the criterion's weight through Pythagorean fuzzy algebra's own exponential rule: support grows, and rejection shrinks, according to this rule. Every cell's score (μ² − ν²) is then taken, and the border approximation area is built, in place of crisp MABAC's geometric mean, as the arithmetic mean of these scores. The border, again, represents neither the best nor the worst; it shows the alternative set's typical score on that criterion.
Distance and total score. Every alternative's score is subtracted from the border's score; this is a plain subtraction, as in crisp MABAC, but it works on values that have already been reduced to a single score, rather than on a two-degree cell. These differences are summed across criteria, and alternatives are ranked from highest to lowest total score.
DecisionMind fixes, in this classical form, the Pythagorean fuzzy scaling and the μ²−ν² score function. Weights are taken from outside, as crisp numbers.
How to Read the Output
The total score is read as in crisp MABAC: a positive score means the alternative sits above the border, a negative score means below it, and this holds only for this particular alternative set. The difference is here: beneath this score lies a two-degree support-rejection pair, and the score carries only this pair's score, not the pair itself. Two alternatives can reach the same score through different support-rejection pairs; one through high support and moderate rejection, another through moderate support and low rejection, arriving at the same difference from the border.
Thus instead of writing:
"According to Pythagorean fuzzy MABAC, this alternative is the most reliable"
the report should read:
"With the given weights, this alternative sits highest above the border approximation area; this position rests on the score that support and rejection degrees jointly produce, not on high support alone"
When to Prefer This over the Base Method
Use this extension where expert pairs' support and rejection degrees sum to more than 1, but the sum of their squares does not. This situation arises where an expert says "I largely agree, but I also have a serious reservation," reporting both strong support and marked rejection. Where the sum does not already exceed 1, intuitionistic fuzzy MABAC is sufficient; moving to Pythagorean fuzzy is not necessary.
Crisp MABAC's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also fully compensatory and will not eliminate anything below a threshold. A measured criterion (price, time) is not opened directly into a support-rejection pair; the Pythagorean fuzzy structure is for the degree of a judgement, not for a quantity itself.
Mistakes Specific to This Extension
Constraint violation. A pair whose μ² + ν² sum exceeds 1 is invalid. This differs from the intuitionistic fuzzy structure's μ + ν ≤ 1 constraint; the two must not be confused.
Calculating hesitancy afterwards and adding it to μ or ν. In this family, hesitancy is a derived margin, not a component mixed into either of the two degrees. μ and ν are taken directly from the expert.
Calculating ν as 1 − μ. In that case the sum always comes to exactly 1, the extra region the Pythagorean constraint offers is never used, and the structure is effectively reduced to crisp data.
The "more advanced" fallacy. Pythagorean fuzzy MABAC does not produce a "more correct" ranking than intuitionistic fuzzy MABAC; it is only the form needed when expert pairs do not fit the intuitionistic constraint.
The governing principle is this:
In Pythagorean fuzzy MABAC, support and rejection degrees are bounded by the sum of their squares; the border approximation area and the distance are both built on a single score derived from these two degrees, and the score alone does not show the support-rejection pair itself.
Cases
The first case is DecisionMind's validation example. It is not carried over, page-numbered, from a published paper; it is a synthetic 3×3 Pythagorean fuzzy table, built so that the formulas can be followed by hand, and it is stamped as such in the manifest. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's validation example): Comparing three software licence quotations
A corporate IT department is comparing three software licence quotations. There are three criteria: fitness for purpose and ease of use (both "higher is better"), and annual fee ("lower is better"). The assessment team has reported, for every quotation, how much support and how much reservation it holds on these criteria, as a Pythagorean fuzzy pair.
| Quotation | Fitness for purpose | Ease of use | Annual fee |
|---|---|---|---|
| Licence A | (0.70; 0.40) | (0.50; 0.50) | (0.60; 0.50) |
| Licence B | (0.80; 0.30) | (0.60; 0.40) | (0.40; 0.60) |
| Licence C | (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 scales every cell by the criterion's weight through the Pythagorean fuzzy rule, takes the μ²−ν² score, builds the border approximation area for each criterion from the mean of these scores, and sums every quotation's difference from this border.
| Quotation | Total score | Rank |
|---|---|---|
| Licence B | 0.3081 | 1 |
| Licence C | -0.1020 | 2 |
| Licence A | -0.2061 | 3 |
The result reads as follows. Licence B has the highest support and the lowest rejection pair, (0.80; 0.30), on the most heavily weighted criterion, fitness for purpose; its higher cost on the annual fee cannot offset this advantage. Licence A finishes last, being best on no criterion.
The team has one hesitation: what would happen if the weight were shifted almost entirely to ease of use, at 0.05/0.90/0.05? Once DecisionMind's engine is independently rerun, Licence C moves to first place at 0.1093, Licence B drops to second at 0.0882, and Licence A remains last. The order between the two leading quotations depends on the relative weight of fitness for purpose against ease of use.
In the report: "With the given weights (0.40/0.35/0.25), Licence B is in the strongest position relative to the border approximation area (0.3081). Once the weight is shifted almost entirely to ease of use, Licence C moves ahead; the preference between these two quotations should be treated as sensitive to the weight distribution."
Source: DecisionMind's PF-MABAC validation example. No single, page-traceable founding application paper exists for Pythagorean fuzzy MABAC; the studies found under this name in the literature (extended, for instance, with the Choquet integral or with rough numbers) carry a different algorithm from the plain five-step MABAC skeleton applied here by DecisionMind. For this reason, no author-year is used in the card's title; the figures were calculated independently by this card's author, running DecisionMind's engine, by applying the Pythagorean fuzzy operators (Yager, 2014) directly to the MABAC steps (Pamučar and Ćirović, 2015).
2. Fishing: A cooperative's choice of cold-chain haulage firm
A fishing cooperative will choose among three haulage firms to transport its catch through the cold chain. There are three criteria: reliability of temperature control, consistency of delivery time (both "higher is better") and haulage fee ("lower is better"). Drawing on their experience over past seasons, cooperative members have reported how much they trust each firm and how much reservation they hold, as a Pythagorean fuzzy pair.
The method scales the three firms, builds the border approximation area for each criterion, and sums every firm's difference from this border. Suppose the firm with the highest support pair on temperature control also charges the highest fee; because temperature control carries a higher weight than fee, it still comes out first.
The cooperative's hesitation is this: if the catch volume rises mid-season, the fee difference grows within the total cost, and the weights may need to be reviewed. The cooperative should tell its members plainly which weight balance the current ranking rests on.
In the report: "With the current weights, the firm with the highest temperature-control reliability is in the strongest position relative to the border approximation area. Its fee is higher than the others'; reviewing the weights is recommended once catch volume rises."
3. What Not to Do
Changing Licence A's ease-of-use pair, (0.50; 0.50), to a value such as (0.80; 0.70) without checking the constraint: the sum of squares comes to 0.64+0.49=1.13, exceeding the constraint. The second mistake is marking the annual-fee criterion as "higher is better"; on this criterion less is better, and the swap must be made in the correct direction before normalisation. The third mistake is reading Licence B's score of 0.3081 as "a hundred per cent suitable quotation"; this score shows only its relative position among these three quotations.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/pf-mabac
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
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