Pythagorean fuzzy WPM

Three things change; the multiplicative logic does not.

Cells. In classical WPM every cell is a single, strictly positive number. Here every cell is a pair: μ, the support degree, and ν, the rejection degree. These two values must satisfy μ² + ν² ≤ 1; a cell that fails this is rejected the moment it is loaded and never enters the calculation. Criterion weights are still taken from outside as crisp numbers; it is only the cells that turn fuzzy, not the weights.

Scale equalisation. Classical WPM scales every column against its own best value. There is no such step here, because support-rejection pairs already lie between 0 and 1. Instead, μ and ν swap places on a cost criterion: (μ, ν) becomes (ν, μ). Because weight remains a crisp number here, there is no need to invert the weight separately as Fuzzy WPM does; only the cell is reversed.

Weighted product and defuzzification. In classical WPM, every column is raised to the power of its weight and multiplied along the row. Here this is done by the Pythagorean weighted geometric aggregation operator (PFWG). The μ component is computed as Πμ_j^{w_j}, a direct weighted product. The ν component is computed as (1 − Π(1−ν_j²)^{w_j})^{1/2}, a complementary product taken over the squares. This aggregation reduces the (μ_j, ν_j) pair on every criterion to a single (μ̃, ν̃) pair; it is not yet a single number. Only at this last step does the combined pair descend to a single number, through the Pythagorean score function (s = μ̃² − ν̃²).

Defuzzification, that is, the score calculation, happens AFTER the row has been aggregated with PFWG, and only once; not cell by cell, but once per alternative. The score can come out negative, meaning ν̃ can exceed μ̃. The kernel applies no correction in this case: it neither clips it to zero, nor takes its absolute value, nor shifts the score. A negative score is a valid result and simply pushes the alternative down the ranking.

DecisionMind holds the PFWG operator and the score function (μ² − ν²) fixed in this extension. Weights are taken from outside; the method does not generate weights.

As in classical WPM, the score only shows a ranking within this particular set of alternatives. It is not a percentage or a probability, and it cannot be compared with a score from a different PF-WPM analysis.

The difference is this. The score is a single difference derived from the combined μ̃ and ν̃, and it carries beneath it a margin of hesitancy (π̃ = √(1−μ̃²−ν̃²)). Two alternatives can have close scores while one carries a narrow π̃ and the other a wide one; the narrower one is the more reliable advantage. A negative score shows that the alternative remained rejection-heavy across that set of criteria; it is not an error.

Thus instead of writing:

"A2's score is positive and A1's negative, so A2 is definitely better"

the report should read:

"A2's score is positive and A1's is negative; this shows that A2's combined support degree exceeds its rejection degree, while A1's does not, and both readings are relative positions within these three alternatives"

Use this extension when experts give a judgement both strong support and marked reservation at once, and the sum of these two degrees exceeds 1. If the sum never exceeds 1, IF-WPM (intuitionistic fuzzy) is enough, and there is no need to move to Pythagorean. If the sum of squares also exceeds 1, Pythagorean falls short and a Fermatean or q-rung orthopair WPM is required. Which constraint fits which situation is shown in the short decision rule on the Pythagorean data-type card.

Classical WPM's severity persists here. Because PFWG's μ component is a direct product, a criterion where μ_j sits near zero drags the combined μ̃ down sharply, even if that criterion's weight is small. Opening up a measured criterion into a μ, ν pair is wrong; in DecisionMind, a matrix must be of a single type throughout.

Violating the value domain. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1; a cell that fails this is rejected on load.

Writing ν as 1 − μ. This makes the sum always exactly 1, zeroes out the margin of hesitancy, and leaves the Pythagorean structure's extra space entirely unused.

Mistaking a negative score for an error. s = μ̃² − ν̃² can come out negative; the kernel does not correct it, because there is nothing to correct. This is a point where PF-WPM behaves differently from SF-WPM: there, a score defuzzified per cell that comes out negative feeds into the next multiplicative step as a base and must be clipped. In PF-WPM, defuzzification happens only once, at the very end, so no such clipping is ever needed.

Forgetting to reverse the cell on a cost criterion. If (μ, ν) is not swapped on a "lower is better" criterion, the worst alternative on that criterion looks like the best.

The governing principle is this:

In PF-WPM, the score is calculated only once, after the row has been aggregated with PFWG, and a negative outcome is a valid result. It should not be confused with extensions that defuzzify per cell and must therefore guard against a negative base.