Pythagorean fuzzy EDAS

Three things change. The decision logic stays the same.

Cells. In crisp EDAS 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. The hesitancy margin, π = √(1 − μ² − ν²), is derived from these two degrees; it is not asked of the expert separately. Criterion weights remain crisp numbers and come from outside.

The average solution and the score. Crisp EDAS takes the numerical average of every column. PF-EDAS first builds the weighted Pythagorean matrix, then computes an average pair for every column. This average is not a single number but a support-rejection pair, found with the Pythagorean weighted geometric mean rule. Both the cells and this average pair are then converted to a single number with the score function, s = μ² − ν². Cells that are already numbers in crisp EDAS are here carried as pairs first, and only turn into scores at the moment of comparison.

Deviation calculation and combination. Crisp EDAS accumulates, in two separate totals, whether every cell stays above or below the average, according to benefit or cost direction. PF-EDAS runs the same logic over the scores: every cell's score is compared with that column's average score, the difference is measured with normalised Euclidean distance, and counted as favourable or unfavourable depending on direction. In the final step, the favourable and unfavourable totals are each divided by their own largest value and combined into a single closure ratio; this step is the same as in crisp EDAS.

Result and defuzzification. The output is a single closure ratio and a rank; it sits on the same 0-1 scale as crisp EDAS's assessment score. Uncertainty collapses to a single number the moment the cells are converted to a score. This differs from PF-TOPSIS, which carries the distance in three components, μ², ν² and π²; PF-EDAS scores first, then applies the remaining steps of crisp EDAS exactly as they stand.

DecisionMind fixes the Pythagorean weighted geometric mean, the score function, and the normalised Euclidean deviation in PF-EDAS. Criterion weights come from outside as crisp numbers; the method does not generate weights.

The closure ratio states how favourably an alternative is positioned relative to the set's own average; this is the same as in crisp EDAS. What differs is that this average is now a Pythagorean average. If, on one criterion, all alternatives' support-rejection pairs sit close to one another, that criterion's scores also come out close together, and the criterion's discriminating power falls; this matches exactly the same phenomenon in crisp EDAS. What conceals the difference is where μ and ν come from: two experts can express the same "middling" judgement with different pairs, and this can shift the average without it being noticed.

Thus instead of writing:

"This alternative was chosen because PF-EDAS found it the most advantageous"

the report should read:

"With these weights and this alternative set, this is the alternative positioned most advantageously relative to the set's average; the report should show on what scale μ and ν were given, and that the average changes together with the set"

This extension is used when experts give a judgement both strong support and marked reservation, and a decision based on position relative to the set's average suits your purpose. If the sum of the two degrees exceeds 1, the intuitionistic fuzzy constraint forces these pairs to shrink, because that constraint keeps the sum at no more than 1. Pythagorean fuzzy solves this problem. A measured value is not turned directly into a support-rejection pair. It is first converted into a judgement, and only then are μ and ν derived from separate sources; the detail is on the data-type card. The table must be of a single data type; part crisp and part Pythagorean is not allowed. The base method'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.

Domain violation. Every cell must satisfy μ ∈ [0,1], ν ∈ [0,1] and μ² + ν² ≤ 1. Entering the calculation without this check invalidates the method.

Writing ν as 1 − μ. In that case the sum always comes to exactly 1. The hesitancy margin is zeroed. The extra region the Pythagorean structure offers over the intuitionistic one is never used.

Changing the score function and expecting the same result. s = μ² − ν² is the canonical choice, but not the only one. Computing the average solution and the deviations with a different score can give a different ranking.

Treating a criterion as unimportant once all alternatives' scores come out close to the average on it. This mistake exists in crisp EDAS too. It is easier to overlook with Pythagorean input, because the same score can come from several different support-rejection pairs.

Turning a measured value directly into μ. Scaling a measured quantity such as price or time to 0-1 and writing it as μ, with ν given as its complement, does not produce a judgement. It only conceals crisp data.

The governing principle is this:

A PF-EDAS result is a summary of the alternatives' position relative to this set's own Pythagorean average. Every alternative added to or removed from the set shifts this average, and the report must show this.