Pythagorean fuzzy MOORA

Cells. In crisp MOORA every cell is a single number. Here every cell is a pair: μ, the support degree, and ν, the rejection degree; it must satisfy μ² + ν² ≤ 1. Criterion weights come from outside as crisp numbers; uncertainty is carried only in the decision matrix's cells.

Scale equalisation. Crisp MOORA divides the cell value directly by the column's magnitude. PF-MOORA first reduces every pair to a single score. The score is support squared minus rejection squared (μ² − ν²); since this value can range from −1 to 1, 1 is added to shift it. The shifted score is normalised, as in crisp MOORA, by dividing it by the root of its own column's magnitude. On a cost criterion the pair is not reversed; the directional difference is instead carried by sign in the following addition-subtraction step, exactly as in crisp MOORA.

Score and the ratio system. The normalised, shifted scores are weighted; the sum over the "lower is better" criteria is subtracted from the sum over the "higher is better" criteria. This is exactly the same operation as crisp MOORA's ratio system; the only difference is that the input has already been reduced to a μ² − ν² score.

Result and defuzzification. The score-subtraction step is itself the defuzzification, and it happens at MOORA's first step, not at the end. The hesitancy margin, π = √(1 − μ² − ν²), never enters this score calculation at all. DecisionMind also lists the reference point approach as a step in this family, but this step simply copies the ratio system's score; it does not produce an independent calculation. The only approach that actually runs is the ratio system.

DecisionMind fixes, for classical PF-MOORA, μ² − ν² as the score function, and this score's calculation at the first step. Weights are taken from outside as crisp numbers; the method does not generate weights.

The output is a net score and a rank, as in crisp MOORA, and it is read the same way. The difference is this: this score in fact rests on the difference between the squares of a judgement's support and rejection degrees, and never sees the hesitancy margin, that is, the portion neither supported nor rejected. Two assessments with the same μ² − ν² difference but different hesitancy margins give the same result in PF-MOORA.

Thus instead of writing:

"PF-MOORA's result is more reliable because it accounts for uncertainty"

the report should read:

"PF-MOORA accounted only for the squared difference between support and rejection; criteria with a high hesitancy margin should be flagged separately in the report"

Use this extension where experts give a judgement both strong support and a marked reservation, and these pairs' sum exceeds the intuitionistic fuzzy constraint. Where the sum does not already exceed 1, intuitionistic fuzzy MOORA (if-moora) is sufficient; moving to Pythagorean adds no information, and only discrimination is lost. Where the sum of squares also exceeds 1, Pythagorean falls short, and q-Rung orthopair is required.

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 table must hold a single data type throughout. The exit condition is the same as for crisp MOORA: 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, and the hesitancy margin is zeroed. The calculation may look like PF-MOORA, but it actually reverts to intuitionistic fuzzy MOORA.

Changing the score function and expecting the same result. μ² − ν² is the canonical choice, but not the only one. A different score can give a different ranking; which score was used must be stated in the report.

Mistaking the reference point approach for a genuine second method. In this extension, the reference point step copies the ratio system's score; it does not perform an independent calculation.

The governing principle is this:

PF-MOORA exists to carry the expert's support and rejection degrees, without clipping them, into the ratio system. Writing ν as 1 − μ, or changing the score function without stating it, undoes this contribution.