Pythagorean fuzzy CoCoSo

Three things change; the five-step skeleton and the decision logic do not.

Cells. In crisp CoCoSo every cell is a single number. Here every cell is a pair: a support degree μ and a rejection degree ν. These two values must satisfy μ² + ν² ≤ 1; their sum (μ+ν) may exceed 1, but the sum of their squares may not. Criterion weights come from outside as crisp numbers; the method does not generate weights.

Score and scale equalisation. Every cell is first reduced to a single number with the Pythagorean fuzzy family's canonical score: s = μ² − ν². On a cost criterion the pair is inverted before this reduction: (μ, ν) is rewritten as (ν, μ). The score is shifted by 1, then pulled onto 0–1 with direction-sensitive min-max scaling, column by column. This does the same job as crisp CoCoSo's column scaling; the difference is that what gets scaled is not the raw criterion value but the score function's output.

Additive and multiplicative measures; the balance coefficient. Over the scaled scores, the weighted-sum-like (P) and weighted-power-sum-like (R) measures are computed, and the three compromise strategies then follow the same formulas as crisp CoCoSo. On one point this extension departs from N-COCOSO and P-COCOSO: the balance coefficient λ in the third compromise strategy is FIXED inside the engine at 0.5 and cannot be changed by the user; in N-COCOSO this coefficient is a parameter supplied from outside.

DecisionMind fixes the score function, the cost-criterion inversion, and λ=0.5 in this extension. Weights come from outside.

As in crisp CoCoSo, the K score is only a ranking within this particular alternative set. Whether the three compromise strategies agree with one another shows robustness.

The difference lies here: μ and ν carry information separately, but once both enter the score function they are reduced to a single difference (μ²−ν²) and are lost from that point on. The same score can be reached both by strong support paired with strong rejection and by weak support paired with weak rejection; the K score does not distinguish between the two.

Thus instead of writing:

"Because Pythagorean fuzzy CoCoSo models the support-rejection balance more finely, the result is more reliable"

the report should read:

"μ and ν carry information only until the score is computed; once the score is computed, it is crisp CoCoSo itself that carries the ranking from there on, so the result's robustness must be tested separately — whether the three strategies agree, and whether the ranking breaks down when a single cell changes"

This extension is used when experts give a judgement both strong support and marked reservation, and the sum of the two exceeds the intuitionistic fuzzy constraint (sum ≤ 1). If the sum does not already exceed 1, intuitionistic fuzzy CoCoSo (if-cocoso) is sufficient; moving to Pythagorean adds no information. If the sum of the squares also exceeds 1, Pythagorean falls short, and q-Rung orthopair CoCoSo (qr-cocoso) is required.

Turning a measured criterion directly into a μ-ν pair is producing uncertainty, not modelling it. The crisp method's exit condition holds here too: the matrix must be of a single type, and if no compromise is acceptable on one criterion, this extension is compensatory as well.

Writing ν as 1 − μ. In that case the sum always comes to exactly 1, and the calculation effectively reverts to intuitionistic fuzzy CoCoSo; the extra region of acceptance the Pythagorean structure provides is never used.

Entering data without checking the domain. Every cell must satisfy μ² + ν² ≤ 1. Entering the calculation without this check invalidates the method.

Assuming λ can be adjusted. In N-COCOSO and P-COCOSO the balance coefficient can be supplied from outside; in PF-COCOSO it is fixed inside the engine at 0.5. Writing in the report that "λ was set to such-and-such a value" is wrong for this extension.

Values that spike when a column's spread narrows. If an alternative's score draws very close to the other values in its column, the min-max denominator shrinks, and the K scores can jump to unrealistic magnitudes; this is also a known weakness of crisp CoCoSo.

The governing principle is this:

In PF-COCOSO, μ and ν carry separate information only until the score is computed; once the score is computed, it is crisp CoCoSo itself that carries the ranking from there on. The balance coefficient is fixed here and cannot be adjusted by the user.