Pythagorean fuzzy CODAS (Peng, 2021)

Four things change. The two-distance comparison logic does not.

Cells. In crisp CODAS 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. Criterion weights are crisp numbers and come from outside; only the cells become fuzzy, not the weights.

Scale equalisation. Crisp CODAS divides the value by the column's largest on a benefit criterion, and sets the column's smallest as a ratio against the value on a cost criterion. PF-CODAS does not do this, because support-rejection pairs already lie between 0 and 1. In its place comes a single operation: on a cost criterion, μ and ν swap places. This operation does not equalise scale; it reverses direction.

Score and weighting. Every cell is first reduced to a single number. Here DecisionMind follows not the in-text equation of Peng's paper, but the "refined score" operation the paper itself uses in its own tables. This operation is: t = (μ² − ν²)·(2 − μ² − ν²). This is not the same as PF-TOPSIS's simple score (μ² − ν²); it gives a different magnitude even when μ and ν are close to one another. The score is computed first, and the criterion weight is then multiplied directly by this single number (R = w · t). In PF-TOPSIS, by contrast, the vector form (μ², ν², π²) is kept all the way to the distance calculation; here the descent to a single number happens far earlier, before weighting.

The negative-ideal and distance. The negative-ideal is the smallest value in the weighted-score column; a single "worst" reference is used, as in crisp CODAS, and no separate ideal point is built, as in TOPSIS. Euclidean and city-block distance are calculated over these single numbers, in exactly the same way as in crisp CODAS. The pairwise-comparison rule (threshold Θ=0.02, Euclidean first, city-block if the difference does not exceed it) and the assessment score are built in exactly the same form as crisp CODAS.

DecisionMind fixes the cost-criterion swap, the refined-score operation, and the threshold value in classical PF-CODAS. Weights are taken from outside as crisp numbers.

Reading the assessment score is the same as in crisp CODAS: it is a relative measure of position, the negative-ideal is rebuilt whenever the alternative set changes, and it cannot be compared with a score from a different analysis. The difference lies here: beneath the score sit both the μ² − ν² difference itself and, implicitly, π² (the hesitancy margin), because the refined score also folds in this component.

In the literature case below, the order between two alternatives can change depending on a small difference stemming from rounding in the paper's own table; this shows how carefully differences sitting close to the threshold value must be read.

Thus instead of writing:

"PF-CODAS found the best alternative with certainty"

the report should read:

"With these weights, this threshold value and this score operation, this is the alternative that moves furthest from the negative-ideal; where two alternatives' scores lie close together, the ranking is sensitive to the threshold and to rounding"

This extension is used where experts give a judgement both strong support and marked reservation, and the sum of the two degrees can exceed 1. This case is further strengthened where there is a real possibility of rivals too close to be told apart by straight-line distance alone. Without this second condition, Pythagorean fuzzy TOPSIS (pf-topsis) may be sufficient; what CODAS adds is the ability to separate close rivals with a second distance measure.

If the sum does not already exceed 1, intuitionistic fuzzy CODAS is sufficient, and there is no need to move to Pythagorean. If the sum of the 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. The table must be of a single data type. The base method's exit condition holds 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.

Confusing the simple score with the refined score. Peng's paper writes t = μ² − ν² in its text. But the paper's own tables use t = (μ² − ν²)(2 − μ² − ν²). DecisionMind follows the tables. Using the two interchangeably without stating in the report which score was used produces a different assessment score.

Carrying PF-TOPSIS's weighted Hamming distance over into this method. PF-CODAS takes the score first, then weights it, and calculates two distances over that single number. Keeping the μ², ν², π² components separate and building a three-component distance as in PF-TOPSIS produces a different method, not PF-CODAS.

Writing ν as 1 − μ. In that case the hesitancy margin is zeroed; the calculation may look like PF-CODAS, but it actually reverts to intuitionistic fuzzy CODAS.

Leaving the threshold value (Θ) unquestioned. The same mistake found in crisp CODAS applies here too.

The governing principle is this:

PF-CODAS exists to reduce the expert's support and rejection degrees to a single number with the refined score, without clipping them, and then compare alternatives with two different distances. Changing the score, borrowing the distance form from PF-TOPSIS, or writing ν as 1 − μ undoes this contribution.

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