Pythagorean fuzzy MARCOS

Three things change; the decision logic does not.

Cells. In crisp MARCOS 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 extended table and scale equalisation. Crisp MARCOS extends the table with two hypothetical rows built from every criterion's observed best and worst value. PF-MARCOS applies the same logic to pairs: for every criterion, the best pair is built from the highest μ and the lowest ν, and the worst pair from the lowest μ and the highest ν. On a cost criterion these two roles swap. Every cell is then ratioed against its own column's best pair; this ratioing corresponds to crisp MARCOS's division operation, except that μ and ν are now ratioed separately.

Combination and row totals. Crisp MARCOS sums the weighted normalised values horizontally. PF-MARCOS performs this summation with the Pythagorean weighted-sum rule; the result is again a single support-rejection pair, not a number. Every alternative's combined pair yields two separate ratios against the ideal and anti-ideal rows' combined pairs. These ratios are turned into a single number by the score function, s = μ² − ν², and then passed through crisp MARCOS's own utility functions.

Result and defuzzification. The output is a single final utility degree, produced by combining the two utility functions; it sits on the same scale as crisp MARCOS. Uncertainty is carried through as the row totals are built by the Pythagorean summation rule; it collapses to a single number only once the score function is applied. This means uncertainty is carried as a pair right up to the last step, and is defuzzified only at the final step.

DecisionMind fixes, for PF-MARCOS, the Pythagorean ratioing rule, the Pythagorean weighted-sum rule, and the score-based utility functions. Criterion weights are taken from outside as crisp numbers; the method does not generate weights.

The final utility degree states where an alternative stands relative to the ideal and anti-ideal references in this set; this is the same as in crisp MARCOS. What differs is that these references are now support-rejection pairs. Where the gap in utility degree between two alternatives is large, this gap generally holds regardless of how much the input pairs' μ and ν shift. Where the gap is small, a small change in μ or ν can easily reverse the order, and this must be shown in the report.

Thus instead of writing:

"PF-MARCOS found the best supplier"

the report should read:

"With these weights and this alternative set, this is the alternative with the most balanced utility ratio relative to the ideal and anti-ideal support-rejection references; how far this advantage holds up against small changes in the input pairs should be tested separately"

Use this extension where experts give a judgement both strong support and a marked reservation, and you want both how much of the ideal has been reached and how far the anti-ideal has been left behind to appear together. If the two degrees sum to more than 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 hold a single data type throughout; 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. Defuzzifying the ideal and anti-ideal ratios with a different score can give a different utility degree.

Reversing the ideal and anti-ideal pair's role on a cost criterion. On a benefit criterion, the ideal pair is built from the highest support and the lowest rejection degree. On a cost criterion this is reversed. Skipping this reversal turns the ranking on its head.

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:

The PF-MARCOS result is a summary of the alternatives' proportional standing relative to this set's own ideal and anti-ideal support-rejection references. These references shift together with the alternative set, and the report must show this.