Pythagorean fuzzy TOPSIS (Zhang and Xu, 2014)

Three things change; the decision logic does not.

Cells. In crisp TOPSIS 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. Only the cells become fuzzy, not the weights.

Scale equalisation. Crisp TOPSIS divides every column by the root of the sum of its squares; this is called vector normalisation. PF-TOPSIS 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. A high rejection degree on a "lower is better" criterion then reads, after the swap, like a low support degree. This operation does not equalise scale; it reverses direction.

Distance, score and combination. First, every column's score is calculated: s = μ² − ν². The ideal and anti-ideal are set from this score; the ideal is the pair with the highest score, the anti-ideal the pair with the lowest. Every alternative's distance to these two references is then measured. Crisp TOPSIS uses Euclidean distance; here a weighted Hamming distance is used instead. This distance is the weighted sum of the differences between the μ², ν² and π² components. The final step does not use the classical "distance to the anti-ideal divided by the sum of the two distances" ratio. In its place comes a measure called revised closeness, ζ. ζ is found by subtracting the ratio of an alternative's distance to the ideal, against the smallest such value in the set, from the ratio of its distance to the anti-ideal, against the largest such value in the set. ζ is always zero or negative. The alternative for which ζ comes out at zero is the one that is simultaneously furthest from the anti-ideal and closest to the ideal. The classical ratio does not guarantee that these two properties meet in the same alternative; revised closeness does.

Result and defuzzification. The output is a single ζ value and a rank. Crisp TOPSIS's score lies between 0 and 1; here ζ is zero or negative. The best alternative is the one closest to zero. Uncertainty does not collapse to a single number at any step; the Hamming distance carries all three components, μ², ν² and π², together throughout.

DecisionMind fixes, for PF-TOPSIS, the score-based selection of the ideal and anti-ideal. It also fixes the weighted Hamming distance and the revised closeness ζ. Criterion weights are taken from outside as crisp numbers; the method does not generate weights.

The ranking logic is the same as in crisp TOPSIS: the alternative that is close to the ideal and far from the anti-ideal comes out ahead. What differs is the scale. A ζ of zero does not mean flawless. It means that, within this set, the alternative deviates least from the ideal and moves furthest from the anti-ideal. The other alternatives' negative ζ values likewise show only their relative position within this set. These values cannot be compared with the ζ values of a different analysis, because the ideal and anti-ideal are built, in every analysis, from that analysis's own data.

Thus instead of writing:

"A ζ of zero shows that this alternative is flawless"

the report should read:

"A ζ of zero means that, within this set, the alternative is both closest to the ideal and furthest from the anti-ideal; this advantage depends on which criterion received how much support and rejection, and can shift once the weights change"

Use this extension where experts give a judgement both strong support and a marked reservation. 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. Where the sum does not already exceed 1, intuitionistic fuzzy is sufficient, and moving to Pythagorean is unnecessary. Where the sum of squares also exceeds 1, Pythagorean falls short, and q-Rung orthopair is required. 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. The calculation may look like PF-TOPSIS, but it actually reverts to intuitionistic fuzzy TOPSIS.

Using, in Pythagorean form, a pair that already fits the intuitionistic constraint. If none of the expert pairs sums to more than 1, moving to Pythagorean adds no information. Only discrimination is lost.

Changing the score function and expecting the same result. s = μ² − ν² is the canonical choice, but not the only one. Setting the ideal and anti-ideal with a different score can give a different ranking. Which score was used must be stated in the report.

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:

PF-TOPSIS exists to carry the expert's support and rejection degrees without clipping them. Three mistakes undo this: writing ν as 1 − μ, using here a pair that already fits the intuitionistic constraint, and turning measured data directly into a degree. All three practices waste the method's one contribution: a wide and honest region of acceptance.