Pythagorean fuzzy PROMETHEE (Zhang et al., 2019)

Three things change; the pairwise comparison and flow logic do not.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell consists of a membership (μ, support) and a non-membership (ν, rejection) degree; the sum of the two may exceed 1, the one constraint being that the sum of their squares (μ²+ν²) does not exceed 1. This allows a wider region of judgement than intuitionistic fuzzy (μ+ν≤1): an alternative can be both strongly supported and strongly rejected at once. Criterion weights remain crisp.

Cost complementing. Crisp PROMETHEE reverses the sign of the difference on a cost criterion. Here, every cell on a cost criterion is first complemented: the support and rejection degrees swap places. Cells on benefit criteria are left as they are.

Difference and preference function. In crisp PROMETHEE the difference is a straightforward subtraction. Here, a score (μ²-ν²) is first calculated for every cell; the signed difference between two alternatives is taken in the direction of whichever score is higher, and its magnitude comes from the actual distance in μ-ν space. This signed difference is then passed through a V-shaped preference function; the threshold is not taken from the user as in crisp PROMETHEE, but derived automatically from the data, based on the largest positive difference observed on that criterion.

Result and defuzzification. The output is, as in crisp PROMETHEE, a single net-flow figure directly. Uncertainty, that is, the support-rejection pair, has already been used up in the score and distance calculation before the pairwise comparison; it is not carried forward in layers into the result.

DecisionMind fixes, in this entry, the score function (μ²-ν²), the data-derived V-shaped threshold, and crisp PROMETHEE's own flow mechanics.

The net flow is read as in crisp PROMETHEE: it is assessed together with the incoming and outgoing flow, and is not compared with another method's score. See the PROMETHEE card.

The difference is here: because the preference function's threshold is derived from the data, adding or removing an alternative from the set can change not only the ranking but the threshold itself; this is a different source of sensitivity from crisp PROMETHEE's fixed, user-set threshold. In the illustrative example below, concentrating the weight on one criterion genuinely changes the order; this has been independently tested in Python.

Thus instead of writing:

"According to PF-PROMETHEE, A2 is the best alternative"

the report should read:

"With the stated weights, A2 is first with a net flow of 0.481; this ranking is sensitive to the weight distribution, and which criterion carries the heaviest weight should be shown in the report"

Use this extension where criteria are expressed as support-rejection pairs, and this pair's sum can exceed the limit intuitionistic fuzzy allows (μ+ν≤1). This is exactly Zhang and colleagues' (2019) own field of application: evaluation panels in which experts report both strong support and strong reservation for an alternative at the same time. Crisp PROMETHEE's exit conditions apply here too: if the number of alternatives exceeds fifty, the pairwise-comparison burden grows large; where an honest partial relation is needed rather than a full ranking, PROMETHEE I's reading should be considered. Where criteria are measured, or where the intuitionistic fuzzy μ+ν≤1 constraint is already sufficient, this extension adds no information; the intuitionistic fuzzy pair remains valid here too, but Pythagorean's extra flexibility goes unused.

Violating the domain. The sum μ²+ν² must not exceed 1; this must not be confused with the μ+ν≤1 constraint, or a pair valid for intuitionistic fuzzy will be treated as valid for Pythagorean too, and the constraint is never actually checked.

Assuming the score function (μ²-ν²) is monotonic everywhere. Some Pythagorean score functions may not behave monotonically when hesitancy (π=1-μ²-ν²) is high; the μ²-ν² used in this extension is monotonic because it is a direct difference of two squares, but this assumption should be checked again if comparing against a different Pythagorean score.

Forgetting that the preference-function threshold is data-derived and reporting it as a fixed user threshold. This threshold is recalculated on every run from the largest positive difference observed on that criterion; if the alternative set changes, so does the threshold.

Defuzzifying first and then running crisp PROMETHEE. Reducing the support-rejection pair to a single number from the outset and applying the crisp method is not this extension; the contribution of the μ-ν distance is erased at the first step.

The governing principle is this:

Pythagorean fuzzy PROMETHEE's contribution is to allow support and rejection degrees to be given independently of one another but bounded quadratically; overlook the data-driven nature of the score function and the threshold, and this flexibility produces a false impression of precision.