Pythagorean Fuzzy

A Pythagorean fuzzy data structure asks the same question as an intuitionistic fuzzy one: how far do you support a judgement (membership, μ), and how far do you reject it (non-membership, ν)? The difference lies in how large the two degrees may be together. In the intuitionistic fuzzy structure, μ + ν can be at most 1; in the Pythagorean structure this sum can exceed 1, provided μ² + ν² does not exceed 1. For the judgement "this supplier is reliable," the pair μ = 0.8, ν = 0.5 is invalid in the intuitionistic structure (the sum is 1.3) but valid in the Pythagorean one (the sum of squares is 0.89).

The hesitation margin is again not taken directly from the expert; it is derived from the two degrees, but the derivation rule works through squares. The (μ, ν) pairs the structure accepts include everything the intuitionistic structure accepts, plus the region where both degrees are high.

Use it where experts report both strong support and marked reservation for the same judgement, and the intuitionistic constraint would force them to shrink their values. In complex projects, risky investments, or candidates with strong upsides and strong downsides together, it is natural for an expert to say "I largely agree, but I have a serious reservation too"; such a statement produces a pair whose sum exceeds 1.

By contrast, where the expert pairs already sum to no more than 1, moving to the Pythagorean structure is not necessary; the intuitionistic fuzzy structure carries the same information and needs fewer rules.

Yes, but two steps are required.

First, turn the criterion into a judgement. A Pythagorean fuzzy pair carries the support and rejection degrees of a proposition, not of a quantity. "Gold is 2,450 Turkish lira" is a measurement; "Gold is a suitable investment for the next six months" is a judgement.

Second, derive μ and ν from separate sources: μ from the share of favourable indicators or experts, ν from the share of unfavourable ones. If the same indicator can signal in both directions, the sum of the two degrees exceeds 1, and this is exactly the situation the Pythagorean structure is meant for.

What must not be done is computing ν as 1 − μ. In that case the sum is always exactly 1, the Pythagorean structure's extra region is never used, and the structure collapses into classical data.

The Pythagorean fuzzy structure accepts more pairs than the intuitionistic one; that does not make it "better." As the accepted region widens, two different judgements become harder to tell apart, and the differences between experts' values carry less weight in the ranking. A wider domain is a gain only where expert pairs genuinely fall within it.

For that reason:

"Pythagorean fuzzy was chosen because it is more general"

is better replaced by:

"Some expert pairs summed to more than 1, so the intuitionistic constraint was insufficient, and the Pythagorean constraint was adopted"

The chief advantage of the Pythagorean fuzzy structure is that it records the expert's actual judgement without clipping it. Under the intuitionistic constraint, an expert who gives (0.8, 0.5) must shrink the pair to something like (0.6, 0.4), which distorts the judgement; the Pythagorean constraint accepts this pair as it stands.

Furthermore, all of the intuitionistic fuzzy structure's tools (scores, distances, aggregation rules) have been adapted to this structure, so moving from the intuitionistic structure requires no great change in the calculation method.

Widening the accepted region reduces discrimination: the difference between two pairs close to one another shows up less clearly in the ranking than it would under the intuitionistic structure.

Because the score and distance functions are defined through squares, the same expert values can give different scores and different rankings under the intuitionistic and Pythagorean structures. Which structure was chosen must be justified in the report. Experts also find the notion of "a pair whose sum exceeds 1" difficult to apply intuitively, so the structure calls for training.

If the sum of the cubes of the expert pairs also exceeds 1 (for example μ = 0.9, ν = 0.7, where the sum of squares is 1.30), the Pythagorean structure falls short too. In that case, the sum of cubes is bounded instead (Fermatean fuzzy), or the exponent in the bound is left free (q-Rung orthopair fuzzy). Intuitionistic (exponent 1), Pythagorean (exponent 2) and Fermatean (exponent 3) are three special cases of the same family; the detail is in the q-Rung card.

The most frequent mistake is choosing the Pythagorean structure for pairs that already fit the intuitionistic one. If none of the expert pairs sums to more than 1, the wider domain is not used, and only discrimination is lost.

The second frequent mistake is writing ν as 1 − μ, which means the Pythagorean structure's extra region never comes into play. Likewise, deriving both degrees from a single source, using a measured quantity directly as μ, and widening the accepted domain "just in case" all lead to methodological problems.

The governing principle is this:

A move to the Pythagorean fuzzy structure is justified by showing that expert pairs genuinely exceed the intuitionistic constraint; the width of the domain alone is not a justification.