q-Rung Orthopair Fuzzy

A q-Rung orthopair fuzzy data structure asks the same question as the intuitionistic and Pythagorean fuzzy structures: how far do you support a judgement (membership, μ), and how far do you reject it (non-membership, ν)? The difference lies in the constraint governing how large the two degrees may be together. In the intuitionistic structure, μ + ν can be at most 1; in the Pythagorean structure, μ² + ν². In the q-Rung structure the exponent is not fixed: μ^q + ν^q must be at most 1, and q is chosen so as to cover the whole of the expert pairs.

The intuitionistic fuzzy structure is the special case q = 1, the Pythagorean fuzzy structure q = 2, and the Fermatean fuzzy structure q = 3. As q grows, the accepted (μ, ν) pairs widen: at q = 1, (0.8, 0.5) is invalid; at q = 2, it becomes valid; (0.9, 0.7) is invalid at q = 2 but valid at q = 4. The hesitation margin, again, is not taken from the expert; it is derived from the two degrees and from q.

Use it where experts report both very strong support and a very strong reservation for the same judgement, and these pairs exceed even the Pythagorean constraint. It is useful in problems where "both very good and very problematic" judgements arise naturally: high-risk, high-return investments; treatments with a strong benefit and a serious side effect; policies that produce both a large benefit and a strong backlash.

By contrast, where the expert pairs already fit the intuitionistic or Pythagorean constraint, there is no need to raise q; a larger q only reduces discrimination.

Yes, but two steps are required.

First, turn the criterion into a judgement. The 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, and only then fix q. Once the pairs across all cells have been gathered, the smallest q that covers all of them is chosen. q is not fixed in advance and the pairs forced to fit it; the pairs come first, and q follows from them.

In DM3, q is not entered by the user; it comes bundled with the method's own definition (for QR-TOPSIS, for instance, q = 3; the interface shows this as "q comes from the method"), and on upload the cells' pairs are checked for validity under that q. The order therefore runs in reverse: first the smallest q covering the expert pairs is established, then the method that works with that q is chosen; the system asks for your data type and recommends a matching method.

What must not be done is computing ν as 1 − μ; in that case the sum is always exactly 1, q = 1 is already enough, and raising q serves no purpose.

As q grows, the structure accepts more pairs; that does not make it more accurate or more powerful. As the accepted region widens, different judgements move closer together: at q = 10, almost every pair is valid, and the difference between (0.9, 0.9) and (0.9, 0.5) barely shows up in the ranking. Using a large q not to record what the expert actually said, but to smooth the ranking, causes a loss of information.

For that reason:

"q = 5 was chosen because a larger q is a more flexible model"

is better replaced by:

"The most extreme expert pair was (0.9, 0.7), so q = 4, the smallest exponent covering this pair, was chosen"

The chief advantage of the q-Rung structure is that it fits the constraint to the data: rather than choosing among the intuitionistic, Pythagorean and Fermatean structures, the required exponent is determined by looking at the expert pairs. None of the expert's judgement is clipped.

Furthermore, because the three special cases sit under one roof, studies that used different constraints for different problems can be compared within the same framework.

Choosing q is itself a decision, and it affects the result: the same pairs can give different scores and different rankings under q = 2 and q = 4. How q was chosen must be stated explicitly in the report.

As the accepted region widens, discrimination falls; at a large q, experts' genuinely different judgements can be reduced to similar scores. The score, distance and aggregation rules also depend on q; different studies use different rules, and results may not be directly comparable.

If the most extreme expert pair exceeds the Pythagorean constraint but its cubes still sum to less than 1 (for example (0.9, 0.6): sum of squares 1.17, sum of cubes 0.95), the Fermatean fuzzy structure (q = 3) is enough. Fermatean is a fixed-exponent member of the q-Rung family, not a separate concept. If the expert pairs also exceed the cube constraint, a q-Rung structure with q = 4 or higher is used.

The most frequent mistake is choosing a large q without looking at the data, "for flexibility." This erases the differences between experts.

The second frequent mistake is using the q-Rung structure for pairs that already fit the intuitionistic or Pythagorean constraint; choosing q = 3 when q = 1 or 2 would do only reduces discrimination. Likewise, writing ν = 1 − μ, deriving both degrees from a single source, choosing a different q cell by cell (a single q is required for the whole matrix), and failing to state q in the report all lead to methodological problems.

The governing principle is this:

q is chosen as the smallest exponent covering the whole of the expert pairs: neither smaller (the pairs get clipped) nor larger (discrimination is lost).