q-Rung orthopair WPM

Three things change. The multiplicative logic does not.

Cells. In crisp WPM every cell is a single, strictly positive number. Here every cell is a pair: support degree μ, rejection degree ν. The sum of the q-th powers of the two degrees cannot exceed 1. The user does not choose q; DecisionMind runs this family at q = 3, this value is fixed, and it is checked at entry. Weights come from outside as crisp numbers.

Scale equalisation. In crisp WPM every column is scaled against its own best value. q-rung cells are already defined between 0 and 1 and on a scale fixed by q itself. No separate scaling step is needed. Instead, the method swaps the support and rejection degree for "less is better" criteria: (μ, ν) becomes (ν, μ). This reversal turns that criterion to face the benefit direction, and the next step proceeds in a single direction.

Weighted product. Crisp WPM's second step raises every column to a power equal to its weight and multiplies. Here this is done with a weighted geometric operator called q-ROFWG: the combined support μ̃ = Πμ_j^{w_j} is a direct weighted product, exactly like crisp WPM itself. The combined rejection, however, is computed as a complementary product over the q-th powers: ν̃ = (1 − Π(1−ν_j^q)^{w_j})^{1/q}. WPM's "weighted product" principle is preserved exactly on the μ side; on the ν side it takes a form that does not break the q-rung constraint.

Result and defuzzification. Once the combined pair (μ̃, ν̃) is obtained, it is reduced to a score by a single score function: S = (1 + μ̃^q − ν̃^q) / 2. Unlike PF-WPM's score of s = μ²−ν², this formula carries a shift. The difference μ̃^q − ν̃^q can range between −1 and 1. Adding 1 and dividing by 2 places the score within 0 and 1 at all times. This shift does not change the ranking, because it is an increasing linear transformation of the difference. It only turns the score, which could otherwise be negative, into a number readable within the 0-1 range. DecisionMind fixes q = 3 and this score function for this family.

The score is read as in crisp WPM: it only ranks this alternative set, it is not a percentage or a probability, it is not compared with another analysis. The difference lies here. Even though the score sits between 0 and 1, this does not mean "how many per cent good"; it is only the μ̃^q − ν̃^q difference carried into a readable range.

Thus instead of writing:

"Because the QR-WPM score falls between 0 and 1, it can be read directly as a percentage"

the report should read:

"The score is the difference between support and rejection shifted into the 0-1 range; a value such as 0.60 does not mean 60 per cent suitability, it only marks a ranking position among these alternatives"

Use this extension when an expert gives a judgement both strong support and a strong reservation at once, and this pair exceeds both the intuitionistic and the Pythagorean bound. Which pair exceeds which bound is shown by the short decision rule on the q-Rung data-type card. If the pair already fits within the intuitionistic or Pythagorean bound there is no need to raise q; a large q reduces discriminating power, it does not raise accuracy.

Converting an already-measured criterion into this form is wrong. In DecisionMind the table must be of a single type throughout. Crisp WPM's severity continues here. Because μ̃ is a direct product, if μ is close to zero on one criterion, that criterion pulls the combined value down sharply even when its weight is small.

Entering data without checking the constraint. In every cell, the sum of the third powers of the support and rejection degrees cannot exceed 1; a cell that fails this is rejected on loading and never enters the calculation.

Confusing q-ROFWG with a simple weighted average. Multiplying μ and ν separately by w_j and summing resembles SAW rather than crisp WPM itself. This can break the q-rung constraint and departs from the method's definition; combination is done only with the product-power formula.

Reversing the cost direction twice. The method swaps the support and rejection degree once for a cost criterion. Repeating this swap after aggregation reverses the direction entirely.

Confusing the shifted score (S) with the raw difference (μ̃^q−ν̃^q) and assigning it a separate meaning. The two give the same ranking, because one is a linear transformation of the other. The shift exists only for readability; a score above 0 does not mean the alternative is "good," nor below 0.5 "bad" — it only shows the order within this alternative set.

The governing principle is this:

In q-rung orthopair WPM, μ̃ carries crisp WPM's weighted product exactly; ν̃ is a complementary product fitted to the q-rung constraint. The final score is this difference shifted into the 0-1 range, not a different computing principle.