q-Rung orthopair SAW

Three things change. The decision logic stays the same.

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

Scale equalisation. In crisp SAW every column is scaled against its own best value; this is to bring criteria measured in different units (currency, days, points) onto the same scale. q-rung orthopair cells are already defined between 0 and 1, and this scale is already fixed by q itself; no separate "scale against the best" step is needed. Instead, the method swaps the support and rejection degree for "less is better" criteria: â_ij = (ν_ij, μ_ij). Every criterion is thereby turned to face the benefit direction, and the next step proceeds in a single direction.

Weighted aggregation. Crisp SAW's second step is a direct sum: every column is multiplied by its weight and summed along the row. The same direct sum cannot be applied here, because the two degrees together must still satisfy the q-rung constraint, and averaging μ and ν separately in linear fashion can break that constraint. Instead the method uses a dedicated weighted-aggregation rule called q-ROFWA: μ̃_i = (1 − ∏_j(1 − μ_ij^q)^{w_j})^{1/q}, ν̃_i = ∏_j ν_ij^{w_j}. This is a product- and power-based aggregation, the counterpart in q-rung space of SAW's "direct and proportional sum" principle; the result is again a valid q-rung orthopair pair.

Result and defuzzification. In crisp SAW the result is already a single summed score. Here the aggregated pair (μ̃_i, ν̃_i) is obtained first, then reduced to a single number by a score function: s_i = μ̃_i^q − ν̃_i^q. In case of a tie, the accuracy degree a_i = μ̃_i^q + ν̃_i^q breaks it. As in SAW, a larger score is better and the ranking runs by descending score; this direction does not change. DecisionMind fixes q = 3 and this score function for this family.

The score is read as in crisp SAW: it is not a percentage, it is not compared with another analysis, a larger score is better. The difference lies here. Beneath the score there now sits a tension between support and rejection, and the single number does not show how sharp that tension is.

Thus instead of writing:

"Because q-rung SAW models uncertainty, the result is more accurate"

the report should read:

"The criteria on which the expert gave both strong support and strong reservation have been flagged separately; the score has turned out sensitive to the weight of these criteria"

Carrying the uncertainty through does not make the score more certain, it only keeps it from staying hidden.

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. A measured criterion is entered as a fixed pair, and that pair carries no uncertainty. SAW's exit condition holds here too: if no compromise is acceptable on one criterion, the SAW family is not the right choice, because full compensation continues here as well.

Entering data without checking the constraint. In every cell, the sum of the third powers of the support and rejection degrees cannot exceed 1. This check is made before the calculation.

Confusing q-ROFWA with a simple weighted average. Multiplying μ and ν separately by w_j and summing looks like crisp SAW's own form. But this can break the q-rung constraint and departs from the method's definition in the literature; aggregation 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.

Defuzzifying first and then running crisp SAW. In the illustrative example below, this path shows the gap between candidates at a different size than its real value; once the rejection degree is set aside, scores rise but the real tension disappears.

The governing principle is this:

q-Rung orthopair SAW exists to carry the tension between support and rejection through to the last step. Any application that disregards the rejection degree or fails to check the constraint erases this extension's own contribution.