q-Rung Orthopair ARAS (Mishra & Rani, 2023)

Four things change; the utility-degree logic does not.

Cells. In crisp ARAS every cell is a single number. Here every cell is a pair: a support degree (μ) and a rejection degree (ν). The sum of these two degrees raised to the q-th power cannot exceed 1. The user does not choose q; it comes bundled with the method's own definition, and DecisionMind holds it fixed. Weights are still supplied from outside as single numbers; the method does not generate weights.

Scale equalisation. Crisp ARAS inverts the cost criterion and divides every column by its own total. No such division happens here, because the support and rejection degrees already lie between 0 and 1 and need no further step to become comparable across columns. For a cost criterion the method inverts the pair instead: the support and rejection degrees swap places. Every criterion is then read in the benefit direction.

Optimal alternative and aggregation. As in crisp ARAS, a hypothetical optimal alternative is built here too: a row made of the highest support and lowest rejection pair observed on each criterion. A weighted aggregation across criteria is then carried out for both the real alternatives and this optimal row. Unlike crisp ARAS's plain sum, this aggregation uses a q-Rung aggregation rule that processes the support and rejection degrees together, and the result is again a single pair (support, rejection).

Score and defuzzification. The aggregated pair is reduced to a single score between 0 and 1. DecisionMind specifically uses a score rule that stays within 0 and 1, to avoid a sign problem that could otherwise arise from q's negative powers, rather than a form of the score that could come out negative. Every real alternative's score is then divided by the optimal alternative's score. This ratio is the utility degree (K) and, as in crisp ARAS, it is read as a percentage benefit against the optimal alternative, between 0 and 1.

DecisionMind holds q and the score rule fixed for this method. Weights are taken from outside as single numbers.

The utility degree K is read exactly as in crisp ARAS: a percentage ratio against the optimal alternative, meaningful only for this alternative set and these weights, and not comparable with a different analysis.

The difference is here: beneath the score now lies a tension between support and rejection. When a judgement is given both high support and high rejection, that is, when an expert reports strong approval and a strong reservation at the same time, this duality enters the calculation but disappears from view inside K. The K gap between two alternatives should be assessed by looking not only at the weights but also at how sharp the input pairs were.

Thus instead of writing:

"According to q-Rung ARAS, A1 is the best alternative"

the report should read:

"Relative to the optimal alternative derived from this alternative set, A1 holds the highest utility degree; this ratio is meaningful only for this alternative set and these weights, and does not show on which criterion the support-rejection tension was highest"

Use this extension when experts give a judgement's support and rejection degrees separately, and these pairs exceed the intuitionistic and Pythagorean fuzzy boundary. It also fits when crisp ARAS's "percentage against the optimum" output is wanted under this kind of uncertainty. How q is chosen, and the boundary that must not be confused with the neighbouring types (intuitionistic, Pythagorean, Fermatean), are covered on the data-type card.

The exit condition is the same as for crisp ARAS. The matrix must be of a single type throughout, and if no compromise is acceptable on one criterion, this extension remains compensatory too.

Choosing q large without looking at the data. As q grows, the accepted pairs widen but discrimination falls; q should be the smallest exponent covering the whole of the expert pairs, chosen once for the entire matrix, not cell by cell.

Computing ν as 1 − μ. In that case the sum is always exactly 1, and there is no need for the q-Rung structure at all; the rejection degree must come from its own separate body of evidence.

Deriving the optimal alternative incorrectly. If the optimal row is built for a cost criterion before the pair is inverted, the best pair is picked in the wrong direction, and every K value is distorted in the same way.

Reading K as a probability. K is only a proportional measure of benefit relative to this alternative set's optimal alternative; it is not a "percentage chance of being the right choice."

The governing principle is this:

The utility degree depends on the optimal alternative derived from the alternative set, on the weights supplied, and on the support-rejection pairs that q covers; change any one of these and K changes too, and the report must show this.