q-Rung orthopair MOORA

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

Scale equalisation. Crisp MOORA divides the cell value directly by the column magnitude. QR-MOORA first reduces each pair to a single score. The score is support cubed minus rejection cubed (μ³ − ν³, because q = 3); since this value can fall between −1 and 1, 1 is added to shift it. The shifted score is then normalised exactly as in crisp MOORA, by dividing by the root of its own column magnitude. For a cost criterion the pair is not reversed; the direction difference is instead carried by sign in the subsequent summing-subtracting step.

Score and ratio system. The normalised shifted scores are weighted; the sum over "more is better" criteria has the sum over "less is better" criteria subtracted from it. This is exactly the same operation as crisp MOORA's ratio system; the only difference is that the input has already been reduced to a μ³ − ν³ score beforehand.

Result and defuzzification. The score-extraction step is itself the defuzzification, and it happens at MOORA's first step, not at the end. DecisionMind also genuinely computes the reference-point approach for this family. A reference point is built from each criterion's best normalised score, and each alternative's largest weighted deviation from that point (Chebyshev distance) is produced as a separate score. The final ranking and the reported score, however, rest solely on the ratio system; the reference-point score stands as an auxiliary, optional cross-check.

DecisionMind fixes q = 3, the score function and vector normalisation for classical QR-MOORA. Weights are taken from outside as crisp numbers; the method does not generate weights.

The output is a net score and a rank, as in crisp MOORA; it is read the same way. The difference is this. This score rests on the cube difference between a judgement's support and rejection degrees, and the tension between the two stays hidden beneath the score. A criterion on which the expert has given both high support and high reservation appears in the report only as a single net difference.

Thus instead of writing:

"Because q-rung MOORA 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 net score has turned out sensitive to the weight of these criteria"

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; in DecisionMind, q is in any case not user-adjustable, it comes bundled with the method itself.

Converting an already-measured criterion into this form is wrong. In DecisionMind the table must be of a single type throughout. The exit condition is the same as for crisp MOORA: if no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold.

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

Changing the score function while expecting the same result. DecisionMind fixes the μ³ − ν³ score. A different score can give a different order; which score was used must be stated in the report.

Mistaking the reference-point score for the final ranking. In this family the reported ranking rests on the ratio system; the reference-point score is a separate, optional cross-check and is not, on its own, the final result.

Defuzzifying first and then running crisp MOORA. Disregarding support and rejection degrees and calculating with only an averaged number is not this extension; the tension is erased at the very first step.

The governing principle is this:

q-Rung orthopair MOORA exists to carry the tension between support and rejection through to the ratio system. Any application that fails to check the constraint, or changes the score function without declaring it, erases this extension's own contribution.