q-Rung orthopair TOPSIS (Pinar and Boran, 2020)

Four things change. The decision logic stays the same.

Cells. In crisp TOPSIS 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; the method runs at q = 3 and this value is fixed in DecisionMind. Weights come from outside as a single number; the method does not generate weights.

Scale equalisation. In crisp TOPSIS columns are divided by the root of the sum of their squares. There is no such division here. For a cost criterion the method reverses the pair: the support and rejection degrees are swapped. Every criterion is thereby read in the benefit direction, and the remaining steps proceed in a single direction.

Distance. After weighting, the best pair and the worst pair are determined for every criterion. Every alternative's distance to these two pairs is measured taking all three components — support, rejection and hesitancy — into account together. In crisp TOPSIS this was a straight-line distance between single numbers. Here the distance has three components, and a hesitancy share also enters it.

Result and defuzzification. The closeness score carries the same definition: the ratio of the distance to the anti-ideal over the sum of the two distances. The result is again a single number between 0 and 1. The method does not discard uncertainty at the outset. Uncertainty is carried through to the distance calculation and descends to a single number there.

DecisionMind fixes the standard q-ROF Euclidean distance for this method. A different distance measure has also been proposed in the literature, and that measure can give a different magnitude, even a different order. Weights are taken from outside as a single number.

The closeness score is read as in crisp TOPSIS: it is not a percentage, it is not compared with another analysis, the ideal point shifts and the order can change once the alternative set changes. The difference lies here. Beneath the score there now also sits a tension between support and rejection, and the score does not say whether it reflects this tension.

The gap between two scores depends on how sharp the input pairs are. If an expert gave the same judgement both high support and high reservation, this duality enters the calculation but becomes invisible in the report. The report should therefore show not only the score but also on which criterion the gap between support and rejection is small.

Thus instead of writing:

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

Carrying the uncertainty through does not make the result 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. If some criteria are measured and some are judgement-based, all of them are written into the same data type. A measured criterion is entered as a fixed pair, and that pair carries no uncertainty. The exit condition of the base TOPSIS holds here too: if no compromise is acceptable on one criterion, the TOPSIS family is not the right choice.

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, not after.

Changing the distance measure while expecting the same result. DecisionMind uses the standard q-ROF Euclidean distance. An alternative distance formula in the literature gives a different score, and in some tables a different order. Which distance was used must be stated in the report.

Reversing the cost direction twice. The method swaps the support and rejection degree for a cost criterion. Repeating this swap once more at the ideal-point selection reverses the direction entirely and turns the ranking on its head.

Defuzzifying first and then running crisp TOPSIS. In the illustrative example below, this path shows A1 and A2 as closer to each other than they really are; the real gap shrinks or grows, but in every case the resulting number is artificial.

The governing principle is this:

q-Rung orthopair TOPSIS 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.