q-Rung orthopair TODIM (Tian, Niu, Zhang, Li and Herrera-Viedma, 2021)

Five things change; the reference-criterion logic continues, but in this extension the reference criterion is always the highest-weighted criterion — the user cannot select another one as reference.

Cells. In crisp TODIM every cell is a single number. Here every cell is a pair: support degree μ and rejection degree ν. The sum of their q-th powers cannot exceed 1. Unlike q-rung TOPSIS, q is not fixed in this extension; it is a numerical value the user can adjust (default 3). This is a departure from the general principle on the q-rung data-type card, which states that q should be set as the smallest sufficient exponent for the data, not raised by hand. Weights are crisp numbers.

Scale equalisation. Crisp TODIM extracts a share by dividing each column by its own total. There is no such division here; because every cell is already a pair within the 0-1 range, it is directly comparable. For a cost criterion the method reverses the pair: the support and rejection degrees are swapped.

Distance. In crisp TODIM the difference is a direct subtraction. Here the distance between two pairs is made up of three parts: the support difference, the rejection difference, and a term of indecision tied to the ratio of the two, called the "herd-mentality" term. This third term can gain considerable weight when the support degrees of two pairs are very close to each other; the paper claims this term always remains bounded, but that claim cannot be verified for every input in DecisionMind's own calculation.

Winner-loser direction. Which alternative "wins" on a criterion is determined by a score function based on the support-minus-rejection difference; if the score is a tie, a second function (support plus rejection) takes over.

θ, q and the distance form. In crisp TODIM the user can only change θ and the reference criterion. In this extension the reference criterion is locked, but θ (the loss-aversion coefficient), q (the rung exponent) and an exponent determining whether the distance is Hamming or Euclidean are all open to the user together. This is the only one of the four extensions considered where θ can be changed from the interface.

DecisionMind fixes the standard q-ROF distance measure (Tian et al., 2021) for this family; θ, q and the distance exponent start at their default values (1, 3, 1) and can be changed.

The global value is read as in crisp TODIM: the lowest overall superiority takes 0, the highest takes 1, and this holds only for this alternative set. The difference is this. Beneath this value there now sits a tension between support and rejection, and this tension enters the distance calculation and descends to a single number only at the final step.

Thus instead of writing:

"Because q-rung TODIM 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 and ranked with the chosen θ and q values; the choice of these two parameters must be justified in the report"

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. It is equally suitable when the intuition that a decision-maker weighs losses more heavily than gains fits the nature of the decision. Which pair exceeds which bound is described by the short decision rule on the q-rung data-type card.

Converting an already-measured criterion into this form is wrong. In DecisionMind the table must be of a single type throughout. TODIM's exit condition applies exactly as it does elsewhere: if no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold. If the loss-aversion assumption does not fit the nature of the decision, a symmetrically compensatory method such as q-rung TOPSIS is simpler.

Raising q without regard to the data. As q grows, the range of accepted pairs widens but discriminating power falls. In the illustrative example below, when q is raised from 3 to 4, the global-value gap between the two closest alternatives narrows markedly; this is not an improvement, it is a loss of information.

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

Changing θ and q at the same time and attributing a single change in the ranking to one of them. Because three parameters (θ, q, the distance exponent) can change together in this extension, which parameter caused a shift in the ranking must be tested separately; changing all of them at once and saying "θ was responsible" builds a false causal claim.

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.

The governing principle is this:

In q-rung orthopair TODIM, the tension between support and rejection is carried through to the last step. q, θ and the distance exponent can be freely changed in this extension, but every change must be separately justified and reported.