q-Rung Orthopair CoCoSo (Kuvvetli, 2023)

Four things change; the five-step skeleton and the decision logic do not.

Cells. In crisp CoCoSo every cell is a single number. Here every cell is a pair: support μ, rejection ν. The sum of these two degrees raised to the q-th power cannot exceed 1. The user does not choose q; DecisionMind works with q=3, just as in QR-TOPSIS. Criterion weights are supplied from outside as crisp numbers.

Score and scale equalisation. Every cell is reduced to a single number with Peng and Huang's (2020) hesitancy-penalised score function: r = μ^q − ν^q − ln(1 + π^q), where π = (1 − μ^q − ν^q)^(1/q) is the hesitancy margin. This score differs from Liu and Wang's (2018) simpler s = μ^q − ν^q score by additionally penalising hesitancy. Crisp CoCoSo scales columns to 0-1 against the raw value; here that scaling is applied on top of the score, through direction-sensitive (benefit/cost) min-max. The cost direction is handled directly at the normalisation step here, rather than by complementing the cell as other family members do.

Degeneracy. A weighted sum (S) and a weighted power-sum (P) are computed over the scaled scores. If an alternative is the worst in the column on at least one criterion, its scaled score on that criterion drops to exactly 0. This can push that alternative's S or P down to exactly zero as well. If S or P is zero, the second compromise strategy, k_b = S/min(S) + P/min(P), stays undefined, because the denominator is zero. In this case DecisionMind does not compute k_b (the published value is left blank) and builds the final K score from the sum of k_a and k_c alone; the normal formula, (k_a·k_b·k_c)^(1/3) + (k_a+k_b+k_c)/3, is not used here. This also occurs in the manifest's own validation example (see the illustrative example below).

Score instead of distance. DecisionMind holds the standard Peng-Huang (2020) score and q=3 fixed. Weights are taken from outside as single numbers.

As in crisp CoCoSo, the K score is only a ranking within this alternative set. Whether the three compromise strategies agree shows robustness; but when k_b stays undefined (see the "Degeneracy" paragraph above), only two strategies (k_a, k_c) remain, and the robustness comparison must be made on that basis.

The difference is here: beneath the score lies both the support-rejection tension and the hesitancy penalty. Two cells with the same μ−ν difference receive different scores if their hesitancy margin π differs; the Peng-Huang penalty additionally punishes a large π.

Thus instead of writing:

"Because q-Rung CoCoSo models the uncertainty, the result is automatically more accurate"

the report should read:

"The score has reduced the support-rejection difference to a single number together with the hesitancy penalty; if an alternative's S or P is zero, the third compromise strategy (k_b) cannot be computed and K is built from only two strategies, and this must be stated in the report"

Use this extension when an expert gives a judgement both strong support and a strong reservation, and this pair exceeds both the intuitionistic and the Pythagorean boundary. If the pair already fits one of these boundaries, there is no need to raise q; a larger q reduces discrimination. Converting a measured criterion into this form manufactures uncertainty that is not there; in DecisionMind the table must be of a single type throughout. The crisp method's exit condition applies here too.

Failing to notice that S or P is exactly zero. In the illustrative example below, the weakest alternative (A3) is the worst in the column on every criterion, so its scaled score is exactly 0 on all three criteria; this in turn pushes its weighted sum (S) down to exactly zero. In this case k_b cannot be computed. DecisionMind does not silently swallow this; it leaves k_b blank and builds K from k_a+k_c. Stating in the report that "all three compromise strategies were computed" is wrong in this case.

Ignoring the hesitancy penalty. The Peng-Huang score (r = μ^q−ν^q−ln(1+π^q)) is a fixed rule. Confusing it with Liu and Wang's (2018) simpler s = μ^q−ν^q score can give a different ranking; which score was used must be stated in the report.

Assuming q can be changed. DecisionMind works with q=3, and the user does not choose it. As q changes, the magnitude of the scores changes too (in the illustrative example, A2's K drops from 1.069 at q=2 to 0.882 at q=5), but this is a parameter DecisionMind holds fixed.

Entering data without checking the constraint. In every cell, the sum of μ^q + ν^q cannot exceed 1; this check is performed before the calculation.

The governing principle is this:

In q-Rung orthopair CoCoSo, an alternative being the exact worst in the column on one criterion can push its additive or multiplicative measure down to exactly zero; in that case the third compromise strategy cannot be computed, and K is built from only two strategies.