q-Rung Orthopair COPRAS

Four things change. The decision logic stays the same.

Cells. In crisp COPRAS 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' third powers cannot exceed 1, because DecisionMind fixes q at 3 for this method. The user does not choose q. Weights are supplied from outside as single numbers.

Scale equalisation. In crisp COPRAS every column is scaled by dividing by its own total. Here every pair is first reduced to a single number: the cube of the support degree minus the cube of the rejection degree. This number is shifted to the positive side by adding 1, and only then divided by the column total. The shift prevents the division from breaking down when the reduced number comes out negative.

Aggregation. Crisp COPRAS sums the benefit criteria, sums the cost criteria separately, and combines the two totals with a formula. The same split is made here too; only now the summed values are the reduced and shifted numbers. If there is no benefit criterion, or no cost criterion, the formula simplifies automatically.

Score and defuzzification. The benefit degree Q keeps the same definition. The largest Q is the best alternative, and it can be converted into a percentage benefit degree. The method does not discard the uncertainty at the outset; the uncertainty is reduced to a single number at the reduction step, and every step after that runs on this number.

DecisionMind holds the reduction fixed with Liu and Wang's 2018 score definition, and the aggregation fixed with Zavadskas and Kaklauskas's classical 1996 formula. The literature also contains another score definition that additionally accounts for the hesitancy margin, and that definition can give a different Q.

The benefit degree Q means the same thing here too. The best alternative is taken as 100, and the others receive a percentage relative to it. The difference is here: although Q appears as a single number, beneath it now lies a tension between support and rejection, and this tension is reduced to a single number, and lost from view, at the reduction step.

If an alternative's support and rejection degrees are both high on a criterion, the reduced number comes out as a middling value. This means there is uncertainty on that criterion, not average performance. The report must make this distinction.

Thus instead of writing:

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

the report should read:

"A1's benefit degree is 100; on the criteria where A1 is strong, the gap between support and rejection is also clear, and does not come from an uncertain middling value"

Reduction makes the calculation easier; it does not remove the uncertainty.

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. Which pair exceeds which boundary is shown by the short decision rule on the q-Rung data-type card. If the pair already fits the intuitionistic or Pythagorean boundary, there is no need to raise q.

Converting a measured criterion into this form is wrong. In DecisionMind the table must be of a single type throughout. If a measured criterion exists, it is entered as a fixed pair, and this pair carries no uncertainty. The base COPRAS exit condition applies here too: if a linear, compensatory aggregation logic does not fit the decision-maker's expectation, another method should be chosen in place of the COPRAS family.

Entering data without checking the constraint. In every cell, the sum of the support and rejection degrees' third powers cannot exceed 1.

Inverting the cost direction at the reduction step too. DecisionMind handles the cost criterion after reduction, only by keeping the benefit and cost totals separate. Also swapping the support and rejection degrees at the reduction step inverts the direction twice and corrupts the total.

Forcing the formula when there is no cost criterion. As in the illustrative example below, if every criterion is in the benefit direction, the cost total is zero and the benefit degree is set directly equal to the benefit total. Skipping this shortcut and applying the formula as it stands produces a division-by-zero error.

Defuzzifying first and ignoring the rejection degree. This path shows A2's benefit degree, in the illustrative example below, as different from its true value.

The governing principle is this:

q-Rung orthopair COPRAS exists to combine the difference between support and rejection honestly into a single benefit degree. Any application that changes this degree arbitrarily, or skips the rejection degree, erases this contribution.