q-Rung Orthopair MARCOS

Four things change. The decision logic stays the same.

Cells. In crisp MARCOS 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. As in crisp MARCOS, the method extends the table with ideal and anti-ideal rows; the difference lies in how these two rows are chosen. On every criterion the ideal row is taken from the real alternative carrying the best pair, according to a score function.

Scale equalisation. In crisp MARCOS every cell is normalised by dividing by the ideal value. No such division happens here. For a cost criterion the method inverts the pair instead: the support and rejection degrees swap places. Every pair is then reduced to a single number, using a score built from the cube of the support degree minus the cube of the rejection degree. This reduction takes the place of the ratio.

Utility ratios. The reduced and weighted numbers are summed along each row; this is done both for the real alternatives and for the ideal and anti-ideal rows. Every alternative's total, divided by the ideal row's total, gives the K+ ratio; divided by the anti-ideal row's total, it gives the K− ratio. In crisp MARCOS these ratios came directly from the measured numbers. Here the ratios are computed from a single number derived from the difference between support and rejection.

Score and defuzzification. The final utility degree keeps the same definition: K+ and K− passed through utility functions and combined. The largest value is the best alternative. 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 has not found a single published numerical example for this method. This extension is a derived construction that combines Yager's 2017 q-ROF definition with classical MARCOS's seven-step skeleton. The illustrative example below therefore does not come from a paper's page, but from recomputing this construction by hand.

The final utility degree means the same thing here too. The largest value shows the best alternative. This value carries both how much of the ideal the alternative has reached and how far it has moved from the anti-ideal, together. The difference is here. Beneath this degree 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 a middling performance. The report must make this distinction.

Thus instead of writing:

"According to q-Rung MARCOS, the best alternative is A2"

the report should read:

"A2's final utility degree is the highest; on the criteria where A2 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 MARCOS exit condition applies here too. If a compensatory logic that speaks in the language of ratios does not fit the decision-maker's expectation, another method should be chosen in place of the MARCOS family.

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

Changing the score function. DecisionMind uses a definition that keeps the score between 0 and 1. A different score definition, for instance one taking values between minus one and one, corrupts the denominator of the K+ and K− ratios, and the result can come out undefined.

Forgetting the reference rows. The method does not work only among the real alternatives; it works relative to the ideal and anti-ideal rows. K+ and K− cannot be computed without these two rows being built.

Ignoring the weight swap. In the illustrative example below, when the weights swap places, the ranking genuinely changes. Presenting the result with a single weight scenario, without seeing that the ranking is sensitive to weight in this table, is misleading.

The governing principle is this:

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