q-Rung orthopair VIKOR (Erdebilli et al., 2023)

Four things change. The decision logic stays the same.

Cells. In crisp VIKOR every cell is a single number. Here every cell is a pair: support degree μ and rejection degree ν. The sum of the third powers of these two degrees cannot exceed 1, because DecisionMind fixes q at 3 in this method. The user does not choose q. Weights and the compromise coefficient v come from outside as single numbers.

Scale equalisation. In crisp VIKOR each criterion's best and worst value is read directly off the column. Here the support and rejection degree are first swapped for a cost criterion, so that every criterion is read in the benefit direction. Then each criterion's ideal pair and anti-ideal pair are chosen, not by comparing single numbers one by one, but by a score function. This score declares a real alternative's pair the best or the worst on that criterion.

Distance. Every alternative's normalised gap on every criterion is measured against that criterion's ideal pair with a three-component distance. This distance is then divided by the range between the ideal and the anti-ideal. In crisp VIKOR this gap was the ratio of a single difference. Here the gap is a distance ratio into which the support, rejection and hesitancy components enter together.

Result and defuzzification. Group utility S, individual regret R and the compromise index Q carry the same definition. A smaller Q is better. The method does not discard uncertainty at the outset; uncertainty is carried through to the gap calculation and descends to a single number when S, R and Q are computed. Both acceptance conditions work exactly as in classical VIKOR.

DecisionMind fixes, for this method, the ideal and anti-ideal pairs as the pair of a real alternative selected by the score function. In the literature it is also possible to take the highest support and lowest rejection degree separately, criterion by criterion. The two approaches give the same result if one alternative dominates on every criterion; if none dominates, they can give a different result.

The triad of Q, S and R is read as in crisp VIKOR. A smaller Q is better, and the gap between the single compromise solution and the compromise set is determined by two conditions. The difference lies here. Beneath the three of them there now sits a tension between support and rejection, and this tension descends to a single number and disappears when S and R are calculated.

If an expert gave both high support and high reservation on a criterion, that alternative's normalised gap on that criterion takes a middling value. The report does not show where this middling value comes from. The report should therefore note not only Q, S and R but also on which criterion the gap between support and rejection is small.

Thus instead of writing:

"According to q-rung VIKOR, the compromise solution is A2"

the report should read:

"With these weights and v = 0.5, A2 is the single compromise solution; on the criteria where A2 is strong, the gap between support and rejection is also large"

Carrying the uncertainty through does not make Q more accurate, 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.

Converting an already-measured criterion into this form is wrong. In DecisionMind the table must be of a single type throughout. If a measured criterion is present, it too is entered as a fixed pair, and that pair carries no uncertainty. The exit condition of the base VIKOR holds here too: if no compromise is acceptable on one criterion, a method working on elimination logic should be preferred over the VIKOR family.

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

Reading Q's direction in reverse. A smaller Q is always better. Some q-ROF VIKOR papers include the phrase "rank from largest to smallest," but this phrase contradicts the paper's own formula. DecisionMind follows classical VIKOR's rule that a smaller Q is better; reversing this rule turns the whole ranking on its head.

Confusing the ideal-pair selection. DecisionMind takes the ideal pair as the pair of a real alternative selected by a score function. Instead, separately collecting the highest support and lowest rejection degree on each criterion to build an artificial pair can give a different result. If none of the alternatives dominates, this difference shows up in the S and R values.

Skipping the two conditions and writing down only the alternative with the smallest Q. VIKOR's defining feature is the compromise set. In the illustrative example below both conditions happen to hold, but this will not be true of every table.

The governing principle is this:

q-Rung orthopair VIKOR exists to carry the tension between support and rejection through to the last step. Any application that disregards the rejection degree, confuses the direction of Q, or skips the two conditions erases this extension's own contribution.