Rough VIKOR (Zhu, Hu, Qi, Gu & Peng, 2015)

Four things change; the decision logic does not.

Cells. In crisp VIKOR every cell is a single number. Here every cell is a rough number: a lower bound and an upper bound, calculated from the group's crisp scores. Criterion weights are also given as rough intervals; there is no requirement here that they sum to 1, because each weight is itself an uncertainty interval.

Scale equalisation. Crisp VIKOR finds each criterion's best and worst value and expresses the distance to the ideal as a ratio over this range. Rough VIKOR follows the same logic, but the best and worst value are now built from the edges of the rough bounds, that is, from the observed crisp best and worst extreme in the column; the bounds are taken without being defuzzified first.

Group utility and individual regret. In crisp VIKOR, S (group utility) and R (individual regret) are each a single number. In rough VIKOR both are rough intervals: they are calculated separately by their lower and upper bound, because the ratio operation between bounds does not simply pair lower bound with lower bound and upper bound with upper bound; which end of a bound affects which end requires its own rule.

Compromise index and defuzzification. In crisp VIKOR, Q is a single number. In rough VIKOR, Q is first built as a rough interval; the rough intervals of S and R are combined with the crisp best/worst edges of these intervals and with coefficient v. Only after this stage does Q come down to a single number, by averaging its lower and upper bound (at v = 0.5). DecisionMind fixes this averaging rule at the final step; a different defuzzification rule can give a different Q.

Crisp VIKOR's two conditions, acceptable advantage and acceptable stability, work exactly the same way in rough VIKOR and are tested on the defuzzified Q. DecisionMind keeps coefficient v at a default of 0.5; the user can change it.

Q is read the same way as in base VIKOR: a smaller value is better, and the real result is not the Q ranking but the decision the two conditions produce. Writing up the alternative with the smallest Q as the "winner" is wrong here too.

The difference is here: beneath Q now lies the group's own disagreement. When two alternatives' defuzzified Q are close, that closeness can depend both on the weights and on how wide the bounds are. An alternative with wide bounds carries a Q that is less reliable than the same-sized Q gap for an alternative with narrow bounds.

Thus instead of writing:

"According to rough VIKOR, the best alternative is A4"

the report should read:

"With these weights and v = 0.5, A4's defuzzified Q is the smallest (0.103); but the gap to A6 (0.200) falls just short of the acceptable-advantage threshold (0.200), so a compromise set of A4 and A6 should be reported rather than a single winner"

Use it when several experts or sources score the same criterion and the disagreement within the group matters for the decision in its own right. With a single expert or a single measurement, a rough number cannot be constructed; base VIKOR is used instead. If the bounds are given directly as "at least, at most" without being calculated, that is a grey number, not a rough number; Grey VIKOR is the better fit there. The base method's exit condition applies exactly as before: if no compromise at all is acceptable on one criterion, no member of the VIKOR family can provide that.

Violating the bound constraint. In every cell, the lower bound cannot exceed the upper bound; if it is violated, the choice of best/worst value, and hence S and R, become meaningless.

Changing the defuzzification rule and expecting the same result. Averaging the lower and upper bound of the Q interval is the canonical choice; taking only the lower bound (pessimistic) or only the upper bound (optimistic) can give a different ranking. Which rule was used must be stated in the report.

Skipping the two conditions and ranking only the defuzzified Q. This is the rough form of the most common mistake in base VIKOR, and it is easier to miss here, because reducing Q from a rough interval to a single number is already a step in itself; it must not be forgotten that the conditions still need to be tested separately on this single number.

Entering bounds by hand rather than from group scores. Inventing bounds around a single expert's score is not a rough number; it violates the basic principle on the data-type card.

The governing principle is this:

Rough VIKOR's result, just as in base VIKOR, is not a ranking but a conditional compromise proposal; the method cannot be considered properly applied unless the rough intervals of S and R, the defuzzification rule and the outcome of the two conditions are all reported together.