Rough TOPSIS (Song, Ming & Wu, 2013)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is a rough number: a lower bound and an upper bound. These bounds are not entered from outside; they are calculated from the group's crisp scores, and how that calculation works is explained on the data-type card. Criterion weights are also given as rough intervals. There is no requirement here that the weights sum to 1, because each weight is itself an uncertainty interval and enters the calculation in that form.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares. Song, Ming and Wu's (2013) method does not work this way. The ideal and anti-ideal point are first built from the best and worst crisp extreme values actually observed in the column; this is the table's own edge, not a hypothetical point. Every alternative's distance to these two edges is then measured separately by its lower and upper bound, and brought into the 0-to-1 range by dividing by the column's widest upper bound. This is a linear scale equalisation, different from crisp TOPSIS's sum-of-squares approach.

Distance. Crisp TOPSIS uses Euclidean distance: it takes the square root of the sum of squared differences. Rough TOPSIS uses a linear difference and carries it in rough form; the distance to the ideal and the distance to the anti-ideal are each an interval, not a single number. The weighted sum is built by adding the lower bounds together and the upper bounds together, separately.

Result and defuzzification. In crisp TOPSIS the closeness score is directly a single number. In rough TOPSIS a risk parameter (alpha) comes into play first. Defuzzifying the distance to the ideal weights the upper side of the interval more heavily, while defuzzifying the distance to the anti-ideal weights the lower side more heavily; the two formulas are not identical. Alpha defaults to 0.5, representing a middling risk stance; above 0.5 reads as optimistic, below it as pessimistic. Defuzzification happens only at this final step.

DecisionMind fixes, for rough TOPSIS, the linear distance, normalisation by dividing by the upper bound, and the alpha = 0.5 default. Weights are taken from outside as rough intervals; the method does not generate weights.

The closeness score is read as in base TOPSIS: it shows how close the alternative sits to the ideal relative to the others in this set, and it cannot be compared with a different analysis.

The difference is here: beneath the score now lies the group's own disagreement, and the choice of alpha determines how that disagreement is read. When two alternatives' scores are close, the gap should be assessed not only against the weights but also against whether alpha was set optimistic or pessimistic. An alternative with wide bounds and one with narrow bounds can reach a similar score; the score itself does not distinguish between them.

Thus instead of writing:

"Rough TOPSIS accounts for uncertainty, so the result is more reliable"

the report should read:

"Because criterion scores are given as intervals derived from the group's disagreement, the bounds have been carried separately through to the final step and defuzzified with alpha = 0.5; A3 leads at 0.577, and this ranking holds unless a particular criterion's weight is changed"

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 TOPSIS 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 TOPSIS is the better fit there. The base method's exit condition applies exactly as before: if no compromise is acceptable on one criterion, no member of the TOPSIS family is suitable.

Violating the bound constraint. In every cell, the lower bound cannot exceed the upper bound. If it is violated, scale equalisation and the choice of ideal/anti-ideal become meaningless.

Entering bounds by hand rather than from group scores. Inventing a lower and upper bound around a single expert's score "for the sake of uncertainty" is not a rough number; it violates the basic principle on the data-type card.

Choosing alpha without justification, or not stating it in the report. The formula that weights the upper side for distance to the ideal and the lower side for distance to the anti-ideal is not symmetric; the ranking can change as alpha moves away from 0.5. Which alpha was used and why must be stated in the report.

Defuzzifying first and then running crisp TOPSIS. Averaging the lower and upper bound at the outset and feeding that into the crisp method causes the ideal and anti-ideal point to be built from the average rather than from the bounds; the disagreement information the rough number carries is erased at the first step.

The governing principle is this:

Rough TOPSIS exists to carry the group's disagreement honestly through to the choice of the ideal and anti-ideal; any application that averages the bounds at the outset, or conceals alpha, destroys the method's one contribution.