Rough SAW (Stević, Pamučar, Zavadskas, Ćirović & Prentkovskis, 2017)

Four things change; the decision logic does not.

Cells. In crisp SAW every cell is a single number. Here every cell is a rough number interval: a lower and an upper approximation. As explained on the data-type card, this interval is calculated from the group's crisp scores, not entered by hand; each expert's score is converted into the averages of the group's scores falling below and above that score.

Scale equalisation. Crisp SAW relates every column to its own best value. Here equalisation is cross-applied: for a benefit criterion, the interval's lower end is divided by the largest of all the upper ends in the column, and its upper end by the largest of all the lower ends in the column. For a cost criterion, the direction reverses and the smallest values are used. This cross-division guarantees that the normalised interval's lower end also stays less than or equal to its upper end.

Weighted sum. Crisp SAW multiplies the equalised columns by the weight and sums them. Here the same operation is done, but with interval arithmetic: the weighted sum of the lower ends is computed with the weights' lower end, and the weighted sum of the upper ends with the weights' upper end. Weights, too, can come as an interval in this extension (from rough BWM, for instance). The result is again a lower-upper interval for every alternative.

Result and defuzzification. Crisp SAW has no defuzzification step because the calculation already starts with a single number. Here defuzzification takes place at the last step, by averaging the interval's lower and upper end. Alternatives are ranked by this single number.

DecisionMind fixes, in classical rough SAW, cross min-max equalisation, weighted summing with interval arithmetic, and midpoint defuzzification. Weights are taken from outside; the method does not generate weights.

The defuzzified score is read like crisp SAW's total score: it produces a ranking only within this alternative set, and it is not a percentage or a probability. The difference is here. Beneath the score now lies an interval, and the width of this interval is a measure of disagreement between experts. If two alternatives' defuzzified scores are close, it is also worth checking whether their intervals are wide or narrow; an alternative with a wide interval suggests a less reliable ranking than its score, even if its midpoint is high, would imply on its own.

Thus instead of writing:

"According to rough SAW, A1 is the best alternative"

the report should read:

"With these weights, A1's defuzzified score is the highest; however, A1's interval is [lower, upper] wide, and this width reflects disagreement between experts"

Where criteria are a single measurement, or a single expert's crisp score, base SAW is sufficient. Where several experts assess the same criterion with crisp scores and the disagreement between them itself matters to the decision, rough SAW is used. You should already have calculated [lower, upper] rough number intervals in hand; these intervals are derived from group scores by the method set out on the data-type card, and are not entered by hand or read from a source, otherwise this becomes a grey or interval number rather than a rough number. The matrix must be of a single type. SAW's exit condition applies here too: if no compromise is acceptable on one criterion, this extension too is compensatory and will not eliminate anything below a threshold.

Entering the interval by hand. A rough number's bounds are calculated from the group's scores. A [lower, upper] interval entered by hand, or read from a source, is a grey or an interval number, not a rough number.

Attempting to build a rough number with a single expert or a single measurement. Without a group, there is no disagreement to derive; the interval either narrows to zero width or becomes meaningless.

Averaging first and only then setting an interval. Taking the average of experts' scores and putting a margin around it is a fabricated interval, not a rough number; the moment the average is taken, disagreement is already erased.

Interpreting without changing the defuzzification method. Midpoint defuzzification is the canonical choice but not the only path; a different defuzzification rule can give a different ranking, and this choice should be stated in the report.

The governing principle is this:

A rough number's interval is a measure of disagreement calculated from the group's scores; the defuzzified single score conceals this disagreement, and the report must also show the width of the interval.