Rough set WASPAS (Stojić, Stević, Antuchevičienė, Pamučar & Vasiljević, 2018)

Four things change; the decision logic does not.

Cells. In crisp WASPAS every cell is a single number. Here every cell is a rough-number interval: a lower approximation and an upper approximation. As explained on the data-type card, this interval is calculated from the group's crisp scores; it is not entered by hand.

Scale equalisation. Equalisation is carried out crosswise: for a benefit criterion, the lower end of the interval is divided by the largest of all the upper ends in the column, and the 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 guarantees that the normalised interval's lower end also stays less than or equal to its upper end.

Additive and multiplicative combination. Both the WSM and WPM components of crisp WASPAS are here calculated as intervals. The WSM-style component is the weighted sum of the equalised intervals; lower ends are summed with the lower end of the weight, upper ends with the upper end of the weight. The WPM-style component is the weighted product of the equalised intervals; a small floor is kept on values at or near zero so that a product is not spoiled by them.

Result, λ and defuzzification. In crisp WASPAS, λ is a fixed number, either chosen by the user or defaulting to 0.5. Here λ is no longer fixed: a lower and an upper λ are calculated from the data itself, from the ratio of the multiplicative component's total magnitude to the sum of the two components. The combined score is built with this λ interval, and only then comes down to a single number by taking the midpoint of the interval.

DecisionMind fixes, for classical rough-set WASPAS, the crosswise min-max equalisation, the interval arithmetic of the two components, the way λ is derived from the data, and the midpoint defuzzification. Weights are taken from outside; the method does not generate weights.

The defuzzified score is read the same way as in crisp WASPAS: it shows an alternative's relative position within this set of alternatives, and it is neither a percentage nor a probability. The difference is here. λ is no longer a number chosen by the decision-maker but an interval calculated from the data itself; testing the result by "changing λ" is therefore meaningless, since λ is already tied to the data. On the other hand, every alternative's score now has an interval beneath it, and the width of that interval shows the disagreement among the experts; a ranking with narrow intervals is more reliable than one with wide intervals.

Thus instead of writing:

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

the report should read:

"With these weights and the λ calculated from the data, A1's defuzzified score is the highest; A1's interval width is [lower, upper], and this width shows the disagreement among the experts"

If the criteria are a single measurement, or a single expert's crisp score, crisp WASPAS is sufficient. If several experts assess the same criterion with crisp scores and the disagreement between them matters for the decision in its own right, rough-set WASPAS is used. You need [lower, upper] rough-number intervals already calculated in hand; these are derived from group scores according to the method on the data-type card, not entered by hand. The matrix must be of a single type and must contain no zero or negative values; the multiplicative component becomes undefined in that case. WASPAS's exit condition applies here too: if no compromise is acceptable on one criterion, this extension is also compensatory and will not eliminate anything below a threshold.

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

Still assuming λ is a fixed number chosen by the user. In crisp WASPAS, λ is in the decision-maker's hands; here λ is an interval calculated from the data. Fixing λ by hand cancels this extension's specific contribution.

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

Trying to build a rough number with a single expert. Without a group, there is no disagreement to derive in the first place.

The governing principle is this:

In rough WASPAS, both the cells and λ are derived from the data; neither is fixed by hand, and both carry the disagreement present in the group's own scores.