Rough COPRAS (Pamučar, Božanić, Lukovac & Komazec, 2018)

Four things change; the decision logic does not.

Cells. In crisp COPRAS every cell is a single number. Here every cell is a rough number: a lower and an upper bound, calculated from the group's crisp scores. In this extension, weights enter COPRAS's own calculation as crisp, single numbers, even when derived one step upstream by a rough method such as rough DEMATEL; bound information is carried only in the decision matrix.

Scale equalisation. Crisp COPRAS divides every column by its own column sum. Rough COPRAS follows the same logic, but the column sum is now a rough interval, and every cell is divided by this rough sum in rough form: lower bound against lower bound, upper bound against upper bound. The result is still a share table with a lower and upper bound.

Benefit and cost sum. In crisp COPRAS the benefit sum and the cost sum are each single numbers. In rough COPRAS both are rough intervals: the weighted shares on the benefit criteria are summed separately by their own lower and upper bounds, and likewise for the cost criteria.

Result and defuzzification. In crisp COPRAS the relative significance value is directly a single number. In rough COPRAS, the rough intervals of the benefit and cost sums are first defuzzified, that is, reduced from a lower and upper bound to a single number. The relative significance value and the percentage relative to the best are then computed from these defuzzified numbers. DecisionMind fixes the defuzzification rule in this extension; a different rule can give a different percentage.

The degree of utility is read exactly as in base COPRAS: the best alternative always scores 100, and the others receive a percentage relative to it; it is not compared with a different analysis.

The difference is here: beneath the percentage now lies the group's own disagreement. If two alternatives' percentages are close, that closeness may depend not only on the weights but also on how wide a bound region the benefit and cost sums were defuzzified from. An alternative with wide bounds has a percentage that is less reliable than the same-sized difference for an alternative with narrow bounds.

Thus instead of writing:

"Rough COPRAS found this alternative 84 per cent successful"

the report should read:

"Because criterion scores are given as intervals derived from group disagreement, the benefit and cost sums have first been carried as rough intervals and then defuzzified; A5 reaches 83.52 per cent of this set's best alternative, and the gap between third and fourth place is sensitive to a small weight change"

Use this extension when several experts or sources score the same criterion and the disagreement within the group itself matters to the decision. Where there is a single expert or a single measurement, a rough number cannot be built; base COPRAS is used. Where bounds are not calculated but given directly as "at least, at most", this is a grey number, not a rough number, and Grey COPRAS is more suitable. The base method's exit condition applies exactly: a negative value in your data must first be transformed, and if no compromise is acceptable on one criterion, no member of the COPRAS family provides that.

Violating the bound constraint. In no cell may the lower bound exceed the upper bound; if it does, the column sum and the share calculation become meaningless.

Changing the defuzzification rule and expecting the same result. Taking the average of the lower and upper bound of the benefit and cost sums is the canonical choice; a different rule, especially for alternatives with wide bounds, can give a different percentage and a different ranking.

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

Reading close percentages as a definite difference without checking bound width. If two alternatives' percentages are close and their bounds are wide, this difference can easily be closed by a small change in the weights; the report should show this.

The governing principle is this:

A rough COPRAS result is a summary of the benefit and cost sums after they have been defuzzified from a rough interval; any calculation that divides by the column sum before defuzzifying the bounds, or that fabricates a bound by hand, destroys the method's one distinctive contribution.