Rough MARCOS (Matić, Marinković, Jovanović, Sremac & Stević, 2022)

Four things change; the decision logic does not.

Cells. In crisp MARCOS 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 MARCOS's own calculation as crisp numbers even when derived one step upstream by a rough method (D-SWARA, for instance); bound information is carried only in the decision matrix and in the ideal/anti-ideal references.

Scale equalisation. Crisp MARCOS relates every cell to the criterion's ideal value. In rough MARCOS the ideal and anti-ideal reference are first built, column by column, from the crisp best and worst end observed for each criterion; these ends come not from a single alternative but from the best/worst bound across all alternatives on that criterion. Every cell is then related to this reference in rough form: lower bound against lower bound, upper bound against upper bound.

Weighting and row sum. The scale-equalised rough table is multiplied by the weight; because the weight is crisp, this multiplication scales the lower and upper bound by the same factor. Each alternative's row is summed, and this sum remains a rough interval; unlike in crisp MARCOS, a single number is not yet reached at this point.

Result and defuzzification. In crisp MARCOS the utility ratios are directly single numbers. In rough MARCOS the ratios to the ideal and the anti-ideal are first divided within the rough interval, and only after that are they reduced to a single number (defuzzified Y+ and Y-). The utility functions and the final degree of utility are built from these defuzzified numbers. DecisionMind performs the division before defuzzification in this extension; this order gives a result that is generally very close to, but for some alternative pairs sensitive enough to change the ranking from, dividing after defuzzifying.

The final degree of utility is read exactly as in base MARCOS: a higher value is better, it is not compared with a different analysis, and the ideal and anti-ideal references change when the alternative set changes.

The difference is here: beneath the degree of utility now lies the group's own disagreement. Two very close degrees can, as in base MARCOS, be sensitive to the weights, but here they can also be sensitive to whether the rough division was performed before or after defuzzification. This sensitivity matters especially when the gap between the top two or three alternatives is on the order of a few parts per thousand.

Thus instead of writing:

"According to rough MARCOS, the best alternative is A11"

the report should read:

"With these weights A11 has the highest final degree of utility (0.7314); the gap with A4 (0.0052) is small, and this gap is sensitive both to the weight distribution and to the order of defuzzification"

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 MARCOS is used. Where bounds are not calculated but given directly as "at least, at most", this is a grey number, not a rough number. The base method's exit condition applies exactly: if no compromise is acceptable on one criterion, no member of the MARCOS family provides that; and with a very small alternative set, the ideal and anti-ideal references remain, here too, sensitive to the alternatives themselves.

Violating the bound constraint. In no cell may the lower bound exceed the upper bound; if it does, the ideal/anti-ideal reference and the scale equalisation become meaningless.

Changing the order of defuzzification and expecting the same result. Dividing within the rough interval and then defuzzifying, versus defuzzifying first and then dividing, give results close to one another for most alternative pairs but can reverse the ranking for pairs with a narrow gap at the top; which order was used should be stated in the report.

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 a gap of a few parts per thousand as a definite advantage. If the gap between the top two places is sensitive to both the weights and the defuzzification order, the report should describe first and second place as "practically close" rather than declaring a single winner.

The governing principle is this:

A rough MARCOS result is a summary of the ideal and anti-ideal references after they have been defuzzified from a rough interval; narrow gaps at the top are sensitive both to the weights and to the order of defuzzification, and the report must show this.