Rough Set

A rough-set data structure expresses an alternative's standing on a criterion not as one number but through two approximations: the lower approximation, the part that can be stated with certainty from the available information, and the upper approximation, the part that cannot be excluded by it. The region between the two is the "boundary region" that the available information cannot resolve.

In decision analysis this structure is most often used as a rough number. When a group of experts scores the same alternative, each score's lower bound is the average of scores less than or equal to it, and its upper bound is the average of scores greater than or equal to it. Every expert's score is thereby converted into a [lower, upper] interval based on its relation to the rest of the group. The interval is derived not from any one expert's own uncertainty but from disagreement within the group.

Rough data are used wherever several experts score the same alternative and the disagreement between them matters to the decision in its own right. This is valuable in group decisions such as design-concept evaluation, supplier selection or quality function deployment, wherever collapsing expert scores to an average would erase that disagreement.

Where only one expert is involved, or where experts give similar scores, a rough number narrows to a tight interval and says nothing that a crisp structure does not already say. Where the interval's source is an expert's own hesitation rather than group disagreement, a hesitant or fuzzy structure is more appropriate.

Yes; crisp expert scores are, in fact, the usual source for this structure. Two steps are needed.

The first is establishing what the value represents: a measurement, or judgements given separately by a group? A single measurement is not carried into a rough structure. Precise scores given by several experts for the same criterion can be converted into a rough number.

The second is deriving the lower and upper bounds from the group's scores alone: the lower bound is the average of scores less than or equal to the one in question, the upper bound the average of those greater than or equal to it. The bounds are calculated, not entered from outside, so they are never arbitrary, and the same set of scores always yields the same interval.

What must not be done is writing a bound by hand around a single expert's score or a single measurement "to allow for uncertainty". Writing 140 mmHg as [135, 145] has nothing to do with rough-set logic.

Both are written as two bounds; the difference lies in where the bounds come from. In a grey number, the bounds are read from a source or given by an expert as "at least, at most". In a rough number, the bounds are not given but derived: they are calculated from the group's score distribution, and if the group changes, so do the bounds.

For this reason:

"The experts gave different scores, so I'll take the lowest and highest as the bounds"

should give way to:

"Each expert's score is converted into an interval using the averages of the group above and below it; the boundary region is the measure of disagreement"

Taking the lowest and highest score gives excessive weight to the extremes; a rough number instead places every score relative to the group as a whole.

The chief advantage of the rough number is that it captures disagreement among experts without a predefined membership function or distributional assumption. In a fuzzy structure, the shape of the triangle is chosen by the analyst; in a rough number, the interval comes entirely from the data itself. It is a way of carrying group uncertainty without adding a subjective scale assumption.

The width of the interval is also directly interpretable: a wide interval shows that experts disagree on this criterion, a narrow one shows agreement, and this information is valuable to the decision report in its own right.

Because a rough number depends on the group's scores, it behaves unsteadily with few experts: with only two or three, the bounds can shift substantially if a single score changes. A single expert giving an extreme score can widen the boundary region unnecessarily.

Since every cell becomes an interval, subsequent steps must use interval arithmetic, and the width grows further; ranking requires either reducing intervals to single values or a rule for comparing intervals, and the choice of rule can affect the outcome. The structure does not carry an expert's own internal hesitation, only the difference between experts.

Where experts give their scores as intervals rather than single figures ("between 6 and 8") and group disagreement is also to be preserved, a rough number is derived separately from the lower end and the upper end of each interval. The result is a structure whose own bounds are intervals (an interval rough number).

This is used in problems where both an expert's own hesitation and disagreement between experts matter at once; the assessment burden and computational complexity rise accordingly.

The most frequent mistake is entering the bounds by hand. A rough number's bounds are calculated; a hand-entered bound is a grey or an interval number, not a rough one.

The second is attempting to build a rough number from a single expert or a single measurement; without a group, there is no disagreement to derive. The third is averaging the experts' scores first and only then deriving a rough number; the moment an average is taken, disagreement is erased and nothing is left for the rough number to preserve.

The governing principle is this:

A rough number's bounds are calculated from a group's scores given for the same criterion; the boundary region is the measure of disagreement between experts and is never entered from outside.

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