Grey MOORA (Stanujkić, Magdalinović, Jovanović & Stojanović, 2012)

Cells. In crisp MOORA every cell is a single number. Here every cell is expressed by two bounds: a minimum and a maximum. No most-likely value or distribution is claimed within the bounds; the information is only the two bounds. Criterion weights are taken as crisp (single) numbers in DecisionMind; grey MOORA's uncertainty is carried only in the cells of the decision matrix.

Scale equalisation. The method first divides every column by that column's largest upper bound. It then reduces each cell's lower and upper bound to a single number by averaging them, that is, it whitenises, and divides this whitenised value by the square root of its own column magnitude, that is, applies vector normalisation, just as in crisp MOORA. The reason for these two stages is this: the first sets the bounds against a common upper reference (1), and the second brings the whitenised values onto the same scale as crisp MOORA.

Whitenisation. No such step exists in crisp MOORA because the cells are already single numbers. In Grey MOORA, a single number is produced by averaging the two bounds; this does not mean "the midpoint is most likely." In a grey-data structure where no point between the bounds is considered more likely than another, the average is only a compromise chosen to carry the calculation forward, and this should be stated in the report.

Ratio system and result. The whitenised and normalised values are weighted; for each alternative, the weighted sum over the "more is better" criteria has the weighted sum over the "less is better" criteria subtracted from it. This is exactly the same operation as crisp MOORA's ratio system; the only difference is that the input has first been whitenised. The output is directly a single net score and a rank, not a grey number. Defuzzification, that is, whitenisation, is done in the middle of the calculation, immediately before the input is normalised.

DecisionMind holds this order fixed in classic Grey MOORA (first dividing by the largest upper bound, then whitenising, then vector normalisation) and uses the arithmetic average of the bounds as the whitenisation rule; an optional reference-point approach can also be computed for cross-checking over the same whitenised matrix.

The output is a net score and a rank, as in crisp MOORA, and it is read the same way: it is not a percentage or a probability, and it cannot be compared with a different analysis. The difference is this. This score in fact rests on the midpoint of a value known only by its bounds. If two alternatives' bounds overlap, that is, if one's lower bound is higher than the other's upper bound, then even if the whitenised averages differ, the true position of these two alternatives is uncertain. A small difference in the net score can conceal this overlap.

Thus instead of writing:

"Grey MOORA found the best alternative with certainty"

the report should read:

"Among the bounds, the alternative with the highest net score according to the whitenised values is this one; if the alternatives' bounds overlap, this ranking is only as certain as the width of that overlap"

Grey MOORA is suitable when only a lower and upper bound are known about the criteria, that is, when there is no most-likely value or distribution: cases with little data, a short track record, or a supplier quoting a price "somewhere between this and that." If a most-likely value is known within the bounds, discarding this information and descending to the grey structure erases information that is available; a fuzzy structure should be preferred. A measured criterion is written in the same matrix in DecisionMind as a grey number with both bounds equal (a single point). The exit condition is the same as crisp MOORA: if a criterion allows no compromise, this extension too is compensatory and will not eliminate anything below a threshold.

Violation of the value space. Every cell's lower bound must be less than or equal to its upper bound; if this order is broken, the calculation becomes invalid.

Presenting whitenisation as a "most-likely value." The average of the bounds is only a computational step; it should not be referred to in the report as "most likely cost" or "expected value," because a grey number does not carry this information.

Building a grey number by adding a percentage around a measured value. The bounds must have their own source (a contract, a past record, an expert statement); a fabricated band such as ±5% of a crisp number breaks grey data's principle of honesty.

Knowingly discarding the information within the bounds. If a most-likely value or a past distribution is actually available and is knowingly ignored, feeding only the bounds into grey MOORA, the method is run with deliberately less information than is available.

The governing principle is this:

Grey MOORA's whitenisation step is a computational convenience, not a claim of information; no "most-likely value" that is not actually available should be loaded into the bounds afterwards.