Grey MABAC

Four things change; the border approximation area logic does not.

Cells. In crisp MABAC every cell is a single number. Here every cell is two numbers: a lower bound and an upper bound. No most-likely point is declared within them; every value between the bounds is treated as equally possible. Criterion weights remain crisp (single numbers) here; DecisionMind's Grey MABAC takes weights as crisp, not grey.

Scale equalisation. Crisp MABAC places every column into the 0–1 range according to its own smallest and largest value. Grey MABAC does this with Weitendorf's direction-sensitive two-branch normalisation. For a benefit criterion, the lower and upper bound are scaled up over the same denominator. For a cost criterion, the bounds swap places; the upper bound becomes the lower output and the lower bound becomes the upper output. This keeps the interval ordered. This method is the spread of crisp MABAC's linear min–max equalisation onto intervals.

Weighting and whitenisation. 1 is added to each bound, then it is multiplied by the criterion's (crisp) weight. This comes from the same technical necessity as the "+1" shift in crisp MABAC; the geometric mean in the next step cannot work with zero or negative numbers. The lower and upper bound are then averaged, reducing the interval to a single number (whitenisation). The border approximation area and the distance are computed from this single number.

Border approximation area and distance. The border is the geometric mean of the whitenised values. This is the same formula as the border definition in crisp MABAC; only the input is now the average of an interval. The distance, too, is a simple difference as in crisp MABAC (value minus border). The grey number undergoes no interval operation here; it proceeds through a single whitenised number.

Result and defuzzification. Defuzzification, that is, whitenisation, is done immediately after the scale-equalisation step, right after weighting finishes. This is the step closest to crisp MABAC. DecisionMind holds this early whitenisation and the (lower+upper)/2 averaging rule fixed. A different whitenisation rule (taking only the upper bound, for instance) can produce a different ranking.

The score is read the same way as in crisp MABAC. A positive score means "above the border," a negative score means "below the border"; this reading is valid only within this set of alternatives. But here the unknown inside the bounds, that is, the absence of a most-likely point, is closed off at an early stage by averaging. If the score difference between two alternatives is small, this difference is sensitive to the width of the bounds (how little is known).

Thus instead of writing:

"Grey MABAC still gave a precise ranking even with little data"

the report should read:

"The bounds were entered as intervals and reduced to a single number by whitenisation; A2 is furthest above the border, and the order between A1 and A3 can change if a single criterion's interval widens"

Use this when only a lower and upper bound are known about a criterion, and no most-likely point can be asserted in between. A typical case is an estimate for a new alternative with little data, for which one can say "at least this much, at most that much" but not "most likely."

If the value is a single measured figure, the base method should be kept. Adding an artificial lower-upper bound to a measured value is not modelling uncertainty, it is manufacturing it. If the table is mixed, DecisionMind requires a single data type; a measured criterion is then also written into an interval, with the lower bound equal to the upper bound. This adds no information, it merely conforms to the format. The exit point is the same as crisp MABAC. This extension, too, is compensatory; if a criterion allows no compromise, the ELECTRE family should be considered.

Using the midpoint of the interval as a most-likely value. A grey number does not carry this information; reporting the midpoint as an "expected value" assumes data that does not exist.

Whitenising while entering the data rather than at the very end. Averaging the bounds first and then running crisp MABAC is not Grey MABAC; the width of the bounds never enters the calculation at all, and score differences are exaggerated.

Entering a lower bound greater than the upper bound. A direction mix-up corrupts normalisation from the start; this is seen most often on a cost criterion when the bound swap is forgotten. A check that lower ≤ upper in every cell should be made before entering the data.

The "more advanced, therefore better" fallacy. Grey MABAC does not produce a "more correct" ranking than crisp MABAC; it is only a more honest representation when the input comes with little information (only bounds).

The governing principle is this:

Grey MABAC exists so that, when only bounds are available, nothing beyond them (a most-likely point, a distribution) is invented; averaging the bounds early, or declaring the midpoint most likely, erases the extension's one contribution.