Grey ARAS (Turskis and Zavadskas, 2010)

Four things change; the decision logic does not.

Cells. In crisp ARAS every cell is a single, strictly positive number. Here every cell is two numbers: a lower bound and an upper bound; no value within the bounds is considered more likely than another. Weights, too, may be given as a grey interval; when a crisp weight is supplied instead, DecisionMind embeds it as an interval with equal lower and upper bounds, adding no width. The method does not combine a group decision on its own.

Scale equalisation. Crisp ARAS first builds the optimal alternative, reverses cost criteria, then divides every column by its own sum. Here, for a cost criterion, every interval [lower; upper] becomes [1/upper; 1/lower] once reversed. This ensures the lower bound stays smaller than the upper bound even after reversal. The method then scales every column crosswise: a cell's lower bound is divided by the sum of all upper bounds in the column, and its upper bound is divided by the sum of all lower bounds in the column (including the optimal-alternative row). This crosswise division guarantees that the two bounds do not cross one another even once normalised. Ordinary division, that is, dividing both bounds by the same sum, gives no such guarantee.

Distance / score / aggregation. ARAS has no distance to a reference point; it takes a weighted sum directly, and that stays the case here, only the summation is carried out bound by bound over the intervals. The method multiplies each equalised interval by the criterion's (grey or crisp) weight, bound by bound (lower bound with lower bound, upper bound with upper bound). It then sums each row's intervals, including the optimal alternative, bound by bound across criteria. This produces a single interval "optimality score" for each row.

Result and defuzzification. The method reduces each row's interval sum to a single number by taking the average of its two bounds, whitening, grey system theory's own term for this. This is the defuzzification, and it happens only at this final step. The degree of utility K carries the same meaning as in crisp ARAS: the ratio of the real alternative's whitened score to the optimal alternative's whitened score.

DecisionMind fixes the midpoint whitening and the crosswise-ratio scale equalisation; intervals are never ranked directly at any stage, ranking is done only on the whitened K.

The degree of utility K carries the same meaning here. The best alternative is treated as 100, and the others receive a percentage relative to it; this percentage is valid only for this alternative set and these bounds. The difference is this: although K appears as a single number, a two-bounded uncertainty lies beneath it. This uncertainty disappears from view in the report because it is reduced to a single number at the whitening step. The K gap between two alternatives can be robust or fragile depending on the width of the input intervals; the same-sized K gap between two alternatives with wide, overlapping intervals is far less reliable than between two alternatives with narrow intervals.

Thus instead of writing:

"According to Grey ARAS, A2 is the best supplier"

the report should read:

"Relative to the optimal supplier derived from this supplier set, A2 has the highest degree of utility (K=0.885); its gap over A3 (0.879) is small relative to the width of the bounds, and the original paper's own table shows these two suppliers' ranking exactly reversed, so the advantage between A2 and A3 should be read as fragile, not decisive"

Use this method when only a lower and an upper bound are known about criterion values, and no most-likely value or distribution is available. Examples: new supplier bids assessed with little data, expert judgements that can state "at least this much, at most that much" but not "most likely," and indicators with a short track record.

If a known most-likely value exists within the bounds, dropping to a grey structure erases information that is already available; a fuzzy structure should be used instead in that case. Opening a measured criterion into a bound afterwards likewise manufactures uncertainty rather than modelling anything. DecisionMind requires a single data type: a measured criterion must be written as an interval with the lower bound equal to the upper bound. Crisp ARAS's exit condition applies here too: when no compromise is acceptable on one criterion, this extension is also fully compensatory and will not eliminate anything below a threshold.

Breaking the interval's order. In every cell the lower bound must be less than or equal to the upper bound, and all bounds must be positive (so that the cost criterion's reciprocal can be taken); if this condition is broken, the crosswise scaling produces a meaningless result.

Reading the whitened midpoint as a "most likely value." Whitening is only a computational step and carries no information about a most likely value within the bounds. The midpoint enters the calculation because of the defuzzification step, not because of the data.

Averaging the bounds first and then running crisp ARAS. In the supplier example below this path happens to give the same ranking (A2 still first), but this is not guaranteed; it erases from the outset the "how little is known" information the interval width carries, and the ranking can change in the next table.

Giving crisp weights and grey cells without saying so. The method embeds a crisp weight as a zero-width interval; this is legitimate, but it should be stated in the report.

The governing principle is this:

Grey ARAS is for taking a ratio against the optimal alternative while honestly preserving the fact that nothing is known beyond the bounds; any implementation that averages the bounds early, or ignores the difference between a narrow and a wide interval, conceals the method's one contribution.