Grey TOPSIS (Zavadskas, Turskis and Bagočius, 2015)

Four things change; the decision logic does not.

Cells. In crisp TOPSIS every cell is a single number. Here every cell consists of two numbers: a lower bound and an upper bound, that is, ⊗[lower, upper]. No most-likely value is declared within the bounds; both ends are equally possible. Criterion weights remain crisp, that is, single numbers. Grey TOPSIS requires the weight to be crisp rather than grey, because a weight is a statement of preference, not missing information.

Scale equalisation. Crisp TOPSIS divides every column by the square root of the sum of its squares; this is called vector normalisation. Grey TOPSIS does not do this. For a benefit criterion, every bound is divided by the column's largest upper bound; for a cost criterion, the column's smallest lower bound is set against the bounds instead. This is linear scale equalisation, and it is simpler than crisp TOPSIS's sum-of-squares approach. It rescales the lower and upper bound separately, without mixing them together.

Order of weighting and defuzzification. The equalised lower and upper bounds are first multiplied by the criterion weight. Up to this point the bounds still stand as two separate numbers. Only after this are the bounds reduced to a single number; this step is called whitenisation, and it is carried out by averaging the lower and upper bound.

Crisp TOPSIS has no defuzzification step, because its values are single numbers from the outset. In Grey TOPSIS, by contrast, whitenisation is one of the last steps, not the first, and this order matters. The ideal and anti-ideal point are first built from the weighted bounds themselves: the largest and smallest of the lower bounds, and the largest and smallest of the upper bounds, are identified. The average is taken only for the distance measurement.

Distance and result. Once whitened, that is, averaged, the values feed into the Euclidean distance and closeness score exactly as in crisp TOPSIS. DecisionMind holds linear scale equalisation, average-based whitenisation and Euclidean distance fixed in this extension. Other whitenisation rules, such as taking the lower bound alone or a weighted average, can change the result; the rule actually used must therefore be stated in the report.

The output is a closeness score and a ranking, as in crisp TOPSIS, and it is read the same way: it is not a percentage, and it cannot be compared with a different analysis.

The difference is here: beneath the score there still lies an interval width that no longer shows. An alternative with narrow bounds and one with wide bounds can share the same average and reach the same score; the score does not distinguish between them. If the score gap between two alternatives is small, look not only at sensitivity to the weights but also at how wide the bounds are. A small gap obtained from wide bounds is less trustworthy than a gap of the same size obtained from narrow bounds.

Thus instead of writing:

"Grey-TOPSIS gives a more reliable result because it takes uncertainty into account"

the report should read:

"Because criterion values are known only by a lower and an upper bound, the bounds have been carried separately through to the choice of ideal and anti-ideal point and averaged only at the end; A2 leads at 0.733, and this ranking does not change unless the weight on Criterion 2 is raised substantially"

Use this extension when nothing beyond the bounds is known about a criterion: new-market or new-technology assessments made with little data, a supplier's quotation given as "somewhere between this and that", cases where an expert can state a minimum and a maximum but not a most-likely value. If a most-likely point within the bounds is known, dropping to the grey structure discards it; the fuzzy extension (Fuzzy TOPSIS or Fuzzy TOPSIS, Chen 2000) then fits better.

The case for staying with the base method is the same as elsewhere: when criteria are measured. Building bounds around a measured value by adding a percentage (turning 2.450 into ⊗[2.400, 2.500], say) breaches the founding principle of the Grey data-type card: bounds must have their own source. If the table is mixed, DecisionMind requires a single data type; a measured criterion is then also written in grey form, with the lower bound equal to the upper bound, giving zero width. If no compromise is acceptable on one criterion, this extension too is compensatory and eliminates nothing below a threshold.

Violating the bound constraint. In every cell, the lower bound cannot exceed the upper bound. If it does, scale equalisation and the choice of ideal/anti-ideal point become meaningless.

Changing the whitenisation rule and expecting the same result. Averaging the lower and upper bound is the canonical choice, but not the only rule; taking the lower bound alone (pessimistic) or the upper bound alone (optimistic) can give a different ranking. The rule actually used must be stated in the report.

Averaging the bounds at the input stage and running crisp TOPSIS. This erases the grey information before the ideal/anti-ideal point is chosen; in the illustrative example the bounds are narrow, so the result changes only slightly, but with wide bounds (large uncertainty) this shortcut can change which alternative the ideal/anti-ideal point is built from, and the difference grows.

Reading the midpoint of a grey number as a "most-likely value". This is the founding principle of the Grey data-type card: a grey number does not carry this information; the average is only a calculation step, not an assumption added to the input from the outset.

The governing principle is this:

Grey-TOPSIS exists to stay honest when only bounds are available; any application that reduces the bounds to a single number before the ideal/anti-ideal point is determined, or treats the midpoint of the bounds as a most-likely value, erases the extension's sole contribution.