Grey WASPAS

Several things change, but the logic of blending sum and product stays the same.

Cells. In crisp WASPAS every cell is a single number. Here every cell is a grey number: only a lower and upper bound (⊗[lower, upper]). No most-likely value or distribution is claimed within the bounds. Weights remain crisp; grey data is used only in the decision-matrix cells.

Whitenisation (early defuzzification). Unlike the other WASPAS extensions (Fuzzy, IF, N), Grey WASPAS consumes uncertainty right at the start, before scale equalisation is even performed: every grey number is reduced to a single number by averaging its lower and upper bound (whitenisation). This should not be confused with the mistake the Grey data-type card warns against, that of taking the midpoint of a grey number as a most-likely value. Here the midpoint is not a claim about likelihood, it is only a step in the calculation, and DecisionMind marks it in the report as a distinct step. But its practical consequence is this: the width of the bounds, that is, the size of the uncertainty, plays no part in any calculation after whitenisation. Only the midpoint's position enters the calculation.

Scale equalisation, sum and product. Once the whitened midpoints are obtained, everything else is identical to crisp WASPAS: for a benefit criterion the column is divided by its largest value, for a cost criterion the column's smallest value is turned into the ratio; the weighted sum (WSM) is the weighted sum of these ratios, and the weighted product (WPM) is the product of the ratios raised to the power of the weight.

Result. The combined score blends the sum and product component with λ. Unlike Fuzzy WASPAS, λ is not computed adaptively here; crisp WASPAS's default value of 0.5 is held fixed. This is not currently offered as a user parameter; anyone wishing to change it would need to make a change at the kernel level.

DecisionMind holds whitenisation fixed as the first step in Grey WASPAS; scale equalisation, the sum and the product are crisp WASPAS's own formulas and operate on the whitened midpoints.

The output is a combined score and a ranking, as in crisp WASPAS, and it is read the same way: it is not a percentage or a probability, and scale equalisation changes, and the ranking can shift, whenever the alternative set changes.

The difference is this. The score reflects only the midpoint of the bounds, not their width. If the score gap between two alternatives is small, and that gap is small relative to the width of the bounds, that is, relative to the actual gap in information, the gap itself is not solid evidence. The report should show which criterion's bounds the gap is sensitive to.

Thus instead of writing:

"According to Grey WASPAS, A2 is definitively the best alternative"

the report should read:

"A2's combined score, computed from the midpoint of its bounds, is the highest; because the width of the bounds does not enter the calculation, the robustness of this advantage has been separately tested by checking whether the ranking changes when the criterion bounds are narrowed and widened"

Use this when only a lower and an upper bound are known about criterion values, with no information on which value within those bounds is more likely. It suits new-project or new-market assessments made with little data, and cases where a supplier quotes "at least–at most" figures. If a most-likely point within the bounds is known, that information is lost by moving to the grey structure; the fuzzy structure is more suitable. If a value has been measured, use the base method instead.

WASPAS's exit condition applies just as it does elsewhere: it is compensatory and unsuitable where no compromise is acceptable on one criterion; if weights rather than a ranking are needed, look to methods such as AHP, BWM, CRITIC or Entropy.

Violating the grey-number constraint (lower ≤ upper). If the lower bound is entered larger than the upper bound, whitenisation and every subsequent step return a meaningless result.

Mistaking whitenisation for a most-likely value. The midpoint, as the Grey data-type card warns, is not a claim about a "most-likely value" within the bounds; it is only the calculation step this family has chosen. It should not be presented in the report as "most likely".

Ignoring width and interpreting only the midpoint. After whitenisation, the score hides the width gap between two alternatives. A wide-bound alternative and a narrow-bound alternative sharing the same midpoint are not equally reliable; the report should state this separately.

Fabricating the source of the bounds. Building a grey bound by adding a percentage to a measured value, the core warning of the Grey data-type card, applies here too; bounds must come from a contract, a past record or expert testimony.

Changing λ without justification. In this family λ is fixed at 0.5. If the WSM and WPM components suggest different rankings on their own, this signals that the combined score is fragile, and keeping λ fixed at 0.5 needs separate justification.

The governing principle is this:

Whitenisation carries only the position of the bounds, not their width; when an alternative's advantage is reported in this extension, whether the ranking changes as the bounds are narrowed and widened must be shown separately.

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