Grey PROMETHEE (Kuang, Kilgour & Hipel, 2015)

Three things change; the logic of the preference function and the flows does not.

Cells. In crisp PROMETHEE every cell is a single number. Here every cell is two numbers: a lower bound and an upper bound. No most-likely value is claimed in between. Weights remain crisp and are taken from outside, summing to 1; the method does not generate weights, and group decisions are not supported by a separate mechanism at this entry point.

Whitenisation (reducing the difference to a single number). In crisp PROMETHEE the difference is a direct subtraction. Here every grey number is first reduced to a single number by the midpoint of its lower and upper bound (whitenisation), and the difference is then taken between these whitenised numbers. This midpoint is not claimed to be a "most-likely value"; it is only a representative number chosen for the calculation. The literature on grey-system methods builds a separate family for this step; because no shared, paper-derived example table exists for this entry point of DecisionMind, the card's verification example is entirely synthetic.

Preference function and threshold. In crisp PROMETHEE the preference threshold (p) and the indifference threshold (q) are usually set by hand. Here DecisionMind does not set this threshold by hand: for each criterion it sets the p threshold equal to the largest observed difference between alternatives on that criterion, that is, the range (largest minus smallest) of the column's whitenised values, and keeps the indifference threshold at zero. The difference is divided by this range and capped at 1, giving a preference degree in the [0, 1] range.

Flows. The reduced and thresholded preference degrees are multiplied by the criterion weight and summed; the entering flow (φ⁺) and leaving flow (φ⁻) are computed with the same averaging definition as in crisp PROMETHEE, net flows are taken, and the ranking follows the same rule.

DecisionMind holds fixed, at this entry point, whitenisation by the midpoint and automatically deriving the threshold from the column range. This automatic threshold depends on the alternative set: adding or removing an alternative changes the range, and so the threshold — an extra source of sensitivity not seen when working with a manually fixed threshold as in crisp PROMETHEE.

The net flow is a value as in crisp PROMETHEE, and it is read the same way: it is assessed together with the entering and leaving flow, and cannot be compared with another method's score.

The difference is here: this net flow rests both on the midpoint of the bounds (whitenisation) and on the automatic threshold derived from the column range. The net-flow gap between two alternatives may be meaningful or meaningless depending on the width of the bounds and on the range of whichever alternative set is currently on the table. This is why the report should show, alongside the net flow, the width of the bounds and whether the ranking stays solid when the alternative set changes (a new candidate is added).

Thus instead of writing:

"According to Grey PROMETHEE, A2 is clear-cut because it also works with little data"

the report should read:

"Criteria were entered only as lower and upper bounds, reduced to a single number by their midpoint, and processed through PROMETHEE's preference function; A2 has the highest net flow, and because the threshold is derived automatically from this alternative set's range, it should be re-tested if a new candidate is added"

Use this when only a lower and upper bound are available for a criterion and no value within the bounds is considered more likely than another: assessments of a new market, a new supplier or a new project made with little data; expert judgements able to say "at least, at most" but not "most likely."

If a most-likely value is known within the bounds, fuzzy PROMETHEE should be preferred over grey; the grey structure erases this information. The base method should be kept when criteria are measured. Adding an unsourced margin around a measured number to build a grey number 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 in grey form, with the lower bound equal to the upper bound. Crisp PROMETHEE's exit conditions apply here in exactly the same way: if the number of alternatives exceeds fifty, the pairwise-comparison burden grows large, and if a full ranking is not needed but an honest partial relation is, PROMETHEE I's reading or the ELECTRE family should be considered.

Interpreting the whitenised midpoint as a "most-likely value." The midpoint of a grey number, computed for instance as 40 for ⊗[30, 50], does not mean that 40 is more likely than other values; there is no defined most-likely point within the bounds, the midpoint is only a representative the calculation needs.

Building a grey number by adding an unsourced bound around a measured figure. Writing 2,450 TRY as ⊗[2,400, 2,500] has no source at all; the bounds must have a source (a contract, a past record, an expert statement).

Treating the automatically derived p threshold as a fixed parameter. This threshold is computed from the range observed among the alternatives currently on the table for that criterion; it changes when a new alternative is added and the range widens or narrows. Assuming without checking that "the threshold was always the same" leads to an inability to explain why the ranking changed when an alternative was added (rank reversal).

The governing principle is this:

Grey PROMETHEE's contribution is to calculate honestly without assuming anything unknown within the bounds (a most-likely value, a distribution); assuming the midpoint is most likely, or forgetting that the threshold depends on the alternative set, breaks this honesty.