Fuzzy PROMETHEE (Goumas & Lygerou, 2000)

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 three numbers: lowest, most likely, highest. Weights remain crisp and are taken from outside, summing to 1; the method does not generate weights. This extension does not support a group decision through a separate mechanism: if more than one expert's scores exist, they are first aggregated into a single triangle, and this single matrix enters the calculation.

Reducing the difference to a single number. In crisp PROMETHEE, two alternatives' difference on a criterion is a direct subtraction. Here, each triangle is first reduced to a single number by its own centroid, that is, the average of the lowest, most likely and highest value; the difference is then taken between these reduced numbers. This step corresponds to TOPSIS's column scaling: it produces a single real number that can be compared with the preference function's threshold values (the indifference threshold q, the preference threshold p). DecisionMind performs this reduction by the centroid, and early, that is, at the very first step.

This early reduction is a simplification of the order used in Goumas and Lygerou's (2000) original paper: the paper defines the preference degree over the fuzzy difference itself, whereas DecisionMind defuzzifies first with the centroid and then runs the same calculation as crisp PROMETHEE. The two routes are not mathematically identical; DecisionMind records this as an engineering choice, and this card does not hide it.

Preference function and flows. The reduced difference passes through the same preference function as crisp PROMETHEE (default: a threshold-linear function, indifference threshold 0, preference threshold 1), is multiplied by the criterion weight and summed; the incoming flow (φ⁺) and outgoing flow (φ⁻) are computed with the same average definition. The fuzziness has affected the difference that feeds the flow, not the flow itself, and leaves no trace beyond this step.

Result and defuzzification. The output is a single net flow value, as in crisp PROMETHEE (φ⁺ − φ⁻), directly; the uncertainty has been consumed at the defuzzification step and is not carried through to the result in fuzzy form. DecisionMind holds the centroid defuzzification and the threshold-linear preference function (default q=0, p=1) fixed in this extension; a different defuzzification rule (only the most-likely value, the most pessimistic end) or a different family of preference functions could be chosen, but the choice must be stated in the report.

The net flow is, as in crisp PROMETHEE, a value between −1 and +1, and it is read the same way: it is assessed together with the incoming and outgoing flows and is not compared with another method's score.

The difference is here: this net flow has reduced the three-cornered uncertainty in the criterion scores to a single point at the first step. The net-flow gap between two alternatives may be meaningful or meaningless depending on the triangles' width; the wider the triangle, the cruder a summary the centroid's single number represents. For this reason, the report should show not only the net flow, but also the triangles' width and whether a weight swap changes the ranking.

So instead of:

"According to Fuzzy PROMETHEE, A2 is the clear winner because uncertainty has been accounted for"

the report should read:

"Criterion scores were entered as triangular fuzzy numbers, reduced to a single number by the centroid, and processed with PROMETHEE's own preference function; A2's net flow is the highest, and this ranking is sensitive to a weight swap"

The centroid of a triangle carries only its centre, not its width; the width must be stated separately in the report.

Use this when criteria are scored verbally or approximately, and reducing these scores to a single number would create an artificial precision. This is Goumas and Lygerou's (2000) own field of application: criteria that cannot be measured directly, such as "environmental impact" or "social acceptance" in energy projects, scored by expert judgement. It is also suitable in areas where PROMETHEE is already frequently used, such as supplier evaluation, investment and project ranking, when one or more of the criteria is scored approximately.

Stay with the base method when criteria are measured. Opening a measured cost or duration into a triangle is not modelling uncertainty but manufacturing it. If the table is mixed, DecisionMind requires a single data type; a measured criterion is also written as a triangle, with all three components equal and a width of zero. 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.

Reporting triangles with three equal components as "uncertainty was carried". If a triangle's lowest, most likely and highest value are the same (zero width), the calculation is indistinguishable from crisp PROMETHEE; the report should not present this as a "fuzzy result" but should state that the input is in fact crisp.

Leaving the threshold values (q, p) at their default without checking the triangles' actual scale. The default thresholds lie between 0 and 1. If the triangles are given on a scale outside 0-1, for example a cost range of thousands of lira, then with the default thresholds almost every difference counts as "full preference", and the preference function effectively collapses into the simplest (absolute) function; this erases the one contribution of the graded-preference idea.

Using only the most-likely value (m) as the defuzzification number instead of the centroid. This silently discards the information at the triangle's ends; the rule DecisionMind holds fixed is the average of the three components, and if a different rule is used, it must be stated in the report.

The governing principle is this:

Fuzzy PROMETHEE's contribution is to carry the approximation in criterion scores honestly through to the centroid; if the input is already crisp, or if the thresholds have not been calibrated to the input's scale, there is no uncertainty left to carry.