PROMETHEE I (Brans, Vincke & Mareschal, 1986)

One thing changes: how the flows are combined. The cells, the scale, the preference function and the weighting are identical to base PROMETHEE.

Result and defuzzification. Base PROMETHEE in DecisionMind subtracts the outgoing flow (φ⁻) from the incoming flow (φ⁺) to produce a single net flow (φ) and builds a complete ranking on this number. PROMETHEE I does not perform this final subtraction; it leaves φ⁺ and φ⁻ separate. The relation between two alternatives then takes one of three forms. If alternative A is both higher on φ⁺ and lower on φ⁻ than B (or equal on one and strictly better on the other), A dominates B (P). If both their φ⁺ and φ⁻ are equal, the two are indifferent (I). If φ⁺ favours one and φ⁻ favours the other, that is, one leaves most rivals behind while at the same time being left behind by several rivals, the two alternatives are declared incomparable (R) and are not positioned relative to each other in the ranking.

An important point: incomparability cannot be understood merely by looking at the direct pairwise preference index (π) between two alternatives. φ⁺ and φ⁻ reflect an alternative's average standing against EVERYONE IN THE SET; because of this, even if A beats B in a direct pairwise comparison, their relationships with a third alternative can make A's and B's total flows conflict, and the two can still turn out incomparable. The second case below demonstrates this with figures.

DecisionMind takes the choice of preference function, the threshold values (p, q, s) and the weights from the user in this extension; all of these are adjustable from the interface just as in crisp PROMETHEE. Only the output form of the fifth step changes.

The output is not a ranking but a table built from P (dominates), I (indifferent) and R (incomparable) relations between alternatives; these relations can be organised into layers (who strictly leaves whom behind), but more than one alternative can sit side by side within a layer. Unlike the net flow (PROMETHEE II), PROMETHEE I does not always answer the question "which is second, which is third"; for some pairs it says "I cannot say this with confidence given the information I have."

Thus instead of writing:

"According to PROMETHEE I, A is second and B is third"

when A and B come out incomparable, the report should read:

"For the pair A and B, φ⁺ and φ⁻ give conflicting signals; A leaves most rivals further behind, but is at the same time left behind by more rivals than B. The method cannot confidently establish dominance between these two alternatives with the information it has; the choice between them should be made on another criterion (such as cost or risk appetite)"

Use this when the decision needs to be a genuine, honest partial relation rather than a strict complete ranking, that is, when leaving some pairs "incomparable" is acceptable, or even preferred. This is particularly valuable when the decision maker wants to see a conflict that the net flow might conceal: two alternatives can come out close on the net flow, but PROMETHEE I distinguishes whether this is "genuinely close" or "actually incomparable."

If a complete ranking is required, that is, if every alternative must be placed somewhere, base PROMETHEE (net flow, PROMETHEE II logic) should be used; PROMETHEE I gives an incomplete answer in that case. Crisp PROMETHEE's exit conditions apply here too: once the number of alternatives exceeds fifty, the pairwise-comparison burden grows large.

Forcing incomparable pairs into a ranking. A decision maker who says "but I need a ranking" and turns an R relation into I (indifferent) or an arbitrary P direction destroys the method's most honest contribution, namely leaving the unknown as unknown. If a ranking is required, PROMETHEE II should be used.

Trying to read incomparability directly from the pairwise preference index (π). As shown above, even if A beats B directly with π(A,B) > π(B,A), the total φ⁺/φ⁻ flows can come out conflicting and render the two incomparable. Incomparability is read only from φ⁺ and φ⁻.

Mistaking the layer order for a complete ranking. If more than one alternative appears within a layer, this does not mean those alternatives are equal; it means they are either incomparable with each other or indifferent. Ranking within a layer requires additional information (for instance, the net flow).

Leaving the preference function and thresholds at their defaults without ever questioning them. This is the general error of crisp PROMETHEE and applies here too: changing a threshold can change both the ranking and which pairs come out incomparable.

The governing principle is this:

PROMETHEE I's contribution is to stay silent wherever it cannot confidently rank with the information it has; forcing this silence into a ranking removes the method's honesty.