PROMETHEE II (Preference Ranking Organisation Method for Enrichment of Evaluations II)

PROMETHEE II is an outranking method for when you already hold a decision table, ranking the alternatives from first to last. Its output, for every alternative, is a positive flow, a negative flow and their difference, the net flow (Φ, between minus one and plus one); the ranking is built from this flow. Unlike TOPSIS and VIKOR, it does not construct an "ideal alternative"; it compares alternatives directly against one another, in pairs. The method was proposed by Brans and Vincke in 1985. Its full-ranking form, PROMETHEE II, was set out clearly in Brans, Vincke and Mareschal's 1986 article. It is one of the most heavily cited outranking methods in the multi-criteria decision literature, applied widely in areas such as supplier selection, project ranking, environmental assessment and energy planning. It does not produce weights; it takes them from outside.

PROMETHEE II's starting point is this: whether an alternative is "good" is measured not by how close it sits to a hypothetical ideal, but by how far it outranks its real rivals. The method takes every pair of alternatives and, on each criterion, filters the difference between them through a "preference function." A small difference can be treated as negligible, a large one can turn into a full preference; the shape of this filter is chosen per criterion. These single-criterion preferences are combined with the weights to build an "overall preference degree" for every pair. The extent to which each alternative outranks the others is then summed; this is called the positive flow. The extent to which it is outranked by the others is also summed; this is called the negative flow.

This structure has a consequence: PROMETHEE II is partly compensatory, but not to the same degree as TOPSIS. Strength on one criterion can offset weakness on another, because all the preferences are combined into a single flow. But the choice of preference function also allows small differences to be discounted entirely; the method can therefore also embody the intuition that "noisy small differences are not a genuine preference." The result is a ranking that rests not on a single ideal point but on the whole set of pairwise comparisons among the alternatives.

The method proceeds through five steps.

First, pairwise differences. For every criterion, the difference is calculated for every pair of alternatives (a,b). For a "higher is better" criterion, b's value is subtracted from a's; for a "lower is better" criterion, the reverse is done. A positive difference thus always means "a is better than b on this criterion."

Second, the preference function. This difference is converted into a degree between 0 and 1 by a preference function chosen for that criterion. DecisionMind asks you to choose one of six classical forms. The simplest is the usual (absolute) type, in which even the smallest positive difference counts as a full preference. Threshold forms treat anything below a given difference as "no difference" and anything above another given difference as "full preference," rising linearly in between. The Gaussian (smooth) form increases the effect of the difference gradually. The choice depends on how precisely the criterion is measured. For a sharply measured criterion, price, say, the usual function counts every difference. For subjective or noisy scoring, a threshold function treats small differences as meaningless.

Third, the overall preference index. The preference degree on each criterion is multiplied by that criterion's weight and summed. This gives, for every pair of alternatives, a single figure: how much a is preferred to b.

Fourth, positive and negative flow. An alternative's average preference degree against every other alternative gives its positive flow (Φ⁺, how far it outranks the others). The other alternatives' average preference degree against it gives its negative flow (Φ⁻, how far it is outranked).

Fifth, net flow and ranking. The net flow is the negative flow subtracted from the positive flow (Φ = Φ⁺ − Φ⁻). Alternatives are ranked from largest to smallest; this is PROMETHEE II's full ranking (PROMETHEE I gives only a partial outranking relation; DecisionMind produces the full ranking here).

The formulas behind each step, the intermediate tables and the citation formats are given on the DecisionMind method page; this card carries no formulas.

The net flow Φ is a value between −1 and +1; a larger value is better. Φ = +1 means an alternative fully outranks every other; Φ = −1 means the opposite. The net flow should be read together with the positive and negative flows, not on its own. A high Φ⁺ shows that an alternative outranks most of its rivals; a low Φ⁻ shows that it is outranked by few of them. Whether an alternative comes out ahead because it "wins a lot" or because it "loses little" should be stated separately in the report. The net flow cannot be compared with the result of a different analysis, the way TOPSIS's closeness score cannot; it is meaningful only for this alternative set and these preference functions.

Thus instead of writing:

"According to PROMETHEE II, A3 is the best alternative"

the report should read:

"With the preference functions and weights chosen, A3 is the alternative most clearly preferred in the pairwise comparisons (Φ = 0.20); this result is sensitive to the type of preference function and to the weights"

PROMETHEE II works with crisp data, one number per cell. Unlike other outranking methods, the criterion scale need not be only ratio or interval; it may also be ordinal. You need alternatives in rows, criteria in columns, one number per cell; direction information for every criterion; weights summing to 1; and a preference-function type for every criterion. If a function type is not specified, DecisionMind assumes the usual (absolute) type. Threshold values (p, q) are needed for threshold functions, and sigma for the Gaussian function. It does not produce weights, it asks for them; they can be drawn from sources such as AHP, BWM, CRITIC or Entropy. DecisionMind holds eighteen PROMETHEE II members alongside the base method. A minimum of two alternatives and two criteria is required. Five to twenty alternatives and three to ten criteria work comfortably. Beyond thirty alternatives, the number of pairwise comparisons grows quadratically and the computational burden becomes noticeable.

PROMETHEE II is a suitable choice if your criteria can be measured numerically or on an ordinal scale, you want a full ranking, and it helps you to define a different "how much of a difference matters" threshold per criterion. Its typical fields are supplier selection, project and investment ranking, environmental assessment, and energy planning.

There are three situations where it should not be used. First, where rank reversal is unacceptable; here, methods closed to reversal, such as SPOTIS or RAFSI, should be considered. Second, where the number of alternatives is very large; beyond fifty, the cost of pairwise comparison rises quickly. Third, where a fully compensatory aggregation is wanted; here TOPSIS or WASPAS is a simpler tool.

Pairwise-comparison logic and a full ranking are wanted → PROMETHEE II

Only a partial outranking relation suffices, a full ranking is not needed → PROMETHEE I

Rank reversal is unacceptable → SPOTIS or RAFSI

Computational burden matters for an alternative set beyond fifty → a lighter ranking method (TOPSIS, SAW)

Not a ranking but elimination or a core set is needed → the ELECTRE family

PROMETHEE II's clear strength is the flexibility of choosing a different preference function per criterion; a sharply measured criterion is not judged with the same rigidity as subjective scoring. Second, it works directly between alternatives without constructing an ideal or anti-ideal point; this makes the result read "relative to real rivals." Third, keeping the positive and negative flows separate makes visible why an alternative comes out ahead, whether it wins a lot or loses little. Fourth, it is one of the most heavily cited outranking methods in the literature, with the widest base of application (Behzadian, Kazemzadeh, Albadvi and Aghdasi, 2010).

Its limitations arise from the same flexibility. First, rank reversal is a known problem: the ranking can change when an alternative is added or removed (Mareschal, De Smet and Nemery, 2008). Second, the choice of preference function and thresholds (p, q) directly affects the result. These choices do not emerge from the data on their own; they require the analyst's or decision-maker's judgement and look arbitrary if left unjustified. Third, the computational burden grows with the square of the number of alternatives (O(m²)), which is costly for large sets. Fourth, criteria are assumed to sit on a comparable scale; if this does not hold, the meaning of the weights is distorted.

The most common mistake is applying the usual (absolute) preference function to every criterion without justification and never asking how treating even the smallest difference as a full preference affects the result. A second mistake is reporting the net flow alone without showing the positive and negative flows; this hides why an alternative came out ahead. A third mistake is adding an alternative once the analysis is finished and being surprised the ranking changes (rank reversal); the alternative set must be fixed before the analysis begins. A fourth mistake is applying the method to a set of more than fifty alternatives without ever considering the computational cost. A fifth mistake is reading the net flow as a percentage, the way TOPSIS's closeness score might be.

The governing principle is this:

The net flow is a product of the preference functions, thresholds and weights chosen; when these choices change, the ranking can change too, and the report must state plainly which functions and which thresholds were used.