PROMETHEE (Preference Ranking Organisation Method for Enrichment of Evaluations)

PROMETHEE is an outranking method for when you already hold a decision table, weighing the alternatives pairwise against each other. Unlike TOPSIS or VIKOR, it does not construct a hypothetical ideal alternative; it compares the real alternatives directly against one another. The family's shared output, for every alternative, is three things: a positive flow showing how far it outranks the others, a negative flow showing how far it is outranked, and a net flow formed from the difference between the two. The method was proposed by Brans and Vincke in 1985. The same team split it, in 1986, into two ranking forms: PROMETHEE I, which gives only a partial outranking relation, and PROMETHEE II, which yields a full ranking based on the net flow. This base PROMETHEE entry in DecisionMind works on net-flow logic and produces a full ranking; it does not produce weights, it takes them from outside. It is one of the most heavily cited outranking families in the multi-criteria decision literature, applied widely in multi-stakeholder fields such as supplier selection, project and investment ranking, environmental assessment, and energy planning.

The PROMETHEE family's starting point is to derive the definition of a "good alternative" not from a hypothetical reference point but from real rivals. The method takes every pair of alternatives and uses the difference on each criterion not in raw form but passed through a preference function. The preference function answers, criterion by criterion, the question "is a difference of this size a genuine preference, or measurement noise." In its simplest form, even the smallest difference counts as a full preference. In threshold forms, anything below a given difference is treated as indifference and anything above another as full preference, rising linearly in between. These filtered preferences are combined with the criterion weights to give, for every pair, a single figure: how much a is preferred to b. Each alternative's degree of outranking the others is then summed; this is called the positive flow. Its degree of being outranked by the others is summed separately; this is called the negative flow.

The family's philosophical divide begins exactly here. The positive and negative flows do not always point to the same ranking. An alternative can outrank most of its rivals, giving it a high positive flow, and at the same time be outranked by several rivals, giving it a high negative flow too. PROMETHEE I keeps these two flows separate and compares only those pairs where both point in the same direction. Where the two conflict, that pair is treated as "incomparable" and is not placed relative to one another in the ranking. This is an honest but incomplete result: the method falls silent where it cannot rank confidently with the information it has. PROMETHEE II resolves this dilemma by combining the positive and negative flows into a single net flow. Every pair becomes comparable, at the cost of losing the "incomparable" option. This base PROMETHEE entry in DecisionMind works on net-flow logic and always produces a full ranking. This means the result is a full ranking rather than a partial outranking, and so shares the same computational logic as PROMETHEE II.

The method proceeds through five steps.

First, choosing the preference function. A preference function, and, where relevant, threshold values (an indifference threshold q, a preference threshold p), are set for every criterion. This choice depends on how precisely the criterion is measured. For a sharply measured criterion, price, say, the simplest function can count every difference. For subjective or noisy scoring, a threshold function can treat small differences as meaningless. Where none is specified, DecisionMind assumes the simplest (absolute) function.

Second, the pairwise preference degree. For every criterion, the difference is calculated for every pair of alternatives; for a "higher is better" criterion, b's value is subtracted from a's, and the reverse for a "lower is better" criterion. This difference is converted into a degree between 0 and 1 by the chosen preference function.

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, 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. The other alternatives' average preference degree against it gives its negative flow.

Fifth, net flow and ranking. The net flow is the negative flow subtracted from the positive flow. In this base PROMETHEE entry, DecisionMind ranks alternatives from largest to smallest net flow; this is PROMETHEE II's full-ranking logic. Where PROMETHEE I's partial outranking relation is wanted, the positive and negative flows' separate rankings are compared, and pairs where they conflict are left incomparable. This is a separate way of reading the result, not the computational step this card describes.

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 is a value between −1 and +1; a larger value is better. It should be read together with the positive and negative flows, not on its own. A high positive flow shows that an alternative outranks most of its rivals; a low negative flow 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 is not the same thing as PROMETHEE I's partial outranking relation: in the net flow every alternative finds a place, whereas in PROMETHEE I some pairs may be deliberately left unranked. The net flow cannot be compared with the result of a different method, 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, A2 is the best alternative"

the report should read:

"With the preference functions and weights chosen, A2 is the alternative most clearly preferred in the pairwise comparisons (Φ = 0.33); this result is sensitive to the type of preference function and to the weights"

PROMETHEE works with crisp data, one number per cell. 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. Where a function type is not specified, DecisionMind assumes the simplest type. Threshold functions also require threshold values. 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 members in this family, including base PROMETHEE and PROMETHEE II, alongside various extensions for different types of uncertainty. A minimum of two alternatives and two criteria is required; three to fifteen alternatives and three to twelve criteria work comfortably. Beyond thirty alternatives, the number of pairwise comparisons grows quadratically and the computational burden becomes noticeable.

PROMETHEE is a suitable choice if your criteria can be measured numerically or on an ordinal scale, pairwise-comparison logic helps your work, and you want 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; it is especially preferred in multi-stakeholder decisions where thresholds are open to debate.

There are three situations where it should not be used. First, where the number of alternatives is very large; beyond fifty, the cost of pairwise comparison rises quickly. Second, where a simple, fully compensatory aggregation suffices; here TOPSIS or WASPAS is a lighter tool. Third, where the decision needs to be not a definitive full ranking but an honest partial relation, that is, where it is not essential that no pair be left "incomparable"; here PROMETHEE I's partial reading, or the ELECTRE family, should be considered.

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

Honestly leaving some pairs "incomparable" is acceptable → PROMETHEE I's partial reading

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

The data is fuzzy, grey or intuitionistic rather than crisp → the relevant extension in the family

The PROMETHEE family'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; the result reads "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, because the family offers both a partial (PROMETHEE I) and a full (PROMETHEE II) reading, it can give the decision-maker two different answers to the question "how confidently can I speak here." This flexibility explains why the family 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; this is costly for large sets. Fourth, because this base PROMETHEE entry produces a full ranking via the net flow, it does not on its own display the honesty PROMETHEE I can offer in saying "these two alternatives are genuinely incomparable"; the user must raise this question separately.

The most common mistake is applying the simplest (absolute) preference function without justification to a continuously scaled criterion (price, duration, say) and treating even small measurement differences as full preferences. This processes measurement noise as though it were a genuine preference. 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. A fourth mistake is setting threshold values (p, q) without ever determining them or consulting anyone; these values require domain-expert judgement. A fifth mistake is reading the net flow as a percentage or a probability.

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.