EXPROM II (Extended PROMETHEE, full ranking)

EXPROM II is an extension of the PROMETHEE family and shares the same computational mechanics as EXPROM I. Given a decision table filled with numbers, it is an outranking method that evaluates alternatives through pairwise comparisons. Its difference from EXPROM I is that, rather than reporting the incomparable pairs found at an intermediate step, it reduces every alternative to a single net-flow score and gives a complete, top-to-bottom ranking. Diakoulaki and Koumoutsos proposed it in 1991, in response to the fact that classical PROMETHEE II holds small and large differences to the same single measure.

The idea behind EXPROM II is the same as EXPROM I's: a small difference forms a weak preference, a very large one a strong preference, and the two are taken into account separately. What creates the difference is what is done with this information. EXPROM I leaves pairs with insufficient evidence incomparable, whereas EXPROM II reduces every alternative's positive and negative flow to a single net figure and assigns each alternative a rank number based on that figure.

This is a philosophical shift. EXPROM I says "do not impose a ranking where the evidence is insufficient"; EXPROM II says "if the decision requires a full order, net flow provides it, but do not forget that some pairs beneath that net flow may in fact be incomparable". EXPROM II is compensatory: in the net flow, a weakness in one pairwise comparison is numerically balanced by strength in another; the full ranking is a consequence of this balancing, not proof of an absolute outranking.

The method proceeds through seven steps; the first six are identical to EXPROM I's, only the final step differs.

First and second step, measuring weak preference. For every criterion and every pairwise comparison, a difference is calculated. Small differences (below the indifference threshold) are disregarded, large differences (above the preference threshold) are counted as full preference, and values in between are proportional. If no threshold is given, every difference above zero is counted as full preference. These degrees are multiplied by the weights and summed to give the weak-preference index.

Third and fourth step, measuring strong preference. The same difference is evaluated again on a separate scale starting from a higher threshold, reaching its full value at a veto threshold. This layer comes into play only when thresholds are defined. Strong-preference degrees are also multiplied by the weights and summed.

Fifth step, combining the flows. For every alternative, the weak- and strong-preference indices against the others are summed and averaged, giving the positive flow; the reverse direction gives the negative flow.

Sixth step, net flow. The negative flow is subtracted from the positive flow.

Seventh step, full ranking. EXPROM II does not apply EXPROM I's partial-comparison rule; it ranks alternatives directly from highest to lowest net flow. Every alternative receives a rank number; there is no pair left incomparable.

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 alternative with the highest net flow ranks first; but this score is the average of that alternative's weak and strong preferences across every pairwise comparison, not an absolute superiority. Alternatives with a net flow close to zero sit in an average position: they are superior to some alternatives and behind others, and the two effects have balanced each other out.

EXPROM II giving a full ranking does not mean every underlying comparison is clear. A small net-flow gap between two alternatives may be the result of forcing, in EXPROM II, a pair that could remain incomparable in EXPROM I into one direction.

Thus instead of writing:

"EXPROM II found the best alternative"

the report should read:

"With these thresholds, the alternative with the highest net flow is this one; where the net-flow gap between neighbouring ranks is small, the underlying comparison may not be clear-cut, and this should be checked together with EXPROM I"

Classical EXPROM II works with crisp data: one number per cell. DecisionMind holds only the base EXPROM II; it has no extensions.

You need alternatives in rows, criteria in columns, one number per cell with no empty cells; direction information for every criterion; and criterion weights summing to 1. EXPROM II does not produce weights, it asks for them. The indifference, preference and veto thresholds are optional; if not given, the method treats every difference above zero as a full preference and the strong-preference layer stays inactive. At least two alternatives and two criteria are required; it works comfortably with up to thirty alternatives.

If your criteria are numerical, you want to weigh small and very large differences separately, and your decision process requires a firm rank number for every alternative (for example, when a priority list must be produced), EXPROM II is a suitable choice. Its typical fields are the same multi-stakeholder decisions as EXPROM I's, but in situations that require presenting a final list.

The situation in which it should not be used is one where pairs of alternatives with insufficient evidence must not be concealed; EXPROM II rounds every pair to one direction and does not, by itself, show the uncertainty beneath that rounding. In such a situation, EXPROM I's partial ranking should be preferred, so as to see which pairs are genuinely clear-cut.

A full order is needed, small and large differences should be distinguished → EXPROM II

Pairs with insufficient evidence must not be concealed, a full order is not essential → EXPROM I

No expert available to set thresholds, a simple additive calculation is enough → WSM, TOPSIS

Positioning against a reference point (ideal/anti-ideal) is wanted → TOPSIS

EXPROM II's most important strength is that it preserves EXPROM I's enriched preference information (the distinction between weak and strong preference) while always giving the decision-maker a usable, applicable full ranking. This is a practical advantage for decisions that require a full order, such as a priority list or a resource-allocation sequence. Because it rests on pairwise comparison, it can bound compensation between criteria with thresholds, more than classical additive methods can.

Its limitations stem from the cost of giving a full ranking. Pairs where the evidence is genuinely insufficient are also rounded to one direction; this rounding can rest on a small numerical gap in the net flow, and if the report does not show this, the order looks more certain than it is. Setting the thresholds requires expert opinion; if no threshold is given, the strong-preference layer never comes into play and EXPROM II effectively approaches classical PROMETHEE II. As a general limitation of the PROMETHEE family, adding a new candidate to the alternative set can change the flows and hence the order (Behzadian et al., 2010). As Roy (1991) emphasises, forcing every pairwise comparison to one direction pushes into the background, within the full ranking, the principle that is outranking's core contribution: making uncertainty visible.

The most common mistake is presenting the full ranking EXPROM II gives as though every underlying pairwise comparison is clear-cut. A second mistake is running the method without ever setting the thresholds and then reporting the result as "this also covers strong preference". A third is treating a small net-flow gap between neighbouring ranks as a firm outranking; in that case EXPROM I should be checked to see whether this pair remains incomparable. A fourth is adding a new alternative once the analysis is finished and being surprised that the order changes.

The governing principle is this:

The full ranking EXPROM II gives is net flow arranged from highest to lowest; where the gap between neighbouring ranks is small, this order should be read together with EXPROM I's partial comparison.