Methods · Outranking
EXPROM I (Extended PROMETHEE, partial ranking)
EXPROM I evaluates the difference between two alternatives on two levels, "somewhat better" and "much better, almost beyond dispute", and leaves pairs that cannot be clearly compared unranked rather than forcing them into an order.
Base method's data type: Classical
What Is the Method?
EXPROM I is an extension of the PROMETHEE family. Given a decision table filled with numbers, it is an outranking method that evaluates alternatives through pairwise comparisons. Its output is a positive flow, a negative flow and their difference, the net flow, for every alternative; but EXPROM I's real distinguishing output is which pairs of alternatives can be firmly compared and which are left incomparable. Diakoulaki and Koumoutsos proposed it in 1991, in response to the criticism that classical PROMETHEE holds small and large differences to the same single preference degree.
The Philosophy Behind It
The idea behind EXPROM I is to accept that the size of a difference carries information in its own right. If the difference between two alternatives is small, this is a "weak" outranking and must be treated with care. If the difference is very large, almost beyond dispute, this is a "strong" outranking and should weigh more heavily in the decision. Classical PROMETHEE melts the two into a single preference degree; EXPROM I calculates them separately and uses both together.
This idea has one philosophical consequence: EXPROM I does not promise a full ranking. As in Roy's outranking approach, if the evidence between two alternatives is not clear enough (one is superior in one respect, the other in another), the method leaves that pair "incomparable". This is not a shortcoming but honesty; rather than forcing every difference into a single number and a place in the order, it openly shows where the evidence is insufficient.
How It Works
The method proceeds through seven steps.
First and second step, measuring weak preference. For every criterion and every pairwise comparison, a difference is calculated. If this difference is small (below a user-set indifference threshold), it is disregarded. If it is large (above a preference threshold), it is accepted as a full weak preference; values in between take a proportional value. If the thresholds are not specified, EXPROM I treats every difference above zero directly as a full preference. These degrees are multiplied by the criterion weights and summed to give a single weak-preference index for every pair.
Third and fourth step, measuring strong preference. The same difference is evaluated again on a separate scale starting from a higher threshold: once this second threshold is exceeded, strong preference begins to rise, reaching its full value at a veto threshold. This layer comes into play only when the user has defined these thresholds; if not, strong preference stays at zero. 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 all other alternatives are summed and averaged over the number of alternatives; this is the positive flow. Reversing the calculation (the preference indices of the other alternatives relative to this one) gives the negative flow.
Sixth step, net flow. The negative flow is subtracted from the positive flow.
Seventh step, partial ranking. EXPROM I accepts that one alternative outranks another only when it is clearly superior on the positive flow and not behind on the negative flow (or the reverse). If this condition is not met, the two alternatives are 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.
How to Read the Output
The alternative with the highest net flow comes out ahead in the overall tendency, but this is not EXPROM I's real message. The report should show which pairs of alternatives were firmly compared and which were left incomparable. A pair left incomparable does not mean "equal"; it means that, since the direction of the evidence is not clear, the method has not forced a side to be chosen.
An alternative with a high positive flow is generally superior to the others. An alternative with a high negative flow is generally behind the others. An alternative with a moderate positive and negative flow alike sits in a "stable middle" position and is likely to remain incomparable with other alternatives.
Thus instead of writing:
"EXPROM I found the best alternative"
the report should read:
"With these thresholds, the alternative with the highest net flow is this one; some pairs of alternatives, however, could not be firmly compared, and this stems not from insufficient thresholds or data but from the outranking evidence being mixed"
Data Type and Inputs
Classical EXPROM I works with crisp data: one number per cell. DecisionMind holds only the base EXPROM I; 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 I 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, which in effect brings EXPROM I close to "ordinary" classical PROMETHEE. At least two alternatives and two criteria are required; it works comfortably with up to thirty alternatives, and the number of pairwise comparisons grows quickly with larger sets.
When to Use It, When Not To
If your criteria are numerical, you want to weigh small and very large differences separately, and you accept that some pairs of alternatives may end up not firmly comparable, EXPROM I is a suitable choice. Its typical fields are complex decisions with multiple stakeholders and conflicting interests, such as institutional choices where different units hold different priorities.
The situation in which it should not be used is one requiring a firm and complete ranking; EXPROM I can leave some pairs incomparable, and this does not fit a requirement of "give every alternative a rank number". If there is no expert opinion available to set the indifference, preference and veto thresholds, the method's distinguishing layer (strong preference) stays inactive and the extra information EXPROM I contributes is lost.
Small and large differences should be distinguished, a full order is not essential → EXPROM I
Same logic, but every alternative needs a firm rank number → EXPROM II
No expert available to set thresholds, a simple additive calculation is enough → WSM, TOPSIS
Positioning against an ideal and anti-ideal point is wanted → TOPSIS
Strengths
EXPROM I's most important strength is that, by treating small and large differences separately, it carries richer preference information than classical PROMETHEE. Leaving pairs of alternatives with insufficient evidence "incomparable" rather than forcing them into an order does not hide the decision's real uncertainty, it shows it. Because it rests on pairwise comparison, compensation between criteria is not compulsory; a strong preference on one criterion comes into play only to a limited degree, depending on the preference thresholds.
Weaknesses
Its limitations stem from the setting of thresholds. Setting the indifference, preference and veto thresholds requires expert opinion; if these thresholds are not given, the method never uses the strong-preference layer, and the feature that distinguishes EXPROM I stays inactive. As a general limitation of the PROMETHEE family, adding a new candidate to the alternative set can change the pairwise comparisons and hence the flows (Behzadian et al., 2010). As Roy (1991) has noted, incomparability in outranking methods is not a loss of information, but the decision-maker still needs a separate explanation of "why these two alternatives could not be compared"; without this explanation the result can look confusing.
Common Mistakes
The most common mistake is running EXPROM I without ever setting the thresholds and then presenting the result as "we also took strong preference into account"; EXPROM I run without thresholds never uses the strong-preference layer. A second mistake is rounding a pair left incomparable arbitrarily to one side and forcing it into the order by net flow; this hides the honest uncertainty the method has provided. A third is reporting an order with a small net-flow gap as a firm outranking. A fourth is adding a new candidate to the alternative set once the analysis is finished and being surprised that the flows change.
The governing principle is this:
The net-flow order EXPROM I gives is a consequence of the thresholds you set (or did not set); with no threshold, the method performs only an ordinary outranking comparison.
Cases
Each case opens with a decision table, describes in words what the method does to it, and shows how to read the result. The first case is DecisionMind's validation example; the figures are confirmed by an engine run. The other cases are illustrative constructions.
1. Decision Science: DecisionMind's validation example (thresholds absent, ordinary preference)
In this example the indifference, preference and veto thresholds are not specified; EXPROM I therefore treats every difference above zero as a full weak preference, and the strong-preference layer stays inactive. Three alternatives (A1, A2, A3) are evaluated on three criteria (all "higher is better"); the weights are 0.40 for C1 and 0.30 each for C2 and C3.
| Alternative | C1 | C2 | C3 |
|---|---|---|---|
| A1 | 3.0 | 2.0 | 5.0 |
| A2 | 1.0 | 5.0 | 4.0 |
| A3 | 4.0 | 3.0 | 3.0 |
| Direction | higher is better | higher is better | higher is better |
| Weight | 0.40 | 0.30 | 0.30 |
For every pairwise comparison, the method checks which criteria one alternative leads on and sums the weights of those criteria. A1 beats A2 on C1 and C3 (weight sum 0.70); A2 beats A1 only on C2 (0.30). A3 beats A1 on C1 and C2 (0.70); A1 beats A3 only on C3 (0.30). A2 beats A3 on C2 and C3 (0.60); A3 beats A2 only on C1 (0.40).
| Alternative | Positive flow | Negative flow | Net flow | Rank |
|---|---|---|---|---|
| A3 | 0.55 | 0.45 | 0.10 | 1 |
| A1 | 0.50 | 0.50 | 0.00 | 2 |
| A2 | 0.45 | 0.55 | −0.10 | 3 |
The result reads as follows. A3 does not have the single highest value on any criterion, but it beats A1 on C1 and C2; this puts it slightly ahead. A1 sits exactly in the middle, with equal positive and negative flow. A2 is third; it is strong only on C2, but its weakness on C1 outweighs this.
The decision-maker hesitates here: the net-flow gap between A3 and A1 is only 0.10, and this gap comes from a run without thresholds, where every outranking is counted in full. If an indifference threshold were defined and small differences (for instance, moderate-sized differences such as the 5.0 against 3.0 on C3) were counted only partially, the gap between A1 and A3 could narrow or reverse direction.
In the report: "In this run without stated thresholds, A3 has the highest net flow (0.10); the gap to A1 is small and may change once indifference/preference thresholds are defined."
Source: this example, which carries the name of Diakoulaki and Koumoutsos (1991), is not the article's own numerical example. The DecisionMind team is reviewing this card; a note in the manifest states that the example is expected to be replaced by a computable example from the article. The figures shown here are an illustrative example validating the internal consistency of DecisionMind's EXPROM I engine and are not matched to a page number in the article.
2. Public Transport: A metropolitan municipality's choice of bus-fleet renewal technology
A metropolitan municipality will choose among three technologies when renewing its bus fleet: diesel-hybrid, fully electric and hydrogen fuel cell. The criteria are cost per vehicle (lower is better), range and charging/fuelling infrastructure readiness. The municipality wants small cost differences to be treated as unimportant below a certain threshold, but very large cost differences to form a strong preference; it therefore favours EXPROM I's two-layer structure.
The method first disregards small differences in every pairwise comparison and treats large ones as full preference; if a cost difference is very large (exceeding the veto threshold) it turns into strong preference. Suppose the fully electric technology takes the highest net flow, thanks to its low operating cost and moderate range; the hydrogen fuel-cell technology falls behind because it is weak on infrastructure readiness.
The municipality hesitates here: if the net-flow gap between the fully electric and diesel-hybrid technologies is small and the two lead each other on different criteria, these two may remain incomparable. In that case the municipality should consider resolving the incomparability, not the net flow, by adding a further criterion, such as maintenance-network coverage.
In the report: "The fully electric technology has the highest net flow; the criteria on which it remains incomparable with diesel-hybrid should be assessed separately once maintenance-network coverage is added."
3. Mining: Choosing a site-rehabilitation method for a mining operation
A mining company will choose one of three methods to rehabilitate a closed site: natural revegetation, engineering-assisted soil stabilisation, and a mixed method. The criteria are rehabilitation time (lower is better), cost (lower is better) and ecological-recovery level. The company wants to separate very large differences in ecological recovery as strong preference and small differences as weak preference.
The method takes weak preferences into account first in the pairwise comparisons, then strong preferences once the threshold is exceeded. Suppose the mixed method gives by far the best result on ecological recovery, so it is supported by strong preference and takes the highest net flow; natural revegetation falls behind because of its long recovery time despite its low cost.
The company hesitates here: the mixed method's lead through strong preference depends on how the veto threshold was set. Had the threshold been set higher, this advantage could fall back to weak preference and the net-flow gap could narrow. The report should therefore state clearly where the threshold that triggers strong preference comes from.
In the report: "The mixed method has the highest net flow thanks to strong preference in ecological recovery; this advantage is sensitive to the choice of the veto threshold."
4. What Not to Do
Presenting an EXPROM I result run without stated thresholds, as in the validation example, as "we also evaluated strong preference" is wrong; the strong-preference layer never comes into play in a run without thresholds. A second error is reporting the net-flow gap of 0.10 between A3 and A1 as a firm outranking; the gap is small and sensitive to the threshold definition. A third error is arbitrarily rounding a pair left incomparable into the order by net flow and saying "EXPROM I ranked these two as well"; the method has deliberately left this pair incomparable.
Sources
For the formulas behind each step, the intermediate tables and citation formats (BibTeX, RIS, APA), see the DecisionMind method page: decisionmind.app/library/exprom-i
Diakoulaki, D., & Koumoutsos, N. (1991). Cardinal ranking of alternative actions: Extension of the PROMETHEE method. European Journal of Operational Research, 53(3), 337–347. DOI: 10.1016/0377-2217(91)90067-6
Brans, J. P., & Vincke, Ph. (1985). A preference ranking organisation method: The PROMETHEE method for MCDM. Management Science, 31(6), 647–656. DOI: 10.1287/mnsc.31.6.647
Behzadian, M., Kazemzadeh, R. B., Albadvi, A., & Aghdasi, M. (2010). PROMETHEE: A comprehensive literature review on methodologies and applications. European Journal of Operational Research, 200(1), 198–215. DOI: 10.1016/j.ejor.2009.01.021
Roy, B. (1991). The outranking approach and the foundations of ELECTRE methods. Theory and Decision, 31(1), 49–73. DOI: 10.1007/bf00134132