FUCA (Flexible and Universal Compromise Analysis)

FUCA is a method that, once you hold a decision table, ranks alternatives into a single order without normalising them, using only their rank on each criterion. On every criterion, alternatives are ranked starting from 1; 1 shows the best alternative on that criterion. These ranks are then multiplied by the criterion weight and summed. The output is a score for every alternative (the weighted sum of its ranks) and a rank produced by that score; the smallest score shows the best alternative. FUCA takes its name from the French phrase "Faire Un Choix Adéquat" (to make a suitable choice) and is one of the few ranking methods that contains no normalisation step.

Most ranking methods first bring raw figures onto the same scale, then compare them. FUCA does the reverse: it is not interested in a figure's own value but in its position relative to the other alternatives. Knowing who finished first and who finished second in a race is simpler than knowing how many seconds each took, and it is independent of the unit of measurement. FUCA likewise ranks alternatives on every criterion and then sums these ranks. The philosophical consequence of this approach is that it avoids entirely the debate over which form of normalisation to use. But precisely because only rank is used, a large difference between two alternatives and a small one count for the same weight.

The method proceeds through two steps.

First, ranking on every criterion. FUCA ranks every criterion column on its own: for a benefit criterion the highest value takes rank 1, for a cost criterion the lowest value takes rank 1. Tied values share the average rank. For instance, if two alternatives are tied on a criterion and would otherwise share rank 2 and rank 3, each is given a rank of 2.5.

Second, calculating the weighted rank sum. FUCA multiplies every alternative's rank on each criterion by that criterion's weight and sums these products. The result is the alternative's overall score; a small score shows a good rank, a large score a poor one. Alternatives are ranked by this score from smallest to largest.

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

The FUCA score shows an alternative's average rank relative to the others under these criteria and weights; it says nothing more. A score of 1.7 does not mean "17 per cent" or "70 per cent short of first place"; it is a weighted average of ranks. The closer the score is to 1, the better the alternative's rank on more criteria. But the score gap between two alternatives does not reflect the gap between their raw values, because FUCA uses ranks, not raw values. Between an alternative that is a clear first on a criterion and one that is second on that same criterion, there is no difference in FUCA score, even though the real figures may be far apart.

Thus instead of writing:

"FUCA found the best alternative"

the report should read:

"With these criteria and weights, this is the alternative with the best average rank; the size of the gap between the raw values is not reflected in this score"

Classical FUCA works with crisp data: a single number in every cell that can be ranked. As of this writing, DecisionMind has no registered FUCA extension alongside the base method.

You need: alternatives in rows, criteria in columns, a number in every cell; direction information for every criterion (benefit or cost); and criterion weights summing to 1. FUCA does not produce weights, it takes them from outside. A minimum of two alternatives and two criteria is required; the computational burden does not grow as the number of alternatives and criteria grows, because FUCA only performs a ranking operation.

If your criteria are in different units and you are wary of a normalisation form affecting the result, and rank information alone is sufficient, FUCA is a suitable choice. Its typical fields are supplier assessment and alternative selection; because its computational burden is low, it also works comfortably with a large number of alternatives.

The situation where it should not be used is one where the magnitude of the gap between raw values matters to the decision; because FUCA uses only rank, it does not distinguish "first by a narrow margin" from "first by a wide margin." If a strong dependency exists between criteria, this needs handling first; FUCA treats criteria as independent.

Avoiding the normalisation debate, rank information alone suffices → FUCA

The magnitude of the raw value matters to the decision → TOPSIS, SAW and other methods that use raw values

Weights, not a ranking, are needed → AHP, BWM, SWARA (subjective), Entropy, CRITIC (objective)

No compromise is acceptable on one criterion → methods that work on elimination logic

FUCA's greatest strength is its simplicity. Because there is no normalisation step, the debate over different normalisation forms changing the result does not apply to FUCA. It can be calculated by hand. Its sensitivity to an outlier (an unusually high or low figure on one criterion) is limited, because an outlier converted to a rank carries only the weight of a single rank. This feature makes FUCA less fragile than other methods in situations where the data is suspected of containing measurement error.

Its limitations stem from this same simplicity. First, converting to rank causes information loss; a large numerical gap between two alternatives and a small one are both reduced to the same rank difference. Second, as the number of alternatives grows, the step between ranks shrinks and FUCA scores draw closer together; this reduces discriminating power. Third, when a new alternative is added, every rank is recalculated and the existing order can change. Fourth, the quality of the weights lies outside the method; a correct rank calculation built on poor weights can still give the wrong priority order.

The most common mistake is reading the FUCA score as a percentage or a distance measure; the score is only a weighted sum of ranks. A second mistake is giving tied values different ranks instead of applying the average-rank rule; this artificially distorts that criterion's discriminating power. A third is using FUCA with very few alternatives (two or three) and presenting the result as a definitive choice; rank gaps with few alternatives are sensitive to small data changes. A fourth is expecting a large gap between raw values to show up in the FUCA score; it does not, because the method works with rank.

The governing principle is this:

A FUCA result is the weighted sum of the alternatives' ranks on each criterion, not their raw values; if the magnitude of the gap between raw values matters to the decision, this information must be reported separately.