Choquet Integral

Classical aggregation methods, such as the weighted average, give every criterion a fixed weight and sum these weights; they cannot see how criteria affect one another. The Choquet integral instead assigns an "importance measure" (a fuzzy measure) not only to individual criteria but to groups of criteria. This way, if two criteria complement each other (becoming more valuable together than either is alone) or overlap with each other (repeating the same information), that relationship is taken into account. Its output is a single aggregated value for every alternative; the classical weighted average is a special case of this value. Building on Michel Sugeno's theory of fuzzy measures, the Choquet integral was adapted to decision analysis by Toshiaki Murofushi and Sugeno in 1989.

The idea behind the Choquet integral is not to ignore situations where criteria are not independent. If two safety measures applied together provide more security than either does alone, these two criteria are synergistic, and their combined weight should be counted as greater than the sum of their individual weights. If two cost items measure the same budget constraint from different angles, these two criteria are redundant, and their combined weight should be counted as smaller than the sum of their individual weights. The Choquet integral sums the "increase in the importance measure," starting from the criterion on which the alternative performs worst and proceeding in order towards the better criteria. The philosophical consequence is that the Choquet integral is compensatory, but it carries out that compensation not on the assumption that criteria are independent, but through the genuine importance of groups of criteria.

The method proceeds through four steps.

First, set the individual importance values. A separate importance value (density) is set for every criterion; these values can come from expert opinion or from the data.

Second, solve for the interaction coefficient. If the sum of the criterion densities is less than 1, there is synergy between the criteria; if it is greater than 1, there is redundancy; if it equals exactly 1, there is no interaction at all. This sum is resolved through a single interaction coefficient that determines the importance measure of every group of criteria.

Third, build the importance measure for every group of criteria. Once the interaction coefficient is found, the importance measure is computed for every group, from individual criteria through pairs and triples up to the full set formed by all the criteria; the importance measure of the full set is always 1.

Fourth, take the integral. An alternative's criterion values are ordered from worst to best. The worst value is multiplied by the importance measure of the full set formed by all the criteria (that is, by 1); the difference to the next value is multiplied by the importance measure of the group formed by the remaining (better) criteria; this continues up to the best criterion, and all these products are summed to give the final value.

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

The Choquet integral value shows an alternative's aggregate performance once the interactions within its groups of criteria are also taken into account; it is not a percentage or a probability, and it is meaningful only for this set of alternatives, these criteria and this importance measure. Because of the structure of the fourth step, the criterion on which an alternative is WEAKEST is always multiplied by the importance measure of the full set (the highest weight); this means the method can respond to a weakness on any given criterion more sensitively than the classical weighted average. Defining synergy between criteria does not mean "the alternative that is strong on the synergistic criteria will always score higher"; the result also depends on which criterion the alternative is weak on.

Thus instead of writing:

"Because synergy has been defined between these criteria, the alternative that is strong on these synergistic criteria will always come out ahead in the Choquet integral"

the report should read:

"A definition of synergy or redundancy pushes the result in a certain direction, but does not guarantee it on its own; which criterion an alternative is weakest on also changes the result, so the outcome should be reported alongside a comparison with the classical weighted average"

The Choquet integral works with crisp data: every cell of the decision table is a single number, and criterion values scaled (normalised) to between 0 and 1 are generally used. DecisionMind currently holds no fuzzy or other-data-type extension of this method; it works only in its base form. You need: alternatives in the rows, criteria in the columns, an individual importance value (density) for every criterion, and these values set through expert opinion in a way that reflects the synergy or redundancy between the criteria. If the individual importance values sum to exactly 1, the method reduces to the classical weighted average and no interaction is captured at all; this is a point often overlooked when setting up the Choquet integral. At least two criteria are required; as the number of criteria grows, the number of groups of criteria multiplies rapidly, and setting the importance values becomes harder.

The Choquet integral is suitable if a known synergy or redundancy relationship exists between your criteria (two criteria reinforce each other, or repeat the same information) and you want that relationship reflected in the aggregation. If your criteria are genuinely independent, or there is no expert opinion available to determine the interaction between them reliably, the Choquet integral adds needless complexity; in that case the classical weighted average is both simpler and rests on fewer assumptions. Where the sum of the individual importance values is very close to 1, the Choquet integral gives a result almost indistinguishable from the classical weighted average; in that case the method's added complexity yields no benefit.

A known synergy or redundancy exists between criteria and must be taken into account → Choquet Integral

Criteria are treated as independent, no interaction information exists → the classical weighted average, or independence-assuming methods such as TOPSIS

Interaction exists only for certain pairs of criteria, not in the same direction throughout → a single-coefficient Choquet integral falls short, a more detailed fuzzy measure is needed

The sum of the individual importance values is very close to 1 → the difference from the weighted average will be small, the added complexity needs justifying

The Choquet integral's greatest advantage is that it can explicitly model interaction between criteria; it accounts for synergy and redundancy relationships that the classical weighted average cannot see. Because a single interaction coefficient determines the importance measure of every group of criteria, there is no need to set a separate value for every group; this requires far fewer parameters than a fully general (Möbius-based) fuzzy measure. When the individual importance values sum to 1, it reduces exactly to the classical weighted average; this makes the method a natural generalisation of the weighted average (Grabisch, 1996).

Its limitations follow from this same single-coefficient structure. A single interaction coefficient imposes the same direction of relationship (all synergistic, or all redundant) across every pair of criteria; it cannot separately capture a case where one pair is synergistic and another pair is redundant, and doing so requires moving to general fuzzy measures that need far more parameters (Grabisch and Labreuche, 2010). Getting the individual importance values and the direction of interaction right through expert opinion is difficult; a wrongly set density can shift the result unexpectedly. Because of the method's structure, an alternative's weakest criterion is always multiplied by the highest weight (the importance measure of the full set), so the result cannot simply be read intuitively as "it rewards synergy"; a correct interpretation requires comparison with the weighted average.

The most common mistake is expecting the Choquet integral to give a different result from the weighted average in a setup where the individual importance values sum to exactly 1; when the sum is exactly 1, the interaction coefficient comes out zero and the method reduces to the classical weighted average, capturing no synergy or redundancy at all. A second mistake is assuming that an alternative strong on criteria with defined synergy will automatically score higher; the result also depends on which criterion that alternative is weakest on, and the direction may not always be the same. A third mistake is trying to define relationships in different directions between different pairs of criteria (some synergistic, some redundant) using a single interaction coefficient; this is something the method's single-coefficient setup cannot do.

The governing principle is this:

The Choquet integral makes interaction between criteria visible, but reading the result simply as "synergy wins, redundancy loses" is wrong; the result must always be interpreted alongside a comparison with the weighted average and with attention to the effect of the weakest criterion.