Bonferroni Mean (BM)

The Bonferroni mean is an aggregation building block that treats criteria in pairs rather than one at a time. It is not a stand-alone decision method like WAM; it is an aggregation step that replaces WAM when criteria are assumed to interact, that is, when a weakness on one criterion combines with a weakness on another to make matters worse. The product of every possible pair of criteria is taken, these products are averaged, and the root of the result is taken. Its output is a single number on the same scale as the inputs.

The method's name comes from the Italian mathematician Carlo Bonferroni, who defined sums of pairwise products in 1950; at the time there was no decision-analysis application, only a pure inequality. The paper that carried this definition into multi-criteria decision-making and set out its weighted form is Beliakov and colleagues' 2010 article; shortly afterwards Xu and Yager extended the operator to intuitionistic fuzzy numbers.

The philosophy of the Bonferroni mean is to capture situations where criteria are not independent of one another. WAM sums each criterion oblivious to the others; one criterion being weak and another strong does not change the result, because the contributions simply add up. The Bonferroni mean, by contrast, pairs every criterion with every other criterion and multiplies them; if two criteria are weak at the same time, that pair yields a low product and drags the average down. Where only a single criterion is weak while the others are strong, this effect is more limited, because that weak criterion only affects the pairs that include it.

This places the Bonferroni mean within a view of decision-making that says "risk spreads between criteria." When assessing a system's security, for instance, a single weak point is a concern, but two weak points existing at the same time is a far greater risk; the Bonferroni mean captures this kind of contagion assumption numerically. WAM cannot see this contagion, because it treats criteria as independent of one another.

The Bonferroni mean requires four inputs: criterion scores, two parameters (p and q), and optional weights.

First, validate the inputs. Check that there are at least two criteria, that p and q are non-negative, and that, if weights are used, they sum to 1.

Second, take the unweighted Bonferroni mean. For every ordered pair of criteria (every pairwise combination in which a criterion is not paired with itself), one criterion's p-th power is multiplied by the other's q-th power. These products are averaged, and then the (p plus q)-th root of that average is taken.

Third, move to the weighted form. Where criteria carry different importance, a weighted average, using the product of each pair's weights, replaces the simple average in the second step.

Fourth, rank. This procedure is repeated for every alternative, and the alternatives are ranked from the highest resulting score to the lowest.

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

The number the Bonferroni mean returns is an overall performance figure that reflects how much mutual weakness overlaps between criteria. A low score can indicate that several criteria are weak at the same time; an isolated weakness on a single criterion does not produce such a sharp drop.

The most notable feature of this figure is that it can distinguish situations WAM cannot. Two alternatives with the same average score, one balanced and one unbalanced, are equal according to WAM; the Bonferroni mean separates the two by giving them different scores and places the unbalanced one slightly behind.

Thus instead of writing:

"The Bonferroni score came out low, so this alternative is generally weak"

the report should read:

"The Bonferroni score came out low; this indicates the alternative may be weak on several criteria at once, and which pairs of criteria contributed to this drop should be examined separately"

The Bonferroni mean works with crisp, non-negative data. DecisionMind holds no separate fuzzy, grey or intuitionistic extension of this building block; it works with crisp numbers.

You need: scores for at least two criteria, on the same scale and comparable with one another; a choice for the p and q parameters (setting both equal to 1 is the most common and most interpretable starting point); and, optionally, weights that sum to 1. As p and q grow larger, the operator becomes more sensitive to extreme values; this choice comes not from the data but from the analyst's view of risk, and it must be stated in the report.

The Bonferroni mean is a suitable choice when criteria genuinely interact, that is, when several criteria being weak at once is considered worse than a single criterion being weak. It has a natural place in fields such as security, risk management and quality control, where the view that "weaknesses combining is dangerous" holds.

Where criteria are genuinely independent of one another, or where there is no information to justify a choice of the p and q parameters, the Bonferroni mean adds unnecessary complexity, and WAM is sufficient. It should also not be used if the interpretation cannot easily be explained to the decision-maker, that is, if you are not prepared for the question "why did these two alternatives with the same average receive different scores."

Criteria interact with one another (risk of simultaneous weakness) → Bonferroni mean

Criteria are independent, no extra complexity is needed → WAM

Not interaction but the wish to penalise a single criterion's extreme low value → WGM or WHM

Interaction is wanted not between pairs but among triples or larger groups → Heronian mean or higher-order generalisations

The degree of interaction should be adjustable through a single parameter → Power Mean, though this family does not capture criterion interaction

The Bonferroni mean's chief advantage is that it makes visible a piece of information WAM cannot see: mutual weakness between criteria. As Beliakov and colleagues show, this feature gives the operator sound mathematical properties, preserving monotonicity and boundary conditions (Beliakov et al., 2010). Furthermore, through the p and q parameters, the operator can be tuned across a wide range, from full compensation (as p and q approach zero) to capturing strong interaction; this flexibility, as Xu and Yager show, also carries over to settings with uncertain and subjective assessments (Xu and Yager, 2011).

The Bonferroni mean's weakness is that its interpretation is far harder than WAM's; explaining to a decision-maker why two alternatives with the same average received different scores takes extra effort. Moreover, the choice of the p and q parameters is largely arbitrary; different p, q pairs can produce different rankings from the same data, and this choice usually comes not from the data but from the analyst's assumption (Beliakov et al., 2010). Finally, as the number of criteria grows, the number of pairs grows rapidly; the computational load is heavier than WAM's, and this performance effect must be borne in mind when working with many criteria.

The most common mistake is leaving the p and q parameters at their default value (usually p = q = 1) without offering any justification; this choice rests on an assumption, not on the data, and must be explained in the report. A second mistake is using the Bonferroni mean when criteria are genuinely independent; in that case the operator adds needless complexity and produces no meaningfully different result from WAM. A third mistake is confusing a small difference between two alternatives' Bonferroni scores with a small difference in the underlying interaction pattern; even where the score gap is small, the criterion pattern beneath it can be very different.

The governing principle is this:

The Bonferroni mean can distinguish two alternatives that WAM shows as equal; but if the result is reported without explaining which pair of criteria the distinction comes from, it remains unintelligible.