Heronian Mean (HM)

The Heronian mean is an aggregation building block from the same family as the Bonferroni mean. It is not an independent decision method in its own right; it is an aggregation step that stands in for WAM when criteria are assumed to interact with one another. It differs from the Bonferroni mean in that it accounts not only for pairs of different criteria but also for each criterion paired with itself (its own square). Its output is a single number on the same scale as the inputs.

Its name comes from Hero of Alexandria, the antiquity-era mathematician who computed a triangle's area from its side lengths; the classical Heron mean is defined for two numbers and sits somewhere between the arithmetic and the geometric mean. The work that carried this classical definition into multi-criteria decision-making and set out its general form with parameters p and q is Yu's 2013 article; Yu applied the Heronian mean to intuitionistic fuzzy numbers to define the IFN-HM operator.

The Heronian mean's philosophy shares its root with the Bonferroni mean's: an interaction exists between criteria, and this interaction can show up not only in pairs of different criteria but also in a criterion's own magnitude. The pair a criterion forms with itself is its own square; this term feeds directly into the average how strong or weak that criterion is on its own. Where the Bonferroni mean leaves these self-paired terms out, the Heronian mean brings them in.

This small difference has an important consequence: the Heronian mean allows a very high criterion score to make a strong contribution to the average through its own square, so it is slightly more tolerant of high extremes than the Bonferroni mean, while remaining just as sensitive to low ones. Both operators say "criteria are not independent," but where the Bonferroni mean rejects that independence, the Heronian mean also accepts a criterion's own magnitude as a signal.

The Heronian mean requires four inputs: non-negative criterion scores, two parameters (p and q), and optional weights.

First, verifying the inputs. It is checked that the criterion scores are non-negative, that there are at least two criteria, and that the parameters p and q are non-negative.

Second, taking the unweighted Heronian mean. For every ordered pair of criteria, including a criterion paired with itself (i less than or equal to j), one criterion's score is raised to the power p and multiplied by the other's score raised to the power q. These products are summed with suitable scaling, and the (p plus q)-th root of this sum is then taken.

Third, moving to the weighted form. Where criteria carry different importance, the simple sum of the second step is replaced by a sum weighted by the product of each pair's weights.

Fourth, ranking. This procedure is repeated for every alternative, and alternatives are ranked by their resulting score from the highest 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 Heronian mean produces is an overall performance figure that reflects both the interaction between criteria and each criterion's own magnitude together. It produces a number close to, but different from, the Bonferroni mean; this difference comes from whether the self-paired terms (a criterion's own square) are included.

A notable feature of this number is that, unlike the Bonferroni mean, it gives slightly more weight to a high criterion score. On the same dataset, the Heronian mean can come out slightly higher than the Bonferroni mean, because the self-squared terms are generally larger than the cross-products.

Thus instead of writing:

"The Heronian mean came out higher than the Bonferroni mean, so it is a more accurate result"

the report should read:

"Because it also accounts for each criterion's own magnitude, the Heronian mean produces a number that differs from, and is generally slightly higher than, the Bonferroni mean; what matters is not which one is correct but whose assumption fits your decision problem"

The Heronian 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 parameters p and q (taking both equal to 1 is the most common starting point); and, optionally, weights that sum to 1. The choice of p and q, as with the Bonferroni mean, comes from the analyst's understanding of risk and should be stated in the report.

The Heronian mean is a sound choice if an interaction exists between criteria and, alongside that interaction, each criterion's own magnitude is also to be treated as valuable in its own right. Its field of use lies very close to the Bonferroni mean's; the choice between the two usually turns on whether or not you want the self-square (a criterion paired with itself) taken into account.

If criteria are independent, or there is no justification supporting an assumption of interaction, the Heronian mean adds needless complexity; WAM is sufficient. If the self-square is not wanted, that is, if only the interaction between different criteria is of interest, the Bonferroni mean is the purer choice.

Interaction exists between criteria, and a criterion's own magnitude is also to be valued → Heronian mean

Only the interaction between different criteria is of interest, the self-square term is not wanted → Bonferroni mean

Criteria are independent, no added complexity is needed → WAM

Not interaction, but the extreme lowness of a single criterion should be penalised → WGM or WHM

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

The Heronian mean's greatest strength is that, in addition to the interaction information the Bonferroni mean captures, it also brings each criterion's own magnitude into the average; this gives the operator a somewhat richer information base. As Yu has shown, this feature allows the operator to be tuned more flexibly in settings of high uncertainty, such as intuitionistic fuzzy environments (Yu, 2013). The Heronian mean also shares core mathematical properties, such as monotonicity and boundary conditions, with the Bonferroni mean, which makes it a reliable building block.

The Heronian mean's weakness is that it carries nearly the same interpretive difficulty as the Bonferroni mean; explaining to a decision-maker why the self-square term is also included takes extra effort. Moreover, the two operators (Bonferroni and Heronian) sit so close to one another that which one is chosen often goes unjustified, which weakens the report's scientific transparency. As seen in the work of Wei, Lu and Gao adapting it to various fuzzy settings, the operator becomes harder to interpret as the number of parameters grows, when extra layers such as picture fuzzy or intuitionistic fuzzy are added (Wei, Lu and Gao, 2018).

The most common mistake is offering no justification when choosing between the Heronian mean and the Bonferroni mean; although the two give very close results, their assumptions differ, and the reason for the choice should be stated in the report. A second mistake is leaving the p and q parameters at their default values without ever discussing this choice. A third is presenting the Heronian mean coming out higher than the Bonferroni mean as "a better result"; the two numbers are products of different assumptions, and neither is superior to the other.

The governing principle is this:

The Heronian mean is an interpretation very close to the Bonferroni mean but one that also accounts for a criterion's own magnitude; which one is chosen, and why, must be clearly stated in the report.