WHM (Weighted Harmonic Mean)

WHM is the correct classical-statistics way of averaging ratios (speed, efficiency, cost per unit, and the like), and it is used in decision analysis as an aggregation building block. It is not an independent decision method in its own right; it comes into play when an alternative's scores across several criteria need collapsing into a single number in a way that foregrounds weakness on any one of them. Its output is one number, on the same scale as the inputs.

WHM's mathematical basis rests on the classical inequality of means: for positive numbers, the harmonic mean is always less than or equal to the geometric mean, and the geometric mean is in turn less than or equal to the arithmetic mean. This ordering is proved in detail in Bullen, Mitrinović and Vasić's comprehensive 1988 work, which defines WHM as the special case of the power-mean family where p equals minus one.

WHM's philosophy is the most limited form of compensation in the family. Under the arithmetic mean, a loss on one criterion is offset exactly by an equal gain on another; under WHM, a low score grows once its reciprocal is taken, and so pulls the average down disproportionately. As a criterion score shrinks, its reciprocal grows rapidly and drags the WHM result towards it; WHM is thus the mean that "feels the weight of the weakest link" most keenly.

This trait makes WHM even more punishing than WGM. WGM punishes a weakness on one criterion through multiplication; WHM punishes that same weakness further by taking its reciprocal. The classical inequality of means fixes this ordering exactly: with the same scores, WHM is always less than or equal to WGM, and WGM is in turn less than or equal to WAM. Where a weakness on one criterion should almost determine the result outright, for instance where a system as a whole is taken to be only as strong as its weakest component, WHM is the correct tool.

WHM takes two inputs: criterion scores greater than zero and weights that sum to 1.

First, validate the inputs. Check that every criterion score is greater than zero and that the weights sum to 1. A score of zero or below makes taking its reciprocal impossible or meaningless.

Second, average the reciprocals with weights. Take the reciprocal of each criterion score (1 divided by the score), multiply each by its own weight, and sum these.

Third, take the reciprocal of the result. The reciprocal of the sum obtained in the second step is taken; this final number is the WHM result.

Fourth, place it within the power-mean family. WHM is the special case of the power-mean family where r equals minus one; the same scores processed with different values of r produce a result somewhere between WHM and WAM.

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

The number WHM returns shows an overall performance dominated by the weakest point among the criteria. A score of 3.4 means the lowest criterion pulls the result down heavily; it reflects the weight of the most fragile criterion rather than, as under WAM, the criteria's average contribution.

WHM's most striking trait is that it can make a balanced but moderately scored alternative more favourable than one that swings between extremes (very high on one criterion, very low on another). This can be the exact opposite of what WAM would give; WAM fully offsets extreme scores, while WHM punishes that same extremity.

Thus instead of writing:

"The WHM score came out low, so this alternative is weak on every criterion"

the report should read:

"The WHM score came out low; this shows that the weakest of the criteria is pulling the result down strongly, and which criterion drove the fall should be examined separately"

WHM works with crisp, strictly positive data; a criterion score of zero makes taking its reciprocal impossible. DecisionMind holds no separate fuzzy, grey or intuitionistic extension of this building block.

You need criterion scores greater than zero for every alternative, on the same scale and comparable with one another, and criterion weights that sum to 1. WHM does not produce weights, it takes them from outside. A minimum of two criteria is required. WHM suits cases where ratio-type quantities (speed, efficiency, cost per unit) need averaging correctly, or where a system's weakest component is taken to determine the whole.

WHM is a sound choice where criteria are ratio-type quantities, where a weakness on one criterion should strongly affect the result, or where the "a chain is only as strong as its weakest link" view is adopted. Its typical use is correctly averaging ratios such as per-unit efficiency or speed.

WHM should not be used where a criterion score can be zero; the result becomes undefined. Where full compensation is wanted, that is, where a weakness on one criterion is expected to be balanced by another, WAM is the better fit; WHM's punishment is harsher even than WGM's, so where a moderate penalty suffices, WGM should be preferred.

Ratio-type quantities, "the weakest link determines it" view → WHM

One of the criterion scores can be zero → WAM (or a pre-processing step that handles the zero first)

Full compensation wanted → WAM

A moderate penalty suffices, WHM's harshness is excessive → WGM

The degree of penalty should be tunable by a single parameter → Power mean, where r equals minus one is equivalent to WHM

WHM's greatest strength is that it correctly averages ratio-type quantities; for quantities such as speed, efficiency or cost per unit, the arithmetic mean systematically gives the wrong answer, while WHM gives the correct one. WHM also does not hide the weakest point of a system or alternative; on the contrary, it reflects that weakness in the result more strongly than any other mean. This trait makes WHM a natural choice in fields such as safety or quality, where "weakest link" logic applies.

WHM's weakness is its extreme severity. If one criterion score approaches zero, the WHM result shrinks rapidly too, which can create an unrealistic harshness in some applications (Bullen, Mitrinović and Vasić, 1988). WHM is also harder to interpret than even WGM for a decision-maker used to an additive average; "take the reciprocal, average it, then take the reciprocal again" is not an intuitive operation, and is misread if not explained carefully. Finally, like the other members of the family, WHM depends on the quality of the weights; the weights come from outside, and a poorly determined set of weights corrupts the result directly.

The most common mistake is using WHM where one criterion score is zero or very close to it; the result then shrinks meaninglessly or becomes undefined. A second mistake is reading a WHM result on the same scale and in the same sense as a WAM result; the two methods can produce very different numbers from the same inputs, and this gap is not an error but the natural consequence of their different assumptions about compensation. A third mistake is using WHM on quantities that are not ratio-type, such as directly additive cost items, where the arithmetic mean is the more correct choice.

The governing principle is this:

WHM correctly averages ratio-type quantities and does not hide the weakest criterion; but this severity can shrink the result unrealistically once one criterion score approaches zero.