WGM (Weighted Geometric Mean)

WGM is an aggregation operator that multiplies criterion scores rather than adding them. It is not an independent decision method in its own right; it is a building block that operates inside subjective weighting methods such as AHP and inside group decisions. In AHP, WGM is the standard way of collapsing several experts' pairwise-comparison judgements into a single group judgement: each expert's judgement is raised to the power of that expert's weight, and the resulting values are multiplied together. The same operation can also collapse an alternative's scores across several criteria into a single result score. Its output is one number, on the same scale as the inputs.

WGM's mathematical basis rests on Aczél and Saaty's 1983 work. That paper proved that the only consistent way of combining ratio-scale judgements is the geometric mean; the arithmetic mean does not preserve this consistency. This result makes WGM, in effect, the required step in AHP's synthesis of group decisions.

WGM's philosophy is partial compensation, and that compensation is multiplicative. Under the arithmetic mean, a loss on one criterion is cancelled exactly by an equal gain on another; under WGM, a low score enters a product and so pulls the result down proportionally. As one criterion score approaches zero, WGM's result approaches zero too, however high the other criteria run. This places WGM close to a decision philosophy that says "I will not accept total collapse on any criterion."

This trait puts WGM at the opposite pole from WAM. WAM compensates a weakness on one criterion fully with strength on another; WGM limits that compensation and punishes weakness proportionally. The gap between the two widens further once ratio judgements are in play (how many times better one thing is than another): as Aczél and Saaty showed, WGM, not WAM, is the consistent way of combining judgements of that kind.

WGM takes two inputs: positive criterion scores and weights that sum to 1.

First, validate the weights. Check that no weight is negative and that they sum to 1; normalise the weights if necessary.

Second, take the weighted product. Each criterion's score is raised to the power of its own weight, and these values are then multiplied together. For numerical stability, this step is usually computed by taking logarithms, summing them, then exponentiating again; the result is the same.

Third, apply it in a group decision. Where several decision-makers have supplied judgements on the same pairwise-comparison matrix, each cell is combined via WGM using its own decision-maker's weight, yielding a single group matrix.

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 WGM returns reflects a proportional balance across criteria; where one criterion scores very low, that low score does not carry through to the result in proportion to its weight but more harshly, because of the nature of multiplication. A result of 4.0 means the criteria's multiplicative contribution reaches 4.0; it is not a simple sum, as under WAM.

WGM's most striking trait is its extreme sensitivity to a score of zero or close to zero. If one criterion is 0, or very close to 0, the WGM result collapses towards 0 too, however high the other criteria run. A low WGM score for an alternative therefore does not mean that alternative is "poor overall"; it can be an amplified reflection of collapse on a single criterion.

Thus instead of writing:

"The WGM score came out low, so this alternative is weak overall"

the report should read:

"The WGM score came out low; this may be the amplified result of an extremely low value on a single criterion via multiplication, and which criterion drove the fall should be checked separately"

WGM works with crisp, positive data; a criterion score of zero or below renders the product undefined or meaningless. DecisionMind holds no separate fuzzy, grey or intuitionistic extension of this building block.

You need positive criterion scores for every alternative, on the same scale and comparable with one another, and criterion weights that sum to 1. WGM does not produce weights, it takes them from outside. A minimum of two criteria is required. WGM is mathematically better suited than WAM specifically where ratio judgements are involved (how many times better one thing is than another), such as in AHP's pairwise-comparison matrices.

WGM is a sound choice where criterion scores sit on a ratio scale (where "how many times more than the other" is meaningful), where several decision-makers' ratio judgements need combining, or where an extreme low on one criterion should pull the result down sharply. Its most common use is building the group pairwise-comparison matrix in AHP.

WGM should not be used where a criterion score can be zero or very close to it; the result collapses meaninglessly to zero. Where full compensation is wanted, that is, where a weakness on one criterion is expected to be fully balanced by another, WAM is the better fit than WGM.

Ratio judgements, group-decision synthesis → WGM

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

Full compensation wanted → WAM

A penalty for weakness harsher even than WGM's is wanted → WHM

The degree of penalty should be tunable by a single parameter → Power mean, which equals WGM as r approaches zero

WGM's greatest strength is its mathematical consistency with ratio judgements. As Aczél and Saaty proved, where ratio-scale judgements are involved, the geometric mean is the only consistent way to combine them; this makes WGM a necessity rather than an arbitrary preference. WGM also does not hide an extreme weakness on one criterion; unlike WAM, it reflects that weak point visibly in the result. This trait, as Forman and Peniwati showed, makes group decisions in AHP less exposed to individual extremes (Forman and Peniwati, 1998).

Its limitations stem from the same structure. WGM's weakness is its extreme sensitivity to zero. If one criterion score is zero or very close to it, the result approaches zero too, however high the other criteria run; this creates an unwanted harshness in some applications (Saaty and Vargas, 2007). WGM is also harder to interpret than WAM; how "good" a multiplicative result is proves less intuitive than an additive one, and takes more effort to explain to a decision-maker. Finally, like WAM, WGM 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 WGM where one criterion score is zero or very close to it; the result then collapses almost automatically to zero, and the information carried by the other criteria is lost. A second mistake is comparing WGM and WAM results directly and asking "which is correct"; the two carry different assumptions, neither is right or wrong, and which fits the decision question must be chosen beforehand. A third mistake is using WGM outside AHP, in a context where an arithmetic sum is more natural, such as directly additive cost items.

The governing principle is this:

WGM is the mathematically consistent way of combining ratio judgements; but where one criterion score sits close to zero, this consistency comes at the cost of pushing the result to an extreme.