WAM (Weighted Arithmetic Mean)

WAM is the most basic way of reducing several criterion scores into a single result figure. It is not an independent decision method in its own right; it is a building block that operates inside many decision methods. This exact operation sits at the heart of simple weighted-sum methods such as SAW: bring criteria onto the same scale, multiply by weights, and sum. The same operation is also used to reduce the scores given by several experts or assessors into a single group score. Its output is a single figure on the same scale as the input scores, and alternatives are ranked by that figure.

The idea of deciding by weighted average traces back to Churchman and Ackoff's 1954 study. They proposed an "approximate measure of value" that broke an investment decision down into criteria and summed each criterion's contribution. This idea resurfaces later as a step inside almost every multicriteria method, from AHP to SAW to TOPSIS; Yager's 1988 paper, in turn, set WAM within a formal framework by comparing it with the rank-based OWA operator as the two ends of the same family.

WAM's philosophy is full compensation. A shortfall on one criterion can be offset by a surplus on another, and this balance is linear. Losing 2 points on one criterion is exactly cancelled by gaining 2 points on another, and as long as the criterion's weight stays fixed there is no curvature between the loss and the gain. This makes WAM the most generous form of compensation within its family.

The family's other members, the geometric mean, the harmonic mean, the power mean, and the Bonferroni and Heronian means, all move away from WAM by bending this linear compensation in one direction or another. WAM is the family's zero point; the other operators are defined relative to it by the question "more generous, or more punishing." If you want a criterion's very low score to have almost no effect on the outcome, WAM is the right choice; if you want a low score to drag the others down as well, another member of the family should be considered.

WAM needs two inputs: the criterion scores and the weights of those criteria. The weights must sum to 1.

First, verifying the weights. It is checked that no weight is negative and that they sum to 1. Working with a weight set that fails to sum to 1 inflates or shrinks the result in a meaningless way.

Second, taking the weighted sum. Each criterion's score is multiplied by its own weight, and these products are then summed. The result is a single figure on the same scale as the inputs; if the scores lie between 0 and 10, the result stays within that range too.

Third, ranking. This operation is repeated for every alternative, and the alternatives are ranked from the highest result figure to the lowest.

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

The figure WAM produces shows an alternative's balanced total performance across criteria; it hides no "secret advantage" on any single criterion independent of the others. A score of 4.0 means the criteria's weighted contributions sum to 4.0 in total; it does not mean that any single criterion on its own deserves a 4.0.

This figure's greatest weakness is that it cannot see balance. An alternative that scores very low on one criterion and very high on another, and an alternative that scores middling on every criterion, are treated as exactly equal by WAM if their weighted sums match. Yet for the decision-maker these two situations may not carry the same level of risk; the unbalanced alternative carries the risk of an unexpected collapse on one criterion. WAM cannot see this difference, because the sum is the only thing it looks at.

Thus instead of writing:

"The alternative with the highest WAM score is the best in every respect"

the report should read:

"With these weights, this is the alternative with the highest total performance; but this score shows only the total, not the balance across criteria"

WAM works with crisp data: a single number for each criterion, and for each alternative a set of weights summing to 1. DecisionMind does not hold a separate fuzzy, grey or intuitionistic extension of this building block; a user who wants to change the data type is directed straight to the aggregation operator suited to that data type (for example, the weighted-average forms used in a fuzzy setting).

You need, for every alternative, criterion scores on the same, comparable scale; and criterion weights that sum to 1. WAM does not produce weights, it takes them from outside. A minimum of two criteria is required; there is no theoretical upper limit on the number of criteria, though WAM generally operates as a single step inside a larger method, such as SAW or a group-decision aggregation.

WAM is a sound choice if the criteria are independent of one another, full compensation between criteria is acceptable, and you do not want an extreme low score on one criterion to collapse the result on its own. Its most common use is as the final summing step of a ranking method such as SAW, and as a way of reducing several experts' scores into a single group score.

Criteria independent, full compensation accepted → WAM

Strong interaction between criteria, "penalise more if two are weak together" wanted → the Bonferroni or Heronian mean

A very low score on one criterion should drive the result close to zero → WGM or WHM

The degree of compensation should be adjustable with a single parameter → the power mean

The goal is to combine votes or rankings, not criterion scores → voting rules such as Borda, Copeland, Condorcet

WAM's greatest strength is its simplicity. Its calculation can be followed even by hand, is easy to explain to anyone, and carries no hidden assumption. The meaning of the weights is direct: if a criterion's weight doubles, its contribution doubles exactly as well. This transparency makes WAM both a usable standalone aggregation tool and a reliable component of larger methods; dozens of methods, SAW foremost among them, use this step as it stands.

WAM's weakness comes from exactly the same source as its strength. Linear compensation ignores interaction between criteria; two criteria being weak at the same time can be riskier than one criterion alone being weak, yet WAM cannot capture this difference (Beliakov et al., 2010). The quality of the weights, which lie outside WAM itself, also determines the result directly; a flawless summing operation built on poor weights still produces a wrong ranking (Triantaphyllou, 2000). If the alternative set is not held fixed, or a criterion's direction (higher is better, lower is better) is marked wrongly, the result becomes entirely meaningless; such fragilities have been examined repeatedly across the wider method family (Wang and Luo, 2009).

The most common mistake is leaving criterion scores on different scales and summing them directly; if a criterion scored 1 to 5 is combined in the same table with one scored 1 to 100, the large-scale criterion dominates automatically. A second mistake is failing to check that the weights sum to 1; if the total exceeds or falls short of 1, every score is artificially inflated or shrunk. A third is reading the WAM result as "superior in every respect" rather than as "a balanced performance summary"; when two alternatives' WAM scores come out equal, they are assumed to be equally reliable, whereas one may be balanced and the other not.

The governing principle is this:

WAM shows only the total, not the balance; if two alternatives' total scores are equal, which one is more balanced must be examined separately.