Geometric Consistency Index (GCI)

In methods such as AHP, the decision-maker compares criteria or alternatives two at a time and gives judgements of the form "A is how many times more important than B." These judgements are gathered into a pairwise comparison matrix. Human judgements are not always perfectly consistent: if A is said to be three times as important as B, and B twice as important as C, then A would be expected to come out six times as important as C, yet the decision-maker may have stated four times directly. The geometric consistency index reduces how well all the judgements in the matrix agree with one another to a single number. A value close to zero indicates high consistency; a large value indicates low consistency. Aguarón and Moreno-Jiménez proposed it in 2003 as an alternative to Saaty's classical consistency ratio (CR).

The idea behind this method is to measure consistency not through probability or an eigenvalue, but directly through the geometric relationship among the judgements themselves. If a judgement matrix is perfectly consistent, every judgement should be derivable from the others; the GCI measures the gap between the actual judgement and this "expected" judgement on a multiplicative (logarithmic) scale. A multiplicative scale is chosen because the comparison judgements themselves are multiplicative (three times, four times); an additive difference would be misleading here. The philosophical consequence is this: the method makes no decision and ranks no criteria; it audits the internal consistency of the decision-maker's own judgements and states whether those judgements need to be revisited.

The method proceeds through three steps.

First, the priority vector. For every row of the pairwise comparison matrix, the geometric mean of all the judgements in that row is taken. This gives a number (a priority) representing each criterion's or alternative's relative importance; all the priorities can be scaled so that they sum to one.

Second, the deviation calculation. Every judgement in the matrix is compared with its "expected" value from the priority vector. The logarithm of the ratio between the actual judgement and the expected judgement is taken and squared; this shows how inconsistent that single judgement is. These squares are summed across all judgement pairs and scaled by the matrix size to produce a single number, the GCI.

Third, comparison against the threshold. The calculated GCI is compared against a threshold value that depends on the matrix size. If the GCI falls below the threshold, the matrix is consistent to an acceptable degree; if it exceeds the threshold, the judgement pair causing the greatest deviation is identified and the decision-maker is asked to revisit that judgement.

The formula behind each step is given on the DecisionMind method page; this card carries no formulas.

The closer the GCI value sits to zero, the more consistent the matrix; zero itself is obtained only under mathematically perfect consistency (where every other judgement can be derived exactly from one), something almost never seen in real judgements. The GCI has a threshold that depends on the matrix size; the acceptable threshold for a three-criterion matrix differs from that of a five-criterion matrix, so GCI values should not be compared directly across matrices of different sizes. A GCI falling below the threshold does not mean "the judgements are correct"; it shows only that the judgements do not contradict one another, that they are logically consistent. A decision-maker can give judgements that are entirely wrong yet internally consistent; the GCI does not catch this. For that reason, instead of writing:

"The GCI came out low, so the weights are correct"

it is correct to write:

"The GCI falls below the threshold, so the judgements do not logically contradict one another; the accuracy of the judgements' content is a separate question"

The geometric consistency index works with crisp data; its input is a reciprocal pairwise comparison matrix. The matrix's diagonal must be one, a_ij and a_ji must be each other's multiplicative reciprocal (a_ij = 1/a_ji), and all values must be positive. At least three criteria or alternatives are required; the concept of consistency is meaningless for a pairwise comparison of two items, because there is no third judgement to contradict it. DecisionMind has no extension of this method; a neighbouring consistency measure such as the Harmonic Consistency Index (HCI) can be applied to the same matrix, and the two cross-validate one another.

GCI is exactly the right tool when, in an AHP study or another weighting method based on pairwise comparison, you want to audit the internal consistency of the decision-maker's judgements. It is particularly useful because Saaty's classical consistency ratio (CR) requires an eigenvalue calculation and slows down on large matrices, whereas GCI rests only on the geometric mean and is simpler and faster to compute.

The cases where it should not be used are as follows. If the matrix is not reciprocal, or its diagonal is not one, the GCI comes out meaningless; the matrix must first satisfy these basic conditions. Consistency cannot be measured on a two-item matrix (containing only one judgement), because there is no third reference point to create a contradiction. If the criterion weights are already derived from data (through objective methods such as Entropy or CRITIC), there is no human judgement present, and so the consistency question GCI measures does not arise either.

Auditing the consistency of a criterion or alternative comparison matrix in AHP → GCI

A second, independent consistency check on the same matrix is wanted → HCI

Criterion weights are derived from data, with no human judgement → Entropy, CRITIC (the consistency question does not apply)

Not consistency but how robust a generated ranking is to a change in weights is the question → Weight Sensitivity Analysis

GCI's greatest strength is its simplicity: it rests only on the geometric mean, needs no heavier operation such as an eigenvalue calculation, and computes quickly even on large matrices. Thresholds adjusted for matrix size have been approximately established in the literature, allowing separate interpretation for matrices of different sizes. It can point directly to the judgement pair causing the greatest deviation, giving the decision-maker a concrete indication of which comparison to reconsider.

Its limitations stem from the nature of the measure itself. GCI is not on the same scale as Saaty's classical consistency ratio (CR); applying a threshold known for CR, such as 0.1, to GCI gives the wrong result (Aguarón and Moreno-Jiménez, 2003). A GCI of zero is obtained only under mathematically perfect consistency; it is almost never seen in real human judgements, which makes how "tight" the threshold should be a matter of debate. The method measures only logical consistency, not the accuracy of the judgements (whether they truly reflect real importance); a consistent but mistaken set of judgements can coexist with a low GCI. Finally, GCI is sensitive to matrix size; GCI values from matrices of different sizes cannot be compared directly with one another, only against their own thresholds.

The most common mistake is applying Saaty's 0.1 threshold for CR to GCI; GCI works on a different scale and needs its own thresholds, which vary with matrix size (Crawford and Williams' (1985) geometric-mean-based approach is the origin of this difference). A second mistake is taking a GCI close to zero as proof that the judgements are accurate; GCI audits only internal consistency, the form of the judgements, not their content. A third is trying to apply GCI to a two-item matrix; the concept of consistency requires at least three items. A fourth is refilling the entire matrix from scratch, without first checking which judgement is causing the most trouble, when a matrix exceeds the threshold; correcting a single judgement is usually enough.

The governing principle is this:

GCI shows whether the judgements contradict one another, not whether the judgements are accurate; these are two separate questions, and the report must draw this distinction clearly.