Harmonic Consistency Index (HCI)

In methods such as AHP, the decision-maker compares criteria or alternatives two at a time and fills in a judgement matrix. The best-known way to audit whether this matrix is consistent is Saaty's consistency ratio (CR), which looks at the matrix's largest eigenvalue. The harmonic consistency index (HCI) asks the same question, but instead of calculating the eigenvalue directly, it derives an "approximate largest eigenvalue" using the geometric-mean priority vector; in this derivation it uses the harmonic mean rather than the arithmetic mean. Stein and Mizzi proposed it in 2007 as an alternative to Saaty's CR that is simpler to calculate and, in some situations, more cautious.

The idea behind this method is to measure consistency approximately, without a full eigenvalue calculation, while still staying faithful to eigenvalue logic. In a perfectly consistent matrix, every row's "actual value divided by priority" ratio comes out equal to the matrix's dimension (the number of criteria). HCI chooses the harmonic mean when averaging these ratios; unlike the arithmetic mean, the harmonic mean does not let large values pull the average upward as much, so it exaggerates the effect of a single, extremely inconsistent row less, but it is more sensitive to small deviations. The philosophical consequence is this: HCI asks the same question as Saaty's CR (how consistent is this matrix) using a different averaging logic, and it generally gives a more cautious, that is a higher, signal of inconsistency than CR.

The method proceeds through three steps.

First, the priority vector. As with GCI, a priority vector is built by taking the geometric mean of each row of the pairwise comparison matrix.

Second, the approximate eigenvalue. The matrix is multiplied by its own priority vector; for each row, the ratio of this product to that row's priority is calculated. In a perfectly consistent matrix, all these ratios come out equal to the matrix's dimension. HCI takes the harmonic mean of these ratios to produce a single "approximate largest eigenvalue."

Third, the index and the threshold. This approximate eigenvalue is converted into the HCI in a manner similar to Saaty's CI calculation: the matrix dimension is subtracted from it, and the result is divided by the dimension minus one. HCI is then compared against a customary threshold (approximately 0.1); if it falls below the threshold, the matrix is consistent to an acceptable degree.

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

The closer the HCI value sits to zero, the more consistent the matrix; zero is obtained only under mathematically perfect consistency. HCI generally comes out somewhat higher (more cautious) than Saaty's CR; so for the same matrix, HCI exceeding the 0.1 threshold is more likely than CR exceeding that same threshold. An HCI falling below the threshold does not mean the content of the judgements is correct; it shows only that the judgements do not logically contradict one another. HCI on its own does not say which judgement is the problem; because of how the approximate eigenvalue is calculated, it does not point to the most deviant row as directly as GCI does, which is why it is generally used together with GCI, as a second, independent check. For that reason, instead of writing:

"HCI is below the threshold, the matrix is perfect"

it is correct to write:

"HCI falls below the threshold, which shows the judgements also pass a more cautious consistency test than Saaty's CR measure; the content of the judgements must be assessed separately"

The harmonic consistency index works with crisp data; its input is the same as GCI's, a reciprocal pairwise comparison matrix. The matrix's diagonal must be one, a_ij and a_ji must be each other's multiplicative reciprocal, and all values must be positive. At least three criteria or alternatives are required. DecisionMind has no extension of this method; the Geometric Consistency Index (GCI) can also be applied to the same matrix, and the two together form two independent consistency tests that validate one another.

HCI is exactly the right tool in an AHP study when, alongside or instead of Saaty's CR, you want a somewhat more cautious consistency test that is simpler to calculate. Used together with GCI in particular, seeing whether two different averaging logics (log-square and harmonic) arrive at the same conclusion lends extra confidence to the consistency decision.

The cases where it should not be used are as follows. If the matrix is not reciprocal, or its diagonal is not one, HCI comes out meaningless. Consistency cannot be measured on a two-item matrix. If Saaty's CR has already been calculated and comes out below the threshold, presenting HCI as though it were an additional requirement on its own (as if CR were not enough) is misleading; HCI is a validation layer, not a replacement for CR.

A simpler-to-calculate consistency test alongside Saaty's CR in AHP is wanted → HCI

The primary, log-square-based consistency test for the same matrix is wanted → GCI

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

HCI's greatest strength is that it produces a consistency signal on the same logic without needing Saaty's eigenvalue calculation, using only the geometric mean and the harmonic mean; this reduces the computational load on large matrices. Because it is reported on the same scale as CR (the matrix dimension subtracted, divided by the dimension minus one), it is easily interpreted by users accustomed to CR. Used together with GCI, two independent methods arriving at the same conclusion lends extra confidence to the consistency decision.

Its limitations stem from its own cautiousness. For the same matrix, HCI generally comes out higher than Saaty's CR; so there can be cases where CR crosses the threshold but HCI does not, and it must be clarified in advance which measure is to be treated as primary (Stein and Mizzi, 2007). Unlike GCI, the method does not directly show which single judgement causes the greatest deviation; it gives only the matrix's overall consistency. The harmonic mean is more sensitive than the arithmetic mean to a single extremely low ratio; in some cases this can cause a single small judgement error to push HCI up disproportionately. Finally, like GCI, the method measures only logical consistency, not the accuracy of the judgements.

The most common mistake is using HCI unconditionally in place of CR; HCI is more cautious, and a matrix that does not cross CR's 0.1 threshold can cross HCI's same threshold, which is a difference between measures, not a calculation error. A second mistake is forgetting that HCI rests on the geometric-mean priority vector and comparing it as though it were a number calculated directly by the eigenvalue method (Saaty's own CR calculation). A third is not looking at GCI at all once HCI comes out below the threshold; the two measures use different mathematical logic and complement one another, neither replaces the other. A fourth is taking a low HCI as proof that the judgements are accurate.

The governing principle is this:

HCI is a consistency test similar to Saaty's CR but more cautious; read together with GCI it lends confidence to the consistency decision, but on its own it says nothing about whether the judgements are accurate.