As part of my current work about AHP scales, here an important finding for the balanced scale:

Salo and Hamalainen [1] pointed out that the integers from 1 to 9 yield local weights, which are not equally dispersed. Based on this observation, they proposed a balanced scale, where local weights are evenly dispersed over the weight range [0.1, 0.9]. They state that for a given set of priority vectors the corresponding ratios can be computed from the inverse relationship

*r* = *w* / (1 – *w*) (1a)

The priorities 0.1, 0.15, 0.2, … 0.8, 0.9 lead, for example, to the scale 1, 1.22, 1.5, 1.86, 2.33, 3.00, 4.00, 5.67 and 9.00. This scale can be computed by

*w*_{bal} = 0.45 + 0.05 *x* (1b)

with *x* = 1 … 9 and

*c* ( resp. 1/*c*) are the entry values in the decision matrix, and *x* the pairwise comparison judgment on the scale 1 to 9.

In fact, eq. 1a or its inverse are the *special case for* *one selected pairwise comparison* of two criteria. If we take into account the complete *n* x *n* decision matrix for *n* criteria, the resulting weights for one criterion, judged as *x*-times more important than all others, can be calculated as:

Eq. 2 simplifies to eq. 1a for *n*=2.

With eq. 2 we can formulate the general case for the balanced scale, resulting in evenly dispersed weights for *n* criteria and a judgment *x* with *x* from 1 to *M*:

(3)

with

(3a)

(3b)

(3c)

We get the general balanced scale (balanced-n) as

(4)

With *n*=2 and *M*=9 it represents the classical balanced scale as given in eq. 1b and 1c. Fig. 1 shows the weights as a function of judgements derived from a case with 7 criteria using the fundamental AHP, balanced and general balanced (bal-n) scale. It can be seen that, for example, a single judgement “*5 – strong more important*” yields to a weight of 45% on the AHP scale, 28% on the balanced scale and 37% on the balanced-n scale.

Figure 1. Weights as function of judgment for the AHP scale, the balanced scale and the corrected balanced scale for 7 decision criteria.

A “strong” criterion is underweighted using the classical balanced scale, and overweighted using the standard AHP scale, compared to the general balanced-n scale. Weights of the balanced-n scale are distributed evenly over the judgment range, and only for *n* = 2 the original proposed balanced scale yields evenly distributed weights.

You can download my complete working paper “*Comparison of Judgment Scales of the Analytical Hierarchy Process – A New Approach*“, submitted for publication.

#### References

[4] Salo, A.,Hämäläinen, R., *On the measurement of preferences in the analytic hierarchy process*, Journal of multi-critria decision analysis,Vol. 6, 309 – 319, (1997).