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)