Why the AHP Balanced Scale is not balanced

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

wbal = 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:






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


Continue reading Why the AHP Balanced Scale is not balanced

Incoming search terms:

  • ahp scaling

AHP-ANP practical Application with Pros and Cons

The analytic hierarchy (AHP) and analytic network process (ANP) are two multi-criteria decision methods (MCDM), originally developed by Prof. Thomas L. Saaty.

ANP is a more general approach, based on the description of the problem by means of a network instead of a hierarchy as in AHP. On the other hand, ANP is also more complex in its application.

In my latest video presentation, pros and cons for both methods are shown, and a few tips for the practical application of AHP,  and setting up a network for ANP are listed.

Download the slides

Link to video in Youtube

Feel free to comment and feedback here!

Incoming search terms:

  • analytical hierarchy process excel
  • ahp excel template
  • ahp in excel
  • anp excel
  • analytical network process excel template