Weight Uncertainties in AHP-OS

It is now possible, to analyse the weight uncertainties in your AHP-OS projects. When you view the results (View Result from the Project Administration Menu), you see the drop-down list for different AHP scales and a tick box var is shown.

Tick var and click on Scale. All priority vectors of your project will display the weight uncertainties with (+) and (-).

For example, “Capital” has a priority of 15.0% with an uncertainty 0f +1.7% and -2.1%.

The diagram for the total result will show in green the calculated priorities, in dark and light grey the possible plus and minus variations. 

Calculation is based on a randomised variation of all judgment inputs by +/- 0.5 on the 1 – 9 judgment scale. For more than 1 participant the variation is reduced by the square root of the number  of participants.

When downloading the results as csv file, uncertainties are listed below the group result.


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Welcome to BPMSG – June 2017

Concepts, Methods and Tools to manage Business Performance

Dear Friends, dear Visitors,

over the last four months I put in a lot of effort to improve the AHP-OS online tool. With several releases a simplified menu structure and new features were introduced.

  • Delete individual participant’s inputs from an existing project.
  • Update a project hierarchy or project description, as long as there is no input.
  • Evaluate your AHP projects using different AHP judgment scales.
  • Analyse weight uncertainties based on small randomised variations of input judgments.

The last two features are based on my recent study about the comparisons of different AHP scales. Up to date there was no recommendation, what scales to use, and I found a new approach to analyse and compare the scales based on simple analytic functions. This study is submitted for publication, and I hope it will not take too long, until it is available. You can find some more information already in my posting here.

The feature of analysing weight uncertainties is an innovative way of doing sensitivity analysis: all judgments are randomly varied by ±0.5 on the judgment scale, and for each variation the maximum and minimum out coming priorities are captured. I use 1000 variations, enough to get a relatively stable margin of errors for each weight. It gives you information, how “precise” a weight or ranking is in your specific project.

Again, a big Thank You to all donors! Please note that the website is a non-commercial website for educational purposes. Your donation is used to cover running costs like web hosting, antispam services etc. PLEASE, help to support this website with a small donation. I spend a lot of time, sharing my knowledge for free. Thank you in advance!

For now, please enjoy your visit on the site and feel free to leave a comment – it is always appreciated.

Klaus D. Goepel,

Singapore, June 2017

BPMSG stands for Business Performance Management Singapore. As of now, it is a non-commercial website, and information is shared for educational purposes. Please see licensing conditions and terms of use.

Please give credit or a link to my site, if you use parts in your work, or make a small donation to support my effort to maintain this website.

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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


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 from researchgate.net or  here


[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).


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