AHP Consensus Indicator

The AHP consensus indicator, based on Shannon beta entropy (e.q. 1.1) for n criteria and k decision makers, was introduced in [1].

(1.1) Shannon beta entropy:

(1.2) Shannon alpha entropy:

(1.3) Shannon gamma entropy:


The similarity measure S (eq. 1.4) depends on the number of criteria, and we used a linear transformation to map it to a range from 0 to 1 (eq. 1.5)


(1.5) Consensus (0% to 100%):

In general Dα min = 1 and Dγ max = n. In the analytic hierarchy process (AHP) Dα min is a function of the maximum scale value M (M = 9 for the fundamental AHP scale) and the number of criteria n (eq. 1.6). The calculation of Dγ max was based on the assumption that respondents compare one distinct criterion M‑times more important than all others (eq. 1.7).



This assumption is actually an unnecessary constrain, because even when the number of decision makers is less than the number of criteria, both  can prioritize a complementing set of criteria as most important and as a result all consolidated criteria weights are equal. Therefore eq. 1.7 can be simplified to:


As a result we get the AHP consensus indicator with:


(1.10) AHP Consensus: Equation (1.10) is used in the latest updated of the AHP excel template and the AHP-OS online software.


[1] Klaus D. Goepel, (2013). Implementing the Analytic Hierarchy Process as a Standard Method for Multi-Criteria Decision Making In Corporate Enterprises – A New AHP Excel Template with Multiple Inputs, Proceedings of the International Symposium on the Analytic Hierarchy Process, Kuala Lumpur 2013

AHP-OS Quick Reference Guide

As I know from my own experience, manuals are seldomly read. On the other hand, a short guideline to complex software can be helpful, to use it effectively. I summarised the main menus of AHP-OS in a four page quick reference guide. The full manual is still available from the AHP-OS entry page (needs update …), and all details regarding methods and calculations are shown in my working paper about the AHP-OS software implemetation.


AHP-OS Hierarchy Evaluation with Partial Inputs from Participants

With the latest update of my AHP online software it is now possible to save judgments (pairwise comparisons) without completing the whole hierarchy evaluation. There are two scenarios, where this could be useful:

  1. You have a complex hierarchy with many nodes to be evaluated. Now participants can start a partial evaluation, save the judgments and complete the remaining nodes at a later time.
  2. As a participant you are only expert for a subset of nodes in the decision hierarchy. As a chair you can now ask participants to give their inputs for a few nodes of their expertise only.

Pairwise comparison input is started as usually: either using the link provided on the project session page, or using the link AHP Group Session on the AHP-OS main entry page. After providong session code and name, in case the participant hasn’t given any input, a message Ok. Group has x participants. Click “Go” to continue will be displayed. Nodes without judgment show the AHP button with red outline.

Continue reading AHP-OS Hierarchy Evaluation with Partial Inputs from Participants

AHP-OS now secured with HTTPS

My online software AHP-OS is mainly used in research. Projects handled with AHP-OS cover a wide range of applications like healthcare, climate, risk assessment, supplier selection, hiring, IT, marketing, environment, transport, project management, manufacturing or quality assurance. Some of these projects could contain sensitive data. Therefore I finally decided to secure the site with HTTPS to protect the site and users.

Continue reading AHP-OS now secured with HTTPS

AHP-OS Update Version 2017-08-31

The latest update of AHP-OS comprises of some minor changes to make the program flow easier to understand for participants w/o background in AHP.

  • The group session input screen does no longer show the headline to login or register, as for participants there is no need to be registered.
  • The text introduction was shortened to two and a half line of text.
  • Menu buttons intended to be clicked are highlighted.

Continue reading AHP-OS Update Version 2017-08-31

Sensitivity Analysis in AHP

Sensitivity analysis is a fundamental concept in the effective use and implementation of quantitative decision models, whose purpose is to assess the stability of an optimal solution under changes in the parameters. (Dantzig)

Weighted sum model (Alternative Evaluation)

In AHP the preference Pi of alternative Ai is calculated using the following formula (weighted sum model):
(1)with  Wj the weight of criterion Cj, and aij the performance measure of alternative Ai with respect to criterion Cj. Performance values  are normalized.


Table 1

Sensitivity analysis will answer two questions:

  • Which is the most critical criterion, and
  • Which is the most critical performance measure,

changing the ranking between two alternatives?

Continue reading Sensitivity Analysis in AHP

Incoming search terms:

  • how evaluate sensitivity analysis in excel from AHP - youtube
  • sensitivity analysis excel

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 Refresh. All priority vectors of your project will display the weight uncertainties with (+) and (-).

Continue reading Weight Uncertainties in AHP-OS

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 Judgment Scales

The original AHP uses ratio scales. To derive priorities, verbal statements (comparisons) are converted into integers from 1 to 9. This “fundamental AHP scale” has been discussed, as there is no thoretical reason to be restricted to these numbers and verbal gradations. In the past several other numerical scales have been proposed [1],[3]. AHP-OS now supports ten different scales:

  1. Standard AHP linear scale
  2. Logarithmic scale
  3. Root square scale
  4. Inverse linear scale
  5. Balanced scale
  6. Balanced-n scale
  7. Adaptive-bal scale
  8. Adaptive scale
  9. Power scale
  10. Geometric scale

Fig. 1 Mapping of the 1 to 9 input values to the elements of the decision matrix.

Continue reading AHP Judgment Scales