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 nine 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. Power scale
  9. Geometric scale

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

Power scale and geometric scale extend the values of matrix elements from 9 to 81 resp. 256. Root square and logarithmic scale reduce the values from 9 down to 3 resp 3.2. Inverse linear and balanced scale keep the values in the original range, but change the weight dispersion. The balanced-n scale is a corrected version of the original balanced scale. The adaptive-bal scale scales the values depending on the number of criteria: for n = 2 criteria it represents the balanced scale, for n = 10 criteria it represents a balanced power scale.

As a result, priority discrimination will be improved using the geometric or power scale, but at the same time the consistency ratio will go up. For the  logarithmic, root square, and inverse linear scales it is the opposite, priorities are more compressed or “equalised” across the criteria, see Fig. 2. At the same time CR improves.

Only the balanced-n scale and adaptive-bal scale will improve (or at least keep) the consistency ratio in a reasonable range and at the same time minimise weight uncertainties and weight dispersion.

Fig. 2 Change of priorities for different scales for an example with eight criteria.

The choice of the appropriate scale is difficult and an often discussed problem. Until today there is no published guideline, when to select which scale. A study on the impact on priorities and consistency ratio (CR) is published in [2]. I have just recently submitted a paper, providing a guideline on the selection of different AHP scales.

How to select different scales in AHP-OS

Open a project with completed judments (participants) from your project list. In the Project menu click on View Result. By default the results are then shown calculated based on the standard AHP 1 to 9 scale. To recalculate for different scales, select the scale in the Group Result menu from the scroll down list and click on Scale.


[1] Ishizaka A., Labib A. Review of the main developments in the analytic hierarchy process, Expert Systems with Applications, 38(11), 14336 - 14345, (2011)

[2] Jiří Franeka, Aleš Krestaa. Judgment scales and consistency measure in AHP, Procedia Economics and Finance, 12, 164 - 173 (2014)

[3] W.W. Kozckodaj. Pairwise Comparison Rating Scale Paradoxon, Cornell University Library, (2015) https://arXiv.org/abs/1511.07540

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AHP-OS Data Download and Import in Excel

Most data generated with AHP-OS can be downloaded as csv files for import into a spreadsheet program and further analysis:

  • From the Hierarchy Input Menu – decision hierarchy and local & global priorities
  • From the Group Result Menu – Priorities by node and consolidated decision matrix
  • From the Project Data Menu – Decision matrices from each participant

For each download you can select “.” or “,” as decimal separator. The downloaded csv (text) file is coded in UTF-8 and supports multi-language characters like Chinese, Korean, Japanese and of course a variety of Western languages.

How to import into excel?

Open Excel, click on “File” -> “New” to have a blank worksheet. Click on “Data“. On the left top you will find the “Get External Data” box.

Click on From Text to select the downloaded cvs file for import. The Text Import Wizzard will open.

Now it is important to select 65001 : Unicode (UTF-8) under File origin.

Then, depending on your decimal separator, select Comma or Semicolon as Delimiters:

When the import is done, your text characters should be displayed correctly.  Save the file “Save as” as Excel workbook (*.xlsx).

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AHP-OS – Editing saved projects

In the project menu of the latest AHP-OS version (2017-05-25), I added a button to edit saved projects. As long as there are no participants’ inputs (completed pairwise comparisons), any saved project’s hierarchy, alternatives or description can be modified.

Open a project from your project list, and click on Edit Project. The project hierarchy page will open with a message on top , indicating that you are modifying an existong project. You can now change the hierarchy, for example add criteria or alternatives. A click on Save/Update in the Hierarchy Input Menu

will overwrite the data of the original project under the same session code. You will see it in a message . Before you click on Go to save,  you  can also update the project short description:

Difference between Use Hierarchy and Edit Project

With Use Hierarchy in the project administration menu, the hierarchy window will open, and you can also modify the hierarchy or alternatives. But in contrast to Edit the modified project will be saved as a new project under a new project session code.

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AHP-OS New Release with simplified project administration

Based on feedback from users, I just released a major update of BPMSG’s AHP online software AHP-OS with simplified menu structure and additional functionality.  Starting the program as registered and logged-in user, the project session  table is displayed, showing your projects.

You can open one of your projects, either using a click on the session code in the project table, or selecting the session code from the session administration menu:

This will bring you to the project summary page, showing

  • Project data
  • Alternatives (if any)
  • Participants (if any)
  • Group input link (to be provided to your project participants)
  • Project Hierarchy and hiearchy definiton (text)

At the bottom you find the new project administration menu:

From here you can:

  • View Result: View the project group result (if there are already participants)
  • Group Input: Start pairwise comparisons
  • Use/Modify Hierarchy: use and modify the project’s hierarchy for a new project
  • Delete selected Participants (a request from many users)
  • Delete the whole project
  • Close the project to go back to the project session table

Due to this new Project Administration menu some of the other menus are simplified. Let me know your experience with the new structure or if you find any bugs. The manual will be updated within the next days.

Deleting participants

On the project summary page select the participants, you want to delete, and click on refresh.

You will then see a message Selected participant(s): Werner. Click on the button to delete the selected user(s). Careful: once deleted, they cannot be recovered and their pairwise comparison data will be lost.


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AHP Group Consensus Indicator – how to understand and interpret?

BPMSG’s AHP excel template and AHP online software AHP-OS can be used for group decision making by asking several participants to give their inputs to a project in form of pairwise comparisons. Aggregation of individual judgments (AIJ) is done by calculating the geometric mean of the elements of all decision matrices using this consolidated decision matrix to derive the group priorities.

AHP consensus indicator

In [1] I proposed an AHP group consensus indicator to quantify the consensus of the group, i.e. to have an estimate of the agreement on the outcoming priorities between participants. This indicator ranges from 0% to 100%. Zero percent corresponds to no consensus at all, 100% to full consensus. This indicator is derived from the concept of diversity based on Shannon alpha and beta entropy, as described in [2].  It is a measure of homogeneity of priorities between the participants and can also be interpreted as a measure of overlap between priorities of the group members.

How to interpret?

If we would categorise group consensus in the three categories low, moderate and high, I would assign the following percentages to these categories:

  • low consensus: below 65%
  • moderate consensus: 65% to 75%
  • high consensus: above 75%

Values below 50% indicate that there is practically no consensus  within the group and a high diversity of judgments. Values in the 80% – 90% range indicate a high overlap of priorites and excellent agreement of judgments from the group members.

AHP Consensus indicator and AHP Consistency Ratio CR

AHP allows for (logical) inconsistencies in judgments; the AHP consistency ratio CR is an indicator for this, and – as a rule of thumb – CR  should not exceed 10% significantly. Please read my posts here and here.

It can be shown that,  given a sufficiently large group size, consistency of the aggregate comparison matrix is guaranteed, regardless of the consistency measures of the individual comparison matrices, if the geometric mean (AIJ) is used to aggregate [3] . In other words, if the group of participants is large enough, the consistency ratio of the consolidated group matrix CR will decrease below 10% and is no longer an issue.

Consensus has to be strictly distinguished from consistency. The consensus is derived from the outcoming priorities and has nothing to do with the consistency ratio. Whether you have a small or a large group, in both cases consensus could be high or low, reflecting the “agreement” between group members. Even if you ask a million people, there could be no agreement (consensus) on a certain topic: half of them have the exact opposite judgment as the other half. As a result, the consensus indicator would be zero: there is no overlap, the total group is divided into two sub-groups having opposite opinions.

Analyzing group consensus – groups and sub-groups

The beauty of the proposed AHP consensus indicator based on Shannon entropy is the possibility to analyse further, and to find out, whether there are  sub-groups (cluster) of participants with high consensus among themself, but with low consensus to other sub-groups. This can be done using the concept of alpha and beta diversity [2]. I have published an excel template to to analyze similarities between the samples based on partitioning diversity in alpha and beta diversity. It can be also be used for your AHP results to analyse group consensus.


[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

[2] Lou Jost, (2006). Entropy and Diversity, OIKOS Vol. 113, Issue 2, pg. 363-375, May 2006

[3] Aull-Hyde, Erdoğan, Duke (2006). An experiment on the consistency of aggregated comparison matrices in AHP, European Journal of Operational Research 171(1):290-295 · February 2006

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