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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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 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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AHP Online Software – Decision Matrices – Update 2016-11-02

With the latest update of my AHP-OS online software you can now view and download the decision matrices of all participant. The group result menu was extended by a selection View Input Data:


From there you reach the Project Detailed Input page, which shows the decision matrix for each individual participant and all nodes of the hierarchy:


The project data menu has the button to download the data in csv format:


The csv file contains the basic project data – Session Code, Project Name, Project description, Author, Date, Type of evaluation, Number of Participants – as well as all input decision matrices sorted by participant and nodes (categories of the hierarchy) or criteria for alternative evaluation.

With the Back link you can go back to the Group Result Page.

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AHP Frequently Asked Questions

Over the year I receive many questions about AHP, my AHP excel template and my AHP online Software. Here a selection of frequently asked questions:

General Questions

Q: I have 15 criteria and 20 alternatives, can you extend your template/software?
A: The excel template can handle up to 10 criteria, my online software is limited to 12 criteria (in one hierarchy level) and 10 alternatives. In principle it could be extended, but the limitation is inherit to the AHP method. Please read my explanations here.

Q: I have more than 10 alternatives, can I use AHP for priority evaluation of criteria and a different method for the evaluation of alternatives?
A: Yes, you can combine AHP for criteria evaluation with another method for alternative evaluation. Alternative evaluation could be done for example using a simple table with a yes/no or applicable/not applicable scale, or any other scale, e.g. Likert scale, how good the individual alternative matches the specific criterion.

Q: I have 150 participants, can I use your excel template/software?
A: My AHP excel template is limited to 20 inputs, my AHP online software can handle a (practically) unlimited number of participants. Use the AHP online software.

Q: How can I resolve the inconsistency (CR>0.1), when participants are done with their pairwise comparisons.
A: Once the pairwise comparison is done and submitted, data can not be changed and consistency ratio is what it is. Ask your decision makers to adjust their judgments  in direction of the most consistent input during the pair-wise comparisons for the highlighted three most inconsistent comparisons. Please see also my posting here.

Q: Can I use the Likert scale instead of the AHP scale?
A: No, AHP is based on the rational scale 1/9 … 1 … 9. It cannot be replaced by the Likert scale.

Q: Do you support Fuzzy AHP?
A: No, I have made no provisions to support Fuzzy AHP, neither in Excel, nor in my online software.

Q: How is the computation done, where do I find the description and formulas?
A: Please download the manual for the excel template from here, and the software description from here.

Q: How can I cite your work, can you give me a reference?
A: Please cite my paper: 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

AHP Excel template

Q: Can I extend the number of participants to more than 20?
A: Though it is possible in principle, the better way is to use my AHP online software with (practically) unlimited number of participants. I will not do a further extension of the template.

Q: Do you have a version of the Excel template w/o multiple inputs?
A: Yes, a simplified version is available on request from the author.

Q: How can I do alternative evaluation using your Excel template.
A: It is not possible. The template can only handle one category of a hierarchy and calculate the priority of one set of criteria.

AHP Online software

Q: If I have a group of decision makers, do they need to register for the online software?
A: No, they don’t need to register. As the owner of a project you get a link for group decision inputs. Simply send them the link, and they can start the pairwise comparisons.

Q: Can I erase/delete inputs from individual participants from the group results?
A: Sorry, at the moment there is no possibility to erase/delete inputs of individual participants. You can open a project with participants’ input and click “Use consol. priorities”. Then “Reset priorities” and “Save hierarchy”. Then you will have the same hierarchy as a new project without participants’ input.

Q: The alternative evaluation is not working?
A: Criteria evaluation (priorities) and alternative evaluation have to be handled as two different projects. Only when you have a decision hierarchy with completed comparisons and evaluated priorities, you can define the alternatives from the group result page clicking on “Use consol. priorities”. Define number and name of alternatives from there and save as new project. Hierarchy evaluation and alternative evaluation projects appear as type “H” for the first and type “A” for the latter in your project list.

Q: Can I get the source code of your online software?
A: Sorry, it is not an open source project.

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Improving AHP consistency

In the implementation of both, AHP excel template as well as AHP online software, inconsistent judgments are highlighted and recommendations for consistent judgments are given.  How it is done, and what is the method behind?

The method is based on Saaty’s article “Decision-making with the AHP: Why is the principal eigenvector necessary“. European Journal of Operational Research 145 (2003) 85–91. He describes three methods how to transform a positive reciprocal matrix to a near consistent matrix.

In my implementation I construct the matrix εij = aij wj/wi to identify the three judgments for which εij is farthest from one. In Saaty’s paper it is shown as table 3, in my excel template it is called Consistency Error Matrix.  As I have to do it on each individual input sheet, I use as an approximation the RGMM (row geometric mean method) results, before calculating the eigenvector solution in the summary sheet.

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AHP online software – a reminder

If you are a registered user of my AHP online software, please keep in mind and be aware:

After 3 months of inactivity, your user account will be deactivated automatically. Please reactivate, using the link provided in the e-mail, if you want to keep your data, otherwise your account and all your data will be deleted 48 hours after deactivation.

This is done to ensure a slim database and help to keep the software fast and responsive. Please also avoid multiple registrations under different user names and e-mail addresses. You can keep up to 20 projects in your account, and it should be sufficient for the majority of users.

AHP online program limits:

  • Number of hierarchy levels: 6 max.
  • Number of hierarchy nodes: 50 max.
  • Number of criteria/node: 15 max. ( 7 to 9 recommended)
  • Number of hierarchy end nodes: 150 max.
  • Number of Alternatives: 10 max.
  • String length for nodes/leafs: 35 char max.
  • Number of characters for hierarchy definition: 6000

Thank you for your cooperation, and PLEASE – as a registered user – help to support this website with a small donation. I do not have a commercial interest, but I spend a lot of time, sharing my knowledge for free, and I have running costs to keep the site alive.

Thank you!

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AHP online software – how it was implemented

When I was asked just now in an e-mail, how my AHP online software operates,   I realized that I have never published a description about it. Therefore, here a short summary:

The AHP-OS software implementation is based on my AHP excel template. I have published one paper about it in 2013. All mathematical formulas used, are shown in this conference paper or in the excel template manual on the last 2 pages. The online version is realized in PHP, using OOP (object oriented programming), running on a Linux server.

The core module is the ahp class. It contains the routines to fill the decision matrix from the pairwise comparisons, to find the dominant eigenvalue using the power method, and to calculate the consistency ratio.

A second important class is the hierarchy class. It translates the text file – defining the hierarchy – into a hierarchical data array. When I implemented the software, I wanted to get a flexible tool, which can be used for all kind of hierarchical problems. This was actually more difficult than the implementation of all mathematical calculations, especially under the aspect of safety and malicious online attacks. The feature also differentiates the online version from the excel implementation. The excel template cannot handle hierarchies at all.

The third important class implements the database and its management. I use SQL with either sqlite or mysql drivers. There are 4 tables:

  • one for user management (registration, password, etc.),
  • one for projects (hierarchy data, description, session code etc.),
  • one for pairwise comparisons (participants, judgments etc.) and
  • one for handling alternatives.

The rest of classes contain supporting functions, like html in/output, excel download and consolidation for group results. I use a basically two open source packages, phpMailer and phpGraphlib, the rest is all realized by myself from scratch.

Please consider a donation, it will help me to maintain the website and program:

Your Donation (1 Singapore Dollar is approx. 0.7 USD):

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How to validate the correct software implementation of AHP?

Sometimes I receive the question, how to validate the correct implementation of my AHP excel template or my AHP online software, i.e. are the results correct and reliable? There are a number of steps to test the correct implementation of the method:

  1. Check for a 2 x 2 decision matrix.
  2. Use “1” for all elements of the decision matrix, i.e. each criterion has the same importance as all other criteria.
  3. Use extremes, i.e. one criterion is extremely (9 times) more important than all other criteria.
  4. Take specific examples from the literature – published research papers containing data of the decision matrix and priorities – and compare.
  5. Compare the results for same input data with the results from another software implementation of the AHP method.

This is what I have done, and in the following I will explain each step in more detail.

1. Check for a 2 x 2 decision matrix

For two criteria there is only one comparison and one solution only: If criterion A is x-times more important than B, the weight w(A) = x/(1+x). w(B) = 1-w(A) as w(A)+w(B) = 1. Eigenvalue = 2; CR is always 0.

Example: Criterion A is 3 times more important than criterion B:
w(A) = 3/4 (75%), w(B) = 1/4 (25%). Check: w(A)/w(B) = 3.

2. Each criterion has the same importance as all other criteria

If all criteria have the same importance, the resulting weight should be 1/n, with n the number of criteria.

Example: Four criteria should result in a weight of 1/4 = 25% for each criterion.

3. One criterion is 9 times more important than all other criteria

If one criterion is 9 times more important than all other criteria, the weights depend on the number of criteria, the maximum weight or maximum priority wmax is always

wmax = M/(n + M – 1)

with M = 9, the maximum of the AHP scale, and n the number of criteria. All other weights should be

wmin = (1 – wmax )/(– 1)

Example: 5 criteria, one criterion 9 times more important than all others

wmax = 9/(5 + 9 – 1) = 9/13 = 69.2%

wmin = (1 – 69.2% )/(5 – 1) = 0.31/4 = 7.7%

4. Specific examples from the literature

Here a practical example comparing the results with  an example (7 criteria) given by Saaty in Int. J. Services Sciences, Vol. 1, No. 1, 2008 (p 86, table 2). The AHP matrix is:

1 9 5 2 1 1  1/2
 1/9 1  1/3  1/9  1/9  1/9  1/9
 1/5 3 1  1/3  1/4  1/3  1/9
 1/2 9 3 1  1/2 1  1/3
1 9 4 2 1 2  1/2
1 9 3 1  1/2 1  1/3
2 9 9 3 2 3 1

The result according Saaty is
(0.177,  0.019, 0.042, 0.116, 0.190, 0.129, 0.327) with consistency ratio of 0,022

My AHP Excel template and my online software should give the same results.

5.  Comparison with other software implementations

As I have implemented AHP under Excel and written in php script language for the online version, I can simply compare the results from both implementations, using the same input data.

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