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

AHP-OS Implementation – working paper

For anyone, who is interested in the implementation of my free AHP-OS online software, and needs a reference:

Implementation of an online software for the Analytical Hierarchy Process

I hope to finalize the paper soon so that I can submit it for publication.

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:

  • sensitivity analysis in ahp
  • sensivity analysis of ahp model

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 priorities scale 6

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

Incoming search terms:

  • AHP scale
  • 9 point pairwise comparison scale
  • ahp ranking scale
  • ahp scale of 1
  • ahp scale square root
  • scale ranges from in ahp

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.

Continue reading AHP-OS Data Download and Import in Excel

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:

Continue reading AHP-OS – Editing saved projects

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

Continue reading AHP-OS New Release with simplified project administration

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.

Continue reading AHP Group Consensus Indicator – how to understand and interpret?