## AHP-OS Implementation – working paper

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

Goepel, K. D. (2017). Implementation of an Online Software Tool for the Analytic Hierarchy Process – Challenges and Practical Experiences.  Working paper prepared for publication, Singapore July 2017, available from http://bpmsg.com/ahp-software/

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.
(2)

Example

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?

#### The most critical criterion

The most critical criterion is defined as the criterion Ck, with the smallest change of the current weight Wk by the amount of  δkij changing the ranking between the alternatives Ai and Aj.

The Absolute-Top (or AT) critical criterion is the most critical criterion with the smallest change δkij changing the ranking of the best (top) alternative.

The Absolute-Any (or AA) critical criterion is the most critical criterion with the smallest change δkij changing any ranking of alternatives.

For each pair of alternatives Ai, Aj, with i = 1 to n and  i < j we calculate
(3)with .

Example

Table 2

• The absolute-top critical criterion is Neighbourhood: a change from 18.8% by -8% will change the ranking between the top alternative A1 (House A) and alternative A2 (House B).
• The absolute-any critical criterion is the same as above, as -8% is the smallest value in the table.

As the weight uncertainty for the criterion Neighbourhood is +1.4% and -1.3%, the solution is stable.

#### The most critical measure of performance

The most critical measure of performance is defined as the minimum change of  the current value of  aij such that the current ranking between alternative Ai  and Aj will change.

For all alternatives Ai and Aj  with ij and  and each criterion we calculate
(4)with .

Example

Table 3

• The absolute-any critical performance measure is found for alternative A3 (House C) under the criterion Financing. A change from 27.9% by 20.4% will change its ranking with alternative A2 (House B), i.e. only a (drastic) change from 27.9% to 48.3% of the evaluation of House C with respect to Financing would change the ranking of House C and House B.

#### Implementation in AHP-OS

For alternative evaluation the method described above is implemented in AHP-OS. On the group result page in the Group Result Menu tick the checkbox var and then click Scale.

Under the headline Sensitivity Analysis TA and AA critical criterion as well as AA critical performance measure will be displayed. You can download the complete tables as csv files with a click on Download.

#### References

Triantaphyllou, E.,  Sánchez, A., A sensitivity analysis approach for some deterministic multi-criteria decision making methods, Decision Sciences, Vol. 28, No. 1, pp. 151-194, (1997).

## 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.

## 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
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.

#### References

`[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. Koczkodaj. Pairwise Comparison Rating Scale Paradox, Cornell University Library, (2015) https://arXiv.org/abs/1511.07540`

#### Incoming search terms:

• x rite singapore loc:SG

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).

#### Incoming search terms:

• EXCEL WIZZARD

## 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.

## 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)
• Project Hierarchy and hiearchy definiton (text)

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.

#### Incoming search terms:

• paperuri:(2f2ca361a5da31c6a7d3bc7b374f9304)

## AHP-OS and Farsi

Thanks to the feedback of a user from Iran, I found out that AHP-OS is working with RTL fonts like Farsi. It is important to add the semicolon to the right end of a node definition, and use latin punctuation marks, for example:

کاربری: مسکونی , تجاری , اداری ;

A simple decision hierarchy example looks like here:

Unfortunately, I am not able to check for a hierarchy with more than one level. If anyone could provide a 2 level hierarchy, I would appreciate. Arabic fonts should work too; kindly contact me, if you have an example.

## Group Decision Making with AHP-OS

My AHP free online software AHP-OS has a feature to involve a group of decision makers to give their inputs to a decision problem. In contrast to my AHP Excel template, in AHP-OS the number of participants is practically unlimited. As of now, I see users having up to 100 participants in one project.

### How to use AHP-OS for Group Decision Making?

As registered user you need to start with a new project by defining your decision hierarchy. In the Project Administration Menu click on New, define your hierarchy, Submit and Save as project. You have the possibility to give a short project description, explaining the project, before it is saved.

Once saved, you will be automatically brought back to your project page, and the project will be shown on top of your project list.

The session code is a unique code identifying the project. When you click on the session code link, the Group Session Input screen will open, and you see a message like “Hierarchy evaluation. Session code EvUhUn Session Chair: Klaus. Project has 0 participants’ inputs. You are participating as Klaus.” Now you can either participate yourself as a decision maker in the project, or get a link to provide to your participants for their inputs.

### Participating by yourself as a member of the group

Simply click on the AHP button to start the pairwise comparisons, and – once completed – click on Submit for group eval to submit your inputs.

### Provide a link to let others participate as members of the group

In the group input menu click on Leave group input mode and confirm ok to leave the input mode without submitting your pairwise comparisons in the pop-up window.

Now you will see the link that you can copy/paste and send to your group members for their participation in the project.

Click on Done in the Active session menu to go back to your project list.

### Participants input

Your participants don’t need to be registered users. If you want to test by yourself, you need to logout from AHP-OS, before you open the group link.

When your participants open the link, they will be asked for their name, before they can start their pairwise comparisons. They will then see the decision hierarchy with a red lined AHP button and  a message “Click on AHP to complete pairwise comparisons. Click on Submit for group eval once completed.”

Now they can start their pairwise comparisons. When completed, all AHP buttons and priorities will be marked green.

After a click on Submit for group eval the AHP button disappears and a message “Ok. Data were submitted. Thank You for your participation!” is displayed.

### Group results

Once you have participants submitted data, a click on the session code link in your project list will automatically bring you to the group result page. As a participant you will get a button View group result in the Group input menu.

The group result page gives you on top the project data and list of all participants. Individual participants can be selected or deselected to analyze the group results in further details.

The next sections of the group result page shows the decision hierarchy with consolidated priorities, a breakdown by nodes (categories) with priorities for the node and the consolidated (aggregated) decision matrix. It also shows the AHP consensus indicator as a measure of group consensus.

Input data for each participant is available under View Input Data in the group result menu:

For you as chair of the group session, it is also possible to download data as csv text file for further processing in a spreadsheet program. With Use consol. priorities you now can define Alternatives with the consolidated priorities of selected participants.