## 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).

## 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

(1c)

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:

(2)

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:

(3)

with

(3a)

(3b)

(3c)

We get the general balanced scale (balanced-n) as

(4)

With n=2 and M=9 it represents the classical balanced scale as given in eq. 1b and 1c. Fig. 1 shows the weights as a function of judgements derived from a case with 7 criteria using the fundamental AHP, balanced and general balanced (bal-n) scale. It can be seen that, for example, a single judgement “5 – strong more important” yields to a weight of 45% on the AHP scale, 28% on the balanced scale and 37% on the balanced-n scale.

`Figure 1. Weights as function of judgment for the AHP scale, the balanced scale and the corrected balanced scale for 7 decision criteria.`

A “strong” criterion is underweighted using the classical balanced scale, and overweighted using the standard AHP scale, compared to the general balanced-n scale. Weights of the balanced-n scale are distributed evenly over the judgment range, and only for n = 2 the original proposed balanced scale yields evenly distributed weights.

You can download my complete working paper “Comparison of Judgment Scales of the Analytical Hierarchy Process – A New Approach” submitted for publication from researchgate.net or  here

#### References

[4] Salo, A.,Hämäläinen, R., On the measurement of preferences in the analytic hierarchy process, Journal of multi-critria decision analysis,Vol. 6, 309 – 319, (1997).

## Welcome to BPMSG – Feb 2017

### Concepts, Methods and Tools to manage Business Performance

Dear friends, dear visitors,

Using Google search for my own website, I could see a message like Your website is not mobile friendy. It was really time to update my website theme in order to make it better readable for visitors using their smart phone or tablet.

Now you see the new design using the standard wordpress theme from 2015. I simplified the menu structure to the categories:

• AHP Articles – Material and Information about the Analytic Hierarchy Process
• Tools – Links to my online tools, like the AHP online system and excel templates
• Other – Posts related to other topics
• Feedback – A place where you can leave your feedback as comments
• Contact – A web form to contact me personally via e-mail

I hope with these changes navigation on the site is much easier, and you can find required information in shorter time.

Again, a big Thank You to all donors! Please note that the website is a non-commercial website for educational purposes. Your donation is used to cover running costs like web hosting, antispam services etc. PLEASE, help to support this website with a small donation. I spend a lot of time, sharing my knowledge for free. Thank you in advance!

Klaus D. Goepel,

Singapore, Februar 2017

BPMSG stands for Business Performance Management Singapore. As of now, it is a non-commercial website, and information is shared for educational purposes. Please see licensing conditions and terms of use.

Please give credit or a link to my site, if you use parts in your work, or make a small donation to support my effort to maintain this website.

#### Incoming search terms:

• AHP 20 ESSA

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.

#### Incoming search terms:

• ahp question
• basic questions in AHP
• no ahp

## AHP Excel Template Version 2016-05-04

Just a small correction in the summary sheet: Now the correct name of the selected scale is displayed.

Please consider a small donation, if you find the template useful for your work.

## 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!

## Welcome to BPMSG – October 2015

### Concepts, Methods and Tools to manage Business Performance

Dear Friends, dear Visitors,

it’s soon coming to five years that my webpage has been online, and things getting slowly a bit messy … There was a need to clean-up the pages, posts and directories, so that the necessary maintainance will take less time, and future updates are possible. As you should know, this project is done in my free time as a hobby, I still have a full-time job for another couple of years.

Unfortunatly, during the whole process, some links were broken and it will take time to repair. Sorry, when you experience missing images or a broken link. Just let me know using my contact page. I am still working on it and plan to simplify the navigation in order  to make it easier for you to find relevant information.

AHP as one category, where you find information about the analytic hierarchy process, will stay in focus. From the start this was the nost frequently visited part of the site, and my AHP excel template slowly reaches 15 thousand (!) downloads. Under  the category Articles you find posts about other topics, mainly the concept of diversity, as I was using it for  different applications. All more personal related  topics under the category Other will move to my new personal blog – Klaus & Lita – running on my own server with topics like multicopter, programming and – maybe – cooking in future.

Klaus D. Goepel,

Singapore, October 2015

BPMSG stands for Business Performance Management Singapore. As of now, it is a non-commercial website, and information is shared for educational purposes. Please see licensing conditions and terms of use.

Please give credit or a link to my site, if you use parts in your work, or make a small donation to support my effort to maintain this website.

#### Incoming search terms:

• bohol philippines
• Bohol Philippinen

## Welcome to BPMSG – May 2014

### Concepts, Methods and Tools to manage Business Performance

Dear Friends, dear Visitors,

The latest beta version of my AHP online software (AHP-OS) has now additional features to manage complete AHP projects. AHP stands for Analytic Hierarchy Process, and is a decision support tool.  To use the full functionality, please register and log in; it’s all free.

You can store complete decision hierarchies, use them to estimate the weights of criteria and sub-criteria and evaluate up to seven decision alternatives.

AHP is also helpful to support group decision making; participants can input their individual judgments and a consolidated group result is calculated. I have prepared a practical example, where you can participate, input your judgments and view the overall group results and consensus. Just click on the link and try it out.

The development is still continuing. I am further optimizing the handling and plan to implement additional analysis, especially for group decision making. Bookmark the page and revisit from time to time to get the latest updates.

Now please enjoy your visit on the site and feel free to give me feedback – it is always appreciated.

Klaus D. Goepel,

Singapore, May 2014

BPMSG stands for Business Performance Management Singapore. As of now, it is a non-commercial website, and information is shared for educational purposes. Please see licensing conditions and terms of use. Please give credit or a link to my site, if you use parts in your website or blog.

## Welcome to BPMSG – Feb 2014

### Concepts, Methods and Tools to manage Business Performance

Dear Friends, dear Visitors,

the development of my AHP online software is still continuing. Most of the recent changes were structural, and not so obvious to users, as I use the software on two different platforms and in two different designs. On the entry page you will now find a short introduction to the analytic hierarchy process, and links to further examples and documentation. From there you can go to the online priority calculator, or the AHP online solution to handle complete projects.

Overall the functionality is complete and the program is stable. Except for group inputs, the priority calculator includes all functions of my AHP excel template. With the AHP online system you can even define complete AHP hierarchies and evaluate alternatives. For both programs the results can now be exported as csv files for further processing in spreadsheet programs in two versions: using either dot or comma as decimal separator.

The next big step would be the implementation of group decision making. My idea is to implement web sessions for different users  on different computers, and an online consolidation of all judgments, to get the consolidated group result. This could be very useful when you are in a meeting and have group discussions, then all participant could do their judgment online, seeing the consolidated group result in real time.

It will still take some more time, so bookmark the page and revisit  from time to time to get the latest updates.

Now please enjoy your visit on the site and feel free to give me feedback – it is always appreciated.

Klaus D. Goepel,

Singapore, Feb 2014

BPMSG stands for Business Performance Management Singapore. As of now, it is a non-commercial website, and information is shared for educational purposes. Please see licensing conditions and terms of use. Please give credit or a link to my site, if you use parts in your website or blog.

## AHP Online System

As I was asked by many visitors, how to evaluate priorities and alternatives for a whole project in AHP, I have realized an online solution, where you can define a hierarchy, get the priorities for all categories and evaluate alternatives. The project is realized in PHP.

 Here the link: BPMSG AHP Online System

### How to use the tool

The online template is easy to use, three steps are necessary:

1. Define the objective and relevant criteria of your decision problem and structure them in a hierarchy.
2. Compare criteria in categories and sub-categories with respect to the objective to find their weights based on pairwise comparisons.,
3. Name a set of alternatives and compare, how good they match your decision criteria. Again pairwise comparisons based on the AHP are used.

#### 1. Define Hierarchy

Hierarchies are defined in a text field using the following simple syntax:

Each branch in the hierarchy is defined by its node (the category) and the node’s leafs (the sub-categories). The node is followed by a colon, leafs are separated by comma, and a branch is closed by a semicolon:

category: sub-category 1, sub-category 2, sub-category 3;

If a sub-category branches out in further sub-categories, you add a line, repeating the sub-category’s name (case sensitive!) as a new node, i.e. followed by a colon:

`sub-category 1: sub-sub-1, sub-sub-2;`

#### 2. Compare Criteria

To find the weight (importance) of criteria, click AHP to start pairwise comparisons. Compare each pair of criteria with respect to the project and category: which criterion in each pair is more important, and how much more on a 1 – 9 scale? Once you have finished click Calculate Result. A table with priorities for each criterion is shown.

It could happen that your pairwise comparison is not consistent, then the most inconsistent judgments are highlighted, and the most consistent judgment is marked light green. In order to improve consistency, check whether you are able to adjust your original mark by plus or minus two on the scale. Click Calculate to re-calculate. Once finished, and you are satisfied with you answers, press Submit Priorities to submit. Completed branches in the hierarchy tree are marked green.

When all comparisons in the hierarchy are done, you might continue to evaluate alternatives. Press Evaluate Alternatives. Note: You only get the selection, if all necessary pairwise comparisons are completed, or the hierarchy text input already contains pre-defined priorities. A table with criteria and alternatives will be displayed.

#### 3. Evaluation of Alternatives

Before you start the evaluation, you might input the number and names of your decision alternatives. Enter the number (1 – 5) and press Go. Then input the names of alternatives and press ok. The procedure of pairwise comparisons is exactly the same as for the criteria. Each completed comparison is highlighted in green. When all evaluations are done, the result is shown.

You might download the data in csv format (comma separated values) for further processing in a spread sheet program. The table starts with a title line; then each row shows one criterion with the columns: (hierarchy level, local priority), global priority, alternatives (local and global priority).

All (decision) matrices are shown below, with category name as heading and criteria for the matrix rows. Alternatives show the heading “Alternatives for” and the respective criterion.

### Program Limitations and Input Restrictions

• Display length of category names: 25 characters max.
• Hierarchy levels 6 max.
• Nodes: 15 max.
• Each node (category) has to have at least 2 branches (sub-categories)
• Endnodes: 100 max.
• Names of categories/sub-categories have to be unique.
• Numbers as category names are not allowed (though names may contain numbers).