## 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).

## 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.

• AHP 20 ESSA

## AHP Online Software – Latest Update 2016-10-29

Based on the request from some users I have modified the result display of my AHP-OS online software. Beside the resulting priorities it is now possible to see the underlying consolidated decision matrix, which results from the aggregation of individual judgments (AIJ) using the geometric mean.

### Decision Hierarchies

For decision hierarchies the result page shows the resulting local and global priorities (overall result) in hierarchical form, and the global priorities in descending order in a bar graph.

Then, in a breakdown by nodes, consolidated priorities, consolidated decision matrix and the priorities of individual participants are shown:

As in this example, the consolidated matrix for one participant is the same as the AHP decision matrix.

### Alternative Evaluation

For alternative evaluation the breakdown is done by criteria. It shows the consolidated preferences for all alternatives with respect to each of the criteria:

In this example the House A from the alternative houses A, B, C has the best match with respect to the criterion “Size”.

### Outlook

I am now working on the extension to let you download the complete set of input values, i.e. all underlying decision matrices, to excel. In the current version only the resulting AHP priorities are available.

## 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.

## A new Consensus Indicator in Group Decision Making with the Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) is one of the multi-criteria decision making methods helping decision makers in rational decision making using a mathematical method. AHP as a practical tool can be especially helpful, when making group decisions.

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 2013

### Group Decision Making

Group decisions are often made because decision problems can become very complex by nature; they could require special expertise and complementing skills, as they cannot be provided by a single person. Another reason could be the wish to spread responsibility or to get a higher commitment from a team for necessary actions as a consequence of the decision to be made.

There are different possible approaches to come to a decision. In the ideal case we get a consensus – an agreement through discussion and debate – but often a decision is a compromise. Group members readjust their opinions and give up some demands. Another way is a majority vote or a single leader’s final decision, based on his position and power.

In any case a possible disadvantage is that during group discussions a strong individual takes the lead, suppressing or ignoring others’ opinions and ideas (dominance), or people don’t want to speak up and conform to whatever is said (conformance).

Table 1: Reasons for group decision making and group decision approach

 Reasons for group decisions Group Decision Approach Special expertise Subject matter experts Complementing skills Different viewpoints/departments Spread of responsibility Board, committee members Higher commitment Team decision Consensus Agreement through discussion and debate Compromise Readjustment, giving up some – demands Majority vote Opinion of majority Single leader’s final decision

### The Analytic Hierarchy Process (AHP) in Group Decision Making

When using AHP with its questionnaire, these problems can be avoided. Each member of the group has to make judgment by doing a pairwise comparison of criteria in the categories and subcategories of the hierarchical structured decision problem. Advantages are:

• It is a structured approach to find weights for criteria and sub-criteria in a hierarchically structured decision problem.
• All participants’ inputs count; no opinion or judgment is ignored and all group members have to fill-out the questionnaire.
• Participants’ evaluation can be weighted by predefined (and agreed) criteria, like expertise, responsibility, or others, to reflect the actual involvement of decision makers.
• The consolidated group result is calculated using a mathematical method; it is objective, transparent and reflects the inputs of all decision makers.

From practical experience, especially the last point results in a usually high acceptance of the group result. Aggregation of individual judgments (AIJ) in AHP can be done using the geometric mean: each matrix element of the consolidated decision matrix is the geometric mean of the corresponding elements of the decision makers’ individual decision matrices. The outcome – consolidated weights or priorities for criteria in a category – can be used as group result for the calculation of global priorities in the decision problem.

### AHP Consensus Indicator

Although mathematically it is always possible to calculate a group result, the question remains, whether a calculated group result makes sense in all cases. For example, if you have two totally opposite judgments for two criteria, an aggregation will result in equal weights (50/50) for both criteria. In fact, there is no consensus, and equal weights may result in a deadlock situation to solve a decision problem.

Therefore, it will be necessary to analyze individual judgments, and find a measure of consensus for the aggregated group result. We use Shannon entropy and its partitioning in two independent components  (alpha and beta diversity) to derive a new AHP consensus indicator. Originating from information theory, the concept of Shannon entropy is well established in biology for the measurement of biodiversity. Instead of relative abundance of species in different habitats, we analyse the priority distribution of criteria among different decision makers.

### Further Reading, References and Examples of Practical Applications

The AHP consensus indicator is calculated in my free AHP Excel template. Group analysis by partitioning of  Shannon entropy in alpha and beta entropy can be done by transferring the calculated priorities (AHP priority vector) from each decision maker to the BPMSG Diversity calculator.

#### Incoming search terms:

• group decision making
• group decision

## Updated AHP Excel Template Version 8.5.2013

In this latest update I followed the several requests to extend the number of participants (decision makers); you now can use the template for up to 20 participants. In addition the weight of individual participants can be adjusted for the aggregation of individual judgments (AIJ). For example, if you have one expert in the group, you might want to give him/her evaluation a  x-time higher importance than the rest of participants. Then you simply change the weight in the input sheet from 1 to x. The calculation is done using the weighted geometric mean:

with cij = element of the consolidated decision matrix, aij(k) element of the decision matrix of participant k.

Kindly let me know in case you find any problem with this new version. Feedback is appreciated always! You can download this latestes version from my AHP template download page.

## Updated AHP Excel Template Version 08.02.13

An updated version of my AHP Excel template for multiple inputs is now available as version 08.02.13. Beside the extension from 8 to 10 criteria and from 7 to 20 participants some new features have been added. In the past it was sometimes difficult for participants to achieve a low consistency ratio. Now inconsistent comparisons in the input sheet will be highlighted, if the required consistency level is exceeded.  The level of consistency needed (“alpha” in the summary sheet) can also be changed from 0.1 (standard rule of thumb from Saaty) to higher values, for example 0.15 or 0.2. In addition another scale for the judgment can be chosen. For my projects I made good experience with the balanced scale.

A new feature is the consensus index. If you have more than 1 participant and do the group aggregation (select participant “0”), the consensus index is an indicator, how homogenous the judgment within the group was done. Zero percent means no consensus, all participants put their preference on different criteria;  100% means full consensus. Here the changes in detail:

## Summary sheet

• Number of criteria increased from 8 to 10
• Number of participants increased from 7 to 20
1. Linear standard scale
2. Log
3. Sqrt
4. InvLin
5. Balanced
6. Power
7. Geom.
• Alpha – allows to adjust consistency threshold (0.1 default)
• Consensus indicator for group aggregation added
• Geometric Consistency Index CGI added

## Input sheets

• Consistency ratio is calculated on each input sheet.
• Priorities are calculated and shown based on RGMM (row geometric mean method)
• Top three inconsistent pairwise comparisons highlighted (if CR>alpha)

## Known Issues

Thanks to feedback from Rick, sometimes there seems to be a problem with the correct display of weights beside the criteria in the summary sheet. If you face this problem, unprotect sheet summary. Select weigths (O18:O27). Click “conditional formating”, “clear rules”,”clear rules from selected cells”. Then the values will be displayed correctly, and you can format them in the way you want. It is a strange effect; it only appears on one of my PCs, on the other it works fine. I uploaded a modified version, but not sure whether it works for everyone.

## Strategic and Operational Business Performance

Decisions can have a short- or long-term impact; they can be strategic or operational. Measuring strategic or operational performance might require a different set of key performance indicators (KPIs). You need both: Operational KPIs help to measure the short term performance of an organization, strategic KPIs help to measure the implementation of a long term strategy.

This new podcast episode #13 explains the differences, using the picture of driving a car.

## BPMSG – One Year On-line

After being on-line with the website for 1 year, it’s time to change the entry page!

Here, on BPMSG.com you will find a couple of interesting articles related to business performance management, from a more practical than theoretical point of view.

Topics covered range from

A new topic was introduced just recently: “Value-driven leadership”. I started with a general article, and added a personality profiling tool (NEO-select). If interested, you might participate and send back the completed questionnaire to get a full report.

In the section other stuff  I post other interesting information, not fitting to the main topic of the site. For example a nice video of the olive-backed sunbird, a review of the Sennheiser Bluetooth Headset etc.

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 also be so kind, if you use parts in your web-site or blog, to give credit and a link to my site. I hope you will find some useful information, let me know your feedback or contact me directly for any suggestions.

Klaus Goepel, Mar 2012