AHP Consensus Indicator

The AHP consensus indicator, based on Shannon beta entropy (e.q. 1.1) for n criteria and k decision makers, was introduced in [1].

(1.1) Shannon beta entropy:

(1.2) Shannon alpha entropy:

(1.3) Shannon gamma entropy:

with

The similarity measure S (eq. 1.4) depends on the number of criteria, and we used a linear transformation to map it to a range from 0 to 1 (eq. 1.5)

(1.4)

(1.5) Consensus (0% to 100%):

In general Dα min = 1 and Dγ max = n. In the analytic hierarchy process (AHP) Dα min is a function of the maximum scale value M (M = 9 for the fundamental AHP scale) and the number of criteria n (eq. 1.6). The calculation of Dγ max was based on the assumption that respondents compare one distinct criterion M‑times more important than all others (eq. 1.7).

(1.6)

(1.7)

This assumption is actually an unnecessary constrain, because even when the number of decision makers is less than the number of criteria, both  can prioritize a complementing set of criteria as most important and as a result all consolidated criteria weights are equal. Therefore eq. 1.7 can be simplified to:

 (1.8)

As a result we get the AHP consensus indicator with:

(1.9)

(1.10) AHP Consensus: Equation (1.10) is used in the latest updated of the AHP excel template and the AHP-OS online software.

Reference

[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

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?

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.

Other articles:

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.

Continue reading Group Decision Making with AHP-OS

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.

About the author

Participate in an AHP group session – AHP practical example

This is a practical example of an AHP group session, using AHP-OS, where you can input your judgment and see the consolidated group result of all participants.

Imagine, you plan to buy a tablet computer. Decide on the importance of criteria, like display size, battery life etc. Click on the image below, input your name, click on AHP and start the pairwise comparisons.

What feature is more important and how much on a scale from 1 to 9? Once finished,  submit your input for group evaluation and see the results.

Result for 176 Participants:

View the detailed group results.

AHP group decision making

The figure below shows, how a group session is conducted to determine group priorities using BPMSG’s AHP online system. The group session chair initiates a group session (You need to be registered and logged in). A six character session code is generated. Participants can use this session code to log into the group session and provide their judgments. Try out a practical example, where you can participate and input your judgments

Example showing the result for two participants. See also my post about group decision making.

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AHP Online Software – Version 2014-03-15

I just released a new version of my AHP Online Software. It includes the possibility to do group sessions: several participants can make the pairwise comparison, and you can view the consolidated group result.

After definition of the hierarchy using the AHP online system, click on “New Group Session” (You need to login  as  registered user). A session code will be provided. Enter your name, start the pairwise comparisons and submit for group evaluation.

Note down the session code, and provide the link, shown under the active group session, to your participants to log in and do their pairwise comparisons.

You can view the group result, once data are submitted, with a click on “View group results”. You might always review results by using the session code and your name using the AHP Group Session link.

Please be aware that this version is still a beta version. Kindly feedback any problems or bugs.

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.

Download (pdf):

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.

Group-DecisionThere 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.

Feedback and Comments are welcome!

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