To make the results easier to interpret, we define a descriptive word scale for the consensus range from zero to unity. For this we analyzed the consensus within 140 hierarchy nodes (a set of criteria or sub-criteria within a decision hierarchy) of 35 AHP group decision projects. It could be shown (Fig. 1) that the consensus SAHP is normal distributed with a mean value of 64 % ± 3 %. With a 99.5% probability the consensus of all projects lies between 28 % and 99 %. Therefore we divided the range of the scale in four equal segments from 50 % to 100 % (going from ‘low’ to ‘very high’), and defined the consensus for values below 50 % as ‘very low’.
Table 1 Qualitative wording scale for AHP consensus indicator
Consensus SAHP
< 0.5
0.5 – 0.625
0.625 – 0.75
0. 75 – 0.87.5
> 0.875
Word Scale
Very low
low
moderate
high
Very high
Switching from the consensus indicator SAHP to the relative homogeneity S shifts the mean value from 64 % to 70 %, which can be explained by the fact that in AHP we have a limited 1 to 9 scale and Hα,min is a function of the maximum scale value.
Figure 1 Distribution of consensus SAHP in blue actual values, in red normal distribution
Since April 2022 a new feature of AHP-OS, Group Consensus Cluster Analysis is available. It can be reached from the AHP-OS main page.
The idea of the program is to cluster a group of decision makers into smaller subgroups with higher consensus. For each pair of decision makers the similarity of priorities is calculated, using Shannon alpha and beta entropy. The result is arranged in a similarity matrix and sorted into clusters of higher similarity based on a consensus threshold.
In order to use the program, you first need to load a priority json file, exported from the AHP-OS Group result menu, containing the priorities of all participants:
Group Result Menu – Export priorities using Priorities (json).
Once downloaded to your computer, you can import this file via the Group Consensus Menu:
Group Consensus Menu
Click on Browse… to select the file; then click Analyze.The result is structured in
Input data
Threshold table
Result for selected node and a
Similarity Matrix
Input Data
Project session code, selected node (default: pTot), number of categories, number of participants and scale are shown. pTot stands for the global priorities of a hierarchy.
Threshold Table
The program calculates the number of clusters and number of unclustered participants based on a similarity threshold in the range between 70% and 97.5% in steps of 2.5%. For each step the values are displayed in the threshold table.
Consensus Threshold Table
Automatically the optimal threshold is determined.
In this case as 0.85 with 2 clusters and no unclustered members. If you want to change, for example the number of clusters to 3, you can enter 0.9 as new threshold in the AHP Group Consensus Menu manually.
Manual Threshold input field in the Group Consensus Menu
In the menu you also find a drop-down selection list for all nodes of the project. With Load new data another json file can be loaded.
Result for selected Node
First the AHP group consensus S* or relative homogeneity S for the whole group is shown, followed by the number of clusters. Next, for each cluster (subgroup) S* or S of the subgroup and the number of members in this cluster are displayed. Individual members are shown with a number and their name. The participants number corresponds to the number displayed on the project result page (Project Participants), so it is easy to select or deselect them by their number on the AHP-OS result page based on the result of the cluster analysis.
Similarity Matrix
The similarity matrix is a visualization of the clusters. Each cell (i,j) contains the AHP consensus S* or relative Homogeneity S for the pair of decision makers i and j in percent. Darker green color means higher values as show in the scale above the matrix. Clusters are always rectangles along the diagonal of the matrix, and are framed by borders.
Similarity Matrix
As you can see in the figure above, the program found two clusters with members 1,3,6,7,10,11,12 respectively 2,4,5,8,9, and one unclustered member 13. In this example the group consensus without clustering is 52.4% (low), the consensus for subgroup 1 is 80.5% (high) and subgroup 2 80.7% (high). This means that within the group there are two individual parties in higher agreement. You can easily go back to the project’s group result page to analyze the consolidated priorities for each group by selecting the individual participants.
Once the number of participants exceeds 40, the similarity matrix is shown without values in order to better fit on the output page.
Example of the similarity matrix with 72 participants. You can clearly identify three clusters.
References
Goepel, K.D. (2022). Group Consensus Cluster Analysis using Shannon Alpha- and Beta Entropy. Submitted for publication.Preprint
Goepel, K.D. (2018). Implementation of an Online Software Tool for the Analytic Hierarchy Process (AHP-OS). International Journal of the Analytic Hierarchy Process, Vol. 10 Issue 3 2018, pp 469-487, https://doi.org/10.13033/ijahp.v10i3.590
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.
If you need a quick calculation of diversity indices from your sample data, you might use my online diversity calculator here. Select the number of categories/classes (2 to 20) and input your samples data (positive integer or decimal numbers). As a result the following parameters and diversity indices will be calculated:
Richness
Berger-Parker Index
Shannon Entropy (nat)
Shannon number equivalent (true diversity of order 1)
The 12th International Symposium on the Analytic Hierarchy Process – Multi-criteria Decision Making – took place under the theme “Better world through better decision making” from June 23rd to 26th in Kuala Lumpur, Malaysia.
Organized by the International Islamic University Malaysia (IIUM), scientists and experts from all continents presented and discussed the latest theoretical developments in AHP and its application in the areas of environment, transportation, CSR, healthcare, SCM, banking and finance, manufacturing, education, IT/IS and group decision making. After the official opening and a welcome speech by Prof. Thomas L. Saaty – connected via video from US – approx. 100 papers were presented in several parallel sessions. The successful meeting ended with a key note speech by Prof. William C. Wedley, “AHP/ANP – Before, Present and Beyond”, and two panel discussions, one about group decision making and the other about publishing AHP/ANP papers.
A half-day tour to interesting places in K.L and the following Gala Dinner with the award giving ceremony gave delegates opportunity for some relaxation and networking.
Many thanks to the organizers, have a look at some impression from the meeting in the video.
Your feedback is always welcome!
You might find my paper, presented on the ISAHP 2013, for download here.
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.
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.
Camera settings depend on the type of recorded video. The Canon XA-10 has only two full HD (1920×1080) recording modes: FXP and MXP. Based on these, most of my videos are recorded in FXP mode as it is a good trade-off between quality and file size.
Recording Modes of Canon XA-10
Edit and Archive
For video editing I use Adobe Premier Elements 10. Editing is done without changing codec (H264), resolution, interlacing, etc. As I have the PAL version of the XA-10, the Adobe project settings are AVCHD Full HD 1080i 25 under PAL. Once the video is edited, I render the clip with the highest quality settings (2 Pass VBR, Render at max. depth, Macro block Adaptive Frame-Field Coding), and a maximum bitrate corresponding to the source (FXP: 17 Mbps) for archiving. In Adobe Premier Elements 10 and for XA-10 FXP mode the settings are:
Archive settings in Adobe Premier Elements 10 for XA-10 FXP mode
Target Media
Depending on the target media the clip to be published is adjusted in codec, resolution etc. I have predefined settings for:
Standard clips to be watched on a PC:
MP4 – PAL DV Widescreen SD – HiQ (576p, VBR 3/6 MBps)
Tablet/phones:
MP4 – PAL DV Widescreen SD – LoQ(576p, VBR 1.3/2.6 Mbps)
Youtube as basic HD clip:
MP4 – HD 720p 25 (720p, VBR 2.5/5 Mbps)
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.
Concepts, Methods and Tools to manage Business Performance
Dear Friends, dear Visitors,
time for an update on my BPMSG welcome page! Being quite busy the last half year, I didn’t work so much on major articles or videos, but at least I tried to keep my site current with some regular updates.
Related to the analytical hierarchy process (AHP), you might find information about the consistency ratio (CR). CR is one of the most critical issue in the practical application of AHP, as it seems to be difficult for many decision makers to fulfill Saaty’s “ten-percent rule-of thumb”. The way out: either you accept higher ratios (up to 0.15 or even 0.2), modify the judgements in the pair-wise comparisons, or you use the balanced scale instead of the standard AHP 1 to 9 scale. All three can be done in my updated AHP template from Februar 2013.
As I received many requests to extend the number of participants to more than 10, here the detailed procedure, how you can do it by yourself. Extending the number of criteria beyond 10 is more complex and not recommended by me. If you actually have more than 10 criteria please try to group in sub-groups. At the moment I don’t have any planes to extend the number of criteria to more than ten.
I also started a new topic: Diversity. Triggered by some business related questions, I found out that the concept of diversity – as applied in ecology – is very universal, and can be applied in many business areas. You can watch my introduction as video:
I already applied the concept in several areas, and even developed a new consensus indicator for group decision making based on the partitioning of the Shannon entropy. A paper is submitted for the ISAHP conference in June, and after the event I will place a copy of the paper on my site for download.
For those of you, interested in the topic of diversity and the partitioning in alpha (within group) and beta (in-between group) components my free BPMSG Diversity Calculator could be a useful tool.
Now please enjoy your visit on the site and feel free to give me feedback –
it’s always appreciated.
As I received many requests to extend the number of participants in my AHP excel template, here a short information how to use it for more than 20 participants. There are two possibilities
Use several templates, each of them for up to 20 participants, and then combine the consolidated results in an additional summary template.
Modify the template.
As the template is quite complex, I strongly recommend to use the first possibility. But if you really want to modify the template itself, follow the step-by-step instruction below. This instruction does not include the AHP consensus indicator calculation.
Unprotect sheet In20; create a copy of the sheet In20 and rename to In21.
Go to “Formulas – Name Manager” and delete name Matrix20 with scope In21.
Mark matrix cells of the decision matrix in In21 (C79:L88), and define new name Matrix21 with scope workbook.
Go to Sheet multInp, unprotect sheet. Add additional matrix, e.g. copy/paste from matrix 20 (2 matrices per rows, same structure as for matrix 1-10).
Mark content cells of new matrix and define new name “m_p21”
Set it {=Matrix21} ( {} = array function, see below).
Mark the consolidated matrix (B9:K18), and modify the formula
{=(M9:V18*B22:K31* …*B74:K83)^(1/N4)} to include the added participant’s matrix.
Go to sheet Summary, unprotect sheet.
Mark matrix starting at line 38, and add new matrix m_p11 in the formula: {=IF(p_sel>0;CHOOSE(p_sel; m_p1; m_p2; … ; m_p20; m_p21);MatrixC)}.
Select field C7 (number of participants). Menu “Data – Data Validation”:
change range from 1 to 20 to 1 to 21.
Continue in the same way for additional participants.
Note: {} is the Excel array function: mark cell area, and use Ctrl-Shift-Enter.
All matrices in the input sheets are named Matrixn, n = 1 to max. number of participants. (Matrix1, Matrix2, etc.)
The matrices in the multInp sheet are named “m_pn” (m_p1, m_p2, etc.)
The 12th International Symposium on the Analytic Hierarchy Process – Multi-criteria Decision Making – took place under the theme “Better world through better decision making” from June 23rd to 26th in Kuala Lumpur, Malaysia.
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.
