Welcome to BPMSG – Oct 2013

Dear Friends, dear Visitors,

Nearly half a year has passed since my last update in May on this page, and the year 2013 is soon coming to an end. In June, I had the opportunity to present my practical experience with the Analytic Hierarchy Process (AHP) on the International Symposium ISAHP 2013 in Kuala Lumpur, Malaysia. You can download my paper here. It was interesting to meet the experts from around the world, dealing with decision making methods, and listening to some of their presentations. A short video shows my impressions of the meeting and the nice touristic spots in K.L.

I was invited to the panel discussion, and could discuss my newly introduced AHP consensus indicator for group decisions. It is based on the concept of diversity. Like AHP, diversity is a very interesting topic, as it can be applied in so many different areas. Originating from information theory (Shannon 1948), it became a well-established concept in ecology and economy. I used the principles to develop a key performance indicator (KPI), describing the diversification of businesses and the quality of growth. Feel free to watch my videos or read my posts on this blog.

What is coming next?

There will be one more video on my YouTube channel, showing the application of Shannon diversity to measure the quality of growth. Looking at diversity and growth over time we can display a growth trajectory of a company, giving a clear picture about the direction, where the company is heading.

The larger project will be the implementation of my AHP template as an AHP online tool. The idea is that you can input your criteria and do the pair-wise comparison online, to get as a result the calculated priorities. So at the moment I am busy to practice some web scripting and become more familiar with php. A first small exercise was my online diversity calculator.

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

Klaus D. Goepel,

Singapore, Oct 2013

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

What is AHP?

AHP stands for analytic hierarchy process, and belongs to the multi-criteria decision making methods (MCDM). In AHP, values like price, weight, or area, or even subjective opinions such as feelings, preferences, or satisfaction, can be translated into measurable numeric relations. The core of AHP is the comparison of pairs instead of sorting (ranking), voting (e.g. assigning points) or the free assignment of priorities.

Read my simple write-up about AHP for people, who have not heared about the method. You might download it as pdf document.

Principia Mathematica Decernendi

or “Mathematical Principles of Decision Making” is the title of a book by Prof. Thomas L. Saaty, in which he describes his method for the mathematical treatment of decision problems, which he developed in the 70s. Called AHP, it is now used around the world in many different fields. AHP stands for analytic hierarchy process, and belongs to the multi-criteria decision making methods (MCDM) group.

In AHP, values like price, weight, or area, or even subjective opinions such as feelings, preferences, or satisfaction, can be translated into measurable numeric relations.

Mathematically, the method is based on the solution of an “eigenvalue problem” – but this is mentioned here only tangentially and is not meant to dissuade anyone.

Analytic hierarchy process

The basic steps in the solution of a decision problem using AHP are quite simple:

  1. Define the goal of the decision – what do I want to decide, for what purpose, and what are my alternatives?
  2. Structure the decision problem in a hierarchy – what are the categories and criteria that figure into my decision?
  3. Pair comparison of criteria in each category – e.g. blue or green? Which do I prefer, and by how much do I prefer one or the other color?
  4. Calculate the priorities and a consistency index – were my comparisons logical and consistent?
  5. Evaluate alternatives according to the priorities identified – what alternative optimum solution is there to the decision problem?

Sometimes alternatives are already implicitly defined by the problem and it is sufficient merely to define the priorities.

The core of AHP is the comparison of pairs instead of sorting (ranking), voting (e.g. assigning points) or the free assignment of priorities. Validation of the method in practical testing shows surprisingly good agreement with actual measured values.

One of the most interesting fields and a very current application of AHP is the identification of suspects by witnesses in criminal cases, where the candidates for identification are not shown all together or sequentially, but in pairs. AHP is then used to evaluate the results. Initial studies show that this increases the reliability of identification from 55 % to 83 % and reduces the false identification rate from 20 % to 17 %, and that the consistency index is a good measure of the reliability of statements by witnesses.

Example

To demonstrate how the method works, let us take a simple example. I want to buy an MP3 player. I have the choice of colors (pink, blue, green, black, red), storage (8, 16, 32, 64 Gbyte), and availability (immediate, 1 week, 1 month). The available models are:

  • Model A – pink, 32 Gbyte, immediate availability, USD 120
  • Model B – blue , 16 Gbyte, immediate availability, USD 120
  • Model C – black , 32 Gbyte, 1 week wait, USD 150
  • Model D – red, 64 Gbyte, 1 month wait, USD 150

We can structure the problem hierarchically as shown in Fig. 1.  In the solution process itself each element is compared by pairs in each category and sub-category, and the criteria are weighted.


Fig. 1 AHP hierarchy for solving a decision problem

Complex decision problems and networks

For complex decision problems a two-layer model can be introduced, in which hierarchies are examined separately by the criteria Benefits (B), Opportunities (O), Costs (C) and Risks (R). This is known as the BOCR model. The problem is then evaluated using the simple formula (B*O)/(C*R) (multiplicative) or (B+O)-(C-R) (additive).

The analytical network process (ANP) is a further development of AHP. In it, the decision problem is modeled not as a hierarchy, but as a network. However, its practical application and mathematical treatment are much more involved.

Applications

AHP has been used successfully in many institutions and companies. Although the method is so universal, it is still simple enough to execute in Excel. One of AHP’s great advantages is the ability to use it for group decisions, in which all participants evaluate pairs and the group result is determined as the mathematically optimum consensus. In practice the solutions arrived at by the method are well accepted, since the results are objective and free of political influence.

Examples of projects in a business context are the weighting of key performance indicators (KPI) or the identification of key strategies for sustained growth.

Klaus D. Goepel – Singapore Aug 2013. 

Calculate priorities using my AHP online calculator or handle a complete AHP project using my AHP online software AHP-OS.

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!

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.

Welcome to BPMSG – May 2013

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.

Klaus D. Goepel,
Singapore, May 2013

How to extend the AHP Excel Template for more Participants?

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 my AHP online Software.
  • 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.

  1. Unprotect sheet In20; create a copy of the sheet In20 and rename to In21.
  2. 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.
  3. 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.
  4. 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.)

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
  • Different scales added:
  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.

I appreciate any feedback! Please download the latest version from my AHP template download page.

Welcome to BPMSG – Dec 2012

Dear Friends, dear Visitors,

yesterday I realized 10000 visits on my website since April 2012, when I implemented the Piwik web statistics. Over the last couple of months the daily visitor frequency was actually increasing, doubling within the last 3 months. On my youtube channel http://www.youtube.com/bpmsg I am now slowly reaching 100,000 video views.

So first to all of you a big thank you, showing interest in the topics of bpmsg.com, and especially to those of you,  giving me feedback, as I can learn and progress from there. For me it also means to stay committed and keep the content interesting and updated.

The topic with the highest interest is AHP – the analytic hierarchy process, and many of you downloaded my AHP excel template. Actually, here I would really like even more feedback about your applications, just to get an idea, in what other areas my template is used. Some of them, as I received, are:

  • Asset management prioritisation
  • BPMSG AHP template as a teaching tool
  • Weights of textual elements that affect difficulty of a given text
  • Environmental quality
  • Threads to biodiversity
  • Green supply chain

In my last update of the template  I improved the accuracy of calculation  significantly, so please always use the latest version, and revisit the site from time to time, to get the latest update. Alternatively you might subscribe to the bpmsg newsfeed; the link is given in the footer of the page.

My latest topic “Diversity index as business KPI – the concept of diversity” seems also to gain some interest. My video on youtube  got in a short time more viewers than the previous video about operational and strategic business performance. For me it was intersting to apply the diversity concept in business performance, as I haven’t seen this before, and the mathematical concept, to measure diversity of species in a habitat (biodiversity), is quite well established . I am thinking to publish a second video, showing more practical applications of the diversity concept in a business context.

After starting my youtube channel in 2009, I gained more and more experience in making videos. You can  clearly see the difference, comparing one of my older videos with the latest ones. Now my camcorder – a Canon XA10 – is with me most of the time on my business trips or vacations. Therefore you also find some video travel impressions on this web site under the topic “others”. My last trip was to the Philippines showing the nice island of Bohol, as well as one of the world’s largest crater lakes on a lake on an island – Lake Taal.

Klaus Goepel,
Singapore, Dec 2012

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

Updated AHP Excel Template Version 11.12.12

AHP IconDue to feedback from several users, I revised the implementation of the power method for the calculation of the Eigenvector and Eigenvalue to improve the accuracy of my AHP excel template. The calculation sheet ‘8×8 in the workbook was completely reworked. My tests show a significant increase in accuracy. As an example see my updated post AHP template – numerical accuracy.

By default the number of iterations is now set to 12.  The check value in sheet ‘8×8 cell B33 shows the sum of all matrix elements solving the Eigenvalue equation (AI*λ) x = 0 with A the Decision matrix, λ = estimated principal Eigenvalue and x = estimated Eigenvector. The ideal check value is zero. With the example numbers given in the template the result is 5E-08.

Please let me know, if  you find any problems in the new version.

For the download of the latest version please go to the AHP template download page .

AHP template – numerical accuracy

Thanks to feedback from Mihail, here a few words about the numerical accuracy when using the AHP excel template.

AHP requires the calculation of the principal Eigenvalue, the weights are derived from the Eigenvector.  In my calculations I use the power method.  It is an iterative method, and  only one of several techniques that can be used to approximate the eigenvalues of a matrix.

Update 11.12.12

The whole calculation is shown in work sheet ’10×10′. I use 12 iterations; at the end of the sheet I do a check (the reverse calculation), using the Eigenvalue equation: (Aλ IX = 0,  with A the AHP matrix; λ the principal Eigenvalue, and X the estimated Eigenvector. The resulting check value in cell B33 shows the sum of all matrix element of the Eigenvalue equation using the iterated Eigenvector and Eigenvalue. Ideally it should be zero.

Update 9.5.14

From version 2014-05-09 onward the template shows the convergence of the power method, when calculating the eigenvalue. In the summary sheet a threshold (squared Euclidean distance d2) can be set, to show how many iterations it takes, until the change of the approximated eigenvector is below the given threshold. By default the value is set to Thresh: 1E-07. As the number of iterations in the template is fixed to 12, care should be taken if the value reaches 12.

Examples

Here a practical example comparing the results from the power method, as now implemented in my template, with  an example (7 criteria) given by Saaty in Int. J. Services Sciences, Vol. 1, No. 1, 2008 (p 86, table 2). The AHP matrix is:

1 9 5 2 1 1  1/2
 1/9 1  1/3  1/9  1/9  1/9  1/9
 1/5 3 1  1/3  1/4  1/3  1/9
 1/2 9 3 1  1/2 1  1/3
1 9 4 2 1 2  1/2
1 9 3 1  1/2 1  1/3
2 9 9 3 2 3 1

The result according Saaty is
(0.177,  0.019, 0.042, 0.116, 0.190, 0.129, 0.327) with consistency ratio of 0,022

The result from my AHP Excel template is
(0.1775, 0.0191, 0.0418, 0.1164, 0.1896, 0.1288, 0.3268) with CR 0f 0.022
exactly the same. The check value in sheet ‘8×8 is 4E-12.

More examples

Latest Excel template download

 

 

 

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