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) Shannon beta entropy:
(1.2) Shannon alpha entropy:
(1.3) Shannon gamma entropy:
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)
(1.5) 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).
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:
As a result we get the AHP consensus indicator with:
(1.10) AHP Consensus: Equation (1.10) is used in the latest updated of the AHP excel template and the AHP-OS online software.
 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
As I know from my own experience, manuals are seldomly read. On the other hand, a short guideline to complex software can be helpful, to use it effectively. I summarised the main menus of AHP-OS in a four page quick reference guide. The full manual is still available from the AHP-OS entry page (needs update …), and all details regarding methods and calculations are shown in my working paper about the AHP-OS software implemetation.
With the latest update of my AHP online software it is now possible to save judgments (pairwise comparisons) without completing the whole hierarchy evaluation. There are two scenarios, where this could be useful:
- You have a complex hierarchy with many nodes to be evaluated. Now participants can start a partial evaluation, save the judgments and complete the remaining nodes at a later time.
- As a participant you are only expert for a subset of nodes in the decision hierarchy. As a chair you can now ask participants to give their inputs for a few nodes of their expertise only.
Pairwise comparison input is started as usually: either using the link provided on the project session page, or using the link AHP Group Session on the AHP-OS main entry page. After providong session code and name, in case the participant hasn’t given any input, a message Ok. Group has x participants. Click “Go” to continue will be displayed. Nodes without judgment show the AHP button with red outline.
Continue reading AHP-OS Hierarchy Evaluation with Partial Inputs from Participants
My online software AHP-OS is mainly used in research. Projects handled with AHP-OS cover a wide range of applications like healthcare, climate, risk assessment, supplier selection, hiring, IT, marketing, environment, transport, project management, manufacturing or quality assurance. Some of these projects could contain sensitive data. Therefore I finally decided to secure the site with HTTPS to protect the site and users.
Continue reading AHP-OS now secured with HTTPS
The latest update of AHP-OS comprises of some minor changes to make the program flow easier to understand for participants w/o background in AHP.
- The group session input screen does no longer show the headline to login or register, as for participants there is no need to be registered.
- The text introduction was shortened to two and a half line of text.
- Menu buttons intended to be clicked are highlighted.
Continue reading AHP-OS Update Version 2017-08-31
For anyone, who is interested in the implementation of my free AHP-OS online software, and needs a reference:
Implementation of an online software for the Analytical Hierarchy Process
I hope to finalize the paper soon so that I can submit it for publication.
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.
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?
Continue reading Sensitivity Analysis in AHP
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It is now possible, to analyse the weight uncertainties in your AHP-OS projects. When you view the results (View Result from the Project Administration Menu), you see the drop-down list for different AHP scales and a tick box var is shown.
Tick var and click on Refresh. All priority vectors of your project will display the weight uncertainties with (+) and (-).
Continue reading Weight Uncertainties in AHP-OS
Concepts, Methods and Tools to manage Business Performance
Dear Friends, dear Visitors,
over the last four months I put in a lot of effort to improve the AHP-OS online tool. With several releases a simplified menu structure and new features were introduced.
- Delete individual participant’s inputs from an existing project.
- Update a project hierarchy or project description, as long as there is no input.
- Evaluate your AHP projects using different AHP judgment scales.
- Analyse weight uncertainties based on small randomised variations of input judgments.
The last two features are based on my recent study about the comparisons of different AHP scales. Up to date there was no recommendation, what scales to use, and I found a new approach to analyse and compare the scales based on simple analytic functions. This study is submitted for publication, and I hope it will not take too long, until it is available. You can find some more information already in my posting here.
The feature of analysing weight uncertainties is an innovative way of doing sensitivity analysis: all judgments are randomly varied by ±0.5 on the judgment scale, and for each variation the maximum and minimum out coming priorities are captured. I use 1000 variations, enough to get a relatively stable margin of errors for each weight. It gives you information, how “precise” a weight or ranking is in your specific project.
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!
For now, please enjoy your visit on the site and feel free to leave a comment – it is always appreciated.
Klaus D. Goepel,
Singapore, June 2017
Please give credit or a link to my site, if you use parts in your work, or make a donation to support my effort to maintain this website.
About the author
The original AHP uses ratio scales. To derive priorities, verbal statements (comparisons) are converted into integers from 1 to 9. This “fundamental AHP scale” has been discussed, as there is no thoretical reason to be restricted to these numbers and verbal gradations. In the past several other numerical scales have been proposed ,. AHP-OS now supports ten different scales:
- Standard AHP linear scale
- Logarithmic scale
- Root square scale
- Inverse linear scale
- Balanced scale
- Balanced-n scale
- Adaptive-bal scale
- Adaptive scale
- Power scale
- Geometric scale
Fig. 1 Mapping of the 1 to 9 input values to the elements of the decision matrix.
Continue reading AHP Judgment Scales