AHP-OS Update July 2019

After many hours of work, I released a new version of my free AHP online system AHP-OS. The whole package was restructured in preparation to make it available for an installation on any other web server.

Beside a lot of under-the-hood improvements, the software now runs under php 7.2 and sqlite 3.10. It should be faster than before. I also prepared to switch from sqlite to MariaDB 10 SQL database, as the number of users is increasing steadily, and the sqlite file getting larger and larger.

New Features

For you, as a user, an additional feature has been implemented. You can now close a project for further pairwise comparison inputs. Once closed, participants can no longer input data.

Message for participants, when the project is closed

The project status can be toggled between open/closed in the project administration menu.

Project Administration Menu: additional button Toggle Project Status on the right.

The project status is shown in your AHP project list as 1 – open, 0 – closed.

As a project author you can now also start pairwise comparisons directly from the Project administration Menu.

Start pairwise comparisons for your project with PWC Input.

New Web Address

AHP-OS can now be easily accessed via the subdomain ahp.bpmsg.com . The package was moved from bpmsg.com/academic to bpmsg.com/ahp. If you have bookmarked the old address, you will be automatically redirected.

Incoming search terms:

  • https://ahp online tool

AHP Alternative Evaluation – Weighted Sum or Weighted Product Model?

In AHP the preference Pi of alternative Ai is usually calculated using the weigthed sum model (WSM), i.e. calculation of global priorities for alternatives results from the additive aggregation of local preferences and criteria weights.

In my online software AHP-OS users can also select the weighted product model (WPM), where alternatives are aggregated using the product instead of the sum (Goepel 2018). We call this – in contrast to the classical (additive) AHP – multiplicative AHP or MAHP.

AHP-OS Group Result Menu: tick the wpm box and refresh to get the MAHP results.

Ishizaka, Balkenborg and Kaplan (2011) have shown that the additive AHP will overrate alternatives with extreme ratings and penalize balanced ones. Bafahm and Sun (2019) examine in their paper some abnormal results of AHP, contradictory to common expectations and basis decision-making logic in very simple cases. These conflicting results can be easily avoided using WPM.

Krejci and Stoklasa (2018) clearly show in their paper the superiority of using the weighted product model over the weighted sum model for the purpose of deriving global priorities of alternatives.

Aggregation of local priorities of alternatives into global priorities in AHP should not be done using the weighted sum model (WSM). Instead, the Weighted Product Method (WSM) should be used.


Bafahm A., Sun M. (2019). Some Conflicting Results in the Analytic Hierarchy Process, International Journal of Information Technology & Decision Making Volume 18, Issue 02 (March 2019) Pages:419–443 https://doi.org/10.1142/S0219622018500517

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

Ishizaka A., Balkenborg D., Kaplan T. (2011). Influence of aggregation and measurement scale on ranking a compromise alternative in AHP. Journal of the Operational Research Society (2011) 62: 700. https://doi.org/10.1057/jors.2010.23

Krejci J., Stoklasa J. (2018). Aggregation in the analytic hierarchy process: Why weighted geometric mean should be used instead of weighted arithmetic mean. Expert Systems with Applications Volume 114, 30 December 2018, Pages 97-106 https://doi.org/10.1016/j.eswa.2018.06.060

AHP-OS and AHP Judgment Scales – Published Articles

My latest articles related to AHP:


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



AHP Judgment scales:

Goepel, K.D. (2018). Comparison of Judgment Scales of the Analytical Hierarchy Process — A New Approach. International Journal of Information Technology & Decision Making, published Dec 11, 2018


AHP Excel Template Update Version 2018-09-15

A new version of of the AHP Excel template with some major updates is now available for download. Based on the work of Tomashevskii (2014, 2015), errors for the resulting priorities/weights are shown.

Calculated weights with error indication

In addition the overall dissonance (ordinal inconsistency) according to Sajid Siraj (2011) is indicated. The zip file for download also contains the updated manual, showing the calculations and references.

Continue reading AHP Excel Template Update Version 2018-09-15

Incoming search terms:

  • project trade-off between criteria excel spreadsheet

AHP Excel Template Update Version 2018-08-22

In this latest version of the template, the balanced scale was replaced by the generalized balanced scale (balanced-n), and the adaptive scale was added. The maximum number of iterations for the power method was increased from 12 to 20.

If you need inputs for more than 20 participants, please contact the author. A version for up to 225 participants is available.

Go to download page.

For more information about AHP scales, please read my paper Comparison of Judgment Scales.

AHP Judgment Scales

A revised version of my paper can now be downloaded:

Goepel, K.D., Comparison of Judgment Scales
of the Analytical Hierarchy Process - A New Approach, Preprint of an article submitted for consideration in International Journal of Information Technology and Decision Making © 2017 World Scientific Publishing Company http://www.worldscientific.com/worldscinet/ijitdm (2017)

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:


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.


[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-OS Quick Reference Guide

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.


AHP-OS Hierarchy Evaluation with Partial Inputs from Participants

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:

  1. 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.
  2. 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

AHP-OS now secured with HTTPS

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