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.
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.
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.
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:
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.
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.
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 aijthe performance measure of alternative Aiwith respect to criterion Cj. Performance values are normalized.