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I am conducting analysis on 'Multi-Attribute Decision Making (MADM)', where I have two attributes (a1, a2) to characterize the quality of m alternative approaches. The first attribute, a1, is measured in percentage (hence does not have a unit), whereas the other has a unit. There are some alternatives that have zero as their first attribute value.

What I am aiming to do is to compare different alternatives and determine the 'best' alternative by assuming equal weighting of the attributes. I am aware that 'Multi-Attribute Decision Making' methods may yield to the problem called 'rank reversal' (i.e. ranking of the alternatives may change by adding new alternatives). However, as far as I understood (and tested), 'Weighted Product Model (1)' does not suffer from this issue. The issue that arise; however, is that I cannot directly use this method as it requires division of the scores of different alternatives.

I thought of increasing all the a1 values by a certain amount and it seems to be working fine. However, if I increase the a1 values more than a value, then rankings start to change. Do you know any other ways to use 'Weighted Product Model' to evaluate alternatives on the presence of zeros?

Reference: (1) Triantaphyllou, E. & Mann, S.H., 1989. An examination of the effectiveness of multi-dimensional decision-making methods: A decision-making paradox. Decision Support Systems, 5(3), pp.303–312.

banbar
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    This thread may also be of interest to you: [Creating an index of quality from multiple variables to enable rank ordering](http://stats.stackexchange.com/questions/9358/). – gung - Reinstate Monica Jun 13 '13 at 16:05
  • I faced the same problem. No answer yet. I have written Dr. Triantaphyllou whose many books contain WPM examples, all with not a single zero. Hoping he will have a solution or a workaround. Please let me know if you have come across one. Thanks Akram Najjar Hi no sooner did I write the comment than I found an article with some provisos. Effectively, we should not include zero's!! Here is the title and authors COMPARISON OF ENVIRONMENTAL CHARACTERISTICS OF FUNCTIONALLY EQUIVALENT DEVICES BY WEIGHTED PRODUCT METHOD L. De Napoli, S. Rizzuti and C. Rocco INTERNATIONAL DESIGN CONFERENCE - DESIGN 20 – Akram Najjar May 23 '17 at 07:01
  • I came up with a workaround, in which values quite close to zero are added to all models and their scores are calculated accordingly. This value is varied within a range of values to observe how the scores change. Here is the paper, which we had practised this exercise: [Spatio-temporal clustering for non-recurrent traffic congestion detection on urban road networks](http://www.sciencedirect.com/science/article/pii/S0968090X14002186) – banbar May 24 '17 at 08:32

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