Abstract
We propose a framework for eliciting and aggregating pairwise preference relations based on the assumption of an underlying fuzzy partial order. We also propose some linear programming optimization methods for ensuring consistency either as part of the aggregation phase or as a pre- or post-processing task. We contend that this framework of pairwise-preference relations, based on the Kemeny distance, can be less sensitive to extreme or biased opinions and is also less complex to elicit from experts. We provide some examples and outline their relevant properties and associated concepts.
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The limits on humans’ ability to accurately assess such information with high granularity has been well documented for decades, e.g. with 7–9 levels seen as the ideal for Likert scales.
In Freson et al. (2010), different transitivity models have been described in terms of the cycle-transitivity framework proposed in Baets and Meyer (2005), Baets et al. (2006). In particular, it is argued that the oft-adopted additive consistency property \(p_{ik} = p_{ij}+p_{jk}-0.5\) should not correctly be referred to as a type of transitivity. In fact, the majority of transitivity conditions are more to do with consistency regarding strength of comparison between transitive 3-tuples. Where possible we will try to distinguish between transitivity of the underlying partial order and such consistency conditions.
We could consider data built from ratios, e.g. from numerical evaluations \(x_i,x_j\) we use the transformation \(p_{ij} = \max (0,\min (1,\log _{1.2} (x_i/x_j)))\) so that if the score for \(x_i\) is 20% higher than \(x_j\) then we have crisp preference. In this case, the transitivity condition still requires Eq. (9) to hold.
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Beliakov, G., James, S. & Wilkin, T. Aggregation and consensus for preference relations based on fuzzy partial orders. Fuzzy Optim Decis Making 16, 409–428 (2017). https://doi.org/10.1007/s10700-016-9258-4
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DOI: https://doi.org/10.1007/s10700-016-9258-4