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Aggregation and consensus for preference relations based on fuzzy partial orders

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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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Notes

  1. 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.

  2. 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.

  3. 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.

References

  • Atanassov, K. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 80, 87–96.

    Article  MATH  Google Scholar 

  • Barrenechea, E., Fernandez, J., Pagola, M., Chiclana, F., & Bustince, H. (2014). Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. Application to decision making. Knowledge-Based Systems, 58, 33–44.

    Article  Google Scholar 

  • Beliakov, G., Bustince, H., & Calvo, T. (2015). A practical guide to averaging functions. Berlin: Springer.

    Google Scholar 

  • Beliakov, G., & James, S. (2015). Unifying approaches to consensus across different preference representations. Applied Soft Computing, 35, 888–897.

  • Beliakov, G., James, S., & Wilkin, T. (2015). Construction and aggregation of preference relations based on fuzzy partial orders. In Proceeddings of FUZZIEEE, Istanbul, Turkey.

  • Bosch, R. (2006). Characterizations of voting rules and consensus measures. Ph.D. Thesis. Universiteit van Tilburg.

  • Bustince, H., Beliakov, G., Pradera, G., Dimuro, B. B., & Mesiar, R. (2016). On the definition of penalty functions in aggregations. Fuzzy Sets and Systems. doi:10.1016/j.fss.2016.09.011.

  • Calvo, T., & Beliakov, G. (2010). Aggregation functions based on penalties. Fuzzy Sets and Systems, 161, 1420–1436.

    Article  MATH  MathSciNet  Google Scholar 

  • Calvo, T., Mesiar, R., & Yager, R. R. (2004). Quantitative weights and aggregation. IEEE Transactions on Fuzzy Systems, 12(1), 62–69.

    Article  Google Scholar 

  • Chiclana, F., Herrera-Viedma, E., Alonso, S., & Herrera, F. (2009). Cardinal consistency of reciprocal preference relations: A characterization of multiplicative transitivity. IEEE Transactions on Fuzzy Systems, 17(1), 14–23.

    Article  Google Scholar 

  • De Baets, B., & De Meyer, H. (2005). Transitivity frameworks for reciprocal relations: Cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems, 152, 249–270.

    Article  MATH  MathSciNet  Google Scholar 

  • De Baets, B., De Meyer, H., De Schuymer, B., & Jenei, S. (2006). Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare, 26, 217–238.

    Article  MATH  MathSciNet  Google Scholar 

  • Fodor, J., & Roubens, M. (1994). Fuzzy preference modelling and multicriteria decision support. Dordrecht: Springer Science+Business Media.

    Book  MATH  Google Scholar 

  • Freson, S., De Meyer, H., De Baets, B. (2010). An algorithm for generating consistent and transitive approximations of reciprocal preference relations. In Proceedings of the 13th international conference on information processing and management of uncertainty (IPMU 2010) (pp. 564–573). Dortmund, Germany.

  • García-Lapresta, J., & Llamazares, B. (2010). Preference intensities and majority decisions based on difference of support between alternatives. Group Decision and Negotiation, 19, 527–542.

    Article  Google Scholar 

  • García-Lapresta, J. L., Pérez-Román, D. (2010). Consensus measures generated by weighted Kemeny distances on weak orders. In Proceedings of ISDA (pp. 463–468). Cairo, Egypt.

  • Grabisch, M., Marichal, J. L., Mesiar, R., & Pap, E. (2009). Aggregation functions. Cambridge: Cambridge University Press.

    Book  MATH  Google Scholar 

  • Herrera-Viedma, E., Herrera, F., & Chiclana, F. (2002). A consensus model for multiperson decision making with different preference structures. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 32, 394–402.

    Article  MATH  Google Scholar 

  • Herrera-Viedma, E., Herrera, F., Chiclana, F., & Luque, M. (2004). Some issues on consistency of fuzzy preference relation. European Journal of Operational Research, 154(1), 98–109.

    Article  MATH  MathSciNet  Google Scholar 

  • Llamazares, B., Pérez-Asurmendi, P., & García-Lapresta, J. (2013). Collective transitivity in majorities based on difference in support. Fuzzy Sets and Systems, 216, 3–15.

    Article  MATH  MathSciNet  Google Scholar 

  • Pérez, I. J., Cabrerizo, F. J., Alonso, S., & Herrera-Viedma, E. (2014). A new consensus model for group decision making problems with non-homogeneous experts. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 44(4), 494–498.

    Article  Google Scholar 

  • Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems, 12, 117–131.

    Article  MATH  MathSciNet  Google Scholar 

  • Tanino, T. (1990). On group decision making under fuzzy preferences. In J. Kacprzyk & M. Fedrizzi (Eds.), Multiperson decision making using fuzzy sets and possibility theory (pp. 172–185). Norwell, MA: Kluwer.

  • Wu, J., & Chiclana, F. (2014). Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building. Knowledge-Based Systems, 71, 187–200.

    Article  Google Scholar 

  • Yager, R. (1988). On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions on Systems, Man and Cybernetics, 18, 183–190.

    Article  MATH  MathSciNet  Google Scholar 

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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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