Fuzzy Measures and Experts’ Opinion Elicitation

An Application to the FEEM Sustainable Composite Indicator

Abstract

To over pass the limits inherent in liner models, suitable aggregation operators are required, taking into account interactions among the criteria. This becomes more and more crucial in Decision Theory, where all the information can be inferred by one or more Experts, using an ad hoc questionnaire. This is the case of the FEEM SI sustainability index, a composite geo-referenced index which aggregates several economic, social and environmental dimensions-structured in a decision tree- into a single number between zero (the worst sustainable country) and one (the best one). Fuzzy measure (non additive measures) are here proposed for the aggregation phase. To this purpose, each intermediate node of the structure combines the values of the sub-nodes using a model based on second-order non additive measure. To infer the value of the measure for each node, a suitable questionnaire has been fulfilled by a set of Experts, and the obtained answers were processed using an optimization algorithm. To guarantee the strict convexity of the algorithm, the questionnaire needs to be carefully designed. The individual measures are subsequently aggregated and the numerical results permitted to compare the sustainability of all the considered territorial units.

Keywords

Fuzzy measures non-additive measures Choquet integral preference structure sustainability aggregation operators 

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References

  1. 1.
    Athanasoglou, S., Weziak-Bialowolska, D., Saisana, M.: Environmental Performance Index, – JRC Analysis and Recommendations, EPI-JRC, pp. 1–33 (2014)Google Scholar
  2. 2.
    Beliakov, G.: Construction of aggregation functions from data using linear programming. Fuzzy Sets and Systems 160, 65–75 (2009)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cardin, M., Giove, S.: Approximation of fuzzy measures using second order measures: Estimation of andness bounds. In: Masulli, F. (ed.) WILF 2013. LNCS (LNAI), vol. 8256, pp. 150–160. Springer, Heidelberg (2013)Google Scholar
  4. 4.
    Decanq, K., Lugo, M.A.: Weights in environmental indices of wellbeing: an overview. Econometric Review 32(1), 7–34 (2013)CrossRefGoogle Scholar
  5. 5.
    Despic, O., Simonovic, S.P.: Aggregation operators for decision making in water resources. Fuzzy Sets and Systems 115(1), 11–33 (2000)MATHCrossRefGoogle Scholar
  6. 6.
    FEEM Sustainability Index Methodological Report 2011, Fondazione Eni Enrico Mattei (2011), http://www.feemsi.org/documents/
  7. 7.
    Jones, D., Mehrdad, T.: Practical Goal Programming, vol. 141. Springer (2010)Google Scholar
  8. 8.
    Ishii, K., Sugeno, M.: A model of human evaluation process using fuzzy measure. International Journal of Man-Machine Studies 67, 242–257 (1996)Google Scholar
  9. 9.
    Grabisch, M., Nguyen, H.T., Walker, E.A.: Fundamentals of uncertainty calculi with applications to fuzzy inference. Kluwer Academic, Dordrecht (1995)CrossRefGoogle Scholar
  10. 10.
    Grabisch, M.: A new algorithm for identifying fuzzy measures and its application to pattern recognition. In: Proceedings of international 4th IEEE Conference on Fuzzy Systems, Yokohama, pp. 145–150 (1995)Google Scholar
  11. 11.
    Grabisch, M.: k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems 92, 167–189 (1997)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Grabisch, M., Roubens, M.: Application of the Choquet integral in multicriteria deci-sion making. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals – Theory and Applications, pp. 348–374. Physica Verlag (2000)Google Scholar
  13. 13.
    Ishii, K., Sugeno, M.: A model of human evaluation process using fuzzy measure. International Journal of Man-Machine Studies 67Google Scholar
  14. 14.
    Kojadinovic, I.: Quadratic distances for capacity and bi-capacity approximation and identification. A Quarterly Journal of Operations Research (in press), doi: 10.1007/s10288-006-0014-4; Marichal, J.-L., Roubens, M.: Determination of weights of interacting criteria from a reference set. European Journal of Operational Research 124, 641–650 (2000)Google Scholar
  15. 15.
    Kojadinovic, I.: Minimum variance capacity identification. European Journal of Operational Research 177, 498–514 (2007)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Marichal, J.: An axiomatic approach of the discrete Choquet integral as a tool to aggregate interacting criteria. The IEEE Transactions on Fuzzy Systems 8(6), 800–807 (2000)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Marichal, J.-L.: Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral. European Journal of Operational Research 155, 771–791 (2004)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Marichal, J.-L.: K-intolerant capacities and Choquet integrals. European Journal of Operational Research 177, 1453–1468 (2007)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Meyer, P., Roubens, M.: Choice, ranking and sorting in fuzzy Multiple Criteria Decision Aid. In: Figueira, J., Greco, S., Ehrgott, M. (eds.) Multiple Criteria Decision Analysis: State of the Art Surveys, pp. 471–506. Springer, New York (2005)Google Scholar
  20. 20.
    Miranda, P., Grabisch, M.: Optimization issues for fuzzy measures. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 7(6), 545–560 (1999)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Mori, T., Murofushi, T.: An analysis of evaluation model using fuzzy measure and the Choquet integral. In: Proceedings of 5th Fuzzy System Symposium, Kobe, Japan, pp. 207–212 (1989) (in Japanese)Google Scholar
  22. 22.
    Murofushi, T.: A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure. In: 2nd Fuzzy Workshop, Nagaoka, pp. 39–48 (1992) (in Japanese)Google Scholar
  23. 23.
    OECD/EC JRC, Handbook on Constructing Composite Indicators: Methodology and User Guide. OECD, Paris (2008)Google Scholar
  24. 24.
    Pinar, M., Cruciani, C., Giove, S., Sostero, M.: Constructing the FEEM sustainability index: A Choquet integral application. Ecological Indicator 39, 189–202 (2014)CrossRefGoogle Scholar
  25. 25.
    Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, Vol. II. Annals of Mathematics Studies, vol. 28, pp. 307–317. Princeton University Press, Princeton (1953)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Mediterranean Center for Climate Change (CMCC)BolognaItaly
  2. 2.Fondazione Eni Enrico MatteiVeniceItaly
  3. 3.University Cà FoscariVeniceItaly

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