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Part of the book series: The Handbooks of Fuzzy Sets Series ((FSHS,volume 1))

Abstract

The problem of aggregating numerical values is addressed in this chapter. The first part deals with the aggregation of criteria into a single one. Properties which are suitable for this case are presented, together with the most common aggregation operators. A special section is devoted to ordered weighted averaging (OWA) operators, and fuzzy integrals. Then, relation between properties, links between operators, characterization of some operators are presented. An important section is devoted to the behavioral analysis of OWA operators and fuzzy integrals, linking values of parameters with the attitude of the decision maker. The last section of this first part is concerned with the problem of identification of operators in a practical problem, a key issue in every application. The second part is devoted to special aspects in aggregation of preferences, in a multiattribute context. The Pareto principle is explained, and then the agreement-discordance principle, which is the basis of ELECTRE III and IV, is addressed.

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References

  • Aczel, J. and Alsina, C. (1987). Synthesizing judgements: a functional equation approach. Math. Modeling.

    Google Scholar 

  • Baas, S. and Kwakernaak, H. (1977). Rating and ranking of multiple-aspect alternatives using fuzzy sets. Automatica, 13:47–58.

    Article  MathSciNet  MATH  Google Scholar 

  • Brans, J., Mareschal, B., and Vincke, P. (1984). Promethee: a new family of outranking methods in multicriteria analysis. In Brans, J., editor, Operational Research′84. North Holland.

    Google Scholar 

  • Brans, J. and Vincke, P. (1985). A preference ranking organization method. Management Science, 31:647–656.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, S. and Hwang, C. (1992). Fuzzy Multiple Attribute Decision Making. Springer-Verlag.

    Google Scholar 

  • Denneberg, D. (1994). Non-Additive Measure and Integral. Kluwer Academic.

    Google Scholar 

  • Dubois, D. and Prade, H. (1984). Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory. In Zimmermann, H.-J., Zadeh, L., and Gaines, B., editors, Fuzzy Sets and Decision Analysis, volume 20 of TIMS Studies in the Management Sciences, pages 209–240.

    Google Scholar 

  • Dubois, D. and Prade, H. (1985). A review of fuzzy set aggregation connectives. Information Sciences, 36:85–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1986). Weighted minimum and maximum operations in fuzzy set theory. Information Sciences, 39:205–210.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1991). Fuzzy sets in approximate reasoning, part 1: inference with possibility distributions. Fuzzy Sets and Systems, 40:143–202.

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois, D., Prade, H., and Testemale, C. (1988). Weighted fuzzy pattern matching. Fuzzy Sets & Systems, 28:313–331.

    Article  MathSciNet  MATH  Google Scholar 

  • Dyckhoff, H. and Pedrycz, W. (1984). Generalized means as model for compensative connectives. Fuzzy Sets & Systems, 14:143–154.

    Article  MathSciNet  MATH  Google Scholar 

  • Filev, D. and Yager, R. (1994). Learning OWA operator weights from data. In Third IEEE International Conference on Fuzzy Systems, pages 468–473, Orlando.

    Google Scholar 

  • Fodor, J., Marichal, J., and Roubens, M. (1995). Characterization of the ordered weighted averaging operators. IEEE Tr. on Fuzzy Systems, 3(2):236–240.

    Article  Google Scholar 

  • Fodor, J. and Ovchinnikov, S. (1995). On aggregation of t-transitive fuzzy binary relations. Fuzzy Sets and Systems, 72:135–145.

    Article  MathSciNet  MATH  Google Scholar 

  • Fodor, J. and Roubens, M. (1994). Fuzzy Preference Modelling and Multi-Criteria Decision Aid. Kluwer Academic Publisher.

    Google Scholar 

  • Fodor, J. and Roubens, M. (1995). On meaningfulness of means. Journal of Computational and Applied Mathematics, 64:103–115.

    Article  MathSciNet  MATH  Google Scholar 

  • Grabisch, M. (Grabisch, a) Alternative representations of discrete fuzzy measures for decision making. Int. J. of Uncertainty, Fuzziness, and Knowledge Based Systems. to appear.

    Google Scholar 

  • Grabisch, M. (Grabisch, b) Fuzzy integral as a flexible and interpretable tool of aggregation. In Bouchon-Meunier, B., editor, Aggregation of evidence under fuzziness. Physica Verlag. to appear.

    Google Scholar 

  • Grabisch, M. (Grabisch, c) k-order additive discrete fuzzy measures and their representation. Fuzzy Sets and Systems. to appear.

    Google Scholar 

  • Grabisch, M. (1994). Fuzzy integrals as a generalized class of order filters. In European Symposium on Satellite Remote Sensing, Roma, Italy.

    Google Scholar 

  • Grabisch, M. (1995a). Fuzzy integral in multicriteria decision making. Fuzzy Sets & Systems, 69:279–298.

    Article  MathSciNet  MATH  Google Scholar 

  • Grabisch, M. (1995b). A new algorithm for identifying fuzzy measures and its application to pattern recognition. In Int. Joint Conf. of the 4th IEEE Int. Conf. on Fuzzy Systems and the 2nd Int. Fuzzy Engineering Symposium, pages 145-150, Yokohama, Japan.

    Google Scholar 

  • Grabisch, M. (1996a). Alternative representations of discrete fuzzy measures for decision making. In 4th Int. Conf. on Soft Computing, Iizuka, Japan.

    Google Scholar 

  • Grabisch, M. (1996b). The application of fuzzy integrals in multicriteria decision making. European J. of Operational Research, 89:445–456.

    Article  MATH  Google Scholar 

  • Grabisch, M. (1996c). k-order additive fuzzy measures. In 6th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), pages 1345-1350, Granada, Spain

    Google Scholar 

  • Grabisch, M. (1997). Alternative representations of OWA operators. In Yager, R. and Kacprzyk, J., editors, The Ordered Weighted Averaging Operators: Theory, Methodology, and Practice, pages 73-85. Kluwer Academic.

    Google Scholar 

  • Grabisch, M., Nguyen, H., and Walker, E. (1995). Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference. Kluwer Academic.

    Google Scholar 

  • Kolmogoroff, A. (1930). Sur la notion de moyenne. Atti delle Reale Accademia Nazionale dei Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez., 12:323–343.

    Google Scholar 

  • Krantz, D., Luce, R., Suppes, P., and Tversky, A. (1971). Foundations of measurement, volume 1: Additive and Polynomial Representations. Academic Press.

    Google Scholar 

  • Mizumoto, M. (1989a). Pictorial representations of fuzzy connectives, part I: Cases of t-norms, t-conorms and averaging operators. Fuzzy Sets & Systems, 31:217–242.

    Article  MathSciNet  Google Scholar 

  • Mizumoto, M. (1989b). Pictorial representations of fuzzy connectives, part II: case of compensatory operators and self-dual operators. Fuzzy Sets & Systems, 32:45–79.

    Article  MathSciNet  MATH  Google Scholar 

  • Mousseau, V. (1992). Analyse et classification de la littérature traitant de l’importance relative des critères en aide multicritère à la décision. Recherche Opérationnelle/Operations Research, 26(4):367–389.

    MATH  Google Scholar 

  • Murofushi, T. (1992). A technique for reading fuzzy measures (I): the Shapley value with respect to a fuzzy measure. In 2nd Fuzzy Workshop, pages 39–48, Nagaoka, Japan. In Japanese.

    Google Scholar 

  • Murofushi, T. and Soneda, S. (1993). Techniques for reading fuzzy measures (III): interaction index. In 9th Fuzzy System Symposium, pages 693–696, Sapporo, Japan. In Japanese.

    Google Scholar 

  • Murofushi, T. and Sugeno, M. (1991). A theory of fuzzy measures. Representation, the Choquet integral and null sets. J. Math. Anal. Appl., 159(2):532–549.

    Article  MathSciNet  MATH  Google Scholar 

  • Murofushi, T. and Sugeno, M. (1993). Some quantities represented by the Choquet integral. Fuzzy Sets & Systems, 56:229–235.

    Article  MathSciNet  MATH  Google Scholar 

  • Nijkamp, P., Rietveld, P., and Voogd, H. (1990). Multicriteria Evaluation in Physical Planning. North Holland.

    Google Scholar 

  • Norwich, A. and Turksen, I. (1984). A model for the measurement of membership and the consequences of its empirical implementation. Fuzzy Sets and Systems, 12:1–25.

    Article  MathSciNet  MATH  Google Scholar 

  • O’Hagan, M. (1990). Using maximum entropy-ordered weighted averaging to construct a fuzzy neuron. In 24th Annual IEEE Asilomar Conf. on Signals, Systems and Computers, pages 618–623, Pacific Grove, CA.

    Google Scholar 

  • Orlovski, S. (1994). Calculus of decomposable properties, fuzzy sets, and decisions. Allerton Press.

    Google Scholar 

  • Ovchinnikov, S. (1990). Means and social welfare functions in fuzzy binary relation spaces. In Kacpzyk, J. and Fedrizzi, M., editors, Multiperson Decision Making Using Fuzzy Sets and Possibility Theory, pages 143–154. Kluwer, dordrecht edition.

    Chapter  Google Scholar 

  • Perny, P. (1992). Modélisation, agrégation et exploitation des préférences floues dans une problématique de rangement. PhD thesis, Univ. Paris-Dauphine.

    Google Scholar 

  • Perny, P. (1996). Multicriteria filtering methods based on agreement and discordance principles. Technical report, LAFORIA, Université Paris VI, France.

    Google Scholar 

  • Perny, P. and Roy, B. (1992). The use of fuzzy outranking relations in preference modelling. Fuzzy Sets & Systems, 49:33–53.

    Article  MathSciNet  MATH  Google Scholar 

  • Roberts, F. (1979). Measurement Theory. Addison-Wesley.

    Google Scholar 

  • Roy, B. (1978). Electre III: un algorithme de classement fondé sur une représentation floue des préférences en présence de critères multiples. Cahiers du Centre d’Etude de Recherche Opérationnelle, 20(1):32–43.

    Google Scholar 

  • Roy, B. and Bouyssou, D. (1994). Aide multicritère à la décision: Méthodes et Cas. Economica.

    Google Scholar 

  • Saaty, T. (1977). A scaling method for priorities in hierarchical structures. J. Math. Psychology, 15:234–281.

    Article  MathSciNet  MATH  Google Scholar 

  • Saaty, T. (1980). The Analytic Hierarchy Process. McGraw Hill, New York.

    MATH  Google Scholar 

  • Shapley, L. (1953). A value for n-person games. In Kuhn, H. and Tucker, A., editors, Contributions to the Theory of Games, Vol. II, number 28 in Annals of Mathematics Studies, pages 307-317. Princeton University Press.

    Google Scholar 

  • Silvert, W. (1979). Symmetric summation: a class of operations on fuzzy sets. IEEE Tr. on Systems, Man, and Cybernetics, 9:657–659.

    Article  MathSciNet  MATH  Google Scholar 

  • Siskos, J. and Yannacopoulos, D. (1985). An ordinal regression method for building additive value functions. Investigação Operacional, 5(1):39–53.

    Google Scholar 

  • Sugeno, M. (1974). Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology.

    Google Scholar 

  • Trillas, E. and Alsina, C. (1992). Some remarks on approximate entailment. Int. J. Approximate Reasoning, 6:525–533.

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R. (Yager, a) Quantifier guided aggregation using OWA operators. International Journal of Intelligent Systems.

    Google Scholar 

  • Yager, R. (1978). Fuzzy decision making using unequal objectives. Fuzzy Sets and Systems, 1:87–95.

    Article  MATH  Google Scholar 

  • Yager, R. (1981). A new methodology for ordinal multiple aspect decisions based on fuzzy sets. Decision Sciences, 12:589–600.

    Article  MathSciNet  Google Scholar 

  • Yager, R. (1983). Quantifiers in the formulation of multiple objective decision functions. Information Sciences, 31:107–139.

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R. (1984). General multiple objective decision making and linguistically quantified statements. Int. J. of Man-Machine Studies, 21:389–400.

    Article  MATH  Google Scholar 

  • Yager, R. (1987). A note on weighted queries in information retrieval systems. J. of the American Society of Information Sciences, 38:23–24.

    Article  Google Scholar 

  • Yager, R. (1988). On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. Systems, Man & Cybern., 18:183–190.

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R. (1991). Connectives and quantifiers in fuzzy sets. Fuzzy Sets & Systems, 40:39–75.

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R. (1993). Families of OWA operators. Fuzzy Sets and Systems, 55:255–271.

    Article  MathSciNet  MATH  Google Scholar 

  • Yager, R. and Filev, D. (1994). Essentials of Fuzzy Modeling and Control. John Wiley and Sons, New York.

    Google Scholar 

  • Yager, R., Filev, D., and Sadeghi, T. (1994). Analysis of flexible structured fuzzy logic controllers. IEEE Transactions on Systems, Man and Cybernetics, 24:1035–1043.

    Article  Google Scholar 

  • Yoneda, M., Fukami, S., and Grabisch, M. (1993). Interactive determination of a utility function represented by a fuzzy integral. Information Sciences, 71:43–64.

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh, L. (1983). A computational approach to fuzzy quantifiers in natural languages. Computing and Mathematics with Applications, 9:149–184.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhukovin, V. (1987). A fuzzy multicriteria decision-making model. In Kacprzyk, J. and Orlovski, S. A., editors, Optimization Models Using Fuzzy Sets and Possibility Theory, pages 203–215. D. Reidei Publ. Co., Dordrecht, Boston, Lancaster, Tokyo.

    Google Scholar 

  • Zimmermann, H.-J. and Zysno, P. (1980). Latent connectives in human decision making. Fuzzy Sets & Systems, 4:37–51.

    Article  MATH  Google Scholar 

  • Zopounidis, C., Despotis, D., and Stavropoulou, E. (1995). Multiattribute evaluation of greek banking performance. Applied Stochastic Models and Data Analysis, 11:97–107.

    Article  Google Scholar 

  • Yager, R. R. and Kacprzyk, J. Eds. (1997) The Ordered Weighted Averaging Operators: Theory and Application Kluwer, Boston.

    MATH  Google Scholar 

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Grabisch, M., Orlovski, S.A., Yager, R.R. (1998). Fuzzy Aggregation of Numerical Preferences. In: Słowiński, R. (eds) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. The Handbooks of Fuzzy Sets Series, vol 1. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5645-9_2

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  • DOI: https://doi.org/10.1007/978-1-4615-5645-9_2

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