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The Use of Fuzzy Sets for the Evaluation and Ranking of Objects

  • Didier Dubois
  • Henri Prade

Abstract

In an article that has since become a classic [4], Bellman and Zadeh proposed the theory of fuzzy sets as a conceptual framework for problems of choice with multiple criteria. The main contribution of that article was to emphasize that objectives and constraints can be represented by fuzzy sets which subsume elements of subjective preference. In this framework, the aggregation of criteria can be viewed as a problem of combining fuzzy sets by means of fuzzy set-theoretic operations. A number of articles—among which we may cite Fung and Fu [14], Yager [31, 33, 34], Zimmerman and Zysno [35, 36, 37], and Dubois and Prade [7, 11]—have been concerned with the axiomatic or practical determination of these aggregative operations. This question is the subject of the first part of this chapter, which summarizes a more detailed survey [38].

Keywords

Decision Maker Membership Function Fuzzy Number Typical Vehicle Aggregation Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    ACZEL, J. (1966). Lectures on Functional Equations and Their Applications. Academic, New York.MATHGoogle Scholar
  2. 2.
    BAAS, S., and KWAKERNAAK, H. (1977). Rating and ranking of multiple aspect alternatives using fuzzy sets. Automatica, 13, 47–58.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    BALDWIN, J.F., and GUILD, N. C. F.(1979). Comparison of fuzzy sets on the same decision space. Fuzzy Sets and, Syst., 2(3), 213–233.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    BELLMAN, R. E., and ZADEH, L. A. (1970). Decision-making in a fuzzy environment. Manage. Sci., 17, B141–B164.MathSciNetCrossRefGoogle Scholar
  5. 5.
    DOMBI, J. (1982). Basic concepts for a theory of evaluation: The aggregative operator. Eur. J. Oper. Res., 10, 282–293.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    DUBOIS, D. (1983). Modèles mathématiques de l’imprécis et de l’incertain en vue d’applications aux techniques d’aide à la décision. Thesis, University of Grenoble.Google Scholar
  7. 7.
    DUBOIS, D., and PRADE, H. (1980). New results about properties and semantics of fuzzy set-theoretic operators. In Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems (P. P. Wang, and S. K. Chang, eds). Plenum Press, New York, pp. 59–75.Google Scholar
  8. 8.
    DUBOIS, D., and PRADE, H. (1983). The use of fuzzy numbers in decision analysis. In Fuzzy Information and Decision Processes (M. M. Gupta, and E. Sanchez, eds). North-Holland, Amsterdam, pp. 309–321 .Google Scholar
  9. 9.
    DUBOIS, D., and PRADE, H. (1983). Ranking fuzzy numbers in the setting of possibility theory. Inf Sci., 30(2), 183–224.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    DUBOIS, D., and PRADE, H. (1983). Two-fold fuzzy sets: An approach to the representation of sets with fuzzy boundaries, based on possibility and necessity measures. Fuzzy Math. (China), 3(4), 53–76.MathSciNetMATHGoogle Scholar
  11. 11.
    DUBOIS, D., and PRADE, H. (1984). Criteria aggregation and ranking of alternatives in the framework of fuzzy set theory. Fuzzy Sets and Decision Analysis (H.-J. Zimmermann, L. A. Zadeh, and B. R. Gaines, eds). TIMS Studies in Management Sciences, Vol. 20, North-Holland, Amsterdam, pp. 209–240.Google Scholar
  12. 12.
    FRANK, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y - F(x, y). Aequationes Math., 19, 194–226.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Freeling, A. N. S. (1980). Fuzzy sets and decision analysis. IEEE Trans. Syst. Man Cybern., 10, 341–354.CrossRefGoogle Scholar
  14. 14.
    FUNG, L. W., and Fu, K. S. (1975). An axiomatic approach to rational decision-making in a fuzzy environment. In Fuzzy Sets and Their Applications to Cognitive and Decision Processes (L. A. Zadeh et al., eds). Academic, New York, pp. 227–256.Google Scholar
  15. 15.
    HAMACHER, H. (1975). Über logische Verknüpfungen Unscharfer Aussagen und deren Zugehörige Bewertungs-funktionen. In Progress in Cybernetics and Systems Research, vol. 3 (R. Trappl, G. J. Klir, and L. Ricciardi, eds). Hemisphere, New York, pp. 276–287.Google Scholar
  16. 16.
    KANDEL, A., and BYATT, W. J. (1978). Fuzzy sets, fuzzy algebra and fuzzy statistics. Proc. IEEE,68, 1619–1639.CrossRefGoogle Scholar
  17. 17.
    KEENEY, R. L., and RAIFFA, H. (1976). Decisions With Multiple Objectives: Preferences and Value Trade-offs. Wiley, New York.Google Scholar
  18. 18.
    LING, C. H. (1965). Representation of associative functions. Publ. Math. Debrecen, 12, 189–212.MathSciNetGoogle Scholar
  19. 19.
    Menger, K. (1942). Statistical metrics, Proc. Natl. Acad. Sci. USA, 28, 535–537.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    OKUDA, T., TANAKA, H., and ASAI, K. (1978). A formulation of fuzzy decision problems with fuzzy information, using probability measures of fuzzy events, Inf Control, 38(2), 135–147.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    ROY, B. (1978). ELECTRE III: Un algorithme de classement fondé sur une représentation floue des préférences en présence de critères multiples. Cah. CERO, 20, 3–24.MATHGoogle Scholar
  22. 22.
    SAATY, T. L. (1978). Exploring the interfaces between hierarchies, multiple objectives, and fuzzy sets, Fuzzy Sets Syst. 1, 57–68.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    SCHWEIZER, B., and SKLAR, A. (1963). Associative functions and abstract semigroups. Publ. Math. Debrecen, 10, 69–81.MathSciNetGoogle Scholar
  24. 24.
    SILVERT, W. (1979). Symmetric summation: A class of operations on fuzzy sets. IEEE Trans. SystMan. Cybern 9, 657–659.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    SISKOS, J., LOCHARD, J., and LOMBARD, J. (1984). A multicriteria decision-making methodology under fuzziness: Application to evaluation of radiological protection in nuclear power-plants. In Fuzzy Sets and Decision Analysis (H. J. Zimmermann, L. A. Zadeh, and B. B. Gaines, eds). TIMS Studies in the Management Sciences, Vol. 20, North-Holland, Amsterdam, pp. 261–283.Google Scholar
  26. 26.
    THOLE, U., ZIMMERMANN, H. J., and ZYSNO, P. (1979). On the suitability of minimum and product operators for the intersection of fuzzy sets. Fuzzy Sets Syst., 2, 167–180.MATHCrossRefGoogle Scholar
  27. 27.
    TRILLAS, E. (1979). Sobre funciones de negacion en la teoria de conjuntos difusos. Stochastica, III(l), 47–59.MathSciNetGoogle Scholar
  28. 28.
    TSUKAMOTO, Y., NIKIFORUK, P. N., and GUPTA, M. M. (1981). On the comparison of fuzzy sets using fuzzy chopping. Proc. 8th Triennal World Congress IFAC, Kyoto, Vol. 5, pp. 46–52.Google Scholar
  29. 29.
    VON NEUMANN, J., and MORGENSTERN, O. (1944). Theory of Games and Economic Behavior. Princeton Univ. Press, Princeton, New Jersey.Google Scholar
  30. 30.
    Watson, S. R., Weiss, J. J., and Donnell, M. (1979). Fuzzy decision analysis. IEEE Trans. Syst. Man. Cybern., 9, 1–9.CrossRefGoogle Scholar
  31. 31.
    YAGER, R. R. (1977). Multiple-objective decision-making using fuzzy sets. Int. J. Man-Machine Stud., 9, 375–382.MATHCrossRefGoogle Scholar
  32. 32.
    YAGER, R. R. (1978). Fuzzy decision-making including unequal objectives. Fuzzy Sets Syst., 1, 85–95.Google Scholar
  33. 33.
    YAGER, R. R. (1980). On a general class of fuzzy connectives. Fuzzy Sets Syst., 4, 235–242.MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    YAGER, R. R. (1982). Some procedures for selecting fuzzy set-theoretic operators. Int. J. Gen. Syst., 8(2), 115–124.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Zimmermann, H. J., and Zysno, P. (1980). Latent connectives in human decision making. Fuzzy Sets Syst., 4(1), 37–51.MATHCrossRefGoogle Scholar
  36. 36.
    ZIMMERMANN, H. J., and ZYSNO, P. (1983). Decisions and evaluations by hierarchical aggregation of information. Fuzzy Sets Syst., 10, 243–260.MATHCrossRefGoogle Scholar
  37. 37.
    ZYSNO, P. (1982). The integration of concepts within judgmental and evaluative processes. Progress in Cybernetics and Systems Research, vol. VIII (R. Trappl, G. Klir, and F. Pichler, eds), Hemisphere, New York, pp. 509–517.Google Scholar
  38. 38.
    DUBOIS, D., and PRADE, H. (1985). A review of fuzzy set aggregation connectives. Inf. Sci, 36, 85–121.MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    DUBOIS, D., and PRADE, H. (1986). Weighted minimum and maximum operations in fuzzy set theory. An addendum to “A review of fuzzy set aggregation connectives.” Inf. Sci. 39, 205–210.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    YAGER, R. R. (1981). A new methodology for ordinal multiple aspect decisions based on fuzzy sets. Decision Sci., 12, 589–600.MathSciNetCrossRefGoogle Scholar
  41. 41.
    YAGER, R. R. (1984). General multiple objective decision functions and linguistically quantified statements. Int. J. Man-Machine Stud., 21, 389–400.MATHCrossRefGoogle Scholar
  42. 42.
    BORTOLAN, G., and DEGANI, R. (1985). A review of some methods for ranking fuzzy numbers. Fuzzy Sets Syst. 15, 1–19.MathSciNetMATHCrossRefGoogle Scholar
  43. ROUBENS, M., VINCKE, P. (1988). Fuzzy possibility graphs and their application to ranking fuzzy numbers. Fuzzy Sets and Systems, to be published.Google Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Didier Dubois
    • 1
  • Henri Prade
    • 1
  1. 1.CNRS, Languages and Computer Systems (LSI)University of Toulouse IIIToulouseFrance

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