Advertisement

Verbally Generated Fuzzy Quantities and Their Aggregation

  • Milan Mareš
  • Radko Mesiar
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 97)

Abstract

The processing of vague data recently becomes one of attractive topics in the fuzzy set theory and its applications. As the vagueness is usually represented by some verbal expressions, this branch of the fuzzy sets is frequently called “computing with words”. Seemingly, but only seemingly, it could be understood as computational processing of fuzzy numbers or fuzzy quantities in the already classical sense. Other authors understand the computing with words rather as a fuzzy logical discipline being near to fuzzy reasoning methods and other related branches. Both approaches are rational and fully acceptable but, in the matter of facts, none of them appears to be complete. Their parallel existence offers a conclusion that the fair approach to computing with words can consist in some kind of their combination. Computing with words has two faces — quantitative and qualitative one — and each of them would be somehow reflected. The fuzzy set theoretical model of verbal variables, their generating and processing suggested in this contribution and in some of the referred papers is intended to offer such combined view on the quantitative — qualitative dualism existing in the “computing with words” and to develop at least elementary methods for manipulation with such dualistic verbal data.

Keywords

Membership Function Fuzzy Number Verbal Variable Fuzzy Subset Verbal Expression 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Aczél and C. Alsina: Characterization of some classes of quasilinear functions with applications to triangular norms and to synthesizing judgments. Methods Oper. Res. 48 (1984), 3–22.MATHGoogle Scholar
  2. 2.
    C. Alsina, E. Trillas and L. Valverde: On some logical connectives for fuzzy sets theory. BUSEFAL 3 (1980), 18–29.Google Scholar
  3. 3.
    D. DeBaets and A. Markovd-Stupnanovd• Analytical expressions for the addition of fuzzy intervals. Fuzzy Sets and Systems 91 (1997), 203–213.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. Dombi: A general class of fuzzy operators, De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets and Systems 8 (1982), 149–163.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    D. Dubois, E. Kerre, R. Mesiar and H. Prade: Fuzzy interval analysis. In: D. Dubois, H. Prade (Eds.): Fundamentals of Fuzzy Sets. Handcontributions on Fuzzy Sets, Vol. 1, Kluwer Acad. Publ., Dordrecht 2000, 483–581.CrossRefGoogle Scholar
  6. 6.
    D. Dubois and H. Prade: New results about properties and semantics of fuzzy set–theoretic operators. In: P. Wang, S. Hcnag, editors, Fuzzy Sets: Theory and Applications to Policy Analysis and Information Sytems. Plenum Press, new York 1980, 59–75.CrossRefGoogle Scholar
  7. 7.
    S. Gottwald: Fuzzy Sets and Fuzzy Logic. Foundations of Application - from a Mathematical Point of View. Mathematical Foundations and Applications. Vieweg, Braunschweig 1993.Google Scholar
  8. 8.
    P. Hâjek: Metamathematics of Fuzzy Logic. Kluwer Acad. Publ, Dordrecht 1998.Google Scholar
  9. 9.
    H. Hamacher: Über logische Aggregationen nicht-binär explicierter Entscheidungskriterien. Rita G. Fischer Verlag, Frankfurt 1978.Google Scholar
  10. 10.
    B. Harman: Sum and product of the modified real fuzzy numbers. Kybernetika, supplement 1992.Google Scholar
  11. 11.
    U. Höhle: Probabilistisch kompakte L-unscharfe Mengen. Manuscripta Math. 26 (1978), 345–356.CrossRefGoogle Scholar
  12. 12.
    D. H. Hong: On shape preserving additions of fuzzy intervals. J. Math. Anal. Appl., in press.Google Scholar
  13. 13.
    J. Jacas, J. Recasens: Fuzzy numbers and equality relations. In: Transactions of 2nd IEEE Internat. Conf. on Fuzzy Systems 1993, IEEE Neural Network Council 1993.Google Scholar
  14. 14.
    E. E. Kerre, M. Mares, R. Mesiar: Orderings of generated fuzzy quantities. In: Proceedings of IPMU’98, La Sorbone, Paris 1998, Vol. I, 250–254.Google Scholar
  15. 15.
    E. E. Kerre, M. Mares, R. Mesiar: Generated fuzzy quantities and their orderings. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems. To appear.Google Scholar
  16. 16.
    E. E. Kerre, X. Wang: Reasonable properties for the ordering of fuzzy quantities. Part I, Part II. Fuzzy Sets and Systems. To appear.Google Scholar
  17. 17.
    E. P. Klement: Fuzzy a-algebras and fuzzy measurable functions. Fuzzy Sets and Systems 4 (1980), 83–93.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    E. P. Klement: Integration of fuzzy valued functions. Revue Roumaine Math. Pures Appl. 30 (1985), 375–384.MathSciNetMATHGoogle Scholar
  19. 19.
    E. P. Klement: Strong law or large numbers for random variables with values in the fuzzy real line. IFSA Commun. Math. Chapt. (1987), 7–11.Google Scholar
  20. 20.
    E. P. Klement, R. Mesiar and E. Pap: Quasi-and pseudo-inverses of monotone functions, and the construction of t-norms. Fuzzy Sets and Systems 104 (1999), 3–13.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    E. P. Klement, R. Mesiar, E. Pap: Triangular Norms. Kluwer Acad. Publ., Dordrecht 2000.Google Scholar
  22. 22.
    E. P. Klement, R. Mesiar, E. Pap: Generated triangular norms. Kybernetika 36 (2000), 363–377.MathSciNetMATHGoogle Scholar
  23. 23.
    G. J. Klir, Bo Yuan (Eds.): Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems. Selected papers by Lotfi A. Zadeh. World Scientific, Singapore, London 1996.Google Scholar
  24. 24.
    A. Kolesârovâ: Additions preserving the linearity of t-norms. Tatra Mt. Math. Publ. 6 (1995), 75–81.MATHGoogle Scholar
  25. 25.
    A. Kolesârovâ: Similarity preserving t-norm based addition of fuzzy numbers. Fuzzy Sets and Systems 91 (1997), 215–229.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    A. Kolesârovâ: Triangular norm-based addition preserving linearity of T-sums of linear fuzzy intervals. Mathware and Soft Computing 5 (1998), 91–98.MathSciNetMATHGoogle Scholar
  27. 27.
    C. M. Ling: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189–212.MathSciNetGoogle Scholar
  28. 28.
    M. Mares: How to handle fuzzy quantities. Kybernetika 13 (1977), 1, 23–40.MathSciNetGoogle Scholar
  29. 29.
    M. Mare: Addition of rational fuzzy quantities: Convolutive approach. Kybernetika 25 (1989), 1, 1–12.MathSciNetGoogle Scholar
  30. 30.
    M. Mares: Computation Over Fuzzy Quantities. CRC—Press, Boca Raton 1994.MATHGoogle Scholar
  31. 31.
    M. Mares: Weak arithmetics of fuzzy numbers. Fuzzy Sets and Systems 91 (1997), 143–154.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    M. Mares: Verbal quantitative information in economic reality. Acta Oeconomica Pragensia 5 (1997), 1, 121–134.Google Scholar
  33. 33.
    M. Mares: Critical path method with verbal inputs. In: Proceedings ICOTA’98, Curtin University of Technology, Perth 1998, Vol. 1, 634–640.Google Scholar
  34. 34.
    M. Mares’: Fuzzy Coalitional Games. Physica—Verlag, Heidelberg. To appear.Google Scholar
  35. 35.
    M. Mares`, R. Mesiar: Composition of shape generators. Acta Mathematica et Informatica Universitatis Ostraviensis 4 (1996), 1, 37–45.MathSciNetMATHGoogle Scholar
  36. 36.
    M. Mare`s, R. Mesiar: Calculation over verbal variables. In: J. Kacprzyk, L. A. Zadeh (Eds.): Computing With Words in Information/Intelligent Systems, Physica—Verlag, Heidelberg 1999, 409–427.Google Scholar
  37. 37.
    M. Mares, R. Mesiar: Vagueness of verbal variable. In: R. Ribeiro, J. Kacprzyk, R. R. Yager, L. A. Zadeh (Eds.): Soft Computing in Financial Engineering. Physica—Verlag, Heidelberg 1999, 3–20.Google Scholar
  38. 38.
    A. Markova: Idempotents of the T-addition of fuzzy numbers. Tatra Mt. Math. Publ. 12 (1997), 67–72.Google Scholar
  39. 39.
    A. Markova: T-sum of L — R fuzzy numbers. Fuzzy Sets and Systems 85 (1996), 379–384.MathSciNetCrossRefGoogle Scholar
  40. 40.
    A. Markovd—Stupnanova: A note to the addition of fuzzy intervals based on the continuous Archimedean t-norm. Fuzzy Sets and Systems 91 (1997), 253–258.MathSciNetCrossRefGoogle Scholar
  41. 41.
    R. Mesiar: Shape preserving additions of fuzzy intervals. Fuzzy Sets and Systems 86 (1997), 73–78.MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    R. Mesiar: Triangular norm-based additions of fuzzy intervals. Fuzzy Sets and Systems 91 (1997), 231–237.MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    H. T. Nguyen and E. Walker: A First Course in Fuzzy Logic. CRC Press, Boca Raton 1997.MATHGoogle Scholar
  44. 44.
    W. Pedrycz: Why triangular membership functions. Fuzzy Sets and Systems 64 (1994), 21–30.MathSciNetCrossRefGoogle Scholar
  45. 45.
    J. Puncochdf, M. Drahos, J. Vrba: Fuzzy number as a product of geometrical construction. Fuzzy Sets and Systems 83 (1996), 43–50.MathSciNetCrossRefGoogle Scholar
  46. 46.
    B. Schweizer, A. Sklar: Statistical metric spaces. Pacific. J. Math. 10 (1960), 313–334.MathSciNetMATHGoogle Scholar
  47. 47.
    B. Schweizer, A. Sklar: Probabilistic Metric Spaces. North—Holland, New York 1983.MATHGoogle Scholar
  48. 48.
    E. Trillas: Sobre functiones de negación en la teoría de conjuntas difusos. Stochastica 3 (1979), 47–60.MathSciNetMATHGoogle Scholar
  49. 49.
    P. Viceník: A note on generators of t-norms. BUSEFAL 75 (1998), 33–38.Google Scholar
  50. 50.
    P. Vicenfk: Generated t-norms and the Archimedean property. Proc. EUFIT’99, Aachen 1999, CD-rom.Google Scholar
  51. 51.
    R. R. Yager: On a general class of fuzzy connectives. Fuzzy Sets and Systems 4 (1980), 235–242.MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    L. A. Zadeh: Fuzzy sets. Information and Control 8 (1965), 3, 338–353.MathSciNetMATHGoogle Scholar
  53. 53.
    L. A. Zadeh: The concept of a linguistic variable and its applications to approximate reasoning. Information Sciences, Part I: 8 (1975), 199–249; Part II: 8 (1975), 301–357; Part III: 9 (1975), 43–80.Google Scholar

Copyright information

© Physica-Verlag Heidelberg 2002

Authors and Affiliations

  • Milan Mareš
    • 1
  • Radko Mesiar
    • 2
  1. 1.Institute of Information Theory and AutomationAcademy of Sciences of the Czech RepublicCzech Republic
  2. 2.Faculty of Civil EngineeringSlovak Technical UniversityBratislavaSlovak Republic

Personalised recommendations