Fuzzy Graphs with Linguistic Inputs-Outputs by Fuzzy Approximation Models

  • Haekwan Lee
  • Hideo Tanaka
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 34)


This paper proposes an approach to construct fuzzy graphs with linguistic inputs-outputs by fuzzy approximation models. Linguistic inputs can be transformed into linguistic outputs through fizzy systems. In this paper, we consider fuzzy approximation models as fuzzy relations to represent linguistic inputs-outputs. In fuzzy regression, two approximation models, i. e. the possibility and necessity models, can be considered. Always there exist a possibility model when a linear system with fuzzy coefficients is considered, but it is not assured to attain a necessity model in a fuzzy linear system. The absence of a necessity model is caused by adopting a model not fitting to the given data Thus we consider polynomials to find a more refined regression model. If we can find a proper necessity model, the necessity and possibility models deserve more credit than the previous models in the timer studies. The measure of fitness is used to gauge the degree of approximation of the obtained models to the given data. The obtained approximation models themselves can be regarded as fizzy graphs. Furthermore, by the obtained approximation models, we can construct another fuzzy graphs which represent linguistic inputs-outputs relations. The possibility and necessity models in fuzzy regression analysis can be considered as the upper and lower approximations in rough sets. Similarities between the fuzzy regression and the rough sets concept are also discussed.


Fuzzy Number Triangular Fuzzy Number Fuzzy Relation Fuzzy Regression Fuzzy Output 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Alefeld and J. Herzberger, Introduction to interval computations, Academic Press, New York, 1983.MATHGoogle Scholar
  2. [2]
    D. Dubois and H. Prade, Twobld fuzzy sets and rough sets–some issues in knowledge representations of expert’s inf’rence models, Int. J. of Fuzzy Sets and Systems, vol. 23, pp. 3–18, 1987.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. Inuiguchi, N. Sakawa, and S. Ushiro, Interval regression based on Minkowski’ s subtraction, Proc. of Fifth IFSA World Congress, Seoul, Korea, pp. 505–508, 1993.Google Scholar
  4. [4]
    H. Ishibuchi and H. Tanaka, A unified approach to possibility and necessity regression analysis with interval regression models, Proc. of Fifth IFSA World Congress, Seoul, Korea, pp. 501–504, 1993.Google Scholar
  5. [5]
    A. Kaufnann and M. M. Gupta, Fuzzy mathematical models in engineering and management science, Elsevier Science Publishers, Amsterdam, 1988.Google Scholar
  6. [6]
    Z. Pawlak, Rough sets, Int. J. of Information and Computer Sciences, vol. 11, pp. 341–356, 1982.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Z. Pawlak, Rough classification, Int. J. of Man-Machine Studies, vol. 20, pp. 469–483, 1984.MATHCrossRefGoogle Scholar
  8. [8]
    M. Sakawa and H. Yano, Multiobjective fuzzy linear regression analysis for fizzzy input-output data, Int. J. of Fuzzy Sets and Systems,vol. 47, pp. 173181, 1992.Google Scholar
  9. [9]
    H. Tanaka, S. Uejima, and K. Asai, Linear regression analysis with fizzy model, IEEE Trans. Systems Man Cybernet, vol. 12, pp. 903–907, 1982.MATHCrossRefGoogle Scholar
  10. [10]
    H. Tanaka, Fuzzy data analysis by possibilistic linear models, Int. J. of Fuzzy Sets and Systems, vol. 24, pp. 363–375, 1987.MATHCrossRefGoogle Scholar
  11. [11]
    H. Tanaka and J. Watada, Possibilistic linear systems and their application to the linear regression model, Int. J. of Fuzzy Sets and Systems, vol. 27, pp. 275–289, 1988.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    H. Tanaka, I. Hayashi, and J. Watada, Possibilistic linear regression analysis for fuzzy data, European J. of Operational Research, vol. 40, pp. 389–396, 1989.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    H. Tanaka and H. Ishibuchi, Identification ofpossibilistic linear systems by quadratic membership Junctions of fuzzy parameters, Int. J. of Fuzzy sets and Systems, vol. 41, pp. 145–160, 1991.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    H. Tanaka and H. Ishibuchi, Possibilistic regression analysis based on linear programming, in J. Kacprzyk and M. Fedrizzi, Eds. Fuzzy Regression Analysis, Omnitech press, Warsaw and Physica-Verlag, Heidelberg, pp. 47–60, 1992.Google Scholar
  15. [15]
    H. Tanaka, K. Koyama, and H. Lee, Interval Regression Analysis based on Quadratic Programming, Proc. of Fifth IEEE Int. Conf. on Fuzzy Systems, New Orleans, USA, pp. 325–329, 1996.Google Scholar
  16. [16]
    H. Tanaka, H. Lee, and T. Mizukami, Identification ofpossibilistic coefficients in fuzzy linear systems, Proc. of Fifth IEEE Int. Conf on Fuzzy Systems, New Orleans, USA, pp. 842–847, 1996.Google Scholar
  17. [17]
    H. Tanaka and H. Lee, Fuzzy linear regression combining central tendency and possibilistic properties, Proc. of Sixth IEEE Int. Conf. on Fuzzy Systems, Barcelona, Spain, pp. 63–68, 1997.Google Scholar
  18. [18]
    L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences, vol. 8, pp. 199–249, 1975.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    L. A. Zadeh, Fuzzy logic = Computing with words, IEEE Trans. Fuzzy Systems, vol. 4, pp. 103–111, 1996.CrossRefGoogle Scholar
  20. [20]
    L. A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Int. J. of Fuzzy sets and Systems, vol. 90, pp. 111–127, 1997.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Haekwan Lee
    • 1
  • Hideo Tanaka
    • 1
  1. 1.Department of Industrial EngineeringOsaka Prefecture UniversitySakai, OsakaJapan

Personalised recommendations