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Correct Solution of Fuzzy Linear System Based on Interval Theory

  • Andrzej Piegat
  • Marcin PietrzykowskiEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 889)

Abstract

In this paper authors would like to make a critical review of the low-dimensional method proposed in [3] by Allahviranloo and Gandavi for solving fuzzy linear system with crisp square matrix and a fuzzy right-hand side vector. The solution presented in the mentioned work is, in general incorrect. The authors prove that given method is incorrect and propose an alternative solution based on multidimensional Relative-Distance-Measure (RDM) interval arithmetic and fuzzy RDM interval arithmetic that gives correct results.

Keywords

Fuzzy linear system Interval linear system Multidimensional Fuzzy RDM arithmetic Multidimensional Interval RDM arithmetic 

References

  1. 1.
    Aliev, R.: Operations on z-numbers with acceptable degree of specificity. Procedia Comput. Sci. 120, 9–15 (2017). 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22–23 August 2017, Budapest, HungaryCrossRefGoogle Scholar
  2. 2.
    Aliev, R., Huseynov, O., Aliyev, R.: A sum of a large number of z-numbers. Procedia Comput. Sci. 120, 16–22 (2017). 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22–23 August 2017, Budapest, HungaryCrossRefGoogle Scholar
  3. 3.
    Allahviranloo, T., Ghanbari, M.: On the algebraic solution of fuzzy linear systems based on interval theory. Appl. Math. Model. 36, 5360–5379 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dymova, L.: Soft Computing in Economics and Finance. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Lodwick, W.A., Dubois, D.: Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst. 281, 227–251 (2015). Special Issue Celebrating the 50th Anniversary of Fuzzy SetsMathSciNetCrossRefGoogle Scholar
  6. 6.
    Mazandarani, M., Pariz, N., Kamyad, A.V.: Granular differentiability of fuzzy-number-valued functions. IEEE Trans. Fuzzy Syst. 26(1), 310–323 (2018)CrossRefGoogle Scholar
  7. 7.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2009)Google Scholar
  8. 8.
    Najariyan, M., Zhao, Y.: Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives. IEEE Trans. Fuzzy Syst. PP(99), 1–15 (2017)Google Scholar
  9. 9.
    Pedrycz, W., Skowron, A., Kreinovich, V.: Handbook of Granular Computing. Wiley-Interscience, New York (2008)CrossRefGoogle Scholar
  10. 10.
    Piegat, A., Landowski, M.: Horizontal membership function and examples of its applications. Int. J. Fuzzy Syst. 17(1), 22–30 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Piegat, A., Landowski, M.: Fuzzy arithmetic type 1 with horizontal membership functions. In: Kreinovich, V. (ed.) Uncertainty Modeling, pp. 233–250. Springer International Publishing, Cham (2017). Dedicated to Professor Boris Kovalerchuk on his AnniversaryCrossRefGoogle Scholar
  12. 12.
    Piegat, A., Landowski, M.: Is an interval the right result of arithmetic operations on intervals? Int. J. Appl. Math. Comput. Sci. 27(3), 575–590 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Piegat, A., Landowski, M.: Is fuzzy number the right result of arithmetic operations on fuzzy numbers? In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K.T., Krawczak, M. (eds.) Advances in Fuzzy Logic and Technology 2017, pp. 181–194. Springer International Publishing, Cham (2018)CrossRefGoogle Scholar
  14. 14.
    Piegat, A., Pluciński, M.: Computing with words with the use of inverse RDM models of membership functions. Int. J. Appl. Math. Comput. Sci. 25(3), 675–688 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Piegat, A., Pluciński, M.: Fuzzy number addition with the application of horizontal membership functions. Sci. World J. 2015, 1–16 (2015)CrossRefGoogle Scholar
  16. 16.
    Piegat, A., Pluciński, M.: Fuzzy number division and the multi-granularity phenomenon. Bull. Pol. Acad. Sci. Tech. Sci. 65(4), 497–511 (2017)Google Scholar
  17. 17.
    Sevastjanov, P., Dymova, L.: A new method for solving interval and fuzzy equations: linear case. Inf. Sci. 179(7), 925–937 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Zeinalova, M.L.: Application of RDM interval arithmetic in decision making problem under uncertainty. Procedia Comput. Sci. 120, 788–796 (2017). 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22–23 August 2017, Budapest, HungaryCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Computer Science and Information TechnologyWest Pomeranian University of TechnologySzczecinPoland

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