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On Fuzzy RDM-Arithmetic

  • Andrzej PiegatEmail author
  • Marek Landowski
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 534)

Abstract

The paper presents notion of horizontal membership function (HMF) and based on it fuzzy, relative distance measure (fuzzy RDM) arithmetic that is compared with standard fuzzy arithmetic (SF arithmetic). Fuzzy RDM-arithmetic possess such mathematical properties which allow for achieving complete fuzzy solution sets of problems, whereas SF-arithmetic, in general, delivers only approximate, partial solutions and sometimes no solutions of problems. The paper explains how to realize arithmetic operations with fuzzy RDM-arithmetic and shows examples of its application.

Keywords

Fuzzy arithmetic Granular computing Fuzzy RDM arithmetic Horizontal membership function Fuzzy HMF arithmetic Multidimensional fuzzy arithmetic 

References

  1. 1.
    Abbasbandy, S., Nieto, J.J., Alavi, M.: Tubing of reacheable set In one dimensional fuzzy differentia inclusion. Chaos, Solitons & Fractals 26(5), 1337–1341 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9(6), 613–626 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dutta, P., Boruah, H., Ali, T.: Fuzzy arithmetic with and without using \(\alpha \)-cut method: a comparative study. Int. J. Latest Trends Comput. 2(1), 99–107 (2011)Google Scholar
  4. 4.
    Dymova, L.: Soft Computing in Economics and Finance. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Hanss, M.: Applied Fuzzy Arithmetic. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  6. 6.
    Kaufmann, A., Gupta, M.M.: Introduction to Fuzzy Arithmetic. Van Nostrand Reinhold, New York (1991)zbMATHGoogle Scholar
  7. 7.
    Klir, G.J., Pan, Y.: Constrained fuzzy arithmetic: Basic questions and answers. Soft Comput. 2(2), 100–108 (1998)CrossRefGoogle Scholar
  8. 8.
    Kosinski, W., Prokopowicz, P., Slezak, D.: Ordered fuzzy numbers. Bull. Polish Acad. Sci. Ser. Sci. Math. 51(3), 327–338 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Markov, S.M., Popova, E.D., Ullrich, C.: On the solution of linear algebraic equations involving interval coefficients. In: Margenov, S., Vassilevski, P. (eds.) Iterative Methods in Linear Algebra, II. IMACS Series in Computational and Applied Mathematics, vol. 3, pp. 216–225 (1996)Google Scholar
  10. 10.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Pedrycz, W., Skowron, A., Kreinovich, V.: Handbook of Granular Computing. John Wiley & Sons, Chichester (2008)CrossRefGoogle Scholar
  12. 12.
    Piegat, A., Landowski, M.: Two interpretations of multidimensional RDM interval arithmetic - multiplication and division. Int. J. Fuzzy Syst. 15(4), 486–496 (2013)MathSciNetGoogle Scholar
  13. 13.
    Piegat, A., Plucinski, M.: Fuzzy number addition with the application of horizontal membership functions. Scient. World J., Article ID: 367214, 1–16 (2015). Hindawi Publishing CorporationGoogle Scholar
  14. 14.
    Piegat, A., Landowski, M.: Horizontal membership function and examples of its applications. Int. J. Fuzzy Syst. 17(1), 22–30 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Piegat, A., Landowski, M.: Aggregation of inconsistent expert opinions with use of horizontal intuitionistic membership functions. In: Atanassov, K.T., et al. (eds.) Novel Developments in Uncertainty Representation and Processing. AISC, vol. 401, pp. 215–223. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-26211-6_18 CrossRefGoogle Scholar
  16. 16.
    Shary, S.P.: On controlled solution set of interval arithmetic of interval algebraic systems. Interval Comput. 6, 66–75 (1992)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Tomaszewska, K., Piegat, A.: Application of the horizontal membership function to the uncertain displacement calculation of a composite massless rod under a tensile load. In: Wiliński, A., Fray, I., Pejaś, J. (eds.) Soft Computing in Computer and Information Science. AISC, vol. 342, pp. 63–72. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-15147-2_6 Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.West Pomeranian University of TechnologySzczecinPoland
  2. 2.Maritime University of SzczecinSzczecinPoland

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