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Horizontal Membership Function and Examples of its Applications

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Abstract

The paper introduces horizontal membership functions (HMFs) which define a fuzzy set not in form of commonly used vertical membership functions of type μ = f 1(x) but in the horizontal form x = f 2(μ). Until now, constructing HMFs had seemed impossible because of horizontal ambiguity of this function. Now, however, it became possible thanks to the multidimensional, RDM-interval arithmetic based on relative-distance-measure variables. HMFs enable direct introducing uncertain, interval or fuzzy variable-values in usual mathematical formulas of type y = f(x 1 ,…,x 2) together with crisp values, without using Zadeh’s extension principle. Thus, a relatively easy aggregation of crisp and uncertain knowledge became possible. The paper shows application of HMFs, first on example of a classical mathematical function y = f(x 1 ,x 2) and next, on example of a computing with words challenge problem.

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References

  1. Aliev, R., Pedrycz, W., Fazlollahi, B., Huseynow, O., Alizadeh, A., Gurimov, B.: Fuzzy logic-based generalized decision theory with imperfect information. Inf. Sci. 189, 18–42 (2012)

    Article  MATH  Google Scholar 

  2. Chen, M.-C., Wang, W.-Y., Su, S.-F., Chien, Y.-H.: Robust T-S fuzzy-neural control of uncertain active suspension systems. Int. J. Fuzzy Syst. 12(4), 321–329 (2010)

    Google Scholar 

  3. Chen, T.-Y.: Remarks on the subtraction and division operations over intuitionistic fuzzy sets and interval-valued fuzzy sets. Int. J. Fuzzy Syst. 9(3), 169–172 (2007)

    MathSciNet  Google Scholar 

  4. Chen, X., Quin, Z.: A new existence and uniqueness theorem for fuzzy differential equations. Int. J. Fuzzy Syst. 13(2), 148–151 (2011)

    MATH  MathSciNet  Google Scholar 

  5. Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. Syst. Sci. 9, 613–626 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dymova, L.: Soft Computing in Economics and Finance. Springer, Berlin (2011)

    Book  Google Scholar 

  7. Ezzati, R., Ziari, S.: Numerical solutions of two-dimensional fuzzy fredholm integral equations of the second kind using fuzzy bivariate Bernstein polynomials. Int. J. Fuzzy Syst. 15(1), 84–89 (2013)

    MathSciNet  Google Scholar 

  8. Gupta, M.: Introduction to fuzzy arithmetic. Van Nostrand Reinhold, New York (1991)

    MATH  Google Scholar 

  9. Hanss, M.: Applied Fuzzy Arithmetic. Springer, Berlin (2005)

    MATH  Google Scholar 

  10. Hsiao, M.-Y., Chen, C.-Y., Li, T.-H.: Interval type-2 adaptive fuzzy sliding-mode dynamic control design for wheeled mobile robots. Int. J. Fuzzy Syst. 10(4), 268–275 (2008)

    MathSciNet  Google Scholar 

  11. Jafarian, A., Nia, S.M.: On the solution of nonlinear fuzzy equation systems by fuzzy neural network method. Int. J. Fuzzy Syst. 15(3), 376–380 (2013)

    Google Scholar 

  12. Jafari, H., Saeidy, M., Vahidi, J.: The homotopy analysis method for solving fuzzy system of linear equations. Int. J. Fuzzy Syst. 11(4), 308–313 (2009)

    MathSciNet  Google Scholar 

  13. Klir, G.J., Yuan, B.: Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems. Selected paper by L. Zadeh. World Scientific, Singapore, New Jersey (1996)

  14. Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets Syst. 91, 165–175 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Landowski, M.: Differences between Moore and RDM Interval Arithmetic. In: Proceedings of the 7th International Conference Intelligent Systems IEEE IS’2014, 24–26 September 2014, Warsaw, Poland, Volume 1: Mathematical Foundations, Theory, Analyses, Intelligent Systems’2014, Advances in Intelligent Systems and Computing, vol. 322, pp. 331–340. Springer, Switzerland (2015)

  16. Marhamati, N., Patel, P., Althobaiti, Y., Khorasani, E.S., Rahimi, S.: Revisiting Linguistic Approximation for Computing with Words. In: Proceedings of the Twenty-Sixth International Florida Artificial Intelligence Research Society Conference. Association for the Advancement of Artificial Intelligence, pp. 327–330 (2013)

  17. Nguyen, H.T.: On foundations of fuzzy theory for soft computing. Int. J. Fuzzy Syst 8(1), 39–45 (2006)

    MathSciNet  Google Scholar 

  18. Pedrycz, W., Gomide, F.: Fuzzy Systems Engineering: Toward Human-Centric Computing. Wiley, Hoboken (2007)

    Book  Google Scholar 

  19. Pedrycz, W., Skowron, A., Kreinovich, V. (eds.): Handbook of Granular Computing. Wiley, Chichester (2008)

    Google Scholar 

  20. Piegat, A.: Fuzzy Modeling and Control. Physica Verlag, Heidelberg (2001)

    Book  MATH  Google Scholar 

  21. Piegat, A., Landowski, M.: Is the conventional interval-arithmetic correct? J. Theor. Appl. Comp. Sci. 6(2), 27–44 (2012)

    Google Scholar 

  22. Piegat, A., Landowski, M.: Multidimensional approach to interval uncertainty calculations. In: Atanassov, K.T., et al. (eds.) New Trends in Fuzzy Sets, Intuitionistic: Fuzzy Sets, Generalized Nets and Related Topics, Volume II: Applications, pp. 137–151. IBS PAN—SRI PAS, Warsaw (2013)

    Google Scholar 

  23. Piegat, A., Tomaszewska, K.: Decision-making under uncertainty using info-gap Theory and a new multidimensional RDM interval arithmetic. Electrotech. Rev. 89(8), 71–76 (2013)

    Google Scholar 

  24. Piegat, A., Landowski, M.: Two interpretations of multidimensional RDM interval arithmetic—multiplication and division. Int. J. Fuzzy Syst. 15(4), 488–496 (2013)

    MathSciNet  Google Scholar 

  25. Piegat, A., Landowski, M.: Correctness-checking of uncertain-equation solutions on example of the interval-modal method. In: K.T. Atanassov et al. (eds.) Modern Approaches in Fuzzy Sets, Intuitionistic: Fuzzy Sets, Generalized Nets and Related Topics, Volume I: Foundations, IBS PAN—SRI PAS, Warsaw, pp. 159–1700 (2014)

  26. Rajati, M.R., Mendel, J.M.: On advanced computing with words using the generalized extension principle for type-1 fuzzy sets. IEEE Trans. Fuzzy Syst. 22(5), 1245–1261 (2014)

    Article  Google Scholar 

  27. Rajati, M.R., Mendel, J.M.: Lower and upper probability calculations using compatibility measures for solving Zadeh’s challenge problem. In: Proceedings WCCI 2012 IEEE World Congress on Computational Intelligence, Brisbane, Australia, 10–15 June 2012

  28. Rajati, M.R., Mendel, J.M.: Solving Zadeh’s Swedes and Italians challenge problem. In: Proceedings of Annual Meeting, North American Fuzzy Information Processing Society, pp. 1–6 (2012)

  29. Rajati, M.R., Wu, D., Mendel, J.M.: On solving Zadeh’s tall Swedes problem. In: Proceedings of World Conference on Soft Computing. (2011)

  30. Rickard, J.T., Ott, A: Cyber-security via Computing with Words. In: Proceedings of 2014 ASE BIGDATA/SOCIALCOM/CYBERSECURITY Conference, Stanford University, 27–31 May 2014

  31. Yeh, C.T.: Reduction of fuzzy linear systems of dual equations. Int. J. Fuzzy Syst. 9(3), 173–178 (2007)

    MathSciNet  Google Scholar 

  32. Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Trans. Fuzzy Syst. 4(2), 103–111 (1996)

    Article  MathSciNet  Google Scholar 

  33. Zadeh, L.A.: From computing with numbers to computing with words. Ann. New York Acad. Sci. 929(1), 221–252 (2001)

    Article  Google Scholar 

  34. Zadeh, L.A.: A new frontier in computation—computing with information described in natural language. IEEE International Conference on Information Reuse and Integration 2007, pp. nil11–nil12. 13–15 Aug 2007, IEEE (2007)

  35. Zadeh, L.A.: Toward a generalized theory of uncertainty (GTU)—an outline. Inf. Sci. 172, 1–40 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Zadeh, L.A.: From computing with numbers to computing with words-from manipulation of measurements to manipulations of perceptions. Int. J.Appl. Math. Comput. Sci. 12, 307–324 (2002)

    MATH  MathSciNet  Google Scholar 

  37. Zhang, X., Liu, P.: Method for multiple attribute decision-making under risk with interval numbers. Int. J. Fuzzy Syst. 12(3), 237–242 (2010)

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Zolnierska 49, 71-210, Szczecin, Poland

    Andrzej Piegat

  2. Maritime University of Szczecin, Waly Chrobrego 1-2, 70-500, Szczecin, Poland

    Marek Landowski

Authors
  1. Andrzej Piegat
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  2. Marek Landowski
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Correspondence to Marek Landowski.

Additional information

In 2014 professor Lotfi Zadeh, creator of fuzzy sets and of Computing with Words idea, celebrated his 93rd birthday. On this occasion a special conference “4th World Conference on Soft Computing” dedicated to his research heritage was organized by University of California in Berkeley. Authors of this paper dedicate it, and especially the novel notion of horizontal membership functions, to professor Zadeh.

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Piegat, A., Landowski, M. Horizontal Membership Function and Examples of its Applications. Int. J. Fuzzy Syst. 17, 22–30 (2015). https://doi.org/10.1007/s40815-015-0013-8

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