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Fuzzy Arithmetic Type 1 with Horizontal Membership Functions

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Uncertainty Modeling

Part of the book series: Studies in Computational Intelligence ((SCI,volume 683))

Abstract

The chapter shortly (because of the volume limitation) presents multidimensional fuzzy arithmetic based on relative-distance-measure (RDM) and horizontal membership functions which considerably facilitate calculations. This arithmetic will be denoted as MD-RDM-F one. It delivers full, multidimensional problem solutions that further enable determining, in an accurate and unique way, various representations of the solutions such as span (maximal uncertainty of the solution), cardinality distribution of possible solution values, center of gravity of the solution granule, etc. It also allows for taking into account relations and dependencies existing between variables, what is absolutely necessary e.g. in calculations with fuzzy probabilities that always should sum up to 1 or in equation system solving.

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References

  1. Abbasbandy, S., Asady, B.: Newton’s method for solving fuzzy nonlinear equations. Applied Mathematics and Computation 159, 349–356 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  2. Abbasbandy, S., Jafarian, A.: Steepest descent method for system of fuzzy linear equations. Applied Mathematics and Computation 175, 823–833 (2006).

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. Aliev, R.A.: Fundamentals of the fuzzy logic-based generalized theory of decisions. Springer-Verlag, Berlin, Heidelberg, 2013.

    Book  MATH  Google Scholar 

  5. Bhiwani, R.J., Patre, B.M.: Solving first order fuzzy equations: a modal interval approach. Proceedings of Second International Conference on Emerging Trends and Technology ICETET-09, IEEE Computer Society, 953–956 (2009).

    Google Scholar 

  6. Boading, L.: Uncertainty theory. 2nd ed., Springer-Verlag, 2007.

    Google Scholar 

  7. Buckley, J.J., Qu, Y.: Solving linear and Quadratic Fuzzy Equations. Fuzzy Sets and Systems, Vol. 38, 43–59 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  8. Buckley, J.J., Feuring, T., Hayashi, Y.: Solving Fuzzy Equations using Evolutionary Algorithms and Neural Nets. Soft Computing - A Fusion of Foundations, Methodologies and Applications, Vol. 6, No. 2, 116–123 (2002).

    Google Scholar 

  9. Chen, M.-C., Wang W.-Y., Su, S.-F., Chien, Y.-H.: Robust T-S fuzzy-neural control of a uncertain active suspension systems. International Journal of Fuzzy Systems, Vol. 12, No. 4, 321–329 (December 2010).

    Google Scholar 

  10. Dubois, D., Prade, H.: Operations on fuzzy numbers. International Journal of Systems Science 9(6), 613–626 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  11. Dubois, D., Fargier, H., Fortin, J.: A generalized vertex method for computing with fuzzy intervals. In: Proceedings of the IEEE International Conference on Fuzzy Systems, 541–546 (2004).

    Google Scholar 

  12. Dutta, P., Boruah, H., Al, T.: Fuzzy arithmetic with and without using \(\alpha \)-cut method: A comparative study. International Journal of Latest Trends in Computing. Vol. 2, Issue 1, 99–107 (March 2011).

    Google Scholar 

  13. Dymova L.: Fuzzy solution of interval nonlinear equations. Parallel Processing and Applied Mathematics, LCNS 6068, Part II, Springer, 418–426 (2010).

    Google Scholar 

  14. Dymova, L.: Soft computing in economics and finance. Springer-Verlag, Berlin, Heidelberg, 2011.

    Book  Google Scholar 

  15. Fodor, J., Bede, B.: Arithmetics with fuzzy numbers: a comparative overview. Proceedings of 4th Hungarian Joint Symposium on Applied Machine Intelligence, Herlany, Slovakia, 54–68 (2006).

    Google Scholar 

  16. Hanss, M.: On the implementation of fuzzy arithmetical operations for engineering problems. Proceedings of the 18th International Conference of the North American Fuzzy Information Processing Society - NAFIPS’99, New York, 462–466 (1999).

    Google Scholar 

  17. Hanss, M.: Applied fuzzy arithmetic. Springer-Verlag, Berlin, Heidelberg, 2005.

    MATH  Google Scholar 

  18. Kaufmann, A., Gupta, M.M.: Introduction to fuzzy arithmetic. Van Nostrand Reinhold, New York, 1991.

    MATH  Google Scholar 

  19. Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets and Systems, 165–175 (1997).

    Google Scholar 

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

    Google Scholar 

  21. Li, Q.X., Liu, Si.F.: The foundation of the grey matrix and the grey input-output analysis. Applied Mathematical Modelling 32, 267–291 (2008).

    Google Scholar 

  22. Moore, R.E., Lodwick, W.: Interval analysis and fuzzy set theory. Fuzzy Sets and Systems 135, 5–9 (2003).

    Google Scholar 

  23. Moore, R.E.: Interval analysis. Prentice-Hall, Englewood Cliffs, NJ, 1966.

    MATH  Google Scholar 

  24. Moore, R.E., Kearfott, R.B., Cloud, J.M..: Introduction to interval analysis. SIAM, Philadelphia, 2009.

    Book  MATH  Google Scholar 

  25. Pedrycz, W., Skowron, A., Kreinovich, V. (eds.): Handbook of granular computing. John Wiley & Sons, Chichester, 2008.

    Google Scholar 

  26. Piegat, A.: Fuzzy modeling and control. Phisica-Verlag, Heidelberg, New York, 2001.

    Book  MATH  Google Scholar 

  27. Piegat, A.: Cardinality Approach to Fuzzy Number Arithmetic. IEEE Transactions on Fuzzy Systems, Vol. 13, No. 2, 204–215 (2005).

    Article  MathSciNet  Google Scholar 

  28. Piegat, A., Landowski, M.: Is the conventional interval-arithmetic correct? Journal of Theoretical and Applied Computer Science, Vol. 6, No. 2, 27–44 (2012).

    Google Scholar 

  29. Piegat, A., Tomaszewska, K.: Decision-Making under uncertainty using Info-Gap Theory and a New Multi-dimensional RDM interval arithmetic. Przeglaad Elektrotechniczny (Electrotechnic Review), R.89, No. 8, 71–76 (2013).

    Google Scholar 

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

    Google Scholar 

  31. Piegat, A., Landowski, M.: Two Interpretations of Multidimensional RDM Interval Arithmetic—Multiplication and Division. International Journal of Fuzzy Systems, vol. 15 no. 4, Dec. 2013, 488–496 (2013).

    Google Scholar 

  32. Piegat, A., Landowski, M.: Correctness-checking of uncertain-equation solutions on example of the interval-modal method. In: Atanassov T. et al., Modern Approaches in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, Volume I: Foundations, SRI PAS - IBS PAN, Warsaw, Poland, 159–170 (2014).

    Google Scholar 

  33. Stefanini, L.: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets and Systems, Vol. 161, Issue 11, 1564–1584 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  34. Stefanini, L.: New tools in fuzzy arithmetic with fuzzy numbers. In: Hullermeier E., Kruse R., Hoffmann F. (eds), IPMU 2010, Part II, CCIS 81, Springer-Verlag, Berlin, Heidelberg, 471–480 (2010).

    Google Scholar 

  35. Tomaszewska, K.: The application of horizontal membership function to fuzzy arithmetic operations. Journal of Theoretical and Applied Computer Science, Vol. 8, No. 2, 3-10 (2014).

    Google Scholar 

  36. Tomaszewska, K., Piegat, A.: Application of the horizontal membership function to the uncertain displacement calculation of a composite massless rod under a tensile load. Proceedings of International Conference Advanced Computer Systems (ACS 2014), Miedzyzdroje, Poland, October 22–24, 2014.

    Google Scholar 

  37. Zadeh, L.A.: Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems 4(2), 103–111 (1996).

    Article  Google Scholar 

  38. Zadeh, L.A.: From computing with numbers to computing with words - from manipulation of measurements to manipulation of perceptions. International Journal of Applied Mathematics and Computer Science 12(3), 307–324 (2002).

    MathSciNet  MATH  Google Scholar 

  39. Zhow, Ch.: Fuzzy-arithmetic-based Lyapunov synthesis in the design of stable fuzzy controllers: a Computing with Words approach. International Journal of Applied Mathematics and Computer Science (AMCS), Vol. 12, No. 3, 411–421 (2002).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Computer Science and Information Systems, 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 Andrzej Piegat .

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Editors and Affiliations

  1. University of Texas at El Paso, Department of Computer Science University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

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Piegat, A., Landowski, M. (2017). Fuzzy Arithmetic Type 1 with Horizontal Membership Functions. In: Kreinovich, V. (eds) Uncertainty Modeling. Studies in Computational Intelligence, vol 683. Springer, Cham. https://doi.org/10.1007/978-3-319-51052-1_14

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  • DOI: https://doi.org/10.1007/978-3-319-51052-1_14

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-51051-4

  • Online ISBN: 978-3-319-51052-1

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