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Aggregation Operator for Ordered Fuzzy Numbers Concerning the Direction

  • Piotr Prokopowicz
  • Samira Malek Mohamadi Golsefid
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8467)

Abstract

Ordered Fuzzy Numbers (OFN) were proposed about 10 years ago [7,8] as a tool for the calculations of imprecise values represented by fuzzy numbers. Calculation methods based on this model shall retain properties of operations known from real numbers. In addition, in contrast to the classic operations on convex fuzzy numbers, making a series of operations in accordance with the OFN model is not doomed to greater and greater imprecision of the results.

Apart from good computational properties, OFNs also offer new possibilities for imprecise information processing by using fuzzy systems. [13,14,18] show examples of systems and the various proposals for methods based on the new model. There is a range of work [20,21,23], which focus on implications or inference operators. In the works [10,23,24,29] various aspects of defuzzification were analyzed. Little attention has been paid to aggregation of premises of rules based on OFN so far. Therefore, the aim of this paper is to propose effective aggregation operator which will generate good results as well as being intuitively consistent with the idea of the new model. Moreover, the proposed solution maintains the expected properties of the aggregate functions [6,16], it takes into account key idea of OFN the direction of components.

Keywords

aggregation operator Ordered Fuzzy Numbers direction in aggregation fuzzy system with Ordered Fuzzy Numbers 

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References

  1. 1.
    Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning, Part I, II, III. Information Sciences 8, 199–249 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Nguyen, H.T.: A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. 64, 369–380 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Driankov, D., Hellendoorn, H., Reinfrank, M.: An Introduction to fuzzy control. Springer, Heidelberg (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Wagenknecht, M., Hampel, R., Schneider, V.: Computational aspects of fuzzy arithmetic based on archimedean t-norms. Fuzzy Sets and Systems 123(1), 49–62 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Calvo, T., Kolesarova, A., Komornikova, M., Mesiar, R.: Aggregation operators: properties, classes and construction methods. In: Aggregation Operators: New Trends and Applications, pp. 3–104. Physica, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Kosiński, W., Prokopowicz, P., Ślęzak, D.: Ordered fuzzy numbers. Bulletin of the Polish Academy of Sciences, Ser. Sci. Math. 51(3), 327–338 (2003)Google Scholar
  8. 8.
    Kosiński, W., Prokopowicz, P., Ślęzak, D.: On algebraic operations on fuzzy numbers. In: Intelligent Information Processing and Web Mining. ASC, vol. 22, pp. 353–362. Springer (2003)Google Scholar
  9. 9.
    Ross, T.J.: Fuzzy Logic for Engineering Applications-, 2nd edn. John Wiley and Sons, UK (2004)Google Scholar
  10. 10.
    Kosiński, W.: On defuzzyfication of ordered fuzzy numbers. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 326–331. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Koleśnik, R., Prokopowicz, P., Kosiński, W.: Fuzzy calculator – useful tool for programming with fuzzy algebra. In: Rutkowski, L., Siekmann, J.H., Tadeusiewicz, R., Zadeh, L.A. (eds.) ICAISC 2004. LNCS (LNAI), vol. 3070, pp. 320–325. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  12. 12.
    Buckley James, J., Eslami, E.: An Introduction to Fuzzy Logic and Fuzzy Sets. Physica-Verlag, A Springer-Verlag Company, Heidelberg (2005)Google Scholar
  13. 13.
    Prokopowicz, P.: Methods based on the ordered fuzzy numbers used in fuzzy control. In: Proc. of the Fifth International Workshop on Robot Motion and Control – RoMoCo 2005, Dymaczewo, Poland, pp. 349–354 (June 2005)Google Scholar
  14. 14.
    Prokopowicz, P.: Using Ordered Fuzzy Numbers Arithmetic in Fuzzy Control. In: Artificial Intelligence and Soft Computing Proc. of the 8th ICAISC, Poland, pp. 156–162. Academic Publishing House EXIT, Warsaw (2006)Google Scholar
  15. 15.
    Kosiński, W.: On fuzzy number calculus. International Journal of Applied Mathematics and Computer Science 16(1), 51–57 (2006)MathSciNetGoogle Scholar
  16. 16.
    Beliakov, G., Pradera, A., Calvo, T.: Aggregation functions: a guide for practitioners. STUDFUZZ, vol. 221. Springer, Berlin (2007)Google Scholar
  17. 17.
    Kosiński, W., Prokopowicz, P.: Fuzziness - Representation of Dynamic Changes, Using Ordered Fuzzy Numbers Arithmetic, New Dimensions in Fuzzy Logic nd Related Technologies. In: Proc. of the 5th EUSFLAT, Ostrava, Czech Republic, vol. I, pp. 449–456. University of Ostrava (2007)Google Scholar
  18. 18.
    Prokopowicz, P.: Adaptation of Rules in the Fuzzy Control System Using the Arithmetic of Ordered Fuzzy Numbers. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2008. LNCS (LNAI), vol. 5097, pp. 306–316. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Kosiński, W., Prokopowicz, P., Kacprzak, D.: Fuzziness – representation of dynamic changes by ordered fuzzy numbers. In: Seising, R. (ed.) Views on Fuzzy Sets and Systems from Different Perspectives. STUDFUZZ, vol. 243, pp. 485–508. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Kosiński, W., Wilczyńska-Sztyma, D.: Defuzzification and implication within ordered fuzzy numbers. In: 2010 IEEE World Congress on Computational Intelligence, WCCI 2010 (2010)Google Scholar
  21. 21.
    Kacprzak, M., Kosiński, W.: On lattice structure and implications on ordered fuzzy numbers. In: Proc. of the 7th EUSFLAT Conference, EUSFLAT 2011 and French Days on Fuzzy Logic and Applications, LFA 2011 (2011)Google Scholar
  22. 22.
    Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E.: Aggregation functions: Means. Information Sciences 181(1), 1–22 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Kacprzak, M., Kosiński, W., Prokopowicz, P.: Implications on Ordered Fuzzy Numbers and Fuzzy Sets of Type Two. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2012, Part I. LNCS (LNAI), vol. 7267, pp. 247–255. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  24. 24.
    Kosiński, W., Rosa, A., Cendrowska, D., Węgrzyn-Wolska, K.: Defuzzification Functionals Are Homogeneous, Restrictive Additive and Normalized Functions. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2012, Part I. LNCS (LNAI), vol. 7267, pp. 274–282. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  25. 25.
    Prokopowicz, P.: Flexible and Simple Methods of Calculations on Fuzzy Numbers with the Ordered Fuzzy Numbers Model. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part I. LNCS (LNAI), vol. 7894, pp. 365–375. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  26. 26.
    Marszałek, A., Burczyński, T.: Modelling Financial High Frequency Data Using Ordered Fuzzy Numbers. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part I. LNCS, vol. 7894, pp. 345–352. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  27. 27.
    Kacprzak, D., Kosiński, W., Kosiński, W.K.: Financial Stock Data and Ordered Fuzzy Numbers. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part I. LNCS (LNAI), vol. 7894, pp. 259–270. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  28. 28.
    Kacprzak, M., Kosiński, W., Węgrzyn-Wolska, K.: Diversity of opinion evaluated by ordered fuzzy numbers. In: Rutkowski, L., Korytkowski, M., Scherer, R., Tadeusiewicz, R., Zadeh, L.A., Zurada, J.M. (eds.) ICAISC 2013, Part I. LNCS (LNAI), vol. 7894, pp. 271–281. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  29. 29.
    Kosiński, W., Prokopowicz, P., Rosa, A.: Defuzzification Functionals of Ordered Fuzzy Numbers. IEEE Transactions on Fuzzy Systems 21(6), 1163–1169 (2013), doi:10.1109/TFUZZ.2013.2243456CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Piotr Prokopowicz
    • 1
  • Samira Malek Mohamadi Golsefid
    • 2
  1. 1.Institute of Mechanics and Applied Computer ScienceKazimierz Wielki UniversityBydgoszczPoland
  2. 2.Department of Industrial EngineeringAmirkabir University of TechnologyTehranIran

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