Abstract
The publication shows the way of implementing arithmetic operations on fuzzy numbers based on Ordered Fuzzy Numbers calculation model [12], [13], [14]. This model allows to perform calculations on fuzzy numbers in a way that the outcomes meet the same criteria as the outcomes of calculations on real numbers. In this text, to the four basic operations with Ordered Fuzzy Numbers, a logarithm and exponentiation was added. Several examples of the calculations are included, the results of which are obvious and typical of real numbers but not achievable with the use of conventional computational methods for fuzzy numbers. From these examples one can see that the use of Ordered Fuzzy Numbers allows to obtain outcomes for real numbers in spite of using the fuzzy values.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338–353 (1965)
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)
Dubois, D., Prade, H.: Operations on fuzzy numbers. Int. J. System Science 9(6), 613–626 (1978)
Nguyen, H.T.: A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. 64, 369–380 (1978)
Kaucher, E.: Interval analysis in the extended interval space IR. Computing, Suppl. 2, 33–49 (1980)
Sanchez, E.: Solutions of fuzzy equations with extended operations. Fuzzy Sets and Systems 12, 237–248 (1984)
Klir, G.J.: Fuzzy arithmetic with requisite constraints. Fuzzy Sets and Systems 91(2), 165–175 (1997)
Wagenknecht, M.: On the approximate treatment of fuzzy arithmetics by inclusion, linear regression and information content estimation. In: Chojcan, J., Łęski, J. (eds.) Fuzzy Sets and their Applications, pp. 291–310. Wydawnictwo Politechniki Śląskiej, Gliwice (2001)
Guanrong, C., Tat, P.T.: Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems. CRS Press, Boca Raton (2001)
Drewniak, J.: Fuzzy numbers. In: Chojcan, J., Łęski, J. (eds.) Fuzzy Sets and their Applications, pp. 103–129. WPŚ, Gliwice (2001)
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)
Kosiński, W., Prokopowicz, P., Ślęzak, D.: Ordered fuzzy number. Bulletin of the Polish Academy of Sciences, Ser. Sci. Math. 51(3), 327–338 (2003)
Kosiński, W., Prokopowicz, P., Ślęzak, D.: On algebraic operations on fuzzy numbers. In: Kłopotek, M., Wierzchoń, S.T., Trojanowski, K. (eds.) Intelligent Information Processing and Web Mining, Proc. of Int. IIS: IIPWM 2003, Conference held in Zakopane, Poland, June 2-5, pp. 353–362. Physica-Verlag (2003)
Kosiński, W., Prokopowicz, P., Ślęzak, D.: Calculus with fuzzy numbers. In: Bolc, L., Michalewicz, Z., Nishida, T. (eds.) IMTCI 2004. LNCS (LNAI), vol. 3490, pp. 21–28. Springer, Heidelberg (2005)
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)
Koleśnik, R., Prokopowicz, P., Kosiński, W.: Fuzzy Calculator – usefull 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)
Buckley James, J., Eslami, E.: An Introduction to Fuzzy Logic and Fuzzy Sets. Physica-Verlag, A Springer-Verlag Company, Heidelberg (2005)
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)
Prokopowicz, P.: Using Ordered Fuzzy Numbers Arithmetic in Fuzzy Control. In: Cader, A., Rutkowski, L., Tadeusiewicz, R., Zurada, J.M. (eds.) Proc. of the 8th International Conference on Artificial Intelligence and Soft Computing, Zakopane, Poland, pp. 156–162. Academic Publishing House EXIT, Warsaw (2006)
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)
Kosiński, W., Prokopowicz, P.: Fuzziness - Representation of Dynamic Changes, Using Ordered Fuzzy Numbers Arithmetic, New Dimensions in Fuzzy Logic and Related Technologies. In: Stepnicka, M., Novak, V., Bodenhofer, U. (eds.) Proc. of the 5th EUSFLAT Conference, Ostrava, Czech Republic, September 11-14, vol. I, pp. 449–456. University of Ostrava (2007)
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. STUDFUZZ, vol. 243, pp. 485–508. Springer, Heidelberg (2009)
Kosiński, W., Wilczyńska-Sztyma, D.: Defuzzification and implication within ordered fuzzy numbers. In: WCCI 2010 IEEE World Congress on Computational Intelligence, July 18-23, pp. 1073–1079. CCIB, Barcelona (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Prokopowicz, P. (2013). 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) Artificial Intelligence and Soft Computing. ICAISC 2013. Lecture Notes in Computer Science(), vol 7894. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38658-9_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-38658-9_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38657-2
Online ISBN: 978-3-642-38658-9
eBook Packages: Computer ScienceComputer Science (R0)