Performing arithmetic operations over the (L–R)-type fuzzy numbers

Authors

DOI:

https://doi.org/10.15587/1729-4061.2020.203590

Keywords:

(L–R)-type fuzzy numbers, compact carrier, rules for performing arithmetic operations.

Abstract

The issue of constructing a system of rules to perform binary operations over fuzzy numbers has been formulated and considered. The set problem has been solved regarding the (L–R)-type fuzzy numbers with a compact carrier. Such a problem statement is predetermined by the simplicity of the analytical notation of these numbers, thereby making it possible to unambiguously set a fuzzy number by a set of values of its parameters. This makes it possible, as regards the (L–R)-type numbers, to reduce the desired execution rules for fuzzy numbers to the rules for simple arithmetic operations over their parameters. It has been established that many cited works provide ratios that describe the rules for performing operations over the (L–R)-type fuzzy numbers that contain errors. In addition, there is no justification for these rules in all cases.

In order to build a correct system of fuzzy arithmetic rules, a set of metarules has been proposed, which determine the principles of construction and the structure of rules for operation execution. Using this set of metarules has enabled the development and description of the system of rules for performing basic arithmetic operations (addition, subtraction, multiplication, division). In this case, different rules are given for the multiplication and division rules, depending on the position of the number carriers involved in the operation, relative to zero. The proposed rule system makes it possible to correctly solve many practical problems whose raw data are not clearly defined. This system of rules for fuzzy numbers with a compact carrier has been expanded to the case involving a non-finite carrier. The relevant approach has been implemented by a two-step procedure. The advantages and drawbacks of this approach have been identified.

Author Biographies

Lev Raskin, National Technical University «Kharkiv Polytechnic Institute» Kyrpychova str., 2, Kharkiv, Ukraine, 61002

Doctor of Technical Sciences, Professor, Head of Department

Department of Distributed Information Systems and Cloud Technologies

Oksana Sira, National Technical University «Kharkiv Polytechnic Institute» Kyrpychova str., 2, Kharkiv, Ukraine, 61002

Doctor of Technical Sciences, Professor

Department of Distributed Information Systems and Cloud Technologies

References

  1. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8 (3), 338–353. doi: https://doi.org/10.1016/s0019-9958(65)90241-x
  2. Kaufman, A. (1975). Theory of Fuzzy subsets. Academic Press, 432.
  3. Dubois, D., Prade, H. (1978). Operations on fuzzy numbers. International Journal of Systems Science, 9 (6), 613–626. doi: https://doi.org/10.1080/00207727808941724
  4. Nahmias, S. (1978). Fuzzy variables. Fuzzy Sets and Systems, 1 (2), 97–110. doi: https://doi.org/10.1016/0165-0114(78)90011-8
  5. Dubois, D., Prade, H. (1988). Théorie des possibilités: application à la représentation des connaissances en informatique. Masson, Paris.
  6. Dubois, D,. Prade, H. (1987). Analysis of Fuzzy Information. Mathematics and logic. Vol. 1. CRC Press, Boca Raton, 3–39.
  7. Kaufman, A., Gupta, Madan M. (1985). Introduction to Fuzzy Arithmetic Theory and Applications. Van Nostrand Reinhold Company, 351.
  8. Raskin, L., Sira, O. (2008). Nechetkaya matematika. Kharkiv: Parus, 352.
  9. Musa, I. M. (2015). Investigation of Basic Concepts of Fuzzy Arithmetic. Eastern Mediterranean University, 49.
  10. Akther, T., Ahmad, S. U. (1970). A computational method for fuzzy arithmetic operations. Daffodil International University Journal of Science and Technology, 4 (1), 18–22. doi: https://doi.org/10.3329/diujst.v4i1.4350
  11. Keskar, A. G. (2016). Presentation on Fuzzy Arithmetic. VNIT-Nagpur, 86.
  12. Garg, H. (2017). Some arithmetic operations on the generalized sigmoidal fuzzy numbers and its application. Granular Computing, 3 (1), 9–25. doi: https://doi.org/10.1007/s41066-017-0052-7
  13. Vahidi, J., Rezvani, S. (2013). Arithmetic Operations on Trapezoidal Fuzzy Numbers. Journal of Nonlinear Analysis and Application, 2013, 1–8. doi: https://doi.org/10.5899/2013/jnaa-00111
  14. Seresht, N. G., Fayek, A. R. (2018). Fuzzy Arithmetic Operations: Theory and Applications in Construction Engineering and Management. Fuzzy Hybrid Computing in Construction Engineering and Management, 111–147. doi: https://doi.org/10.1108/978-1-78743-868-220181003
  15. Gerami Seresht, N., Fayek, A. R. (2019). Computational method for fuzzy arithmetic operations on triangular fuzzy numbers by extension principle. International Journal of Approximate Reasoning, 106, 172–193. doi: https://doi.org/10.1016/j.ijar.2019.01.005
  16. Rotshteyn, A. P. (1999). Intellektual'nye tehnologii identifikatsii: nechetkie mnozhestva, geneticheskie algoritmy, neyronnye seti. Vinnytsia: UNIVERSUM-Vinnytsia, 295.
  17. Alim, A., Johora, F. T., Babu, S., Sultana, A. (2015). RETRACTED: Elementary Operations on L-R Fuzzy Number. Advances in Pure Mathematics, 05 (03), 131–136. doi: https://doi.org/10.4236/apm.2015.53016
  18. Grinyaev, Yu. V. (2008). Teoriya nechetkih mnozhestv. Tomsk, 196.
  19. Asady, B., Akbari, M., Keramati, M. A. (2011). Ranking of fuzzy numbers by fuzzy mapping. International Journal of Computer Mathematics, 88 (8), 1603–1618. doi: https://doi.org/10.1080/00207160.2010.518757
  20. Leonenkov, A. V. (2003). Nechetkoe modelirovanie v srede Matlab i fuzzyTech. Sankt-Peterburg: BHV – Peterburg, 736.
  21. Ibragimov, V. A. (2010). Elementy nechetkoy matematiki. Baku: AGNA, 394.
  22. Raskin, L., Sira, O. (2016). Fuzzy models of rough mathematics. Eastern-European Journal of Enterprise Technologies, 6 (4 (84)), 53–60. doi: https://doi.org/10.15587/1729-4061.2016.86739
  23. Raskin, L., Sira, O. (2016). Method of solving fuzzy problems of mathematical programming. Eastern-European Journal of Enterprise Technologies, 5 (4 (83)), 23–28. doi: https://doi.org/10.15587/1729-4061.2016.81292

Downloads

Published

2020-06-30

How to Cite

Raskin, L., & Sira, O. (2020). Performing arithmetic operations over the (L–R)-type fuzzy numbers. Eastern-European Journal of Enterprise Technologies, 3(4 (105), 6–11. https://doi.org/10.15587/1729-4061.2020.203590

Issue

Vol. 3 No. 4 (105) (2020): Mathematics and Cybernetics - applied aspects

Section

Mathematics and Cybernetics - applied aspects

License

Copyright (c) 2020 Lev Raskin, Oksana Sira

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

The consolidation and conditions for the transfer of copyright (identification of authorship) is carried out in the License Agreement. In particular, the authors reserve the right to the authorship of their manuscript and transfer the first publication of this work to the journal under the terms of the Creative Commons CC BY license. At the same time, they have the right to conclude on their own additional agreements concerning the non-exclusive distribution of the work in the form in which it was published by this journal, but provided that the link to the first publication of the article in this journal is preserved.

A license agreement is a document in which the author warrants that he/she owns all copyright for the work (manuscript, article, etc.).
The authors, signing the License Agreement with TECHNOLOGY CENTER PC, have all rights to the further use of their work, provided that they link to our edition in which the work was published.
According to the terms of the License Agreement, the Publisher TECHNOLOGY CENTER PC does not take away your copyrights and receives permission from the authors to use and dissemination of the publication through the world's scientific resources (own electronic resources, scientometric databases, repositories, libraries, etc.).
In the absence of a signed License Agreement or in the absence of this agreement of identifiers allowing to identify the identity of the author, the editors have no right to work with the manuscript.
It is important to remember that there is another type of agreement between authors and publishers – when copyright is transferred from the authors to the publisher. In this case, the authors lose ownership of their work and may not use it in any way.