PERFORMING ARITHMETIC OPERATIONS OVER THE (L–R)-TYPE FUZZY NUMBERS
Description
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
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References
- Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8 (3), 338–353. doi: https://doi.org/10.1016/s0019-9958(65)90241-x
- Kaufman, A. (1975). Theory of Fuzzy subsets. Academic Press, 432.
- 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
- Nahmias, S. (1978). Fuzzy variables. Fuzzy Sets and Systems, 1 (2), 97–110. doi: https://doi.org/10.1016/0165-0114(78)90011-8
- Dubois, D., Prade, H. (1988). Théorie des possibilités: application à la représentation des connaissances en informatique. Masson, Paris.
- Dubois, D,. Prade, H. (1987). Analysis of Fuzzy Information. Mathematics and logic. Vol. 1. CRC Press, Boca Raton, 3–39.
- Kaufman, A., Gupta, Madan M. (1985). Introduction to Fuzzy Arithmetic Theory and Applications. Van Nostrand Reinhold Company, 351.
- Raskin, L., Sira, O. (2008). Nechetkaya matematika. Kharkiv: Parus, 352.
- Musa, I. M. (2015). Investigation of Basic Concepts of Fuzzy Arithmetic. Eastern Mediterranean University, 49.
- 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
- Keskar, A. G. (2016). Presentation on Fuzzy Arithmetic. VNIT-Nagpur, 86.
- 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
- 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
- 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
- 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
- Rotshteyn, A. P. (1999). Intellektual'nye tehnologii identifikatsii: nechetkie mnozhestva, geneticheskie algoritmy, neyronnye seti. Vinnytsia: UNIVERSUM-Vinnytsia, 295.
- 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
- Grinyaev, Yu. V. (2008). Teoriya nechetkih mnozhestv. Tomsk, 196.
- 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
- Leonenkov, A. V. (2003). Nechetkoe modelirovanie v srede Matlab i fuzzyTech. Sankt-Peterburg: BHV – Peterburg, 736.
- Ibragimov, V. A. (2010). Elementy nechetkoy matematiki. Baku: AGNA, 394.
- 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
- 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