Geodesic
Convergence of series of fuzzy numbers with unimodal membership function
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 1, pp. 11-17.

Voir la notice de l'article provenant de la source Math-Net.Ru

Solving the applied problems by the methods of fuzzy mathematics frequently generates the need to conduct operations on fuzzy numbers. The calculation of such expressions requires quite complex manipulations and a serious effort. For example, addition and subtraction formulae can be obtained by $L-R$ fuzzy numbers, but this approach enables to calculate multiplication and division only approximately. $t$-norms and interval mathematics are used to implement the arithmetics of trapezoidal numbers. We present fuzzy numbers with unimodal membership functions that can be used for fuzzy analysis in such subject fields as ecology and chemical technology. Some knowledge of behavior of these series enable us to analyze such mathematical models more effectively. The associativity of addition enables to analyze number series effectively. The problem of convergence of series of fuzzy numbers with unimodal membership function is considered. The formulae for calculating arithmetic operations on sequences of fuzzy numbers are obtained. We generalize the addition formula for sequences of fuzzy numbers. We investigate convergence of series of fuzzy numbers. The conditions for divergence of a series are given. It is shown that calculations with large amount of data may cause an indefinite result. The reason of this lies in the fact that a membership function of a sum of the series is identically equal to unit. This means the complete indefiniteness of the result and enables to make a conclusion about divergence of the series. The obtained results for evaluation of the arithmetic operations enable to use fuzzy analysis for investigation of complex systems in, for instance, ecology and chemical technology. The proposed approach is quite general and can be used for rather large class of studies using the methods of fuzzy analysis. In this case it makes sense to limit the length of sequences of fuzzy numbers on the compromise of calculation accuracy and indefiniteness of the result.
Keywords: fuzzy numbers, arithmetical operations, associativity, series of fuzzy numbers, convergence of a series. 
@article{VVGUM_2018_21_1_a1,
     author = {I. V. Germashev and E. V. Derbisher and V. E. Derbisher and N. Yu. Kulikova},
     title = {Convergence of series of fuzzy numbers with unimodal membership function},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {11--17},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a1/}
}
TY  - JOUR
AU  - I. V. Germashev
AU  - E. V. Derbisher
AU  - V. E. Derbisher
AU  - N. Yu. Kulikova
TI  - Convergence of series of fuzzy numbers with unimodal membership function
JO  - Matematičeskaâ fizika i kompʹûternoe modelirovanie
PY  - 2018
SP  - 11
EP  - 17
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a1/
LA  - ru
ID  - VVGUM_2018_21_1_a1
ER  - 
%0 Journal Article
%A I. V. Germashev
%A E. V. Derbisher
%A V. E. Derbisher
%A N. Yu. Kulikova
%T Convergence of series of fuzzy numbers with unimodal membership function
%J Matematičeskaâ fizika i kompʹûternoe modelirovanie
%D 2018
%P 11-17
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a1/
%G ru
%F VVGUM_2018_21_1_a1
I. V. Germashev; E. V. Derbisher; V. E. Derbisher; N. Yu. Kulikova. Convergence of series of fuzzy numbers with unimodal membership function. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 1, pp. 11-17. http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a1/

[1] F. P. Vasilyev, Numerical methods of extremum problems solution, Nauka Publ., Moscow, 1988, 552 pp.

[2] I. V. Germashev, E. V. Derbisher, V. E. Derbisher, “Calculation of arithmetic operations on fuzzy numbers for analysis of complex systems”, Collected Works of the 29 th International Academic Conference, Mathematical Methods in Engineering and Technology – MMTT-29, in 12 vols. (Saint Petersburg, May 31 – June 3, 2016), v. 2, SPbGTI Publ., Saint Petersburg, 2016, 63–65

[3] I. V. Germashev, G. Yu. Klinkova, “Model of estimation of successional age of acquatic ecosystem”, Collected Works of the 27 th International Academic Conference, Mathematical Methods in Engineering and Technology, in 12 vols. (Tambov, June 3–5, 2014), v. 7, TGTU Publ., Tambov, 2014, 5–7

[4] I. V. Germashev, V. E. Derbisher, T. F. Morozenko, S. A. Orlova, “The Assessment of Technical Objects Using Fuzzy Sets”, Industrial Laboratory. Diagnostic of Materials, 67:1 (2001), 65–68

[5] D. Dubois, H. Prade, “Operations on fuzzy numbers”, Int. J. Systems Sci, 9:6 (1978), 613–626 | DOI | MR | Zbl

[6] A. Alim, F. T. Johora, S. Babu, A. Sultana, “Elementary Operations on $L-R$ Fuzzy Number”, Advances in Pure Mathematics, 2015, no. 5, 131–136 | DOI

[7] I. V. Germashev, V. E. Derbisher, “Fuzzy Optimization of Polymer Compositions”, Theoretical Foundations of Chemical Engineering, 35:4 (2001), 418–421 | DOI

[8] I. V. Germashev, V. E. Derbisher, “Properties of unimodal membership functions in operations with fuzzy sets”, Russian Mathematics (Iz. VUZ), 51:3 (2007), 73–76 | DOI | MR

[9] M. V. Koroteev, P. V. Terelyanskii, V. A. Ivanyuk, “Arithmetic of fuzzy numbers in generalized trapezoidal form”, Journal of Mathematical Sciences, 216:5 (2016), 696–701 | DOI | MR | Zbl