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

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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/}
}
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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/