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## Archive of Issues

Russia Chelyabinsk
Year
2016
Volume
26
Issue
1
Pages
87-94
 Section Mathematics Title Comparison of fuzzy numbers in decision-making problems Author(-s) Ukhobotov V.I.a, Mikhailova E.S.a Affiliations Chelyabinsk State Universitya Abstract The paper deals with decision-making problems, when a decision maker receives information about possible pay-off as a result of a strategy selection. This information can be given as a fuzzy number and the problem of its comparison appears. A specific character of the problem is a main factor to choose the method of the fuzzy numbers comparison. In this paper an approach of comparing fuzzy numbers has been proposed, it’s based on the comparison of $\alpha$-cuts. These $\alpha$-cuts are segments. During the comparison of the segments, each segment can contain a merit value; one of the decision-making criteria is chosen (Wald's maximin model, Regret theory models, Routh-Hurwitz stability criterion etc.). The results of the comparison are averaged out. Fuzzy numbers are compared according to these mean values. According to geometrical interpretation which has been given, the comparison of fuzzy numbers is equivalent to the comparison of figures' areas. These areas are formed by graphics of membership functions of the fuzzy numbers. As an example trapezoidal and bell-shaped fuzzy numbers are examined. Keywords fuzzy number, membership function, level set UDC 519.816 MSC 03B52, 68T37 DOI 10.20537/vm160108 Received 25 February 2016 Language Russian Citation Ukhobotov V.I., Mikhailova E.S. Comparison of fuzzy numbers in decision-making problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 87-94. References Zadeh L.A. Fuzzy sets, Information and Control, 1965, vol. 8, no. 3, pp. 338-353. Dutta P., Boruah H., Ali T. Fuzzy arithmetic with and without using alpha-cut method: a comparative study, International Journal of Latest Trends in Computing, 2011, vol. 2, no. 1, pp. 99-107. Bansal A. Trapezoidal fuzzy numbers (a, b, c, d): arithmetic behavior, International Journal of Physical and Mathematical Sciences, 2011, vol. 2, no. 1, pp. 39-44. Chen S.-J.J., Hwang C.L. Fuzzy multiple attribute decision making: methods and applications, New York: Springer, 1992, 552 p. Ukhobotov V.I., Shchichko P.V. An approach to ranking fuzzy numbers, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program., 2011, vol. 10, pp. 54-62 (in Russian). Ukhobotov V.I. Izbrannye glavy teorii nechetkikh mnozhestv: uchebnoe posobie (The selected chapters of the theory of fuzzy sets: study guide), Chelyabinsk: Chelyabinsk State University, 2011, 245 p. Gallyamov E.R., Ukhobotov V.I. Computer implementation of operations with fuzzy numbers, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Vychisl. Mat. Inform., 2014, vol. 3, no. 3, pp. 97-108 (in Russian). Ukhobotov V.I., Mihailova E.S. An approach to the comparison of fuzzy numbers in decision-making problems, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Mekh. Fiz., 2015, vol. 7, no. 1, pp. 32-37 (in Russian). Zhukovskii V.I., Kudryavtsev K.N. Uravnoveshivanie konfliktov i prilozheniya (Equilibrating conflicts and applications), Moscow: Lenand, 2012, 304 p. Smirnov V.I. Kurs vysshei matematiki. Tom 2 (The course of higher mathematics. Vol. 2), Moscow: Nauka, 1974, 226 p. Full text