Abstract
In the class of decision-making problems with fuzzy information concerning criterion values, the problem of comparing fuzzy numbers is relevant. There are various approaches to solving it. They are determined by the specific character of the problem under consideration. This paper proposes one approach to comparing fuzzy numbers. The proposed approach is as follows. At first, a rule is constructed for comparing a real number with a level set of a fuzzy number. Then, with the help of a procedure for constructing the exact lower approximation for the collection of sets, a fuzzy set is constructed. This fuzzy set determine the rule for comparing a real number with a fuzzy number. Using this rule and the approach based on separating two fuzzy numbers with a real number, the procedure is chosen for comparing two fuzzy numbers. As an example, fuzzy numbers with trapezoidal membership functions are considered, and the geometric interpretation of the results being given.
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References
Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1963)
Zadeh, L.A.: Fuzzy logic. Computer 21(4), 83–93 (1988)
Zimmermann, H.J.: Fuzzy Set Theory - and its Applications. Kluwer Academic Publishers, New York (1996)
Kosko, B.: Fuzzy systems as universal approximators. IEEE Trans. Comput. 43(11), 1329–1333 (1994)
Li, L., Lai, K.K.: A fuzzy approach to the multiobjective transportation problem. Comput. Oper. Res. 27(1), 43–57 (2000)
Chen, C.T., Lin, C.T., Huang, S.F.: A fuzzy approach for supplier evaluation and selection in supply chain management. Int. J. Prod. Econ. 102(2), 289–301 (2006)
Bahri, O., Talbi, E.G., Amor, N.B.: A generic fuzzy approach for multi-objective optimization under uncertainty. Swarm Evol. Comput. 40, 166–183 (2018)
Korzhov, A.V., Korzhova, M.E.: A method of accounting for fuzzy operational factors influencing 6 (10) kV power cable insulation longevity. In: 2016 2nd International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM), pp. 1–4. IEEE, Chelyabinsk (2016). https://doi.org/10.1109/ICIEAM.2016.7911429
Larbani, M.: Non cooperative fuzzy games in normal form: a survey. Fuzzy Sets Syst. 160(22), 3184–3210 (2009)
Kudryavtsev, K.N., Stabulit, I.S., Ukhobotov, V.I.: A bimatrix game with fuzzy payoffs and crisp game. In: CEUR Workshop Proceedings 1987, pp. 343–349 (2017)
Kudryavtsev, K.N., Stabulit, I.S., Ukhobotov, V.I.: One approach to fuzzy matrix games. In: CEUR Workshop Proceedings 2098, pp. 228–238 (2018)
Verma, T., Kumar, A.: Ambika methods for solving matrix games with Atanassov’s intuitionistic fuzzy payoffs. IEEE Trans. Fuzzy Syst. 26(1), 270–283 (2018)
Dutta, P., Boruah, H., Ali, T.: Fuzzy arithmetic with and without using \(\alpha \)-cut method: a comparative study. Int. J. Latest Trends Comput. 2(1), 99–107 (2011)
Bansal, A.: Trapezoidal fuzzy numbers (a, b, c, d): arithmetic behavior. Int. J. Phys. Math. Sci. 2(1), 39–44 (2011)
Gallyamov, E.R., Ukhobotov, V.I.: Computer implementation of operations with fuzzy numbers. Bull. South Ural State Universit. 3(3), 97–108 (2014). Series “Vychislitelnaya Matematika i Informatika”. (in Russian)
Yager, R.R.: A procedure for ordering fuzzy subsets of the unit interval. Inf. Sci. 24(2), 143–161 (1981)
Ibanez, L.M.C., Munoz, A.G.: A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst. 29(2), 145–153 (1989)
Chen, S.J., Hwang, C.L.: Fuzzy multiple attribute decision making methods. In: Fuzzy Multiple Attribute Decision Making, pp. 289–486. Springer, Heidelberg (1992). https://doi.org/10.1007/978-3-642-46768-4_5
Ukhobotov, V.I., Shchichko, P.V.: An approach to ranking fuzzy numbers. Bull. South Ural State Universit. 10, 54–62 (2011). Series “Matematicheskoe modelirovanie I programmirovanie”. (in Russian)
Ukhobotov, V.I., Mikhailova, E.S.: An approach to the comparison of fuzzy numbers in decision-making problems. Bull. South Ural State University 71, 32–37 (2015). Series “Mathematics. Mechanics. Physics”. (in Russian)
Ukhobotov, V.I., Mikhailova, E.S.: Comparison of fuzzy numbers in decision-making problems. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki 26(1), 87–94 (2016). https://doi.org/10.20537/vm160108. (in Russian)
Ukhobotov, V.I., Stabulit, I.S., Kudryavtsev, K.N.: Comparison of triangular fuzzy numbers. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki 29(2), 197–210 (2019). https://doi.org/10.20537/vm190205. (in Russian)
Ukhobotov, V.I., Stabulit, I.S., Kudryavtsev, K.N.: On decision making under fuzzy information about an uncontrolled factor. Procedia Comput. Sci. 150, 524–531 (2019). https://doi.org/10.1016/j.procs.2019.02.088
Ramik, J., Vlach, M.: Generalized Concavity in Fuzzy Optimization and Decision Analysis. Kluwer Academic Publishers, Boston (2001)
Ukhobotov, V.I.: The Selected Chapters of Fuzzy Set Theory: Study Guide. Chelyabinsk State University, Chelyabinsk (2011). (in Russian)
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Ukhobotov, V., Stabulit, I., Kudryavtsev, K. (2019). On the Issue of Comparison of Fuzzy Numbers. In: Bykadorov, I., Strusevich, V., Tchemisova, T. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2019. Communications in Computer and Information Science, vol 1090. Springer, Cham. https://doi.org/10.1007/978-3-030-33394-2_45
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