The Foundations of Homotopic Fuzzy Sets

Fuzzy sets are determined by membership functions. Many methods have been developed when determining the membership function of a fuzzy set. However, a fuzzy set can be specified with more than one membership function. Therefore, the membership function fitting problem is a well-known problem in fuzzy set theory. In this article, we have introduced the concepts of topologically continuous fuzzy set and homotopic fuzzy set whose membership functions are topologically continuous and homotopic, using the basic concepts of topology to overcome this problem. We have studied its basic structural properties. Finally, we proposed a solution method to the membership function fitting problem in fuzzy set theory using the homotopic fuzzy set concept.

Keywords:

Fuzzy set, Homotopy, Homotopic fuzzy set Topology,

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[1] Chen, J.E., Otto, K.N., Constructing membership functions using interpolation and measurement theory. Fuzzy Set Syst. 1995;73:313–327.
[2] Civanlar, M. R., Trussel, H. J. Constructing membership functions using statistical data. Fuzzy Set Syst. 1986;18:1–14.
[3] Dombi, J. Membership function as an evaluation. Fuzzy Set Syst. 1990;35:1–22.
[4] Dubois, D., Prade, H. Fuzzy sets and systems : theory and applications. New York: Academic Press; 1980.
[5] Furukawa, M., Yamakawa, T. The design algorithms of membership functions for a fuzzy neuron. Fuzzy Set Syst. 1995;71(3):329–343.
[6] Giles, R. The concept of grade of membership. Fuzzy Set Syst. 1988;25:297–323.
[7] Hersh, H., Carmazza, A., Brownell, H. H. Effects of context on fuzzy membership functions. In M. M. Gupta, R. M. Ragade, R. R. Yager, editors, Advances in Fuzzy Set Theory, Amsterdam: North Holland; 1979. 389–408.
[8] Lambert, J. The fuzzy set membership problem using the hierarchy decision method, Fuzzy Set Syst. 1992;48(3):323–330.
[9] Ying-Ming, L., Mao-Kang, L. Fuzzy Topology. Singapore: World Scientific Publishing Co. Pte. Ltd.; 1997.
[10] Mabuchi, S. An interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators .1. case of type1 fuzzy sets. Fuzzy Set Syst. 1992;49(3):271–283.
[11] Munkres, J. R. Topology 2nd Ed. Upper Saddle River, NJ: Prentice Hall, Inc.; 2000.
[12] Norwich, A. M., T¨urks¸en, I. B. The construction of membership functions. In R. R. Yager editor, Fuzzy Sets and Possibility Theory: Recent Developments, New York: Pergamon Press; 1982. 61–67.
[13] Norwich, A. M., T¨urks¸en, I. B. The fundamental measurement of fuzziness. In R. R. Yager editor, Fuzzy Sets and Possibility Theory: Recent Developments, New York: Pergamon Press; 1982. 49–60.
[14] Norwich, A. M., T¨urks¸en, I. B. A model for the measurement of membership and the consequences of its empirical implementation, Fuzzy Set Syst. 1984;12:1–25.
[15] Patty, C. W. Foundations of topology. Boston: PWS-KENT Pub.; 1992.
[16] Pedrycz, W. Why triangular membership functions?. Fuzzy Set Syst. 1994;64(1):21–30.
[17] Saaty, T. L. Scaling the membership function. Eur. J. Oper. Res. 1986;25:320–329.
[18] Triantaphyllou, E., Mann, S. H. An evaluation of the eigenvalue approach for determining the membership values in fuzzy sets. Fuzzy Set Syst. 1990;35:295–301.
[19] Turks¸en, I. B. Measurement of membership functions and their acquisition. Fuzzy Set Syst. 1991;40:5–38.
[20] Yamakawa, T., Furukawa, M. A design algorithm of membership functions for a fuzzy neuron using example based learning. Proceedings of the First IEEE Conference on Fuzzy Systems; San Diego: 1992. p. 75–82.
[21] Zadeh, L. A. Fuzzy sets, Inform. Contr. 1965;8:338–353.
[22] Zimmermann, H. J. Fuzzy set theory and its applications 2nd Ed. Boston: Kluwer Academic Publishers; 1991.
[23] Zysno, P. Modeling membership functions. In B. B. Rieger, editor, Empirical Semantics I, Vol. 1 of Quantitative Semantics, Vol. 12, Bochum: StudienverlagBrockmeyer; 1982. 350–375.

Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2013
Yayıncı: Mehmet Zeki SARIKAYA

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