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JACIII Vol.20 No.4 pp. 535-542
doi: 10.20965/jaciii.2016.p0535
(2016)

Paper:

# An Objective Approach for Constructing a Membership Function Based on Fuzzy Harvda-Charvat Entropy and Mathematical Programming

## Takashi Hasuike* and Hideki Katagiri**

*Faculty of Science and Technology, Waseda University
3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

**Faculty of Engineering, Kanagawa University
3-27-1 Rokkakubashi, Kanagawa-ku, Yokohama, Kanagawa 221-8686, Japan

December 25, 2015
Accepted:
March 26, 2016
Published:
July 19, 2016
Keywords:
membership function, constructing approach with subjectivity and objectivity, fuzzy entropy, mathematical programming
Abstract
This paper proposes an objective approach to the construction of an appropriate membership function that extends to our previous studies. It is important to set a membership function with subjectivity and objectivity to obtain a reasonable optimal solution that complies with the decision maker’s feelings in real-world decision making. To ensure objectivity and subjectivity of the obtained membership function, an entropy-based approach based on mathematical programming is integrated into the interval estimation considered by the decision maker. Fuzzy Harvda-Charvat entropy, which is a natural extension of fuzzy Shannon entropy, is introduced as general entropy with fuzziness. The main steps of our proposed approach are to set intervals with membership values 0 and 1 to enable a decision maker to judge confidently, and to solve the proposed mathematical programming problem strictly using nonlinear programming. In this paper, the given membership function is assumed to be a piecewise linear membership function as an approximation of nonlinear functions, and each intermediate value of partial linear function is optimally obtained.
T. Hasuike and H. Katagiri, “An Objective Approach for Constructing a Membership Function Based on Fuzzy Harvda-Charvat Entropy and Mathematical Programming,” J. Adv. Comput. Intell. Intell. Inform., Vol.20 No.4, pp. 535-542, 2016.
Data files:
References
1. [1] S. Gottwald, “A note on measures of fuzziness,” Elektron Informationsverarb Kybernet, Vol.15, pp. 221-223, 1979.
2. [2] B. Bharathi and V.V.S. Sarma, “Estimation of fuzzy membership from histograms,” Information Sciences, Vol.35, pp. 43-59, 1985.
3. [3] M. R. Civanlar and H. J. Trussell, “Constructing membership functions using statistical data,” Fuzzy Sets and Systems, Vol.18, pp. 1-13, 1986.
4. [4] H. D. Cheng and Y. H. Cheng, “Thresholding based on fuzzy partition of 2D histogram,” Proc. of IEEE Int. Conf. on Pattern Recognition, Vol.2, pp. 1616-1618, 1997.
5. [5] G. Nieradka and B. Butkiewicz, “A method for automatic membership function estimation based on fuzzy measures,” Proc. of IFSA2007, LNAI4529, pp. 451-460, 2007.
6. [6] H.D. Cheng and J.R. Cheng, “Automatically determine the membership function based on the maximum entropy principle,” Information Sciences, Vol.96, pp. 163-182, 1997.
7. [7] T. Hasuike, H. Katagiri, and H. Tsubaki, “Constructing an appropriate membership function integrating fuzzy Shannon entropy and human’s interval estimation,” ICIC Express Letters, Vol.8, No.3, pp. 809-813, 2014.
8. [8] J.L. Chameau and J.C. Santamarina, “Membership function I: Comparing methods of Measurement,” Int. J. of Approximate Reasoning, Vol.1, pp. 287-301, 1987.
9. [9] A. Yoshikawa, “Influence of procedure for interacive identification method on forms of identified membership functions,” Japan Society for Fuzzy Theory and Intelligent Informatices, Vol.19, No.1, pp. 69-78, 2007 (in Japanese).
10. [10] T. Hasuike, H. Katagiri, and H. Tsubaki, “A constructing algorithm for appropriate piecewise linear membership function based on statistics and information theory,” Procedia Computer Science, Vol.60, pp. 994-1003, 2015.
11. [11] K. Fujimoto and M. Sugeno, “Obtaining admissible preference orders using hierarchical bipolar Sugeno and Choquet integrals,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.17, No.4, pp. 493-503, 2013.
12. [12] K. Hayashi, A. Otsumo, and K. Shiranita, “Fuzzy control using piecewise linear membership functions based on knowledge of tuning a PID Controller,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.5, No.1, pp. 71-77, 2001.
13. [13] T. Hasuike, H. Katagiri, and H. Tsubaki, “An interactive algorithm to construct an appropriate nonlinear membership function using information theory and statistical method,” Procedia Computer Science, Vol.61, pp. 32-37, 2015.
14. [14] J.H. Havrada and F. Charvat, “Quantification method of classification process: Concept of structural α-entropy,” Kybernetika, Vol.3, pp. 30-35, 1967.
15. [15] S. Kumar and A. Choudhary, “Some coding theorems on generalized Havrda-Charvat and Tsallis’s entropy,” Tamkang J. of Mathematics, Vol.43, No.3, pp. 437-444, 2012.
16. [16] C. Tsallis, “Possible generalization of Boltzmann-Gibbs statistics,” J. of Statistical Physics, 52, 1988.
17. [17] R.K. Bajaj and D.S. Hooda, “On some new generalized measures of fuzzy information,” World Academy of Science, Engineering and Technology, Vol.38, pp. 747-753, 2010.
18. [18] A. Kumar, S. Mahajan, and R. Kumar, “Some generalized measures of fuzzy entropy,” Int. J. of Mathematical Sciences and Applications, Vol.1, No.2, pp. 821-829, 2011.
19. [19] T. Kumar, R.K. Bajaj, and N. Gupta, “On some parametric generalized measures of fuzzy information, directed divergence and information improvement,” Int. J. of Computer Applications, Vol.30, No.9, pp. 5-10, 2011.
20. [20] S. Al-sharhan, F. Karray, and O. Basir, “Fuzzy entropy: a brief survey,” Proc. of FUZZ-IEEE2001, Vol.3, pp. 1135-1138, 2001.
21. [21] N.R. Pal and J.C. Bezdek, “Measuring fuzzy uncertainty,” IEEE Trans. on Fuzzy Systems, Vol.2, No.2, pp. 107-118, 1994.

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