A Method for Constructing Non-Isosceles Triangular Fuzzy Numbers Using Frequency Histogram and Statistical Parameters

Document Type : Regular Article

Authors

1 Ph.D. Student, Faculty of Science and Engineering, Curtin University, Kent St, Bentley WA 6102, Australia

2 Senior lecturer, Faculty of Science and Engineering, Curtin University, Kent St, Bentley WA 6102, Australia

Abstract

The philosophy of fuzzy logic was formed by introducing the membership degree of a linguistic value or variable instead of divalent membership of 0 or 1. Membership degree is obtained by mapping the variable on the graphical shape of fuzzy numbers. Because of simplicity and convenience, triangular membership numbers (TFN) are widely used in different kinds of fuzzy analysis problems. This paper suggests a simple method using statistical data and frequency chart for constructing non-isosceles TFN when we are using direct rating for evaluating a variable in a predefined scale. In this method, the relevancy between assessment uncertainties and statistical parameters such as mean value and the standard deviation is established in a way that presents an exclusive form of triangle number for each set of data. The proposed method with regard to the graphical shape of the frequency chart distributes the standard deviation around the mean value and forms the TFN with the membership degree of 1 for mean value. In the last section of the paper modification of the proposed method is presented through a practical case study.

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[1]     Ross TJ, Booker JM, Parkinson WJ. Front Matter. Fuzzy Log. Probab. Appl., Society for Industrial and Applied Mathematics; 2002, p. i–xxiii. doi:10.1137/1.9780898718447.fm.
[2]     Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci (Ny) 1975;8:199–249. doi:10.1016/0020-0255(75)90036-5.
[3]     Zadeh LA. Fuzzy logic and approximate reasoning. Synthese 1975;30:407–28. doi:10.1007/BF00485052.
[4]     Zadeh LA. A new direction in AI: Toward a computational theory of perceptions. AI Mag 2001;22:73. doi:10.1609/aimag.v22i1.1545.
[5]     Zadeh LA. Information and control. Fuzzy Sets 1965;8:338–53.
[6]     Castillo, Oscar, Melin P. Type-2 Fuzzy Logic: Theory and Applications. vol. 223. Berlin, Heidelberg: Springer Berlin Heidelberg; 2008. doi:10.1007/978-3-540-76284-3.
[7]     Dubois D, Prade H. Possibility theory: An approach to computerized processing of uncertainty Plenum New York MATH Google Scholar 1988.
[8]     Bede B. Fuzzy Numbers. In: Bede B, editor., Berlin, Heidelberg: Springer Berlin Heidelberg; 2013, p. 51–64. doi:10.1007/978-3-642-35221-8_4.
[9]     Dubois DJ. Fuzzy sets and systems: theory and applications. vol. 144. Academic press; 1980.
[10]    Stefanini L, Sorini L. Fuzzy Arithmetic with Parametric LR Fuzzy Numbers. IFSA/EUSFLAT Conf., 2009, p. 600–5.
[11]    Melin P, Castillo O. Modelling, simulation and control of non-linear dynamical systems: an intelligent approach using soft computing and fractal theory. CRC Press; 2001.
[12]    Bilgic T, Turksen IB. Elicitation of membership functions: how far can theory take us? Proc. 6th Int. Fuzzy Syst. Conf., vol. 3, IEEE; 1997, p. 1321–5. doi:10.1109/FUZZY.1997.619736.
[13]    Bilgiç T, Türk┼čen IB. Measurement of Membership Functions: Theoretical and Empirical Work. In: Dubois D, Prade H, editors., Boston, MA: Springer US; 2000, p. 195–227. doi:10.1007/978-1-4615-4429-6_4.
[14]    Krantz DH, Suppes P, Luce RD. Foundations of measurement. Academic Press; 1971.
[15]    Wierman MJ. An introduction to the mathematics of uncertainty. Creight Univ 2010:149–50.
[16]    Saaty TL. The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.
[17]    Zadeh LA. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst 1999;100:9–34. doi:10.1016/S0165-0114(99)80004-9.
[18]    Dubois D, Prade H. Unfair coins and necessity measures: Towards a possibilistic interpretation of histograms. Fuzzy Sets Syst 1983;10:15–20. doi:10.1016/S0165-0114(83)80099-2.
[19]    KLIR GJ. A PRINCIPLE OF UNCERTAINTY AND INFORMATION INVARIANCE*. Int J Gen Syst 1990;17:249–75. doi:10.1080/03081079008935110.
[20]    Civanlar MR, Trussell HJ. Constructing membership functions using statistical data. Fuzzy Sets Syst 1986;18:1–13. doi:10.1016/0165-0114(86)90024-2.
[21]    Pham TD, Valliappan S. Constructing the Membership Function of a Fuzzy Set with Objective and Subjective Information. Comput Civ Infrastruct Eng 1993;8:75–82. doi:10.1111/j.1467-8667.1993.tb00194.x.
[22]    Chen JE, Otto KN. Constructing membership functions using interpolation and measurement theory. Fuzzy Sets Syst 1995;73:313–27. doi:10.1016/0165-0114(94)00322-X.
[23]    Pedrycz W. Why triangular membership functions? Fuzzy Sets Syst 1994;64:21–30. doi:10.1016/0165-0114(94)90003-5.
[24]    Medasani S, Kim J, Krishnapuram R. An overview of membership function generation techniques for pattern recognition. Int J Approx Reason 1998;19:391–417. doi:10.1016/S0888-613X(98)10017-8.
[25]    Sancho-Royo A, Verdegay JL. Methods for the construction of membership functions. Int J Intell Syst 1999;14:1213–30.
[26]    Sivanandam SN, Sumathi S, Deepa SN. Introduction to Fuzzy Logic using MATLAB. vol. 1. Berlin, Heidelberg: Springer Berlin Heidelberg; 2007. doi:10.1007/978-3-540-35781-0.
[27]    Azar A, Faraji H. Fuzzy Management Science. Manag Product Study Cent Iran 2008.
[28]    Asgharpour M. Multi Criteria Decision Makings, Tehran University publications; 2009, p. 332–46.
[29]    NORWICH AM, TURKSEN IB. A MODEL FOR THE MEASUREMENT OF MEMBERSHIP AND THE CONSEQUENCES OF ITS EMPIRICAL IMPLEMENTATION. Readings Fuzzy Sets Intell. Syst., Elsevier; 1993, p. 861–73. doi:10.1016/B978-1-4832-1450-4.50091-2.
[30]    Amini A. Evaluating the effective parameters in prioritizing urban roadway bridges for maintenance operation using fuzzy logic. M.Sc. Thesis, Science and research branch of Tehran Azad University, 2010.
[31]    Friedman M. The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance. J Am Stat Assoc 1937;32:675–701. doi:10.1080/01621459.1937.10503522.
[32]    Mabuchi S. An approach to the comparison of fuzzy subsets with an alpha -cut dependent index. IEEE Trans Syst Man Cybern 1988;18:264–72. doi:10.1109/21.3465.
[33]    Fathizadeh A. Evaluating the effective parameters in urban roadway bridges prioritization. M.Sc. Thesis, Science and research branch of Tehran Azad University, 2010.