Construction of the method for building analytical membership functions in order to apply operations of mathematical analysis in the theory of fuzzy sets - Dykhta, Leonid; Kozub, Nataliia; Malcheniuk, Alexander; Novosadovskyi, Oleksii; Trunov, Alexander; Khomchenko, Anatolii (Eastern-European Journal of Enterprise Technologies, 2018)
This paper considers four methods for finding parameters of the analytical expressions of sigmoids, data on which are given numerically. We have conducted a comparative analysis of the approximation effectiveness using sigmoids by applying the least squares method, by the direct calculation of constants based on values at the equilibrium and saturation threshold points, by the Taylor expansion and splines using an example with different thresholds of equilibrium, sensitivity, saturation. It has been demonstrated that the direct calculation of two constants based on the threshold points of equilibrium, sensitivity, or saturation, could easily, in terms of an algorithm, find two coefficients. It has been shown that when approximating with sigmoids employing the method of least squares the error of the approximating function depends on the symmetrical selection of grid points relative to the equilibrium threshold. We have investigated construction algorithms of membership functions based on two base functions – sigmoid functions of two types of flash and recession. We have built a set of standard membership functions of triangle, trapezoid, rectangle in the form of a product operation. The conditions have been formulated under which the curved shapes of the membership functions are formed, as well as the influence of approximation coefficients on the magnitude of deviations; the properties of completeness and sufficiency have been examined. It has been demonstrated that such a procedure aimed at forming membership functions based on the totality of numerical values as the approximation spline does not make it possible to meet the requirement for the limit of interval of the value domain. We have derived a general solution to the optimization problem using the analytical membership functions and compared it to the results of its solution in the Bellman-Zadeh statement. We have analyzed the properties of transformed operations on fuzzy sets using the example of an optimization problem. It has been demonstrated that the solution in this new statement has two advantages. First, it is derived by applying an optimum search operation employing methods of classical mathematical analysis, using the conditions for a stationary point and conditions for the unchanged signs of second derivatives. Second, it is searched for using the operations of differentiation and root derivation, even under conditions for non-linearity, by commonly known methods by newton-kantorovich or recurrent approximation