**Author details**

Ali Sadollah

many MFs are needed (e.g., low, med, and high MF) and also choosing the intervals of MFs.

In addition, looking at the distribution of the data is a good idea. Although, trial and error method is often used for MF shape, because there is no exact method for choosing the MFs. The shape of MFs depends on how one believes in a given linguistic variable. It is more a question of intuition then criteria. The only condition a MF must really satisfy is that it must vary between 0 and 1. The function itself can be an arbitrary curve whose shape we can define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency. Therefore, the type of MF doesnot play a crucial role in shaping how the model performs.

However, the number of MF has greater influence as it determines the computational time. Hence, the optimum model can be determined by varying the number/type of MFs for achieving best system performance. Ref. [1] discusses which shape is best if one uses fuzzy logic as a universal approximator. Also, a constrained interpolations scheme was developed for fitting

There are many references giving directions of how to choose MF [3–6]. The basic problem with modeling a situation, is to break the 0–1 modeling. This can be done by using triangular MF. However, if the situation is complex and deep, we might need a special type of MF. For instance, if the problem at hand is a quantum mechanics problem, then a special MF is needed. In order to make the best choice, one needs a lot of "experience" with the given situation. This experience will tune up and best fit, the subjective choice of the researcher with the given reality. There is no objective way to do so. Thus, a high fidelity intuition based on

Generally speaking, triangular MF is one of the most encountered MF in practice. Of highly applied MFs, the triangular MFs are formed using straight lines. These straight line membership functions have the advantage of simplicity. Gaussian MFs are popular methods for specifying fuzzy sets because of their smoothness and concise notation. These curves have the

It is advisable to use the symmetric triangular MF with 50% overlap, and then apply tuning procedure during which we can either change the left and/or right spread and/or overlapping. This is to be continued till we get satisfactory results. Same approach can be attempted

Triangular shapes represent fuzzy numbers, while trapezoid shapes represent fuzzy intervals. These are the simplest shapes. Other different shapes can be obtained from transformations of the triangle induced by linguistic modifiers, truth-functional modifiers, compositions,

In fact, the selection of MF shape is problem specific. Based on extensive review on many literatures, it can be concluded that the triangular MF is widely used because of its simplicity. Using various MF for given problems, usually Gaussian and triangular MFs are found to be closely performing well and better than other types of MF. In specific, the triangular MF is found to be better than Gaussian MF. Zhao and Bose [7] compared the response of the system

with various MFs and conveyed that the triangular MF is superior to any other MFs.

a MF to a finite number of known membership values [2].

sufficient experience will give an acceptable answer.

advantage of being smooth and nonzero at all points.

projections, and other operations.

for other shapes such as trapezoidal, bell-shape, and so forth.

These two factors also have a great impact on the outcome of a fuzzy logic system.

4 Fuzzy Logic Based in Optimization Methods and Control Systems and Its Applications

Address all correspondence to: ali\_sadollah@yahoo.com

School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
