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Satvir Singh (Shaheed Bhagat Singh College of Engineering & Technology, India), J. S. Saini (Deenbandhu Chhotu Ram University of Science & Technology, India) and Arun Khosla (Dr. B. R. Ambedkar National Institute of Technology, India)

Source Title: Machine Learning Algorithms for Problem Solving in Computational Applications: Intelligent Techniques

Copyright: © 2012
|Pages: 19
DOI: 10.4018/978-1-4666-1833-6.ch013

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TopFLSs are being used successfully in an increasing number of application areas where system response can be described far easily using linguistic variables and rules than using their mathematical models. One of the most important considerations in designing any FLS is the generation of the fuzzy rules and Membership Function (MF) for each involved Fuzzy Set (FS). In most existing applications, the fuzzy rules are extracted from the experts' knowledge (Mendel, 2001, Khosla et al., 2005). With an increasing number of variables, the possible number of rules for the system increases exponentially, which makes it difficult for experts to define a complete rule set for modeling a system with good performance. An automated way to design fuzzy systems might be preferable (Shi et al., 1999, Saini et al., 2004, Khosla et al., 2005).

The design of an FLS can be formulated as a search problem in high-dimensional search-space where each point is an FLS, i.e., collection of FSs, and rulebase, etc. Given some performance measurement criteria, the performance of the system forms a hyper-surface in the space. Developing the optimal FLS design is equivalent to finding the optimal location of this hyper-surface. The hyper-surface can be (Shi et al., 1999):

*•**infinitely large*since the number of possible FSs for each variable can be unbounded.*•**non-differentiable*since changes in the FSs can be discrete and can have a discontinuous effect on the fuzzy systems performance.*•**complex*and*noisy*since the mapping from a fuzzy rule set to its performance is indirect and dependent on the evaluation method used.*•**multi-modal*since different fuzzy rule sets and/or membership functions may have similar performance.*•**deceptive*since almost similar fuzzy rule sets and membership functions may have quite different performances.

All these characteristics make evolutionary algorithms a better option to search optimal designs from the hyper-surface search-space than the conventional hill-climbing search methods.

TopEach input is expressed linguistically and depicted in graphical shapes (S, Z, Triangular, Trapezoidal, and Gaussian, etc.) called FSs or MFs. If membership value for a particular FS is crisp (certain) for some crisp input over the Universe of Discourse (UOD), the FS is called T1 FS and denoted as . It may be expressed in two different mathematical formats as in (1) and (2)

If the membership value for a particular FS is not one crisp value for a crisp input from the UOD , i.e., , but is another single FS or are multiple FSs (called secondary MFs) of any shape, then such an FS is called T2 FS. Figure 1 shows a T2 FS, with all triangular secondary MFs, in 3D and 2D representations. Such a T2 FS is denoted, by , and is characterized by a T2 MF as , where and , where is the primary membership of where for all . can be expressed mathematically in either of the following forms, i.e., (3) and (4).

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