Abstract
In this chapter, the authors expand the notion of type-2 fuzzy sets. An introduction to standard and interval (type-2) fuzzy sets and systems is explained in the early part of the discussion. The chapter also covers the ideas of hybrid type-2 fuzzy system. Next, the authors study the applicability of type-2 fuzzy logic (FL) system in student’s performance in oral presentation as it is clearly new field of research topic and have an excellent opportunity to combine several fuzzy set method developed in the recent years. The proposed application shows the linkage of type-2 fuzzy system with TOPSIS. The present chapter also highlights the possible future directions for type-2 FL system research. By the end of the chapter, the authors hope that even those with little previous experience of fuzzy logic should be enabled to apply these methods in their own application areas and/or begin research in this fascinating and exciting area.
Top1. Introduction
The concept was first proposed by (Zadeh, 1975). Subsequently Mizumoto and Tanaka (1976) developed and discussed some properties of type-2 fuzzy sets but no work was done to further develop it into a useful and practical tool. Karnik et al. (1999) introduced the concept of “Footprint of Uncertainty” and the upper and lower membership functions to describe type-2 fuzzy sets. Using interval type-2 fuzzy sets they developed the singleton and non-singleton type-2 fuzzy architectures for practical applications in engineering. A simple and straightforward treatment of s was given by Mendel and John (2002). The three dimensional nature of type-2 fuzzy sets suggest that uncertainties could be better accommodated compared to the two dimensional type-1 fuzzy sets.
The overall work plan of the proposed chapter is as follows:
- A.
An introduction to standard and interval (type-2) fuzzy sets.
- B.
Basic operations of type-2 fuzzy sets.
- C.
Introduction to type-2 fuzzy system.
- D.
Introduction to various hybrid type-2 FL system:
- E.
A case study on the applicability of Students’ Performance in Oral Presentation Using Interval Type-2 Fuzzy Approach.
- F.
Future directions for type-2 FL system research and applications.
The objectives of this chapter is to: (i) review the literature; (ii) perspective on the issues and problems related to computational intelligence using interval type-2 fuzzy logic based on the application to Students’ Performance in Oral Presentation research; and, (iii) identify future research directions. This chapter is organized as follows. Section 2 introduces detail definitions and discussions of type-2 fuzzy sets and incorporate views of others. In section 3, concepts of type-2 fuzzy system are introduced. A brief introduction and logic of various hybrid type-2 fuzzy systems is stated in section 4. A case study on the applicability of Students’ Performance in Oral Presentation using Interval Type-2 Fuzzy Approach will be framed in Section 5. Section 6 puts emphasis on the issues of future directions for type-2 FL system. The conclusions to this chapter are given in Section 7.
Top2. Background
A type-2 fuzzy set is characterized by a fuzzy membership function, i.e., the membership grade for each element of this set is a fuzzy set in [0,1], unlike a type-1 set where the membership grade is a crisp number in [0,1]. Such sets can be used in situations where there is uncertainty about the membership grades themselves, e.g., an uncertainty in the shape of the membership function or in some of its parameters. Consider the transition from ordinary sets to fuzzy sets. When we cannot determine the membership of an element in a set as 0 or 1, we use fuzzy sets of type-1. Similarly, when the situation is so fuzzy that we have trouble determining the membership grade even as a crisp number in [0,1], we use fuzzy sets of type-2. This does not mean that we need to have extremely fuzzy situations to use type-2 fuzzy sets. There are many real-world problems where we cannot determine the exact form of the membership functions, e.g., in time series prediction because of noise in the data. Another way of viewing this is to consider type-1 fuzzy sets as a first order approximation to the uncertainty in the real world. Then type-2 fuzzy sets can be considered as a second order approximation. Of course, it is possible to consider fuzzy sets of higher types but the complexity of the fuzzy system increases very rapidly. For this reason, we will only consider very briefly type-2 fuzzy sets.
Key Terms in this Chapter
Type-2 Fuzzy Set Defuzzification: Defuzzification of a type-2 fuzzy sets consists of two stages: type-reduction, creating a type-1 fuzzy set, and defuzzification of the resultant type-1 fuzzy set. The second stage is easily implemented; the first stage is more problematic. We defuzzify the type-reduced set to get a crisp output from a type-2 FLS. Having both the defuzzified output and its associated type-reduced set is analogous to having both the mean and its associated standard deviation for a random variable.
Footprint of Uncertainty (FOU): To indicate a graph of type-2 fuzzy set more easily, the concept of the footprint of uncertainty is used. The membership function of a general type-2 fuzzy set, Ã, is three-dimensional ( Figure 1 ), where the third dimension is the value of the membership function at each point on its two-dimensional domain that is called its footprint of uncertainty (FOU). The footprint of uncertainty (FOU) of a type-2 fuzzy set à is a region with boundaries covering all the primary membership points of elements x, and is defined as follows: The more (less) area in the FOU the more (less) is the uncertainty.
Type-2 Fuzzy TOPSIS: Fuzzy TOPSIS method, a combination of ordinary TOPSIS method and Fuzzy theory, could heal some of shortcomings of uncertainties and ordinary TOPSIS in decision-making. However there are still lots of occasions in which decision-making is faced with lots of shadowiness making the Fuzzy TOPSIS method not sufficiently receptive. As a sensible response to this drawback, is to utilize a brand-new extension to TOPSIS and Fuzzy TOPSIS methods, based on type-2 fuzzy notions with ability to cope with type-2 fuzzy environment and data incorporating much more fuzziness in decision-making.
Type Reduction: Type reduction is one phase to defuzzify type-2 fuzzy sets. It means that by this method, we can transform a type-2 fuzzy set into a type-1 fuzzy set. To go from an interval type-2 fuzzy set to a number (usually) requires two steps ( Figure 3 ). The first step, called type-reduction, is where an interval type-2 fuzzy set is reduced to an interval-valued type-1 fuzzy set. Type-reduction methods are “extended” versions of type-1 defuzzification methods, and compute the centroid of the type-2 output fuzzy set Type-reduction is an extension of the type-1 defuzzification procedure for type-2 fuzzy sets.
Gaussian Membership Function: A Gaussian type-1 MF (in a type-1 FLS) it would be characterized by two parameters—its mean and standard deviation. On the other hand, if that Gaussian MF acted as the primary MF for an interval type-2 FS, and we only assumed uncertainty about its mean value, then the resulting interval type-2 MF would be characterized by three parameters—the two end-points of the interval of uncertainty for the mean and the standard deviation.
T-Conorm: A T-conorm, is an operation whose order is reversed against T-norm in the interval [0, 1]. This kind of operation can be used to stand for a disjunction in fuzzy logic and a union in fuzzy set theory, such as maximum T-conorm.
Type-2 Fuzzy Set & System: A type-2 fuzzy set maps elements in a crisp domain to type-1 fuzzy numbers bounded in the range [0,1]. We denote a fuzzy set as type-2 by placing a tilde character above the name of the set. Since the value at each point in a type-2 fuzzy set is given as a function, type-2 fuzzy sets are three-dimensional. Such sets can be used in situations where there is uncertainty about the membership grades themselves, e.g., an uncertainty in the shape of the membership function or in some of its parameters. Type-2 fuzzy sets and systems generalize (type-1) fuzzy sets and systems so that more uncertainty can be handled.
Interval Type-2 Fuzzy Set & Logic: An interval type-2 fuzzy set is one in which the membership grade of every domain point is a crisp set whose domain is some interval contained in [0,1]. That is, interval type-2 fuzzy sets let us model and minimizes the effects of uncertainties in rule-base interval type-2 fuzzy logic systems (IT2FLS). This came about initially by restricting type-2 fuzzy sets to a special category known as interval type-2 fuzzy sets in which the third-dimension was restricted to values of either zero or one. Interval type-2 fuzzy logic includes fuzzifier, fuzzy rule base, fuzzy inference engine and output processing where output processing contains type-reducer and defuzzifier.
General Type-2 Fuzzy Set: The general type-2 FLS will have the ability to model uncertainty more accurately than interval type-2 sets, which, in turn, will result in the potential for a superior control performance in comparison to type-1 and interval type-2 FLSs. they are no longer a single number from 0 to 1, but are instead a continuous range of values between 0 and 1, say [a, b] (some people call this a blurring of the membership function value). one can either assign the same weighting or a variable weighting to the interval of membership function values [a, b]. When the latter is done, the resulting type-2 fuzzy set is called a general type-2 fuzzy set.
T-norm: In the field of mathematics, is respected as an operation in the interval [0, 1], which is always utilized in fuzzy logic. A T-norm can be extended to be a conjunction in fuzzy logic and an intersection in fuzzy set theory, such as a Minimum T-norm and a Product T-norm, both of which are involved in the operations on type-2 fuzzy sets.