Type-2 fuzzy alpha-cuts

Type-2 fuzzy logic systems make use of type-2 fuzzy sets. To be able to deliver useful type-2 fuzzy logic applications we need to be able to perform meaningful operations on these sets. These operations should also be practically tractable. However, type-2 fuzzy sets suffer the shortcoming of being complex by definition. Indeed, the third dimension, which is the source of extra parameters, is in itself the origin of extra computational cost. The quest for a representation that allow practical systems to be implemented is the motivation for our work. In this paper we define the alpha-cut decomposition theorem for type- 2 fuzzy sets which is a new representation analogous to the alpha-cut representation of type-1 fuzzy sets and the extension principle. We show that this new decomposition theorem forms a methodology for extending mathematical concepts from crisp sets to type-2 fuzzy sets directly. In the process of developing this theory we also define a generalisation that allows us to extend operations from interval type-2 fuzzy sets or interval valued fuzzy sets to type-2 fuzzy sets. These results will allow for the more applications of type-2 fuzzy sets by expiating the parallelism that the research here affords

Join and Meet Operations for Type-2 Fuzzy Sets With Nonconvex Secondary Memberships

In this paper, we will present two theorems for the join and meet operations for general type-2 fuzzy sets with arbitrary secondary memberships, which can be nonconvex and/or nonnormal type-1 fuzzy sets. These results will be used to derive the join and meet operations of the more general descriptions of interval type-2 fuzzy sets presented in a paper by Bustince Sola et al. ('Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wider view on their relationship,' IEEE Trans. Fuzzy Syst., vol. 23, pp. 1876-1882, 2015), where the secondary grades can be nonconvex. Hence, this study will help to explore the potential of type-2 fuzzy logic systems which use the general forms of interval type-2 fuzzy sets which are not equivalent to interval-valued fuzzy sets. Several examples for both general type-2 and the more general forms of interval type-2 fuzzy sets are presented

Operations on Concavoconvex Type-2 Fuzzy Sets

Concavoconvex fuzzy set is the result of the com-bination of the concepts of convex and concave fuzzy sets. This paper investigates concavoconvex type-2 fuzzy sets. Basic operations, union, intersection and complement on concavoconvex type-2 fuzzy sets us-ing min and product t-norm and max t-conorm are studied and some of their algebraic properties are explored

Type-2 Fuzzy Alpha-cuts

Systems that utilise type-2 fuzzy sets to handle uncertainty have not been implemented in real world applications unlike the astonishing number of applications involving standard fuzzy sets. The main reason behind this is the complex mathematical nature of type-2 fuzzy sets which is the source of two major problems. On one hand, it is difficult to mathematically manipulate type-2 fuzzy sets, and on the other, the computational cost of processing and performing operations using these sets is very high. Most of the current research carried out on type-2 fuzzy logic concentrates on finding mathematical means to overcome these obstacles. One way of accomplishing the first task is to develop a meaningful mathematical representation of type-2 fuzzy sets that allows functions and operations to be extended from well known mathematical forms to type-2 fuzzy sets. To this end, this thesis presents a novel alpha-cut representation theorem to be this meaningful mathematical representation. It is the decomposition of a type-2 fuzzy set in to a number of classical sets. The alpha-cut representation theorem is the main contribution of this thesis. This dissertation also presents a methodology to allow functions and operations to be extended directly from classical sets to type-2 fuzzy sets. A novel alpha-cut extension principle is presented in this thesis and used to define uncertainty measures and arithmetic operations for type-2 fuzzy sets. Throughout this investigation, a plethora of concepts and definitions have been developed for the first time in order to make the manipulation of type-2 fuzzy sets a simple and straight forward task. Worked examples are used to demonstrate the usefulness of these theorems and methods. Finally, the crisp alpha-cuts of this fundamental decomposition theorem are by definition independent of each other. This dissertation shows that operations on type-2 fuzzy sets using the alpha-cut extension principle can be processed in parallel. This feature is found to be extremely powerful, especially if performing computation on the massively parallel graphical processing units. This thesis explores this capability and shows through different experiments the achievement of significant reduction in processing time.The National Training Directorate, Republic of Suda

Constrained interval type-2 fuzzy sets

In many contexts, type-2 fuzzy sets are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation there is no restriction on the shape of the footprint of uncertainty and the embedded sets that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the type-2 fuzzy set and the concept it models. As a consequence, the interpretability of some of the embedded sets and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 fuzzy sets have been proposed. However, no formal definitions for some of their key components (e.g. acceptable embedded sets) and constrained operations have been given. The goal of this paper is to provide some theoretical underpinning for the definition of constrained type-2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real world cases and briefly compared to the standard interval type-2 inference and defuzzification method

A Comparison of Type-1 and Type-2 Fuzzy Logic Controllers in Robotics: A review

Most real world applications face high levels of uncertainties that can affect the operations of such applications. Hence, there is a need to develop different approaches that can handle the available uncertainties and reduce their effects on the given application. To date, Type-1 Fuzzy Logic Controllers (FLCs) have been applied with great success to many different real world applications. The traditional type-1 FLC which uses crisp type-1 fuzzy sets cannot handle high levels of uncertainties appropriately. Nevertheless it has been shown that a type-2 FLC using type-2 fuzzy sets can handle such uncertainties better and thus produce a better performance. As such, type-2 FLCs are considered to have the potential to overcome the limitations of type-1 FLCs and produce a new generation of fuzzy controllers with improved performance for many applications which require handling high levels of uncertainty. This paper will briefly introduce the interval type-2 FLC and its benefits. We will also present briefly some of the type-2 FLC real world applications

Fuzzy in 3-D: Two Contrasting Paradigms

DIGITS The full text of this article can be read via open access on the publisher's page.Type-2 fuzzy sets and complex fuzzy sets are both three dimensional extensions of type-1 fuzzy sets. Complex fuzzy sets come in two forms, the standard form, postulated in 2002 by Ramot et al., and the 2011 innovation of pure complex fuzzy sets, proposed by Tamir et al.. In this paper we compare and contrast both forms of complex fuzzy set with type-2 fuzzy sets, as regards their rationales, applications, definitions, and structures. In addition, pure complex fuzzy sets are compared with type-2 fuzzy sets in relation to their inferencing operations. Complex fuzzy sets and type-2 fuzzy sets differ in their roles and applications; complex fuzzy sets are pertinent to inferencing where there is seasonality, and type-2 fuzzy sets are applicable to reasoning under uncertainty. Their definitions differ also, though there is equivalence between those of a pure complex fuzzy set and a type-2 fuzzy set. Structural similarity is evident between these three- dimensional sets. Complex fuzzy sets are represented by a 3–D line, and type- 2 fuzzy sets by a 3–D surface, but a surface is simply a generalisation of a line. This similarity is particularly apparent between pure complex fuzzy sets and type- 2 fuzzy sets, which are both mappings from the domain onto the unit square. However type-2 fuzzy sets were found not to be isomorphic to pure complex fuzzy sets. The mechanisms by which complex fuzzy sets model and quantify periodicity, and type-2 fuzzy sets model and quantify uncertainty are discussed. A type-2 fuzzy set can be represented as the union of its type-2 embedded set. An embedded type-2 fuzzy set is a type-2 fuzzy set in itself, whose geomet- rical representation is a 3-D line. Thus, geometrically an embedded type-2 fuzzy set can be seen as equivalent to a pure complex fuzzy set and therefore a type-2 fuzzy set can be represented as the union of a collection pure complex fuzzy sets, which in turn can be regarded as embedded complex fuzzy sets of a type-2 fuzzy set. This relationship is exploited to provide a complex definition of a type-2 fuzzy set

Geometric Fuzzy Logic Systems

There has recently been a significant increase in academic interest in the field oftype-2 fuzzy sets and systems. Type-2 fuzzy systems offer the ability to model and reason with uncertain concepts. When faced with uncertainties type-2 fuzzy systems should, theoretically, give an increase in performance over type-l fuzzy systems. However, the computational complexity of generalised type-2 fuzzy systems is significantly higher than type-l systems. A direct consequence of this is that, prior to this thesis, generalised type-2 fuzzy logic has not yet been applied in a time critical domain, such as control. Control applications are the main application area of type-l fuzzy systems with the literature reporting many successes in this area. Clearly the computational complexity oftype-2 fuzzy logic is holding the field back. This restriction on the development oftype-2 fuzzy systems is tackled in this research. This thesis presents the novel approach ofdefining fuzzy sets as geometric objects - geometric fuzzy sets. The logical operations for geometric fuzzy sets are defined as geometric manipulations of these sets. This novel geometric approach is applied to type-I, type-2 interval and generalised type-2 fuzzy sets and systems. The major contribution of this research is the reduction in the computational complexity oftype-2 fuzzy logic that results from the application of the geometric approach. This reduction in computational complexity is so substantial that generalised type-2 fuzzy logic has, for the first time, been successfully applied to a control problem - mobile robot navigation. A detailed comparison between the performance of the generalised type-2 fuzzy controller and the performance of the type-l and type-2 interval controllers is given. The results indicate that the generalised type-2 fuzzy logic controller outperforms the other robot controllers. This outcome suggests that generalised type-2 fuzzy systems can offer an improved performance over type-l and type-2 interval systems

A fuzzy AHP multi-criteria decision-making approach applied to combined cooling, heating and power production systems

Most of the real-world multi-criteria decision-making (MCDM) problems contain a mixture of quantitative and qualitative criteria; therefore quantitative MCDM methods are inadequate for handling this type of decision problems. In this paper, a MCDM method based on the Fuzzy Sets Theory and on the Analytic Hierarchy Process (AHP) is proposed. This method incorporates a number of perspectives on how to approach the fuzzy MCDM problem, as follows: (1) combining quantitative and qualitative criteria (2) expressing criteria pair-wise comparison in linguistic terms and performance of the alternative on each criterion in linguistic terms or exact values when criterion is qualitative or quantitative, respectively, (3) converting all the assessments into trapezoidal fuzzy numbers, (4) using the difference minimization method to calculate the local weight of criteria, employing the algebraic operations of fuzzy numbers based on the concept of α-cuts, (4) calculating the global weight of criteria and the global performance of each alternative using geometric mean and the weighted sum, respectively, (5) using the centroid method to rank the alternatives. Finally, an illustrative example on evaluation of several combined cooling, heat and power production systems is used to demonstrate the effectiveness of the proposed methodology

Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quantifiers

The OWA operator proposed by Yager has been widely used to aggregate experts' opinions or preferences in human decision making. Yager's traditional OWA operator focuses exclusively on the aggregation of crisp numbers. However, experts usually tend to express their opinions or preferences in a very natural way via linguistic terms. These linguistic terms can be modelled or expressed by (type-1) fuzzy sets. In this paper, we define a new type of OWA operator, the type-1 OWA operator that works as an uncertain OWA operator to aggregate type-1 fuzzy sets with type-1 fuzzy weights, which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights. The procedure for performing type-1 OWA operations is analysed. In order to identify the linguistic weights associated to the type-1 OWA operator, type-2 linguistic quantifiers are proposed. The problem of how to derive linguistic weights used in type-1 OWA aggregation given such type of quantifier is solved. Examples are provided to illustrate the proposed concepts. Crown Copyright © 2008