N-Ary aggregation operators on function spaces: perspective of construction

Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes.

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.

Managing ambiguous and asymmetric types of information is a very challenging task under the consideration of classical data. Furthermore, Aczel-Alsina aggregation operators are the new developments in fuzzy sets theory. However, when decision-makers need to use these structures in fuzzy rough structures, these operators fail to deal with such types of values, as fuzzy rough structures use lower and upper approximation spaces. Thus, an encasement of an intuitionistic fuzzy set has a chance of data loss, whereas an intuitionistic fuzzy rough set can resolve the problem of data loss. Motivated by the notion of intuitionistic fuzzy rough sets and new aggregation operators i.e., intuitionistic fuzzy Aczel-Alsina operators, this paper firstly initiates some basic Aczel-Alsina operational rules for intuitionistic fuzzy rough numbers. Secondly, based on these newly defined operational rules, we have developed some new aggregation operators, such as intuitionistic fuzzy rough Aczel-Alsina weighted average (IFRAAWA), intuitionistic fuzzy rough Aczel-Alsina ordered weighted average (IFRAAOWA), and intuitionistic fuzzy rough Aczel-Alsina hybrid average (IFRAAHA) aggregation operators. Moreover, the properties of these aggregation operators have been initiated. These operators can help in evaluating awkward and asymmetric information in real-life problems. The use of aggregation operators in medical diagnosis and MADM is an efficient method that can help in healthcare and decision-making applications. To present an effective use of these developed operators in medical diagnosis and the selection of the best next-generation firewall, we have established an algorithm along with a numerical example to provide authenticity and clarity to the established work. Furthermore, a comparative analysis of the introduced work shows the superiority of the introduced approach.

The Wang-Mendel Approach (WMA) focuses on combining the numerical as well as linguistic information for achieving greater explainability for inference models. The standard WMA models the linguistic information using type-1 (T1) fuzzy sets (FSs), which have a reduced capability to model the semantics of linguistic information. Therefore, we propose a novel Enhanced WMA, which models the linguistic information using the type-2 (T2) FSs. Further, our Enhanced T2 FS-based WMA can be modified to reflect the use of interval type-2 (IT2) FSs, for modelling linguistic uncertainty. IT2 FSs are suitable when better uncertainty handling capabilities are required compared to T1 FSs, however, at a computational cost lesser than the T2 FSs. Performance of Enhanced WMA is demonstrated through a real-world crop-yield prediction problem in smart agriculture and an additional exemplar application on users’ satisfaction ratings. Further, we have compared our approach with the performance obtained from the T1 FS-based WMA and the original estimations given in the original data. We found that our Enhanced WMA provides more precise estimates than the other two with 95% confidence level. To the best of our knowledge, this is the first proposal of a T2 FSs method for enhancing the modeling of linguistic uncertainty in the WMA.

Emerging trends in soft set theory and related topics.

This paper employs the concept of fuzzify upgrade operator to describe the relationship between crisp sets, fuzzy sets and higher order fuzzy sets, such as type-2 fuzzy sets. Theory analysis and case studies demonstrate that a crisp set can be represented by a type-1 fuzzy set, a type-1 fuzzy set can be represented by a type-2 fuzzy set, and a type-n (n>=2) fuzzy set can be represented by a type-(n+1) (n>=2) fuzzy set. The fuzzify upgrade operator makes crisp sets, type-1 fuzzy sets, type-2 fuzzy sets and any fuzzy sets much more accessible to all readers. The relationship between crisp sets and fuzzy sets are explored.

This chapter formally introduces type-1 fuzzy sets and fuzzy logic. It is the backbone for Chap. 3 and provides the foundation upon which type-2 fuzzy sets and systems are built in later chapters. Its coverage includes: crisp sets, type-1 fuzzy sets and associated concepts [including a short biography of Prof. Zadeh (the father of fuzzy sets and fuzzy logic)], type-1 fuzzy set defined, linguistic variables, returning to linguistic variables from a numerical value of a membership function, set theoretic operations for crisp and type-1 fuzzy sets, crisp and fuzzy relations and compositionson the same or different product spaces , compositions of a type-1 fuzzy set with a type-1 fuzzy relation, hedges, the Extension Principle (which is about functions of fuzzy sets), α-cuts (which are a powerful way to represent a type-1 fuzzy set in terms of intervals), functions of type-1 fuzzy sets computed by using α-cuts, multivariable membership functions and Cartesian products, crisp logic, going from crisp logic to fuzzy logic, Mamdani (engineering) implications, some final remarks, and an appendix about properties/laws of type-1 fuzzy sets. 35 examples are used to illustrate this chapter’s important concepts.

To manage ambiguity and uncertainty information in real life problems, certain intellectuals have initiated numerous modifications of fuzzy sets. But, in certain situations, the decision-makers have been neglected to take a feasible decision. For instance, when an intellectual provides information in the form of a two-dimension or type-1 fuzzy set, then the prevailing theories have been neglected. For this, in this study, we elaborated the new idea of type-2 picture fuzzy sets (T2-PFS) and their algebraic laws have been given. Moreover, by using the T2-PFS, Hamacher aggregation operators are proposed and discussed their important properties. The Minkowski sorts of measures under the T2-PFS are also initiated. In the current study, we aim to utilize the matrix game in a type-2 picture fuzzy environment. By using the proposed operators and measures based on the Hausdorff metric. The elaborated idea has been defined with the biogas-plant implementation dilemma to verify the pertinence and authenticity. Lastly, to discuss the sensitivity analysis and geometrical shown of the initiated works, certain examples are illustrated to determine the supremacy and consistency of the proposed approaches.

This article provides a high-level retrospective of type-2 fuzzy sets and fuzzy logic systems. It explains how type-2 fuzzy sets can be used to model membership function uncertainties, and how by doing this smoother performance can be obtained than by using type-1 fuzzy sets. It also summarizes the notation that should be used for type-2 fuzzy sets, describes four important mathematical representations for these fuzzy sets, explains the differences between type-1 and type-2 fuzzy logic systems and which of the four representations is most useful when designing an optimal type-2 fuzzy logic system, provides a very useful strategy for optimal designs of fuzzy logic systems – one that guarantees performance improvement as one goes from a type-1 fuzzy logic system to a type-2 fuzzy logic system design – , and describes four methods for simplifying the designs of type-2 fuzzy logic systems. Finally, it explains why type-2 fuzzy sets can capture two kinds of linguistic uncertainties simultaneously (the uncertainty of an individual and the uncertainties of a group about a word), whereas type-1 fuzzy sets cannot, and that such type-2 fuzzy set word models are what should be used to implement Zadeh’s Computing With Words paradigm.

Assessment method sends messages to students to define and priorities what is important to learn and ultimately how they spend their time leaning it. Traditional grading methods are largely based on human judgments, which tend to be subjective. In addition, it is based on sharp criteria instead of fuzzy criteria and suffers from erroneous scores assigned by indifferent or inexperienced examiners, which represent a rich source of uncertainties, which might impair the credibility of the system. In an attempt to reduce uncertainties and provide more objective, reliable, and precise grading, a sophisticated assessment approach based on type-2 fuzzy set theory is developed. In this paper, interval type-2 (IT2) fuzzy sets, which are a special case of the general T2 fuzzy sets, are used. The transparency and capabilities of type-2 fuzzy sets in handling uncertainties is expected to provide an evaluation system able to justify and raise the quality and consistency of assessment judgments. A simplified implementation of interval type-2 fuzzy system using the basic knowledge of type-1 fuzzy is presented. A comparison between the use of type-1, interval type-2 fuzzy systems and the simplified IT2 fuzzy systems in reducing uncertainties and providing more transparent and fair assessment that can reflect needs of individual students and foster development is presented.

Finite-interval-valued Type-2 Gaussian fuzzy numbers applied to fuzzy TODIM in a healthcare problem

In many contexts, type-2 fuzzy sets (T2 FS) 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 (ESs) that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the T2 FS and the concept it models. As a consequence, the interpretability of some of the ESs and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 (CT2) fuzzy sets have been proposed. However, no formal definitions for some of their key components [e.g., acceptable ESs (AESs)] and constrained operations have been given. In this article, we provide some theoretical underpinning for the definition of CT2 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.

Tolerance-based intuitionistic fuzzy-rough set approach for attribute reduction

Type-2 fuzzy sets (T2FSs) exhibit evident merits when it comes to representing the complex and high uncertainty, which has been encountered in fuzzy optimization and multiple criteria decision making (MCDM) problems. Since T2FSs theory was proposed by Zadeh in 1975, then a series of theories and methods were investigated by more scholars. Type-2 fuzzy aggregation operators, contributing to the fundamental information fusion theory, have been paid more attention and applied to different areas during the last two decades. In this paper, a survey of type-2 fuzzy aggregation and application for MCDM is carried out. We first review some basic knowledge including definitions, operations, type reduction and ranking methods of T2FSs. Then the definitions and properties of some main aggregation operators are introduced. Furthermore, some application categories under type-2 fuzzy environment are given. Finally, we identify some existing shortcomings and point at future research directions on this topic.

As tools for dealing with universal fuzziness in human thinking, traditional fuzzy sets, namely type-1 fuzzy sets, are not able to directly model uncertainties about membership functions, however type-2 fuzzy sets can make up for this deficiency. Interval type-2 fuzzy sets, the simplified cases of general type-2 fuzzy sets, are practical because its membership functions and operations are greatly simplified. The inclusion measure and entropy of fuzzy sets are two important parameters with application prospects, the former describes the degree to which a fuzzy set is included in another, the latter shows the quantity of fuzzy information in a fuzzy set. It is important to make in-depth research on the two measures. In this paper, we mainly discuss the relationship of the inclusion measure and entropy of interval type-2 fuzzy sets proposed by Zheng et al., and present a theorem that converts the entropy to the inclusion measure of interval type-2 fuzzy sets. The conclusion provides the basis for further research.