Fuzzy types: a framework for handling uncertainty about types of objects

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.

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.

Recently, fuzzy linguistic variables have gained a great deal of attention from researchers in many decision-making problems. In this problems, type-2 fuzzy sets have been used to better cover linguistic data uncertainty. However, many of research works in this regard have been performed in type-2 fuzzy domain and in static mode. Since the decision making problems in the real world usually fluctuate over time, so it need to use decision making models in multi period of time. In the present research, a Multi-Period (Dynamic) Multi-Attribute Group Decision-Making (MPMADM) method is presented based on type-2 fuzzy sets where decision-making attributes are first expressed in linguistic terms and then incorporated, as interval type-2 fuzzy numbers, into problem solving where a new integrating operator called Multi-Period Trapezoidal Interval Type-2 Fuzzy Number Weighted Arithmetic averaging (MPTIT2FNWA) is defined on type-2 interval fuzzy numbers to integrate decision-making information in multiple periods of time. Once finished with explaining the proposed method, a numerical example is given to evaluate the proposed method in terms of effectiveness and applicability, with the results compared to those of other methods.

Nowadays, most people can get enough energy to maintain one-day activity, while few people know whether they eat healthily or not. It is quite important to analyze nutritional facts of foods eaten for those who are losing weight or suffering chronic diseases such as diabetes. However, diet is a problem with a high uncertainty, and it is widely pointed out that classical ontology is not sufficient to deal with imprecise and vague knowledge for some real-world applications like diet. On the other hand, a fuzzy ontology can effectively help handle and process uncertain data and knowledge. This chapter proposes a type-2 fuzzy set and fuzzy ontology for diet application and uses the type-2 fuzzy markup language (T2FML) to describe the knowledge base and rule base of the diet, including ingredients and the contained servings of six food categories of some common foods in Taiwan. The experimental results show that type-2 fuzzy logic system (FLS) performs better than type-1 FLS, proving that type-2 FLS can provide a powerful paradigm to handle the high level of uncertainties present in diet.

In this paper a theoretical and practical comparative study is proposed, between types 1 and 2 fuzzy systems for the monitoring of a gas turbine process. Firstly, the type 1 and type 2 fuzzy sets are presented followed by the description of the two fuzzy systems, to understand the differences between there inference systems, then a theoretical comparative study is presented, and finally the contribution of this work in the practical part by the application of the two fuzzy control approaches type-1 and type-2 in a fuel valve positioning of a gas turbine process, the obtained results shows the advantage of the used time in fuzzy control type-2, which is a very important factor in industrial applications.

Knowledge Based System (KBS) is a problem solving approach that makes use of human knowledge in decision strategies. Modeling and representing imperfect human knowledge associated with uncertainty is an important task in KBS development. There are various types of uncertainty, and randomness and fuzziness are among the most important. Handling hybrid uncertainty in one KBS is critical to support real world applications. Knowware System (KWS) is an intelligent tool designed to support application developers in constructing customized hybrid KBS without requiring developers being familiar with relevant intelligent techniques. It is essential for KWS to construct corresponding inference structure in resulting KBS and process the inference with hybrid uncertainty. To fulfill this requirement the extended Truth Value Flow Inference (TVFI) and Interval-Valued Confidence (IVC) have been defined and developed as ambedded mechanisms of KWS, and the hybrid logic has been adopted for the framework of handling hybrid uncertainty.This chapter discusses the IVC for the inference in hybrid KBS constructed by KWS with hierarchical knowledge representation. The knowledge content (precise or imprecise) represented in multiple units of knowledge hierarchy and the confidence obtained during inference process are treated as at two levels separately but simultaneously based on the extended TVFI. With the basic form of IVC, a fuzzy truth value is represented as a fuzzy number defined as a three-parametric triangular type-1 fuzzy set on the unit interval [0, 1]. The inference with both fuzzy truth and probability results an Extended Interval-Valued Confidence (EIVC) which is an interval type-2 fuzzy set on [0, 1] having the probability as an uncertainty measure on the fuzzy truth. The main focus of discussion will be put on interval-valued confidence. To provide the background of discussion, the KWS scheme, the extended TVFI and the concepts of hybrid logic will be briefly introduced.KeywordsUncertaintyRandomnessFuzzinessKnowwareSystemHybridLogicInterval-Valued Confidence

The main aim of this paper is to introduce the definition of intuitionistic fuzzy set and operations on it that will prove fruitful to conclude to a solution on problems of real life and how fuzzy logic and Graphics Processing unit can be used together to solve different domain problems for faster responses. Various comprehensive survey of Type-2 fuzzy set and the intuitionistic fuzzy set theories have been described here. IFS is nothing but a simple extension of the fuzzy set defined in the domain of discourse. Graphics Processing Unit (GPU) has been used to enhance the speed of execution. The intuitionistic fuzzy set is utilized as a proper element for representing the degree of hesitation concerning both the non-membership and membership degrees of an element of a set. Recent trends show how GPU is used for various computational applications that are run parallel to reduce overall execution time. Both the fuzzy domains seek application in tackling of the situation of vagueness with their respective characteristic feature sets to the fields concerned to different engineering domains of interdisciplines. This paper highlights the overview of the two eminent fuzzy categories like Fuzzy type 2 and intuitionistic fuzzy theory along with their application on greater scale.

Fuzzy set theory is particularly appropriate approach when data include imprecise. Type-2 fuzzy set theory captures ambiguity that associates the uncertainty of membership functions by incorporating footprints and models high level uncertainty. If the quality characteristic is a binary classification into conforming/non-conforming of product, this decision depends on human subjectivity that have ambiguity or vague. In this situation, monitoring the process with statistical control charts based on interval type-2 fuzzy sets, a special case of type-2 fuzzy sets, is more suitable due to the human imprecise judgments on quality characteristics. In this paper, interval type-2 fuzzy p-control chart is developed into the literature for the first time. Due to the interval type-2 fuzzy sets modelled more uncertainty for defining membership functions, in this paper interval type-2 fuzzy fraction nonconforming numbers used for handling more uncertainty in process. Real word application is implemented with developed fuzzy control chart.

This paper presents a comparative study between the firefly algorithm (FA) and the galactic swarm optimization (GSO) method, where the performance of both methods is observed and tested in the optimization of a fuzzy controller for path tracking of an autonomous mobile robot. The main contribution of this work is finding the best method that generates an optimal vector of values for the membership function optimization of the fuzzy controller. This with the goal of improving the performance of the controller and thus the trajectory generated by the autonomous robot is closer to the desired trajectory. It should be noted that the fuzzy controller that is optimized is an interval type-2 fuzzy controller, which has a greater capability for managing uncertainty than a type-1 fuzzy controller. In this case, the limiting membership functions in the interval type-2 fuzzy sets are themselves type-1 fuzzy sets that define the footprint of uncertainty. Type-2 fuzzy controllers have been shown in previous works to handle better the control of robotic systems under noisy and dynamic conditions and this is why their optimal design is very important. Simulation results show that GSO outperforms FA in the optimal design of interval type-2 fuzzy controllers.

The Ball and Beam system (BBS) is a nonlinear and unstable system which resembles with many real-time complicated systems. Providing an appropriate beam angle to give the stability to the ball on the beam in a specific position, is a challenging task for the control system researchers. In this paper a robust interval type-2 fuzzy logic controller (IT2FLC) is designed. The dimension of the rule base is reduced by using signed distance method. This signed distance method makes the IT2FLC to a Single input Interval Type-2 Fuzzy logic Controller (SIIT2FLC). The type-2 fuzzy sets resolve the problem of determining membership functions in type-1 fuzzy systems. The membership function of a type-2 fuzzy set is three dimensional, where third dimension is the value of membership function at each point on its two dimensional domain that is called its footprint of uncertainty (FOU). The ability of FOU to represent more uncertainties enables one to cover the input and output domains with less number of fuzzy sets. This SIIT2FLC gives the smooth 2-D control surface and robustness to the system. The simulation work is carried out in simulink environment of MATLAB (7.8.0) software. The simulation results are also validated in the real-time implementation of the BBS, designed by Googol Technology. Experimental results show that the performance of the proposed controller is better than the single input type-1 fuzzy logic controller in terms of transient and steady state response. The ability of handling uncertainty (robustness) of proposed controller has been checked by applying a disturbance signal to the position sensor’s output and also by parameter variation of the BBS.

A type-2 fuzzy set, which is characterized by a fuzzy membership function, involves more uncertainties than the type-1 fuzzy set. As the most widely used type-2 fuzzy set, interval type-2 fuzzy set is a very useful tool to model the uncertainty in the process of decision making. As a special case of interval type-2 fuzzy set, trapezoidal interval type-2 fuzzy set can express linguistic assessments by transforming them into numerical variables objectively. The aim of this paper is to investigate the multiple attribute group decision-making problems in which the attribute values and the weights take the form of trapezoidal interval type-2 fuzzy sets. First, we introduce the concept of trapezoidal interval type-2 fuzzy sets and some arithmetic operations between them. Then, we develop several trapezoidal interval type-2 fuzzy aggregation operators for aggregating trapezoidal interval type-2 fuzzy sets and examine several useful properties of the developed operators. Furthermore, based on the proposed operators, we develop two approaches to multiple attribute group decision making with linguistic information. Finally, a practical example is given to illustrate the feasibility and effectiveness of the developed approach.

Internet of things is projected to make its way into all spheres of human life in the near future. This has been compounded with the growing demand for contactless solutions in the wake of the recent pandemic. A potential solution could involve a privacy-preserving gesture-based control system that could control a wide range of appliances. Implementing such gesture-based control systems is mainly conducted using opaque box Artificial Intelligence (AI) models. Systems based on such opaque box AI models have shown high-performance metrics on in-distribution data in a lab environment. However, they are prone to failure when exposed to real-world out-of-distribution data where they cannot be tuned or calibrated due to their complexity and opaqueness. Interval Type-2 Fuzzy Logic-based explainable AI models offer an alternative to opaque box models showing comparable performance on lab in-distribution data. In contrast, in the real world, out-of-distribution data, the type-2 fuzzy models could be easily calibrated and tuned (thanks for their explainability) to provide similar performance to those achieved on the lab in-distribution data.

From the beginning of human history to today, all of the decisions that human beings have made in order to make their assets more efficient and valuable can be considered as investment decisions. The aim is always to maximize the return. For this purpose, researchers have developed a wide range of investment theories and decision methodologies. Multi-Attribute Decision Making techniques have found effective use for mathematical optimization to obtain both theoretical and practical results for researchers. In recent years, many extensions have been proposed that use Type-2 fuzzy sets for classical mathematical optimization techniques to address the uncertainty in nature. In this study, we used the Type-2 fuzzy TOPSIS method for the evaluation of alternative stocks in an investment decision. And a case study for the stocks selected from Borsa Istanbul (BIST) is performed to show the applicability of the proposed methodology. The results of the case study are examined, and suggestions for future studies are provided.KeywordsPortfolio selection problemInvestment decisionFuzzy TOPSISInterval Type-2 fuzzy numbers

In this paper we will present two theorems for the\njoin and meet operations for general type-2 fuzzy sets with\narbitrary secondary memberships, which can be non-convex\nand/or non-normal type-1 fuzzy sets. These results will be used\nto derive the join and meet operations of the more general\ndescriptions of interval type-2 fuzzy sets presented in [1], where\nthe secondary grades can be non-convex. Hence, this work will\nhelp to explore the potential of type-2 fuzzy logic systems which\nuse the general forms of interval type-2 fuzzy sets which are\nnot equivalent to interval valued fuzzy sets. Several examples\nfor both general type-2 and the more general forms of interval\ntype-2 fuzzy sets are presented.

A narrow band interval type-2 fuzzy approach for image segmentation