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• Preface xiii 1 Fuzzy Fractals in Cervical Cancer 1 T. Sudha and G. Jayalalitha 1.1 Introduction 2 1.1.1 Fuzzy Mathematics 2 1.1.1.1 Fuzzy Set 2 1.1.1.2 Fuzzy Logic 2 1.1.1.3 Fuzzy Matrix 3 1.1.2 Fractals 3 1.1.2.1 Fractal Geometry 4 1.1.3 Fuzzy Fractals 4 1.1.4 Cervical Cancer 5 1.2 Methods 7 1.2.1 Fuzzy Method 7 1.2.2 Sausage Method 11 1.3 Maximum Modulus Theorem 15 1.4 Results 18 1.4.1 Fuzzy Method 19 1.4.2 Sausage Method 20 1.5 Conclusion 21 References 23 2 Emotion Detection in IoT-Based E-Learning Using Convolution Neural Network 27 Latha Parthiban and S. Selvakumara Samy 2.1 Introduction 28 2.2 Related Works 30 2.3 Proposed Methodology 31 2.3.1 Students Emotion Recognition Towards the Class 31 2.3.2 Eye Gaze-Based Student Engagement Recognition 31 2.3.3 Facial Head Movement-Based Student Engagement Recognition 34 2.4 Experimental Results 35 2.4.1 Convolutional Layer 35 2.4.2 ReLU Layer 35 2.4.3 Pooling Layer 36 2.4.4 Fully Connected Layer 36 2.5 Conclusions 42 References 42 3 Fuzzy Quotient-3 Cordial Labeling of Some Trees of Diameter 5-Part III 45 P. Sumathi and J. Suresh Kumar 3.1 Introduction 46 3.2 Related Work 46 3.3 Definition 47 3.4 Notations 47 3.5 Main Results 48 3.6 Conclusion 71 References 71 4 Classifying Fuzzy Multi-Criterion Decision Making and Evolutionary Algorithm 73 Kirti Seth and Ashish Seth 4.1 Introduction 74 4.1.1 Classical Optimization Techniques 74 4.1.2 The Bio-Inspired Techniques Centered on Optimization 75 4.1.2.1 Swarm Intelligence 77 4.1.2.2 The Optimization on Ant Colony 78 4.1.2.3 Particle Swarm Optimization (PSO) 82 4.1.2.4 Summary of PSO 83 4.2 Multiple Criteria That is Used for Decision Making (MCDM) 83 4.2.1 WSM Method 86 4.2.2 WPM Method 86 4.2.3 Analytic Hierarchy Process (AHP) 87 4.2.4 TOPSIS 89 4.2.5 VIKOR 90 4.3 Conclusion 91 References 91 5 Fuzzy Tri-Magic Labeling of Isomorphic Caterpillar Graph J62,3,4 of Diameter 5 93 P. Sumathi and C. Monigeetha 5.1 Introduction 93 5.2 Main Result 95 5.3 Conclusion 154 References 154 6 Fuzzy Tri-Magic Labeling of Isomorphic Caterpillar Graph J6 2,3,5 of Diameter 5 155 P. Sumathi and C. Monigeetha 6.1 Introduction 155 6.2 Main Result 157 6.3 Conclusion 215 References 215 7 Ceaseless Rule-Based Learning Methodology for Genetic Fuzzy Rule-Based Systems 217 B. Siva Kumar Reddy, R. Balakrishna and R. Anandan 7.1 Introduction 218 7.1.1 Integration of Evolutionary Algorithms and Fuzzy Logic 219 7.1.2 Fuzzy Logic-Aided Evolutionary Algorithm 220 7.1.3 Adaptive Genetic Algorithm That Adapt Manage Criteria 220 7.1.4 Genetic Algorithm With Fuzzified Genetic Operators 220 7.1.5 Genetic Fuzzy Systems 220 7.1.6 Genetic Learning Process 223 7.2 Existing Technology and its Review 223 7.2.1 Techniques for Rule-Based Understanding with Genetic Algorithm 223 7.2.2 Strategy A: GA Primarily Based Optimization for Computerized Built FLC 223 7.2.3 Strategy B: GA-Based Optimization of Manually Created FLC 224 7.2.4 Methods of Hybridization for GFS 225 7.2.4.1 The Michigan Strategy-Classifier System 226 7.2.4.2 The Pittsburgh Method 229 7.3 Research Design 233 7.3.1 The Ceaseless Rule Learning Approach (CRL) 233 7.3.2 Multistage Processes of Ceaseless Rule Learning 234 7.3.3 Other Approaches of Genetic Rule Learning 236 7.4 Findings or Result Discussion so for in the Area of GFS Hybridization 237 7.5 Conclusion 239 References 240 8 Using Fuzzy Technique Management of Configuration and Status of VM for Task Distribution in Cloud System 243 Yogesh Shukla, Pankaj Kumar Mishra and Ramakant Bhardwaj 8.1 Introduction 244 8.2 Literature Review 244 8.3 Logic System for Fuzzy 246 8.4 Proposed Algorithm 248 8.4.1 Architecture of System 248 8.4.2 Terminology of Model 250 8.4.3 Algorithm Proposed 252 8.4.4 Explanations of Proposed Algorithm 254 8.5 Results of Simulation 257 8.5.1 Cloud System Numerical Model 257 8.5.2 Evaluation Terms Definition 258 8.5.3 Environment Configurations Simulation 259 8.5.4 Outcomes of Simulation 259 8.6 Conclusion 260 References 266 9 Theorems on Fuzzy Soft Metric Spaces 269 Qazi Aftab Kabir, Ramakant Bhardwaj and Ritu Shrivastava 9.1 Introduction 269 9.2 Preliminaries 270 9.3 FSMS 271 9.4 Main Results 273 9.5 Fuzzy Soft Contractive Type Mappings and Admissible Mappings 278 References 282 10 Synchronization of Time-Delay Chaotic System with Uncertainties in Terms of Takagi-Sugeno Fuzzy System 285 Sathish Kumar Kumaravel, Suresh Rasappan and Kala Raja Mohan 10.1 Introduction 285 10.2 Statement of the Problem and Notions 286 10.3 Main Result 291 10.4 Numerical Illustration 302 10.5 Conclusion 312 References 312 11 Trapezoidal Fuzzy Numbers (TrFN) and its Application in Solving Assignment Problem by Hungarian Method: A New Approach 315 Rahul Kar, A.K. Shaw and J. Mishra 11.1 Introduction 316 11.2 Preliminary 317 11.2.1 Definition 317 11.2.2 Some Arithmetic Operations of Trapezoidal Fuzzy Number 318 11.3 Theoretical
• Part 319 11.3.1 Mathematical Formulation of an Assignment Problem 319 11.3.2 Method for Solving an Assignment Problem 320 11.3.2.1 Enumeration Method 320 11.3.2.2 Regular Simplex Method 321 11.3.2.3 Transportation Method 321 11.3.2.4 Hungarian Method 321 11.3.3 Computational Processor of Hungarian Method (For Minimization Problem) 323 11.4 Application With Discussion 325 11.5 Conclusion and Further Work 331 References 332 12 The Connectedness of Fuzzy Graph and the Resolving Number of Fuzzy Digraph 335 Mary Jiny D. and R. Shanmugapriya 12.1 Introduction 336 12.2 Definitions 336 12.3 An Algorithm to Find the Super Resolving Matrix 341 12.3.1 An Application on Resolving Matrix 344 12.3.2 An Algorithm to Find the Fuzzy Connectedness Matrix 347 12.4 An Application of the Connectedness of the Modified Fuzzy Graph in Rescuing Human Life From Fire Accident 349 12.4.1 Algorithm to Find the Safest and Shortest Path Between Two Landmarks 352 12.5 Resolving Number Fuzzy Graph and Fuzzy Digraph 356 12.5.1 An Algorithm to Find the Resolving Set of a Fuzzy Digraph 360 12.6 Conclusion 362 References 362 13 A Note on Fuzzy Edge Magic Total Labeling Graphs 365 R. Shanmugapriya and P.K. Hemalatha 13.1 Introduction 365 13.2 Preliminaries 366 13.3 Theorem 367 13.3.1 Example 368 13.4 Theorem 370 13.4.1 Example 371 13.4.1.1 Lemma 374 13.4.1.2 Lemma 374 13.4.1.3 Lemma 374 13.5 Theorem 374 13.5.1 Example as Shown in Figure 13.5 Star Graph S(1,9) is FEMT Labeling 374 13.6 Theorem 376 13.7 Theorem 377 13.7.1 Example 378 13.8 Theorem 380 13.9 Theorem 381 13.10 Application of Fuzzy Edge Magic Total Labeling 383 13.11 Conclusion 385 References 385 14 The Synchronization of Impulsive Time-Delay Chaotic Systems with Uncertainties in Terms of Takagi-Sugeno Fuzzy System 387 Balaji Dharmalingam, Suresh Rasappan, V. Vijayalakshmi and G. Suseendran 14.1 Introduction 387 14.2 Problem Description and Preliminaries 389 14.2.1 Impulsive Differential Equations 389 14.3 The T-S Fuzzy Model 391 14.4 Designing of Fuzzy Impulsive Controllers 393 14.5 Main Result 394 14.6 Numerical Example 400 14.7 Conclusion 410 References 410 15 Theorems on Soft Fuzzy Metric Spaces by Using Control Function 413 Sneha A. Khandait, Chitra Singh, Ramakant Bhardwaj and Amit Kumar Mishra 15.1 Introduction 413 15.2 Preliminaries and Definition 414 15.3 Main Results 415 15.4 Conclusion 429 References 429 16 On Soft ( , )-Continuous Functions in Soft Topological Spaces 431 N. Kalaivani, E. Chandrasekaran and K. Fayaz Ur Rahman 16.1 Introduction 432 16.2 Preliminaries 432 16.2.1 Outline 432 16.2.2 Soft -Open Set 432 16.2.3 Soft Ti Spaces 434 16.2.4 Soft ( , s)-Continuous Functions 436 16.3 Soft ( , )-Continuous Functions in Soft Topological Spaces 438 16.3.1 Outline 438 16.3.2 Soft ( , )-Continuous Functions 438 16.3.3 Soft ( , )-Open Functions 444 16.3.4 Soft ( , )-Closed Functions 447 16.3.5 Soft ( , )-Homeomorphism 450 16.3.6 Soft ( , s)-Contra Continuous Functions 450 16.3.7 Soft ( , )-Contra Continuous Functions 455 16.4 Conclusion 459 References 459 Index 461.
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Fuzzy Intelligent Systems: Methodologies, Techniques and Applications comprises state-of-the-art chapters detailing how expert systems are built and the fuzzy logic resembling human reasoning powering them. Hybrid and neuro-fuzzy intelligent systems are discussed along with Evolutionary and, in particular, Genetic Algorithms. This approach has been extended by using Multiobjective Evolutionary Algorithms, which can consider multiple conflicting objectives instead of a single one. The book also discusses the hybridization between Multiobjective Evolutionary Algorithms and Fuzzy Systems which is known as Multiobjective Evolutionary Fuzzy Systems.
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• Each chapters ends with a problem set Fuzzy Set Theory: Classical Set Theory Fuzzy Set Theory Interval Arithmetic Operations on Fuzzy Sets: Fuzzy Logic Theory Classical Logic Theory The Boolean Algebra Multi-Valued Logic Fuzzy Logic and Approximate Reasoning Fuzzy Relations Some Applications of Fuzzy Logic Product Quality Evaluation Decision Making for Investment Performance Evaluation Miscellaneous Examples Fuzzy Rule Base and Fuzzy Modeling Fuzzy Rule Base Fuzzy Modeling Fuzzy Control Systems Classical Programmable Logic Control Fuzzy Logic Control: A General Model-Free Approach Fuzzy PID Control Systems Conventional PID Controllers Fuzzy PID Controllers (Type 1) Fuzzy PID Controllers (Type 2) Fuzzy PID Controllers: Stability Analysis Computational Verb Fuzzy Controllers Computational Verbs and Verb Numbers Verb Rules and Verb Inference Computational Verb-Based Fuzzy PID Controllers References Solutions Index...
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"Introduction to Fuzzy Systems" provides students with a self-contained introduction that requires no preliminary knowledge of fuzzy mathematics and fuzzy control systems theory. Simplified and readily accessible, it encourages both classroom and self-directed learners to build a solid foundation in fuzzy systems. After introducing the subject, the authors move directly into presenting real-world applications of fuzzy logic, revealing its practical flavor. This practicality is then followed by basic fuzzy systems theory. The book also offers a tutorial on fuzzy control theory, based mainly on the well-known classical Proportional-Integral-Derivative (PID) controllers theory and design methods. In particular, the text discusses fuzzy PID controllers in detail, including a description of the new notion of generalized verb-based fuzzy-logic control theory. "Introduction to Fuzzy Systems" is primarily designed to provide training for systems and control majors, both senior undergraduate and first year graduate students, to acquaint them with the fundamental mathematical theory and design methodology required to understand and utilize fuzzy control systems.
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• Introduction
• Types of Fuzzy Systems
• Formal Models for Fuzzy Networks
• Basic Operations in Fuzzy Networks
• Structural Properties of Basic Operations
• Advanced Operations in Fuzzy Networks
• Feedforward Fuzzy Networks
• Feedback Fuzzy Networks
• Evaluation of Fuzzy Networks
• Conclusion.

This book introduces the novel concept of a fuzzy network whose nodes are rule bases and the connections between the nodes are the interactions between the rule bases in the form of outputs fed as inputs. The concept is presented as a systematic study for improving the feasibility and transparency of fuzzy models by means of modular rule bases whereby the model accuracy and efficiency can be optimised in a flexible way. The study uses an effective approach for fuzzy rule based modelling of complex systems that are characterised by attributes such as nonlinearity, uncertainty, dimensionality and structure. The approach is illustrated by formal models for fuzzy networks, basic and advanced operations on network nodes, properties of operations, feedforward and feedback fuzzy networks as well as evaluation of fuzzy networks. The results are demonstrated by numerous examples, two case studies and software programmes within the Matlab environment that implement some of the theoretical methods from the book. The book shows the novel concept of a fuzzy network with networked rule bases as a bridge between the existing concepts of a standard fuzzy system with a single rule base and a hierarchical fuzzy system with multiple rule bases.

• Powerset operators in categories with fuzzy relations dened by monads.- Improved Fuzzy Q-Learning with Replay Memory.- The ulem package: underlining for emphasis.- A Dynamic Hierarchical Genetic-Fuzzy Sugeno Network.- Fuzzy Mathematical Morphology and Applications in Image Processing.
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This book describes how to use expert knowledge-which is often formulated by using imprecise (fuzzy) words from a natural language. In the 1960s, Zadeh designed special "fuzzy" techniques for such use. In the 1980s, fuzzy techniques started controlling trains, elevators, video cameras, rice cookers, car transmissions, etc. Now, combining fuzzy with neural, genetic, and other intelligent methods leads to new state-of-the-art results: in aerospace industry (from drones to space flights), in mobile robotics, in finances (predicting the value of crypto-currencies), and even in law enforcement (detecting counterfeit banknotes, detecting online child predators and in creating explainable AI systems). The book describes these (and other) applications-as well as foundations and logistics of fuzzy techniques. This book can be recommended to specialists-both in fuzzy and in various application areas-who will learn latest techniques and their applications, and to students interested in innovative ideas.
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• Introduction.- Stability and Stabilization of CTIT2FMBSs.- Output-feedback Control of CTIT2FMBSs.- Sampled-Data Control of CTIT2FMBSs.- Output Tracking Control of CTIT2FSs.- Switched Output-feedback Control of CTIT2FMBSs.- Filter Design of CTIT2FMBSs.- Fault Detection of CTIT2FMBSs.- Model Reduction of CTIT2FMBSs.- Optimal Control of DTIT2FMBSs.- State-feedback Control of DTIT2FMBSs.- Static Output-feedback Control of DTIT2FMBSs.- Guaranteed Cost Output Tracking Control of DTIT2FMBSs.
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This book develops a set of reference methods capable of modeling uncertainties existing in membership functions, and analyzing and synthesizing the interval type-2 fuzzy systems with desired performances. It also provides numerous simulation results for various examples, which fill certain gaps in this area of research and may serve as benchmark solutions for the readers. Interval type-2 T-S fuzzy models provide a convenient and flexible method for analysis and synthesis of complex nonlinear systems with uncertainties.
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• Introduction to Takagi-Sugeno Fuzzy Systems
• Stabilization of Takagi-Sugeno Fuzzy Systems with Constrained Controls
• Static Output Feedback Control for Fuzzy Systems
• Stabilization of Discrete-time Takagi-Sugeno Fuzzy Positive Systems
• Stabilization of Delayed T-S Fuzzy Positive Systems
• Robust Control of T-S Fuzzy Systems with Time-varying Delay
• Robust Output H[infinity] Fuzzy Control.-Stabilization of Discrete-time T-S Fuzzy Positive Systems with Multiple Delays
• Stabilization of Two Dimensional T-S Fuzzy Systems.

This monograph puts the reader in touch with a decade's worth of new developments in the field of fuzzy control specifically those of the popular Takagi-Sugeno (T-S) type. New techniques for stabilizing control analysis and design based on multiple Lyapunov functions and linear matrix inequalities (LMIs), are proposed. All the results are illustrated with numerical examples and figures and a rich bibliography is provided for further investigation. Control saturations are taken into account within the fuzzy model. The concept of positive invariance is used to obtain sufficient asymptotic stability conditions for the fuzzy system with constrained control inside a subset of the state space. The authors also consider the non-negativity of the states. This is of practical importance in many chemical, physical and biological processes that involve quantities that have intrinsically constant and non-negative sign: concentration of substances, level of liquids, etc. Results for linear systems are then extended to linear systems with delay. It is shown that LMI techniques can usually handle the new constraint of non-negativity of the states when care is taken to use an adequate Lyapunov function. From these foundations, the following further problems are also treated: · asymptotic stabilization of uncertain T-S fuzzy systems with time-varying delay, focusing on delay-dependent stabilization synthesis based on parallel distributed controller (PDC); · asymptotic stabilization of uncertain T-S fuzzy systems with multiple delays, focusing on delay-dependent stabilization synthesis based on PDC with results obtained under linear programming; · design of delay-independent, observer-based, H-infinity control for T-S fuzzy systems with time varying delay; and · asymptotic stabilization of 2-D T-S fuzzy systems. Advanced Takagi-Sugeno Fuzzy Systems provides researchers and graduate students interested in fuzzy control systems with further approaches based LMI and LP.

• Chapter 1: An Introduction to Fuzzy and Fuzzy Control Systems.-
• Chapter 2: Classification of Adaptive Fuzzy Controllers.-
• Chapter 3: Type-2 Fuzzy Systems.-
• Chapter 4: Training Interval Type-2 Fuzzy Systems Based on Error Backpropagation.
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This book explains the basic concepts, theory and applications of fuzzy systems in control in a simple unified approach with clear ex-amples and simulations in the MATLAB programming language. Fuzzy systems, especially, type-2 neuro-fuzzy systems, are now used extensively in various engineering fields for different purposes. In plain language, this book aims to practically explain fuzzy sys-tems and different methods of training and optimizing these systems. For this purpose, type-2 neuro-fuzzy systems are first analyzed along with various methods of training and optimizing these systems through implementation in MATLAB. These systems are then em-ployed to design adaptive fuzzy controllers. The authors aim at pre-senting all the well-known optimization methods clearly and code them in the MATLAB language.
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• Toward Explainable Artificial Intelligence through Fuzzy Systems.- An Overview of Fuzzy Systems.- Interpretability Constraints and Criteria for Fuzzy Systems.- Revisiting Indexes for Assessing Interpretability of Fuzzy Systems.- Designing Interpretable Fuzzy Systems.
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The importance of Trustworthy and Explainable Artificial Intelligence (XAI) is recognized in academia, industry and society. This book introduces tools for dealing with imprecision and uncertainty in XAI applications where explanations are demanded, mainly in natural language. Design of Explainable Fuzzy Systems (EXFS) is rooted in Interpretable Fuzzy Systems, which are thoroughly covered in the book. The idea of interpretability in fuzzy systems, which is grounded on mathematical constraints and assessment functions, is firstly introduced. Then, design methodologies are described. Finally, the book shows with practical examples how to design EXFS from interpretable fuzzy systems and natural language generation. This approach is supported by open source software. The book is intended for researchers, students and practitioners who wish to explore EXFS from theoretical and practical viewpoints. The breadth of coverage will inspire novel applications and scientific advancements.
(source: Nielsen Book Data)

• 1. Introduction: visual uncertainty, categorization, objectivity and practices andvalues of imprecision.- 2. From ordinary to mathematical categorization in the visual world... of words, pictures and practices.- 3. Vagueness and fuzziness in words and predication.- 4. Representations: from words to images.- 5. Epistemology, aesthetics and pragmatics of scientific and other images:visualization, representation and reasoning.- 6. Visual representation: from perceptions to pictures.- 7. Vague pictures: epistemology, aesthetics and pragmatics of fuzziness, from fuzzy perception to fuzzy pictures.- 8. Blur as vagueness: seeing images vaguely and seeing vague images
• perception and representation.- 9. Vague pictures as pictures.- 10. Vague pictures as vague representations and representing.- 11. The cognitive values of imprecision: towards a scientific epistemology, aesthetics and pragmatics of fuzziness, contextual lessons from the history of picture-making practices.- 12. Introduction: fuzzy-set representation and processing of fuzzy images
• nonlinguistic vagueness as scientific practice
• scientific epistemology, aesthetics, methodology and technology of fuzziness.- 13. Application of mathematics in the representation of images: from geometry to set theory.- 14. Cognitive framework of set-theoretic methodology of analysis and synthesis: categorization, classification and many faces of digital geometry.- 15. Conceptual resources and philosophical grounds in set-theoretic models of vagueness: fuzzy, rough and near sets.- 16. Analytic and synthetic forms of vague categorization.- 17. From intrinsic to extrinsic vague categorization and content.- 18. Pictorial representation and simplicity of categorization.- 19. Fuzzy visual thinking: interpreting and thinking with fuzzy pictures and fuzzydata.- 20. Pictorial approximation: pictorial accuracy, vagueness and fuzziness.- 21. Pictorial vagueness as scientific practice: picture-making and the mathematical practice of fuzzy categorization.- 22. Conclusion.
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This book presents a comprehensive discussion on the characterization of vagueness in pictures. It reports on how the problem of representation of images has been approached in scientific practice, highlighting the role of mathematical methods and the philosophical background relevant for issues such as representation, categorization and reasoning. Without delving too much into the technical details, the book examines and defends different kinds of values of fuzziness based on a complex approach to categorization as a practice, adopting conceptual and empirical suggestions from different fields including the arts. It subsequently advances criticisms and provides suggestions for interpretation and application. By describing a cognitive framework based on fuzzy, rough and near sets, and discussing all of the relevant mathematical and philosophical theories for the representation and processing of vagueness in images, the book offers a practice-oriented guide to fuzzy visual reasoning, along with novel insights into the field of interpreting and thinking with fuzzy pictures and fuzzy data.
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• Introduction
• Introduction to Fuzzy Logic
• Sustainability Indicators
• Fuzzy Assessment
• Sustainability of Organisations
• Index.
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People from many disciplines talk about sustainable development (SD) without having a concrete definition or idea of what it is. Politicians and decision-makers, biologists and environmentalists, engineers and scientists, philosophers and sociologists often mean different things when they mention SD. To take an extreme example, a neoclassical economist might define SD as continual economic growth without regard to the environmental situation, whereas at the other end of the spectrum, a deep ecologist might define it as the preservation and restoration of the ecosystem, ignoring the effect on the economy completely. It has become clear throughout the last few decades leading up to the time this book was written that our society is facing environmental problems of global magnitude such as species extinction, population explosion, global warming, water shortages, exhaustion of fisheries, and deforestation, among others, which have economic and social repercussions. Many people doubt that the world we shall bequeath to our descendants will be all healthy, supportive, and beautiful as the one we inherited from our parents. The fundamental questions then of the SD discussion are: Can we continue our present course without destroying part or most of our environment and society? And if we have to change course (which is currently the consensus opinion) how we do it? This book addresses one of the most fundamental questions of SD: How do we define and measure in the mathematical sense sustainability? In effect it summarises our research in this field for the last 15 years. The mathematical model which we use to evaluate sustainability serves both as a definition and as a measuring scheme. This is done with the help of fuzzy logic, which is well suited to perform reasoning in fields where concrete mathematical models do not exist and concepts such as sustainability are multifaceted and often subjective. Fuzzy reasoning, in some respects, emulates human thinking and relies on expert opinion and knowledge as well as subjective evaluations. This multistage model is developed step-by-step, in the end providing sustainability assessments of various aspects of a society on a scale from zero to one, such as the environmental or societal situation of a country. All nations of the world for which reliable data exist are then ranked according to their sustainability as measured by the model. A sensitivity analysis reveals the most important components of sustainability for each country. The book is intended for undergraduate students, graduate students, and practitioners working in the field of sustainability. It is based on a course Yannis Phillis has taught at the University of California at Los Angeles (UCLA) and the Technical University of Crete (TUC) at the senior/graduate level. The mathematical knowledge needed to follow the book is minimal and is all contained in the text. Computational guidance is provided for those who would like to write their own model.
(source: Nielsen Book Data)

• List of Contributors.Foreword.Preface.Part I Fundamentals.1 Fundamentals of Fuzzy Clustering (Rudolf Kruse, Christian Doring and Marie-Jeanne Lesot).1.1 Introduction.1.2 Basic Clustering Algorithms.1.3 Distance Function Variants.1.4 Objective Function Variants.1.5 Update Equation Variants: Alternating Cluster Estimation.1.6 Concluding Remarks.Acknowledgements.References.2 Relational Fuzzy Clustering (Thomas A. Runkler).2.1 Introduction.2.2 Object and Relational Data.2.3 Object Data Clustering Models.2.4 Relational Clustering.2.5 Relational Clustering with Non-spherical Prototypes.2.6 Relational Data Interpreted as Object Data.2.7 Summary.2.8 Experiments.2.9 Conclusions.References.3 Fuzzy Clustering with Minkowski Distance Functions (Patrick J.F. Groenen, Uzay Kaymak and Joost van Rosmalen).3.1 Introduction.3.2 Formalization.3.3 The Majorizing Algorithm for Fuzzy C-means with Minkowski Distances.3.4 The Effects of the Robustness Parameter.3.5 Internet Attitudes.3.6 Conclusions.References.4 Soft Cluster Ensembles (Kunal Punera and Joydeep Ghosh).4.1 Introduction.4.2 Cluster Ensembles.4.3 Soft Cluster Ensembles.4.4 Experimental Setup.4.5 Soft vs. Hard Cluster Ensembles.4.6 Conclusions and Future Work.Acknowledgements.References.Part II Visualization.5 Aggregation and Visualization of Fuzzy Clusters Based on Fuzzy Similarity Measures (Janos Abonyi and Balazs Feil).5.1 Problem Definition.5.2 Classical Methods for Cluster Validity and Merging.5.3 Similarity of Fuzzy Clusters.5.4 Visualization of Clustering Results.5.5 Conclusions.Appendix 5A.1 Validity Indices.Appendix 5A.2 The Modified Sammon Mapping Algorithm.Acknowledgements.References.6 Interactive Exploration of Fuzzy Clusters (Bernd Wiswedel, David E. Patterson and Michael R. Berthold).6.1 Introduction.6.2 Neighborgram Clustering.6.3 Interactive Exploration.6.4 Parallel Universes.6.5 Discussion.References.Part III Algorithms and Computational Aspects.7 Fuzzy Clustering with Participatory Learning and Applications (Leila Roling Scariot da Silva, Fernando Gomide and Ronald Yager).7.1 Introduction.7.2 Participatory Learning.7.3 Participatory Learning in Fuzzy Clustering.7.4 Experimental Results.7.5 Applications.7.6 Conclusions.Acknowledgements.References.8 Fuzzy Clustering of Fuzzy Data (Pierpaolo D'Urso).8.1 Introduction.8.2 Informational Paradigm, Fuzziness and Complexity in Clustering Processes.8.3 Fuzzy Data.8.4 Fuzzy Clustering of Fuzzy Data.8.5 An Extension: Fuzzy Clustering Models for Fuzzy Data Time Arrays.8.6 Applicative Examples.8.7 Concluding Remarks and Future Perspectives.References.9 Inclusion-based Fuzzy Clustering (Samia Nefti-Meziani and Mourad Oussalah).9.1 Introduction.9.2 Background: Fuzzy Clustering.9.3 Construction of an Inclusion Index.9.4 Inclusion-based Fuzzy Clustering.9.5 Numerical Examples and Illustrations.9.6 Conclusions.Acknowledgements.Appendix 9A.1.References.10 Mining Diagnostic Rules Using Fuzzy Clustering (Giovanna Castellano, Anna M. Fanelli and Corrado Mencar).10.1 Introduction.10.2 Fuzzy Medical Diagnosis.10.3 Interpretability in Fuzzy Medical Diagnosis.10.4 A Framework for Mining Interpretable Diagnostic Rules.10.5 An Illustrative Example.10.6 Concluding Remarks.References.11 Fuzzy Regression Clustering (Mikal Sato-Ilic).11.1 Introduction.11.2 Statistical Weighted Regression Models.11.3 Fuzzy Regression Clustering Models.11.4 Analyses of Residuals on Fuzzy Regression Clustering Models.11.5 Numerical Examples.11.6 Conclusion.References.12 Implementing Hierarchical Fuzzy Clustering in Fuzzy Modeling Using the Weighted Fuzzy C-means (George E. Tsekouras).12.1 Introduction.12.2 Takagi and Sugeno's Fuzzy Model.12.3 Hierarchical Clustering-based Fuzzy Modeling.12.4 Simulation Studies.12.5 Conclusions.References.13 Fuzzy Clustering Based on Dissimilarity Relations Extracted from Data (Mario G.C.A. Cimino, Beatrice Lazzerini and Francesco Marcelloni).13.1 Introduction.13.2 Dissimilarity Modeling.13.3 Relational Clustering.13.4 Experimental Results.13.5 Conclusions.References.14 Simultaneous Clustering and Feature Discrimination with Applications (Hichem Frigui).14.1 Introduction.14.2 Background.14.3 Simultaneous Clustering and Attribute Discrimination (SCAD).14.4 Clustering and Subset Feature Weighting.14.5 Case of Unknown Number of Clusters.14.6 Application
• 1: Color Image Segmentation.14.7 Application
• 2: Text Document Categorization and Annotation.14.8 Application
• 3: Building a Multi-modal Thesaurus from Annotated Images.14.9 Conclusions.Appendix 14A.1.Acknowledgements.References.Part IV Real-time and Dynamic Clustering.15 Fuzzy Clustering in Dynamic Data Mining - Techniques and Applications (Richard Weber).15.1 Introduction.15.2 Review of Literature Related to Dynamic Clustering.15.3 Recent Approaches for Dynamic Fuzzy Clustering.15.4 Applications.15.5 Future Perspectives and Conclusions.Acknowledgement.References.16 Fuzzy Clustering of Parallel Data Streams (Jurgen Beringer and Eyke Hullermeier).16.1 Introduction.16.2 Background.16.3 Preprocessing and Maintaining Data Streams.16.4 Fuzzy Clustering of Data Streams.16.5 Quality Measures.16.6 Experimental Validation.16.7 Conclusions.References.17 Algorithms for Real-time Clustering and Generation of Rules from Data (Dimitar Filev and Plamer Angelov).17.1 Introduction.17.2 Density-based Real-time Clustering.17.3 FSPC: Real-time Learning of Simplified Mamdani Models.17.4 Applications.17.5 Conclusion.References.Part V Applications and Case Studies.18 Robust Exploratory Analysis of Magnetic Resonance Images using FCM with Feature Partitions (Mark D. Alexiuk and Nick J. Pizzi).18.1 Introduction.18.2 FCM with Feature Partitions.18.3 Magnetic Resonance Imaging.18.4 FMRI Analysis with FCMP.18.5 Data-sets.18.6 Results and Discussion.18.7 Conclusion.Acknowledgements.References.19 Concept Induction via Fuzzy C-means Clustering in a High-dimensional Semantic Space (Dawei Song, Guihong Cao, Peter Bruza and Raymond Lau).19.1 Introduction.19.2 Constructing a High-dimensional Semantic Space via Hyperspace Analogue to Language.19.3 Fuzzy C-means Clustering.19.4 Word Clustering on a HAL Space - A Case Study.19.5 Conclusions and Future Work.Acknowledgement.References.20 Novel Developments in Fuzzy Clustering for the Classification of Cancerous Cells using FTIR Spectroscopy (Xiao-Ying Wang, Jonathan M. Garibaldi, Benjamin Bird and Mike W. George).20.1 Introduction.20.2 Clustering Techniques.20.3 Cluster Validity.20.4 Simulated Annealing Fuzzy Clustering Algorithm.20.5 Automatic Cluster Merging Method.20.6 Conclusion.Acknowledgements.References.Index.
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This is a comprehensive, coherent, and in depth presentation of the state of the art in fuzzy clustering. Fuzzy clustering is now a mature and vibrant area of research with highly innovative advanced applications. Encapsulating this through presenting a careful selection of research contributions, this book addresses timely and relevant concepts and methods, whilst identifying major challenges and recent developments in the area.Split into five clear sections, Fundamentals, Visualization, Algorithms and Computational Aspects, Real Time and Dynamic Clustering, and Applications and Case Studies, the book covers a wealth of novel, original and fully updated material, and in particular offers: a focus on the algorithmic and computational augmentations of fuzzy clustering and its effectiveness in handling high dimensional problems; distributed problem solving and uncertainty management; presentations of the important and relevant phases of cluster design, including the role of information granules; fuzzy sets in the realization of human centricity facet of data analysis, as well as system modelling demonstrations of how the results facilitate further detailed development of models; and enhance interpretation aspects a carefully organized illustrative series of applications and case studies in which fuzzy clustering plays a pivotal role. This book will be of key interest to engineers associated with fuzzy control, bioinformatics, data mining, image processing, and pattern recognition, while computer engineers, students and researchers, in most engineering disciplines, will find this an invaluable resource and research tool.
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• Basic Types of Fuzzy Rule Based Systems.- Rule Base Reduction Methods for Fuzzy Systems.- Formal Presentation of Fuzzy Rule Based Systems.- Formal Manipulation of Fuzzy Rule Based Systems.- Formal Manipulation with Special Rule Bases.- Formal Transformation of Fuzzy Rule Based Systems.- Formal Transformation of Feedback Rule Bases.- Formal Simplification of Fuzzy Rule Based Systems.- Conclusion.
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This book presents a systematic study on the inherent complexity in fuzzy systems, resulting from the large number and the poor transparency of the fuzzy rules. The study uses a novel approach for complexity management, aimed at compressing the fuzzy rule base by removing the redundancy while preserving the solution. The compression is based on formal methods for presentation, manipulation, transformation and simplification of fuzzy rule bases.
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• Preface.1 Introduction.1.1 Digital communities and a fundamental quest for human-centric systems.1.2 A historical overview: towards a non-Aristotelian perspective of the world.1.3 Granular Computing.1.4 Quantifying information granularity: generality versus specificity.1.5 Computational Intelligence.1.6 Granular Computing and Computational Intelligence.1.7 Conclusions.Exercises and problems.Historical notes.References.2 Notions and Concepts of Fuzzy Sets.2.1 Sets and fuzzy sets: a departure from the principle of dichotomy.2.2 Interpretation of fuzzy sets.2.3 Membership functions and their motivation.2.4 Fuzzy numbers and intervals.2.5 Linguistic variables.2.6 Conclusions.Exercises and problems.Historical notes.References.3 Characterization of Fuzzy Sets.3.1 A generic characterization of fuzzy sets: some fundamental descriptors.3.2 Equality and inclusion relationships in fuzzy sets.3.3 Energy and entropy measures of fuzziness.3.4 Specificity of fuzzy sets.3.5 Geometric interpretation of sets and fuzzy sets.3.6 Granulation of information.3.7 Characterization of the families of fuzzy sets.3.8 Fuzzy sets, sets, and the representation theorem.3.9 Conclusions.Exercises and problems.Historical notes.References.4 The Design of Fuzzy Sets.4.1 Semantics of fuzzy sets: some general observations.4.2 Fuzzy set as a descriptor of feasible solutions.4.3 Fuzzy set as a descriptor of the notion of typicality.4.4 Membership functions in the visualization of preferences of solutions.4.5 Nonlinear transformation of fuzzy sets.4.6 Vertical and horizontal schemes of membership estimation.4.7 Saaty's priority method of pairwise membership function estimation.4.8 Fuzzy sets as granular representatives of numeric data.4.9 From numeric data to fuzzy sets.4.10 Fuzzy equalization.4.11 Linguistic approximation.4.12 Design guidelines for the construction of fuzzy sets.4.13 Conclusions.Exercises and problems.Historical notes.References.5 Operations and Aggregations of Fuzzy Sets.5.1 Standard operations on sets and fuzzy sets.5.2 Generic requirements for operations on fuzzy sets.5.3 Triangular norms.5.4 Triangular conorms.5.5 Triangular norms as a general category of logical operators.5.6 Aggregation operations.5.7 Fuzzy measure and integral.5.8 Negations.5.9 Conclusions.Exercises and problems.Historical notes.References.6 Fuzzy Relations.6.1 The concept of relations.6.2 Fuzzy relations.6.3 Properties of the fuzzy relations.6.4 Operations on fuzzy relations.6.5 Cartesian product, projections and cylindrical extension of fuzzy sets.6.6 Reconstruction of fuzzy relations.6.7 Binary fuzzy relations.6.8 Conclusions.Exercises and problems.Historical notes.References.7 Transformations of Fuzzy Sets.7.1 The extension principle.7.2 Compositions of fuzzy relations.7.3 Fuzzy relational equations.7.4 Associative Memories.7.5 Fuzzy numbers and fuzzy arithmetic.7.6 Conclusions.Exercises and problems.Historical notes.References.8 Generalizations and Extensions of Fuzzy Sets.8.1 Fuzzy sets of higher order.8.2 Rough fuzzy sets and fuzzy rough sets.8.3 Interval-valued fuzzy sets.8.4 Type-2 fuzzy sets.8.5 Shadowed sets as a three-valued logic characterization of fuzzy sets.8.6 Probability and fuzzy sets.8.7 Probability of fuzzy events.8.8 Conclusions.Exercises and problems.Historical notes.References.9 Interoperability Aspects of Fuzzy Sets.9.1 Fuzzy set and its family of s-cuts.9.2 Fuzzy sets and their interfacing with the external world.9.3 Encoding and decoding as an optimization problem of vector quantization.9.4 Decoding of a fuzzy set through a family of fuzzy sets.9.5 Taxonomy of data in structure description with shadowed sets.9.6 Conclusions.Exercises and problems.Historical notes.References
• .10. Fuzzy Modeling: Principles and Methodology.10.1 The architectural blueprint of fuzzy models.10.2 Key phases of the development and use of fuzzy models.10.3 Main categories of fuzzy models: an overview.10.4 Verification and validation of fuzzy models.10.5 Conclusions.Exercises and problems.Historical notes.References.11 Rule-based Fuzzy Models.11.1 Fuzzy rules as a vehicle of knowledge representation.11.2 General categories of fuzzy rules and their semantics.11.3 Syntax of fuzzy rules.11.4 Basic Functional Modules: Rule base, Database, and Inference scheme.11.5 Types of Rule-Based Systems and Architectures.11.6 Approximation properties of fuzzy rule-based models.11.7 Development of Rule-Based Systems.11.8 Parameter estimation procedure for functional rule-based systems.11.9 Design issues of rule-based systems - consistency, completeness, and the curse of dimensionality.11.10 The curse of dimensionality in rule-based systems.11.11 Development scheme of fuzzy rule-based models.11.12 Conclusions.Exercises and problems.Historical notes.References.12 From Logic Expressions to Fuzzy Logic Networks.12.1 Introduction.12.2 Main categories of fuzzy neurons.12.3 Uninorm-based fuzzy neurons.12.4 Architectures of logic networks.12.5 The development mechanisms of the fuzzy neural networks.12.6 Interpretation of the fuzzy neural networks.12.7 From fuzzy logic networks to Boolean functions and their minimization through learning.12.8 Interfacing the fuzzy neural network.12.9 Interpretation aspects - a refinement of induced rule-based system.12.10 Reconciliation of perception of information granules and granular mappings.12.11 Conclusions.Exercises and problems.Historical notes.References
• .13. Fuzzy Systems and Computational Intelligence.13.1 Computational Intelligence.13.2 Recurrent neurofuzzy systems.13.3 Genetic fuzzy systems.13.4 Coevolutionary hierarchical genetic fuzzy system.13.5 Hierarchical collaborative relations.13.6 Evolving fuzzy systems.13.7 Conclusions.Exercises and problems.Historical notes.References
• .14. Granular Models and Human Centric Computing.14.1 The cluster-based representation of the input - output mappings.14.2 Context-based clustering in the development of granular models.14.3 Granular neuron as a generic processing element in granular networks.14.4 Architecture of granular models based on conditional fuzzy clustering.14.5 Refinements of granular models.14.6 Incremental granular models.14.7 Human-centric fuzzy clustering.14.8 Participatory learning in fuzzy clustering.14.9 Conclusions.Exercises and problems.Historical notes.References
• .15. Emerging Trends in Fuzzy Systems.15.1 Relational ontology in information retrieval.15.2 Multiagent fuzzy systems.15.3 Distributed fuzzy control.15.4 Conclusions.Exercises and problems.Historical notes.References.Appendix A: Mathematical Prerequisites.Appendix B: Neurocomputing.Appendix C: Biologically Inspired Optimization.Index.
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This is a self-contained treatment of fuzzy systems engineering, offering conceptual fundamentals, design methodologies, development guidelines, and carefully selected illustrative material. Forty years have passed since the birth of fuzzy sets, in which time a wealth of theoretical developments, conceptual pursuits, algorithmic environments, and other applications have emerged. Now, this reader-friendly book presents an up-to-date approach to fuzzy systems engineering, covering concepts, design methodologies, and algorithms coupled with interpretation, analysis, and underlying engineering knowledge. The result is a holistic view of fuzzy sets as a fundamental component of computational intelligence and human-centric systems.Throughout the book, the authors emphasize the direct applicability and limitations of the concepts being discussed, and historical and bibliographical notes are included in each chapter to help readers view the developments of fuzzy sets from a broader perspective. A radical departure from current books on the subject, "Fuzzy Systems Engineering" presents fuzzy sets as an enabling technology whose impact, contributions, and methodology stretch far beyond any specific discipline, making it applicable to researchers and practitioners in engineering, computer science, business, medicine, bioinformatics, and computational biology. Additionally, three appendices and classroom-ready electronic resources make it an ideal textbook for advanced undergraduate- and graduate-level courses in engineering and science.
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