The purpose of this book is to provide the reader who is interested in applications of fuzzy set theory, in the first place with a text to which he or she can refer for the basic theoretical ideas, concepts and techniques in this field and in the second place with a vast and up to date account of the literature. Although there are now many books about fuzzy set theory, and mainly about its applications, e. g. in control theory, there is not really a book available which introduces the elementary theory of fuzzy sets, in what I would like to call "a good degree of generality." To write a book which would treat the entire range of results concerning the basic theoretical concepts in great detail and which would also deal with all possible variants and alternatives of the theory, such as e. g. rough sets and L-fuzzy sets for arbitrary lattices L, with the possibility-probability theories and interpretations, with the foundation of fuzzy set theory via multi-valued logic or via categorical methods and so on, would have been an altogether different project. This book is far more modest in its mathematical content and in its scope.

Rule-based fuzzy modeling has been recognised as a powerful technique for the modeling of partly-known nonlinear systems. Fuzzy models can effectively integrate information from different sources, such as physical laws, empirical models, measurements and heuristics. Application areas of fuzzy models include prediction, decision support, system analysis, control design, etc. Fuzzy Modeling for Control addresses fuzzy modeling from the systems and control engineering points of view. It focuses on the selection of appropriate model structures, on the acquisition of dynamic fuzzy models from process measurements (fuzzy identification), and on the design of nonlinear controllers based on fuzzy models. To automatically generate fuzzy models from measurements, a comprehensive methodology is developed which employs fuzzy clustering techniques to partition the available data into subsets characterized by locally linear behaviour. The relationships between the presented identification method and linear regression are exploited, allowing for the combination of fuzzy logic techniques with standard system identification tools. Attention is paid to the trade-off between the accuracy and transparency of the obtained fuzzy models. Control design based on a fuzzy model of a nonlinear dynamic process is addressed, using the concepts of model-based predictive control and internal model control with an inverted fuzzy model. To this end, methods to exactly invert specific types of fuzzy models are presented. In the context of predictive control, branch-and-bound optimization is applied. The main features of the presented techniques are illustrated by means of simple examples. In addition, three real-world applications are described. Finally, software tools for building fuzzy models from measurements are available from the author.

Fuzzy systems and soft computing are new computing techniques that are tolerant to imprecision, uncertainty and partial truths. Applications of these techniques in nuclear engineering present a tremendous challenge due to its strict nuclear safety regulation. The fields of nuclear engineering, fuzzy systems and soft computing have nevertheless matured considerably during the last decade. This book presents new application potentials for Fuzzy Systems and Soft Computing in Nuclear Engineering. The root of this book can be traced back to the series of the first, second and third international workshops on Fuzzy Logic and Intelligent Technologies in Nuclear Science (FUNS), which were successfully held in Mol, September 14-16, 1994 (FLINS'94), in Mol, September 25-27, 1996 (FLINS'96), and in Antwerp, September 14-16, 1998 (FLINS'98). The conferences were organised by the Belgian Nuclear Research Centre (SCKeCEN) and aimed at bringing together scientists, researchers, and engineers from academia and industry, at introducing the principles of fuzzy logic, neural networks, genetic algorithms and other soft computing methodologies, to the field of nuclear engineering, and at applying these techniques to complex problem solving within nuclear industry and related research fields. This book, as its title suggests, consists of nuclear engineering applications of fuzzy systems (Chapters 1-10) and soft computing (Chapters 11-21). Nine pertinent chapters are based on the extended version of papers at FLINS'98 and the other 12 chapters are original contributions with up-to-date coverage of fuzzy and soft computing applications by leading researchers written exclusively for this book."

The volume "Fuzziness in Database Management Systems" is a highly informative, well-organized and up-to-date collection of contributions authored by many of the leading experts in its field. Among the contributors are the editors, Professors Patrick Bose and Janusz Kacprzyk, both of whom are known internationally. The book is like a movie with an all-star cast. The issue of fuzziness in database management systems has a long history. It begins in 1968 and 1971, when I spent my sabbatical leaves at the IBM Research Laboratory in San Jose, California, as a visiting scholar. During these periods I was associated with Dr. E.F. Codd, the father of relational models of database systems, and came in contact with the developers ofiBMs System Rand SQL. These associations and contacts at a time when the methodology of relational models of data was in its formative stages, made me aware of the basic importance of such models and the desirability of extending them to fuzzy database systems and fuzzy query languages. This perception was reflected in my 1973 ffiM report which led to the paper on the concept of a linguistic variable and later to the paper on the meaning representation language PRUF (Possibilistic Relational Universal Fuzzy). More directly related to database issues during that period were the theses of my students V. Tahani, J. Yang, A. Bolour, M. Shen and R. Sheng, and many subsequent reports by both graduate and undergraduate students at Berkeley.

Fuzzy Logic and Soft Computing contains contributions from world-leading experts from both the academic and industrial communities. The first part of the volume consists of invited papers by international authors describing possibilistic logic in decision analysis, fuzzy dynamic programming in optimization, linguistic modifiers for word computation, and theoretical treatments and applications of fuzzy reasoning. The second part is composed of eleven contributions from Chinese authors focusing on some of the key issues in the fields: stable adaptive fuzzy control systems, partial evaluations and fuzzy reasoning, fuzzy wavelet neural networks, analysis and applications of genetic algorithms, partial repeatability, rough set reduction for data enriching, limits of agents in process calculus, medium logic and its evolution, and factor spaces canes. These contributions are not only theoretically sound and well-formulated, but are also coupled with applicability implications and/or implementation treatments. The domains of applications realized or implied are: decision analysis, word computation, databases and knowledge discovery, power systems, control systems, and multi-destinational routing. Furthermore, the articles contain materials that are an outgrowth of recently conducted research, addressing fundamental and important issues of fuzzy logic and soft computing.

Advances in Computational Intelligence and Learning: Methods and Applications presents new developments and applications in the area of Computational Intelligence, which essentially describes methods and approaches that mimic biologically intelligent behavior in order to solve problems that have been difficult to solve by classical mathematics. Generally Fuzzy Technology, Artificial Neural Nets and Evolutionary Computing are considered to be such approaches.

The Editors have assembled new contributions in the areas of fuzzy sets, neural sets and machine learning, as well as combinations of them (so called hybrid methods) in the first part of the book. The second part of the book is dedicated to applications in the areas that are considered to be most relevant to Computational Intelligence.

Fuzzy Control of Industrial Systems: Theory and Applications presents the basic theoretical framework of crisp and fuzzy set theory, relating these concepts to control engineering based on the analogy between the Laplace transfer function of linear systems and the fuzzy relation of a nonlinear fuzzy system. Included are generic aspects of fuzzy systems with an emphasis on the many degrees of freedom and its practical design implications, modeling and systems identification techniques based on fuzzy rules, parametrized rules and relational equations, and the principles of adaptive fuzzy and neurofuzzy systems. Practical design aspects of fuzzy controllers are covered by the detailed treatment of fuzzy and neurofuzzy software design tools with an emphasis on iterative fuzzy tuning, while novel stability limit testing methods and the definition and practical examples of the new concept of collaborative control systems are also given. In addition, case studies of successful applications in industrial automation, process control, electric power technology, electric traction, traffic engineering, wastewater treatment, manufacturing, mineral processing and automotive engineering are also presented, in order to assist industrial control systems engineers in recognizing situations when fuzzy and neurofuzzy would offer certain advantages over traditional methods, particularly in controlling highly nonlinear and time-variant plants and processes.

In 1982, Professor Pawlak published his seminal paper on what he called "rough sets" - a work which opened a new direction in the development of theories of incomplete information. Today, a decade and a half later, the theory of rough sets has evolved into a far-reaching methodology for dealing with a wide variety of issues centering on incompleteness and imprecision of information - issues which playa key role in the conception and design of intelligent information systems. "Incomplete Information: Rough Set Analysis" - or RSA for short - presents an up-to-date and highly authoritative account of the current status of the basic theory, its many extensions and wide-ranging applications. Edited by Professor Ewa Orlowska, one of the leading contributors to the theory of rough sets, RSA is a collection of nineteen well-integrated chapters authored by experts in rough set theory and related fields. A common thread that runs through these chapters ties the concept of incompleteness of information to those of indiscernibility and similarity.

This book presents the main tools for aggregation of information given by several members of a group or expressed in multiple criteria, and for fusion of data provided by several sources. It focuses on the case where the availability knowledge is imperfect, which means that uncertainty and/or imprecision must be taken into account. The book contains both theoretical and applied studies of aggregation and fusion methods in the main frameworks: probability theory, evidence theory, fuzzy set and possibility theory. The latter is more developed because it allows to manage both imprecise and uncertain knowledge. Applications to decision-making, image processing, control and classification are described.

It is known that many control processes are characterized by both quantitative and qualitative complexity. Tbe quantitative complexity is usually expressed in a large number of state variables, respectively high dimensional mathematical model. Tbe qualitative complexity is usually associated with uncertain behaviour, respectively approximately known mathematical model. If the above two aspects of complexity are considered separately, the corresponding control problem can be easily solved. On one hand, large scale systems theory has existed for more than 20 years and has proved its capabilities in solving high dimensional control problems on the basis of decomposition, hierarchy, decentralization and multilayers. On the other hand, the fuzzy linguistic approach is almost at the same age and has shown its advantages in solving approximately formulated control problems on the basis of linguistic reasoning and logical inference. However, if both aspects of complexity are considered together, the corresponding control problem becomes non-trivial and does not have an easy solution. Modem control theory and practice have reacted accordingly to the above mentioned new cballenges of tbe day by utilizing the latest achievements in computer technology and artificial intelligence distributed computation and intelligent operation. In this respect, a new field has emerged in the last decade, called " Distributed intelligent control systems" . However, the majority of the familiar works in this field are still either on an empirical or on a conceptual level and this is a significant drawback.

"Statistical Modeling, Analysis and Management of Fuzzy Data," or SMFD for short, is an important contribution to a better understanding of a basic issue -an issue which has been controversial, and still is though to a lesser degree. In substance, the issue is: are fuzziness and randomness distinct or coextensive facets of uncertainty? Are the theories of fuzziness and random ness competitive or complementary? In SMFD, these and related issues are addressed with rigor, authority and insight by prominent contributors drawn, in the main, from probability theory, fuzzy set theory and data analysis com munities. First, a historical perspective. The almost simultaneous births -close to half a century ago-of statistically-based information theory and cybernetics were two major events which marked the beginning of the steep ascent of probability theory and statistics in visibility, influence and importance. I was a student when information theory and cybernetics were born, and what is etched in my memory are the fascinating lectures by Shannon and Wiener in which they sketched their visions of the coming era of machine intelligence and automation of reasoning and decision processes. What I heard in those lectures inspired one of my first papers (1950) "An Extension of Wiener's Theory of Prediction," and led to my life-long interest in probability theory and its applications to information processing, decision analysis and control."

Fuzzy logic in narrow sense is a promising new chapter of formal logic whose basic ideas were formulated by Lotfi Zadeh (see Zadeh 1975]a). The aim of this theory is to formalize the "approximate reasoning" we use in everyday life, the object of investigation being the human aptitude to manage vague properties (as, for example, "beautiful," "small," "plausible," "believable," etc. ) that by their own nature can be satisfied to a degree different from 0 (false) and I (true). It is worth noting that the traditional deductive framework in many-valued logic is different from the one adopted in this book for fuzzy logic: in the former logics one always uses a "crisp" deduction apparatus, producing crisp sets of formulas, the formulas that are considered logically valid. By contrast, fuzzy logical deductive machinery is devised to produce a fuzzy set of formulas (the theorems) from a fuzzy set of formulas (the hypotheses). Approximate reasoning has generated a very interesting literature in recent years. However, in spite of several basic results, in our opinion, we are still far from a satisfactory setting of this very hard and mysterious subject. The aim of this book is to furnish some theoretical devices and to sketch a general framework for fuzzy logic. This is also in accordance with the non Fregean attitude of the book."

This book presents new approaches to constructing fuzzy models for model-based control. Simulated examples and real-world applications from chemical and process engineering illustrate the main methods and techniques. Supporting MATLAB and Simulink files create a computational platform for exploration of the concepts and algorithms.

Information is precious. It reduces our uncertainty in making decisions. Knowledge about the outcome of an uncertain event gives the possessor an advantage. It changes the course of lives, nations, and history itself. Information is the food of Maxwell's demon. His power comes from know ing which particles are hot and which particles are cold. His existence was paradoxical to classical physics and only the realization that information too was a source of power led to his taming. Information has recently become a commodity, traded and sold like or ange juice or hog bellies. Colleges give degrees in information science and information management. Technology of the computer age has provided access to information in overwhelming quantity. Information has become something worth studying in its own right. The purpose of this volume is to introduce key developments and results in the area of generalized information theory, a theory that deals with uncertainty-based information within mathematical frameworks that are broader than classical set theory and probability theory. The volume is organized as follows."

The advent of the computer age has set in motion a profound shift in our perception of science -its structure, its aims and its evolution. Traditionally, the principal domains of science were, and are, considered to be mathe matics, physics, chemistry, biology, astronomy and related disciplines. But today, and to an increasing extent, scientific progress is being driven by a quest for machine intelligence - for systems which possess a high MIQ (Machine IQ) and can perform a wide variety of physical and mental tasks with minimal human intervention. The role model for intelligent systems is the human mind. The influ ence of the human mind as a role model is clearly visible in the methodolo gies which have emerged, mainly during the past two decades, for the con ception, design and utilization of intelligent systems. At the center of these methodologies are fuzzy logic (FL); neurocomputing (NC); evolutionary computing (EC); probabilistic computing (PC); chaotic computing (CC); and machine learning (ML). Collectively, these methodologies constitute what is called soft computing (SC). In this perspective, soft computing is basically a coalition of methodologies which collectively provide a body of concepts and techniques for automation of reasoning and decision-making in an environment of imprecision, uncertainty and partial truth."

Fuzzy rule systems have found a wide range of applications in many fields of science and technology. Traditionally, fuzzy rules are generated from human expert knowledge or human heuristics for relatively simple systems. In the last few years, data-driven fuzzy rule generation has been very active. Compared to heuristic fuzzy rules, fuzzy rules generated from data are able to extract more profound knowledge for more complex systems. This book presents a number of approaches to the generation of fuzzy rules from data, ranging from the direct fuzzy inference based to neural net works and evolutionary algorithms based fuzzy rule generation. Besides the approximation accuracy, special attention has been paid to the interpretabil ity of the extracted fuzzy rules. In other words, the fuzzy rules generated from data are supposed to be as comprehensible to human beings as those generated from human heuristics. To this end, many aspects of interpretabil ity of fuzzy systems have been discussed, which must be taken into account in the data-driven fuzzy rule generation. In this way, fuzzy rules generated from data are intelligible to human users and therefore, knowledge about unknown systems can be extracted."

As of today, Evolutionary Computing and Fuzzy Set Computing are two mature, wen -developed, and higbly advanced technologies of information processing. Bach of them has its own clearly defined research agenda, specific goals to be achieved, and a wen setUed algorithmic environment. Concisely speaking, Evolutionary Computing (EC) is aimed at a coherent population -oriented methodology of structural and parametric optimization of a diversity of systems. In addition to this broad spectrum of such optimization applications, this paradigm otTers an important ability to cope with realistic goals and design objectives reflected in the form of relevant fitness functions. The GA search (which is often regarded as a dominant domain among other techniques of EC such as evolutionary strategies, genetic programming or evolutionary programming) delivers a great deal of efficiency helping navigate through large search spaces. The main thrust of fuzzy sets is in representing and managing nonnumeric (linguistic) information. The key notion (whose conceptual as weH as algorithmic importance has started to increase in the recent years) is that of information granularity. It somewhat concurs with the principle of incompatibility coined by L. A. Zadeh. Fuzzy sets form a vehic1e helpful in expressing a granular character of information to be captured. Once quantified via fuzzy sets or fuzzy relations, the domain knowledge could be used efficiently very often reducing a heavy computation burden when analyzing and optimizing complex systems.

This introduction to fuzzy set theory and its multitude of applications seeks to balance the character of the book with the dynamic nature of the research. This edition includes new chapters on possibility theory, fuzzy logic and approximate reasoning, expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. Existing material has been updated, and extended exercises are included.

This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets."

The past fifteen years has witnessed an explosive growth in the fundamental research and applications of artificial neural networks (ANNs) and fuzzy logic (FL). The main impetus behind this growth has been the ability of such methods to offer solutions not amenable to conventional techniques, particularly in application domains involving pattern recognition, prediction and control. Although the origins of ANNs and FL may be traced back to the 1940s and 1960s, respectively, the most rapid progress has only been achieved in the last fifteen years. This has been due to significant theoretical advances in our understanding of ANNs and FL, complemented by major technological developments in high-speed computing. In geophysics, ANNs and FL have enjoyed significant success and are now employed routinely in the following areas (amongst others): 1. Exploration Seismology. (a) Seismic data processing (trace editing; first break picking; deconvolution and multiple suppression; wavelet estimation; velocity analysis; noise identification/reduction; statics analysis; dataset matching/prediction, attenuation), (b) AVO analysis, (c) Chimneys, (d) Compression I dimensionality reduction, (e) Shear-wave analysis, (f) Interpretation (event tracking; lithology prediction and well-log analysis; prospect appraisal; hydrocarbon prediction; inversion; reservoir characterisation; quality assessment; tomography). 2. Earthquake Seismology and Subterranean Nuclear Explosions. 3. Mineral Exploration. 4. Electromagnetic I Potential Field Exploration. (a) Electromagnetic methods, (b) Potential field methods, (c) Ground penetrating radar, (d) Remote sensing, (e) inversion.

Introduction to Fuzzy Reliability treats fuzzy methodology in hardware reliability and software reliability in a relatively systematic manner. The contents of this book are organized as follows. Chapter 1 places reliability engineering in the scope of a broader area, i.e. system failure engineering. Readers will find that although this book is confined to hardware and software reliability, it may be useful for other aspects of system failure engineering, like maintenance and quality control. Chapter 2 contains the elementary knowledge of fuzzy sets and possibility spaces which are required reading for the rest of this book. This chapter is included for the overall completeness of the book, but a few points (e.g. definition of conditional possibility and existence theorem of possibility space) may be new. Chapter 3 discusses how to calculate probist system reliability when the component reliabilities are represented by fuzzy numbers, and how to analyze fault trees when probabilities of basic events are fuzzy. Chapter 4 presents the basic theory of profust reliability, whereas Chapter 5 analyzes the profust reliability behavior of a number of engineering systems. Chapters 6 and 7 are devoted to probist reliability theory from two different perspectives. Chapter 8 discusses how to model software reliability behavior by using fuzzy methodology. Chapter 9 includes a number of mathematical problems which are raised by applications of fuzzy methodology in hardware and software reliability, but may be important for fuzzy set and possibility theories.

This book provides a comprehensive, up-to-date account on recent applications of fuzzy sets and possibility theory in reliability and safety analysis. Various aspects of system's reliability, quality control, reliability and safety of man-machine systems fault analysis, risk assessment and analysis, structural, seismic, safety, etc. are discussed. The book provides new tools for handling non-probabilistic aspects of uncertainty in these problems. It is the first in this field in the world literature.

Fundamentals of Fuzzy Sets covers the basic elements of fuzzy set theory. Its four-part organization provides easy referencing of recent as well as older results in the field. The first part discusses the historical emergence of fuzzy sets, and delves into fuzzy set connectives, and the representation and measurement of membership functions. The second part covers fuzzy relations, including orderings, similarity, and relational equations. The third part, devoted to uncertainty modelling, introduces possibility theory, contrasting and relating it with probabilities, and reviews information measures of specificity and fuzziness. The last part concerns fuzzy sets on the real line - computation with fuzzy intervals, metric topology of fuzzy numbers, and the calculus of fuzzy-valued functions. Each chapter is written by one or more recognized specialists and offers a tutorial introduction to the topics, together with an extensive bibliography.

Although the use of fuzzy control methods has grown nearly to the level of classical control, the true understanding of fuzzy control lags seriously behind. Moreover, most engineers are well versed in either traditional control or in fuzzy control-rarely both. Each has applications for which it is better suited, but without a good understanding of both, engineers cannot make a sound determination of which technique to use for a given situation.

A First Course in Fuzzy and Neural Control is designed to build the foundation needed to make those decisions. It begins with an introduction to standard control theory, then makes a smooth transition to complex problems that require innovative fuzzy, neural, and fuzzy-neural techniques. For each method, the authors clearly answer the questions: What is this new control method? Why is it needed? How is it implemented? Real-world examples, exercises, and ideas for student projects reinforce the concepts presented.

Developed from lecture notes for a highly successful course titled The Fundamentals of Soft Computing, the text is written in the same reader-friendly style as the authors' popular A First Course in Fuzzy Logic text. A First Course in Fuzzy and Neural Control requires only a basic background in mathematics and engineering and does not overwhelm students with unnecessary material but serves to motivate them toward more advanced studies.

Aggregation plays a central role in many of the technological tasks we are faced with. The importance of this process will become even greater as we move more and more toward becoming an information-cent.ered society, us is happening with the rapid growth of the Internet and the World Wirle Weh. Here we shall be faced with many issues related to the fusion of information. One very pressing issue here is the development of mechanisms to help search for information, a problem that clearly has a strong aggregation-related component. More generally, in order to model the sophisticated ways in which human beings process information, as well as going beyond the human capa bilities, we need provide a basket of aggregation tools. The centrality of aggregation in human thought can be be very clearly seen by looking at neural networks, a technology motivated by modeling the human brain. One can see that the basic operations involved in these networks are learning and aggregation. The Ordered Weighted Averaging (OWA) operators provide a parameter ized family of aggregation operators which include many of the well-known operators such as the maximum, minimum and the simple average."