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This book presents an extension of fuzzy set theory allowing for multi-polar information, discussing its impact on the theoretical and practical development of multi-criteria decision making. It reports on set of hybrid models developed by the authors, and show how they can be adapted, case by case, to the lack of certainty under a variety of criteria. Among them, hybrid models combining m-polar fuzzy sets with rough, soft and 2-tuple linguistic sets, and m-polar hesitant fuzzy sets and hesitant m-polar fuzzy are presented, together with some significant applications. In turn, outranking decision-making techniques such as m-polar fuzzy ELECTRE I, II, III and IV methods, as well as m-polar fuzzy PROMETHEE I and II methods, are developed. The efficiency of these decision-making procedures, as well as other possible extensions studied by the authors, is shown in some real-world applications. Overall, this book offers a guide on methodologies to deal with the multi-polarity and fuzziness of the real-world problems, simultaneously. By including algorithms and computer programming codes, it provides a practice-oriented reference guide to both researchers and professionals working at the interface between computational intelligence and decision making.

This book provides an examination of major problems facing the world using mathematics of uncertainty. These problems include climate change, coronavirus pandemic, human tracking, biodiversity, and other grand challenges. Mathematics of uncertainty is used in a modern more general sense than traditional mathematics. Since accurate data is impossible to obtain concerning human tracking and other global problems, mathematics of uncertainty is an ideal discipline to study these problems. The authors place several scientific studies into different mathematical settings such as nonstandard analysis and soft logic. Fuzzy differentiation is used to model the spread of diseases such as the coronavirus. The book uses fuzzy graph theory to examine the problems of human tracking and illegal immigration. The book is an excellent reference source for advanced under-graduate and graduate students in mathematics and the social sciences as well as for researchers and teachers.

This book presents the theory and methods of flexible and generalized uncertainty optimization. Particularly, it describes the theory of generalized uncertainty in the context of optimization modeling. The book starts with an overview of flexible and generalized uncertainty optimization. It covers uncertainties that are both associated with lack of information and are more general than stochastic theory, where well-defined distributions are assumed. Starting from families of distributions that are enclosed by upper and lower functions, the book presents construction methods for obtaining flexible and generalized uncertainty input data that can be used in a flexible and generalized uncertainty optimization model. It then describes the development of the associated optimization model in detail. Written for graduate students and professionals in the broad field of optimization and operations research, this second edition has been revised and extended to include more worked examples and a section on interval multi-objective mini-max regret theory along with its solution method.

Covering a wide range of notions concerning hesitant fuzzy set and its extensions, this book provides a comprehensive reference to the topic. In the case where different sources of vagueness appear simultaneously, the concept of fuzzy set is not able to properly model the uncertainty, imprecise and vague information. In order to overcome such a limitation, different types of fuzzy extension have been introduced so far. Among them, hesitant fuzzy set was first introduced in 2010, and the existing extensions of hesitant fuzzy set have been encountering an increasing interest and attracting more and more attentions up to now. It is not an exaggeration to say that the recent decade has seen the blossoming of a larger set of techniques and theoretical outcomes for hesitant fuzzy set together with its extensions as well as applications.As the research has moved beyond its infancy, and now it is entering a maturing phase with increased numbers and types of extensions, this book aims to give a comprehensive review of such researches. Presenting the review of many and important types of hesitant fuzzy extensions, and including references to a large number of related publications, this book will serve as a useful reference book for researchers in this field.

This book introduces readers to the fundamentals of transportation problems under the fuzzy environment and its extensions. It also discusses the limitations and drawbacks of (1) recently proposed aggregation operators under the fuzzy environment and its various extensions; (2) recently proposed methods for solving transportation problems under the fuzzy environment; and (3) recently proposed methods for solving transportation problems under the intuitionistic fuzzy environment. In turn, the book proposes simplified methods to overcome these limitations.

This book introduces the basic concepts of fuzzy collaborative forecasting and clustering, including its methodology, system architecture, and applications. It demonstrates how dealing with disparate data sources is becoming more and more popular due to the increasing spread of internet applications. The book proposes the concepts of collaborative computing intelligence and collaborative fuzzy modeling, and establishes several so-called fuzzy collaborative systems. It shows how technical constraints, security issues, and privacy considerations often limit access to some sources. This book is a valuable source of information for postgraduates, researchers and fuzzy control system developers, as it presents a very effective fuzzy approach that can deal with disparate data sources, big data, and multiple expert decision making.

This book chiefly presents a novel approach referred to as backward fuzzy rule interpolation and extrapolation (BFRI). BFRI allows observations that directly relate to the conclusion to be inferred or interpolated from other antecedents and conclusions. Based on the scale and move transformation interpolation, this approach supports both interpolation and extrapolation, which involve multiple hierarchical intertwined fuzzy rules, each with multiple antecedents. As such, it offers a means of broadening the applications of fuzzy rule interpolation and fuzzy inference. The book deals with the general situation, in which there may be more than one antecedent value missing for a given problem. Two techniques, termed the parametric approach and feedback approach, are proposed in an attempt to perform backward interpolation with multiple missing antecedent values. In addition, to further enhance the versatility and potential of BFRI, the backward fuzzy interpolation method isextended to support α-cut based interpolation by employing a fuzzy interpolation mechanism for multi-dimensional input spaces (IMUL). Finally, from an integrated application analysis perspective, experimental studies based upon a real-world scenario of terrorism risk assessment are provided in order to demonstrate the potential and efficacy of the hierarchical fuzzy rule interpolation methodology.

The use of fuzzy logic has become prominent in a variety of fields and applications. By implementing these logic sets, problems and uncertainties are more effectively resolved. Emerging Research on Applied Fuzzy Sets and Intuitionistic Fuzzy Matrices is a pivotal reference source for the latest scholarly perspectives on the interdisciplinary use of fuzzy logic theory, focusing on the application of sets and matrices. Highlighting theoretical framework and empirical research findings, this book is ideally designed for academics, practitioners, upper-level students, and professionals interested in an innovative overview of fuzzy logic sets and matrices.

The book showcases cutting-edge concepts and methods, and presents the principle of imprecise-information processing. It also proposes a new theory and technology for imprecise-information processing that differs from fuzzy technology, thus providing a platform for related applications and laying the theoretical basis for further research.Imprecise-information processing – a type of processing based on flexible linguistic values and quantifiable rigid linguistic values – is an important component of intelligence science and technology. This book offers an easy-to-understand overview of the basic principles and methods of imprecise-information processing, allowing readers to develop related applications or pursue further research.

In this book, a series of granular algorithms are proposed. A nature inspired granular algorithm based on Newtonian gravitational forces is proposed. A series of methods for the formation of higher-type information granules represented by Interval Type-2 Fuzzy Sets are also shown, via multiple approaches, such as Coefficient of Variation, principle of justifiable granularity, uncertainty-based information concept, and numerical evidence based. And a fuzzy granular application comparison is given as to demonstrate the differences in how uncertainty affects the performance of fuzzy information granules.

Frontiers of Higher Order Fuzzy Sets, provides a unified representation theorem for higher order fuzzy sets. The book elaborates on the concept of gradual elements and their integration with the higher order fuzzy sets. This book also is devoted to the introduction of new frameworks based on general T2FSs, IT2FSs, Gradual elements, Shadowed sets and rough sets. Such new frameworks will provide more capable frameworks for real applications. Applications of higher order fuzzy sets in various fields will be discussed. In particular, the properties and characteristics of the new proposed frameworks would be studied. Such frameworks that are the result of the integration of general T2FSs, IT2FSs, gradual elements, shadowed sets and rough sets will be shown to be suitable to be applied in the fields of bioinformatics, business, management, ambient intelligence, medicine, cloud computing and smart grids.

Real-world information is imperfect and is usually described in natural language (NL). Moreover, this information is often partially reliable and a degree of reliability is also expressed in NL. In view of this, the concept of a Z-number is a more adequate concept for the description of real-world information. The main critical problem that naturally arises in processing Z-numbers-based information is the computation with Z-numbers. Nowadays, there is no arithmetic of Z-numbers suggested in existing literature.This book is the first to present a comprehensive and self-contained theory of Z-arithmetic and its applications. Many of the concepts and techniques described in the book, with carefully worked-out examples, are original and appear in the literature for the first time.The book will be helpful for professionals, academics, managers and graduate students in fuzzy logic, decision sciences, artificial intelligence, mathematical economics, and computational economics.

This short book provides a unified view of the history and theory of random sets and fuzzy random variables, with special emphasis on its use for representing higher-order non-statistical uncertainty about statistical experiments. The authors lay bare the existence of two streams of works using the same mathematical ground, but differing form their use of sets, according to whether they represent objects of interest naturally taking the form of sets, or imprecise knowledge about such objects. Random (fuzzy) sets can be used in many fields ranging from mathematical morphology, economics, artificial intelligence, information processing and statistics per se, especially in areas where the outcomes of random experiments cannot be observed with full precision. This book also emphasizes the link between random sets and fuzzy sets with some techniques related to the theory of imprecise probabilities.This small book is intended for graduate and doctoral students in mathematics or engineering, but also provides an introduction for other researchers interested in this area. It is written from a theoretical perspective. However, rather than offering a comprehensive formal view of random (fuzzy) sets in this context, it aims to provide a discussion of the meaning of the proposed formal constructions based on many concrete examples and exercises. This book should enable the reader to understand the usefulness of representing and reasoning with incomplete information in statistical tasks. Each chapter ends with a list of exercises.

This book provides the readers with a thorough and systematic introduction to hesitant fuzzy theory. It presents the most recent research results and advanced methods in the field. These includes: hesitant fuzzy aggregation techniques, hesitant fuzzy preference relations, hesitant fuzzy measures, hesitant fuzzy clustering algorithms and hesitant fuzzy multi-attribute decision making methods. Since its introduction by Torra and Narukawa in 2009, hesitant fuzzy sets have become more and more popular and have been used for a wide range of applications, from decision-making problems to cluster analysis, from medical diagnosis to personnel appraisal and information retrieval. This book offers a comprehensive report on the state-of-the-art in hesitant fuzzy sets theory and applications, aiming at becoming a reference guide for both researchers and practitioners in the area of fuzzy mathematics and other applied research fields (e.g. operations research, information science, management science and engineering) characterized by uncertain ('hesitant') information. Because of its clarity and self contained explanations, the book can also be adopted as a textbook from graduate and advanced undergraduate students.

Since its inception 20 years ago the theory of fuzzy sets has advanced in a variety of ways and in many disciplines. Applications of this theory can be found in artificial intelligence, computer science, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, robotics and others. Theoretical advances, too, have been made in many directions, and a gap has arisen between advanced theoretical topics and applications, which often use the theory at a rather elementary level. The primary goal of this book is to close this gap - to provide a textbook for courses in fuzzy set theory and a book that can be used as an introduction. This revised book updates the research agenda, with the chapters of possibility theory, fuzzy logic and approximate reasoning, expert systems and control, decision making and fuzzy set models in operations research being restructured and rewritten. Exercises have been added to almost all chapters and a teacher's manual is available upon request.

Intelligent decision support is based on human knowledge related to a specific part of a real or abstract world. When the knowledge is gained by experience, it is induced from empirical data. The data structure, called an information system, is a record of objects described by a set of attributes. Knowledge is understood here as an ability to classify objects. Objects being in the same class are indiscernible by means of attributes and form elementary building blocks (granules, atoms). In particular, the granularity of knowledge causes that some notions cannot be expressed precisely within available knowledge and can be defined only vaguely. In the rough sets theory created by Z. Pawlak each imprecise concept is replaced by a pair of precise concepts called its lower and upper approximation. These approximations are fundamental tools and reasoning about knowledge. The rough sets philosophy turned out to be a very effective, new tool with many successful real-life applications to its credit. It is worthwhile stressing that no auxiliary assumptions are needed about data, like probability or membership function values, which is its great advantage. The present book reveals a wide spectrum of applications of the rough set concept, giving the reader the flavor of, and insight into, the methodology of the newly developed disciplines. Although the book emphasizes applications, comparison with other related methods and further developments receive due attention.

Fuzzy Set Theory - And Its Applications, Third Edition is a textbook for courses in fuzzy set theory. It can also be used as an introduction to the subject. The character of a textbook is balanced with the dynamic nature of the research in the field by including many useful references to develop a deeper understanding among interested readers. The book updates the research agenda (which has witnessed profound and startling advances since its inception some 30 years ago) with 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. All chapters have been updated. Exercises are included.

First of all, I would like to congratulate Gabriella Pasi and Gloria Bordogna for the work they accomplished in preparing this new book in the series'Study in Fuzziness and Soft Computing'.'Recent Issues on the Management of Fuzziness in Databases'is undoubtedly a token of their long-lasting and active involvement in the area of Fuzzy Information Retrieval and Fuzzy Database Systems. This book is really welcome in the area of fuzzy databases where they are not numerous although the first works at the crossroads of fuzzy sets and databases were initiated about twenty years ago by L. Zadeh. Only five books have been published since 1995, when the first volume dedicated to fuzzy databases published in the series'Study in Fuzziness and Soft Computing'edited by J. Kacprzyk and myself appeared. Going beyond books strictly speaking, let us also mention the existence of review papers that are part of a couple of handbooks related to fuzzy sets published since 1998. The area known as fuzzy databases covers a bunch of topics among which: -flexible queries addressed to regular databases, -the extension of the notion of a functional dependency, -data mining and fuzzy summarization, -querying databases containing imperfect attribute values represented thanks to possibility distributions.

Decision makers in managerial and public organizations often encounter de­ cision problems under conflict or competition, because they select strategies independently or by mutual agreement and therefore their payoffs are then affected by the strategies of the other decision makers. Their interests do not always coincide and are at times even completely opposed. Competition or partial cooperation among decision makers should be considered as an essen­ tial part of the problem when we deal with the decision making problems in organizations which consist of decision makers with conflicting interests. Game theory has been dealing with such problems and its techniques have been used as powerful analytical tools in the resolution process of the decision problems. The publication of the great work by J. von Neumann and O. Morgen­ stern in 1944 attracted attention of many people and laid the foundation of game theory. We can see remarkable advances in the field of game theory for analysis of economic situations and a number of books in the field have been published in recent years. The aim of game theory is to specify the behavior of each player so as to optimize the interests of the player. It then recommends a set of solutions as strategies so that the actions chosen by each decision maker (player) lead to an outcome most profitable for himself or her­ self.

This volume presents the results of approximately 15 years of work from researchers around the world on the use of fuzzy set theory to represent imprecision in databases. The maturity of the research in the discipline and the recent developments in commercial/industrial fuzzy databases provided an opportunity to produce this survey. In this introduction we will describe briefly how fuzzy databases fit into the overall design of database systems and then overview the organization of the text. FUZZY DATABASE LANDSCAPE The last five years have been witness to a revolution in the database research community. The dominant data models have changed and the consensus on what constitutes worthwhile research is in flux. Also, at this time, it is possible to gain a perspective on what has been accomplished in the area of fuzzy databases. Therefore, now is an opportune time to take stock of the past and establish a framework. A framework should assist in evaluating future research through a better understanding of the different aspects of imprecision that a database can model [ 1 l.

Fuzzy Set Theory and Advanced Mathematical Applications contains contributions by many of the leading experts in the field, including coverage of the mathematical foundations of the theory, decision making and systems science, and recent developments in fuzzy neural control. The book supplies a readable, practical toolkit with a clear introduction to fuzzy set theory and its evolution in mathematics and new results on foundations of fuzzy set theory, decision making and systems science, and fuzzy control and neural systems. Each chapter is self-contained, providing up-to-date coverage of its subject. Audience: An important reference work for university students, and researchers and engineers working in both industrial and academic settings.

This volume contains the proceedings of the Second Joint IFSA-EC and EURO-WGFS Workshop on Progress in Fuzzy Sets in Europe held on April 6 -8, 1989 in Vienna, Austria. The workshop was organized by Prof. Dr. Wolfgang H. Janko from the University of Economics in Vienna under the auspices of IFSA-EC, the European chapter of the International Fuzzy Systems Association, and EURO-WGFS, the working group on Fuzzy Sets of the Association of Eu­ ropean Operational Research Societies. The workshop gathered more than 30 participants coming from Western European countries (Austria, Bel­ gium, England, Germany, Finland, France, Hungary, Italy, Scotland and Spain) Eastern European countries (Bulgaria, the German Federal Repu­ blic, Hungary and Poland) and non-European countries such as China and Japan. The 15 selected and refereed papers included in the volume are in prin­ ciple the author's own versions, with limited editorial changes and small corrections. They are arranged in alphabetical order. I wish to thank all the contributors for their valuable papers and an outstan­ ding cooperation in the editorial project. I also would like to express my sincere thanks to Professor Dr. H. J. Zimmermann for the cooperation in the refereeing procedure.

In the two decades since its inception by L. Zadeh, the theory of fuzzy sets has matured into a wide-ranging collection of concepts, models, and tech­ niques for dealing with complex phenomena which do not lend themselves to analysis by classical methods based on probability theory and bivalent logic. Nevertheless, a question which is frequently raised by the skeptics is: Are there, in fact, any significant problem areas in which the use of the theory of fuzzy sets leads to results which could not be obtained by classical methods? The approximately 5000 publications in this area, which are scattered over many areas such as artificial intelligence, computer science, control engineering, decision making, logic, operations research, pattern recognition, robotics and others, provide an affirmative answer to this question. In spite of the large number of publications, good and comprehensive textbooks which could facilitate the access of newcomers to this area and support teaching were missing until recently. To help to close this gap and to provide a textbook for courses in fuzzy set theory which can also be used as an introduction to this field, the first volume ofthis book was published in 1985 [Zimmermann 1985 b]. This volume tried to cover fuzzy set theory and its applications as extensively as possible. Applications could, therefore, only be described to a limited extent and not very detailed.

Since their inception, fuzzy sets and fuzzy logic became popular. The reason is that the very idea of fuzzy sets and fuzzy logic attacks an old tradition in science, namely bivalent (black-or-white, all-or-none) judg­ ment and reasoning and the thus resulting approach to formation of scientific theories and models of reality. The idea of fuzzy logic, briefly speaking, is just the opposite of this tradition: instead of full truth and falsity, our judgment and reasoning also involve intermediate truth values. Application of this idea to various fields has become known under the term fuzzy approach (or graded truth approach). Both prac­ tice (many successful engineering applications) and theory (interesting nontrivial contributions and broad interest of mathematicians, logicians, and engineers) have proven the usefulness of fuzzy approach. One of the most successful areas of fuzzy methods is the application of fuzzy relational modeling. Fuzzy relations represent formal means for modeling of rather nontrivial phenomena (reasoning, decision, control, knowledge extraction, systems analysis and design, etc.) in the pres­ ence of a particular kind of indeterminacy called vagueness. Models and methods based on fuzzy relations are often described by logical formulas (or by natural language statements that can be translated into logical formulas). Therefore, in order to approach these models and methods in an appropriate formal way, it is desirable to have a general theory of fuzzy relational systems with basic connections to (formal) language which enables us to describe relationships in these systems.

The book deals with some of the fundamental issues of risk assessment in grid computing environments. The book describes the development of a hybrid probabilistic and possibilistic model for assessing the success of a computing task in a grid environment

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.

In the last 25 years, the fuzzy set theory has been applied in many disciplines such as operations research, management science, control theory, artificial intelligence/expert system, etc. In this volume, methods and applications of crisp, fuzzy and possibilistic multiple objective decision making are first systematically and thoroughly reviewed and classified. This state-of-the-art survey provides readers with a capsule look into the existing methods, and their characteristics and applicability to analysis of fuzzy and possibilistic programming problems. To realize practical fuzzy modelling, it presents solutions for real-world problems including production/manufacturing, location, logistics, environment management, banking/finance, personnel, marketing, accounting, agriculture economics and data analysis. This book is a guided tour through the literature in the rapidly growing fields of operations research and decision making and includes the most up-to-date bibliographical listing of literature on the topic.

The purpose of this book is to present a methodology for designing and tuning fuzzy expert systems in order to identify nonlinear objects; that is, to build input-output models using expert and experimental information. The results of these identifications are used for direct and inverse fuzzy evidence in forecasting and diagnosis problem solving.The book is organised as follows: Chapter 1 presents the basic knowledge about fuzzy sets, genetic algorithms and neural nets necessary for a clear understanding of the rest of this book. Chapter 2 analyzes direct fuzzy inference based on fuzzy if-then rules. Chapter 3 is devoted to the tuning of fuzzy rules for direct inference using genetic algorithms and neural nets. Chapter 4 presents models and algorithms for extracting fuzzy rules from experimental data. Chapter 5 describes a method for solving fuzzy logic equations necessary for the inverse fuzzy inference in diagnostic systems. Chapters 6 and 7 are devoted to inverse fuzzy inference based on fuzzy relations and fuzzy rules. Chapter 8 presents a method for extracting fuzzy relations from data. All the algorithms presented in Chapters 2-8 are validated by computer experiments and illustrated by solving medical and technical forecasting and diagnosis problems. Finally, Chapter 9 includes applications of the proposed methodology in dynamic and inventory control systems, prediction of results of football games, decision making in road accident investigations, project management and reliability analysis.

Table of contents: Stochastic methods in nonlinear structural dynamics.- Stochastic models of uncertainties in computational structural dynamics and structural acoustics.- The tale of stochastic linearization techniques: over half a century of progress.- Comprehensive modeling of uncertain systems using fuzzy set theory.- Bounding uncertainty in civil engineering: theoretical background and applications.- Combined methods in nondeterministic mechanics.In this book the current state of the art of nondeterministic mechanics in its various forms is presented. The topics range from stochastic problems to fuzzy sets; from linear to nonlinear problems; from specific methodologies to combinations of various techniques; from theoretical considerations to practical applications. It is specially designed to illuminate the various aspects of the three methodologies (probabilistic or stochastic modelling, fuzzy sets based analysis, antioptimization of structures) to deal with various uncertainties and deepen the discussion of their pros and cons.

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.

An Introduction to Fuzzy Logic Applications in Intelligent Systems consists of a collection of chapters written by leading experts in the field of fuzzy sets. Each chapter addresses an area where fuzzy sets have been applied to situations broadly related to intelligent systems. The volume provides an introduction to and an overview of recent applications of fuzzy sets to various areas of intelligent systems. Its purpose is to provide information and easy access for people new to the field. The book also serves as an excellent reference for researchers in the field and those working in the specifics of systems development. People in computer science, especially those in artificial intelligence, knowledge-based systems, and intelligent systems will find this to be a valuable sourcebook. Engineers, particularly control engineers, will also have a strong interest in this book. Finally, the book will be of interest to researchers working in decision support systems, operations research, decision theory, management science and applied mathematics. An Introduction to Fuzzy Logic Applications in Intelligent Systems may also be used as an introductory text and, as such, it is tutorial in nature.

The primary aim of this monograph is to provide a formal framework for the representation and management of uncertainty and vagueness in the field of artificial intelligence. It puts particular emphasis on a thorough analysis of these phenomena and on the development of sound mathematical modeling approaches. Beyond this theoretical basis the scope of the book includes also implementational aspects and a valuation of existing models and systems. The fundamental ambition of this book is to show that vagueness and un­ certainty can be handled adequately by using measure-theoretic methods. The presentation of applicable knowledge representation formalisms and reasoning algorithms substantiates the claim that efficiency requirements do not necessar­ ily require renunciation of an uncompromising mathematical modeling. These results are used to evaluate systems based on probabilistic methods as well as on non-standard concepts such as certainty factors, fuzzy sets or belief functions. The book is intended to be self-contained and addresses researchers and practioneers in the field of knowledge based systems. It is in particular suit­ able as a textbook for graduate-level students in AI, operations research and applied probability. A solid mathematical background is necessary for reading this book. Essential parts of the material have been the subject of courses given by the first author for students of computer science and mathematics held since 1984 at the University in Braunschweig.

On the basis of fuzzy sets and some of their relevant generalizations, this book systematically presents the fundamental principles and applications of group decision making under different scenarios of preference relations. By using intuitionistic knowledge as the field of discourse, this work investigates by utilizing innovative research means the fundamental principles and methods of group decision making with various different intuitionistic preferences: Mathematical reasoning is employed to study the consistency of group decision making; Methods of fusing information are applied to look at the aggregation of multiple preferences; Techniques of soft computing and optimization are utilized to search for satisfactory decision alternatives. Each chapter follows the following structurally clear format of presentation: literature review, development of basic theory, verification and reasoning of principles, construction of models and computational schemes, and numerical examples, which cover such areas as technology, enterprise competitiveness, selection of airlines, experts decision making in weather-sensitive enterprises, etc. In terms of theoretical principles, this book can be used as a reference for researchers in the areas of management science, information science, systems engineering, operations research, and other relevant fields. It can also be employed as textbook for upper level undergraduate students and graduate students. In terms of applications, this book will be a good companion for all those decision makers in government, business, and technology areas.

This book presents a mathematically-based introduction into the fascinating topic of Fuzzy Sets and Fuzzy Logic and might be used as textbook at both undergraduate and graduate levels and also as reference guide for mathematician, scientists or engineers who would like to get an insight into Fuzzy Logic. Fuzzy Sets have been introduced by Lotfi Zadeh in 1965 and since then, they have been used in many applications. As a consequence, there is a vast literature on the practical applications of fuzzy sets, while theory has a more modest coverage. The main purpose of the present book is to reduce this gap by providing a theoretical introduction into Fuzzy Sets based on Mathematical Analysis and Approximation Theory. Well-known applications, as for example fuzzy control, are also discussed in this book and placed on new ground, a theoretical foundation. Moreover, a few advanced chapters and several new results are included. These comprise, among others, a new systematic and constructive approach for fuzzy inference systems of Mamdani and Takagi-Sugeno types, that investigates their approximation capability by providing new error estimates.

This book starts with the basic concepts of fuzzy sets and progresses through a normative view on possibility distributions and OWA operators in multiple criteria decisions. Five applications (that all build on experience from solving complex real world problems) of possibility distributions to strategic decisions about closing/not closing a production plant using fuzzy real options, portfolio selection with imprecise future data, predictive probabilities and possibilities for risk assessment in grid computing, fuzzy ontologies for process industry, and design (and implementation) of mobile value services are presented and carefully discussed. It can be useful for researchers and students working in soft computing, real options, fuzzy decision making, grid computing, knowledge mobilization and mobile value services.

This book comprehensively studies fuzzy temporal and spatial information, starting from the basics on fuzzy set theory and temporal/spatial reasoning, the development of a new model to represent fuzzy temporal/spatial information, the study of efficient and complete reasoning algorithms, and their application in an information retrieval context. This useful volume presents the first approach that goes beyond merely representing information, by thoroughly addressing a variety of reasoning tasks. It also focuses on applications in the domain of information retrieval, and demonstrating the practical importance of the proposed framework.

This book is a tribute to Etienne E. Kerre on the occasion of his retirement on October 1st, 2010, after being active for 35 years in the field of fuzzy set theory. It gathers contributions from researchers that have been close to him in one way or another during his long and fruitful career. Besides a foreword by Lotfi A. Zadeh, it contains 13 chapters on both theoretical and applied topics in fuzzy set theory, divided in three parts: 1) logics and connectives, 2) data analysis, and 3) media applications. The first part deals with fuzzy logics and with operators on (extensions of) fuzzy sets. Part 2 deals with fuzzy methods in rough set theory, formal concept analysis, decision making and classification. The last part discusses the use of fuzzy methods for representing and manipulating media objects, such as images and text documents. The diversity of the topics that are covered reflect the diversity of Etienne's research interests, and indeed, the diversity of current research in the area of fuzzy set theory.

Explains for the first time how'computing with words'can aid in making subjective judgments Lotfi Zadeh, the father of fuzzy logic, coined the phrase'computing with words'(CWW) to describe a methodology in which the objects of computation are words and propositions drawn from a natural language. Perceptual Computing explains how to implement CWW to aid in the important area of making subjective judgments, using a methodology that leads to an interactive device—a'Perceptual Computer'—that propagates random and linguistic uncertainties into the subjective judgment in a way that can be modeled and observed by the judgment maker. This book focuses on the three components of a Perceptual Computer—encoder, CWW engines, and decoder—and then provides detailed applications for each. It uses interval type-2 fuzzy sets (IT2 FSs) and fuzzy logic as the mathematical vehicle for perceptual computing, because such fuzzy sets can model first-order linguistic uncertainties whereas the usual kind of fuzzy sets cannot. Drawing upon the work on subjective judgments that Jerry Mendel and his students completed over the past decade, Perceptual Computing shows readers how to: Map word-data with its inherent uncertainties into an IT2 FS that captures these uncertainties Use uncertainty measures to quantify linguistic uncertainties Compare IT2 FSs by using similarity and rank Compute the subsethood of one IT2 FS in another such set Aggregate disparate data, ranging from numbers to uniformly weighted intervals to nonuniformly weighted intervals to words Aggregate multiple-fired IF-THEN rules so that the integrity of word IT2 FS models is preserved Free MATLAB-based software is also available online so readers can apply the methodology of perceptual computing immediately, and even try to improve upon it. Perceptual Computing is an important go-to for researchers and students in the fields of artificial intelligence and fuzzy logic, as well as for operations researchers, decision makers, psychologists, computer scientists, and computational intelligence experts.

Regarding the set of all feature attributes in a given database as the universal set, this monograph discusses various nonadditive set functions that describe the interaction among the contributions from feature attributes towards a considered target attribute. Then, the relevant nonlinear integrals are investigated. These integrals can be applied as aggregation tools in information fusion and data mining, such as synthetic evaluation, nonlinear multiregressions, and nonlinear classifications. Some methods of fuzzification are also introduced for nonlinear integrals such that fuzzy data can be treated and fuzzy information is retrievable.The book is suitable as a text for graduate courses in mathematics, computer science, and information science. It is also useful to researchers in the relevant area.

The book contains ten chapters as follows, Prepare Knowledge, Regression and Self-regression Models with Fuzzy Coefficients; Regression and Self-regression Models with Fuzzy Variables, Fuzzy Input/output Model, Fuzzy Cluster Analysis and Fuzzy Recognition, Fuzzy Linear Programming, Fuzzy Geometric Programming, Fuzzy Relative Equation and Its Optimizing, Interval and Fuzzy Differential Equations and Interval and Fuzzy Functional and Their Variation. It can not only be used as teaching materials or reference books for under-graduates in higher education, master graduates and doctor graduates in the courses of applied mathematics, computer science, artificial intelligence, fuzzy information process and automation, operations research, system science and engineering, and the like, but also serves as a reference book for researchers in these fields, particularly, for researchers in soft science.

In our new century, the theory of fuzzy sets and systems is in the core of'Soft Computing'and'Computational Intelligence'and has become a normal scientific theory in the fields of exact sciences and engineering and it is well on its way to becoming normal in the soft sciences as well. This book is a collection of the views of numerous scholars in different parts of the world who are involved in various research projects concerning fuzziness in science, technology, economic systems, social sciences, logics and philosophy. This volume demonstrates that there are many different views of the theory of fuzzy sets and systems and of their interpretation and applications in diverse areas of our cultural and social life.

In conventional mathematical programming, coefficients of problems are usually determined by the experts as crisp values in terms of classical mathematical reasoning. But in reality, in an imprecise and uncertain environment, it will be utmost unrealistic to assume that the knowledge and representation of an expert can come in a precise way. The wider objective of the book is to study different real decision situations where problems are defined in inexact environment. Inexactness are mainly generated in two ways – (1) due to imprecise perception and knowledge of the human expert followed by vague representation of knowledge as a DM; (2) due to huge-ness and complexity of relations and data structure in the definition of the problem situation. We use interval numbers to specify inexact or imprecise or uncertain data. Consequently, the study of a decision problem requires answering the following initial questions: How should we compare and define preference ordering between two intervals?, interpret and deal inequality relations involving interval coefficients?, interpret and make way towards the goal of the decision problem? The present research work consists of two closely related fields: approaches towards defining a generalized preference ordering scheme for interval attributes and approaches to deal with some issues having application potential in many areas of decision making.

It is well known that “fuzziness”—informationgranulesand fuzzy sets as one of its formal manifestations— is one of important characteristics of human cognitionandcomprehensionofreality. Fuzzy phenomena existinnature and are encountered quite vividly within human society. The notion of a fuzzy set has been introduced by L. A., Zadeh in 1965 in order to formalize human concepts, in connection with the representation of human natural language and computing with words. Fuzzy sets and fuzzy logic are used for mod- ing imprecise modes of reasoning that play a pivotal role in the remarkable human abilities to make rational decisions in an environment a?ected by - certainty and imprecision. A growing number of applications of fuzzy sets originated from the “empirical-semantic” approach. From this perspective, we were focused on some practical interpretations of fuzzy sets rather than being oriented towards investigations of the underlying mathematical str- tures of fuzzy sets themselves. For instance, in the context of control theory where fuzzy sets have played an interesting and practically relevant function, the practical facet of fuzzy sets has been stressed quite signi?cantly. However, fuzzy sets can be sought as an abstract concept with all formal underpinnings stemming from this more formal perspective. In the context of applications, it is worth underlying that membership functions do not convey the same meaning at the operational level when being cast in various contexts.

Fuzzy Logic in Action: Applications in Epidemiology and Beyond, co-authored by Eduardo Massad, Neli Ortega, Laécio Barros, and Cláudio Struchiner is a remarkable achievement. The book brings a major paradigm shift to medical sciences exploring the use of fuzzy sets in epidemiology and medical diagnosis arena. The volume addresses the most significant topics in the broad areas of epidemiology, mathematical modeling and uncertainty, embodying them within the framework of fuzzy set and dynamic systems theory. Written by leading contributors to the area of epidemiology, medical informatics and mathematics, the book combines a very lucid and authoritative exposition of the fundamentals of fuzzy sets with an insightful use of the fundamentals in the area of epidemiology and diagnosis. The content is clearly illustrated by numerous illustrative examples and several real world applications. Based on their profound knowledge of epidemiology and mathematical modeling, and on their keen understanding of the role played by uncertainty and fuzzy sets, the authors provide insights into the connections between biological phenomena and dynamic systems as a mean to predict, diagnose, and prescribe actions. An example is the use of Bellman-Zadeh fuzzy decision making approach to develop a vaccination strategy to manage measles epidemics in São Paulo. The book offers a comprehensive, systematic, fully updated and self- contained treatise of fuzzy sets in epidemiology and diagnosis. Its content covers material of vital interest to students, researchers and practitioners and is suitable both as a textbook and as a reference. The authors present new results of their own in most of the chapters. In doing so, they reflect the trend to view fuzzy sets, probability theory and statistics as an association of complementary and synergetic modeling methodologies.

Many biological systems and objects are intrinsically fuzzy as their properties and behaviors contain randomness or uncertainty. In addition, it has been shown that exact or optimal methods have significant limitation in many bioinformatics problems. Fuzzy set theory and fuzzy logic are ideal to describe some biological systems/objects and provide good tools for some bioinformatics problems. This book comprehensively addresses several important bioinformatics topics using fuzzy concepts and approaches, including measurement of ontological similarity, protein structure prediction/analysis, and microarray data analysis. It also reviews other bioinformatics applications using fuzzy techniques.

There are many good AI books. Usually they consecrate at most one or two chapters to the imprecision knowledge processing. To our knowledge this is among the few books to be entirely dedicated to the treatment of knowledge imperfection when bui- ing intelligent systems. We consider that an entire book should be focused on this important aspect of knowledge processing. The expected audience for this book - cludes undergraduate students in computer science, IT&C, mathematics, business, medicine, etc., graduates, specialists and researchers in these fields. The subjects treated in the book include expert systems, knowledge representation, reasoning under knowledge Imperfection (Probability Theory, Possibility Theory, Belief Theory, and Approximate Reasoning). Most of the examples discussed in details throughout the book are from the medical domain. Each chapter ends with a set of carefully pe- gogically chosen exercises, which complete solution provided. Their understanding will trigger the comprehension of the theoretical notions, concepts and results. Chapter 1 is dedicated to the review of expert systems. Hence are briefly discussed production rules, structure of ES, reasoning in an ES, and conflict resolution. Chapter 2 treats knowledge representation. That includes the study of the differences between data, information and knowledge, logical systems with focus on predicate calculus, inference rules in classical logic, semantic nets and frames.

In trying to make a satisfactory decision when imprecise and multicriteria situations are involved, a decision maker has to use a fuzzy multicriteria decision making method.'Fuzzy Multi-Criteria Decision Making'(MCDM) presents fuzzy multiattribute and multiobjective decision-making methodologies by distinguished MCDM researchers. In summarizing the concepts and results of the most popular fuzzy multicriteria methods, using numerical examples, this work examines all the fuzzy multicriteria methods recently developed, such as fuzzy AHP, fuzzy TOPSIS, interactive fuzzy multiobjective stochastic linear programming, fuzzy multiobjective dynamic programming, grey fuzzy multiobjective optimization, fuzzy multiobjective geometric programming, and more. Each of the 22 chapters includes practical applications along with new developments/results. This book may be used as a textbook in graduate operations research, industrial engineering, and economics courses. It will also be an excellent resource, providing new suggestions and directions for further research, for computer programmers, mathematicians, and scientists in a variety of disciplines where multicriteria decision making is needed.

In the late eighties of the twentieth century I encountered a paper referring to mathematics dealing with imprecision applied to medical diagnosis. I was working in the area of medical statistics at that time, and obviously I was interested in the contents of the paper. After I had read it I became fascinated by new possibilities of medical data interpretation and processing proposed by the author. The new world of fuzziness, originated by Professor Lotfi Zadeh, seemed to open up to me and I started reading all accessible material about fuzzy set theory. Many years have passed since then. We are now living in the information so- ety and we do not experience troubles in reaching scientific material. Lately I have read many papers and books about treating medical tasks solved by using fuzzy ideas. I am still keen on tracing applications in medicine, and by myself I have been contributing to some concepts in this subject that has motivated me to p- pare my own book. The objective of writing such a book has been a little parti- lar, namely, I have intended to present the subject of fuzzy tools and techniques in medicine for eventual users. These, maybe representatives of medical or phar- cological staffs, are not expected to possess a large amount of mathematical knowledge.

This book proposes a set of models to describe fuzzy multi-objective decision making (MODM), fuzzy multi-criteria decision making (MCDM), fuzzy group decision making (GDM) and fuzzy multi-objective group decision-making problems, respectively. It also gives a set of related methods (including algorithms) to solve these problems. One distinguishing feature of this book is that it provides two decision support systems software for readers to apply these proposed methods. A set of real-world applications and some new directions in this area are then described to further instruct readers how to use these methods and software in their practice.