Studies in Fuzziness and Soft Computing 292

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Studies in Fuzziness and Soft Computing

292

Editor-in-Chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6

01-447 Warsaw Poland

E-mail: [email protected]

For further volumes:

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Maciej Wygralak

Intelligent Counting

under Information

Imprecision

Applications to Intelligent Systems

and Decision Support

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Maciej Wygralak, Assoc. Prof.

Faculty of Mathematics and Computer Science Adam Mickiewicz University

Umultowska 87 61-614 Pozna´n Poland

E-mail: [email protected]

ISSN 1434-9922 ISSN 1860-0808 (electronic) ISBN 978-3-642-34684-2 ISBN 978-3-642-34685-9 (eBook) DOI 10.1007/978-3-642-34685-9

Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013937947

c

Springer-Verlag Berlin Heidelberg 2013

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.

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Preface

This monograph begins with a presentation of the concept of and selected issues in fuzzy sets and fuzzy logic, interval-valued fuzzy sets, and I-fuzzy sets - Atanassov’s intuitionistic fuzzy sets. However, that study is only of preparatory character. The main subject of the book will be intelligent counting under imprecision of information (about the objects of counting). Why is this worth investigating and deliberating?

It seems that counting belongs to the most basic and frequent mental activities of human beings as its results are a basis for coming to a decision in a lot of situations. One should distinguish, however, between two very different cases occurring in practice. First, the objects of counting can be precisely specified and, then, the counting process collapses to the trivial task of counting in a set by means of the natural numbers. Second, those objects can be imprecisely (fuzzily) specified and just this much more advanced and sophisticated case of counting requiring intelligence will be the subject of our investigations. Speaking formally, that intelligent counting collapses to counting in a fuzzy set or - whenever imprecision is combined with incompleteness of information - to counting in an interval-valued fuzzy set or I-fuzzy set. Theoretical aspects as well as applications of intelligent counting will be discussed. Especially, we mean applications to intelligent systems and decision support. The emphasis will be on showing that the presented methods of intelligent counting and the resulting cardinalities are human-consistent, i.e. are reflections and formalizations of real, human counting methods under imprecision possibly combined with incompleteness of information. It is self-evident that our main interest will be in counting in finite fuzzy sets, finite interval-valued fuzzy sets and I-fuzzy sets. Nevertheless, for completeness, the infinite case will be concisely discussed, too.

The monograph is divided into two parts and eleven chapters. The first one is of introductory character, whereas Chapters 2-6, forming Part I, are devoted to those elements of fuzzy sets and fuzzy logic which are relevant from the viewpoint of the main aim of this book. We will present operations on and basic characteristics of fuzzy sets, negations, triangular norms and conorms, fuzzy numbers and linguistic variables, aggregation operators, fuzzy relations, and an introduction to approximate reasoning and fuzzy rule-based systems. Moreover, interval-valued fuzzy sets and I-fuzzy sets will be discussed as tools for modeling incompletely known fuzzy sets. Chapters 7-11 constitute Part II, the key part of this book, devoted to methods of intelligent counting and related cardinalities of fuzzy sets, interval-valued fuzzy sets and I-fuzzy sets. Both the scalar and fuzzy approaches to these questions will be discussed in detail, including human-consistency and giving the reader a novel and up-to-date image of the subject matter.

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viii Preface

some basic knowledge of mathematics with special reference to set theory, mathe-matical logic, analysis, and general algebra.

This monograph is intended for computer and information scientists, researchers, engineers and practitioners, applied mathematicians, and postgraduate students interested in dealing with information imprecision and incompleteness.

The most pleasant moment of each book project is to write acknowledgments. I am grateful to all who supported me. Especially, I would like to thank my wife Renata and our daughters Karolina and Agata for their continuous and reliable support, understanding and patience.

As to the financial side, this book project has been partially supported by a research grant from the National Science Centre (NCN), and this support is greatly appreciated.

Last but not least, let me thank my collaborators, Dr. Krzysztof Dyczkowski and Dr. Anna Stachowiak, for kind technical assistance and valuable suggestions.

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Part I Elements of Fuzzy Sets and Their Extensions

2 Basic Notions of the Language of Fuzzy Sets . . . . 19

2.1 What Are Fuzzy Sets? . . . 19

2.1.1 The Concept of a Fuzzy Set . . . 19

2.1.2 Examples and Interpretations . . . 21

2.1.3 Remarks on Many-Valued Roots of Fuzzy Sets . . . 25

2.2 Operations on Fuzzy Sets ! The Standard Approach . . . 28

2.2.1 Basic Definitions . . . 28

2.2.2 Properties of Operations . . . 30

2.2.3 Looking through Many-Valued Logic . . . 32

2.3 Main Characteristics of Fuzzy Sets . . . 33

2.3.1 Core, Support, t-Cuts . . . . 33

2.3.2 Decompositions and Maps of Fuzzy Sets . . . 35

2.3.3 Convexity . . . 39

2.3.4 Fuzziness Measures . . . 40

2.4 Flexible Framework for Operations on Fuzzy Sets . . . 45

2.4.1 Negations . . . 45

2.4.2 Complements Based on Negations . . . 48

2.4.3 Triangular Norms . . . 49

2.4.4 Operations Based on Triangular Norms and Conorms . . . 55

2.4.5 Implication Operators . . . 57

2.4.6 Logical Background . . . 59

2.5 Fuzzy Numbers and Linguistic Variables . . . 59

2.5.1 Fuzzy Numbers and Their Types . . . 60

2.5.2 The Extension Principle and Operations on Fuzzy Numbers 63 2.5.3 Comparisons of Fuzzy Numbers . . . 66

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x Contents

3 Further Aspects of Triangular Norms - A Study Inspired

by Flexible Querying . . . . 71

3.1 Flexible Querying in Databases . . . 71

3.1.1 Constructing the Answer to a Flexible Query . . . 72

3.1.2 Unequally Important Elementary Conditions . . . 76

3.2 The Case of Hotel Simenon ! More Advanced Aspects of Triangular Norms . . . 78

3.2.1 Classes of Triangular Norms . . . 79

3.2.2 Continuous and Archimedean Triangular Norms . . . 81

3.2.3 Generators . . . 83

3.2.4 Induced Negations and Complementarity . . . 87

4 Aggregation of Information and Aggregation Operators . . . . 93

4.1 The Issue of Information Aggregation . . . 93

4.2 Aggregation Operators . . . 94

4.3 Aggregation Operators Involving Triangular Norms and Conorms 95 4.3.1 Compensatory Operators . . . 95

4.3.2 Soft Triangular Norms and Conorms . . . 96

4.4 Averaging Operators . . . 96

4.4.1 Means . . . 96

4.4.2 Weighted Means . . . 98

4.4.3 OWA Operators . . . 98

4.5 Conclusions and Systematization . . . 100

4.6 Applications to Decision Making in a Fuzzy Environment . . . . 101

4.6.1 Bellman-Zadeh Model . . . 102

4.6.2 Computational Examples . . . 103

4.7 Two Issues Related to Aggregation . . . 105

4.7.1 Mean Values of Aggregations . . . 105

4.7.2 Averaging with Respect to Triangular Norms and Conorms 106 5 Fuzzy Relations, Approximate Reasoning, Fuzzy Rule-Based Systems . . . . 111

5.1 Fuzzy Relations . . . 111

5.1.1 The Concept of a Fuzzy Relation . . . 111

5.1.2 Composition of Fuzzy Relations . . . 114

5.1.3 Types of Fuzzy Relations . . . 115

5.1.4 Similarity Measures and Similarity Classes . . . 116

5.1.5 Cardinality-Based Similarity Measures for Sets . . . 119

5.1.6 Inclusion and Equality Measures for Fuzzy Sets . . . 121

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Contents xi

5.3 Fuzzy Control and Fuzzy Rule-Based Systems . . . 128

5.3.1 Computational Approach to Fuzzy Rules . . . 128

5.3.2 Fuzzy Controller . . . 130

5.3.3 How Does It Work? . . . 133

6 Modeling Incompletely Known Fuzzy Sets . . . . 139

6.1 Incompletely Known Sets and Their Modeling . . . 139

6.2 Interval-Valued Fuzzy Sets . . . 142

6.2.1 The Concept of an Interval-Valued Fuzzy Set . . . 143

6.2.2 General Properties of Uncertainty Degrees . . . 144

6.2.3 Operations on IVFSs . . . 145

6.3 I-Fuzzy Sets . . . 147

6.3.1 Basic Notions . . . 148

6.3.2 General Properties of Hesitation Degrees . . . 153

6.3.3 Operations on IFSs . . . 154

6.3.4 Model Examples . . . 156

Part II Methods of Intelligent Counting under

Information Imprecision

7 General Remarks and Motivations . . . . 163

8 Scalar Approach . . . . 167

8.1 Sigma f-Counts and Counting in Fuzzy Sets . . . . 167

8.1.1 Sigma f-Counts . . . . 167

8.1.2 Main Weighting Functions and Cases of Sigmaf-Counts . 170 8.1.3 The Eight-Bottle Example Continued . . . 173

8.2 Arithmetic Aspects . . . 175

8.2.1 Addition of Sigma f-Counts . . . . 176

8.2.2 Multiplication . . . 178

8.3 Relative Cardinality . . . 179

8.4 Counting in IVFSs and IFSs . . . 180

8.4.1 Main Properties . . . 181

8.4.2 Relativization . . . 183

8.4.3 The Eight-Bottle Example Once Again . . . 185

9 Fuzzy Approach . . . . 187

9.1 Related Methods of Counting in Fuzzy Sets . . . 187

9.1.1 MCAC and the Basic Fuzzy Count . . . 188

9.1.2 FGCount and FLCount . . . 193

9.1.3 FECount . . . 198

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xii Contents

9.2 Three Looks at FECounts . . . 202

9.2.1 Looking through Similarity Measures . . . 202

9.2.2 Rule-Based Optics . . . 204

9.2.3 FECounts as Fuzzy Decisions . . . 206

9.3 Generalizations and Extensions . . . 208

9.3.1 Fuzzy Cardinalities with Triangular Norms . . . 208

9.3.2 Extensions to IVFSs and IFSs . . . 211

9.4 Equipotency, Comparisons of and Operations on Fuzzy Cardinalities 216 9.4.1 Equipotent Fuzzy Sets . . . 217

9.4.2 Inequalities . . . 219

9.4.3 Operations on Fuzzy Cardinalities . . . 222

9.5 Dealing with Cardinalities of IFSs and IVFSs . . . 228

10 Selected Applications . . . . 231

10.1 Cardinality-Based Measures for Fuzzy Sets . . . 231

10.2 Time Series Analysis . . . 237

10.3 Linguistic Quantification . . . 238

10.4 Decision Making in a Fuzzy Environment through Counting . . . 243

10.5 Group Decision Making . . . 245

10.5.1 Direct Approach to GDM . . . 248

10.5.2 Indirect Approach . . . 255

10.6 Bibliographical References to Applications . . . 259

11 Cardinalities of Infinite Fuzzy Sets . . . . 261

11.1 Notation and Introductory Remarks . . . 261

11.2 Fuzzy Cardinalities of Infinite Fuzzy Sets . . . 262

11.3 References to the Continuum Hypothesis . . . 265

11.4 Arithmetic Operations . . . 267

11.5 Axiomatic Approaches to Fuzzy Cardinalities . . . 268

11.6 Using IVFSs and IFSs . . . 270

References . . . . 271