Studies in Fuzziness and Soft Computing

Editor-in-Chief

Prof. Janusz KacprzykSystems Research InstitutePolish Academy of Sciencesul. Newelska 601-447 WarsawPolandE-mail: kacprzyk@ibspan.waw.pl

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279

Zeshui Xu

Intuitionistic FuzzyAggregation and Clustering

123

Zeshui XuCollege of SciencesPLA University of Science

and TechnologyNanjingPeople’s Republic of China

ISSN 1434-9922 ISSN 1860-0808 (electronic)ISBN 978-3-642-28405-2 ISBN 978-3-642-28406-9 (eBook)DOI 10.1007/978-3-642-28406-9Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012953466

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Preface

The concept of intuitionistic fuzzy set (IFS) was originally introduced by Atanassov(1983) to extend the concept of the traditional fuzzy set. Each element in an IFS isexpressed by an ordered pair which is called an intuitionistic fuzzy value (IFV) (orintuitionistic fuzzy number (IFN)), and each IFV is characterized by a membershipdegree, a nonmembership degree, and a hesitancy degree. The sum of the mem-bership degree, the nonmembership degree, and the hesitancy degree of each IFV isequal to one. IFVs can describe the fuzzy characters of things comprehensively, andthus are a powerful and effective tool in expressing uncertain or fuzzy informationin actual applications. Recently, a lot of research work has been done on theaggregation and cluster analysis. Since 2006, my research group has been focusingon the investigation of these interesting and important topics, and achieved fruitfulresearch results which have been published in some well-known peer-reviewedprofessional journals.

This book offers a systematic introduction to the latest research work of mygroup on information aggregation and cluster analysis under intuitionistic fuzzyenvironments, including the various algorithms for clustering intuitionistic fuzzyinformation and the intuitionistic fuzzy aggregation techniques, and their appli-cations in multi-attribute decision making, such as supply chain management,military system performance evaluation, project management, venture capital,information system selection, building materials classification, and operationalplan assessment, and so on. We organized this book as below:

Chapter 1 introduces the intuitionistic fuzzy aggregation techniques. We firstgive a survey of the existing methods for ranking IFVs, and then introduce variousoperational laws of IFVs. On the basis of these ranking methods and operationallaws, we present varieties of the intuitionistic fuzzy power aggregation operators,the intuitionistic fuzzy geometric Bonferroni means, the intuitionistic fuzzyaggregation operators based on Archimedean t-conorm and t-norm, the generalizedintuitionistic fuzzy aggregation operators based on Hamacher t-conorm and t-norm, the generalized intuitionistic fuzzy point aggregation operators, and theirgeneralizations in interval-valued intuitionistic fuzzy environments and theapplications in multi-attribute decision making.

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Chapter 2 introduces the clustering algorithms of IFSs. The chapter first definesthe concept of intuitionistic fuzzy similarity degree, and constructs the intuition-istic fuzzy similarity matrix and the intuitionistic fuzzy equivalence matrix. Then,the chapter defines the compound operational law of intuitionistic fuzzy similaritymatrix, and gives an approach to transforming the intuitionistic fuzzy similaritymatrices into the intuitionistic fuzzy equivalence matrices. After that, the chapterdefines the k-cutting matrices of the intuitionistic fuzzy similarity matrix and theintuitionistic fuzzy equivalence matrix, based on which an approach is presentedfor clustering IFSs. Moreover, the chapter defines the concept of association andequivalent association matrix, and introduces some methods for calculating theassociation coefficients of IFSs. Then, based on the association matrix, the chapterintroduces a clustering algorithm for IFSs, and extends the algorithm to clusterinterval-valued IFSs. Additionally, some other clustering algorithms, such as theintuitionistic fuzzy hierarchical clustering algorithms, the intuitionistic fuzzyorthogonal clustering algorithm, the intuitionistic fuzzy C-means clusteringalgorithms, the intuitionistic fuzzy minimum spanning tree (MST) clusteringalgorithm, the intuitionistic fuzzy clustering algorithm based on Boole matrix andassociation measure, the intuitionistic fuzzy netting clustering method, and thedirect cluster analysis based on intuitionistic fuzzy implication are also introduced.

This book can be used as a reference for researchers and practitioners working inthe fields of fuzzy mathematics, operations research, information science, man-agement science and engineering, and so on. It can also be used as a textbook forpostgraduate and senior undergraduate students.

This work was supported by the National Natural Science Foundation of Chinaunder Grant 71071161.

Nanjing, April 2012 Zeshui Xu

vi Preface

Contents

1 Intuitionistic Fuzzy Aggregation Techniques . . . . . . . . . . . . . . . . . 11.1 Rankings of Intuitionistic Fuzzy Values . . . . . . . . . . . . . . . . . . 2

1.1.1 Intuitionistic Fuzzy Values . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Methods for Ranking IFVs. . . . . . . . . . . . . . . . . . . . . . 21.1.3 The Application of Ranking IFVs Using

the Similarity Measure and the Accuracy Degreein Multi-Attribute Decision Making . . . . . . . . . . . . . . . 12

1.2 Intuitionistic Fuzzy Power Aggregation Operators . . . . . . . . . . . 171.2.1 Power Aggregation Operators . . . . . . . . . . . . . . . . . . . . 171.2.2 Some Operational Laws of IFVs . . . . . . . . . . . . . . . . . . 181.2.3 Power Aggregation Operators for IFVs . . . . . . . . . . . . . 201.2.4 Approaches to Multi-Attribute Group Decision

Making with Intuitionistic Fuzzy Information . . . . . . . . . 251.2.5 Practical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Interval-Valued Intuitionistic Fuzzy PowerAggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.3.1 Interval-Valued Intuitionistic Fuzzy Values . . . . . . . . . . 331.3.2 Power Aggregation Operators for IVIFVs . . . . . . . . . . . 361.3.3 Approaches to Multi-Attribute Group Decision Making

with Interval-Valued Intuitionistic Fuzzy Information . . . 411.4 Intuitionistic Fuzzy Geometric Bonferroni Means . . . . . . . . . . . 47

1.4.1 Geometric Bonferroni Mean . . . . . . . . . . . . . . . . . . . . . 471.4.2 Intuitionistic Fuzzy Geometric Bonferroni Mean. . . . . . . 481.4.3 The Weighted Intuitionistic Fuzzy Geometric

Bonferroni Mean and Its Applicationin Multi-Attribute Decision Making . . . . . . . . . . . . . . . 56

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1.5 Generalized Intuitionistic Fuzzy Bonferroni Means . . . . . . . . . . 651.5.1 Generalized Bonferroni Means . . . . . . . . . . . . . . . . . . . 651.5.2 Generalized Intuitionistic Fuzzy Weighted

Bonferroni Mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.5.3 Generalized Intuitionistic Fuzzy Weighted

Bonferroni Geometric Mean . . . . . . . . . . . . . . . . . . . . . 741.6 Intuitionistic Fuzzy Aggregation Operators Based

on Archimedean t-conorm and t-norm . . . . . . . . . . . . . . . . . . . 801.6.1 Intuitionistic Fuzzy Operational Laws Based

on t-conorm and t-norm. . . . . . . . . . . . . . . . . . . . . . . . 801.6.2 Intuitionistic Fuzzy Aggregation Operators

Based on Archimedean t-conorm and t-norm . . . . . . . . . 861.6.3 An Approach to Intuitionistic Fuzzy Multi-Attribute

Decision Making. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941.7 Generalized Intuitionistic Fuzzy Aggregation Operators

Based on Hamacher t-conorm and t-norm. . . . . . . . . . . . . . . . . 991.8 Point Operators for Aggregating IFVs . . . . . . . . . . . . . . . . . . . 1231.9 Generalized Point Operators for Aggregating IFVs . . . . . . . . . . 130

2 Intuitionistic Fuzzy Clustering Algorithms . . . . . . . . . . . . . . . . . . 1592.1 Clustering Algorithms Based on Intuitionistic

Fuzzy Similarity Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1602.2 Clustering Algorithms Based on Association Matrices . . . . . . . . 1772.3 Intuitionistic Fuzzy Hierarchical Clustering Algorithms . . . . . . . 1922.4 Intuitionistic Fuzzy Orthogonal Clustering Algorithm . . . . . . . . 1992.5 Intuitionistic Fuzzy C-Means Clustering Algorithms . . . . . . . . . 2092.6 Intuitionistic Fuzzy MST Clustering Algorithm . . . . . . . . . . . . . 2212.7 Intuitionistic Fuzzy Clustering Algorithm Based

on Boole Matrix and Association Measure . . . . . . . . . . . . . . . . 2292.7.1 Intuitionistic Fuzzy Association Measures . . . . . . . . . . . 2292.7.2 Intuitionistic Fuzzy Clustering Algorithm. . . . . . . . . . . . 2322.7.3 Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . 2342.7.4 Interval-Valued Intuitionistic Fuzzy

Clustering Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 2412.8 A Netting Method for Clustering Intuitionistic

Fuzzy Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2472.8.1 An Approach to Constructing Intuitionistic

Fuzzy Similarity Matrix . . . . . . . . . . . . . . . . . . . . . . . . 2472.8.2 A Netting Clustering Method . . . . . . . . . . . . . . . . . . . . 2502.8.3 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 251

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2.9 Direct Cluster Analysis Based on IntuitionisticFuzzy Implication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2592.9.1 The Intuitionistic Fuzzy Implication Operator

and Intuitionistic Fuzzy Products . . . . . . . . . . . . . . . . . 2592.9.2 The Applications of Two Intuitionistic Fuzzy Products . . . 2622.9.3 The Application of the Intuitionistic Fuzzy

Triangle Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2622.9.4 The Application of the Intuitionistic Fuzzy

Square Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2642.9.5 A Direct Intuitionistic Fuzzy Cluster Analysis Method. . . 265

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Contents ix