This book summarizes years of research in the field of fuzzy relational programming, with a special emphasis on geometri

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*61*
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*English*
*Pages XIII, 245
[253]*
*Year 2020*

- Author / Uploaded
- Bing-Yuan Cao
- Ji-Hui Yang
- Xue-Gang Zhou
- Zeinab Kheiri
- Faezeh Zahmatkesh
- Xiao-Peng Yang

*Table of contents : Front Matter ....Pages i-xiii Basic Theory of Fuzzy Set (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 1-27 Fuzzy Relation (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 29-43 Fuzzy Relational Equations/Inequalities (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 45-66 Fuzzy Relational Linear Programming (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 67-104 Fuzzy Relation Geometric Programming (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 105-143 Relational Geometric Programming with Fuzzy Coefficient (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 145-175 Fuzzy Relational of Non-linear Optimization (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 177-207 (+, \(\wedge \)) Fuzzy Relational Inequality and Its Network Optimization (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 209-242 Research Progress of Fuzzy Relational Geometric Programming (Bing-Yuan Cao, Ji-Hui Yang, Xue-Gang Zhou, Zeinab Kheiri, Faezeh Zahmatkesh, Xiao-Peng Yang)....Pages 243-245*

Studies in Fuzziness and Soft Computing

Bing-Yuan Cao · Ji-Hui Yang · Xue-Gang Zhou · Zeinab Kheiri · Faezeh Zahmatkesh · Xiao-Peng Yang

Fuzzy Relational Mathematical Programming Linear, Nonlinear and Geometric Programming Models

Studies in Fuzziness and Soft Computing Volume 389

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing, which include fuzzy sets, rough sets, neural networks, evolutionary computation, probabilistic and evidential reasoning, multi-valued logic, and related ﬁelds. The publications within “Studies in Fuzziness and Soft Computing” are primarily monographs and edited volumes. They cover signiﬁcant recent developments in the ﬁeld, both of a foundational and applicable character. An important feature of the series is its short publication time and world-wide distribution. This permits a rapid and broad dissemination of research results. Indexed by ISI, DBLP and Ulrichs, SCOPUS, Zentralblatt Math, GeoRef, Current Mathematical Publications, IngentaConnect, MetaPress and Springerlink. The books of the series are submitted for indexing to Web of Science.

More information about this series at http://www.springer.com/series/2941

Bing-Yuan Cao Ji-Hui Yang Xue-Gang Zhou Zeinab Kheiri Faezeh Zahmatkesh Xiao-Peng Yang •

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Fuzzy Relational Mathematical Programming Linear, Nonlinear and Geometric Programming Models

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Bing-Yuan Cao University of Foshan Foshan, China University of Guangzhou Guangzhou, China Guangzhou Vocational and Technical University of Science and Technology Guangzhou, Guangdong, China Xue-Gang Zhou School of Financial Mathematics and Statistics Guangdong University of Finance Guangzhou, China

Ji-Hui Yang College of Science Shenyang Agricultural University Shenyang, Liaoning, China Zeinab Kheiri Higher Education Mega Center Guangzhou University Guangzhou, Guangdong, China Xiao-Peng Yang Department of Mathematics and Statistics Hanshan Normal University Chaozhou, Guangdong, China

Faezeh Zahmatkesh Higher Education Mega Center Guangzhou University Guangzhou, Guangdong, China

ISSN 1434-9922 ISSN 1860-0808 (electronic) Studies in Fuzziness and Soft Computing ISBN 978-3-030-33784-1 ISBN 978-3-030-33786-5 (eBook) https://doi.org/10.1007/978-3-030-33786-5 © Springer Nature Switzerland AG 2020 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. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

In 1987, I began to pay attention to the study of fuzzy relational equations, then, in 1998, I tried to introduce fuzzy relation to fuzzy geometric programming, and in 2004, while working in Shantou University, I enrolled my ﬁrst Ph.D. (Ji-Hui Yang), who was the ﬁrst student of mine researching for fuzzy geometric Programming. He began to study the programming, and in particular, the relationship between fuzzy relational inequalities and geometric programming, and put it as topic of his doctoral work. In 2005, we proposed for the ﬁrst time fuzzy relational geometric programming at the Fuzzy Systems, IEEE International Conference held in the United States. Since then, my Ph.D. and postdoctoral students have been studying fuzzy relational geometric programming as their major topics. We have combined our research on fuzzy geometric programming with researches on fuzzy relational linear programming, achieving interesting results. In this book, we describe our own research work, and refer to papers by other scholars such as S. C. Fang, Y. K. Wu, etc. The writing tasks have been distributed as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Basic Theory of Fuzzy Set; Fuzzy Relation; Fuzzy Relational Equations/Inequalities; Fuzzy Relational Linear Programming; Fuzzy Relational Geometric Programming; Relational Geometric Programming with Fuzzy Coefﬁcient; Fuzzy Relational Non-linear Programming; (+, ^) Fuzzy Relational Inequality and Its Network Optimization; Research Progress of Fuzzy Relational Geometric Programming.

Professor Bing-Yuan Cao from Foshan University, Guangzhou University and Guangzhou Vocational and Technical University of Science and Technology wrote Sects. 1.1–1.5, 3.2, 3.3, 4.3, 8.1, 8.2, Chap. 9 and was also responsible for compiling the book; Dr. Ji-Hui Yang, associate professor of Shenyang Agricultural University, took care of Chap. 2, and Sects. 5.3, 5.4, 6.1, 6.2.; Dr. Xue-Guang Zhou, associate professor of Guangdong University of Finance, wrote Sects. 3.1, 5.1, 5.2 and v

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Preface

Chap. 7. F. Zahmatkesh, China Ph.D. candidates, at Guangzhou University wrote Sects. 4.1, 4.2; Z. Kheiri, China Ph.D. Candidates at Guangzhou University took care of Sects. 1.6, 4.4, 6.3 and Dr. Xiao-Peng Yang, associate prof. of Hanshan Teachers College, completed Sects. 4.5, 8.1, 8.3. We thank the Springer Publishing House for distributing this book worldwide. We also thank for is support by National Natural Science Foundation of China (No. 70771030, No. 61877014), the Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province, China (No. S2013040012506), China Postdoctoral Science Foundation Funded Project (2014M562152), the Innovation Capability of Independent Innovation to Enhance the Class of Building Strong School Projects of Colleges of Guangdong Province (2015KQNCX094, 2015KTSCX095), the General Fund Project of the Ministry of Education and Social Science Research (16YJAZH081) and the Natural Science Foundation of Guangdong Province (2016A030313552, 2016A030307037). At present, research on fuzzy relational programming, especially the theory and applications of fuzzy relational geometric programming, is rising worldwide, achieving gratifying results. However three issues of fuzzy geometric programming, remains to be resolved. I hope that its publication enables researchers in fuzzy relational programming to move on in the conjecture of solutions to the three guesses. Guangzhou, China 2018 Chinese New Year

Bing-Yuan Cao

Acknowledgements Heartfelt thanks to Mrs. Pei-Hua Wang, from Guangzhou University and Guangzhou Vocational and Technical University of Science and Technology, China, for the whole English proofreading.

List of Books Published By Cao Bing-Yuan

Vol. 1: Cao Bing-Yuan, Fuzzy Geometric Programming Kluwer Acadmic Publishers, 2002.10 Vol. 2: Cao Bing-Yuan, Optimal Models and Methods with Fuzzy Quantity Springer Science Business, 2010.1 Vol. 3: Cao Bing-Yuan, Application of Fuzzy Mathematics and Systems Science Press in China, 2005.10 Vol. 4: Cao Bing-Yuan, chief editor. Applied Probability and Statistics Course Science Press in China, 2005.8. Vol. 5: Hao-Ran Lin, Cao Bing-Yuan and Yun-Zhang Liao, Fuzzy Sets Theory Preliminary Springer Science Business, 2018. Vol. 6: Seyed Hadi Nasseri, Ali Ebrahimnejad and Cao Bing-Yuan, Fuzzy Linear Programming: Solution Techniques and Applications Springer Science Business, 2019. Cao Bing-Yuan is Professor, Doctoral and Postdoctoral Supervisor of School of Mathematics and Information Science, Guangzhou University and Second-level Chair Professor of Lingnan of Foshan University and Dean and Professor of School of Finance and Economics, Guangzhou Vocational and Technical University of Science and Technology, China.

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Contents

1 Basic Theory of Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fuzzy Sets and Membership Functions . . . . . . . . . . . . . . . 1.1.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Convex Fuzzy Sets and Decomposition Theorem . . . . . . . . 1.2.1 a–Cut Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Convex Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lattice and Fuzzy Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Distance of Fuzzy Sets and t-Norm . . . . . . . . . . . . . . . . . . 1.4.1 Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 t-Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Expansion Principle and Six Type Fuzzy Numbers . . . . . . . 1.5.1 Expansion Principle . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Six-Type Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . 1.6 Expansion of Fuzzy Number—Intuitionistic Fuzzy Number References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.1.5 3.1.6 3.2 ð_; Þ 3.2.1 3.2.2 3.2.3

Algorithm and Example . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type Fuzzy Relational Equation . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solubility of the M-PFRE and Theorem for Maximum Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Solubility of the M-PFRE and Theorem for Minimum Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Comparing in Algorithm . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Application in Business Management . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Fuzzy Relational Linear Programming . . . . . . . . . . . . . . . 4.1 ð_; ^Þ Fuzzy Relational Linear Programming . . . . . . . . 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Properties of ð_; ^Þ Composition and Two Sub-problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Rules for Reducing and Solving the Problem . . 4.1.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . 4.1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 ð_; Þ Fuzzy Relational Linear Programming . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Properties of ð_; Þ Composition and Two Sub-problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Reduction Procedures and Solving the Problem . 4.2.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . 4.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Lattice Linear Programming with ð_; Þ Composition Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Nature of Optimal Solution . . . . . . . . . . . . . . . 4.3.3 Method to Finding an Optimal Solution . . . . . . 4.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fuzzy Relational Linear Programming with Fuzzy Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Fuzzy Numbers . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . 4.4.4 Solution Procedure . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.5 Application of Fuzzy Relational Linear Programming . . . . . . . 4.5.1 Application of ð_; ^Þ FRLP in Three-Tier Multimedia Streaming Services . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Application of ð_; Þ FRLP in WiFi Terminal System . 4.5.3 Application of ð þ ; ^Þ FRLP in BitTorrent-Like Peer-to-Peer File Sharing System . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Fuzzy Relation Geometric Programming . . . . . . . . . . . . . . 5.1 Posynomial Geometric Programming Subject to ð_; ^Þ Fuzzy Relation Equations . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 An Algorithm and Its Computational Complexity 5.1.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . 5.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 ð_; ^Þ Fuzzy Relation Geometric Programming . . . . . . . 5.2.1 Structure of Solution Set on Model (5.2.1) . . . . . 5.2.2 Solving Solution on Model (5.2.2) . . . . . . . . . . . 5.2.3 Algorithm of Model (5.2.2) . . . . . . . . . . . . . . . . 5.2.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . 5.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Monomial Geometric Programming with Fuzzy Relation Equation Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Proposition of the Question . . . . . . . . . . . . . . . . 5.3.3 Structure of Solution Set on Eq. (5.3.1) . . . . . . . 5.3.4 Solution to Optimization (5.3.2) . . . . . . . . . . . . . 5.3.5 Numerical Example . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 ð_; Þ Fuzzy Relation Geometric Programming . . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Structure of Solution Set On Equation . . . . . . . . 5.4.3 Solving Solution on Model . . . . . . . . . . . . . . . . 5.4.4 Algorithm to Model . . . . . . . . . . . . . . . . . . . . . . 5.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Relational Geometric Programming with Fuzzy Coefﬁcient 6.1 Posynomial Fuzzy Relational Geometric Programming with Fuzzy Coefﬁcient and Variable . . . . . . . . . . . . . . . 6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Deﬁnition of Posynomial Fuzzy Relational Geometric Programming . . . . . . . . . . . . . . . . . .

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6.1.3 Structure of Solution Set on Fuzzy Relational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Solving Method to Posynomial Fuzzy Relational Geometric Programming with Clear Objective . . . . . . . 6.1.5 Solving Method to Posynomial Fuzzy Relational Geometric Programming with Fuzzy Objective . . . . . . 6.1.6 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Geometric Programming with Intuitionistic Fuzzy Coefﬁcient . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Intuitionistic Fuzzy Sets and Geometric Programming . 6.2.3 Geometric Programming with Intuitionistic Fuzzy Coefﬁcient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Solution of Geometric Programming with Intuitionistic Fuzzy Coefﬁcient . . . . . . . . . . . . . . . . . . 6.2.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Posynomial Geometric Programming with Intuitionistic Fuzzy Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Posynomial Geometric Programming with Intuitionistic Fuzzy Coefﬁcient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Fuzzy Relational of Non-linear Optimization . . . . . . . . 7.1 Quadratic Programming with ð_; Þ Fuzzy Relational Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 ð_; Þ Fuzzy Relational Inequalities . . . . . . . . 7.1.4 Properties and Algorithms . . . . . . . . . . . . . . 7.1.5 Numerical Examples . . . . . . . . . . . . . . . . . . 7.1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Special Nonlinear Programming with ð_; ^Þ Fuzzy Relational Inequalities Constraint . . . . . . . . . . . . . . . 7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Numerical Example . . . . . . . . . . . . . . . . . . . 7.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

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8 (+, ^) Fuzzy Relational Inequality and Its Network Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 (+, ^) Fuzzy Relational Inequality and P2P File Sharing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 BT-Like P2P File Sharing System Exported (+, ^) Fuzzy Relational Inequality . . . . . . . . . . . . . . . . . . . . . 8.1.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Operating Condition of the Data Transmission in a BT-Like P2P File Sharing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 System Solutions and Their Properties . . . . . . . . . . . . . 8.2.3 Algorithm to (8.1.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Optimization Model with ð þ ; ^Þ Fuzzy Relational Inequalities Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Min-Max Programming Problem Subject to ð þ ; ^Þ Fuzzy Relational Inequalities . . . . . . . . . . . . . . . . . . . . 8.3.3 Resolution of Problem (8.3.2) . . . . . . . . . . . . . . . . . . . . 8.3.4 Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Research Progress of Fuzzy Relational Geometric Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Conjectures of Fuzzy Relational Geometric Programming . 9.2 Fuzzy Relational Geometric Programming with Fuzzy Coefﬁcients and Variables . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Basic Theory of Fuzzy Set

This chapter represents theory of fuzzy sets, including fuzzy sets and membership functions, convex fuzzy sets and decomposition theorem, fuzzy lattice, t-norm, expansion principle and five-type fuzzy numbers, with basic knowledge provided for introduction of the book.

1.1 Fuzzy Sets and Membership Functions 1.1.1 Concept The so-called universe meaning that all of the objects involved, is an ordinary set, usually by writing English alphabets X , Y , Z and etc. to represent universe. Since the fuzzy sets differ from classic ones with a strict mathematics definition, we give its mathematics description [1] as follows. Definition 1.1.1 Let X be universe. If X to a real number close to interval [0, 1] arbitrary mapping μ A˜ (x), μ A˜ : X −→ [0, 1], x → μ A˜ (x) determines a fuzzy subset A˜ of x, then we call a membership function in fuzzy set ˜ A˜ in set X is ˜ and call μ A˜ (x) a membership degree from point x to A. A, A˜ = {(μ A˜ (x), x)|x ∈ X }, a fuzzy subset, fuzzy subsets are also often called fuzzy sets. © Springer Nature Switzerland AG 2020 B.-Y. Cao et al., Fuzzy Relational Mathematical Programming, Studies in Fuzziness and Soft Computing 389, https://doi.org/10.1007/978-3-030-33786-5_1

1

2

1 Basic Theory of Fuzzy Set

Thus, in the fuzzy sets there exist few next conclusions: (1) The concept of fuzzy sets is an expansion concept of classical sets. If F (X ) means all fuzzy sets on X , i.e., ˜ A˜ is a fuzzy set on X }, F (X ) = { A| then P(X ) ⊂ F (X ), where P(X ) is the power sets on X , i.e., P(X ) = {A|Ais a classic set on X }, that is, if the membership function of fuzzy set A˜ is only taken by 0 and 1, two values, then A˜ is exuviated into the classic sets of X . (2) The concept of a membership function is an expansion of the characteristic function concept. When A ∈ P(X ) is an ordinary subset in X , the characteristic function of A is 1, x ∈ A (membership degree of x for A is 1), χA = 0, x ∈ / A (membership degree of x for A is 0). This means, in fuzzy sets, the nearer the membership degree μ A˜ (x) in fuzzy set A˜ is to 1, the bigger x belonging to A˜ degree is; whereas, the nearer μ A˜ (x) is to 0, the smaller x belonging to A˜ degree is. If the value region of μ A˜ (x) is {0, 1}, then fuzzy set A˜ is an ordinary set A, but membership function μ A˜ (x) is characteristic function χ A (x). (3) We call fuzzy sets in F (X )P(X ) true fuzzy ones. Several representation methods to fuzzy sets are shown as follows. 10

A representation method to fuzzy set by Zadeh If set X is a finite set, let universe X = {x1 , x2 , . . . , xn }. The fuzzy set is μ ˜ (xn ) μ A˜ (xi ) μ ˜ (x1 ) μ A˜ (x2 ) + + ··· + A = , A˜ = A x1 x2 xn xi i=1 n

μ (x )

here symbol “Σ” is no longer a numerical sum, A˜xi i is not a fraction; it only has the sign meaning, that is, only membership degree of the point xi with respect to fuzzy set A˜ is μ A˜ (xi ). If X is an infinite set, a fuzzy set on X is A˜ =

x∈X

μ A˜ (x) . x

1.1 Fuzzy Sets and Membership Functions

3

Similarly, the sign “ ” is not an integral any more, only means an infinite μ (x) logic sum. But the meaning of A˜x is in accordance with the finite case. 20 When the universe X is a finite set, the fuzzy set represented in Definition 1.1.1 is A˜ = {(μ A˜ (x1 ), x1 ), (μ A˜ (x2 ), x2 ), . . . , (μ A˜ (xn ), xn )}. 30

When the universe X is a finite set, the fuzzy set represented according to a vector form is A˜ = (μ A˜ (x1 ), μ A˜ (x2 ), . . . , μ A˜ (xn )). Remarkably, X and φ also can be seen as fuzzy set in X , if membership functions μ A˜ (x) ≡ 1 and μ A˜ (x) ≡ 0, then A˜ is a complete set X and an empty set φ, respectively.

An element which the membership degree is 1 definitely belongs to this fuzzy set, while an element which the membership degree is 0 does not belong to this fuzzy set definitely. But the membership function value in (0, 1) forms a distinct boundary, which is also called distinct subsets of fuzzy sets. When a fuzzy object is described by using the fuzzy set, it is important to choose its membership function for it. Now we give three membership functions basically: 1. Partial minitype (abstains up) −1 , when x c, 1 + (a(x − c))b μ A˜ (x) = 1, when x < c, where c ∈ X is an arbitrary point, a > 0, b > 0 are two parameters. 2. Partial large-scale (abstains down) 0, when x c, −1 μ A˜ (x) = , when x > c, 1 + (a(x − c))−b where x ∈ X is an arbitrary point, a > 0, b > 0 are two parameters. Obviously, Type 1 and 2 is dual, and its meaning shows clear at a glance. Type 3 is ˜ which is “sufficiently near to number set of a”, then this membership a fuzzy set A, function in A˜ is defined on a center type according to the definition. Example 1.1.1 Suppose X = {1, 2, 3, 4}, these four elements constitute a small number set. Obviously, element 1 is standardly a small number, it should belong to the set, and its membership degree is 1; element 4 is not a small number, and it should not belong to this set, its membership degree being 0. Element 2 “also returns small” or makes “eighty percent small”, its membership degree being 0.8; element 3 probably is “force small”, or makes “two percent small”; its membership degree ˜ its elements still is 1, 2, 3, 4, being 0.2. The fuzzy sets written in small numbers as A, at the same time, and a membership degree of element in A˜ is given, denoted by

4

1 Basic Theory of Fuzzy Set

1 0.8 0.2 0 + + . Zadeh’s representation is A˜ = + 1 2 3 4 An order dual representation is A˜ = {(1, 1), (0.8, 2), (0.2, 3), (0, 4)}. A vector method simply shows as A˜ = (1, 0.8, 0.2, 0).

1.1.2 Operations Because the value region in fuzzy set membership function corresponding to clearsubset characteristic function is extended from {0, 1} to [0, 1], similarly to the characteristic function to demonstrate the relation between a distinctive subset, we have ˜ B˜ ∈ F (X ). If ∀x ∈ X , we have Definition 1.1.2 Let A, A˜ ⊆ B˜ ⇐⇒ μ A˜ (x) μ B˜ (x) (Inclusion). A˜ = B˜ ⇐⇒ μ A˜ (x) = μ B˜ (x) (Equality). ˜ That is to say, the incluFrom Definition 1.1.2, A˜ = B˜ ⇐⇒ A˜ ⊆ B˜ and B˜ ⊆ A. sion relation is a binary relation on fuzzy power set F (X ) with following properties, i.e., (1) A˜ ⊆ A˜ (reflection). (2) A˜ ⊆ B˜ and B˜ ⊆ A˜ =⇒ A˜ = B˜ (symmetry). (3) A˜ ⊆ B˜ and B˜ ⊆ C˜ =⇒ A˜ ⊆ C˜ (transitivity). Since relation “⊆” constitutes an order relation on F (X ), (F (X ), ⊆) stands for a partially ordered set. Again as φ, X ∈ F (X ), hence F (X ) contains maximum element X and minimum element φ. ˜ ˜ Definition 1.1.3 Let A, B ∈ F (x). Then we define A˜ B˜ (Union), whose membership function is (x) = μ (x) μ (x) = max{μ A˜ (x), μ B˜ (x)}. μ( A˜ B) ˜ A˜ B˜

A˜ B˜ (Intersection), whosemembership function is μ B˜ (x) = min{μ A˜ (x), μ B˜ (x)}. (μ A˜ B) ˜ (x) = μ A˜ (x) c ˜ A (Complement), whose membership function is μ A˜ c (x) = 1 − μ A˜ (x). Comparing operation of union, intersection and complement in distinctive set, we discover immediately that the fuzzy sets operation is exactly a parallel definition of the distinct set

operation, A˜ B˜ is a minimum fuzzy set embodying A˜ and embodied ˜ A˜ B˜ is a maximum fuzzy set embodying A˜ and embodied again in B. ˜ again in B. According to the two kinds of cases, where the universe X is finite or infinite, the calculation formula of union, intersection and complement in fuzzy sets A˜ and B˜ can be represented, respectively, like the following: n μ A˜ (xi ) n , B˜ = i=1 (1) The universe is X = {x1 , x2 , . . . , xn }, and A˜ = i=1 xi μ B˜ (xi ) , then xi

1.1 Fuzzy Sets and Membership Functions

A˜ A˜

B˜ = B˜ =

A˜ c =

5

n μ ˜ (xi ) ∨ μ ˜ (xi ) A

i=1 n i=1 n i=1

(2) X is an infinite set, and A˜ =

x∈X

B

xi

,

μ A˜ (xi ) ∧ μ B˜ (xi ) , xi 1 − μ A˜ (xi ) . xi μ A˜ (x) , x

B˜ =

x∈X

μ B˜ (x) , then x

μ A˜ (x) ∨ μ B˜ (x) , x x∈X μ A˜ (x) ∧ μ B˜ (x) , A˜ B˜ = x x∈X μ ˜ (x) . A˜ c = 1− A x x∈X A˜

B˜ =

Example 1.1.2 Suppose X = {x1 , x2 , x3 , x4 , x5 }; A˜ = 0.5 0.3 0.1 0.7 + + + , then x1 x2 x4 x5

0.2 0.7 1 0.5 ˜ + + + ;B = x1 x2 x3 x5

0.2 ∨ 0.5 0.7 ∨ 0.3 1 ∨ 0 0 ∨ 0.1 0.5 ∨ 0.7 + + + + B˜ = x1 x2 x3 x4 x5 0.5 0.7 1 0.1 0.7 = + + + + . x1 x2 x3 x4 x5 0.2 ∧ 0.5 0.7 ∧ 0.3 1 ∧ 0 0 ∧ 0.1 0.5 ∧ 0.7 + + + + A˜ B˜ = x1 x2 x3 x4 x5 0.2 0.3 0.5 = + + . x1 x2 x5 1 − 0.2 1 − 0.7 1 − 1 1 − 0 1 − 0.5 + + + + A˜ c = x1 x2 x3 x4 x5 0.8 0.3 1 0.5 = + + + . x1 x2 x4 x5 A˜

Example 1.1.3 Let X ⊆ R + (R + is a non-negative real number set). Regard age as universe and take X = [0, 100]. Zadeh gave “oldness” O˜ and “youth” Y˜ , these two membership functions respectively are

6

1 Basic Theory of Fuzzy Set

μ O˜ (x) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

1+

0, x − 50 5 1,

−2 −1

0 x 50, , 50 < x 100, x > 100,

and ⎧ 1, 0 x 25, ⎪ ⎪ ⎪ 2 −1 ⎨ x − 25 μY˜ (x) = 1+ , 25 < x 100, ⎪ 5 ⎪ ⎪ ⎩ 0, x > 100. If some person’s age is 28, then his membership degree belongings to “youth” or “oldness” respectively is

1+

28 − 25 5

2 −1 = 0.735 and 0.

If some person’s age is 55, then his membership degree belongings to “youth” or “oldness” respectively is

1+

55 − 25 5

2 −1 = 0.027

and

1+

55 − 50 5

−2 −1

= 0.5.

Definition 1.1.4 Suppose T to be index set A˜ t ∈ F (X) (t ∈ T ), then μ A˜ t (x) = t∈T

Obviously,

μ A˜ t (x) = sup μ A˜ t (x), x ∈ X, t∈T

t∈T

μ A˜ t (x) = t∈T

μ A˜ t (x) = inf μ A˜ t (x), x ∈ X. t∈T

t∈T

t∈T

A˜ t ,

t∈T

In particular, when T is a finite set,

A˜ t ∈ F (X ).

1.1 Fuzzy Sets and Membership Functions

7

μ A˜ t (x) = max μ A˜ t (x), x ∈ X, t∈T

t∈T

μ A˜ t (x) = min μ A˜ t (x), x ∈ X. t∈T

t∈T

Theorem 1.1.1 (F (X ), , , c) satisfies the following properties:

˜ A˜ A˜ = A˜ (Idempotent law). (1) A˜ A˜ = A,

˜ A˜ B˜ = B˜ (2) A˜ B˜ = B˜ A, A˜ (Commutative law). (3) ˜ ˜ ( A˜ B) C˜ = A˜ ( B˜ C),

˜ ˜ ˜ ˜ ˜ ˜ ( A B) C = A ( B C) (Associativelaw). (4) ( A˜ (5)

˜ B)

˜ ( A˜ A˜ = A,

˜ B)

A˜ = A˜ (Absorptive law).

˜ ˜ ˜ ( A˜ B) C˜ = ( A˜ C) ( B˜ C),

˜ ˜ ˜ (Distributivelaw). ( A˜ B) C˜ = ( A˜ C) ( B˜ C)

(6)

˜ A˜ φ = φ, A˜ X = A, A˜ X = X, A˜ φ = A˜ (0 − 1 law).

(7) ( A˜ c )c = A˜ (Restore

original

law). ˜ c = A˜ c B˜ c (Dual law). ˜ ˜ c = A˜ c B˜ c , ( A˜ B) (8) ( A B) Proof Proved by taking Property (8) for example, the rest can be verified directly. From ∀x ∈ X , we have μ( A˜ B) ˜ c (x) = 1 − μ A˜ B˜ (x) = 1 − max{μ A˜ (x), μ B˜ (x)} = min{1 − μ A˜ (x), 1 − μ B˜ (x)} = min{μ A˜ c (x), μ B˜ c (x)} = μ A˜ c B˜ c (x).

Hence

( A˜

Similarly, we can prove

( A˜

˜ c = A˜ c B) ˜ c = A˜ c B)

B˜ c . B˜ c .

It is pointed out that the operation in a fuzzy set no longer satisfies the excludedmiddle law. Namely, under circumstance generally, we have A˜ But we have

A˜ c = X, A˜

A˜ c = φ.

8

1 Basic Theory of Fuzzy Set

A˜

1 1 A˜ c . A˜ c , A˜ 2 2

Example 1.1.4 If μ A˜ (x) ≡ 0.5, μ A˜ c (x) ≡ 0.5, then μ A˜ A˜ c (x) = max{0.5, 0.5} = 0.5 = 1, μ A˜ A˜ c (x) = min{0.5, 0.5} = 0.5 = 0.

1.2 Convex Fuzzy Sets and Decomposition Theorem 1.2.1 α–Cut Set Definition 1.2.1 Suppose A˜ ∈ F (X ), ∀α ∈ [0, 1], we write ˜ α = Aα = {x|μ A˜ (x) α}, ( A) ˜ Again, we write then Aα is said to be an α–cut set of fuzzy set A. ˜ α = Aα = {x|μ A˜ (x) > α}, ( A) · · ˜ α a confidence level, and Aα· is called a strong α–cut set of fuzzy set A, ˜ ˜ 0 = A0 = {x|μ A˜ (x) > 0} = supp A, ( A) · · ˜ A0· is called a support of fuzzy set A. If this support supp A˜ = {x} is a single point set, then A˜ is called a fuzzy point on X . Audio-visually, the meaning in Aα is that if x to the membership degree of A˜ attains or exceeds the level α, at last it has the qualified member. Since all of these qualified members constitute Aα , it is a classical subset in X . 0.1 0.3 0.8 0.6 1 Example 1.2.1 Suppose A˜ = + + + + , then x1 x2 x3 x4 x5 at at at at at at

α = 1, α = 0.8, α = 0.6, α = 0.3, α = 0.1, α = 0,

A1 = {x5 }, A0.8 = {x3 , x5 }, A0.6 = {x3 , x4 , x5 }, A0.3 = {x2 , x3 , x4 , x5 }, A0.1 = {x1 , x2 , x3 , x4 , x5 }, A0 = {x1 , x2 , x3 , x4 , x5 },

α-cut set has the following properties.

A1 = φ, A0.8 = {x5 }, A0.6 = {x3 , x5 }, A0.3 = {x3 , x4 , x5 }, A0.1 = {x2 , x3 , x4 , x5 }, A0 = {x1 , x2 , x3 , x4 , x5 }.

1.2 Convex Fuzzy Sets and Decomposition Theorem

9

Property 1.2.1

˜ α = Aα ˜ α = Aα B) B) Bα , ( A˜ Bα .

˜ α = Aα ˜ α = Aα Bα· , ( A˜ Bα· . (2) ( A˜ B) B) · · · ·

(1) ( A˜

Proof We prove only the first formula in (1).

( A˜

˜ α = {x|μ A˜ B˜ (x) α} = {x|μ A˜ (x) ∨ μ B˜ (x) α} B) = {x|μ A˜ (x) α} {x|μ B˜ (x) α} = Aα Bα .

Proof of the other formulas is the same. Property 1.2.2

( A˜ t )α , ( A˜ t )α = ( A˜ t )α , ( A˜ c )α = (A1−α )c . t∈T t∈T t∈T t∈T

(2) ( A˜ t )α. = ( A˜ t )α , ( A˜ t )α. ⊆ ( A˜ t )α. , ( A˜ c )α. = (A1−α )c .

(1) (

A˜ t )α ⊇

t∈T

t∈T

t∈T

t∈T

Proof in Property 1.2.2 is easy, readers themselves can prove it. It must be pointed out that the first formula in (1) and the second formula in (2) can’t be changed for the equation. Example 1.2.2 Let μ A˜ n (x) ≡ (

∞

1 1 ∞ (1 − ), n = 1, 2, . . .. Then μ 2 n

A˜ n

n=1

A˜ n )0.5 = X. But

n=1

( A˜ n )0.5 = φ (n 1),

such that

∞

( A˜ n )0.5 = φ.

n=1

Therefore

(

∞

A˜ n )0.5 =

n=1

Similarly, let μ B˜ n (x) ≡

∞

( A˜ n )0.5 .

n=1

1 1 (1 + ), n = 1, 2, . . . . We can prove 2 n (

∞

n=1

B˜ n )0.5 =

∞ n=1

( B˜ n )0.5 .

(x) ≡

1 , so that 2

10

1 Basic Theory of Fuzzy Set

Definition 1.2.2 Suppose A˜ ∈ F (X), set Ker A˜ = {x|μ A˜ (x) = 1} is called a kernel of fuzzy set A˜ and A˜ is a normal fuzzy set if Ker A˜ = φ.

1.2.2 Convex Fuzzy Sets Suppose X = R n to be n-dimensional Euclidean space, A is an ordinary subset in X . If ∀x1 , x2 ∈ A, and ∀λ ∈ [0, 1], recall the first concept of ordinary convex sets, we have λx1 + (1 − λ)x2 ∈ A, then call A convex sets. Before introduction of the convex fuzzy set concepts, we prove first result below. Theorem 1.2.1 Suppose A˜ to be a fuzzy set in X , if α ∈ [0, 1], Aα = {x|μ A˜ (x) α} are all convex sets if and only if ∀x1 , x2 ∈ X, λ ∈ [0, 1], there is μ A˜ (λx1 + (1 − λ)x2 ) μ A˜ (x1 ) ∧ μ A˜ (x2 ).

(1.2.1)

Proof If we have already known α ∈ [0, 1], Aα are all convex sets, ∀x1 , x2 ∈ X might as well suppose μ A˜ (x2 ) μ A˜ (x1 ) = α0 , then μ A˜ (x1 ) ∧ μ A˜ (x2 ) = α0 . Because Aα0 is a convex set, ∀x1 , x2 ∈ Aα0 , and ∀λ ∈ [0, 1], we have λx1 + (1 − λ)x2 ∈ Aα0 , hence μ A˜ (λx1 + (1 − λ)x2 ) α0 . Therefore μ A˜ (λx1 + (1 − λ)x2 ) μ A˜ (x1 ) μ A˜ (x2 ). Conversely, if we have already known ∀x1 , x2 ∈ X, α ∈ [0, 1], there exist μ A˜ (λx1 + (1 − λ)x2 ) μ A˜ (x1 ) ∧ μ A˜ (x2 ), then, if α ∈ [0, 1], x1 , x2 ∈ Aα , hence μ A˜ (x1 ) α, μ A˜ (x2 ) α, such that μ A˜ (x1 ) ∧ μ A˜ (x2 ) α, so μ A˜ (λx1 + (1 − λ)x2 ) μ A˜ (x1 ) ∧ μ A˜ (x2 ) α, hence λx1 + (1 − λ)x2 ∈ Aα . Therefore, Aα is a convex set. Definition 1.2.3 Suppose X = R n to be n-dimensional Euclidean space, A˜ is a fuzzy set in X . If ∀α ∈ [0, 1], Aα are all convex sets, call fuzzy set A˜ a convex fuzzy set.

1.2 Convex Fuzzy Sets and Decomposition Theorem

11

From Theorem 1.2.1 we know that A˜ is a convex set if and only if ∀λ ∈ [0, 1], x1 , x2 ∈ X , there is μ A˜ (λx1 + (1 − λ)x2 ) μ A˜ (x1 ) ∧ μ A˜ (x2 ). ˜ Theorem 1.2.2 If A˜ and B˜ are convex sets, so is A˜ ∩ B. Proof ∀x1 , x2 ∈ X, ∀λ ∈ [0, 1], μ A∩ ˜ B˜ λx 1 + (1 − λ)x 2 = μ A˜ λx 1 + (1 − λ)x 2 ∧ μ B˜ λx 1 + (1 − λ)x 2 μ A˜ (x1 ) ∧ μ A˜ (x2 ) ∧ μ B˜ (x1 ) ∧ μ B˜ (x2 ) = μ A˜ (x1 ) ∧ μ B˜ (x1 ) ∧ μ A˜ (x2 ) ∧ μ B˜ (x2 ) = μ A∩ ˜ B˜ (x 1 ) ∧ μ A∩ ˜ B˜ (x 2 ). Therefore, A˜ ∩ B˜ denotes a convex fuzzy set. ˜ B, ˜ ˜ ∈ F (X ). Then a convex combination with respect to Definition 1.2.4 Let A, ˜ B; ˜ ), ˜ of A˜ and B˜ is a fuzzy set, denoted by ( A, ˜ with its membership function being ˜ ˜ μ( A, ˜ B; ˜ ) ˜ (x) = (x)μ A˜ (x) + 1 − (x) μ B˜ (x), ∀x ∈ X. ˜ i ∈ F (X )(1 i m) and Generally, if A˜ i ,

m

˜ i (x) = 1(∀x ∈ X ), then a

i=1

˜ i of A˜ i is written as convex combination with respect to μ( A˜ 1 , A˜ 2 ,..., A˜ m ;˜ 1 ,˜ 2 ,...,˜ m ) (x) =

m

˜ i (x)μ A˜ i (x), ∀x ∈ X.

i=1

Definition 1.2.5 Suppose A˜ ∈ F (X ), if ∀α ∈ [0, 1], Aα is to be all bounded sets in X , then A˜ is called a bounded fuzzy set in X . Theorem 1.2.3 Both union and intersection of two bounded fuzzy sets are bounded fuzzy sets, respectively. It is easy to prove it by property and Definition 1.2.5 of α-cut sets.

1.3 Lattice and Fuzzy Lattice Property 1.3.1 Let α, β and γ be real numbers. (1) α ∨ α = α, α ∧ α = α (Power law); (2) α ∨ β = β ∨ α, α ∧ β = β ∧ α (Commutative law); (3) α ∨ (β ∨ γ) = (α ∨ β) ∨ γ, α ∧ (β ∧ γ) = (α ∧ β) ∧ γ (Associative law);

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1 Basic Theory of Fuzzy Set

(4) (α ∧ β) ∨ α = α, (α ∨ β) ∧ α = α (Absorptivity); (5) (α ∨ β) ∧ γ = (α ∧ γ) ∨ (β ∧ γ), (α ∧ β) ∨ γ = (α ∨ γ) ∧ (β ∨ γ) bution law).

(Distri-

Definition 1.3.1 Suppose the binary operation “∨,” “∧” is defined on a nonempty set L, if the property is satisfied (1), (2), (3) and (4), then (L , ∧, ∨) is called lattice. Definition 1.3.2 Call (K , ) a subset, and if the relationship on k satisfies the following conditions: (1) α α (Self-reflexivity); (2) α β, β γ =⇒ α γ ( Transitivity). Call ( p, ) a partial ordered set, if it is a sub-ordered set and satisfies: (3) α β, β α =⇒ α = β (Symmetry), where “ p −→ q” indicates that both “ p and q” are established. In the partial order, there may not be one of α β and β α. For arbitrary α, β ∈ P, there must be α β or β α, callin P a linear order set. The lattice can be equivalently defined by the partial order, so we can use (L , ) to represent a lattice in the future. ∀a, b, define a ∧ b = inf{a, b}, a ∨ b = sup{a, b}, where inf and inf mean respectively greatest lower bound and least upper bound. Complete lattice Suppose (L , ) is a partial order, for any subset of L, if there is constant existence of sup A and inf A, then we call (L , ) a complete lattice. Distributive lattice: Suppose (L , ∧, ∨) is a lattice, and for ∀a, b, c ∈ L, satisfies distribution law (5), then (L , ∧, ∨) is called distributive lattice. Fuzzy lattice: A class of special complete distributive lattices. Because of this lattice is closely related to the structure of fuzzy mathematics, people call it fuzzy lattice. Definition 1.3.3 Suppose L to be a lattice, a ∈ L is called a union irreducible element, if for arbitrary element x and y ∈ L, when a x ∨ y, there is a = x or a = y. The non-zero union irreducible element of L is called a molecule. Because of the completely distributive lattice with sufficient number of molecules, therefore, it is often said that the completely distributive lattice is a molecular lattice. If the L is a molecular lattice with a corresponding with reverse order convolution, i.e., there is a mapping N : L −→ L satisfies: (1) If a ≤ b, N (a) ≥ N (b), (2) N (N (a)) = a. Then L is fuzzy lattice. Let f : L 1 −→ L 2 . If (3) (4)

f is guaranteed and mapped; f −1 is an inverse mapping, i.e., ∀b ∈ L 2 , there is f −1 (N (b)) = N ( f −1 (b)); f −1 b = ∨x ∈ L 1 f (x) ≤ b;

then mapping f is called the order homomorphism from the fuzzy lattice L 1 to L 2 .

1.4 Distance of Fuzzy Sets and t-Norm

13

1.4 Distance of Fuzzy Sets and t-Norm 1.4.1 Distance Definition 1.4.1 Suppose X = φ, If mapping ρ : X × Y → R such that the conditions as follows: ∀x, y, z ∈ X : (1) ρ(x, y) = 0 ⇔ x = y (N or malit y); (2) ρ(x, y) = ρ(y, x) (Symmetr y); (3) ρ(x, z) ρ(x, y) + ρ(y, z) (Atriangleinequalit y). Then we call ρ a distance function on X , ρ(x, y) a distance from x to y on X , and (X, ρ) a metric space. If the mapping ρ satisfies (1), (3) and (1’) ρ(x, x) = 0. Then we call ρ a quasi distance function on X , and ρ(x, y) quasi distance on X from x to y. Following next is a common (quasi) distance formula between fuzzy sets. We use F[a, b] to represent the whole of a fuzzy set in membership function integrable on [a, b](a < b). Difference Module Distance of Union and Intersection ˜ B˜ ∈ F (X ), let Suppose that X = (x1 , x2 , . . . , xn ), A, ρUi =

n n 1 ˜ ˜ i )) − 1 ˜ i ) ∧ B(x ˜ i )). ( A(xi ) ∨ B(x ( A(x n i=1 n i=1

(1.4.1)

˜ B˜ ∈ F([a, b]) ⊆ Then ρUi is a distance function on F (X ). When X = [a, b], A, F ([a, b]). Let ρUi =

n n 1 ˜ ˜ i )) − 1 ˜ i ) ∧ B(x ˜ i )). ( A(xi ) ∨ B(x ( A(x n i=1 n i=1

Then ρUi is a difference module distance function on F (X ).

1.4.2 t-Norm T -Norm is related to the transaction, also known as a triangular norm.

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1 Basic Theory of Fuzzy Set

Definition 1.4.2 T -Norm is defined on the binary function, satisfying the following condition: (1) t (0, 0) = 0; t (μ A˜ (x), 1) = t (1, μ A˜ (x)) = μ A˜ (x), x ∈ X.

(1.4.2)

(2) Monotonicity If μ A˜ (x) μC˜ (x), μ B˜ (x) μ D˜ (x), then t (μ A˜ (x), μ B˜ (x)) t (μC˜ (x), μ D˜ (x)).

(1.4.3)

t (μ A˜ (x), μ B˜ (x)) = t (μ B˜ (x), μ A˜ (x)).

(1.4.4)

(3) Exchangeability

(4) Associativity t (μ A˜ (x), t (μ B˜ (x), μC˜ (x))) = t (t (μ A˜ (x), t (μ B˜ (x)), μC˜ (x)).

(1.4.5)

1.5 Expansion Principle and Six Type Fuzzy Numbers 1.5.1 Expansion Principle Theorem 1.5.1 (Extension Principle I, L.A. Zadeh [1, 2]) Let f : X → Y be an ordinary point function, A˜ ∈ F(X ). Two mappings can be induced by f and f −1 as follows: f : F(X ) → F(Y ), ˜ ∈ F(Y ), A˜ → f ( A)

f −1 : F(Y ) → F(X ), ˜ ∈ F(X ), B˜ → f −1 ( B)

whose membership functions are denoted by ˜ f ( A)(y)

⎧ ˜ ⎨ A(x),

f −1 (y) = φ,

⎩ 0,

f −1 (y) = φ,

f (x)=y

˜ ˜ f (x)), y = f (x), f −1 ( B)(x) B( ˜ an inverse image of ˜ is called an image of A˜ under f and f −1 ( B) respectively. f ( A) ˜ B. The representation of a cut-set in extension principle. Theorem 1.5.2 (Extension Principle II) Let mapping f : X → Y be extended as mapping f : F(X ) → F(Y ) and f −1 : F(Y ) → F(X ). Then ∀α ∈ [0, 1], A˜ ∈ F(X ), B˜ ∈ F(Y ), we have

1.5 Expansion Principle and Six Type Fuzzy Numbers

10 20 30 ˜ α Here, f ( A)

15

˜ α = f (Aα ), f ( A) · · −1 ˜ α f ( B) · = f −1 (Bα· ), ˜ α = f −1 (Bα ). f −1 ( B) ˜ α. is a simplification of ( f ( A))

Proof Only 10 is proved and the others can be verified in a similar way. Because ˜ α ⇐⇒ f ( A)(y) ˜ y ∈ f ( A) >α · ˜ ⇐⇒ A(x) >α f (x)=y

˜ ⇐⇒ ∃ : x ∈ X, satisfying f (x) = y, such that A(x) >α ⇐⇒ ∃ : x ∈ X, satisfying f (x) = y, such that x ∈ Aα· ⇐⇒ y ∈ f (Aα· ). 10 is proved.

1.5.2 Six-Type Fuzzy Numbers In this section, we shall discuss properties of six types of fuzzy numbers including interval-type fuzzy numbers, (·, c)-type, T -type, L-R-type, flat-type and Triangulartype Fuzzy Numbers [3–5], two membership functions. ˜ Because number 0 is a special example in interval number 0¯ and fuzzy number 0, in this Section, 0 denotes a number by adopting all the same marks. 1. Interval-type Fuzzy Numbers Definition 1.5.1 Let R denote a real number set. We call c, d interval numbers, written as c, d ∈ IR , where IR = {[ci , di ]|ci < di , ci , di ∈ R, (i = 1, 2)} is a set consisting of all interval numbers. If c = [c1 , c2 ], d = [d1 , d2 ], the operation of defined interval numbers is as follows: c + d = [c1 + d1 , c2 + d2 ], c − d = [(c1 − d1 ) ∧ (c2 − d2 ), (c1 − d1 ) ∨ (c2 − d2 )], c · d = [c1 d1 ∧ c1 d2 ∧ c2 d1 ∧ c2 d2 , c1 d1 ∨ c1 d2 ∨ c2 d1 ∨ c2 d2 ], c ÷ d = [c1 /d1 ∧ c1 /d2 ∧ c2 /d1 ∧ c2 /d2 , c1 /d1 ∨ c1 /d2 ∨ c2 /d1 ∨ c2 /d2 ], c ∨ d = [c1 ∨ d1 , c2 ∨ d2 ], c ∧ d = [c1 ∧ d1 , c2 ∧ d2 ]. Theorem 1.5.3 Given c, d ∈ IR , then c ∗ d ∈ IR , where “∗” denotes algebra operations {+, −, ·, ÷, ∨, ∧} on R. Fuzzy numbers are obtained by applying the extension principle. From now on, F(R) represents the set of real fuzzy numbers.

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Definition 1.5.2 Given that c, ˜ d˜ denote fuzzy numbers, written as c, ˜ d˜ ∈ F(R), with α cuts of c˜ and d˜ being cα = [c1 (α), c2 (α)], dα = [d1 (α), d2 (α)], α ∈ [0, 1] respectively, so that the operations of a fuzzy number are defined as follows:

c˜ + d˜ = = c˜ − d˜ = = c˜ · d˜ = =

α∈[0,1] α∈[0,1]

α∈[0,1] α∈[0,1]

α∈[0,1]

=

α[c1 (α) + d1 (α), c2 (α) + d2 (α)]; α(cα − dα ) α[(c1 (α) − d1 (α)) ∧ (c2 (α) − d2 (α)),

(c1 (α) − d1 (α)) ∨ (c2 (α) − d2 (α))]; α(cα · dα )

α∈[0,1]

c˜ ÷ d˜ =

α(cα + dα )

α[c1 (α)d1 (α) ∧ c1 (α)d2 (α) ∧ c2 (α)d1 (α) ∧ c2 (α)d2 (α),

α∈[0,1] α∈[0,1]

c1 (α)d1 (α) ∨ c1 (α)d2 (α) ∨ c2 (α)d1 (α) ∨ c2 (α)d2 (α)]; α(cα ÷ dα ) α[c1 (α)/d1 (α) ∧ c1 (α)/d2 (α) ∧ c2 (α)/d1 (α) ∧ c2 (α)/d2 (α),

c (α)/d1 (α) ∨ c1 (α)/d2 (α) ∨ c2 (α)/d1 (α) ∨ c2 (α)/d2 (α)]; 1 α(cα ∨ dα ) α∈[0,1] = α[c1 (α) ∨ d1 (α), c2 (α) ∨ d2 (α)], α∈[0,1] α(cα ∧ dα ) c˜ ∧ d˜ = α∈[0,1] = α[c1 (α) ∧ d1 (α), c2 (α) ∧ d2 (α)].

c˜ ∨ d˜ =

α∈[0,1]

Theorem 1.5.4 Let c, ˜ d˜ ∈ F(R). Then c˜ ∗ d˜ ∈ F(R). It is easy to prove the two theorems above similar to the corresponding theorems in Refs. [6–8]. Definition 1.5.3 Suppose that c˜ ∈ F(R) is called a fuzzy number, where R represents the set of whole real numbers. If (i) c˜ is normal, i.e., x0 ∈ R exists, such that c(x ˜ 0 ) = 1. (ii) ∀α ∈ (0, 1], cα is a closed interval. Theorem 1.5.5 Let c˜ ∈ F(R) be a fuzzy number. Then (i) c˜ is fuzzy convex.

1.5 Expansion Principle and Six Type Fuzzy Numbers

17

(ii) If c(x ˜ 0 ) = 1, then c(x) ˜ is nondecreasing for x x0 and c(x) ˜ nonincreasing for x x0 . Proof Because cα (α ∈ (0, 1]) is the closed interval, c0 = R, i.e., ∀α ∈ [0, 1], cα is a convex set. c˜ can be proved to be fuzzy convex according to Theorem 1.2.1. ˜ 1 ). Since c(x ˜ 0 ) = 1, then [x1 , x0 ] ⊂ cα , Now, take x1 < x2 x0 and let α = c(x ˜ α, i.e., c(x ˜ 1 ) c(x ˜ 2 ). hence x2 ∈ cα , such that c(x) ˜ 1 ) can be proved if x0 x1 < x2 . Similarly, c(x ˜ 2 ) c(x Overall, the theorem holds. Theorem 1.5.6 Let c˜ ∈ F(R) and sup c˜ be bounded. Then c˜ is a fuzzy number ⇔ ∃ : interval [c1 , c2 ], such that ⎧ ⎪ x ∈ [c1 , c2 ] = φ, ⎨1, c(x) ˜ = L(x), x < c1 , ⎪ ⎩ R(x), x > c2 ,

(1.5.1)

where L(x) represents an increasing function of right continuance (0 L(x) < 1); R(x) represents a decreasing one of left continuance (0 R(x) < 1). Proof Necessity. Let c˜ ∈ F(R). Then ˜ = 1 on [c1 , c2 ]. Obvi(1) Because c˜1 is a closed convex set, c˜1 = [c1 , c2 ] and c(x) ously c(x) ˜ < 1 for x ∈ / [c1 , c2 ]. (2) Because c˜ ∈ F(R), ∀α ∈ [0, 1], cα is a closed interval, we assume cα = [c1α , c2α ] ⊂ [0, 1], then c˜ =

α∈(0,1]

αcα =

α[c1α , c2α ].

α∈(0,1]

As for x < c1 , α ∧ χ[c1α ,c2α ] (x) L(x) = c(x) ˜ = α∈(0,1] = {α|x ∈ [c1α , c2α )} = {α|c1α x < c2α }, α∈(0,1]

α∈(0,1]

where χ represents a characteristic function. Therefore, 0 L(x) < 1. ˜ 1 ) c(x ˜ 2 ), otherwise, c(x ˜ 1 ) > c(x ˜ 2 ). Again, x1 < x2 If x1 < x2 c1 , then c(x c1 ⇒ x2 ∈ (x1 , c1 + ε) ⇒ ∃ : λ ∈ [0, 1], such that x2 = λx1 + (1 − λ)(c1 + ε), c1 + ε ∈ (c1 , c2 ). Since c˜ represents a convex fuzzy set on [c1 , c2 ],

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1 Basic Theory of Fuzzy Set

c(x ˜ 2 ) = c(λx ˜ 1 + (1 − λ)(c1 + ε)) c(x ˜ 1 ) ∧ c(c ˜ 1 + ε), ˜ 2) c(x ˜ 1 ) > c(x (3)

which is a contradiction. Therefore, L(x) is an increasing function. L(x) continues on the right, otherwise there exists x < c1 , xn → x, then lim L(xn ) = α > L(x).

xn →x ∗

Since xn ∈ cα and cα is closed, then x ∈ cα , such that c(x) ˜ = L(x) α, a contradiction. For the same reason, c(x) ˜ = R(x) is a continuously decreasing function on the left for x > c2 , with 0 R(x) < 1. Sufficiency. Let c˜ satisfy the condition in the theorem. (1) c˜ is obviously normal. (2) Prove cα = [c1α , c2α ], ∀α ∈ (0, 1]. c(x) ˜ = L(x) for x < c1 , so we select c1α = min{x|L(x) α} and c(x) ˜ = R(x) for x > c2 , such that we select c2α = max{x|R(x) α}. Obviously, cα ⊂ [c1α , c2α ]. Now, we prove [c1α , c2α ] ⊂ cα , only [c1α , c1 ) ⊂ cα (because we can prove (c2 , c2α ] ⊂ cα for the same reason). Again, we prove only c1α ∈ cα due to the monotonicity of L(x). Select xn → c1α , then L(c1α ) = lim (L(xn )) α, such that c1α ∈ cα . xn →c1α

2. Type (·, c) Fuzzy Numbers Definition 1.5.4 c˜ = (α, c) is defined as a type (·, c) fuzzy number on a product space α1 × α2 × · · · × α J ; its membership function is c(a) ˜ = min[c˜ j (a j )], j

⎧ ⎨1 − |α j − a j | , α − c a α + c , j j j j j cj c(a ˜ j) = ⎩ 0, otherwise,

(1.5.2)

where α = (a1 , a2 , . . . , a J )T , c = (c1 , c2 , . . . , c J )T ; α denotes the center of c, ˜ c the extension of c, ˜ with c j > 0. Coming next are special cases. 3. Type L-R Fuzzy Numbers Definition 1.5.5 L is called a reference function of fuzzy numbers, if L satisfies (i) L(x) = L(−x); (ii) L(0) = 1; (iii) L(x) is a nonincreasing and piecewise continuous function at [0, +∞).

1.5 Expansion Principle and Six Type Fuzzy Numbers

19

Definition 1.5.6 Let L , R be reference functions of fuzzy numbers c, ˜ called a type L-R fuzzy number. If ⎧ c−x ⎪ ⎪ , x c, c > 0, ⎨L c c(x) ˜ = x −c ⎪ ⎪ ⎩R , x c, c > 0, c

(1.5.3)

we write c˜ = (c, c, c) L R , where c is a mean value; c and c are called the left and the right spreads of c, ˜ respectively. L is called a left reference and R a right reference. Operations properties in type L-R fuzzy number. Let c˜ = (c, c, c) L R , d˜ = (d, d, d) L R , p˜ = ( p, p, p) R L be an L-R fuzzy number. Then (1) c˜ + d˜ =(c + d, c + d, c + d) L R . (kc, kc, kc) L R , when k 0 (2) k · c˜ = (k ∈ R). (kc, −kc, −kc) R L , when k < 0 Let (−1)c˜ = −c˜ for k = −1. Then −c˜ = (−c, c, c) R L . (3) c˜ − p˜ = (c − p, c + p, c + p) L R for L = R. (4) c˜ · d˜ ≈ (cd, cd + dc, cd + dc) L R . c pc + c p pc + c p (5) c˜ ÷ p˜ ≈ ( , , ) L R , p = 0, c˜ and p˜ can not be divided p p2 p2 for L = R. ˜ ≈ (c ∨ d, c ∧ d, c ∨ d) L R , (6) max(c, ˜ d) ˜ ≈ (c ∧ d, c ∨ d, c ∧ d) L R . min(c, ˜ d) (7) c˜ d˜ ⇐⇒ c d, c d, c d; ˜ c˜ ⊆ d˜ ⇐⇒ c + c d − d, or c˜ = d. 4. Type T Fuzzy Numbers If we take c to be variable x, then x˜ = (x, ξ, ξ) L R represents T -fuzzy variables. Definition 1.5.7 If L and R are functions satisfying T (x) =

1 − |x|, if − 1 x 1, 0, otherwise,

(1.5.4)

then we call c˜ = (c, c, c)T T -fuzzy numbers, T (R) representing T -fuzzy numbers sets. If Take c to be variable x, and x˜ = (x, ξ, ξ)T represents T -fuzzy variables. Operations properties appear in type T -fuzzy numbers. If c˜1 = (c1 , c1 , c1 )T , c˜2 = (c2 , c2 , c2 )T , then ˜ c˜2 = (c1 + c2 , c1 + c2 , c1 + c2 )T ; (1) c˜1 + ˜ c˜2 = (c1 − c2 , c1 + c2 , c1 + c2 )T ; (2) c˜1 −

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1 Basic Theory of Fuzzy Set

(3) λc˜ = λ(c, c, c)T =

(λc, λc, λc)T , (λc, −λc, −λc)T ,

∀λ > 0, ∀λ < 0.

−2 −2 (4) c˜−1 = (c, c, c)−1 T ≈ (1/c, cc , cc )T .

5. Type Flat Fuzzy Numbers Definition 1.5.8 Let L , R be reference functions and the quadruple c˜ = (c− , c+ , σc− , σc+ ) L R be called a type L-R flat fuzzy numbers. Then we have ⎧ − c −x ⎪ ⎪ L , x c− , σc− > 0 ⎪ ⎪ ⎨ σc− x − c+ c(x) ˜ = R , x c+ , σc+ > 0 ⎪ + ⎪ σ ⎪ c ⎪ ⎩ 1, otherwise

(1.5.5)

satisfying ∃ : (c− , c+ ) ∈ R, c− < c+ , with c(x) ˜ = 1. Especially, c˜ = (c− , c+ , σc− , σc+ ) is said to be a flat fuzzy number, where ⎧ c− − x ⎪ − − − ⎪ 1 − ⎪ − , if c − σc x c , ⎪ σ ⎪ c ⎪ ⎨1, if c− < x < c+ , c(x) ˜ = + ⎪ ⎪1 − x − c , if c+ x c+ + σ + , ⎪ c ⎪ ⎪ σc+ ⎪ ⎩ 0, otherwise.

(1.5.6)

If we take interval (c− , c+ ) to be fuzzy interval, ∀x˜ ∈ [c− , c+ ], then x˜ = ((x − , x + ), ξ, ξ) L R and x˜ = ((x − , x + ), ξ, ξ) represent L-R fuzzy variables and flat fuzzy ones, respectively. Definition 1.5.9 Suppose that “∗” represents an arbitrary ordinary binary operation in R, such that ∀c, ˜ d˜ ∈ F(R) and we define ˜ ˜ c˜ ∗ d = c(x) ˜ ∧ d(y)/x ∗ y, x,y∈R

that is, ∀z ∈ R,

˜ c˜ ∗ d(z) =

˜ (c(x) ˜ ∧ d(y)),

x∗y=z

where “∗” represents arithmetic operations +, −, ·, ÷. Accordingly, we can define the operations of type L-R, T and flat fuzzy numbers. Operation properties exist in type flat fuzzy numbers. Let c˜ = (c− , c+ , σc− , σc+ ) and d˜ = (d − , d + , σd− , σd+ ) be flat fuzzy numbers. Then

1.5

Expansion Principle and Six Type Fuzzy Numbers

21

(1) c˜ + d˜ =(c− + d − , c+ + d + , σc− + σd− , σc+ + σd+ ). (kc− , kc+ , kσc− , kσc+ ), for k > 0, (2) k · c˜ = (kc+ , kc− , −kσc− , −kσc+ ), for k 0. 6. Type Triangular Fuzzy Numbers Definition 1.5.10 If A˜ ∈ F(R), and its membership function A can be expressed as ⎧ x − AL ⎪ L C ⎪ ⎪ ⎨ AC − A L , A ≤ x ≤ A , x − AR μ(x) = ⎪ , AC ≤ x ≤ A R , ⎪ ⎪ ⎩ AC − A R 0, otherwise.

(1.5.7)

Then A˜ is called a triangular fuzzy number, which is denoted by A˜ = (A L , AC , A R ). Here A L , AC and A R are called three parameter variables. Property 1.5.1 Let A˜ = (A L , AC , A R ), B˜ = (B L , B C , B R ). Then (1) A˜ + B˜ = (A L + B L , AC + B C , A R + B R ); (2) A˜ − B˜ = (A L − B R , AC − B C , A R − B L ); (3) k A˜ =

(k A L , k AC , k A R ), k ≥ 0, (k A R , k AC , k A L ), k < 0.

1.6 Expansion of Fuzzy Number—Intuitionistic Fuzzy Number We quote several different definitions of triangular and trapezoidal intuitionistic fuzzy (IF) numbers. Definition 1.6.1 An intuitionistic fuzzy set (IFS) A˜ I in X is given by A˜ I = {x, μ A˜ I (x), ν A˜ I (x)|x ∈ X }, where the functions μ A˜ I (x) : X → [0, 1] and ν A˜ I (x) : X → [0, 1], with the condition 0 ≤ μ A˜ I (x) + ν A˜ I (x) ≤ 1, define respectively, the degree of membership and degree of non-membership of the element x ∈ X to the set A˜ I which is a subset of X. For each A˜ I in X , we can compute the intuitionistic index of the element in x to the set A˜ I , which is defined as follows: π A˜ I (x) = 1 − μ A˜ I (x) − ν A˜ I (x),

22

1 Basic Theory of Fuzzy Set

where π A˜ I (x) is also called a hesitancy degree of x to A˜ I . Obviously, x ∈ X, 0 ≤ π A˜ I (x) ≤ 1 [9]. Definition 1.6.2 An intuitionistic fuzzy subset A˜ I = {(x, μ A˜ I (x), ν A˜ I (x))|x ∈ R} of the real line is called an intuitionistic fuzzy number (IFN) if: (1) A is IF-normal, if there exist at least two points x0 , x1 ∈ X such that μ A˜ I (x0 ) = 1, and ν A˜ I (x1 ) = 0, it is easily seen that the given intuitionistic fuzzy set A˜ I is IFnormal if there exists at least one point that surely belongs to A and at least one point which does not belong to A˜ I . (2) A˜ I is IF-convex, an IFS A˜ I = {(x, μ A˜ I (x), ν A˜ I (x))|x ∈ R} of the real line is called IF-convex, if ∀x1 , x2 ∈ R, ∀λ ∈ [1, 0], μ A˜ I (λx1 + (1 − λ)x2 ) ≥ min{μ A˜ I (x1 ), μ A˜ I (x2 )}, ν A˜ I (λx1 + (1 − λ)x2 ) ≤ max{ν A˜ I (x1 ), ν A˜ I (x2 )}. Thus A means IF-convex if its membership function is fuzzy convex and its non membership function is fuzzy concave. (3) μ A˜ I (x) is upper semicontinuous and ν A˜ I (x) is lower semicontinuous. (4) A˜ I = {(x ∈ R}|ν A˜ I (x) < 1} is bounded. Definition 1.6.3 A generalized triangular intuitionistic fuzzy number (GTIFN) a˜ I = (a, lμ , rμ , ωa ), (a, lν , rν , υa ) is a special intuitionistic fuzzy set on a real number set R, whose membership function and non-membership functions are defined as follows:

μa˜ I (x) =

and

⎧ x −a +l μ ⎪ ωa , a − lμ ≤ x ≤ a, ⎪ ⎪ ⎪ l μ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ωa , x = a, ⎪ ⎪ ⎨ a + rμ − x ⎪ ⎪ ωa , a ≤ x ≤ a + rμ , ⎪ ⎪ rμ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ other wise ⎪ ⎩ 0,

1.6

Expansion of Fuzzy Number—Intuitionistic Fuzzy Number

νa˜ I (x) =

⎧ (a − x) + υa (x − a + lν ) ⎪ ⎪ , ⎪ ⎪ lν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ υa , x = a, ⎪ ⎪ ⎨ (x − a) + υa (a + rν − x) ⎪ ⎪ , ⎪ ⎪ rν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1, other wise, ⎪ ⎩

23

a − lν ≤ x ≤ a,

a ≤ x ≤ a + rμ ,

where lμ , rμ , lν , rν are called the spreads of membership and non-membership function respectively and a is called mean value. ωa and υa represent the maximum degree of membership and minimum degree of non-membership respectively such that they satisfy the conditions 0 ≤ ωa ≤ 1, 0 ≤ υa ≤ 1 and 0 ≤ ωa + υa ≤ 1. If a − lν ≥ 0, then GTIFN a˜ I is called positive GTIFN and if a + rν ≤ 0 is called negative GTIFN. When ωa = 1, υa = 0 is called a normal intuitionistic fuzzy number, namely traditional fuzzy number. Definition 1.6.4 An IFN a˜ I = (a1 , a2 , γμ , τμ , ωa ), (a1 , a2 , γν , τν , υa ) is said to be a GTrIFN if its membership and non-membership function are respectively given by ⎧ x −a +γ 1 μ ⎪ ωa , a1 − γμ ≤ x ≤ a1 , ⎪ ⎪ ⎪ γμ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a1 ≤ x ≤ a2 , ⎪ ωa , ⎪ ⎨ μa˜ I (x) = a2 + τμ − x ⎪ ⎪ ωa , a2 ≤ x ≤ a2 + τμ , ⎪ ⎪ τμ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ other wise ⎪ 0, ⎩ and

νa˜ I (x) =

⎧ (a1 − x) + υa (x − a1 + γν ) ⎪ ⎪ , a1 − γν ≤ x ≤ a1 , ⎪ ⎪ γν ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a1 ≤ x ≤ a2 , ⎨ υa , ⎪ ⎪ (x − a2 ) + υa (a2 + τν − x) ⎪ ⎪ , ⎪ ⎪ ⎪ τν ⎪ ⎪ ⎪ ⎪ ⎩ 1, other wise,

a 2 ≤ x ≤ a 2 + τν ,

where a1 ≤ a2 , γμ , τμ ≥ 0, γμ , τμ , γν , τν are called the spreads of memberships and non-membership functions respectively, such that γμ ≤ γν and τμ ≤ τν . ωa and

24

1 Basic Theory of Fuzzy Set

υa represent a maximum degree of membership and a minimum degree of nonmembership respectively, satisfying 0 ≤ ωa ≤ 1, 0 ≤ υa ≤ 1 and 0 ≤ ωa + υa ≤ 1. When ωa = 1, υa = 0 is called normal IFN, namely traditional fuzzy number. Generally, there is (a1 , a2 , γμ , τμ ) = (a1 , a2 , γν , τν ) intuitionistic trapezoidal fuzzy number a˜ I , here, denoted as a˜ I = (a1 , a2 , γμ , τμ ); ωa , υa When a1 = a2 , the intuitionistic trapezoidal fuzzy number becomes intuitionistic triangular fuzzy number. A GTIFN is called positive if a1 − γν ≥ 0. Definition 1.6.5 Scalar multiplication: (1) If a˜ I = (a, lμ , rμ , ωa ), (a, lν , rnu , υa ) is a GTIFN, then

K a˜I =

⎧ (ka, klμ , krμ , ωa ), (ka, klν , krν , υa ), ⎪ ⎪ ⎨ (ka, −krμ , −klμ , ωa ), (ka, −krν , −klν , υa ), ⎪ ⎪ ⎩

f or k > 0, f or k < 0.

(2) If a˜ I = (a1 , a2 , γμ , τμ , ωa ), (a1 , a2 , γν , τν , υa ) be GTrIFN, then

k a˜ I =

⎧ (ka1 , ka2 , kγμ , kτμ , ωa ), (ka1 , ka2 , kγν , kτν , υa ), ⎪ ⎪ ⎨ (ka1 , ka2 , −kτμ , −kγμ , ωa ), (ka1 , ka2 , −kτν , −kγν , υa ), ⎪ ⎪ ⎩

f or k > 0, f or k < 0.

Definition 1.6.6 An IFN a˜ I in R is said to be a symmetric trapezoidal intuitionistic fuzzy number (STrIFN) if there exists real numbers a1 , a2 , h, h´ where a1 ≤ a2 , h, h´ ≥ 0 and h ≤ h´ such that the membership and non-membership functions are as follows: ⎧ x − a1 + h ⎪ ⎪ , a1 − h ≤ x ≤ a1 , ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a1 ≤ x ≤ a2 , ⎨ 1, μa˜ I (x) = ⎪ ⎪ a2 + h − x ⎪ ⎪ , a2 ≤ x ≤ a2 + h, ⎪ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, other wise,

νa˜ I (x) =

⎧ (a1 − x) ⎪ ⎪ , a1 − h´ ≤ x ≤ a1 , ⎪ ⎪ ´ h ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ a1 ≤ x ≤ a2 , ⎨ 0, ⎪ ⎪ (x − a2 ) ⎪ ´ ⎪ , a2 ≤ x ≤ a2 + h, ⎪ ⎪ ´ ⎪ h ⎪ ⎪ ⎪ ⎪ ⎩ 1, other wise.

1.6

Expansion of Fuzzy Number—Intuitionistic Fuzzy Number

25

Definition 1.6.7 A (α, β)-cut set of GTIFN a˜ I = (a, lμ , rμ , ωa ), (a, lν , rν , υa ) is defined as I = {x : μa(x) ≥ α, νa˜ (x) ≤ β}, a˜ α,β ˜ where 0 ≤ α ≤ ωa , υa ≤ β ≤ 1. A α-cut set of GTIFN a˜ I is a crisp subset of R, which is defined as aˆ α = [a L (α), a R (α)] = [(a − lμ ) +

lμ α rν α , (a + rν ) − ]. ωa ωa

According to Definitions 1.6.3 and 1.6.7, by using membership function, μa˜ (x) ≥ α, then x − a + lμ lμ α ωa ≥ α → x ≥ (a − lμ ) + , lμ ωa and

a + rμ − x rν α ωa ≥ α → x ≤ (a + rν ) − , rμ ωa

we gain [a L (α), a R (α)] = [(a − lμ ) +

lμ α rν α , (a + rν ) − ]. ωa ωa

Similarly a β-cut of GTIFN a˜ I is defined as aˆ β = [a L (β), a R (β)] = [(a − lν ) +

(1 − β)lν (1 − β)rν , (a + rν ) − ]. 1 − υa 1 − υa

Definition 1.6.8 The (α, β)-cut set of GTrIFN a˜ I = (a1 , a2 , γμ , τμ , ωa ), (a1 , a2 , γν , τν , υa ) is defined as usually, by I = {x : μa˜ (x) ≥ α, νa˜ (x) ≤ β}, a˜ α,β

where 0 ≤ α ≤ ωa , υa ≤ β ≤ 1. A α-cut set of GTIFN a˜ I is a crisp subset of R, defined as aˆ α = [a L (α), a R (α)] = [(a1 − γμ ) +

γμ α τμ α , (a2 + τμ ) − ]. ωa ωa

Similarly a β-cut of GTrIFN a˜ is defined as aˆ β = [a L (β), a R (β)] = [(a1 − γν ) +

(1 − β)γν (1 − β)τν , (a2 + τν ) − ]. 1 − υa 1 − υa

´ where Definition 1.6.9 A (α, β)-Cut Set of STrIFN a˜ I = (a1 , a2 , h), (a1 , a2 , h), ´ ´ a1 ≤ a2 , h, h ≥ 0 and h ≤ h is defined as

26

1 Basic Theory of Fuzzy Set

aˆ α = [a L (α), a R (α)] = [(a1 − h) + hα, (a2 + h) − hα]. Similarly a (β)-cut of STrIFN a˜ is defined as ´ + (1 − β)h, ´ (a2 + h) ´ − (1 − β)h]. ´ aˆ β = [a L (β), a R (β)] = [(a1 − h) Theorem 1.6.1 Let a˜ I be any GTIFN or GTrIFN. For any α ∈ [0, ωa ] and β ∈ [υa , 1], where 0 ≤ α + β ≤ 1 the following equality is valid: I = aˆ α ∩ aˆ β . a˜ α,β

(1.6.1)

Proof See [9]. According to this theorem and definition of the intersection between aˆ α and aˆ β , we have following result (1.6.2) a˜ α,β = [a L , a R ], where a L = max{a L (α), a L (β)},

(1.6.3)

a R = min{a R (α), a R (β)}.

(1.6.4)

and Theorem 1.6.2 For each a > 0, the exponential function f (x) = a x , is continuous. Note 1: Multiplication of two continuous functions is continuous. Ishibuchi and Tanaka defined three definitions to rank intervals. In this section, according to our approach we just introduce the order relation is determined by left and right limits of an interval. Definition 1.6.10 Let A = [a L , a R ] and B = [b L , b R ] be two closed intervals. The order relation between two closed intervals as A≤B

i f f a L ≤ b L and a R ≤ b R .

(1.6.5)

Definition 1.6.11 (Interval-valued function) Let a > 0, b > 0 and consider the interval [a, b]. From a mathematical point of view, any real number can be represented on a line. Similarly, we can represent an interval by a function. If the interval is of the form [a, b], the interval-valued function is taken as h(ρ) = a (1−ρ) bρ f or ρ ∈ [0, 1].

(1.6.6)

The choice of the parameter ρ reflects some attitude on the part of the decision maker. Lemma 1.6.1 For given [a, b], a > 0, b > 0, then h(ρ) = a (1−ρ) bρ for ρ ∈ [0, 1] is a strictly monotone increasing continuous function.

1.6 Expansion of Fuzzy Number—Intuitionistic Fuzzy Number

27

Proof According to Theorem 1.6.2 and Note 1.6.1, h(ρ) is continuous. Since 0 ≤ ρ ≤ 1, then d(h(ρ)) 1 = ρ(1 − ρ) ρ (1−ρ) ≥ 0, dρ a b then h(ρ) is monotone increasing and the proof completes. Lemma 1.6.2 Let A = [a L , a R ] and B = [b L , b R ] are two closed intervals. If A ≤ B, then for ρ ∈ [0, 1], h A (ρ) ≤ h B (ρ). Proof From Definition 1.6.10, a L ≤ b L and a R ≤ b R , since ρ ∈ [0, 1], we obtain two following inequalities (1−ρ) (1−ρ) ≤ bL , (1.6.7) aL ρ

ρ

aR ≤ bR . (1−ρ) ρ aR

Then we have a L

(1−ρ) ρ bR ,

≤ bL

(1.6.8)

hence, h A (ρ) ≤ h B (ρ).

References 1. Yang, J.H., Cao, B.Y.: Monomial geometric programming with fuzzy relation equation constraints. Fuzzy Optim. Decis. Mak. 6(4), 337–349 (2007) 2. Yang, J.H., Cao, B.Y., Lv, J.: The global optimal solutions for fuzzy relation quadratic programming. Fuzzy Syst. Math. 27(6), 154–161 (2013) 3. Cao, B.Y.: Optimal Models and Methods with Fuzzy Quantities. Springer, Berlin (2010) 4. Diamond P.: Fuzzy least squares. Inform. Sci. 46, 141–157 (1988). In Proceeding of the IFSA Congress July 20–July 25, Tokyo, vol. I, pp. 329-332 (1987) 5. Dubois, D., Prade, H.: Operations on fuzzy number. Int. J. Syst. Sci. 9(6), 613–626 (1978) 6. Loetamonphong, J., Fang, S.C.: An efficient solution procedure for fuzzy relational equations with max-product composition. IEEE Trans. Fuzzy Syst. 7(4), 441–445 (1999) 7. Loetamonphong, J., Fang, S.C., Young, R.E.: Multi-objective optimization problems with fuzzy relation equation constraints. Fuzzy Sets Syst. 127, 141–164 (2002) 8. Loia, V., Sessa, S.: Fuzzy relation equations for coding/decoding processes of images and videos. Inf. Sci. 171, 145–172 (2005) 9. Atanassov, K.T.: Intuitionistic Fuzzy Sets. Springer, Berlin (1999)

Chapter 2

Fuzzy Relation

Fuzzy relation is the basis of fuzzy relational programming. This chapter introduces its basic knowledge, including concept, operations, composition, properties and several commonly-used membership functions. This chapter, later by using a set, represents fuzzy relationships, without misunderstanding, for simplicity, and in ˜ ˜ or μ A˜ = A. convention, marked as μ A˜ (x) = A(x)

2.1 The Concept of Fuzzy Relation Definition 2.1.1 Suppose X × Y to be a Cartesian product in X and Y, R˜ is a fuzzy set of X × Y , its membership function μ R˜ (x, y)(x ∈ X, y ∈ Y ) determines a fuzzy relation R˜ in X and Y . Beacause in μ R˜ (x, y) value is only taken in the closed interval [0, 1], we call the matrix of elements a fuzzy matrix by taking value in the closed interval [0, 1], marked as r˜ = μ R˜ (x, y) ∈ [0, 1], and call B˜ = (˜r ) or B˜ = (˜ri j ), r˜i j ∈ [0, 1], i = 1, 2, . . . ; j = 1, 2, . . . fuzzy matrix. Example 2.1.1 Suppose X = {x1 , x2 , x3 } denotes three kinds of energy resource sets {electricity, coal, petroleum}, Y = {y1 , y2 , y3 , y4 } denotes the set of four factories {Factory 1, Factory 2, Factory 3, Factory 4}, Table 2.1 denotes fuzzy relations R˜ between factories and each energy resource, r˜i j denotes dependence degree from factory i to energy resource j, where 0 ≤ r˜i j ≤ 1(i = 1, 2, 3; j = 1, 2, 3, 4).

© Springer Nature Switzerland AG 2020 B.-Y. Cao et al., Fuzzy Relational Mathematical Programming, Studies in Fuzziness and Soft Computing 389, https://doi.org/10.1007/978-3-030-33786-5_2

29

30

2 Fuzzy Relation

Table 2.1 Fuzzy relations between factories and energy resource R˜ Factory 1 Factory 2 Factory 3

Factory 4

Electricity Coal Petroleum

r˜11 r˜21 r˜31

r˜12 r˜22 r˜32

r˜13 r˜23 r˜33

It is easy to see that the upper table forms relational matrix, i.e., ⎛ r˜11 r˜12 r˜13 B˜ = ⎝r˜21 r˜22 r˜23 r˜31 r˜32 r˜33

r˜14 r˜24 r˜34

a 4 × 3 matrix B, called the fuzzy ⎞ r˜14 r˜24 ⎠ . r˜34

Fuzzy relation R˜ and fuzzy relational matrix B are always one-to-one correspondence. Example 2.1.2 Suppose that X = Y is a real number set, Cartesian product X × Y is the whole plane. R: “x > y” is an ordinary relation, in this case, the ordinary relation R corresponds to the Boolean matrix, meaning an ordinary matrix. But we consider the relation as follows: ˜ “x y”, that is “x is much greater than y”, which is a fuzzy relation; write R, and we define its membership function as ⎧ ⎨ 0, μ R˜ (x, y) = ⎩ 1+

100 −1 , (x − y)2

x y, x > y.

In this case, the fuzzy relation R˜ corresponds to the fuzzy matrix. From here we can know the following: Fuzzy relation R˜ from X to Y is a fuzzy set in Cartesian product X × Y . Because Cartesian product with order relevant, i.e., X × Y = Y × X , R˜ is also with order relevant. ˜ y) in fuzzy 20 If two values {0, 1} are taken from the membership function R(x, relation only, then R˜ confirms an ordinary set in X × Y , so the fuzzy relation is extended to an ordinary relation.

10

In Example 2.1.2, R˜ is fuzzy relation between the same universe. If the conditions X = Y are satisfied, we call R˜ fuzzy relation in X . Example 2.1.3 Suppose X = {x1 , x2 , x3 } denotes three persons’ sets, R˜ denotes fuzzy relation in three persons’ trust each other, i.e.,

2.1 The Concept of Fuzzy Relation

R˜ =

31

1 0.7 0.5 0.9 1 + + + + (x1 , x1 ) (x1 , x2 ) (x1 , x3 ) (x2 , x1 ) (x2 , x2 ) 0.5 0.1 1 0.4 + + + . + (x2 , x3 ) (x3 , x1 ) (x3 , x2 ) (x3 , x3 )

μ R˜ (xi , xi ) = 1 expresses that everybody trusts most himself. μ R˜ (x2 , x1 ) = 0.1 indicates x2 to x1 “distrust basically”. Following next is a few special fuzzy relations to be introduced, which is often used in later fuzzy relational operations, and suppose R˜ to be fuzzy relation in X . Definition 2.1.2 Inverse fuzzy relation of fuzzy relation R˜ denotes R˜ −1 , its membership function being μ R˜ −1 (x, y) = μ R˜ (y, x), ∀x, y ∈ X . Example 2.1.4 In Example 2.1.3, inverse relation of R˜ is R˜ −1 =

0.9 0.5 0.7 1 0.1 1 + + + + + (x1 , x1 ) (x1 , x2 ) (x1 , x3 ) (x2 , x1 ) (x2 , x2 ) (x2 , x3 ) +

0.4 1 0.5 + + . (x3 , x1 ) (x3 , x2 ) (x3 , x3 )

Definition 2.1.3 If fuzzy relation R˜ satisfies μ R˜ −1 (x, y) = μ R˜ (x, y), ∀x, y ∈ X , then R˜ is called symmetry. Example 2.1.5 The “friend relation” is symmetric, while “paternity relation” and “consequence relation” are not symmetric. Definition 2.1.4 Fuzzy relation I˜ on X called identical relation means that I˜ represents an ordinary relation with its membership function being 1, x = y, μ I˜ (x, y) = ∀x, y ∈ X . 0, x = y, Definition 2.1.5 Zero relation O˜ and the whole relation X˜ are μ O˜ (x, y) = 0, μ X˜ (x, y) = 1, ∀x, y ∈ X . Definition 2.1.1 can be expanded into fuzzy relations between finite, even an infinite universe. Since fuzzy relation R˜ is given through a set R˜ in Cartesian product set X × X , then some operations and properties of fuzzy relations are all those of fuzzy sets. In addition, the fuzzy relation R˜ is a special fuzzy set, R˜ must have some special operations of its own.

32

2 Fuzzy Relation

2.2 The Operations of Fuzzy Relation Definition 2.2.1 Assume that both R˜ 1 and R˜ 2 are fuzzy relation from X to Y , if R˜ 1 (x, y) ≤ R˜ 2 (x, y), for all x ∈ X, y ∈ Y , then R˜ 1 is included in R˜ 2 . We write R˜ 1 ⊆ R˜ 2 . Definition 2.2.2 Assume that both R˜ 1 and R˜ 2 are fuzzy relation from X to Y , if R˜ 1 ⊆ R˜ 2 and R˜ 2 ⊆ R˜ 1 , then R˜ 1 and R˜ 2 are identical. We write R˜ 1 = R˜ 2 . Definition 2.2.3 The union R˜ 1 ∪ R˜ 2 of two fuzzy relations R˜ 1 and R˜ 2 is a new fuzzy relation from X to Y , R˜ 1 ∪ R˜ 2 has the following membership function: ( R˜ 1 ∪ R˜ 2 )(x, y) = R˜ 1 (x, y) ∨ R˜ 2 (x, y). Definition 2.2.4 The intersection R˜ 1 ∩ R˜ 2 of two fuzzy relations R˜ 1 and R˜ 2 is a new fuzzy relation from X to Y , R˜ 1 ∩ R˜ 2 has membership function as follows: ( R˜ 1 ∩ R˜ 2 )(x, y) = R˜ 1 (x, y) ∧ R˜ 2 (x, y). Definition 2.2.5 The complement of fuzzy relation R˜ is denoted by R˜ C and is a new fuzzy relation from X to Y , R˜ C has the next membership function: ˜ y). R˜ C (x, y) = 1 − R(x, Theorem 2.2.1 (Decomposition Theorem) If R˜ is a fuzzy relation from X to Y , then R˜ can be represented in the form below: R˜ =

λRλ ,

λ∈[0,1]

˜ where Rλ = {(x, y)| R(x, y) ≥ λ, x ∈ X, y ∈ Y }.

2.3 The Composition of Fuzzy Relation Definition 2.3.1 Suppose R˜ 1 to be a fuzzy relation from X to Y , R˜ 2 is a fuzzy relation from Y to Z , then composition R˜ 1 ◦ R˜ 2 of R˜ 1 and R˜ 2 is a fuzzy relation from X to Z ; its membership function confirms as follows:

2.3 The Composition of Fuzzy Relation

μ( R˜ 1 ◦ R˜ 2 ) (x, z) =

33

[μ R˜ 1 (x, y) ∧ μ R˜ 2 (y, z)], ∀(x, z) ∈ X × Z

(2.3.1)

y∈Y

where x ∈ X, z ∈ Z . If R1 , R2 are two ordinary relations, according to method in ordinary set, its composition denotes R1 ◦ R2 = {(x, z)|(x, z) ∈ X × Z , ∃y ∈ Y, s.t. (x, y) ∈ R1 , (y, z) ∈ R2 }.

(2.3.2)

From here, as an ordinary relation R1 and R2 , its composition (2.3.1) and (2.3.2) should be accordant. In fact, at this time, composition (2.3.1) of R1 and R2 also can take only two values {0, 1}. It is easy to prove that (2.3.1) is equivalent to (2.3.2). The above composition of fuzzy relation is based on operator {∨, ∧}, the {∨, ∧} is usually called max-min operators, of course, we can define again composition of fuzzy relation based on operator {∨, ·}, similar results can still be obtained, which are no longer discussed. If there is no special explanation, the subsequent composition of fuzzy relation will be considered based on {∨, ∧}. Example 2.3.1 Suppose R˜ 1 to be a fuzzy relation in X and Y , its membership function 2 is μ R˜ 1 (x, y) = e−k(x−y) and R˜ 2 is a fuzzy relation in Y and Z , its membership 2 function is μ R˜ 2 (y, z) = e−k(y−z) (k 1, constant), then its composition R˜ 1 ◦ R˜ 2 is a fuzzy relation in X and Z , its membership function is μ( R˜ 1 ◦ R˜ 2 ) (x, z) =

y∈Y

=e

[e−k(x−y)

2

−k x−

e−k(y−z) ] 2

x − z 2 x + z 2 −k 2 2 =e .

Example 2.3.2 Let R˜ indicate the relationship between color of tomatoes and degree of ripeness, where color universe is X = {gr een, yellow, r ed}, degree universe of ripeness is Y = {ver dant, hal f − matur e, matur e}, taste universe is Z = {sour, taste − less, sweet}. We denote the fuzzy relation R˜ 1 and R˜ 2 as follows: R˜ 1 =

0.5 0 1 + + (gr een, ver dant) (gr een, hal f − matur e) (gr een, matur e) +

1 0.4 0.3 + + (yellow, ver dant) (yellow, hal f − matur e) (yellow, matur e)

+

0.2 1 0 + + . (r ed, ver dant) (r ed, hal f − matur e) (r ed, matur e)

34

2 Fuzzy Relation

R˜ 2 =

0.2 0 1 + + (ver dant, sour ) (ver dant, taste − less) (ver dant, sweet) +

1 0.7 + (hal f − matur e, sour ) (hal f − matur e, taste − less)

+

0 0.7 0.3 + + (hal f − matur e, sweet) (matur e, sour ) (matur e, taste − less)

+

1 . (matur e, sweet)

If the (∨, ∧) operator is selected, then R˜ 3 = R˜ 1 ◦ R˜ 2 =

0.5 0.3 1 + + (gr een, sour ) (gr een, taste − less) (gr een, sweet) +

1 0.4 0.7 + + (yellow, sour ) (yellow, taste − less) (yellow, sweet)

+

0.7 1 0 + + . (r ed, sour ) (r ed, taste − less) (r ed, sweet)

If the (∨, ·) operator is selected, then R˜ 4 = R˜ 1 ◦ R˜ 2 =

0.5 0.15 1 + + (gr een, sour ) (gr een, taste − less) (gr een, sweet) +

1 0.4 0.7 + + (yellow, sour ) (yellow, taste − less) (yellow, sweet)

+

0.7 1 0.14 + + . (r ed, sour ) (r ed, taste − less) (r ed, sweet)

Definition 2.3.2 If R˜ is fuzzy relation on X , then the composition of k fuzzy relations can be denoted as ˜k ˜ ˜ ˜ R ◦ R ◦· · · ◦ R = R .

(2.3.3)

k

Proposition 2.3.1 The composition of fuzzy relations have associativity, namely ( R˜ 1 ◦ R˜ 2 ) ◦ R˜ 3 = R˜ 1 ◦ ( R˜ 2 ◦ R˜ 3 ). Proof Because [( R˜ 1 ◦ R˜ 2 )(x, z) R˜ 3 (z, w)] [( R˜ 1 ◦ R˜ 2 ) ◦ R˜ 3 ](x, w) = z∈X

(2.3.4)

2.3 The Composition of Fuzzy Relation

35

{ [ R˜ 1 (x, y) R˜ 2 (y, z)] R˜ 3 (z, w)} z∈X y∈X [ ( R˜ 1 (x, y) R˜ 2 (y, z) R˜ 3 (z, w))] = y∈X z∈X { R˜ 1 (x, y) [ ( R˜ 2 (y, z) R˜ 3 (z, w))]} = y∈X z∈X [ R˜ 1 (x, y) ( R˜ 2 ◦ R˜ 3 )(y, w)] = =

y∈X

= [ R˜ 1 ◦ ( R˜ 2 ◦ R˜ 3 )](x, w), therefore, the proof is finished. ˜ we have Proposition 2.3.2 For arbitrarily fuzzy relation R, ˜ O˜ ◦ R˜ = R˜ ◦ O˜ = O. ˜ I˜ ◦ R˜ = R˜ ◦ I˜ = R, ˜ Proposition 2.3.3 If S˜ ⊆ T˜ , then R˜ ◦ S˜ ⊆ R˜ ◦ T˜ , S˜ ◦ R˜ ⊆ T˜ ◦ R. ˜ Proposition 2.3.4 For arbitrarily a tuft fuzzy relation { R˜ i }i∈I and fuzzy relation R, we have ˜ (2) ( R˜ i ) ◦ R˜ = R˜ ◦ R˜ i ; R˜ i ◦ R. (1) R˜ ◦ ( R˜ i ) = i∈I

i∈I

i∈I

i∈I

Proof Only prove (1). ∀(x, , z) ∈ X × X μ R◦( {μ R˜ (x, y) [ μ R˜ i (y, z)]} ˜ μ R˜ ) (x, z) = i y∈X i∈I i∈I {μ R˜ (x, y) [ μ R˜ i (y, z)]} = { [μ R˜ (x, y) μ R˜ i (y, z)]} = y∈X i∈I i∈I y∈X μ( R◦ μ( R◦ = ˜ R˜ i ) (x, z) = ˜ R˜ i ) (x, z). i∈I

i∈I

Therefore (1) holds. Proposition 2.3.5 (1) R˜ ◦ (

i∈I

(2) (

R˜ i ) ⊆

i∈I

R˜ i ) ◦ R˜ ⊆

i∈I

R˜ ◦ R˜ i ;

˜ R˜ i ◦ R.

i∈I

Proof Onlyprove (1). ∀i ∈ I, R˜ i ⊆ R˜ i , hence ∀(x, z) ∈ X × X, ∀i ∈ I , from Proposition 2.3.3, then i∈I

μ[ R◦( ˜ ˜ R˜ i ) (x, z), R˜ i )] (x, z) μ( R◦ i∈I

hence μ[ R◦( ˜ R˜ i )] (x, z) i∈I

Therefore (1) holds.

μ( R◦ ˜ R˜ i ) (x, z).

36

2 Fuzzy Relation

Proposition 2.3.6 ( R˜ 1 ◦ R˜ 2 )−1 = R˜ 2−1 ◦ R˜ 1−1 . Proof ∀(x, z) ∈ X × X , we have μ( R˜ 1 ◦ R˜ 2 )−1 (x, z) = μ( R˜ 1 ◦ R˜ 2 ) (z, x) = =

[μ R˜ 1 (z, y) ∧ μ R˜ 2 (y, x)]

y∈X

[μ R˜ 1−1 (y, z) ∧ μ R˜ 2−1 (x, y)]

y∈X

=

[μ R˜ 2−1 (x, y) ∧ μ R˜ 1−1 (y, z)]

y∈X

= μ( R˜ 2−1 ◦ R˜ 1−1 ) (x, z). Hence ( R˜ 1 ◦ R˜ 2 )−1 = R˜ 2−1 ◦ R˜ 1−1 . Proposition 2.3.7 (1) (

R˜ i )−1 =

i∈I

i∈I

R˜ i−1 ;

(2) (

i∈I

R˜ i )−1 =

i∈I

R˜ i−1 .

Proof Only prove (1). ∀(x, y) ∈ X × X , μ( R˜ i )−1 (x, y) = μ( R˜ i ) (y, x) = μ R˜ i (y, x) i∈I

i∈I

= μ R˜ i−1 (x, y) = μ( R˜ i−1 ) (x, y). i∈I

i∈I

i∈I

Therefore (1) holds.

2.4 The Properties of Fuzzy Relation ˜ then Definition 2.4.1 Suppose R˜ to be a fuzzy relation on X . If R˜ satisfies I˜ ⊆ R, R˜ is called a reflexive fuzzy relation. ˜ then Definition 2.4.2 Given that R˜ is a fuzzy relation on X . If R˜ satisfies R˜ −1 = R, ˜ R is called a symmetry fuzzy relation. ˜ then R˜ Definition 2.4.3 Let R˜ be a fuzzy relation on X . If R˜ satisfies R˜ ◦ R˜ ⊆ R, is called a transitivity fuzzy relation. Notice, if R is an ordinary relation on X ; R is transitive if and only if (x, y)∈ R and (y, z) ∈ R, then (x, z)∈ R. It is easy to understand transitivity in the Definition 2.4.3, when R˜ degenerates into the ordinary relation, with the ordinary transitivity being the same. Proposition 2.4.1 still symmetric.

The union and intersection of symmetric fuzzy relation also are

Proposition 2.4.2

The intersection of transitive fuzzy relation is transitive.

2.4 The Properties of Fuzzy Relation

Proposition 2.4.3

37

˜ we have the following: To arbitrarily fuzzy relation R,

Existence of inclusive R˜ is the least reflexive fuzzy relation, that is the reflexive ˜ recorded as r ( R). ˜ closure of R, ˜ (2) Existence of inclusive R is the least symmetric fuzzy relation, that is the symmetric ˜ recorded as S( R). ˜ closure of R, ˜ that is the transitive (3) Existence of the least transitive fuzzy relation contains R, ˜ ˜ closure of R, recorded as T ( R). (1)

Proof Now, we only prove (2). ˜ because Use Q˜ to denote all sets of containments symmetric fuzzy relation R, ˜ ˜ ˜ the whole relation X is symmetric on X , i.e., X ∈ Q, as a result not empty. Let ˜ S˜ ∈ Q} ˜ from Proposition 2.4.1. Then S˜0 is the least symmetric relation S˜0 = { S| ˜ containing R. ˜ = R˜ ∪ I˜. Proposition 2.4.4 Given that R˜ to be a fuzzy relation on X , then r ( R) ˜ = R˜ ∪ R˜ −1 . Proposition 2.4.5 Suppose R˜ to be a fuzzy relation on X , then S( R) ∞

˜ = Proposition 2.4.6 Let R˜ be a fuzzy relation on X , then T ( R)

R˜ k .

k=1

Assume that the cardinality of X is |X | = n, when n is relatively large, solv˜ based on Proposition 2.4.6 will produce great amount of calculation, ing the T ( R) in general, we employ so called a square method, through successive calculation ˜ = R˜ k will be obtained, R˜ 2 , R˜ 4 , . . . , R˜ 2k , . . ., when R˜ 2k = R˜ k first appeared, T ( R) as many as k = [logn] + 1 steps, you can get its result, the computational efficiency is very high. Proposition 2.4.7 Suppose R˜ 1 and R˜ 2 to be a symmetric fuzzy relation, then R˜ 1 ◦ R˜ 2 is symmetric ⇐⇒ R˜ 1 ◦ R˜ 2 = R˜ 2 ◦ R˜ 1 . Proof “=⇒” Because R˜ 1 ◦ R˜ 2 is symmetric, then R˜ 1 ◦ R˜ 2 = ( R˜ 1 ◦ R˜ 2 )−1 = R˜ 2−1 ◦ R˜ 1−1 = R˜ 2 ◦ R˜ 1 . “⇐=” If R˜ 1 ◦ R˜ 2 = R˜ 2 ◦ R˜ 1 , then ( R˜ 1 ◦ R˜ 2 )−1 = R˜ 2−1 ◦ R˜ 1−1 = R˜ 2 ◦ R˜ 1 = R˜ 1 ◦ R˜ 2 . Therefore R˜ 1 ◦ R˜ 2 is symmetric. Proposition 2.4.8 If R˜ denotes transitive, then R˜ −1 is transitive. ˜ then from Proposition 2.4.6, ∀(x, y) ∈ X × X , hence Proof Because R˜ ◦ R˜ ⊆ R, μ( R˜ −1 ◦ R˜ −1 ) (x, y) = μ( R◦ ˜ R) ˜ −1 (x, y) = μ( R◦ ˜ R) ˜ (y, x) μ R˜ (y, x) = μ R˜ −1 (x, y), that is, R˜ −1 means transitive.

38

2 Fuzzy Relation

Definition 2.4.4 Suppose R˜ to be a fuzzy relation on X . If R˜ satisfies reflexive and symmetry, then R˜ is called a fuzzy similarity relation. Definition 2.4.5 Let R˜ be a fuzzy relation on X . If R˜ satisfies reflexive, symmetry and transitivity, then R˜ is called fuzzy equivalence relation. ˜ then R˜ Theorem 2.4.1 Suppose R˜ is a fuzzy similarity relation on X , and R˜ 2 ⊆ R, must be a fuzzy equivalence relation. ˜ furthermore, R˜ is a fuzzy equivaProof Obviously, the R˜ is transitive by R˜ 2 ⊆ R, lence relation by the condition of theorem. Example 2.4.1 Suppose X = {x1 , x2 , x3 , x4 , x5 } denotes five persons’ sets, including the father, mother, son, daughter, neighbor, the degree of similarity can be got by comparing with each other, the following fuzzy relation R˜ can be obtained, i.e., R˜ =

0.8 0.6 0.1 0.2 1 + + + + (x1 , x1 ) (x1 , x2 ) (x1 , x3 ) (x1 , x4 ) (x1 , x5 ) +

1 0.8 0.2 0.85 0.8 + + + + (x2 , x1 ) (x2 , x2 ) (x2 , x3 ) (x2 , x4 ) (x2 , x5 )

+

0.8 1 0 0.9 0.6 + + + + (x3 , x1 ) (x3 , x2 ) (x3 , x3 ) (x3 , x4 ) (x3 , x5 )

+

0.2 0 1 0.1 0.1 + + + + (x4 , x1 ) (x4 , x2 ) (x4 , x3 ) (x4 , x4 ) (x4 , x5 )

+

0.85 0.9 0.1 1 0.2 + + + + . (x5 , x1 ) (x5 , x2 ) (x5 , x3 ) (x5 , x4 ) (x5 , x5 )

Obviously, R˜ has reflexivity and symmetry, R˜ is a fuzzy similarity relation. However, we were able to notice that μ R˜ (x1 , x2 ) = 0.8, μ R˜ (x2 , x5 ) = 0.85, but μ R˜ (x1 , x5 ) = 0.2 < min(0.8, 0.85). This shows that R˜ does not have transitivity, so R˜ is not a fuzzy equivalence relation. By calculation, the result R˜ 4 = R˜ 2 can be obtained, obviously,

2.4 The Properties of Fuzzy Relation

R˜ 2 =

39

0.8 0.8 0.2 0.8 1 + + + + (x1 , x1 ) (x1 , x2 ) (x1 , x3 ) (x1 , x4 ) (x1 , x5 ) +

0.8 1 0.85 0.2 0.85 + + + + (x2 , x1 ) (x2 , x2 ) (x2 , x3 ) (x2 , x4 ) (x2 , x5 )

+

0.85 1 0.2 0.9 0.8 + + + + (x3 , x1 ) (x3 , x2 ) (x3 , x3 ) (x3 , x4 ) (x3 , x5 )

+

0.2 0.2 0.2 1 0.2 + + + + (x4 , x1 ) (x4 , x2 ) (x4 , x3 ) (x4 , x4 ) (x4 , x5 )

+

0.85 0.9 0.2 1 0.8 + + + + . (x5 , x1 ) (x5 , x2 ) (x5 , x3 ) (x5 , x4 ) (x5 , x5 )

is a fuzzy equivalence relation. As for a series of propositions concerning fuzzy relation, the above is considered all for X fuzzy relations. We can throw away this restraint actually, that is, abovementioned proposition holds as long as the composition exists.

2.5 Several Commonly Used Membership Function For any set A, a membership function on A is any function from A to the real unit interval [0, 1]. Membership functions on A represent fuzzy subsets of A. The ˜ is usually denoted by μ A . For membership function, which represents a fuzzy set A, an element x ∈ A, the value μ A (x) is called the membership degree of x in the fuzzy ˜ The membership degree μ A (x) quantifies the grade of membership of the set A. ˜ The value 0 means that x is not a member of the fuzzy element x to the fuzzy set A. set; the value 1 means that x is fully a member of the fuzzy set. The values between 0 and 1 characterize fuzzy members, which belong to the fuzzy set only partially. If a is a real number, a1 , a2 are left and right endpoints of an interval [a1 , a2 ], n denotes a natural number. Some common fuzzy membership functions are introduced as follows, and these membership functions will have important applications in practical problems. 1. Partial minitype (abstains up) (1) Lower semi-rectangular distribution function ˜ A(x) = (2) Lower semi-Γ distribution function

1, x a, 0, a < x.

40

2 Fuzzy Relation

˜ A(x) =

1, x a, e−k(x−a) , a < x, k > 0.

(3) Lower semi-normal distribution function

˜ A(x) =

1, x a, −k(x−a)2 , a < x, k > 0. e

(4) Lower semi-Cauchy distribution function ˜ A(x) =

⎧ ⎨

x a, 1 , a < x, α > 0, β > 0. ⎩ 1 + α(x − a)β 1,

(5) Lower semi-trapezoid distribution function ⎧ 1, x a1 , ⎪ ⎨ a −x 2 ˜ , a1 < x a2 , A(x) = ⎪ a2 − a1 ⎩ 0, a2 < x. (6) Lower mountainous distribution function ⎧ 1, x a1 , ⎪ ⎨1 1 + a a π 2 1 ˜ A(x) = (x − + sin ), a1 < x a2 , ⎪ a2 − a1 2 ⎩2 2 0, a2 < x. 2. Partial large-scale (abstains down) (1) Uper semi-rectangular distribution function ˜ A(x) =

0, x a, 1, a < x.

(2) Uper semi-Γ distribution function ˜ A(x) =

0, x a, 1 − e−k(x−a) , a < x, k > 0.

(3) Uper semi-normal distribution function ˜ A(x) =

0, x a, −k(x−a)2 , a < x, k > 0. 1−e

2.5 Several Commonly Used Membership Function

41

(4) Uper semi-Cauchy distribution function ˜ A(x) =

⎧ ⎨

x a, 1 , a < x, α > 0, β > 0. ⎩ 1 + α(x − a)β 0,

(5) Uper semi-trapezoid distribution function

˜ A(x) =

⎧ 0, ⎪ ⎨ x −a

x a1 , 1

⎪ ⎩ a2 − a1 1,

, a1 < x a2 , a2 < x.

(6) Uper mountainous distribution function ⎧ 0, x a1 , ⎪ ⎨1 1 π a2 + a1 ˜ A(x) = + sin ), a1 < x a2 , (x − ⎪ a2 − a1 2 ⎩2 2 1, a2 < x. 3. Normal type (middle type) (1) Rectangular distribution function ⎧ x a − b, ⎨ 0, ˜ A(x) = 1, a − b < x a + b, ⎩ 0, a + b < x. (2) Peak Γ distribution function ˜ A(x) =

x a, ek(x−a) , e−k(x−a) , a < x, k > 0.

(3) Normal distribution function 2 ˜ A(x) = e−k(x−a) , k > 0.

(4) Cauchy distribution function ˜ A(x) =

1 , α > 0. 1 + α(x − a)β

where β is a positive even number. (5) Trapezoid distribution function

42

2 Fuzzy Relation

⎧ 1, x a − a2 , ⎪ ⎪ ⎪ a2 + x − a ⎪ ⎪ ⎪ ⎪ a − a , a − a2 < x a − a1 , ⎨ 2 1 ˜ 1, a − a1 < x a + a1 , A(x) = ⎪ a2 − x + a ⎪ ⎪ ⎪ , a + a1 < x a + a2 , ⎪ ⎪ ⎪ ⎩ a2 − a1 0, a + a2 < x. (6) Mountainous distribution function ⎧ 0, ⎪ ⎪ ⎪ ⎪1 ⎪ ⎪ + ⎪ ⎨2 ˜ A(x) = 1, ⎪ ⎪ ⎪1− ⎪ ⎪ ⎪ ⎪2 ⎩ 0,

x −a2 , 1 π a2 + a1 sin ), −a2 < x −a1 , (x − 2 a2 − a1 2 −a1 < x a1 , 1 π a2 + a1 sin ), a1 < x a2 , (x − 2 a2 − a1 2 a2 < x.

4. Others type (1) Lower semi-k times parabolic distribution function ⎧ 1, x a, ⎪ ⎨ b−x k ˜ A(x) = ( ) , a < x b, ⎪ b−a ⎩ 0, b < x.

(2.5.1)

(2) Uper semi-k times parabolic distribution function ⎧ 0, x a, ⎪ ⎨ b−x k ˜ A(x) = ( ) , a < x b, ⎪ b−a ⎩ 1, b < x. (3) k times parabolic distribution function ⎧ 0, x a, ⎪ ⎪ ⎪ x −a k ⎪ ⎪ ⎪ ( ) , a < x b, ⎪ ⎨ b−a ˜ 1, b < x c, A(x) = ⎪ ⎪ d−x k ⎪ ⎪ ) , c < x d, ( ⎪ ⎪ ⎪ ⎩ d −c 0, d < x. (4) ∀x ∈ X, f (x) is a real bounded function defined on X , and its infimum and supremum are written as inf( f ) and sup( f ), respectively, such that we define

2.5 Several Commonly Used Membership Function

M˜ f (x) =

f (x) − inf( f ) sup( f ) − inf( f )

43

n (2.5.2)

calling M˜ f : X → [0, 1] a maximal set of f , where M˜ f (x) = 0. (5) Regard D˜ 0 = {x ∈ m | f 0 (x) z 0 } as a fuzzy objective set and assume a membership function of D˜ 0 as follows: ⎧ ⎨ 0, n if f 0 (x) z 0 − d0 , A˜ 0 (x) = b0 − t0 d0 , if f 0 (x) = z 0 − t0 , 0 t0 d0 , ⎩ 1, if f 0 (x) z 0 ,

(2.5.3)

where d0 0 is a maximum flexible index of f 0 (x) and z 0 an objective value. (6) If D˜ i = {x ∈ m | f i (x) bi } (1 i p) is a fuzzy constraint set corresponding to fuzzy constraint inequations f i (x) bi , then the membership functions of D˜ i are ⎧ ⎨ 0, n if f i (x) bi + di , μi (x) = bi − ti di , if f i (x) = bi + ti , 0 ti di , ⎩ 1, if f i (x) bi ,

(2.5.4)

where di ∈ ( is a real number set) denotes a maximum flexible index of f i (x).

Chapter 3

Fuzzy Relational Equations/Inequalities

Fuzzy relational equations (FRE) were firstly proposed by Sanchez [1]. Fuzzy relational equations/inequalities (FRE/FRI) have played an important role in fuzzy set theory and fuzzy logic systems, and many researchers have discussed them based on different fuzzy relational compositions [2–9]. The complete solution set of continuous t-norm fuzzy relational equations can completely be determined by a unique maximum solution and a finite number of minimal solutions [10]. The maximum solution is easily solved, but computing all the minimal solutions to FRE/FRI remains as a challenging problem [11–13]. Even though Li and Fang discuss the classification and the solvability of general fuzzy relational equations with various compositions [14].

3.1 (∨, ∧) Fuzzy Relational Inequalities 3.1.1 Introduction In this section, we recall some basic concepts and important properties of (∨, ∧) Fuzzy relational inequalities (FRI). We prove that the solution set of (∨, ∧) FRI can be completely determined by a unique maximum solution and a finite number of minimal solutions. The maximum solution can be easy computed by using simple formula. However, finding the minimum solution set of (∨, ∧) FRI is an NP problem. Consequently, for obtaining all the solutions, it is enough to get all the minimal solutions by an FRI path which is to be proposed here to do the job.

© Springer Nature Switzerland AG 2020 B.-Y. Cao et al., Fuzzy Relational Mathematical Programming, Studies in Fuzziness and Soft Computing 389, https://doi.org/10.1007/978-3-030-33786-5_3

45

46

3 Fuzzy Relational Equations/Inequalities

3.1.2 Model We discuss the following (∨, ∧) fuzzy relational inequalities: x ◦ A b, x ◦ B d,

(3.1.1)

where the operation “◦” denotes (∨, ∧) composition, A = (ai j )(0 ai j 1), B = (bik )(0 bik 1) are respectively m × n and m × l-dimensional fuzzy matrices, b = (b1 , b2 , . . . , bn )(0 b j 1), d = (d1 , d2 , . . . , dl )(0 dk 1) are respectively n and l dimensional vectors, i ∈ I = {1, . . . , m}, j ∈ J = {1, . . . , n}, k ∈ K = {1, . . . , l} are index sets. Without loss of generality, assume that problem (3.1.1) satisfies the following inequality: 1 d1 d2 · · · dl 0. {x|x ◦ A b, xi ∈ [0, 1], i ∈ I }, S2 = {x|x ◦ B d, xi ∈ [0, 1], i ∈ I }, Let S1 = S = S1 S2 . It implies that, for any x ∈ S, we have m i=1 m

(xi (xi

ai j ) b j ,

j ∈ J,

bik ) dk ,

k ∈ K.

i=1

Readers may refer to Refs. [15–19] for a rather complete overview of (∨, ∧) fuzzy relational inequalities/equations. Definition 3.1.1 Solution xˆ ∈ S is called a maximum one if x xˆ for all x ∈ S. Also, a solution xˇ ∈ S is called a minimal solution if x x, ˇ for any x ∈ S, implies x = x. ˇ Let x 1 = (x11 , x21 , . . . , xm1 ), x 2 = (x12 , x22 , . . . , xm2 ) ∈ S. We write x 1 x 2 if and only if xi1 xi2 , for all i ∈ I . x 1 < x 2 if and only if xi1 xi2 and there exists some i ∈ I such that xi1 < xi2 . Obviously, the operator ‘’ forms a partial order relation on S and (S, ) becomes a lattice. We also write x 1 x 2 (x 1 > x 2 ) if and only if x 2 x 1 (x 2 < x 1 ).

3.1.3 Theorem for Maximum Solution Define xˆ = (xˆ1 , xˆ2 , . . . , xˆm ) by using the following operation: xˆi =

1, if ai j b j for any j ∈ J, min{b j |ai j > b j , j ∈ J }, otherwise.

(3.1.2)

3.1 (∨, ∧) Fuzzy Relational Inequalities

47

Lemma 3.1.1 Assume that S = φ, x ∈ [0, 1]m . If there exists some i 0 ∈ I such that / S. xi0 > xˆi0 then x ∈ Proof It follows from xi0 > xˆi0 that xˆi0 < 1 and there exists some j ∈ J such that ai0 j > b j , implying that there exists some j0 ∈ J such that ai0 j0 > b j0 , xˆi0 = b j0 = min{b j |ai0 j > b j , j ∈ J } < 1 and xˆi0 ∧ ai0 j0 = b j0 ∧ ai0 j0 = b j0 . So, xi0 ∧ ai0 j0 > m / S. b j0 and (xi ai j0 ) > b j0 , which implies x ◦ A b. Thus, x ∈ i=1

Lemma 3.1.2 S = φ if and only if xˆ ◦ B d. Proof Assume that S = φ and xˆ ◦ B d. It follows that there exists some k0 ∈ K m such that (xˆi bik0 ) < dk0 . Then, for any x ∈ [0, 1]m such that x x, ˆ we have m i=1

(xi

i=1

bik0 )

m

(xˆi

bik0 ) < dk0 . That is, x ∈ / S if x ∈ [0, 1]m and x x. ˆ If x ∈

i=1

/ S by Lemma 3.1.1., [0, 1]m and there exists some i 0 ∈ I such that xi0 > xˆi0 , then x ∈ implying S = φ. There is a contradiction here. So, xˆ ◦ B d if S = φ. Assume that xˆ ◦ B d. In order to prove S = φ, we only show that xˆ ◦ A b. On the basis of the definition of x, ˆ for any i ∈ I , xˆi = 1 or xˆi = min{b j |ai j > b j , j ∈ J }. If xˆi = 1, then ai j b j for any j ∈ J . Therefore, xˆi ∧ ai j b j for each j ∈ J . If xˆi = min{b j |ai j > b j , j ∈ J }, then ai j b j or ai j > b j for any j ∈ J . If ai j b j , then xˆi ∧ ai j b j . If ai j > b j , then xˆi b j and xˆi ∧ ai j b j . Thus, for any j ∈ J , m (xi ai j ) b j . Therefore, xˆ ∈ S, that is, S = φ.

i=1

By Lemmas 3.1.1 and 3.1.2, we can obtain the following result. Lemma 3.1.3 If S = φ, then x ∈ S and xˆ is a unique maximum solution of (3.1.1). Note 3.1.1 If 1 b1 b2 · · · bn , then, for obtaining the maximum solution of problem (3.1.1), we can adopt the following procedure: for i = 1 to m for j = n to 1 if ai j > b j then xˆi = b j break; end end end

3.1.4 Theorem for Minimal Solution Define index sets Ik = {i ∈ I | min{bik , xˆi } dk } for any k ∈ K and Λ = I1 × I2 × · · · × Il . A vector p = ( p1 , p2 , . . . , pl ) ∈ Λ if and only if pk ∈ Ik , ∀k ∈ K . For any p ∈ Λ, define

48

3 Fuzzy Relational Equations/Inequalities

K ip = {k ∈ K | pk = i}, i ∈ I

(3.1.3)

and F : Λ −→ R m such that, Fi ( p) =

max dk , if K ip = φ, k∈K ip

0,

∀i ∈ I.

if K ip = φ,

(3.1.4)

Lemma 3.1.4 If S = φ, then Ik = φ for any k ∈ K , that is, Λ = φ. Proof Assume that there exists some k0 ∈ K such that Ik0 = φ. It follows from the m definition of Ik that we have min{bik0 , xˆi } < dk0 for any i ∈ I . So, (xˆi bik0 ) < i=1

/ S. That is, S = φ. Hence, if S = φ, then Ik = φ for any k ∈ K . dk0 , implying xˆ ∈ Theorem 3.1.1 Suppose that S = φ. Then we have the following results. (1) For any p ∈ Λ, we have F( p) ∈ S. (2) For any x ∈ S, there exists some p ∈ Λ such that F( p) x. Proof (1) Let p ∈ Λ and x = F( p) where xi = Fi ( p) for any i ∈ I . Given i ∈ I . If K ip = φ, then xi = 0 xˆi . If K ip = φ, then xi = max dk . If k ∈ K ip , then we have k∈K ip

xˆi dk since pk ∈ Ii and Ik = {i ∈ I | min{bik , xˆi } dk }. It implies that xi xˆi . So, we can obtain x = F( p) x, ˆ i.e., x ◦ A xˆ ◦ A b. For any k0 ∈ K , there exists i 0 ∈ I such that pk0 = i 0 ∈ Ik0 and xi0 = Fi0 ( p) = max dk dk0 . From pk0 = i 0 ∈ Ik0 , we have min{bi0 k0 , xˆi0 } dk0 . It follows that i

k∈K p0

bi0 k0 dk0 . Therefore, bi0 k0 ∧ xi0 dk0 . It implies

m

(xi

bik0 ) dk0 . So, x ◦

i=1

B d. So, for any p ∈ Λ, we have F( p) ∈ S. m m (2) For any x ∈ S, we have x xˆ and (xˆi bik ) (xi bik ) dk . Therei=1 i=1 fore, for any k ∈ K , there exists some i 0 ∈ I such that xˆi0 bi0 k xi0 bi0 k dk , i.e., i 0 ∈ Ik and xi0 dk . We get pk = i 0 . Then p = ( p1 , p2 , . . . , pl ) ∈ Λ. Hence, Fi0 ( p) = max dk xi0 . So, for any x ∈ S, there exists some p ∈ Λ such that i

k∈K p0

F( p) x.

Definition 3.1.2 Vector p = ( p1 , p2 , . . . , pl ) is called a general path or G-path of (3.1.1), if it satisfies p ∈ Λ. Let the set of all the G-paths of (3.1.1) be GP. Let p ∈ G P. p p p Solution x p = (x1 , x2 , . . . , xm ) is called a quasi-minimal one corresponding to Gp path p, where, for any i ∈ I , xi = Fi ( p). p is called a corresponding G-path of x p. Then we can obtain the following result by Theorem 3.1.1. ˆ p ∈ G P}. Theorem 3.1.2 If S = φ, then S = {x | x p x x,

3.1 (∨, ∧) Fuzzy Relational Inequalities

49

Definition 3.1.3 Vector p = ( p1 , p2 , . . . , pl ) is called an FRI path of (3.1.1) if it satisfies that ⎧ ⎨ ∈ I1 , k = 1, pk ∈ Ik , Ik { p1 , p2 , . . . , pk−1 } = φ, k ∈ K , k > 1, ⎩ = 0, otherwise. Denote the set of all the FRI paths of (3.1.1) by FRIP. Lemma 3.1.5 Suppose S = φ and x is a quasi-minimal solution of (3.1.1). There exists an FRI path p of (3.1.1) satisfying x p x, where x p is computed by (3.1.4). Proof Since x is a quasi-minimal solution of (3.1.1), there exists at least one corresponding G-path q such that x = x q . A vector p is defined by ⎧ ⎨ q1 , k = 1, pk = qk , Ik {q1 , q2 , . . . , qk−1 } = φ, k ∈ K , k > 1, ⎩ 0, otherwise.

(3.1.5)

Obviously, p is an FRI path. p p q p For any i ∈ I , if xi = 0, it follows that 0 = xi xi = x i . If xi = 0, from the p p definition of x , there exists some k ∈ K satisfying pk = i and xi = dk . By using q p (3.1.5), we have pk = qk = i. Obviously, xi = max{dk | pk = i} dk = xi . Proposition 3.1.1 (1) Let p = ( p1 , p2 , . . . , pl ) be an FRI path of (3.1.1). If pk1 = pk2 for any k1 , k2 ∈ K and k1 = k2 , then we have pk1 = pk2 = 0. (2) Assume that, d1 > d2 > · · · > dl > 0. If p, q ∈ F R I P and p = q, then x p = x q , and x p is a minimal solution for any given FRI path p. Proof (1) It can be easily verified by the definition of FRI path. (2) If p = q, then there exists some k0 ∈ K satisfying pk0 = qk0 . Without loss of generality, suppose pk0 = i = 0. From the definition of FRI path, we see pk = i for any k ∈ K and k = k0 . Then we have p

xi = max{dk | pk = i} = dk0 . q

p

If qk = i for any k ∈ K , then xi = 0 = xi . If there exists some k1 = k0 such that qk1 = i, then we have dk0 = dk1 and q

xi = max{dk |qk = i} = dk1 = dk0 . q

p

So, xi = xi , i.e., x p = x q . Assume that p is an FRI path of (3.1.1), x ∈ S and x < x p . Then, for any i ∈ I , p p xi xi and there exists some i 0 such that xi0 < xi0 = 0. So, there is some k1 ∈ K p such that pk1 = i 0 and xi0 = dk1 > xi0 . Since x ∈ S, we have maxi∈I (xi ∧ bik1 ) dk1 . Then there exists a i = i 0 such that xˆi ∧ bik1 xi ∧ bik1 dk1 . Consequently,

50

3 Fuzzy Relational Equations/Inequalities

xi dk1 and i ∈ Ik1 . It follows from the definition of FRI path that for any k < k1 , pk = i. It implies that p xi = max{dk |qk = i} < dk1 . p

Thus, one has xi dk1 > xk . This conclusion contradicts to the assumption that x < x p . It implies that x p is a minimal solution to any given FRI path p. Example 3.1.1 We consider the following (∨, ∧) fuzzy relational inequalities x ◦ A b, x ◦ B d,

(3.1.6)

where b = (0.9, 0.8, 0.7, 0.2), d = (0.85, 0.6, 0.5, 0.1), x = (x1 , x2 , x3 , x4 , x5 , x6 ), I = {1, 2, 3, 4, 5, 6}, J = K = {1, 2, 3, 4}, ⎡

0.4 ⎢0.3 T A =⎢ ⎣0.8 0.2

0.7 0.3 0.75 0

0.95 0.4 0.2 1.0 0.3 0.2 0 0.2

0.9 0.2 0.2 0

⎡ ⎤ 0.5 0.5 0.8 ⎢0.2 0.2 0.85⎥ T ⎥, B = ⎢ ⎣0.8 0.4 0.2 ⎦ 0 0 0

0.9 0.1 0.1 0.1

0.3 0.95 0.8 0

0.85 0.1 0.1 0

⎤ 0.4 0.8⎥ ⎥. 0.1⎦ 0

By using Note 3.1.1, the maximum solution of (3.1.6) is xˆ = [0.7, 0.7, 0.9, 0.8, 1.0, 0.8]T . For any k ∈ K , we compute the Ik by Ik = {i ∈ I | min{bik , xˆi } dk }, one has 4 I1 = {3, 5}, I2 = {4, 6}, I3 = {1, 4}, I4 = {3}. So, Λ = Ik = {3, 5} × {4, 6} × {1, 4} × {3}. It follows that all the G-paths are

k=1

p 1 = [1, 3, 3, 4]T , p 2 = [3, 3, 4, 4]T , p 3 = [1, 3, 3, 6]T , p 4 = [3, 3, 4, 6]T , p 5 = [1, 3, 4, 5]T , p 6 = [3, 4, 4, 5]T , p 7 = [1, 3, 5, 6]T , p 8 = [3, 4, 5, 6]T . p

By xi =

l k=1

x1 x3 x5 x7

{dk | pk = i} , i ∈ I, compute the corresponding solutions:

= [0.5, 0, 0.85, 0.6, 0, 0]T , = [0.5, 0, 0.85, 0, 0, 0.6]T , = [0.5, 0, 0.1, 0.6, 0.85, 0]T , = [0.5, 0, 0.1, 0, 0.85, 0.6]T ,

x2 x4 x6 x8

= [0, 0, 0.85, 0.6, 0, 0]T , = [0, 0, 0.85, 0.5, 0, 0.6]T , = [0, 0, 0.1, 0.6, 0.85, 0]T , = [0, 0, 0.1, 0.5, 0.85, 0.6]T .

It follows that the set of minimal solutions of (3.1.6) is Xˇ = {x 2 , x 3 , x 4 , x 6 , x 7 , x 8 }. All the FRI paths are p¯ 1 = [0, 0, 3, 4]T , p¯ 2 = [0, 1, 3, 6]T , p¯ 3 = [0, 3, 4, 6]T , p¯ 4 = [0, 3, 4, 5]T , p¯ 5 = [1, 3, 5, 6]T , p¯ 6 = [3, 4, 5, 6]T . The corresponding solutions are

3.1 (∨, ∧) Fuzzy Relational Inequalities

51

xˇ 1 = [0, 0, 0.85, 0.6, 0, 0]T , xˇ 2 = [0.5, 0, 0.85, 0, 0, 0.6]T , 3 T xˇ = [0, 0, 0.85, 0.5, 0, 0.6] , xˇ 4 = [0, 0, 0.1, 0.6, 0.85, 0]T , xˇ 5 = [0.5, 0, 0.1, 0, 0.85, 0.6]T , x 6 = [0, 0, 0.1, 0.5, 0.85, 0.6]T . Then we can obtain the following result by Theorem 3.1.2 and Proposition 3.1.1. Theorem 3.1.3 If S = φ, then S = {x | x p x x, ˆ p ∈ G P} = {x | x p x x, ˆ p ∈ F R I P}. Note 3.1.2 Let p be an FRI path of (3.1.1). It follows from Proposition 3.1.1 that if pk1 = pk2 for any k1 , k2 ∈ K , then pk1 = pk2 = 0. Therefore, for any i ∈ I, p xi

=

dk , 0,

if there exists pk = i, otherwise.

Note 3.1.3 From Proposition 3.1.1, in order to obtain all minimal solution to (3.1.1), we just need to find all FRI paths to (3.1.1). If k1 > k2 and Ik1 ⊇ Ik2 , then deleting Ik1 from Λ does not affect the minimal solution sets of (3.1.1) based on Definition 3.1.3.

3.1.5 Algorithm and Example Based on the idea of FRI paths, we now propose an algorithm to compute the solution set of (3.1.1). Algorithm to Fuzzy Relational Inequalities (3.1.1)

Step 1. If d satisfies 1 d1 d2 · · · dl 0, then goes to Step 2. Otherwise, arrange the order of constraints so that the dk ’s are in decreasing order. Step 2. Compute the potential maximum feasible solution of xˆ by using (3.1.2). If xˆ ◦ B d, then go to Step 2. Otherwise, the feasible domain is empty, stop. Step 3. Compute the index Ik = {i ∈ I | min{xˆi , bik } dk } for any k ∈ K , and l Λ= Ik . k=1

Step 4. Reduce Λ by Note 3.1.3. Step 5. Generate all the FRI path p and F R I P. Step 6. For any p ∈ F R I P, we compute the corresponding minimal solution x p . ˆ p ∈ F R I P}. Generate S = {x | x p x x, Note 3.1.4 If A = B, then (3.1.1) will be reduced to the following fuzzy relational inequalities: d x ◦ A b. (3.1.7) If A = B and b = d, then (3.1.1) can be simplified to the following fuzzy relation equations:

52

3 Fuzzy Relational Equations/Inequalities

x ◦ A = b.

(3.1.8)

Therefore, we can utilize algorithm from (3.1.1) to solve (3.1.7) and (3.1.8). Example 3.1.2 We consider the following (∨, ∧) fuzzy relational inequalities: d x ◦ A b, where x = (x1 , x2 , x3 , x4 ), b = (0.9, 0.8, 0.7, 0.3, 0.4), d = (0.9, 0.7, 0.6, 0.3, 0.2), ⎡

0.8 ⎢0.9 A=⎢ ⎣0.0 1.0

1.0 0.7 0.9 0.5

0.5 0.6 0.6 0.7

0.3 0.2 0.2 0.3

⎤ 0.3 0.4⎥ ⎥. 0.1⎦ 0.0

Step 1. It is obvious that 1 d1 d2 d3 d4 0, then go to Step 2. Step 2. Based on (3.1.2), we have xˆ = (0.8, 1.0, 0.8, 0.9). It is clear that xˆ ◦ A d, go to Step 3. Step 3. For any k = 1, 2, 3, 4, 5, we compute the following index Ik by using Ik = {i ∈ I | min{xˆi , aik } dk }: I1 = {2, 4}, I2 = {1, 2, 3}, I3 = {2, 3, 4}, I4 = {1, 4}, I5 = {1, 2}. Thus, Λ =

5

Ik = {2, 4} × {1, 2, 3} × {2, 3, 4} × {1, 4} × {1, 2}.

k=1

Step 4. Since 3 > 1 and I1 ⊆ I3 , delete I3 from Λ. So, Λ = I1 × I2 × I4 × I5 = {2, 4} × {1, 2, 3} × {1, 4} × {1, 2}. Step 5. Generate all the following FRI paths p: p 1 = (2, 0, 1, 0), p 2 = (2, 0, 4, 0), p 3 = (4, 1, 0, 0), p 4 = (4, 2, 0, 0), p 5 = (4, 3, 0, 1), p 6 = (4, 3, 0, 2), and F R I P = { p 1 , p 2 , p 3 , p 4 , p 5 , p 6 }. Step 6. For all p ∈ F R I P, by using Note 3.1.2, the corresponding quasi-minimal solutions are xˇ 1 = (0.3, 0.9, 0.0, 0.0), xˇ 2 = (0.0, 0.9, 0.0, 0.3), xˇ 3 = (0.7, 0.0, 0.0, 0.9), xˇ 4 = (0.0, 0.7, 0.0, 0.9), xˇ 5 = (0.2, 0.0, 0.7, 0.9), xˇ 6 = (0.0, 0.2, 0.7, 0.9). Therefore, the solution set is S =

6 i=1

{xˇ i x x}. ˆ

3.1 (∨, ∧) Fuzzy Relational Inequalities

53

Example 3.1.3 We consider the following (∨, ∧) fuzzy relational equations: x ◦ A = b, where x = (x1 , x2 , . . . , x9 ), b = (0.72, 0.70, 0.64, 0.56, 0.55, 0.52, 0.48, 0.45, 0.42), ⎡

0.65 ⎢0.75 ⎢ ⎢0.82 ⎢ ⎢0.43 ⎢ AT = ⎢ ⎢0.23 ⎢ 0.7 ⎢ ⎢0.35 ⎢ ⎣0.45 0.42

0.92 0.9 0.61 0.56 0.56 0.72 0.68 0.46 0.43

0.72 0.76 0.67 0.56 0.71 0.45 0.43 0.48 0.40

0.61 0.32 0.65 0.57 0.62 0.54 0.7 0.42 0.20

0.53 0.95 0.8 0.81 0.8 0.7 0.40 0.38 0.42

0.78 0.61 0.63 0.59 0.93 0.9 0.55 0.45 0.8

0.82 0.49 0.54 0.8 0.55 0.34 0.45 0.43 0.33

0.62 0.64 0.76 0.56 0.55 0.52 0.25 0.32 0.42

⎤ 0.73 0.7 ⎥ ⎥ 0.64⎥ ⎥ 0.47⎥ ⎥ 0.38⎥ ⎥. 0.52⎥ ⎥ 0.48⎥ ⎥ 0.22⎦ 0.26

Step 1. Obviously 1 > b1 > b2 > · · · > b9 > 0, then go to Step 2. Step 2. Based on (3.1.2), we have xˆ = (0.52, 0.43, 0.45, 0.48, 0.52, 0.42, 0.56, 0.64, 0.72). It is clear that xˆ ◦ A d, goto Step 3. Step 3. For any k ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9}, we compute the following index Ik by using Ik = {i ∈ I | min{xˆi , aik } dk }: I1 = {9}, I2 = {9}, I3 = {8, 9}, I4 = {7, 8}, I5 = {7, 8}, I6 = {1, 5, 8, 9}, I7 = {4, 9}, I8 = {1, 3}, I9 = {1, 2, 6, 8}. Thus, Λ =

9

Ik .

k=1

Step 4. Since I2 ⊇ I1 , I3 ⊇ I1 , I6 ⊇ I1 , I7 ⊇ I1 and I5 ⊇ I4 , then we delete I2 , I3 , I5 , I6 , I7 from Λ. Therefore, Λ = I1 × I4 × I8 × I9 = {9} × {7, 8} × {1, 3} × {1, 2, 6, 8}. Step 5. Generate all the following FRI paths p: p 1 = (9, 7, 1, 0), p 2 = (9, 7, 3, 1), p 3 = (9, 7, 3, 2), p 4 = (9, 7, 3, 6), p 5 = (9, 7, 3, 8), p 6 = (9, 8, 1, 0), p 7 = (9, 8, 3, 0), and F R I P = { p 1 , p 2 , p 3 , p 4 , p 5 , p 6 , p 7 }. Step 6. For all p ∈ F R I P, by using Note 3.1.2, the corresponding quasi-minimal solutions are xˇ 1 xˇ 2 xˇ 3 xˇ 4 xˇ 5 xˇ 6 xˇ 6

= (0.45, 0.0, 0.0, 0.0, 0.0, 0.0, 0.56, 0.0, 0.72), = (0.42, 0.0, 0.45, 0.0, 0.0, 0.0, 0.56, 0.0, 0.72), = (0.0, 0.42, 0.45, 0.0, 0.0, 0.0, 0.56, 0.0, 0.72), = (0.0, 0.0, 0.45, 0.0, 0.0, 0.42, 0.56, 0.0, 0.72), = (0.0, 0.0, 0.45, 0.0, 0.0, 0.0, 0.56, 0.42, 0.72), = (0.45, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.56, 0.72), = (0.0, 0.0, 0.45, 0.0, 0.0, 0.0, 0.0, 0.56, 0.72).

Therefore, the solution set is S =

7 i=1

{xˇ i x x}. ˆ

54

3 Fuzzy Relational Equations/Inequalities

3.1.6 Conclusion In this section, we consider (∨, ∧) fuzzy relational inequality, proving its maximal and minimal solution theorem. Besides, we demonstrate the proposed algorithm by numerical examples. As for the (∨, ·) operator, we will discuss the next section.

3.2 (∨, ·) Type Fuzzy Relational Equation 3.2.1 Introduction We study the existence of solutions to fuzzy relational equations in (∨, ·) and the theorems for maximum and minimum solutions before we give a short-circuit for solution. And at the same time, with this relational equations, we solve the influence factor for economical benefits in enterprize of commerce, the result of which tallies with practice basically.

3.2.2 Model Let U = {x1 , x2 , . . . , x p }, V = {Y1 , Y2 , . . . , Yq }( p ≥ m, q ≥ n) be finite field, and fuzzy relation A˜ ∈ F(U × V ), x ∈ F(U ), B˜ ∈ F(V ). Then consider the generalized fuzzy relational equations: ˜ A˜ ◦ X˜ = B, (3.2.1) where “◦” represents (∨, ·) operation, that is operator (∨, ·), we call (3.2.1) a (∨, ·) fuzzy relational equations (M-PFRE). Here B˜ = (b1 , b2 , . . . , bn )T and ⎛

A˜ 1 (x1(1) ) A˜ 2 (x2(1) ) · · · ˜ ⎝ ··· ··· ··· A= A˜ 1 (x1(n) ) A˜ 1 (x2(n) ) · · ·

⎞ A˜ m (xm(1) ) ⎠, ··· (n) A˜ m (xm )

(3.2.2)

where “T” represents transpose.

3.2.3 Solubility of the M-PFRE and Theorem for Maximum Solution Definition 3.2.1 ai j α −1 bi

bi ai j

1,

,

ai j > bi , ∀ai j , bi ∈ [0, 1], ai j bi ,

(3.2.3)

3.2 (∨, ·) Type Fuzzy Relational Equation

55

where α −1 is an operator defined at [0, 1]. And let kj

n

ai j α −1 bi ( j = 1, 2, . . . , m).

(3.2.4)

i=1

Then k˜ = (k1 , k2 , . . . , km )T ∈ X˜ is a maximum element in X˜ . Proposition 3.2.1 Proof

b c ⇒ aα −1 b aα −1 c.

a > b ⇒ a > c, from (3.2.3), aα −1 b =

b a

c a

= aα −1 c.

a b ⇒ aα −1 b = 1, but aα −1 c 1, hence aα −1 b aα −1 c. aα −1 (b ∨ c) aα −1 c.

Corollary 3.2.1 Proposition 3.2.2 Proof

a · (aα −1 b) = a ∧ b; aα −1 (a · b) b.

10 a > b ⇒ aα −1 b =

b a

⇒ a · (aα −1 b) = b.

a b ⇒ aα −1 b = 1 ⇒ a · (aα −1 b) = a · 1 = a. So

a · (aα −1 b) = a ∧ b. 20

When a > ab ⇒ aα −1 (a · b) = b; a ab ⇒ aα −1 (a · b) = 1,

then aα −1 (a · b) b. Theorem 3.2.1 There exists a solution x˜ = (x1 , x2 , . . . , xm )T to fuzzy relational equations (3.2.1) if and only if ai j x j bi (i n, j m) and for each ji , there exists ji , such that ai ji · x ji = bi . Proof Sufficiency is certified. Now let us prove the necessity. If X˜ = (x1 , x2 , . . . xm )T is the solution to (3.2.1), then ai j · xi bi (i n, j m). Otherwise, if there exists i, j, such that ai j · x j > bi , then (ai1 · x1 ) ∨ · · · ∨ (ai j · x j ) ∨ · · · ∨ (aim · xm ) > bi . Contradictory, therefore, (3.2.5) holds.

(3.2.5)

56

3 Fuzzy Relational Equations/Inequalities

At the same time, if there exists a solution in (3.2.5), and ai j x j bi (i n; j m), there must exist ji for each i n, such that ai ji x ji = bi (i n). Otherwise: 10 We have proved it impossible that if for each i, there exists ji such that ai ji x ji > bi . 20 If for each j, there exists i, such that ai j x j < bi , then (ai1 x1 ) ∨ (ai2 x2 ) ∨ · · · ∨ (aim xm ) < bi , which is in contradictory with no solution to (3.2.1). In practical application, (3.2.1) probably has no solution, but small alteration may be always given to A˜ for ε and B˜ for δ, so that A˜ ε ◦ x˜ = B˜

or A˜ ◦ x˜ = B˜ δ

has a steady solution. So the following is always assumed to have a solution to (3.2.1). If B˜ in (3.2.1) is arranged in standardization, then b1 b2 · · · bn (or b1 b2 · · · bn ). For short, let bi still stand for bi and (ai j ) for (ai j ) correspondingly, then: If there exists solution X = φ to (3.2.1), then k˜ is its maximum

Theorem 3.2.2 solution.

Proof Because X = φ, then {i, ai j > bi } = φ (1 i n; 1 j m). Hence when A˜ ◦ k˜ = (bi ; i n), then bi

n n n −1 = [ai j · ( (ai j α bi )] = [ai j · (ai j α −1 bi )], i=1

n i=1

(ai j ·

i=1

bi ) = bi (1 i n). ai j

Again x, ˜ k˜ ∈ X , then,

i=1

3.2 (∨, ·) Type Fuzzy Relational Equation

k j0 =

57

n n (ai j0 α −1 bi ) = (ai j0 α −1 bi ) i=1

i=1

n n n = [ai j0 α −1 ( ai j0 · x j0 )] [ai j0 α −1 (ai j0 · x j0 )] i=1

i=1

i=1

−1

= aio j0 α (aio j0 · x j0 ) x j0 . ˜ Hence x˜ ⊆ k. ˜ then k˜ T is its maximum solution. Corollary 3.2.2 If we have solution to x˜ ◦ R˜ = B, Proof

Because

and

x˜ ◦ R˜ = B˜ ⇔ R˜ T ◦ x˜ T = B˜ T ⇒ x˜ T ⊂ [( R˜ T )T α −1 B˜ T ] = k˜ R˜ T ◦ k˜ = B˜ T ⇔ x˜ ⊂ k˜ T

and

˜ k˜ T ◦ R˜ = B,

where α −1 represents the compound operation of α −1 . So, the solution introduced in this section is suitable for the inverse problem of comprehensive decision.

3.2.4 Solubility of the M-PFRE and Theorem for Minimum Solution Definition 3.2.2

Stipulate ai∗j

= ai j β

−1

bi

kj, 0,

ai j k j = bi , others.

(3.2.6)

If k j is definition by Definition 3.2.1, it’s impossible to make ai j k j > bi . Then we can get a definition equal to Definition 3.2.2. Definition 3.2.3

Stipulate ai∗j

= ai j β

−1

bi

kj, 0,

ai j k j = bi , ai j k j < bi .

(3.2.7)

Definition 3.2.4 Matrix A˜ ∗ = (ai∗j ), as nonzero element is the element of solution ˜ then we call A˜ ∗ a matrix of solution being chosen in (3.2.1) and the set of each k, row element in A˜ ∗ is called a row element set, written as follows

58

3 Fuzzy Relational Equations/Inequalities

a∗ a∗ a∗ A˜ i∗ = ( i1 + i2 + · · · + im ) (1 i n). x1 x2 xm Definition 3.2.5 Stipulate an operator ri ri r j , If ∃ : i = j, xi P( )( )= xi xj all multiplication of sum, otherwise. Proposition 3.2.3

1in

P ˜ ∗ , where R ˜∗ = A˜ i∗ ⇐⇒ R

is one of k j (1 j m).

1in

˜∗ = R i

ri ( xi ), and ri i

i

Proof From Definition 3.2.5 and by law of set operation, It is easy to obtain

A˜ i∗ ⇐⇒

in

ri ( ). xi i i

As ai∗j = 0 is omitted in the course of P operation and also nonzero repeatedly removable element ai∗j has to be rejected when in the application of absorptive law and so on. Hence, reserve ri is one of k j . Obviously, when ai∗j > 0, k˜ = A˜ i∗ . We reject the repeatedly removable ele˜ then, x˜i∗ is obtained. ment in k,

1in

Let X = φ. Then xi∗ ∈ X is minimum solution to (3.2.1) at

Theorem 3.2.4 ai j k j = bi .

Proof As X = φ, then { j : ai j k j = bi } = φ(1 j ∈ m). Hence at A˜ · x˜i∗ = (bi ; 1 j m), we know the following from Definition 3.2.5,

bi =

n n P (ai j · r j ) ⇐⇒ (ai j · k j ) = bi (1 i n). i=1

i=1

So x˜i∗ is a solution to (3.2.1), such that it is a minimum solution. Otherwise, if we have another x˜i∗ ⊂ X , there exists (i 0 , j0 ), such that r j0 < r j0 , then n

(ai j · r j ) = ai0 j0 r j0