Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning 9789819951536, 9789819951543

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Hua Shi Hu-Chen Liu

Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning

Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning

Hua Shi · Hu-Chen Liu

Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning

Hua Shi School of Materials Shanghai Dianji University Shanghai, China

Hu-Chen Liu School of Economics and Management Tongji University Shanghai, China

ISBN 978-981-99-5153-6 ISBN 978-981-99-5154-3 (eBook) https://doi.org/10.1007/978-981-99-5154-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

With the development of artificial intelligence, developing expert systems to simulate human thinking has become a hot research topic nowadays. Expert system is an intellectual programming system that uses the knowledge captured from experts to solve specific problems reaching the level of experts. The crucial issues in developing an expert system are the representation of the obtained expert knowledge, the acquisition of experts’ professional knowledge, and the reasoning process of knowledge rules. So far, many knowledge representation methods have been introduced in the literature. Among them, the fuzzy Petri nets (FPNs) are a promising modelling tool for expert systems and have a couple of attractive advantages. Combining fuzzy sets and Petri nets, the FPNs are a graphical and mathematical model tool for representing imprecise information and supporting fuzzy reasoning in expert systems. An FPN is a marked graphical system containing places and transitions, where graphically circles represent places, bars depict transitions, and directed arcs denote the relationships between places and transitions. The main features of an FPN are that it supports visualized representation of information and provides a unified form to deal with imprecise and uncertain knowledge information. Due to these characteristics, the FPN method has been applied to many industrial fields for knowledge representation and reasoning in expert systems. Although the FPN model is a useful mathematical technique for knowledge representation and reasoning, it is plagued by a number of shortcomings when applied in real-life situations. Therefore, how to enhance the performance of FPNs has attracted considerable attention from both academics and practitioners. In this book, we provide an in-depth and systematic introduction to different types of new FPN models to enhance the performance of traditional methods and implement the rulebased reasoning intelligently. In addition, various engineering problems in different industries are provided to demonstrate the applicability and effectiveness of the developed FPN models. The strengths and practicality of the proposed models are further validated by using comparative analysis with existing methods. The book is organized into the following four parts, which comprise 21 chapters.

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Part I consists of three chapters (Chaps. 1–3) which review the literature applying FPNs for knowledge representation and reasoning and introduce a method to determine the truth degrees of input places in FPNs. Concretely speaking, Chap. 1 presents a comprehensive review of the current improved FPN models from the perspectives of reasoning algorithms, knowledge representations, and FPN models. Chapter 2 conducts a bibliometric analysis of the FPN studies to generate a global picture of developments, focus areas, and research trends in this field. Chapter 3 proposes a group decision-making method using hesitant 2-tuple linguistic term sets to obtain the initial truth values of FPNs. Part II consists of eight chapters (Chaps. 4–11) which introduce different types of improved FPNs for knowledge representation and acquisition. Specifically, Chap. 4 provides a knowledge representation and acquisition approach using the fuzzy evidential reasoning (FER) approach and the dynamic adaptive fuzzy Petri nets (DAFPNs). Chapter 5 proposes a knowledge representation and acquisition framework based on the linguistic interval 2-tuples and the interval-valued intuitionistic FPNs (IVIFPNs). Chapter 6 reports the picture fuzzy Petri nets (PFPNs) to represent and acquire imprecise and uncertain expert knowledge. Chapter 7 develops the R-numbers Petri nets (RPNs) to represent and acquire expert knowledge more accurately. Chapter 8 introduces the bipolar fuzzy Petri nets (BFPNs) to improve the performance of FPNs in knowledge representation and acquisition. Chapter 9 is concerned with the linguistic Z-number Petri nets (LZPNs), Chap. 10 is dedicated to the spherical linguistic Petri nets (SLPNs), and Chap. 11 is about the grey reasoning Petri nets (GRPNs) for knowledge representation and acquisition in the large group environment. Part III consists of six chapters (Chaps. 12–17) which introduce a variety of improved FPNs for knowledge representation and reasoning. In Chap. 12, we propose the intuitionistic fuzzy Petri net (IFPN) model to improve the performance of conventional FPNs. In Chap. 13, we develop the linguistic reasoning Petri net (LRPN) model for knowledge representation and reasoning. In Chap. 14, we describe the dynamic adaptive fuzzy Petri net (DAFPN) method for knowledge representation and reasoning. In Chap. 15, the two-dimensional uncertain linguistic Petri nets (2DULPNs) are introduced for knowledge representation and reasoning. In Chap. 16, the cloud reasoning Petri nets (CRPNs) are provided for uncertain knowledge representation and reasoning. In Chap. 17, the Pythagorean fuzzy Petri nets (PFPNs) are offered for knowledge representation and reasoning in the large group context. Part IV consists of four chapters (Chaps. 18–21) which applied the improved FPNs to different fields. Specifically, Chap. 18 proposes a fault diagnosis and cause analysis model using the FER approach and the DAFPNs. Chapter 19 constructs a failure mode and effects analysis (FMEA) model based on FER method and FPNs. Chapter 20 puts forward an FMEA model by integrating probabilistic linguistic term sets (PLTSs) and FPNs for the risk assessment and prioritization of failure modes. Chapter 21 proposes an FMEA model using interval type-2 fuzzy sets and FPNs to improve the effectiveness of the traditional FMEA.

Preface

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This book is very interesting for practitioners and academics working in the fields of knowledge management, artificial intelligence, industrial and production engineering, and management science and engineering, etc. It can be considered as the guiding document for knowledge engineers to effectively acquire and represent professional knowledge from domain experts and conduct efficient knowledge inference in developing expert systems, so as to increase and sustain the competitive advantages of knowledge-intensive organizations. This book can also serve as a useful reference source for postgraduate and senior undergraduate students in courses related to the areas indicated above. The book contains a large number of illustrations. These will help the reader to understand otherwise difficult concepts, models, and methods. We would like to acknowledge support from the major project of National Social Science Fund of China (No. 21ZDA024), the National Natural Science Foundation of China (No. 61773250), and the Fundamental Research Funds for the Central Universities (No. 22120230184). Shanghai, China June 2023

Hua Shi Hu-Chen Liu

Contents

Part I 1

2

Literature Review and Truth Determination of FPNs

FPNs for Knowledge Representation and Reasoning: A Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 FPRs and FPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 FPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 FPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Improvements of FPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Reasoning Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 New FPN Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Applications of FPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Operational Management . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Fault Diagnosis and Risk Assessment . . . . . . . . . . . . . . . . 1.4.3 Wireless Sensor Networks . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Transportation Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Biological and Healthcare Systems . . . . . . . . . . . . . . . . . . 1.4.6 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Observations and Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 4 5 7 7 8 11 11 12 14 14 15 16 17 18 18

FPNs for Knowledge Representation and Reasoning: A Bibliometric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Publication Trend in the FPN Field . . . . . . . . . . . . . . . . . . 2.3.2 Cooperation Network Analysis in the FPN Field . . . . . . 2.3.3 Co-Citation Analysis in the FPN Field . . . . . . . . . . . . . . . 2.3.4 Keyword Analysis in the FPN Field . . . . . . . . . . . . . . . . . 2.3.5 Application Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 28 28 29 34 37 39 ix

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2.4 Suggestions for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 41 42

Determining Truth Degrees of Input Places in FPNs . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Hesitant 2-Tuple Linguistic Term Sets . . . . . . . . . . . . . . . 3.2.2 Interval 2-Tuple Linguistic Model . . . . . . . . . . . . . . . . . . . 3.3 The Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Assess the Truth Degrees of Input Places . . . . . . . . . . . . . 3.3.2 Determine the Weight Vector of Experts . . . . . . . . . . . . . . 3.3.3 Compute the Truth Values of Input Places . . . . . . . . . . . . 3.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 46 48 50 50 52 53 54 58 58

Part II 4

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Improved FPNs for Knowledge Representation and Acquisition

Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Fuzzy Evidential Reasoning Approach . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Acquisition of Rule-Based Knowledge Using Belief Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Group Belief Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Defuzzification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Dynamic Adaptive Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Definition of DAFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 DAFPN Representations for WFPRs . . . . . . . . . . . . . . . . . 4.3.3 Execution Rules of DAFPNs . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Concurrent Reasoning Algorithm of DAFPNs . . . . . . . . . 4.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval-Valued Intuitionistic FPNs for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Interval 2-Tuple Linguistic Variables . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Interval-Valued Intuitionistic Fuzzy Petri Nets . . . . . . . . . . . . . . . . 5.3.1 Definition of IVIFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Representations of IVIFPRs . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Inference Algorithm of IVIFPNs . . . . . . . . . . . . . . . . . . . . 5.4 The Proposed KRA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 65 65 67 68 69 69 70 71 72 74 81 82 85 85 87 88 88 90 90 93

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Empirical Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5.2 Knowledge Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5.3 Knowledge Representation and Reasoning . . . . . . . . . . . 102 5.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6

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Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Picture Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Defuzzification of PFNs . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Picture Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Definition of PFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 PFPN Representations of WPFPRs . . . . . . . . . . . . . . . . . . 6.3.3 Knowledge Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Execution Rules of PFPNs . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Reasoning Algorithm Based on PFPNs . . . . . . . . . . . . . . 6.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Comparisons and Discussions . . . . . . . . . . . . . . . . . . . . . . 6.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 109 111 111 113 114 114 114 116 118 118 119 120 126 128 128

R-Numbers Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 R-Numbers Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Definition of RPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Weighted R-numbers Production Rules . . . . . . . . . . . . . . 7.3.3 Inference Algorithm of RPNs . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Knowledge Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Reasoning Algorithm of RPNs . . . . . . . . . . . . . . . . . . . . . . 7.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 133 135 135 136 138 139 140 141 141 150 152 152

Bipolar Fuzzy Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

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Bipolar Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Definition of BFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Weight Bipolar Fuzzy Production Rules . . . . . . . . . . . . . . 8.3.3 Execution Rules of BFPNs . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Reasoning Algorithm of BFPNs . . . . . . . . . . . . . . . . . . . . 8.4 The Proposed Knowledge Acquisition Method . . . . . . . . . . . . . . . 8.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159 159 160 161 162 165 168 168 175 176 176

Linguistic Z-Number Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Linguistic Scale Functions . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Linguistic Z-Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Proposed LZPN Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Definition of LZPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Linguistic Z-Number Production Rules . . . . . . . . . . . . . . 9.3.3 Execution Rules of LZPNs . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Simplification Method of LZPNs . . . . . . . . . . . . . . . . . . . . 9.3.5 Reasoning Algorithm of LZPNs . . . . . . . . . . . . . . . . . . . . 9.4 The Large Group Knowledge Acquisition Method . . . . . . . . . . . . 9.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 181 181 182 183 183 184 184 185 186 187 189 189 190 197 198 200 200

10 Spherical Linguistic Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Spherical Linguistic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Spherical Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Definition of SLPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Knowledge Representation . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Execution Rules of SLPNs . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 Knowledge Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.5 Reasoning Algorithm of SLPNs . . . . . . . . . . . . . . . . . . . . . 10.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Comparison and Discussions . . . . . . . . . . . . . . . . . . . . . . .

203 203 205 206 206 207 208 208 212 214 214 221

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10.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 11 Grey Reasoning Petri Nets for Knowledge Representation and Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Grey Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Grey Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Grey Production Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The Proposed GRPN Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Definition of GRPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Execution Rules of GRPNs . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Reasoning Algorithm of GRPNs . . . . . . . . . . . . . . . . . . . . 11.4 Knowledge Parameter Determination . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Assess the Knowledge Parameters of GRPNs . . . . . . . . . 11.4.2 Cluster the Grey Assessments of Experts . . . . . . . . . . . . . 11.4.3 Determine the Values of Knowledge Parameters . . . . . . . 11.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Comparison and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227 227 229 229 230 231 232 232 234 235 236 236 237 238 239 239 249 250 251

Part III Improved FPNs for Knowledge Representation and Reasoning 12 Intuitionistic Fuzzy Petri Nets for Knowledge Representation and Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 The IFSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 The OWA Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Intuitionistic Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Definition of IFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 IFPN Representations of WFPRs . . . . . . . . . . . . . . . . . . . . 12.3.3 Execution Rules of IFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 Reasoning Algorithm of IFPNs . . . . . . . . . . . . . . . . . . . . . 12.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Comparison and Discussions . . . . . . . . . . . . . . . . . . . . . . . 12.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 257 257 259 260 260 261 262 264 265 265 269 271 271

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13 Linguistic Reasoning Petri Nets for Knowledge Representation and Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Linguistic 2-Tuple Representation Model . . . . . . . . . . . . . . . . . . . . 13.3 Ordered Weighted Linguistic Reasoning Technique . . . . . . . . . . . 13.4 Linguistic Reasoning Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.1 Definition of LRPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Execution Rules of LRPNs . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.3 Reasoning Algorithm Based on LRPNs . . . . . . . . . . . . . . 13.5 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.2 Comparative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273 273 274 278 280 280 282 283 285 285 292 294 294

14 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation and Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 DAFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Definition of DAFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 DAFPN Representations for WFPRs . . . . . . . . . . . . . . . . . 14.2.3 Execution Rules of DAFPNs . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Concurrent Reasoning Algorithm of DAFPNs . . . . . . . . . 14.3 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

297 297 299 299 300 301 302 304 310 311

15 Two-Dimensional Uncertain Linguistic Petri Net for Knowledge Representation and Reasoning . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 2-Dimensional Uncertain Linguistic Variables . . . . . . . . 15.2.2 Choquet Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 The 2-Dimensional Uncertain Linguistic Choquet Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 The 2DULPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 Definition of 2DULPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 2DULPN Representation of LPRs . . . . . . . . . . . . . . . . . . . 15.3.3 Implementation Rules of 2DULPNs . . . . . . . . . . . . . . . . . 15.3.4 Reasoning Algorithm of 2DULPNs . . . . . . . . . . . . . . . . . . 15.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Comparison Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 313 315 315 316 317 318 318 318 319 320 321 322 326 328 328

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16 Cloud Reasoning Petri Nets for Knowledge Representation and Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Cloud Model Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Conversion Between Linguistic Terms and Clouds . . . . . 16.3 The Proposed CRPN Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Definition of a CRPN Model . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 CRPN Representations of CRPRs . . . . . . . . . . . . . . . . . . . 16.3.3 Implementation Rules of CRPNs . . . . . . . . . . . . . . . . . . . . 16.3.4 Reasoning Algorithm Based on CRPNs . . . . . . . . . . . . . . 16.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Comparisons and Discussions . . . . . . . . . . . . . . . . . . . . . . 16.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331 331 333 333 336 337 337 337 339 339 340 341 349 350 350

17 Pythagorean Fuzzy Petri Nets for Knowledge Representation and Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Pythagorean Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Pythagorean Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Definition of PFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 PFPN Representations of PFPRs . . . . . . . . . . . . . . . . . . . . 17.3.3 Execution Rules of PFPNs . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.4 Reasoning Algorithm of PFPNs . . . . . . . . . . . . . . . . . . . . . 17.3.5 Determining Truth Degrees of Input Places . . . . . . . . . . . 17.4 Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.1 Application of PFPNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353 353 355 357 357 358 358 360 361 362 363 370 371 372

Part IV Applications of Improved FPNs 18 Fault Diagnosis and Cause Analysis Using Dynamic Adaptive Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 Fuzzy Evidential Reasoning Approach . . . . . . . . . . . . . . . 18.2.2 Dynamic Adaptive Fuzzy Petri Nets . . . . . . . . . . . . . . . . . 18.3 Reversed DAFPN and FDCA Algorithms . . . . . . . . . . . . . . . . . . . . 18.3.1 Definition of Reversed DAFPNs . . . . . . . . . . . . . . . . . . . . 18.3.2 FDCA Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.1 Fault Judgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377 377 379 379 379 380 380 380 382 383

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18.4.2 Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.3 Cause Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4.4 Compassions to Other FPN-Based Fault Diagnosis Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

385 387

19 Failure Mode and Effect Analysis Using Fuzzy Petri Nets . . . . . . . . . 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.1 Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2.2 Execution Rules of FPNs . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 The Proposed FMEA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.1 Background and Problem Description . . . . . . . . . . . . . . . . 19.4.2 Illustration of the Proposed FMEA . . . . . . . . . . . . . . . . . . 19.4.3 Comparison and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

391 391 393 393 394 395 399 399 401 406 408 409

20 Failure Mode and Effect Analysis Using Probabilistic Linguistic Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2.1 Probabilistic Linguistic Term Sets . . . . . . . . . . . . . . . . . . . 20.2.2 Probabilistic Linguistic Fuzzy Petri Nets . . . . . . . . . . . . . 20.2.3 PLPN Representations of WFPRS . . . . . . . . . . . . . . . . . . . 20.2.4 Implementation Rules of PLPNS . . . . . . . . . . . . . . . . . . . . 20.3 The Proposed FMEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.4.3 Comparison and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411 411 414 414 415 416 417 418 421 421 426 427 430 430

21 Failure Mode and Effect Analysis Using Interval Type-2 Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Interval Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Interval Type-2 Fuzzy Petri Nets . . . . . . . . . . . . . . . . . . . . 21.2.3 IT2FPN Representations of WFPRs . . . . . . . . . . . . . . . . . 21.2.4 Execution Rules of IT2FPNs . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Proposed FMEA Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 433 435 435 439 440 440 442 444

388 389 389

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21.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4.2 Comparison and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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444 455 457 457

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

Abbreviations

2DULPN 2DULV AFPN AHP APN BFEWA BFHWA BFN BFPN BFR BFS BFWA BM CRPN CRPR CW DAFPN DFPN DTFPN DWFPN ECG EFPN ER ERP FDCA FER FIPN FMEA FPN FPPN FPR

2-Dimensional uncertain linguistic Petri net 2-Dimensional uncertain linguistic variable Adaptive fuzzy Petri net Analytic hierarchical process Associative Petri net Bipolar fuzzy Einstein weighted average Bipolar fuzzy Hamacher weighted average Bipolar fuzzy number Bipolar fuzzy Petri net Breaker failure protective relays Bipolar fuzzy set Bipolar fuzzy weighted average Bonferroni mean Cloud reasoning Petri net Cloud reasoning production rule Computing with word Dynamic adaptive fuzzy Petri net Dynamic fuzzy Petri net Dynamic timed fuzzy Petri net Dynamic weighting fuzzy Petri net Electrocardiograms Extended fuzzy Petri net Evidential reasoning Enterprise resource planning Fault diagnosis and cause analysis Fuzzy evidential reasoning Fuzzy inference Petri net Failure mode and effect analysis Fuzzy Petri net Fuzzy predicate Petri net Fuzzy production rule xix

xx

FRPN GBM GFPN GIFOWA GOWA GPFWA GPR GRA GRP GRPN GWA GWBM HFLTS HLFPN ICHA ICOWA IED IF-FMEA IFN IFOWA IFPN IFS IFWA IIICPN IT2F IT2FGWA IT2FN IT2FOWA IT2FPN IT2FS IT2WFPR ITL-GRA ITWA ITWG IVIFPN IVIFPR IVIFS IVIFV KRA LPR LPR LRPN LZPN LZPR LZWA

Abbreviations

Fuzzy reasoning Petri net Grey Bonferroni mean Generalized fuzzy Petri net Generalized IFOWA Generalized OWA Generalized Pythagorean fuzzy weighted averaging Grey production rule Grey correlation analysis Grey relational projection Grey reasoning Petri net Grey weighted averaging Grey weighted Bonferroni mean Hesitant 2-tuple linguistic term set High-level fuzzy Petri net Interval cloud hybrid averaging Interval cloud ordered weighted averaging Intelligent electronic device Intuitionistic fuzzy FMEA Intuitionistic fuzzy number Intuitionistic fuzzy OWA Intuitionistic fuzzy Petri net Intuitionistic fuzzy set Intuitionistic fuzzy weighted averaging Interval intuitionistic integrated cloud Petri net Interval type-2 fuzzy Interval type-2 fuzzy generalized weighted averaging Interval type-2 fuzzy number Interval type-2 fuzzy ordered weighted averaging Interval type-2 fuzzy Petri net Interval type-2 fuzzy set Interval type-2 weighted fuzzy production rule Interval 2-tuple linguistic GRA Interval 2-tuple weighted average Interval 2-tuple weighted geometric Interval-valued intuitionistic FPN Interval-valued intuitionistic FPR Interval-valued intuitionistic fuzzy set Interval-valued intuitionistic fuzzy value Knowledge representation and acquisition Linguistic production rule Local backup protective relays Linguistic reasoning Petri net Linguistic Z-number Petri net Linguistic Z-number production rule Linguistic Z-number weighted average

Abbreviations

MANET MCDM MGL-COPRAS MPR OWA OWLR PFN PFPN PFPN PFPR PFPWA PFS PFV PFWA PFWG PLC PLPN PLTS PPN RDAFPN RFPN RMSM RPN RPN RPV SAODV SFPN SFPN SFPR SLN SLPN SLS SNPN SPR T2FS tFPN TNA TOPSIS TOWA TRFPN TWA UCG UDHHLPN WBFPR WFPN

xxi

Mobile ad hoc network Multi-criteria decision making Multi-granular linguistic complex proportional assessment Main protective relays Ordered weighted averaging Ordered weighted linguistic reasoning Picture fuzzy number Picture fuzzy Petri net Pythagorean fuzzy Petri net Pythagorean fuzzy production rule Pythagorean fuzzy power weighted averaging Picture fuzzy set Pythagorean fuzzy value Pythagorean fuzzy weighted averaging Picture fuzzy weighted geometric Programmable logic controllers Probabilistic linguistic Petri net Probabilistic linguistic term set Probabilistic Petri net Reversed DAFPN Reversed fuzzy Petri nets R-numbers Maclaurin symmetric mean Risk priority number R-numbers Petri net Risk priority value Secure ad hoc on-demand distance vector Spherical fuzzy Petri net Synergy-effect-incorporated fuzzy Petri net Spherical fuzzy production rule Spherical linguistic number Spherical linguistic Petri net Spherical linguistic set Simplified neutrosophic Petri net Secondary backup protective relays Type-2 fuzzy set Timed fuzzy Petri net 2-Tuple normalized average Technique for order preference by similarity to an ideal solution 2-Tuple ordered weighted averaging Temporal reasoning fuzzy Petri net 2-Tuple weighted average Uncertain causality graph Unbalance double hierarchy hesitant linguistic Petri net Weight bipolar fuzzy production rule Weighted fuzzy Petri net

xxii

WFPR WFRPN WoS WPFPR WPLPR WRPR

Abbreviations

Weighted FPR Weighted fuzzy reasoning Petri net Web of Science Weight picture fuzzy production rule Weighted probabilistic linguistic production rule Weighted R-numbers production rule

List of Figures

Fig. 1.1 Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7 Fig. 2.8 Fig. 2.9 Fig. 2.10 Fig. 2.11 Fig. 2.12 Fig. 2.13 Fig. 2.14 Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 4.1 Fig. 4.2 Fig. 4.3 Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 5.1 Fig. 5.2 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 7.1

FPN representations of WFPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . The review process of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . Annual number of publications on FPNs . . . . . . . . . . . . . . . . . . . . Author cooperation network in the FPN field . . . . . . . . . . . . . . . . Institution cooperation network in the FPN field . . . . . . . . . . . . . Country/region cooperation network in the FPN field . . . . . . . . . Distribution of articles in regard to continents . . . . . . . . . . . . . . . Geographic distribution of the FPN documents . . . . . . . . . . . . . . Journal co-citation network of the FPN area . . . . . . . . . . . . . . . . . Author co-citation network in the FPN area . . . . . . . . . . . . . . . . . Document co-citation network in the FPN area . . . . . . . . . . . . . . Timeline view of document clusters in the FPN area . . . . . . . . . . Keyword co-occurrence analysis of the FPN area . . . . . . . . . . . . The keywords with the strongest citation bursts . . . . . . . . . . . . . . Distribution of publications by application areas . . . . . . . . . . . . . Flowchart of the proposed model . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregation steps by using ITWA and ITWG . . . . . . . . . . . . . . . The FPN model of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy membership function for linguistic terms . . . . . . . . . . . . . . Interval fuzzy grades set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of type 1 rule . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of type 2 rule . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of type 3 rule . . . . . . . . . . . . . . . . . . . . . . DAFPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Knowledge representation based on IVIFPNs . . . . . . . . . . . . . . . IVIFPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PFPN representation of Type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . PFPN representation of Type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . PFPN representation of Type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . PFPN model for the gene network . . . . . . . . . . . . . . . . . . . . . . . . . RPN model of Type 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 27 28 29 31 33 33 34 35 36 37 38 38 39 40 51 54 55 66 67 71 71 71 75 91 104 115 115 115 121 137 xxiii

xxiv

Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 9.1 Fig. 9.2 Fig. 9.3 Fig. 9.4 Fig. 10.1 Fig. 10.2 Fig. 10.3 Fig. 10.4 Fig. 10.5 Fig. 10.6 Fig. 11.1 Fig. 11.2 Fig. 11.3 Fig. 11.4 Fig. 11.5 Fig. 11.6 Fig. 12.1 Fig. 12.2 Fig. 12.3 Fig. 12.4 Fig. 13.1 Fig. 13.2 Fig. 13.3 Fig. 13.4 Fig. 13.5 Fig. 14.1 Fig. 14.2 Fig. 14.3 Fig. 14.4 Fig. 15.1 Fig. 15.2 Fig. 15.3 Fig. 15.4 Fig. 16.1 Fig. 16.2 Fig. 16.3 Fig. 16.4

List of Figures

RPN model of Type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RPN model of Type 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RPN of the electric vehicle motor . . . . . . . . . . . . . . . . . . . . . . . . . BFPN representation of type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . BFPN representation of type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . BFPN representation of type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . An BFPN divided into two layers . . . . . . . . . . . . . . . . . . . . . . . . . BFPN for the risk index evaluation system . . . . . . . . . . . . . . . . . . LZPN representation of LZPRs . . . . . . . . . . . . . . . . . . . . . . . . . . . LZPN for the security risk assessment . . . . . . . . . . . . . . . . . . . . . Simplified LZPN model for the security risk assessment . . . . . . . Local weight results of sensitivity analysis . . . . . . . . . . . . . . . . . . SLPN model of type 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SLPN model of type 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SLPN model of type 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of a clustering process . . . . . . . . . . . . . . . . . . . . . . . . . . SLPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution of places after backward reasoning . . . . . . . . . . . . . . GRPN representation of type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . GRPN representation of type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . GRPN representation of type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . GRPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clustering pedigree chart of certainty factors . . . . . . . . . . . . . . . . Clustering pedigree chart of global weights . . . . . . . . . . . . . . . . . IFPN representation of type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . . IFPN representation of type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . . IFPN representation of type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . . IFPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LRPN representation of type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . LRPN representation of type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . LRPN representation of type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . LRPN representation of type 4 rule . . . . . . . . . . . . . . . . . . . . . . . . LRPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of type 1 rule . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of type 2 rule . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of type 3 rule . . . . . . . . . . . . . . . . . . . . . . DAFPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2DULPN representation of type 1 LPR . . . . . . . . . . . . . . . . . . . . . 2DULPN representation of type 2 LPR . . . . . . . . . . . . . . . . . . . . . 2DULPN representation of type 3 LPR . . . . . . . . . . . . . . . . . . . . . 2DULPN of the example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CRPN model of Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CRPN model of Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CRPN model of Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The structure of CRPN for the line L0910 . . . . . . . . . . . . . . . . . . . .

137 138 142 161 161 161 163 169 185 191 195 198 209 209 209 210 215 221 233 233 234 240 243 244 262 262 262 266 281 282 282 282 287 300 301 301 305 319 319 320 323 338 338 338 342

List of Figures

xxv

Fig. 17.1 Fig. 17.2 Fig. 17.3 Fig. 17.4 Fig. 18.1 Fig. 18.2 Fig. 18.3

358 359 359 364 381 384

Fig. 19.1 Fig. 19.2 Fig. 19.3 Fig. 19.4 Fig. 19.5 Fig. 19.6 Fig. 20.1 Fig. 20.2 Fig. 20.3 Fig. 20.4 Fig. 20.5 Fig. 20.6 Fig. 21.1 Fig. 21.2 Fig. 21.3 Fig. 21.4 Fig. 21.5 Fig. 21.6

PFPN representation of Type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . PFPN representation of Type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . PFPN representation of Type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . PFPN of the security risk assessment . . . . . . . . . . . . . . . . . . . . . . DAFPN representation of WFPRs . . . . . . . . . . . . . . . . . . . . . . . . . Overall DAFPN model of the example . . . . . . . . . . . . . . . . . . . . . Fault diagnosis DAFPN model and cause analysis RDAFPN model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy ratings for O and their membership functions . . . . . . . . . . Fuzzy ratings for S and their membership functions . . . . . . . . . . Fuzzy ratings for D and their membership functions . . . . . . . . . . Fuzzy ratings for the risk of failure modes and their membership functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fuzzy inference reasoning of the proposed method . . . . . . . . . . . Fuzzy ratings for risk factor weights and their membership functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PLPN representation of Type 1 rule . . . . . . . . . . . . . . . . . . . . . . . . PLPN representation of Type 2 rule . . . . . . . . . . . . . . . . . . . . . . . . PLPN representation of Type 3 rule . . . . . . . . . . . . . . . . . . . . . . . . Flowchart of the proposed FMEA model . . . . . . . . . . . . . . . . . . . The PLPN of the ith failure mode . . . . . . . . . . . . . . . . . . . . . . . . . The PLPN of the first failure mode . . . . . . . . . . . . . . . . . . . . . . . . The membership function of a trapezoidal IT2FN . . . . . . . . . . . . IT2FPN representation of Type 1 rule . . . . . . . . . . . . . . . . . . . . . . IT2FPN representation of Type 2 rule . . . . . . . . . . . . . . . . . . . . . . IT2FPN representation of Type 3 rule . . . . . . . . . . . . . . . . . . . . . . The IT2FPN of the hth failure mode . . . . . . . . . . . . . . . . . . . . . . . The IT2FPN of the first failure mode . . . . . . . . . . . . . . . . . . . . . .

384 396 397 397 398 398 401 417 417 417 418 420 425 436 440 441 441 443 453

List of Tables

Table 1.1 Table 2.1 Table 2.2 Table 2.3 Table 2.4 Table 3.1 Table 3.2 Table 3.3 Table 3.4 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 5.8 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 7.1

Advantages and disadvantages of the two types of knowledge reasoning algorithms . . . . . . . . . . . . . . . . . . . . . . . Top 10 productive authors in the FPN field . . . . . . . . . . . . . . . . Top 10 prolific institutions in the FPN field . . . . . . . . . . . . . . . . The top 10 highly cited journals on FPN . . . . . . . . . . . . . . . . . . Top 10 most cited authors in the FPN area . . . . . . . . . . . . . . . . . Linguistic assessments on input places provided by the decision makers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval 2-tuple vector group R˜ 1 . . . . . . . . . . . . . . . . . . . . . . . . . Collective interval 2-tuple vector group . . . . . . . . . . . . . . . . . . . Interval 2-tuple truth vector and transformed fuzzy values . . . . Ratings for linguistic terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weight assessment information of fuzzy rules . . . . . . . . . . . . . . Certainty factor assessment information of fuzzy rules . . . . . . . Threshold assessment information of fuzzy rules . . . . . . . . . . . Propositions and their meanings . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic term sets adopted by the KRA team . . . . . . . . . . . . . Linguistic evaluations on input thresholds . . . . . . . . . . . . . . . . . Linguistic evaluations on output thresholds . . . . . . . . . . . . . . . . Linguistic evaluations on weights . . . . . . . . . . . . . . . . . . . . . . . . Linguistic evaluations on certainty factors . . . . . . . . . . . . . . . . . Group evaluations on input thresholds and weights . . . . . . . . . . Group evaluations on output thresholds and weights . . . . . . . . . Places of the PFPN model and their propositions . . . . . . . . . . . Linguistic term set defined by PFNs . . . . . . . . . . . . . . . . . . . . . . Local weights on six WPFPRs by five experts . . . . . . . . . . . . . . Global weights on six WPFPRs by five experts . . . . . . . . . . . . . Certain factors on six WPFPRs by five experts . . . . . . . . . . . . . Threshold values on six WPFPRs by five experts . . . . . . . . . . . Rankings of p7 , p8 and p9 by PFPNs, IFPNs and FPNs . . . . . . . Places of the RPN and their propositions . . . . . . . . . . . . . . . . . .

17 31 32 35 36 56 57 57 58 66 76 77 78 97 98 99 99 100 100 101 103 121 121 122 123 123 124 127 143 xxvii

xxviii

Table 7.2 Table 7.3 Table 7.4 Table 7.5 Table 7.6 Table 7.7 Table 8.1 Table 8.2 Table 8.3 Table 8.4 Table 9.1 Table 9.2 Table 9.3 Table 10.1 Table 10.2 Table 10.3 Table 11.1 Table 11.2 Table 11.3 Table 11.4 Table 11.5 Table 11.6 Table 12.1 Table 12.2 Table 13.1 Table 13.2 Table 13.3 Table 15.1 Table 15.2 Table 15.3 Table 16.1 Table 16.2 Table 17.1 Table 17.2 Table 17.3 Table 17.4 Table 18.1 Table 18.2 Table 19.1 Table 19.2 Table 19.3

List of Tables

Linguistic terms for assessing knowledge parameters . . . . . . . . Local weights of five WRPRs by five experts . . . . . . . . . . . . . . Input thresholds of five WRPRs by five experts . . . . . . . . . . . . . Certainty factors of five WRPRs by five experts . . . . . . . . . . . . Output thresholds of five WRPRs by five experts . . . . . . . . . . . Reasoning results by different methods . . . . . . . . . . . . . . . . . . . Places of the BFPN and their propositions . . . . . . . . . . . . . . . . . Linguistic terms defined by BFNs . . . . . . . . . . . . . . . . . . . . . . . . Bipolar fuzzy evaluation vectors of global weights given by the experts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rankings of intermediate and terminating places by different FPN models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Places of the LZPN and their propositions . . . . . . . . . . . . . . . . . Linguistic Z-number evaluation vectors of local weights . . . . . Reasoning results by different FPN methods . . . . . . . . . . . . . . . Places in the SLPN model and their propositions . . . . . . . . . . . Clustering results of Certainty factors . . . . . . . . . . . . . . . . . . . . . Reasoning results by the compared models . . . . . . . . . . . . . . . . Places in the GRPN model and their propositions . . . . . . . . . . . Global weight assessment information . . . . . . . . . . . . . . . . . . . . Certainty factor assessment information . . . . . . . . . . . . . . . . . . . Clustering results and experts’ weights of certainty factors . . . Clustering results and experts’ weights of global weights . . . . . Comparison results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Places in the IFPN model and their propositions . . . . . . . . . . . . Reasoning results with different Input/output operators . . . . . . Places in the LRPN model and their propositions . . . . . . . . . . . Linguistic truth values of the intermediate and goal places . . . . Linguistic truth values of the consequent places . . . . . . . . . . . . Places in 2DULPN model and their propositions . . . . . . . . . . . . Fuzzy measure of each place and place combination . . . . . . . . . Comparison analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time constrains of alarm events . . . . . . . . . . . . . . . . . . . . . . . . . Truth values of middle and goal places . . . . . . . . . . . . . . . . . . . . Linguistic terms for rating the truth degrees of input places . . . Places in the PFPN model and their propositions . . . . . . . . . . . . Truth assessments of the input places . . . . . . . . . . . . . . . . . . . . . Truth degrees of places for the comparative FPN models . . . . . Fuzzy ratings for linguistic terms . . . . . . . . . . . . . . . . . . . . . . . . Initial truth values of abnormal events provided by the five team members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Considered failure modes for the ship fire-safety system . . . . . Evaluate information of failure modes provided by the experts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collective risk assessment information . . . . . . . . . . . . . . . . . . . .

143 143 144 144 144 151 170 170 171 175 191 192 199 216 217 222 241 241 241 244 244 249 266 269 287 292 293 323 325 327 348 349 361 364 365 370 385 385 400 402 404

List of Tables

Table 19.4 Table 19.5 Table 20.1 Table 20.2 Table 20.3 Table 20.4 Table 20.5 Table 20.6 Table 21.1 Table 21.2 Table 21.3 Table 21.4 Table 21.5 Table 21.6 Table 21.7 Table 21.8 Table 21.9

xxix

Ranking result by the proposed FMEA method . . . . . . . . . . . . . Comparison of the risk rankings . . . . . . . . . . . . . . . . . . . . . . . . . Failure modes for the oil spill case . . . . . . . . . . . . . . . . . . . . . . . Assessed information on failure modes by the FMEA team . . . Group probabilistic linguistic evaluation matrix . . . . . . . . . . . . Results obtained by the proposed FMEA . . . . . . . . . . . . . . . . . . Risk rankings of failure modes with respect to different scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ranking comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic terms for rating failure modes . . . . . . . . . . . . . . . . . . Linguistic evaluations of failure modes provided by the FMEA team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linguistic evaluations of risk factor weights provided by the FMEA team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The IT2F group assessment matrix . . . . . . . . . . . . . . . . . . . . . . . The IT2F group weight matrix . . . . . . . . . . . . . . . . . . . . . . . . . . The normalized weight matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . Places in the IT2FPN model and their propositions . . . . . . . . . . Results obtained by the proposed approach . . . . . . . . . . . . . . . . Ranking comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407 407 422 424 425 427 428 429 445 446 447 448 450 452 453 455 456

Part I

Literature Review and Truth Determination of FPNs

Chapter 1

FPNs for Knowledge Representation and Reasoning: A Literature Review

Fuzzy Petri nets (FPNs) are a potential modeling technique for knowledge representation and reasoning of rule-based expert systems. To date, many studies have focused on the improvement of FPNs and various new models have been proposed in the literature to enhance the modeling power and applicability of FPNs. Giving this evolving research area, this chapter presents an overview of the improved FPN theories and models from the perspectives of reasoning algorithms, knowledge representations, and FPN models. In addition, a survey of the applications of FPNs is provided for solving practical problems in various fields. Finally, some findings from the literature review regrading reasoning algorithms and FPN models are discussed.

1.1 Introduction Fuzzy Petri nets (FPNs) are a modification of classical Petri nets for dealing with imprecise, vague or fuzzy information in knowledge-based systems (Looney 1988; Chen et al. 1990), which have been extensively used to model fuzzy production rules (FPRs) and formulate fuzzy rule-based reasoning automatically. An FPN is a marked graphical system containing places and transitions, where graphically circles represent places, bars depict transitions, and directed arcs denote the incidence relationships from places to transitions or from transitions to places. The main characteristics of an FPN are that it supports structural organization of information, provides visualization of knowledge reasoning, and facilitates design of efficient fuzzy inference algorithms. All these render FPNs a potential modeling methodology for knowledge representation and reasoning in expert systems (Yeung and Tsang 1994; Gao et al. 2003; Ha et al. 2007). Since the introduction of FPNs, they have received a great deal of attention from academics and practitioners for supporting approximate reasoning in a fuzzy rulebased system (Zhou and Zain 2016; Yu et al. 2022b). However, the earlier FPNs are plagued by a number of shortcomings and not suitable for increasingly complex © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_1

3

4

1 FPNs for Knowledge Representation and Reasoning: A Literature Review

knowledge-based systems. Therefore, a variety of alternative models have been put forward in the literature to enhance the knowledge representation power of FPNs and implement the rule-based reasoning more intelligently (Liu et al. 2017). Besides, FPNs have been widely used by researchers and practitioners to manage different kinds of engineering problems in many fields (Jiang et al. 2022). Therefore, this chapter aims to summarize and analyze the existing studies to enhance the performance of FPNs, and further introduce the applications of FPNs to solve real-world problems. Related articles published in international journals between 2000 and 2022 are gathered and reviewed. The specific objectives of this chapter are: (1) To establish sources of improvements around FPNs and identify those aspects that attract the most attention in the FPN literature. (2) To describe the development of FPNs and find the approaches that are prevalently applied. (3) To uncover gaps and trends in the FPN literature and highlight future directions for research. This chapter also provides a spur to further study this area in depth and develop richer knowledge on FPNs to help practitioners build effective expert systems for intelligent decision making. The rest of this chapter is organized in the following way. Section 1.2 presents the background knowledge regarding FPRs and FPNs. Section 1.3 reviews the improved FPN approaches from the perspectives of reasoning algorithms, knowledge representations, and FPN models. In Sect. 1.4, the applications of FPNs in different engineering areas are introduced. In Sect. 1.5, some observations and findings of this literature review are described. Finally, Sect. 1.6 summarizes this chapter.

1.2 FPRs and FPNs 1.2.1 FPRs FPRs have been comprehensively used to represent, capture and store vague expert knowledge in decision systems. Each rule is expressed in the form of a fuzzy if–then rule in which both the antecedent and the consequent are fuzzy terms expressed by fuzzy sets. If an FPR consists of either AND or OR connectors, then it is called a composite or compound FPR (Chen 1996). To enhance the representation and reasoning capabilities of FPRs, the weight parameter (Yeung and Tsang 1997; Tsang et al. 2004) has been incorporated into fuzzy if–then rules, obtaining the weighted FPRs (WFPRs). Let R be a set of WFPRs, i.e., R = {R1 , R2 , . . . , Rn }, the form of the ith rule can be presented as Ri : IF a THEN c (CF = μ), T h, w

(1.1)

1.2 FPRs and FPNs

5

where a and c are the antecedent and consequent parts of the rule, respectively, which comprise one or more propositions with fuzzy variables. The parameter μ(μ ∈ [0, 1]) is the certainty factor indicating the belief strength of the rule, T h = {λ1 , λ2 , ..., λm } is a set of threshold values specified for each of the propositions in the antecedent, and w = {w1 , w2 , ..., wm } is a set of weights assigned to all propositions in the antecedent, showing the relative importance of each proposition in the antecedent contributing to the consequent. In general, WFPRs can be divided into five types as listed below (Yeung and Ysang 1998; Chen 2002; Liu et al. 2013a): Type 1 A simple weighted fuzzy production rule. R: IF a THEN c (μ; λ; w). Type 2 A composite weighted fuzzy conjunctive rule in the antecedent. R: IF a1 AND a2 AND…AND am THEN c (μ; λ1 , λ2 , ..., λm ; w1 , w2 , ..., wm ). Type 3 A composite weighted fuzzy conjunctive rule in the consequent. R: IF a THEN c1 AND c2 AND…AND cm (μ; λ; w). Type 4 A composite weighted fuzzy disjunctive rule in the antecedent. R: IF a1 OR a2 OR…OR am THEN c (μ; λ1 , λ2 , ..., λm ; w1 , w2 , ..., wm ). Type 5 A composite weighted fuzzy disjunctive rule in the consequent. R: IF a THEN c1 OR c2 OR…OR cm (μ; λ; w). In many practical applications, the rules of Types 4 and 5 are not allowed to appear in a knowledge base since they can be transferred into several rules of Type 1.

1.2.2 FPNs To deal with uncertainty in knowledge representation and reasoning, FPNs have been developed from the Petri net theory, where tokens representing the state of propositions are marked by a truth value between 0 and 1. By applying a Petri net formalism to fuzzy rule-based systems, it is able to visualize the structure of an expert system and express its dynamic logic reasoning behavior efficiently. For an FPN, a transition is enabled if all of its input places are marked by a token and its real value is greater than or equal to a threshold value. The reasoning process of an FPN is executed by firing the rules and updating the truth degree vector at each reasoning step. In 1988, Looney (1988) pioneered the concept of FPNs to represent the FPRs of a rule-based decision making system. In a later work, Chen et al. (1990) proposed a more generic FPN model to model knowledge representation and described a fuzzy algorithm to perform knowledge reasoning automatically. According to Chen et al. (1990), an FPN structure is defined as an 8-tuple: FPN = (P, T , D, I, O, f, α, β) where

(1.2)

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

(1) P = { p1 , p2 , ..., pm } is a finite set of places, (2) T = {t1 , t2 , ..., tn } is a finite set of transitions, (3) D = {d1 , d2 , ..., dm } is a finite set of propositions with P ∩ T ∩ D = ∅, |P| = |D|, (4) I : T → P ∞ denotes the input function, a mapping from transitions to the bags of places, (5) O : T → P ∞ denotes the output function, a mapping from transitions to the bags of places, (6) f : T → [0, 1] denotes an association function, a mapping from transitions to real values between 0 and 1, (7) α : P → [0, 1] is an association function, a mapping from places to real values between 0 and 1, (8) β : P → D is an association function, a objective mapping between places and propositions. To capture more information of WFPRs, Yeung and Ysang (1998) improved the above FPN model by introducing the threshold and weight parameters, and an improved FPN can be presented as follows: FPN = (P, T , D, T h, I, O, F, W, f, α, β, γ , θ ),

(1.3)

where P, T , D, I, O, f and β are defined as in Eq. (1.2). T h = {λ1 , λ2 , ..., λm } is a set of threshold values, F = {f 1 , f 2 , ..., f m } is  a set of fuzzy sets, W = w1, w2 , ..., wm is a set of weights of WFPRs, α : P → F is an association function which assigns a fuzzy set to a place, γ : P → T h is an association function, a mapping from places to threshold values, (7) θ : P → W is an association function which assigns a weight to a place. (1) (2) (3) (4) (5) (6)

In an FPN, propositions are represented by places; the certainty factor of a rule is associated with its corresponding transition; the mutual causality interconnections between the propositions and reasoning rules are expressed by the arcs between places and transitions. A place may or may not contain a token associated with a truth value between 0 and 1. The token is pictorially represented by a dot. The knowledge reasoning processes are modeled through the firing of the transitions in FPNs. Based on the above specification, the three types of WFPRs can be graphically represented with FPN structures as depicted in Fig. 1.1.

1.3 Improvements of FPNs

7

Fig. 1.1 FPN representations of WFPRs

1.3 Improvements of FPNs In this section, we present the results of a comprehensive literature search on FPNs for knowledge representation and reasoning. The database used for our study is Scopus in which the articles published between 2000 to 2022 were searched. The database search was limited to peer-reviewed articles appearing in academic journals. Furthermore, we only include articles that report on an algorithm or a model to handle vague knowledge and approximate reasoning, or studies applying the FPN models for dealing with practical problems. Also, the researches based on high-level FPNs (Scarpelli et al. 1996) are not considered within this study, since they are another trend of research on Petri nets. The literature analysis begins by identifying 549 studies, which are then distilled down to 206 papers satisfying the selection criteria. To enhance the performance of FPNs, a number of studies have been conducted through improving knowledge representation and reasoning abilities or developing new FPN models. Therefore, we propose a framework for classifying the reviewed papers from the perspectives of reasoning algorithm, knowledge representation, and new FPN model. Next, we more specifically go into the critical references and show what has been done.

1.3.1 Reasoning Algorithms (1) Reasoning algorithm improvement

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

In Chen (2000), the author extended the work of Chen et al. (1990) to describe a fuzzy backward reasoning algorithm for knowledge-based systems. In Fryc et al. (2004), the authors proposed an algebraic representation of FPNs and provided a parallel algorithm for the fuzzy reasoning of expert systems. Suraj (2013) extended the FPNs by introducing three operators in the form of triangular norms as substitutes of min, max and algebraic product operators, and proposed the generalized FPNs (GFPNs) for knowledge representation and inexact reasoning in decision support systems. To overcome the issue of state space explosion, Zhou et al. (2015) proposed a biphasic decomposition algorithm that includes a backward search stage and a forward strategy for the FPN model. This algorithm can divide a large-scale FPN into a series of sub-FPNs via an index function and an incidence matrix. Zhou et al. (2019) proposed an equivalent generating algorithm to model FPNs for knowledgebased system. The proposed algorithm can produce final FPRs by investigating the inner-inference paths between them, followed by a transformation algorithm that automatically generates an equivalent FPN for the corresponding expert system. (2) Knowledge representation Chen (2002) presented a weighted FPN (WFPN) model and a weighted fuzzy reasoning algorithm for the rule-based systems using WFPNs. Ha et al. (2007) developed two types of knowledge representation parameters, i.e., input weights and output weights, and introduced a GFPN method to enhance the representation capability of WFPRs. Shih et al. (2007) reported an associative Petri net (APN) model to represent the associative production rules of a rule-based system, in which an associated parameter of every production rule was introduced by measuring the associative degree between different propositions. Liu et al. (2013c) introduced a knowledge acquisition and representation approach using the fuzzy evidential reasoning (FER) approach and the dynamic adaptive FPNs (DAFPNs). Based on the work in Liu et al. (2013a), Liu et al. (2013c) further presented an improved DAFPN model for knowledge representation and reasoning, in which distinct threshold values are assigned to both antecedent and consequent propositions of a composite production rule.

1.3.2 New FPN Models (1) FPNs combing Petri nets and fuzzy logic By combining Petri net theory and fuzzy sets, Koriem (2000) presented a modified FPN model for automated modeling and verification of ruled-based decision-making systems. Bostan-Korpeoglu and Yazici (2007) proposed an FPN model to represent imprecise knowledge, which can deal with both active and deductive rules along with the compositions and perform required computations in addition to the sup-min composition. Sobrino et al. (2021) introduced the fuzzy stochastic timed Petri nets as a graphical tool to represent time, co-occurrence, looping, and imprecision in causal flow. Cheng et al. (2019) reported a fuzzy spatio-temporal Petri net model as an

1.3 Improvements of FPNs

9

extension of Petri nets for representing fuzzy spatio-temporal knowledge and Ding et al. (2018) provided an intelligent Petri net method to model self-adaptive software systems by incorporating fuzzy rules to a regular Petri net. (2) FPNs considering time factor Suraj and Fryc (2006) proposed a new class of timed approximate Petri nets by combining FPNs with time and uncertain information for representing uncertain knowledge and evaluating inexact reasoning in decision systems. For knowledge representation of chemical abnormality, a temporal version of FPNs, called designated timed FPNs (tFPNs), was proposed in Liu et al. (2011). In the tFPN approach, a timing factor was assigned to each transition and a reliability degree was associated with each place to capture the dynamic nature of fuzzy knowledge pertaining to abnormal events. To deal with dynamic time delays between correlated variables, Yang and Li (2018) suggested a dynamic timed fuzzy Petri net (DTFPN) approach based on the dynamic time delay analysis, in which a colored graph describing dynamic time delays between correlated variables was created using data mining techniques. (3) FPNs using neural networks Li and Lara-Rosano (2000) formulated an FPN model called adaptive FPNs (AFPNs) and developed its weight learning algorithm for dynamic knowledge representation and inference. In Li et al. (2000), the authors relaxed the restrictions of the AFPN model and introduced a modified back propagation learning algorithm for knowledge learning. In Amin and Shebl (2014), the authors developed an adaptive fuzzy higher order Petri net considering the weight changes of the arc in fuzzy reasoning process, which has the learning ability as neural networks and can be used for knowledge representation and dynamic reasoning. Instead of using neural networks, Wang et al. (2014) proposed a dynamic representation of fuzzy knowledge model based on FPNs and particle swarm optimization for knowledge representation and inference. In addition, neural network-based FPNs also appeared in other researches such as Konar et al. (2005). (4) FPNs based on matrix operations In Wang et al. (2001), the authors defined an extended FPN model based on generating rules of knowledge base and presented two concurrent reasoning algorithms based on multitask schedule by considering both forward reasoning and backward reasoning. Gao et al. (2003) elaborated upon a fuzzy reasoning Petri net (FRPN) model to represent a fuzzy rule-based system, in which the algorithm exhibits parallel reasoning ability via the operators of max-algebra. Lehocki et al. (2008) proposed the logical Petri nets and FPNs as models for knowledge representation, and introduced a matrix-based algorithm for knowledge propagation in decision support systems. In Hu et al. (2011), the authors proposed the reversed Petri nets for solving backward reasoning problems, and presented a max-algebra based iterative algorithm such that the backward reasoning can be implemented efficiently.

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

Liu et al. (2016b) proposed a linguistic reasoning Petri net (LRPN) model where the truth degrees of the proposition appearing in the rules are represented by linguistic 2-tuples. Liu et al. (2016a) presented a new type of FPNs, namely intuitionistic FPNs (IFPNs) by using intuitionistic fuzzy sets and ordered weighted averaging operators. Liu et al. (2022) presented the 2-dimensional uncertain linguistic Petri nets (2DULPNs) based on 2-dimensional uncertain linguistic variables and Choquet integral for knowledge representation and reasoning. Besides, some Choquet integral aggregated operators were proposed for the approximate reasoning to capture the interactions among antecedent propositions. Liu et al. (2018b) proposed the cloud reasoning Petri nets (CRPNs) based on interval clouds and a hybrid averaging operator for knowledge representation and reasoning considering both local and ordered weight coefficients. Yue et al. (2023) proposed an unbalance double hierarchy hesitant linguistic Petri net (UDHHLPN) model for the superheat degree recognition of aluminum electrolysis cell. Yue et al. (2022) also proposed a simplified neutrosophic Petri net (SNPN) for the identification of superheat degree of aluminum electrolysis cell. Specifically, an extended technique for order preference by similarity to an ideal solution (TOPSIS) was used for knowledge fusion. Besides, Yue et al. (2021) put forward the interval-valued intuitionistic FPNs based on interval-valued intuitionistic fuzzy sets and an extended TOPSIS method, Yue et al. (2020) defined the linguistic Petri nets based on interval type-2 fuzzy sets and an extended TOPSIS method, and Yue et al. (2019) proposed the self-learning interval type-2 FPNs using interval type-2 fuzzy sets and an extended TOPSIS method. To deal with the synergy effects among events during their reasoning process, Wang et al. (2022b) defined a synergy-effect-incorporated fuzzy Petri net (SFPN) which can capture the depict catalyst and inhibitor effects in fuzzy rule-based expert systems. To address the limitations of the existing FPNs in cause analysis, Li et al. (2022) presented an enhanced grey reasoning Petri net (GRPN) model which can improve the reliability of the knowledge reasoning process based on matrix operations. (5) FPNs for knowledge acquisition and representation Shi et al. (2022) developed the linguistic Z-number Petri nets (LZPNs) for knowledge acquisition and representation in the large group environment. The linguistic Z-number production rules were introduced for knowledge representation, and a knowledge acquisition approach was proposed to obtain the knowledge parameters of LZPNs based on a large group of experts. Mou et al. (2022) introduced the spherical linguistic Petri nets (SLPNs) for knowledge representation and reasoning. The spherical linguistic sets were applied to represent the uncertainty of experts’ judgements and a large group knowledge acquisition approach was developed to determine knowledge parameters. Mou et al. (2021) proposed the R-numbers Petri nets (RPNs) for knowledge representation and acquisition. Based on R-numbers, expert knowledge was depicted in the form of weighted R-number production rules. In addition, the conflict opinions of experts were handled with the RPN model to obtain more precise knowledge parameters.

1.4 Applications of FPNs

11

In Liu et al. (2021b), the Pythagorean fuzzy Petri nets (PFPNs) were developed for knowledge representation and reasoning in the large group context. The Pythagorean fuzzy sets were used to capture imprecise knowledge and a large group truth determination method was proposed to obtain the truth degrees of input places. In Xu et al. (2020), the bipolar fuzzy Petri nets (BFPNs) were suggested for knowledge representation and acquisition considering non-cooperative behaviors. Because of the increasing scale of expert systems, a large group expert weighting method was proposed for knowledge acquisition by taking experts’ non-cooperative behaviors into account. In Liu et al. (2020), the GRPN model was established for knowledge representation and reasoning, in which grey numbers were used to represent knowledge parameters and a large group decision-making method was introduced for obtaining the knowledge parameters of GRPNs. Besides, Xu et al. (2019) presented the picture fuzzy Petri nets for knowledge representation and acquisition by considering conflicting opinions. Li et al. (2018) developed a theoretical model based on linguistic interval 2-tuples and interval-valued intuitionistic FPNs (IVIFPNs) for acquiring and representing tacit knowledge.

1.4 Applications of FPNs Due to the graphical representation and dynamic processing ability, the FPNs have been employed to address various engineering problems over the past decades. Therefore, practical applications of FPNs have been investigated in this section.

1.4.1 Operational Management (1) Disassembly process planning Gao et al. (2004) utilized an FRPN model to represent the disassembly rules in a product with uncertainty, which can attain the next operation on the product at each disassembly step based on current status and disassembly rules. Considering human factors in manufacturing systems, Tang et al. (2006) developed a fuzzy attributed Petri net model to represent the uncertainty in disassembly process due to a large amount of human intervention. Tang and Turowski (2007) proposed a fuzzy disassembly Petri net model for modelling uncertain product conditions, and designed an adaptive fuzzy system to estimate their impact on a disassembly process. Tang (2009) introduced an FPN model to represent and analyze uncertainty in the disassembly process, in which a self-adaptive disassembly process planner was designed to accumulate the past experience and exploit the “knowledge” captured in the data. In addition, Zhao et al. (2014) applied the FRPNs to disassembly sequence decision making for the endof-life product remanufacturing, and Hsu (2017) developed a fuzzy attributed and

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

timed predicate/transition net for the knowledge-based disassembly process planning considering varying skill levels of operators and product condition. (2) Operation planning Wu et al. (2002) presented a modified FPN for the optimal operation planning with resource constraints, which can model the knowledge for operation selection and yield feasible operation plans. Kasirolvalad et al. (2006) presented an AND/OR net approach for the planning of a CNC machining operation and employed the AFPNs with learning capability to improve the product and machining process quality with CNC machine tools. Kang et al. (2022) employed the multi-level variable weight FPN model to analyze the potential leakage and explosion accident process for the safe and stable operation of hydrogen refueling stations. (3) Other operational management problems Qiao et al. (2011) proposed an FPN-based model for describing the rescheduling strategy problem and discussed a fuzzy reasoning approach for rescheduling startup decision making. Ye et al. (2011) reported a knowledge-based hybrid exception handling approach for workflow management using the typed FPNs extended by process knowledge. Zhou et al. (2012) developed an FRPN framework to deal with the uncertainty, complexity, and dynamics associated with user experience modeling for product ecosystem design. Recently, Wang et al. (2019) presented a dynamic adaptive fuzzy reasoning Petri net to determine the machine energy saving state of a discrete stochastic manufacturing system.

1.4.2 Fault Diagnosis and Risk Assessment (1) Electric power system fault diagnosis Sun et al. (2004) used the FPNs for fault diagnosis of electric power systems, which can diagnose faults when alarm information of protective relays and circuit breakers is incomplete and uncertain. Luo and Kezunovic (2008) implemented the FRPNs for power system fault section estimation and He et al. (2014) employed the AFPNs to solve a power system fault-section estimation problem. In Zhang et al. (2016), the authors investigated the temporal constraint between event occurrences in power systems and introduced a temporal reasoning FPN approach for fault diagnosis. In Cheng et al. (2015a), a fault diagnosis method based on FPNs was presented to diagnose faults of power supply system devices considering service feature of information source devices. Chen et al. (2015) employed the FPNs to locate nontechnical losses and outage events in the microdistribution systems dealing with power utilities. Yang and Huang (2002) introduced an FPN-based knowledge representing approach to achieve the on-line service-restoration plan of distribution systems, and Chen et al. (2018) combined FPNs with the neural network for the fault-line selection of a small current neutral grounding system.

1.4 Applications of FPNs

13

Lin et al. (2022b) suggested an approach to power system fault diagnosis based on the fuzzy temporal order Petri nets, in which the temporal order of alarms information was combined with a hierarchical Petri net model. In Yuan et al. (2021), a fault diagnosis method based on the time sequence hierarchical fuzzy Petri nets was proposed for distribution network, in which Gaussian function was introduced to optimize the fault probability and improve the accuracy of fault diagnosis results. In Kiaei and Lotfifard (2020), a fault diagnosis model using multi-source data based on the FPNs was established for fault section identification in smart distribution systems. (2) Manufacturing system fault diagnosis To address the impact of solar array anomalies, Wu et al. (2011) established a model using fault tree analysis and FRPNs to perform reliability analysis of a solar array mechanical system. Wu et al. (2012) also developed a reliability apportionment approach by combining fuzzy comprehensive evaluation with FRPNs to accomplish the reliability apportionment of spacecraft solar array. Wu and Hsieh (2012) explored a real-time FPN approach to diagnose progressive faults in the programmable logic controllers (PLC)-based discrete manufacturing systems. Liu et al. (2013b) presented a fault diagnosis and cause analysis approach based on the FER approach and the DAFPNs, which can capture different types of abnormal event information and identify both root causes and consequences of abnormal events. Zhou (2020) gave an enhanced WFPN method for fault analysis considering factor influences. Shi et al. (2020) presented a failure mode and effect analysis (FMEA) method based on FER and FPNs to improve the classical FMEA. In this model, the FPN model was employed to determine the risk priority of the failure modes identified in FMEA. Li et al. (2019c) constructed an FMEA model using probabilistic linguistic term sets and FPNs for the risk assessment and prioritization of failure modes and Li et al. (2019b) developed an FMEA model that combines interval type-2 fuzzy sets and FPNs to improve the effectiveness of the traditional FMEA. Zhao et al. (2023) proposed a FMEA method using evidential reasoning and picture fuzzy Petri nets to address the deficiencies of existing methods in information aggregation and risk decision-making. Wang et al. (2022a) introduced an intelligent adjustment height method of shearer drum based on adaptive FRPNs, which could represent rules that had non-linear and attribute mapping relationships and adjust the parameters adaptively to improve the accuracy of output. Zhao et al. (2021) presented a method by combining the FPNs with an adaptive arc and deep belief network to realize sensor fault diagnosis. Sun and Wang (2018) constructed an intuitionistic fuzzy fault Petri net model to deal with the large amount of uncertain information in the fault diagnosis of gas turbine. (3) Risk assessment applications Guo et al. (2016) established a comprehensive risk evaluation framework based on FPNs for long-distance oil and gas transportation pipelines. Bharathi et al. (2017) developed an FPN model to assess the risks during enterprise resource planning (ERP) adoption in small and medium enterprises. da Rocha et al. (2023) utilized the DAFPN model to assess the risk of electrical fires to simulate potential failure modes

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

and their responses in various scenarios. Zhou and Reniers (2020) utilized the WFPNs with inhibitor arcs to model relationships between risk factors and established a risk assessment structure considering veto factors, and Zhou et al. (2017) proposed a WFPN-based approach for security risk assessment in the chemical industry. Li et al. (2019a) employed the layered FPN modelling and reasoning method for process equipment failure risk assessment. In Zhang et al. (2022), the authors proposed a risk assessment method based on interval intuitionistic integrated cloud Petri nets (IIICPNs). As a proof, a subway fire accident model was adopted to confirm the feasibility of the proposed method. In Yu et al. (2022a), the authors suggested a probabilistic Petri net (PPN) method based on intuitionistic fuzzy evidential reasoning to determine the failure probability of basic events and evaluate the probability of submarine pipeline leakage failure. In Selvaraj and Ramalingam (2022), a risk assessment method based on FPNs was put forward for the assessment of sago preparation process with uncertain and inadequate information. Besides, the FPNs have been used for railway safety risk assessment (Zhang et al. 2020), information security risk assessment (Pramod and Bharathi 2018), and deepwater drilling riser risk assessment (Chang et al. 2018).

1.4.3 Wireless Sensor Networks Khoukhi et al. (2014) proposed a model based on FPNs to implement traffic adaptation in wireless mesh network characterized by information uncertainty and imprecision. Based on the secure ad hoc on-demand distance vector (SAODV), Pouyan and Yadollahzadeh Tabari (2017) used FPNs to introduce a secure routing protocol in mobile ad hoc network (MANET), in which a type of bidirectional node-to-node fuzzy security verification was carried out for sending and receiving packets between each pair of nodes. Chiang et al. (2009) focused on a dynamic knowledge inference approach using AFPNs to discover the best routing path for multicast routing protocols in a highly bandwidth-scarce environment. The work in Tan et al. (2015) designed a trust based routing mechanism to defend against attacks in both data plane and routing plane in optimized link state routing-based MANETs, in which a trust reasoning model based on FPNs was used to evaluate trust values of mobile nodes. Wang et al. (2020) proposed a trust reasoning model based on cloud model and FPNs to enhance the security of MANETs.

1.4.4 Transportation Systems Fay (2000) developed a fuzzy knowledge-based system for use in railway operation control systems and described an FPN notion to model expert knowledge in the dispatching support system. Cheng and Yang (2009) utilized an FPN approach to formulate the decision processes based on the train dispatching rules transformed

1.4 Applications of FPNs

15

from dispatchers in the case of disturbance. In Milinkovi´c et al. (2013), an FPN model with characteristics of hierarchy, color, time, and fuzzy reasoning was proposed to simulate traffic processes in a railway system for estimating train delays. Yang and Feng (2021) established a PFPN-based security assessment model for civil aviation airport security inspection information system. Based on the characteristics of PFPNs, the propositions credibility reasoning algorithm and the security situation fuzzy reasoning algorithm were further designed. M’Hala (2021) proposed a monitoring approach based on the stochastic fuzzy Petri nets for railway transport networks.

1.4.5 Biological and Healthcare Systems (1) Gene regulatory networks Hamed et al. (2010a) proposed an FPN approach to design genetic regulatory networks and describe the dynamical behavior of gene, and Hamed et al. (2010b) presented a fuzzy reasoning model based on FPNs for modelling gene regulatory networks. In (Hamed and Ahson 2011), an FPN approach was proposed to predict the confidence values for each base in DNA sequencing. In (Hamed 2018), the FPNs were used for the quantitative modeling of gene networks of biological systems. In Assaf et al. (2022), the authors employed the color FPNs for modelling membrane systems enriched by fuzzy kinetic parameters and introduced a workflow for simulating general biological systems. Besides, Liu et al. (2021a) presented a methodology and workflow utilizing fuzzy continuous Petri nets to achieve hybrid modelling of biological systems and (Liu et al. 2018a) proposed a class of fuzzy continuous Petri nets for modeling biological systems with uncertain kinetic data. Li et al. (2017) applied an extended FPN model, which integrated reverse reasoning into FPNs, to model gene regulatory network and predict the change in expression level of target. (2) Disease assessment and diagnosis Hamed (2015) developed an AFPN reasoning algorithm as a prognostic system to predictive the risk degree for esophageal cancer based on the serum concentrations of C-reactive protein and albumin. Chen et al. (2014) used FPNs to propose a rule-based diagnosis system to evaluate arteriovenous shunt stenosis for long-term hemodialysis treatment of patients. Chiang (2015) created a rule-based reasoning model by combining fuzzy computing and APNs for electrocardiograms (ECG)-based mental stress assessment, and Chiang and Pao (2016) described an EEG-based fuzzy probability model using fuzzy logic and APN method for the early diagnosis of Alzheimer’s disease. Majma and Babamir (2020) used hierarchical fuzzy colored Petri nets for the runtime monitoring and adaptation of pacemaker behavior and Majma et al. (2017) employed hierarchical fuzzy colored Petri nets for the runtime verification of pacemaker functionality. Rosdi et al. (2019) proposed an FPN-based classification method for the speech intelligibility detection of children with speech impairments.

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

1.4.6 Other Applications Luo et al. (2023) designed an indicator system for determining data security requirements and identify differentiated data security protection schemes. Then, based on this system, a classification method of data security requirements was put forward based on hierarchical fuzzy Petri nets. Guo and Wang (2023) analyzed the uncertainty propagation path of fire-induced domino effect based on an approach of layered FPNs, and Zhou and Reniers (2017) analyzed the emergency response actions for preventing fire-induced domino effects by using an approach of reversed FPNs. In (Lin et al. 2022a), the high-level FPN model was applied to develop a karaoke system which can automatically calibrate karaoke songs with lyrics. In (Yang et al. 2019), the high-level FPN model was used for video shot boundary detection in news streams. Since impreciseness and uncertainty are often involved in computing with words (CWs), Cao and Chen (2010) developed a concurrency computational model of CW by exploiting FPNs. Chen et al. (2005) applied a dynamic FPN (DFPN) model to web learning systems to increase the flexibility of the tutoring agent’s behavior and provide a dynamic learning content structure for a lecture course. In (Huang et al. 2008), a complete course generation platform was developed to facilitate efficient course design and management in e-Learning, in which the DFPN was adopted to organize courses for lecturers dynamically. Cheng et al. (2015b) presented a fuzzy semantic-based automatic Web service composition method, in which the fuzzy predicate Petri net (FPPN) was applied to model a Horn clause set and a T-invariant technique was used to determine the composite services fulfilling user input/output requirements. Ivasic-Kos et al. (2015) defined a fuzzy knowledge representation method based on the FPN formalism to represent knowledge concepts in an image and put forward an intelligent system for multi-layered image annotation. Zhou et al. (2022) constructed a cognitive reasoning and decision-making model for the formal description of attribute information granules based on the granular FPNs, in which place and transition nodes were represented by attribute information granules and qualitative maps, respectively. Muni et al. (2022) used a fuzzy embedded neural network approach for the motion planning of humanoid robots, in which a Petri-net controller was embedded with the neuro-fuzzy controller to perform dynamic path analysis. Sun et al. (2019) applied a shared control method based on fused FPNs for combining the robot automatic control and the brain-actuated control in brain-computer interface systems. Kim and Yang (2018) adopted the FPNs to simulate, assess, and communicate the process and reasoning of the self-navigating robot algorithm. Zhang et al. (2017) proposed a method of fuzzy inference Petri nets (FIPNs) to represent the human–machine hybrid control system comprising a Mamdani-type fuzzy model of operator functional state and a logical switching controller in a unified framework. Yarava and Bindu (2022) developed a trust inference model in online social networks using the DAFPNs to compute the user direct and indirect trust values based on twitter user data set. The direct trust of a user was evaluated by using his social activities with a fuzzy inference model and the indirect trust

1.5 Observations and Findings

17

was evaluated through the DAFPN model. In addition, the FPNs have been used for network attack path selection (Wu et al. 2021a), web user trust evaluation (Wu et al. 2021b), uncertain data processing of Phasor measurement unit (Juneja et al. 2021), JavaScript malware detection (Lin et al. 2020), and sleep quality assessment (Chiang and Wu 2018).

1.5 Observations and Findings (1) Reasoning algorithms For the convenience of reading and comparison, we summarize the reviewed reasoning algorithms in the reviewed articles, where the knowledge reasoning algorithms are generally classified into two types, i.e., reasoning based on the reachability tree and the algebraic representation. The two reasoning mechanisms have different advantages and disadvantages and the details are given in Table 1.1. (2) New FPN models Besides, different categories of new FPN models have been presented in previous studies. Based on above literature review, it can be seen that lots of alternative FPN modes have been developed for enhancing the knowledge representation and logic reasoning of FPNs, and each approach has its own characteristics. Especially, some reviewed studies introduced time factors into FPNs so as to represent the dynamic nature of uncertain knowledge (Liu et al. 2011; Suraj and Fryc 2006), and some Table 1.1 Advantages and disadvantages of the two types of knowledge reasoning algorithms Advantages

Disadvantages

Reachability tree

A complex expert system reasoning path can be reduced to a simple sprouting tree A graphical representation of the inference process can be given for visual appraisal Easy to follow and find inference path

Large reachability sets and adjacent places tables may result for a complex expert system The sprouting tree becomes complex as the number of places and transitions increase Reasoning speed and efficacy is low Hard to be stored and processed by computer

Algebra representation

A complex fuzzy expert system reasoning path can be reduced to simple matrix operations Ease in computer processing Reasoning speed and efficacy is high

The dimensions of these matrices and vectors are increased with the growing of the scale of the FPN model

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1 FPNs for Knowledge Representation and Reasoning: A Literature Review

papers presented dynamic FPN frameworks which have the learning ability in realworld applications (Amin and Shebl 2014; Konar et al. 2005; Li and Lara-Rosano 2000; Li et al. 2000; Wang et al. 2014).

1.6 Chapter Summary In this chapter, we conducted a systematic literature review of the literature on FPN models and their applications from 2000 to 2022 to present the available body of knowledge and analyze the trends in considering FPNs as expert systems involving fuzzy-based reasoning. This chapter provides a framework for the FPN literature as an aid to the categorization of studies in this area. Overall, the FPN-based artificial intelligence field is growing and maturing. Significant room still exists for development given the small number of reviewed articles and that there are only 206 papers relatively close-related. We believe that this number will continue to increase given the solid foundation provided by the existing researches, a foundation that did not exist a decade ago. Particularly, opportunities abound for additional research in the formal modeling of FPNs with practical applications.

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Pouyan AA, Yadollahzadeh Tabari M (2017) FPN-SAODV: using fuzzy Petri nets for securing AODV routing protocol in mobile Ad hoc network. Int J Commun Syst 30(1):e2935 Pramod D, Bharathi SV (2018) Developing an information security risk taxonomy and an assessment model using fuzzy Petri nets. J Cases Inform Technol 20(3):48–69 Qiao F, Wu QD, Li L, Wang ZT, Shi B (2011) A fuzzy Petri net-based reasoning method for rescheduling. Trans Inst Measur Control 33(3–4):435–455 Rosdi F, Salim SS, Mustafa MB (2019) An FPN-based classification method for speech intelligibility detection of children with speech impairments. Soft Comput 23(7):2391–2408 Scarpelli H, Gomide F, Yager RR (1996) A reasoning algorithm for high-level fuzzy Petri nets. IEEE Trans Fuzzy Syst 4(3):282–294 Selvaraj P, Ramalingam S (2022) Integrated risk assessment in sago preparation process using fuzzy Petri net model. J Food Process Eng 45(8):e14046 Shi H, Wang L, Li XY, Liu HC (2020) A novel method for failure mode and effects analysis using fuzzy evidential reasoning and fuzzy Petri nets. J Ambient Intell Hum Comput 11(6):2381–2395 Shi H, Liu HC, Wang JH, Mou X (2022) New linguistic Z-number Petri nets for knowledge acquisition and representation under large group environment. Int J Fuzzy Syst 24(8):3483–3500 Shih DH, Chiang HS, Lin B (2007) A generalized associative Petri net for reasoning. IEEE Trans Knowl Data Eng 19(9):1241–1251 Sobrino A, Garrido-Merchán EC, Puente C (2021) Fuzzy stochastic timed Petri nets for causal properties representation. New Math Nat Comput 17(3):633–653 Sun XL, Wang N (2018) Gas turbine fault diagnosis using intuitionistic fuzzy fault Petri nets. J Intell Fuzzy Syst 34(6):3919–3927 Sun J, Qin SY, Song YH (2004) Fault diagnosis of electric power systems based on fuzzy Petri nets. IEEE Trans Power Syst 19(4):2053–2059 Sun F, Zhang W, Chen J, Wu H, Tan C, Su W (2019) Fused fuzzy Petri nets: a shared control method for brain–computer interface systems. IEEE Trans Cognit Dev Syst 11(2):188–199 Suraj Z (2013) A new class of fuzzy Petri nets for knowledge representation and reasoning. Fund Inform 128(1):193–207 Suraj Z, Fryc B (2006) Timed approximate Petri nets. Fund. Inform 71(1):83–99 Tan SS, Li XP, Dong QK (2015) Trust based routing mechanism for securing OSLR-based MANET. Ad Hoc Netw 30:84–98 Tang Y (2009) Learning-based disassembly process planner for uncertainty management. IEEE Trans Syst Man Cybern A Syst Hum 39(1):134–143 Tang Y, Turowski M (2007) Adaptive fuzzy system for disassembly process planning with uncertainty. J Chin Instit Ind Eng 24(1):20–29 Tang Y, Zhou M, Gao M (2006) Fuzzy-Petri-net-based disassembly planning considering-human factors. IEEE Trans Syst Man Cybern A Syst Hum 36(4):718–726 Tsang EC, Yeung DS, Lee JW, Huang DM, Wang XZ (2004) Refinement of generated fuzzy production rules by using a fuzzy neural network. IEEE Trans Syst Man Cybern B Cybern 34(1):409–418 Wang H, Jiang C, Liao S (2001) Concurrent reasoning of fuzzy logical Petri nets based on multi-task schedule. IEEE Trans Fuzzy Syst 9(3):444–449 Wang WM, Peng X, Zhu GN, Hu J, Peng YH (2014) Dynamic representation of fuzzy knowledge based on fuzzy Petri net and genetic-particle swarm optimization. Exp Syst Appl 41(4):1369– 1376 Wang J, Fei Z, Chang Q, Li S (2019) Energy saving operation of manufacturing system based on dynamic adaptive fuzzy reasoning Petri net. Energies 12(11):2216 Wang X, Zhang P, Du Y, Qi M (2020) Trust routing protocol based on cloud-based fuzzy Petri net and trust entropy for mobile Ad hoc network. IEEE Access 8:47675–47693 Wang W, Wang S, Zhao S, Lu Z, He H (2022a) Novel intelligent adjustment height method of Shearer drum based on adaptive fuzzy reasoning Petri net. J Intell Fuzzy Syst 42(3):1767–1781 Wang X, Lu F, Zhou M, Zeng Q (2022b) A synergy-effect-incorporated fuzzy Petri net modeling paradigm with application in risk assessment. Exp Syst Appl 199:117037

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Wu ZH, Hsieh SJ (2012) A realtime fuzzy Petri net diagnoser for detecting progressive faults in PLC based discrete manufacturing system. Int J Adv Manuf Technol 61(1–4):405–421 Wu RR, Ma L, Mathew J, Duan GH (2002) Optimal operation planning using fuzzy Petri nets with resource constraints. Int J Comput Integr Manuf 15(1):28–36 Wu J, Yan S, Xie L (2011) Reliability analysis method of a solar array by using fault tree analysis and fuzzy reasoning Petri net. Acta Astron 69(11–12):960–968 Wu J, Yan S, Xie L, Gao P (2012) Reliability apportionment approach for spacecraft solar array using fuzzy reasoning Petri net and fuzzy comprehensive evaluation. Acta Astron 76:136–144 Wu R, Gong J, Tong W, Fan B (2021a) Network attack path selection and evaluation based on Q-learning. Appl Sci 11(1):1–13 Wu Z, Tian L, Zhang Y, Wang Z (2021b) Web user trust evaluation: a novel approach using fuzzy Petri net and behavior analysis. Symmetry 13(8):1487 Xu XG, Shi H, Xu DH, Liu HC (2019) Picture fuzzy Petri nets for knowledge representation and acquisition in considering conflicting opinions. Appl Sci 9(5):983 Xu XG, Xiong Y, Xu DH, Liu HC (2020) Bipolar fuzzy Petri nets for knowledge representation and acquisition considering non-cooperative behaviors. Int J Mach Learn Cybern 11:2297–2311 Yang H, Feng Y (2021) A Pythagorean fuzzy Petri net based security assessment model for civil aviation airport security inspection information system. Int J Intell Syst 36(5):2122–2143 Yang HT, Huang CM (2002) Distribution system service restoration using fuzzy Petri net models. Int J Electr Power Energy Syst 24(5):395–403 Yang B, Li H (2018) A novel dynamic timed fuzzy Petri nets modeling method with applications to industrial processes. Exp Syst Appl 97:276–289 Yang SH, Lin YN, Chiou GJ, Chen MK, Shen VRL, Tseng HY (2019) Novel shot boundary detection in news streams based on fuzzy Petri nets. Appl Artif Intell 33(12):1035–1057 Yarava A, Bindu CS (2022) An efficient trust inference model in online social networks using fuzzy Petri nets. Concurr Comput Pract Exp 35(6):1–18 Ye Y, Jiang Z, Diao X, Du G (2011) Extended event-condition-action rules and fuzzy Petri nets based exception handling for workflow management. Exp Syst Appl 38(9):10847–10861 Yeung DS, Tsang ECC (1994) Fuzzy knowledge representation and reasoning using Petri nets. Exp Syst Appl 7(2):281–289 Yeung DS, Tsang ECC (1997) Weighted fuzzy production rules. Fuzzy Sets Syst 88(3):299–313 Yeung DS, Ysang ECC (1998) A multilevel weighted fuzzy reasoning algorithm for expert systems. IEEE Trans Syst Man Cybern A Syst Hum 28(2):149–158 Yu J, Zeng Q, Yu Y, Wu S, Ding H, Gao H, Yang J (2022a) An intuitionistic fuzzy probabilistic Petri net method for risk assessment on submarine pipeline leakage failure. Ocean Eng 266:112788 Yu Y-X, Gong H-P, Liu H-C, Mou X (2022b) Knowledge representation and reasoning using fuzzy Petri nets: a literature review and bibliometric analysis. Artif Intell Rev 56(7):6241–6265 Yuan C, Liao Y, Kong L, Xiao H (2021) Fault diagnosis method of distribution network based on time sequence hierarchical fuzzy Petri nets. Electr Power Syst Res 191:106870 Yue W, Gui W, Chen X, Zeng Z, Xie Y (2019) Knowledge representation and reasoning using self-learning interval type-2 fuzzy Petri nets and extended TOPSIS. Int J Mach Learn Cybern 10(12):3499–3520 Yue W, Gui W, Xie Y (2020) Experiential knowledge representation and reasoning based on linguistic Petri nets with application to aluminum electrolysis cell condition identification. Inform Sci 529:141–165 Yue W, Liu X, Li S, Gui W, Xie Y (2021) Knowledge representation and reasoning with industrial application using interval-valued intuitionistic fuzzy Petri nets and extended TOPSIS. Int J Mach Learn Cybern 12(4):987–1013 Yue W, Wan X, Li S, Ren H, He H (2022) Simplified neutrosophic Petri nets used for identification of superheat degree. Int J Fuzzy Syst 24(8):3431–3455 Yue W, Hou L, Wan X, Chen X, Gui W (2023) Superheat degree recognition of aluminum electrolysis cell using unbalance double hierarchy hesitant linguistic Petri nets. IEEE Trans Instrum Measur 72:1–15

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Chapter 2

FPNs for Knowledge Representation and Reasoning: A Bibliometric Analysis

FPNs are a powerful modeling tool for the knowledge representation and reasoning of expert systems. During the past decades, a variety of models and methods have been developed to improve the performance of FPNs. However, there is a lack of a comprehensive bibliometric analysis of the literature on the FPN topic. The objective of this chapter is to conduct a bibliometric analysis of the FPN studies to generate a global picture of developments, focus areas, and research trends in this field. To achieve this goal, 287 journal articles extracted from the Web of Science over the period of 2000–2022 were analyzed with respect to cooperation network, co-citation network, and keyword co-occurrence network. According to the current research trends, possible future research directions concerning FPNs are revealed. This chapter provides important reference for scholars and practitioners to grasp the research status, hot topics, and future research agenda of the FPN domain.

2.1 Introduction Combining fuzzy sets and Petri nets, the FPNs are a graphical and mathematical model tool for representing imprecise information and supporting fuzzy reasoning in expert systems (Looney 1988; Chen et al. 1990; Yeung and Tsang 1994). An FPN is a bipartite directed graph in which places represent propositions, transitions represent FPRs, and directed arcs represent the relationships between places and transitions (Li and Lara-Rosano 2000; Shi et al. 2022). The main features of an FPN are that it supports visualized representation of information and offers dynamic knowledge inference process (Liu et al. 2013; Li et al. 2022). Because of its characteristics and competencies, the FPN method has been applied to many industrial fields for knowledge management (Guo and Wang 2023; Luo et al. 2023; Zhao et al. 2023). Although the FPN model is a useful mathematical technique for knowledge representation and reasoning, it suffers from various deficiencies when applied in real-life expert systems (Suraj 2013; Zhou et al. 2015; Liu et al. 2016b). Therefore, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_2

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how to enhance the performance of FPNs has attracted considerable attention from both academics and practitioners, and many extended FPNs have been proposed in the literature (Zhou et al. 2019; Liu et al. 2020b; Mou et al. 2022). Up to now, some literature reviews of FPNs have been performed in previous studies. For instance, Liu et al. (2017) conducted a comprehensive review of FPN models from the perspectives of reasoning algorithm, knowledge representation, and practical application. In Zhou and Zain (2016), the FPN development and its background, formalisms, reasoning algorithm, and industrial applications were reviewed. In Kabir and Papadopoulos (2019), the applications of FPNs in system safety, reliability and risk assessments were investigated. Liu et al. (2020a) reviewed the FPNs that have been used for modelling uncertain biological systems, and classified them into three categories, i.e., basic fuzzy Petri nets, fuzzy quantitative Petri nets, and Petri nets with fuzzy kinetic parameters. Jiang et al. (2022) reviewed recent developments of the FPN model from the perspectives of knowledge representation, reasoning mechanism, and industrial application, and discussed some modeling and reasoning methods to solve the ‘state-explosion problem’ of FPNs in knowledge reasoning. The above literature researches provided valuable insights and research agendas on FPNs. But these researches mainly adopted subjective categorization and qualitative analysis to summarize FPN studies (Yu et al. 2023). Compared to general literature reviews, the bibliometric analysis is an objective and quantitative evaluation method to obtain high-level insights in a large body of academic literature regarding a given research area (Hou et al. 2021; Huang et al. 2022). Therefore, the aim of this chapter is to systematically review the FPN articles based on the bibliometric analysis method. In this review, a total of 287 journal papers published between 2000 and 2022 were identified from the Web of Science (WoS) database. The selected documents were analyzed in regard to cooperation network, co-citation network, and keyword co-occurrence network. This chapter aims to identify: (1) the most prolific authors, institutions, and countries/regions, and their cooperation relationships, (2) the most influential journals, authors, and articles according to their co-citations, and (3) the hot research topics and future research directions in the FPN field. Additionally, research gaps and opportunities are discussed according to the analysis of existing FPN studies. The remaining part of this chapter is structured as follows. Section 2.2 describes the research methodology and the review process. Section 2.3 presents the results of bibliometric analysis considering the listed research aims. Additionally, keyword analysis and application field analysis are performed in this section. In Sect. 2.4, we indicate the blind spots missed by researchers and suggestions for the future work. Finally, conclusions of this chapter are given in Sect. 2.5.

2.2 Research Methodology

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2.2 Research Methodology In this chapter, scientific publications were collected from the WoS database to conduct the bibliometric review. To ensure the quality data, only peer reviewed journal papers are considered in this chapter, excluding book chapters, letters and conference proceedings. The term “fuzzy Petri nets” was used as the topic search query to search papers in the “article title, abstracts, and keywords” field. Additionally, the time span for literature search is from 2000 to 2022, and only academic articles written in English were taken into account in this review. The search was completed in December 2022, and a preliminary list of 516 publications were identified in line with the search strategy described above. This review only focuses on the articles which developed new FPN models or applied FPNs for knowledge representation and reasoning. We manually removed the unrelated data based on titles, abstracts and full-texts. Eventually, 287 papers related to FPNs were acquired for further analysis. The detailed review process of this chapter is shown in Fig. 2.1. This chapter employed bibliometric analysis to analyze the relationship among academic journals and evaluate the research trends in the area of FPNs (Huang et al. 2020; Akbari et al. 2022). To visualize knowledge structure of the retrieved literature, the CiteSpace (Chen 2006) was utilized to realize the bibliometric analysis. Fig. 2.1 The review process of this chapter

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2 FPNs for Knowledge Representation and Reasoning: A Bibliometric …

2.3 Results and Discussions 2.3.1 Publication Trend in the FPN Field Figure 2.2 depicts the annual numbers of published papers related to FPNs from 2000 to 2022. As can be seen from Fig. 2.2, the development of FPN studies can be broadly divided into three stages. The first stage from 2000 to 2010 can be viewed as a preliminary exploration stage. The annual number of publications during this stage was less than ten and the number of publications decreased to four articles in 2007. These indicate that there was relatively less interest of researchers in the initial stage. From 2011 to 2017, the number of studies in this field began to increase and showed a stepwise upward trend, which is a more rapid development compared to the first stage. From 2018 onwards, the researches on FPNs turned into a rapid development stage. The quantity of articles in this field has increased rapidly until the number of publications increased to 29 in 2022. This shows that there was a strong interest for this research topic in recent years. Therefore, it is expected that the FPN topic will attract growing attention from scholars and the quantity of related publications will continually increase over the next decade.

Fig. 2.2 Annual number of publications on FPNs

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2.3.2 Cooperation Network Analysis in the FPN Field In this section, the cooperation network analyses of authors, institutions, and countries are conducted to obtain observed features of evolution in the FPN area and identify prolific authors, representative institutions, and major countries. (1) Author cooperation network analysis The author cooperation network of FPN publications is displayed in Fig. 2.3, in which the thicknesses of links reflect the cooperation intensity of two authors and the color reflects the year that they were firstly co-authored. As shown in Fig. 2.3, the authors who published at least three papers on the topic can be segmented into six research groups. For instance, Hu-Chen Liu has cooperated with Jian-Xin You for four times since 2016, with Guang-Dong Tian twice since 2018, and with Qing-Lian Lin for three times since 2013 in #1. For another cooperation network #5, Kai-Qing Zhou has authored with Azlan Mohd Zain for four times since 2019, and with Li-Ping Mo for four times since 2018. According to the above discussions, a specific analysis on two representative research groups is conducted here. Group #1 has published 23 documents between 2011 and 2022, and Hu-Chen Liu, Jia-Ning Wu, and Jian-Xin You made the most contributions based on the number of publications. There were two research subgroups in #1, and the core authors are Hu-Chen Liu and Jia-Ning Wu. The first subgroup mainly centered around how to overcome the limitations of the traditional FPNs in knowledge representation, acquisition, and reasoning. For example, Liu et al. (2020b) proposed a grey reasoning Petri net model in which grey numbers were applied to depict human expert knowledge and a large group decision-making method was introduced for obtaining knowledge parameters from experts. In Liu et al. (2016b), linguistic 2-tuples were employed to describe the truth values of

Fig. 2.3 Author cooperation network in the FPN field

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2 FPNs for Knowledge Representation and Reasoning: A Bibliometric …

places, an ordered weighted linguistic reasoning algorithm was utilized for knowledge reasoning, and both local and global weights were combined with the inference process to enhance the effectiveness of approximate reasoning. Liu et al. (2016a) introduced an intuitionistic FPN model in which intuitionistic fuzzy sets were applied to deal with the vague knowledge information provided by experts, and a maxalgebra-based inference algorithm was employed to conduct the knowledge inference. According to interval clouds and a hybrid averaging operator, Liu et al. (2018) put forward a cloud reasoning fuzzy Petri net model to improve the knowledge representation and reasoning capacity of traditional FPNs. In addition, the spherical linguistic Petri nets (Mou et al. 2022), the 2-dimensional uncertain linguistic Petri nets (Liu et al. 2022), the R-numbers Petri nets (Mou et al. 2021), and the Pythagorean fuzzy Petri nets (Liu et al. 2021) were proposed by the first subgroup for knowledge representation and reasoning. The second subgroup focused on the application of FPNs. For instance, Wu and Yan (2014) introduced fuzzy reasoning Petri nets (FRPNs) to predict the product count reliability of a new product during the early design stage. For addressing the anomalies of solar array mechanical system, FRPNs were established by Wu et al. (2011) to conduct reliability analysis of the system to derive its fault mechanisms. A method which combines the fuzzy comprehensive evaluation with FRPNs was proposed in Wu et al. (2012) to implement the reliability allocation of spacecraft solar array. Group #3 has published 11 research articles in the FPN field since 2006, and Victor R. L. Shen contributed the most based on the number of publications. This group proposed a high-level fuzzy Petri net (HLFPN) model to represent the FPRs of an expert system and focused on the application of HLFPNs. For instance, Shen et al. (2012) presented a learning evaluation model using the HLFPNs and a fuzzy reasoning method to evaluate the students’ learning achievement. Shen et al. (2015) adopted the HLFPNs for the analysis and identification of normalaction, exercising, and falling down for the elderly. In Shen et al. (2016), a fuzzy blood pressure verification system for home care was constructed using the fuzzy analytic hierarchical process (AHP) and HLFPNs. In Shen et al. (2018a), the HLFPN model was integrated with keypoint matching to improve the precision of shot boundary detection by removing false shots. In addition, the HLFPNs have been applied to stock market prediction system (Shen et al. 2018b), shot boundary detection in news streams (Yang et al. 2019), JavaScript malware detection (Lin et al. 2020), and investment decision (Chiang et al. 2021). Table 2.1 summaries the top 10 productive authors in the FPN area according to the total number of published documents. It can be observed that Hu-Chen Liu from Tongji University is the most productive author with 21 publications, followed by Victor R. L. Shen from National Taipei University with eight articles. (2) Institution cooperation network analysis Figure 2.4 portrays the institution cooperation network of the selected published articles, in which the size of nodes reflects the quantity of publications, the line between the nodes represents cooperative relationship, and the thickness of the lines means the

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Table 2.1 Top 10 productive authors in the FPN field No

Publication

Author

Institution

Country/ region

1

21

Hu-Chen Liu

Tongji University

China

2

8

Victor R. L. Shen

National Taipei University

Taiwan

3

7

Kai-Qing Zhou

Shangdong University of Science and Technology

China

4

6

Azlan Mohd Zain

University Teknologi Malaysia

Malaysia

5

5

Jian-Xin You

Tongji University

China

6

5

Fei Liu

Wuhan University

China

7

4

Chen-Ying Yang

University of Taipei

Taiwan

8

4

Jia-Ning Wu

Georgia Institute of Technology

USA

9

4

Li-Ping Mo

Jishou University

China

10

4

M. K. Tiwari

National Institute of Foundry and Forge Technology

India

cooperative intensity. Only the institutions with over three published documents are displayed. It can be observed that Tongji University is the centrality with the most cooperation published documents; Shanghai University, China Jiliang University, New Jersey Institute of Technology, and Macau University of Science and Technology keep a close cooperated with it. Besides, the National Taipei University has cooperated with Ming Chi University of Technology, Chaoyang University of Technology, University of Taipei, Shih Hsin University, and Tunghai University due to the geographical vicinity.

Fig. 2.4 Institution cooperation network in the FPN field

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Table 2.2 illustrates the most productive institutions based on the quantity of publications on FPNs. The ten institutions published 94 documents at the rate of 34.56% of total documents. It can be observed that the most prolific institution is Tongji University (19 articles), followed by Shanghai University (15 articles), and National Taipei University (12 articles). This may because that Hu-Chen Liu and Jian-Xin You affiliate to Tongji University and Shanghai University, and Victor R. L. Shen belongs to National Taipei University. (3) Country cooperation network analysis Figure 2.5 portrays the cooperation networks among countries/regions in the FPN field and only the countries/regions with more than three publications are displayed. In this figure, the size of the circles reflects the quantity of publications, the curve between the circles represents cooperative relationship, and the thickness of the connection represents the cooperation strength. As can be seen, there exist strong collaborations between Malaysia and Iran, Canada and Australia, Canada and Poland, China and Japan. That is to say, these countries played a particularly important role in connecting the cross-country cooperation in the FPN area. Moreover, China makes the most contributions, publishing 109 articles, followed by USA (37 articles), and the Taiwan region (34 articles). This signifies that these countries and regions are vital to drive and direct FPN researches. Figure 2.6 displays the distribution of published documents on the basis of continents. It can be observed that 96.32% of the considered published documents are dated from Asia, Europe, and North America. According to the number of articles, Asia topped the list with 173 articles. Figure 2.7 portrays the geographic distribution of the published documents by countries, where the quantity of the published documents is denoted by a graduated color, i.e., the darker the color, the large the number of published documents. As one can see, China has the highest concern towards the researches concerning FPNs in Asia. USA and Canada have published the maximum Table 2.2 Top 10 prolific institutions in the FPN field No

Institution

Publication

Country/region

1

Tongji University

19

China

2

Shanghai University

15

China

3

National Taipei University

12

Taiwan

4

Shandong University of Science and Technology

10

China

5

New Jersey Institute of Technology

8

USA

6

National Cheng Kung University

6

Taiwan

7

Tsinghua University

6

China

8

China Jiliang University

6

China

9

Rzeszow University of Technology

6

Poland

10

Central South University

6

China

2.3 Results and Discussions

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Fig. 2.5 Country/region cooperation network in the FPN field

number of articles on the FPN topic in North America. South America is the continent with the fewest countries involved in the publication of FPN articles during this period. Fig. 2.6 Distribution of articles in regard to continents

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2 FPNs for Knowledge Representation and Reasoning: A Bibliometric …

Fig. 2.7 Geographic distribution of the FPN documents

2.3.3 Co-Citation Analysis in the FPN Field Co-citation is defined as the frequency with which two or more journals, authors, and documents are cited together by the third document (Hou et al. 2021). Cocitation networks are performed to identify the structure of science in certain aspects, including journals, authors, and documents. (1) Journal co-citation analysis Figure 2.8 displays the journal co-citation network of the FPN area, in which the size of circles reflects the number of journal citations and the line between them indicates the co-citation frequency between journals. It can be observed that the highest cited journals are IEEE Transactions on Systems Man and Cybernetics Part BCybernetics (138 citations), IEEE Transactions on Systems Man and Cybernetics Part A-Systems and Humans (96 citations), and IEEE Transactions on Knowledge and Data Engineering (93 citations). Table 2.3 summarizes the most significant journals on FPNs. With respect to the number of links, IEEE Transactions on Knowledge and Data Engineering (14 links) has the most citation with others in the FPN domain, followed by Information Sciences (13 links), IEEE Transactions on Systems Man and Cybernetics Part A-Systems and Humans (12 links), and IEEE Transactions on Fuzzy Systems (12 links). (2) Author co-citation analysis Figure 2.9 displays the author co-citation network in the FPN area. Note that the size of circles represents the quantity of author citations and the connection between them reflects co-citation relationship between authors, and the thickness of different links reflects the frequency of different authors being cited together. As shown in Fig. 2.9, the top-cited author is Shyi-Ming Chen from the National Taiwan University of Science with 103 citations, who devoted to the research of fuzzy reasoning algorithm

2.3 Results and Discussions

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Fig. 2.8 Journal co-citation network of the FPN area

Table 2.3 The top 10 highly cited journals on FPN No

Journal

Citation

1

IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics

138

6.22

2

IEEE Transactions on Systems Man and Cybernetics Part A-Systems and Humans

96

2.183

Impact factor

3

IEEE Transactions on Knowledge and Data Engineering

93

9.235

4

Expert Systems with Applications

91

8.665

5

IEEE Transactions on Fuzzy Systems

89

12.253

6

Fuzzy Sets and Systems

60

4.462

7

Information Sciences

55

8.233

8

Computers and Industrial Engineering

48

7.18

9

IEEE Transactions on Cybernetics

40

19.12

10

IEEE Transactions on Systems Man Cybernetics-Systems

37

11.47

for rule-based systems. Carl G. Looney from the University of Nevada and Tadao Murata from the University of Illinois at Chicago rank the second and the third in terms of the number of citations, and they received 74 and 59 citations, respectively. The FPN theory was originally introduced by Looney (1988) for rule-based decisionmaking in a fuzzy context. Tadao Murata mainly focused on the research of fuzzy timing in high-level Petri nets (Zhou and Murata 1999; Zhou et al. 2000). The ten most cited authors also include Hu-Chen Liu (57 citations, Tongji University), Lotfi A. Zadeh (55 citations, University of California), Mei-Mei Gao (55 citations, Seton Hall University) (cf. Table 2.4). According to the number of links, Carl G. Looney (18 links) has co-citated with others most in the FPN field, followed by Shyi-Ming Chen (14 links) and Alberto J. Bugarin (13 links).

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2 FPNs for Knowledge Representation and Reasoning: A Bibliometric …

Fig. 2.9 Author co-citation network in the FPN area

Table 2.4 Top 10 most cited authors in the FPN area Author

Citation

1

Shyi-Ming Chen

103

2

Carl G. Looney

74

No

Institution National Taiwan University of Science and Technology University of Nevada

3

Tadao Murata

59

University of Illinois at Chicago

4

Hu-Chen Liu

57

Tongji University

5

Lotfi A Zadeh

55

University of California

6

Mei-Mei Gao

55

Seton Hall University

7

Witold Pedrycz

40

University of Alberta

8

Daniel S. Yeung

30

South China University of Technology

9

Alberto J. Bugarin

25

Universidad de Santiago de Compostela

10

Victor R. L. Shen

25

National Taipei University

(3) Document co-citation analysis Figure 2.10 shows the document co-citation network of the FPN field. Each node reflects a document and its size represents the number of citations of a linked reference. The connection between nodes denotes co-citation relationship between different publications and the color indicates the first year the connection was made. The node publications with more than four co-citations are displayed in the figure. It can be seen that the three most co-cited documents were published by Liu et al. (2017), Liu et al. (2016a) and Liu et al. (2018), which received 23, 15, and 12 co-citations, respectively. The timeline view displays the hot research topics during different periods, which helps people better identify the development trend of a given field (Wang et al. 2021). Figure 2.11 displays the timeline view of document clusters, and shows that

2.3 Results and Discussions

37

Fig. 2.10 Document co-citation network in the FPN area

639 references of the selected documents can be clustered into six clusters in the FPN area. The line between different nodes represents co-citation relationships, and the thickness of a line stands for the frequency of different references being cited together. The horizontal line reflects changes over time of the research hotspots, and the size of the nodes represents their hot degree in a certain period. From Fig. 2.11, we can see that six clusters are formed by the researches in the FPN field. The oldest cluster is #3 (disassembly planning), followed by #1 (expert system) and #4 (risk assessment). This demonstrates that academics started to mathematically represent uncertainty in disassembly operations (Tang et al. 2006; Tang and Turowski 2007; Tang 2009). Then, researchers focused on developing improved FPN models to accurately represent increasingly complex expert systems (Liu et al. 2017) and applying FPNs for risk assessment in different fields (Guo et al. 2016; Bharathi et al. 2017; Zhou et al. 2017; Chang et al. 2018). In addition, #2 (fault diagnosis), #5 (failure mode and effect analysis), and #6 (knowledge representation) appeared to have publications in recent years. Researches paid attention to the application of FPNs to address real problems in many areas, e.g., fault diagnosis (Cheng et al. 2015; Zhang et al. 2016; Sun and Wang 2018) and failure mode and effect analysis (Li et al. 2019a, b; Shi et al. 2020). Thus, the topics of “fault diagnosis”, “failure mode and effect analysis”, and “knowledge representation” may be possible research directions for further study.

2.3.4 Keyword Analysis in the FPN Field In this section, a keyword co-occurrence analysis is employed to determine common keywords in the documents of FPNs. Moreover, a keyword burst analysis is applied for monitoring critical research topics and rising trends in the FPN area. Figure 2.12

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2 FPNs for Knowledge Representation and Reasoning: A Bibliometric …

Fig. 2.11 Timeline view of document clusters in the FPN area

displays the keyword co-occurrence network of FPN documents. The label font size denotes the co-occurrence frequency of a keyword. In the keyword co-occurrence analysis, “fuzzy Petri net” and “Petri net” are ignored because they were used as the search terms and cannot contribute to the keyword analysis. From Fig. 2.12, we can see that “system”, “knowledge representation”, “expert system”, “model”, “fault diagnosis”, and “fuzzy reasoning” are the prominent keywords. Furthermore, it can be seen that “fuzzy set”, “neural network”, and “cloud model” are the methods applied for knowledge representation. The keywords “fault diagnosis”, “risk assessment”, “power system”, “failure mode and effect analysis”, and “supply chain management” are the application areas of FPNs. The keywords “fuzzy timed Petri net”, “high-level Petri net”, “weighted fuzzy Petri net”, “colored Petri net”, and “associative Petri net” are improvement versions of the traditional FPN model. In addition, the “expert system”, “knowledge representation”, “fault diagnosis”, and “fuzzy reasoning” can be considered as four important research topics.

Fig. 2.12 Keyword co-occurrence analysis of the FPN area

2.3 Results and Discussions

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Fig. 2.13 The keywords with the strongest citation bursts

Based on the keyword burst analysis, Fig. 2.13 demonstrates eight keywords with the strongest citation bursts between 2000 and 2022, where red line reflects the certain time period of their active years. It can be observed that the term “genetic algorithm” and “risk assessment” burst for the maximum duration, but their intensities are different. The intensity of “genetic algorithm” is not very high. By contrast, “risk assessment” is the strongest keyword, which burst from 2017 to 2022. The keywords “reasoning algorithm”, “representation”, “inference”, “model”, and “network” are critical portions of FPNs for knowledge management. Lastly, “fault diagnosis” is the term in connection with the application of the FPN model with the burst duration of 2018–2022. This indicates that scholars not only focused on how to improve the performance of FPNs but also intended to apply the FPN models to solve practical problems (e.g., risk assessment and fault diagnosis) in recent years.

2.3.5 Application Field Analysis For the FPN model, we here further analyze its application fields based on the selected articles. Figure 2.14 shows the application field distribution of FPNs with their publication numbers and percentages. It can be observed that the published documents are mostly related to information technology (accounting for 26.47%), which indicates that FPNs are a vital knowledge management tool to tackle expert knowledge in the information technology industry. In addition, the FPN models have been commonly used in the mechanical and manufacture, the power, and the biology and healthcare industries.

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2 FPNs for Knowledge Representation and Reasoning: A Bibliometric …

Fig. 2.14 Distribution of publications by application areas

2.4 Suggestions for Future Research Although breakthroughs have been made in the research of FPNs, there still exists significant room to further refine the FPNs for knowledge management. According to the results of bibliometric analysis, the potential directions for further study are summarized as follows. (1) In many situations, knowledge information is usually vague, uncertain, or even incomplete due to time pressure, lack of data, and experts’ limited expertise related to the problem domain. It has been found that various extended fuzzy methods such as cloud model theory, spherical linguistic sets, and twodimension linguistic uncertain variables have been employed to deal with the subjective and ambiguous knowledge information of experts. In the future, it is suggested to apply new and advanced uncertainty theories to effectively manipulate and represent the uncertain and ambiguous expert knowledge and experience. (2) Many tasks in FPNs (e.g., acquiring knowledge parameters) heavily relies on expert judgments. Due to the differences in terms of experts’ experience, backgrounds, and organizations, conflict opinions are inevitable in the knowledge acquisition of FPNs. Hence, as another direction for future studies, consensus reaching methods are supposed to be introduced to solve the conflict opinions given by domain experts, which will lead to efficiency improvement in the knowledge acquisition processes of FPNs.

2.5 Chapter Summary

41

(3) Along with the vigorous development of knowledge-based systems, more advanced knowledge inference algorithms are suggested to be employed in the future research. New algorithms that enable FPNs to perform better in complex environment are supposed to be explored to reduce the reasoning complexity of large expert systems. Another direction for the future work is to develop inference algorithms that can learn from the data and execute knowledge reasoning effectively. (4) Different FPN models have their own advantages and disadvantages. The current works did not make sufficient comparisons among various FPN methods. As a result, it is not easy to choose an appropriate FPN model in practical applications. Thus, it is necessary to compare the advantages and drawbacks of different FPN models in detail to help practitioners in choosing the most appropriate one for a given application. (5) New emerging technologies have large room for enhancing the performance of FPNs. So, another possible direction for future work is to explore new techniques such as artificial intelligence tools to support the implementation of FPNs in sophisticated rule-based systems. For instance, deep learning algorithms can be employed to learn the knowledge parameters of FPNs from the evaluation data of experts. There are many other research directions in the FPN field and the bibliometric analysis conducted in this chapter enables us to grasp hot topics and predict the trends from many aspects. To balance, the FPN models need to be enhanced as a whole to deal with changeable application environment and complex expert system modelling issues.

2.5 Chapter Summary In this chapter, we conducted a bibliometric review and a visualized analysis of the literature on FPNs published between 2000 and 2022. A total of 287 journal articles were identified for the bibliometric analysis and comprehensive review. By the cooperation network analysis, one can observed that the most prolific authors in the area are Hu-Chen Liu from Tongji University and Victor R. L. Shen from National Taipei University. The co-citation network analysis indicated that Shyi-Ming Chen and Carl G. Looney are the most significant authors in this field. The three most cited documents are those published by Liu et al. (2017), Liu et al. (2016a) and Liu et al. (2018).The document cluster analysis showed that “fault diagnosis”, “failure mode and effect analysis”, and “knowledge representation” are the major directions for further study. The keyword network analysis indicated that “expert system”, “knowledge representation”, “fault diagnosis” are important research topics in the FPN literature. In conclusion, this chapter provides valuable insights for both academics and practitioners to capture current research hotspots and potential research directions in the FPN domain.

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Liu HC, Xue L, Li ZW, Wu J (2018) Linguistic Petri nets based on cloud model theory for knowledge representation and reasoning. IEEE Trans Knowl Data Eng 30(4):717–728 Liu F, Heiner M, Gilbert D (2020a) Fuzzy Petri nets for modelling of uncertain biological systems. Brief Bioinform 21(1):198–210 Liu HC, Luan X, Lin W, Xiong Y (2020b) Grey reasoning Petri nets for large group knowledge representation and reasoning. IEEE Trans Fuzzy Syst 28(12):3315–3329 Liu HC, Xu DH, Duan CY, Xiong Y (2021) Pythagorean fuzzy Petri nets for knowledge representation and reasoning in large group context. IEEE Trans Syst Man Cybern Syst 51(8):5261–5271 Liu HC, Luan X, Zhou M, Xiong Y (2022) A new linguistic Petri net for complex knowledge representation and reasoning. IEEE Trans Knowl Data Eng 34(3):1011–1020 Looney CG (1988) Fuzzy Petri nets for rule-based decision-making. IEEE Trans Syst Man Cybern 18(1):178–183 Luo X, He L, Wei X, Zhu M, Li Z (2023) Security requirement classification of electricity trading data based on hierarchical fuzzy Petri network. Energy Rep 9:189–199 Mou X, Zhang QZ, Liu HC, Zhao J (2021) Knowledge representation and acquisition using Rnumbers Petri nets considering conflict opinions. Exp Syst 38(3):e12660 Mou X, Mao LX, Liu HC, Zhou M (2022) Spherical linguistic Petri nets for knowledge representation and reasoning under large group environment. IEEE Trans Artif Intell 3(3):402–413 Shen VRL, Yang CY, Wang YY, Lin YH (2012) Application of high-level fuzzy Petri nets to educational grading system. Exp Syst Appl 39(17):12935–12946 Shen VRL, Lai HY, Lai AF (2015) The implementation of a smartphone-based fall detection system using a high-level fuzzy Petri net. Appl Soft Comput 26:390–400 Shen VRL, Wang YY, Yu LY (2016) A novel blood pressure verification system for home care. Comput Stand Interf 44:42–53 Shen RK, Lin YN, Juang TTY, Shen VRL, Lim SY (2018a) Automatic detection of video shot boundary in social media using a hybrid approach of HLFPN and keypoint matching. IEEE Trans Comput Soc Syst 5(1):210–219 Shen RK, Yang CY, Shen VRL, Li WC, Chen TS (2018b) A stock market prediction system based on high-level fuzzy Petri nets. Int J Uncert Fuzziness Knowl Based Syst 26(5):771–808 Shi H, Wang L, Li XY, Liu HC (2020) A novel method for failure mode and effects analysis using fuzzy evidential reasoning and fuzzy Petri nets. J Ambient Intell Hum Comput 11(6):2381–2395 Shi H, Liu HC, Wang JH, Mou X (2022) New linguistic Z-number Petri nets for knowledge acquisition and representation under large group environment. Int J Fuzzy Syst 24:3483–3500 Sun XL, Wang N (2018) Gas turbine fault diagnosis using intuitionistic fuzzy fault Petri nets. J Intell Fuzzy Syst 34(6):3919–3927 Suraj Z (2013) A new class of fuzzy Petri nets for knowledge representation and reasoning. Fundam Inform 128(1):193–207 Tang Y (2009) Learning-based disassembly process planner for uncertainty management. IEEE Trans Syst Man Cybern A Syst Hum 39(1):134–143 Tang Y, Turowski M (2007) Adaptive fuzzy system for disassembly process planning with uncertainty. J Chin Instit Ind Eng 24(1):20–29 Tang Y, Zhou M, Gao M (2006) Fuzzy-Petri-net-based disassembly planning considering-human factors. IEEE Trans Syst Man Cybern A Syst Hum 36(4):718–726 Wang X, Xu Z, Su SF, Zhou W (2021) A comprehensive bibliometric analysis of uncertain group decision making from 1980 to 2019. Inform Sci 547:328–353 Wu JN, Yan S (2014) An approach to system reliability prediction for mechanical equipment using fuzzy reasoning Petri net. Proc Instit Mech Eng O J Risk Reliabil 228(1):39–51 Wu J, Yan S, Xie L (2011) Reliability analysis method of a solar array by using fault tree analysis and fuzzy reasoning Petri net. Acta Astron 69(11–12):960–968 Wu J, Yan S, Xie L, Gao P (2012) Reliability apportionment approach for spacecraft solar array using fuzzy reasoning Petri net and fuzzy comprehensive evaluation. Acta Astron 76:136–144

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Yang SH, Lin YN, Chiou GJ, Chen MK, Shen VRL, Tseng HY (2019) Novel shot boundary detection in news streams based on fuzzy Petri nets. Appl Artif Intell 33(12):1035–1057 Yeung DS, Tsang ECC (1994) Fuzzy knowledge representation and reasoning using Petri nets. Exp Syst Appl 7(2):281–289 Yu YX, Gong HP, Liu HC, Mou X (2023) Knowledge representation and reasoning using fuzzy Petri nets: a literature review and bibliometric analysis. Artif Intell Rev 56:6241–6265 Zhang Y, Zhang Y, Wen F, Chung CY, Tseng CL, Zhang X, Zeng F, Yuan Y (2016) A fuzzy Petri net based approach for fault diagnosis in power systems considering temporal constraints. Int J Electr Power Energy Syst 78:215–224 Zhao XK, Zhu XM, Bai KY, Zhang RT (2023) A novel failure model and effect analysis method using a flexible knowledge acquisition framework based on picture fuzzy sets. Eng Appl Artif Intell 117:105625 Zhou Y, Murata T (1999) Petri net model with fuzzy timing and fuzzy-metric temporal logic. Int J Intell Syst 14(8):719–745 Zhou KQ, Zain AM (2016) Fuzzy Petri nets and industrial applications: a review. Artif Intell Rev 45(4):405–446 Zhou Y, Murata T, DeFanti TA (2000) Modeling and performance analysis using extended fuzzytiming Petri nets for networked virtual environments. IEEE Trans Syst Man Cybern B 30(5):737– 756 Zhou KQ, Zain AM, Mo LP (2015) A decomposition algorithm of fuzzy Petri net using an index function and incidence matrix. Exp Syst Appl 42(8):3980–3990 Zhou J, Reniers G, Zhang L (2017) A weighted fuzzy Petri-net based approach for security risk assessment in the chemical industry. Chem Eng Sci 174:136–145 Zhou KQ, Mo LP, Jin J, Zain AM (2019) An equivalent generating algorithm to model fuzzy Petri net for knowledge-based system. J Intell Manuf 30(4):1831–1842

Chapter 3

Determining Truth Degrees of Input Places in FPNs

As one type of high-level Petri nets, FPNs have attracted a lot of attention over the recent decade due to its adequacy for knowledge representation and logic reasoning. However, in the literature, the truth degrees of input places are usually given directly or supposed by researchers. No or little research has been performed on the determination of initial marking vector for a specific FPN. In this chapter, we introduce a group decision-making model using hesitant 2-tuple linguistic term sets to obtain the initial truth values of FPNs based on domain experts’ knowledge. As is illustrated by a numerical example, the proposed framework can well capture domain experts’ diversity judgements and derive initial truth degrees for an FPN under different types of uncertainties.

3.1 Introduction FPNs are a dynamic and marked graphical tool to handle fuzzy reasoning problems of expert systems (Looney 1988; Chen et al. 1990; Yeung and Tsang 1994). It is used for representing FPRs in a knowledge base system, and executing fuzzy reasoning process to evaluate the truth degrees of goal propositions. Along with the rapid advance of expert systems, the descriptions of FPRs are more and more complex. Thus, scholars conducted their researches with extended FPN theory and developed many improved FPN models, which include linguistic Z-number Petri nets (Shi et al. 2022), spherical linguistic Petri Nets (Mou et al. 2022), 2-dimensional uncertain linguistic Petri nets (Liu et al. 2022), R-numbers Petri nets (Mou et al. 2021), Pythagorean fuzzy Petri nets (Liu et al. 2021), and so on (Li et al. 2022; Yu et al. 2023; Yue et al. 2022). In addition, due to their characteristics and capabilities, FPNs have been widely applied to various industrial fields, such as security requirement classification (Luo et al. 2023), subway fire risk assessment (Zhang et al. 2022), submarine pipeline leakage risk assessment (Yu et al. 2022), equipment failure risk analysis (Wang et al. 2022), and power system fault diagnosis (Lin et al. 2022). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_3

45

46

3 Determining Truth Degrees of Input Places in FPNs

Although FPNs have received considerable attention of academics and practitioners in recent years, there is little investigation on determining truth degrees of input places in FPNs (Liu et al. 2017b). In the literature, the initial marking vector was usually assigned directly, which affect the effectiveness and limit the realization of the FPN model. Setting appropriate truth degrees of input places according to actual situations and experts’ opinions is of significance to the knowledge reasoning of FPNs and the following recommended actions (Liu et al. 2017a). In other words, if the initial marking vector is not correctly determined, the rule-based expert system molded by FPNs may give biased or misleading decision results, even if the inference algorithm and process are perfect. Therefore, there is a need for systematic and effective methods to guide decision makers in determining the truth values of FPN input places and making right decisions (Gao et al. 2003; Liu et al. 2013; Xu et al. 2020). Focusing on the above challenge, in this chapter, we develop a group decisionmaking model by using hesitant 2-tuple linguistic term sets (HFLTSs) to compute the initial truth values of FPNs. This model is preferable because experts are allowed to evaluate closer to natural language and it has exact characteristic in linguistic information processing. In order to do so, the remainder of this chapter is structured as follows. Section 3.2 introduces the basic concepts concerning HFLTSs. Section 3.3 proposes a group decision-making methodology for evaluating the truth degrees of FPN input places. In Sect. 3.4, a numerical example is provided to illustrate the effectiveness and advantages of the proposed model. Finally, conclusions are discussed in Sect. 3.5.

3.2 Preliminaries 3.2.1 Hesitant 2-Tuple Linguistic Term Sets The HFLTSs were introduced by Rodriguez et al. (2012) to deal with the situations where decision makers have a doubt among several possible linguistic terms. In the following, some basic definitions of HFLTSs are given. { } Definition 3.1 (Rodriguez et al. 2012; Liu et al. 2017b) Let S = s0 , s1 , ..., sg be a fixed set of linguistic term set. An HFLTS associated with S, H S , is an ordered finite subset of the consecutive linguistic terms of S. The empty and full HFLTSs for a linguistic variable ϑ are defined as HS (ϑ) = ∅ and HS (ϑ) = S, respectively. } { Definition 3.2 (Rodriguez et al. 2012; Liu et al. 2017b) Let S = s0 , s1 , ..., sg be a linguistic term set, a context-free grammar is a 4-tuple GH = (V N , V T , I, P), where V N indicates a set of nonterminal symbols, V T is a set of terminal symbols, I is the starting symbol, and P denotes the production rules. The elements of GH are defined as follows:

3.2 Preliminaries

47

VN = {, , , , }; } { VT = lower than, greater than, at least, at most, between, and, s0 , s1 , ..., sg ; I ∈ VN ; P = {I :: =| ::=|

| ::= s0 | s1 |...|sg :: = lower than|greater than|at least|at most :: = between :: = and}. Definition 3.3 (Rodriguez et al. 2012; Liu et al. 2017b) Let E GH be a function that transforms the comparative linguistic expressions generated by GH into an HFLTS H S of the linguistic term set S. The obtained linguistic expressions can be converted into HFLTSs according to the following ways: (1) (2) (3) (4) (5)

E G H (greater than si ) = {sk |sk ∈ S and sk > si }; E G H (lower than si ) = {sk |sk ∈ S and sk < si }; E G H (at least si ) = {sk |sk ∈ S and sk ≥ si }; E G H (at ( most si ) = {sk |s)k ∈ {S and sk ≤ si }; } E G H between si and s j = sk |sk ∈ S and si ≤ sk ≤ s j .

} { Definition 3.4 (Rodriguez et al. 2012; Liu et al. 2017b) Let S = s0 , s1 , ..., sg be a linguistic term set. The envelope of an HFLTS, i.e., env(H S ), is a linguistic interval whose limits are determined by its upper bound HS+ and lower bound HS− , shown as follows: ] [ env(HS ) = HS− , HS+ ,

HS− ≤ HS+ ,

(3.1)

where HS+ = max(si ) = s j , si ≤ s j and si ∈ HS , ∀i,

(3.2)

HS− = min(si ) = s j , si ≥ s j and si ∈ HS , ∀i.

(3.3)

48

3 Determining Truth Degrees of Input Places in FPNs

3.2.2 Interval 2-Tuple Linguistic Model As an extension to the 2-tuple linguistic model (Herrera and Martínez 2000), the interval 2-tuple linguistic representation approach (Zhang 2012) was proposed to manage uncertain linguistic expressions. } { Definition 3.5 (Zhang 2012) Suppose that S = s0 , s1 , ..., sg is a linguistic term set. An interval 2-tuple linguistic variable is denoted by [(sk , α1 ), (sl , α2 )], where (sk , α1 ) ≤ (sl , α2 ), sk (sl ) and α1 (α2 ) represent the linguistic label from S and symbolic translation, respectively. The [ interval ]( 2-tuple that expresses the) equivalent information to an interval value β L , β U β L , β U ∈ [0, 1], β L ≤ β U can be derived by ⎧ ( ) sk , k = round( β L · g ) ⎪ ⎪ ⎪ ⎪ ⎨ sl , l = round β U · g[ [ L U] ( ∆ β , β = [(sk , α1 ), (sl , α2 )] with α = β L − k , α ∈ − 1 , 1 1 1 ⎪ g 2g 2g ⎪ ( [ ⎪ ⎪ ⎩ α = βU − l , α ∈ − 1 , 1 . 2

g

2

2g

2g

(3.4) where round(·) is the usual round operation. Conversely, there exists an inverse function ∆[−1 which ] can transform an interval 2-tuple [(sk , α1 ), (sl , α2 )] into its interval value β L , β U : ∆−1 [(sk , α1 ), (sl , α2 )] =

[

] [ ] l k + α1 , + α2 = β L , β U . g g

(3.5)

Specifically, a 2-tuple linguistic variable can be regarded as a special case of the interval 2-tuple linguistic variable, i.e., (sk , α1 ) = [(sk , α1 ), (sk , α1 )]. Definition 3.6 (Liu et al. 2014a) Consider any three interval 2-tuples a˜ = [(r, α), (t, ε)], a˜ 1 = [(r1 , α1 ), (t1 , ε1 )] and a˜ 2 = [(r2 , α2 ), (t2 , ε2 )], and let λ ∈ [0, 1], then their basic operational laws are given as follows: a˜ 1 ⊗ a˜ 2 =[(r1 , α1 ), (t1 , ε1 )] ⊗ [(r2 , α2 ), (t2 , ε2 )] [ ] =∆ ∆−1 (r1 , α1 ) · ∆−1 (r2 , α2 ), ∆−1 (t1 , ε1 ) · ∆−1 (t2 , ε2 ) ; a˜ 1 ⊕ a˜ 2 =[(r1 , α1 ), (t1 , ε1 )] ⊕ [(r2 , α2 ), (t2 , ε2 )] (2) [ ] =∆ ∆−1 (r1 , α1 ) + ∆−1 (r2 , α2 ), ∆−1 (t1 , ε1 ) + ∆−1 (t2 , ε2 ) ; ] [( )λ ( )λ (3) a˜ λ = ([(r, α), (t, ε)])λ = ∆ ∆−1 (r, α) , ∆−1 (t, ε) ; [ ] (4) λa˜ = λ[(r, α), (t, ε)] = ∆ λ∆−1 (r, α), λ∆−1 (t, ε) . (1)

Definition 3.7 (Zhang 2012; Liu et al. 2015) Let a˜ i [(ri , αi ), (ti , εi )](i = 1, 2, ..., n) be a set of interval 2-tuples and w

= =

3.2 Preliminaries

49

∑n wi = 1. (w1 , w2 , ..., wn )T be an associated weight vector, with wi ∈ [0, 1], i=1 The interval 2-tuple weighted averaging (ITWA) operator is defined as: n

ITWAw (a˜ 1 , a˜ 2 , ..., a˜ n ) = ⊕ (wi a˜ i ) i=1 [ n ] n ∑ ∑ −1 −1 wi ∆ (ri , αi ), wi ∆ (ti , εi ) . =∆ i=1

(3.6)

i=1

Definition 3.8 (Liu et al. 2017b) Let a˜ i = [(ri , αi ), (ti , εi )](i = 1, 2, ..., n) be a set w = (w1 , w2 , ..., wn )T be their associated weight vector, of interval 2-tuples ∑and n with wi ∈ [0, 1], i=1 wi = 1. The interval 2-tuple weighted geometric (ITWG) operator is defined as: n

ITWGw (a˜ 1 , a˜ 2 , ..., a˜ n ) = ⊗ (a˜ i )wi i=1 [ n ] n || ( ( −1 )wi || )wi −1 ∆ (ri , αi ) , ∆ (ti , εi ) =∆ . i=1

(3.7)

i=1

Definition 3.9 (Liu et al. 2014b; You et al. 2015) Suppose that a˜ 1 = [(r1 , α1 ), (t1 , ε1 )] and a˜ 2 = [(r2 , α2 ), (t2 , ε2 )] are two interval 2-tuples, then the Hamming distance between them is computed by ] | | −1 |) 1 (|| −1 −1 −1 | | | ∆ (r1 , α1 ) − ∆ (r2 , α2 ) + ∆ (t1 , ε1 ) − ∆ (t2 , ε2 ) . d(a˜ 1 , a˜ 2 ) = ∆ 2 (3.8) [

[ ] Definition 3.10 (Liu et al. 2017b) Let R˜ = r˜i j m×n be an interval 2-tuple matrix ) ( )] [( and r˜i j = ri j , αi j , ti j , εi j , then σ j2 =

m (]2 1 ∑[ ( d r˜i j , r˜ j , m i=1

(3.9)

where σ j2 denotes the variance of r˜ j , r˜ j represents an arbitrary column of interval ˜ and r˜ j represent the arithmetic means of the jth column interval 2-tuples in R, 2-tuples.

50

3 Determining Truth Degrees of Input Places in FPNs

3.3 The Proposed Model In extant FPNs, the truth values of input places are restricted to be fuzzy values between 0 and 1, which are usually determined by decision makers (or users) according to the data gathered from the considered system. However, in practice, it is natural for decision makers to express their judgements by linguistic labels. Moreover, they may evaluate the truth degrees of FPN input places using the form of uncertain linguistic terms due to time pressure, lack of data, and complexity of the problem. To overcome these drawbacks, this section develops a group decisionmaking methodology that allows for deriving the truth degrees of input places, so as to support the uncertain knowledge reasoning of FPNs. The flowchart of the proposed approach consisting of three stages is depicted in Fig. 3.1. Next, the procedure of the proposed model to determine the initial truth degrees of FPNs is described in detail.

3.3.1 Assess the Truth Degrees of Input Places In the practical application, the truth degrees of input places can be affected by a single factor or multiple factors. We here consider a general knowledge reasoning problem under the hesitant 2-tuple linguistic environment. Assume that there are l decision makers DMk (k = 1, 2, ..., l) in an expert committee responsible for the assessment of a set of m input places I Pi (i = 1, 2, ..., m). Each input place is evaluated against to a finite set of influence factors Fi j ( j = 1, 2, ..., Ni ), where N i is the number of factors considered for IPi . In our proposal, the decision makers give their assessments with respect } by means of the context-free grammar approach. Suppose { to each factor that S = s0 , s1 , ..., sg is the adopted linguistic term set, the hesitant linguistic assessment vector group of the kth decision maker is given by ⎤ h k11 h k12 · · · h k1N1 ⎢ hk hk · · · hk ⎥ 2N2 ⎥ ⎢ 21 22 Hk = ⎢ . .. . ⎥ ⎣ .. . · · · .. ⎦ h km1 h km2 · · · h km Nm ⎡

where Ni /= N j , k = 1, 2, ..., l.

(3.10)

where h ikj is the linguistic expression provided by DM k over the input place IPi against the factor F ij . The condition of Ni /= N j is going to give us a degree of flexibility in the implementation as it would not impose the same factors over each input place. Step 1 Transform hesitant linguistic assessments into interval 2-tuples. In the truth assessment processes, the experts may give single linguistic terms or need to elicit comparative linguistic expressions due to hesitation. So, to homogenize all types of judgements, hesitant linguistic assessment vector groups are provided by decision makers for each influence factor with the context-free grammar GH .

3.3 The Proposed Model

51

Fig. 3.1 Flowchart of the proposed model

After converting into corresponding HFLTSs according to the transformation funclinguistic expressions h i j can be transformed into tion E GH , each decision [ maker’s ]

linguistic intervals rikj , tikj , where rikj , tikj ∈ S and rikj ≤ tikj , by calculating the envelope of each HFLTS. Then, linguistic intervals can be represented using the [( ( ( the(] k k interval 2-tuple as ri j , 0 , ti j , 0 . As a result, the linguistic assessment information given by all the experts can be expressed via the interval 2-tuple vector group as seen below

52

3 Determining Truth Degrees of Input Places in FPNs

⎡

k r˜11 ⎢ r˜ k ⎢ 21 ˜ Rk = ⎢ . ⎣ .. k r˜m1

⎤ k k r˜12 · · · r˜1N 1 k k ⎥ r˜22 · · · r˜2N 2 ⎥ .. .. ⎥, . ··· . ⎦ k r˜m2 · · · r˜mk Nm

k = 1, 2, ..., l.

(3.11)

3.3.2 Determine the Weight Vector of Experts During the initial truth determination process, decision makers should be assigned different weights because of their different domain knowledge and expertise. In general, decision makers’ assessments may be scattered in the expert group, and some of them can even be far away from the group’s overall assessment. Therefore, it is essential to adjust expert weights to reach a group consensus. Because the variance is used to estimate the deviation degree between random variables and their mathematical expectations, it is a useful tool to measure the inconsistent degree of a decision maker to the group. Thus, to determine the weight vector of experts, a minimized variance model is proposed here. Step 2 Compute the weights of decision makers. First, each interval 2-tuple vector group R˜ k is transform into an equivalent vector ˜ k as ] [ k k k k k , ..., r˜1N , r˜21 , ..., r˜2N , ..., r˜m1 , ..., r˜mk Nm ˜ k = r˜11 1 2 [ ] = t˜1k , t˜2k ..., t˜Lk

(3.12)

∑m where L = i=1 Ni . That is, we can use a single matrix to represent all the assessment information of the decision makers by ⎡

[ ] T˜ = t˜kp l×L

t˜11 ⎢ t˜2 ⎢ 1 =⎢ . ⎣ .. t˜1l

t˜21 t˜22 .. . t˜2l

⎤ · · · t˜L1 · · · t˜L2 ⎥ ⎥ .. ⎥. ··· . ⎦

(3.13)

· · · t˜Ll

We assume that if a group of decision makers has a high consistence with the overall assessment, then the assessment values in each column of T˜ will have low degrees of discreteness. Hence, we assign each decision maker of the expert group a weight and then minimize the sum of all variances of the weighted influence factors. Then, the variance of the assessments provided by the l decision makers for the kth factor is compute as

3.3 The Proposed Model

53

σk2 =

L (]2 1 ∑[ ( d t˜kp , t˜ p , L p=1

(3.14)

where t˜ p indicates the interval 2-tuple arithmetic mean of the weighted assessments in the kth column of T˜ . At last, the weight vector of the decision makers w = (w1 , w2 ..., wl )T can be computed by using the following equation: 1 − σk2 ( ), 2 k=1 1 − σk

wk = ∑l

where wk > 0 and

l ∑

k = 1, 2, ..., l,

(3.15)

wk = 1.

k=1

3.3.3 Compute the Truth Values of Input Places The following steps present how to obtain the collective assessments for each input place of an FPN. Step 3 Establish the collective interval 2-tuple vector group. Considering the degrees of relative agreement and the importance weights of decision makers, the ITWA operator is utilized to aggregate all the individual decision vector groups[ R˜ k](k = 1, 2, ..., l) to establish the collective interval 2-tuple vector group R˜ = r˜i j m×Ni . Using the weight vector w = (w1 , w2 , ..., wl )T of the l experts, ) ( )] [( the collective interval 2-tuples r˜i j = ri j , αi j , ti j , εi j for i = 1, 2, ..., m and j = 1, 2, ..., n can be derived by ( ) ) l ( r˜i j =ITWAw r˜i1j , r˜i2j , ..., r˜il j = ⊕ wk r˜ikj k=1 ] [ l l ∑ ( ) ∑ ( ) −1 k k −1 k k wk ∆ ri j , αi j , wk ∆ ti j , εi j . =∆ k=1

(3.16)

k=1

Step 4 Compute the interval 2-tuple truth values of input places. ˜ this step utilizes the After obtaining the collective interval 2-tuple vector group R, [( ) ( )] ITWG operator to compute the interval 2-tuple truth value r˜i = r j , α j , t j , ε j (i = 1, 2, ..., m) of the input place I Pi , Ni ( )ω r˜i =ITWGωi j (˜ri1 , r˜i2 , ..., r˜in ) = ⊗ r˜i j i j j=1

54

3 Determining Truth Degrees of Input Places in FPNs

Fig. 3.2 Aggregation steps by using ITWA and ITWG

⎡

⎤ Ni Ni || [ −1 ( )]ωi j || [ −1 ( )]ωi j ⎦, ∆ ri j , αi j ∆ ti j , εi j =∆⎣ , j=1

(3.17)

j=1

where ωi j ( j = 1, 2, ..., Ni ) are the weights of influence factors regarding the input place I Pi . As a result, an interval 2-tuple truth vector R˜ V = (˜r1 , r˜2 , ..., r˜m )T can be generated. The complete operation of the above aggregation steps with the interval 2-tuple vector groups R˜ k (k = 1, 2, ..., l) is described as Fig. 3.2. Step 5 Derive the fuzzy truth degrees of input places. This step is converting the interval 2-tuple truth values into fuzzy values between 0 and 1. Such transformation is not unique because different methods are possible. With the resulting interval 2-tuple vector R˜ V = (˜r1 , r˜2 , ..., r˜m )T for each input place, the transformation could be performed according to the following cases: (1) Pessimistic case. We consider only the lower values of the given interval 2tuples to arrange elements of the initial marking vector θ = (θ1 , θ2 , ..., θm )T , where θi ∈ [0,(1] means ) the truth degree of the input place I Pi (i = 1, 2, ..., m) and θi = ∆−1 t j , ε j ; ˜ (2) Optimistic case. We apply the upper values of the interval( 2-tuple ) vector RV to determine the initial marking vector θ . That is θi = ∆−1 r j , α j ; (3) Average value. The interval 2-tuple truth values of R˜ V are averaged into single (and the )] initial marking vector θ is generated by θi = [ −1 ( fuzzy) values, ∆ r j , α j + ∆−1 t j , ε j /2.

3.4 Illustrative Example In this section, we use a fault diagnosis example from (Liu et al. 2013) to illustrate the presented model. Gas turbine is one of the most important types of power equipment and has been widely used in the aviation and marine propulsion systems, electric

3.4 Illustrative Example

55

Fig. 3.3 The FPN model of the example

power station, and natural gas transportation. Thus, Liu et al. (2013) developed a fault diagnosis tool and applied it to guarantee the gas turbine to run efficiently under a safe reliable condition. The turbine fault diagnosis system simulated by FPNs is shown in Fig. 3.3. It is observed that the places p1 , p2 , p3 , p4 and p5 are input places (I P1 , I P2 , ..., I P5 ), and their truth degrees need to be determined based on the data gathered from the gas turbine system. For this system, we assume that 2, 3, 1, 4 and 3 factors are considered in evaluating the truth degrees of the five input places, respectively. An expert committee comprising five decision makers is built to provide their judgements on each of the input places against their influence factors. In practical situations, the expert assessment data employed in multi-factor truth degree determination are generally linguistic terms. So, for any input place, all the factors are to be evaluated by means of grammar-free expressions over a seven-point linguistic term set S, i.e., ⎧ ⎫ ⎪ ⎨ s0 = Very Low (VL), s1 = Low (L), s2 = Medium Low (ML),⎪ ⎬ S = s3 = Medium (M), s4 = Medium High (MH), s5 = High (H), . ⎪ ⎪ ⎩ ⎭ s6 = Very High (VH) As a result, the linguistic assessments of the five input places given by the expert group are summarized in Table 3.1. Next, the proposed model is applied to derive the truth degrees of the input places for knowledge reasoning and fault diagnosis. First, the individual linguistic expressions of decision makers are converted into HFLTSs by applying the transformation function E GH . Then, the linguistic intervals are transformed by calculating the envelope of each obtained HFLTS and the interval 2-tuple vector group R˜ k (k = 1, 2, ..., 5)

56

3 Determining Truth Degrees of Input Places in FPNs

Table 3.1 Linguistic assessments on input places provided by the decision makers Input places IP1

IP2

IP3

IP4

IP5

Decision makers

Fi1

Influence factors Fi2

DM1

H

Greater than M

DM2

Between MH and VH

MH

DM3

H

Between MH and H

DM4

At least MH

MH

DM5

H

At least M

DM1

ML

At most H

Lower than M

DM2

ML

MH

M

DM3

Between ML and M

MH

M

DM4

Lower than M

Between MH and H

M

DM5

ML

H

Between ML and M

DM1

Lower than MH

DM2

M

DM3

ML

DM4

Between M and MH

DM5

At most H

DM1

M

At most MH

ML

At least ML

DM2

M

MH

Between ML and M

M

DM3

Greater than L

MH

ML

ML

DM4

Between M and MH

Lower than H

ML

Between ML and M

DM5

H

MH

At most M

ML

DM1

VH

At least M

DM2

H

MH

DM3

Greater than MH

MH

DM4

VH

Greater than MH

DM5

Between MH and VH

MH

Fi3

Fi4

3.4 Illustrative Example

57

of every decision maker is constructed. For example, the interval 2-tuple vector group R˜ 1 is presented in Table 3.2. In Step 2, we apply the concept of statistical variance to determine the weights of decision makers. By Eqs. (3.14) and (3.15), the weight vector of the five decision makers is computed as w = (0.161, 0.228, 0.218, 0.209, 0.184)T . In Step 3, we utilize the ITWA operator to aggregate the opinions [ ] of the decision makers to form the collective interval 2-tuple vector group R˜ = r˜i j 5×N ii , as shown in Table 3.3. In Step 4, we adopt the ITWG operator to generate the interval 2-tuple truth vector R˜ V = (˜r1 , r˜2 , r˜3 , r˜4 )T . Note that the influence factors regarding each of the input places are given equal importance coefficients in this example. The produced results based on the above calculations are gave in Table 3.4. Finally, the truth values of input places in the FPN are obtained by transforming interval 2-tuple truth values into fuzzy values according to the pessimistic, optimistic, and average methods. The detailed computed results are shown in Table 3.4. As we can observe, depending on the particular transformation method used, the fuzzy truth degrees of input places may be different, thus leading to different reasoning results. For example, the initial truth values of p1 , p2 , p3 , p4 , and p5 are 0.778, 0.451, 0.425, 0.481 and 0.818, respectively, with the average method. In contrast, when the method of (Liu et al. 2013) is applied, the corresponding truth degrees are 0.739, 0.454, 0.427, 0.427 and 0.813, respectively. It can be seen that the two sets of initial truth values have a very high similarity. Thus, the proposed method is validated. But, Table 3.2 Interval 2-tuple vector group R˜ 1 Input places

Influence factors Fi1

Fi2

Fi3

Fi4

IP1

[(s5 , 0), (s5 , 0)]

[(s4 , 0), (s6 , 0)]

IP2

[(s2 , 0), (s2 , 0)]

[(s0 , 0), (s5 , 0)]

[(s0 , 0), (s2 , 0)]

IP3

[(s0 , 0), (s3 , 0)]

IP4

[(s3 , 0), (s3 , 0)]

[(s0 , 0), (s4 , 0)]

[(s2 , 0), (s2 , 0)]

[(s2 , 0), (s6 , 0)]

IP5

[(s6 , 0), (s6 , 0)]

[(s3 , 0), (s6 , 0)]

Fi3

Fi4

Table 3.3 Collective interval 2-tuple vector group Input places

Influence factors Fi1

Fi2

IP1

∆[0.761,0.906]

∆[0.636,0.818]

IP2

∆[0.264,0.37]

∆[0.59,0.759]

∆[0.389,0.473] ∆[0.272,0.402]

IP3

∆[0.291,0.56]

IP4

∆[0.525,0.705]

∆[0.436,0.736]

IP5

∆[0.864,0.962]

∆[0.675,0.79]

∆[0.371,0.514]

58

3 Determining Truth Degrees of Input Places in FPNs

Table 3.4 Interval 2-tuple truth vector and transformed fuzzy values Input places

Interval 2-tuple truth values

Fuzzy truth values Pessimistic

Optimistic

Neutral

IP1

[(s4 , 0.029), (s5 , 0.028)]

0.695

0.861

0.778

IP2

[(s2 , 0.059), (s3 , 0.010)]

0.392

0.510

0.451

IP3

[(s2 , −0.042), (s3 , 0.060)]

0.291

0.560

0.425

IP4

[(s2 , 0.057), (s3 , 0.072)]

0.390

0.572

0.481

IP5

[(s5 , −0.070), (s5 , 0.038)]

0.764

0.872

0.818

the propoed model has exact characteristic in linguistic computing and considers decision-makers’ risk attitudes in determining the truth degrees of FPN input places. From the analysis above, it can be concluded that the proposed decision-making method provides a useful, practical and effective way for the determination of initial marking vector in FPNs. It could not only increase the flexibility and reliability of initial truth degree evaluations, but also preserve linguistic information in the computing process. Moreover, the proposed model permits to take different scenarios into consideration and thus provides more complete information for knowledge reasoning and decision-making.

3.5 Chapter Summary This chapter introduced a group decision-making model based on HFLTSs to obtain the initial truth values of FPNs based on domain experts’ knowledge and experience. The case study and simulation results showed the feasibility and effectiveness of the proposed model. In summary, the main advantages of the proposed framework are as follows: (1) The diversity and uncertainty of experts’ subjective assessment information can be well reflected and modeled. (2) The relative importance weights of decision makers are considered and determined objectively using the assessment information available. (3) Multiple factors can be considered in evaluating the truth degrees of input places, which makes the proposed method more realistic and more flexible. (4) It is possible to deal with the attitudinal character of decision makers and provide a more complete picture for knowledge inference.

References Chen SM, Ke JS, Chang JF (1990) Knowledge representation using fuzzy Petri nets. IEEE Trans Knowl Data Eng 2(3):311–319 Gao MM, Zhou MC, Huang XG, Wu ZM (2003) Fuzzy reasoning Petri nets. IEEE Trans Syst Man Cybern A Syst Hum 33(3):314–324

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Herrera F, Martínez L (2000) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752 Li L, Xie Y, Cen L, Zeng Z (2022) A novel cause analysis approach of grey reasoning Petri net based on matrix operations. Appl Intell 52(1):1–18 Lin Z, Zhang J, Chen Y, Tian Q, Lin Z, Huang G (2022) A new approach to power system fault diagnosis based on fuzzy temporal order Petri nets. Energy Rep 8:969–978 Liu HC, Lin QL, Ren ML (2013) Fault diagnosis and cause analysis using fuzzy evidential reasoning approach and dynamic adaptive fuzzy Petri nets. Comput Ind Eng 66(4):899–908 Liu HC, Lin QL, Wu J (2014a) Dependent interval 2-tuple linguistic aggregation operators and their application to multiple attribute group decision making. Int J Uncert Fuzziness Knowl Based Syst 22(5):717–735 Liu HC, You JX, You XY (2014b) Evaluating the risk of healthcare failure modes using interval 2-tuple hybrid weighted distance measure. Comput Ind Eng 78:249–258 Liu HC, Li P, You JX, Chen YZ (2015) A novel approach for FMEA: combination of interval 2-tuple linguistic variables and grey relational analysis. Qual Reliabil Eng Int 31(5):761–772 Liu HC, You JX, Li ZW, Tian G (2017a) Fuzzy Petri nets for knowledge representation and reasoning: a literature review. Eng Appl Artif Intell 60:45–56 Liu HC, You JX, Tian G (2017b) Determining truth degrees of input places in fuzzy Petri nets. IEEE Trans Syst Man Cybern Syst 47(12):3425–3431 Liu HC, Xu DH, Duan CY, Xiong Y (2021) Pythagorean fuzzy Petri nets for knowledge representation and reasoning in large group context. IEEE Trans Syst Man Cybern Syst 51(8):5261–5271 Liu HC, Luan X, Zhou M, Xiong Y (2022) A new linguistic Petri net for complex knowledge representation and reasoning. IEEE Trans Knowl Data Eng 34(3):1011–1020 Looney CG (1988) Fuzzy Petri nets for rule-based decision-making. IEEE Trans Syst Man Cybern 18(1):178–183 Luo X, He L, Wei X, Zhu M, Li Z (2023) Security requirement classification of electricity trading data based on hierarchical fuzzy Petri network. Energy Rep 9:189–199 Mou X, Zhang QZ, Liu HC, Zhao J (2021) Knowledge representation and acquisition using Rnumbers Petri nets considering conflict opinions. Exp Syst 38(3):e12660 Mou X, Mao LX, Liu HC, Zhou M (2022) Spherical linguistic Petri nets for knowledge representation and reasoning under large group environment. IEEE Trans Artif Intell 3(3):402–413 Rodriguez RM, Martinez A, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20(1):109–119 Shi H, Liu HC, Wang JH, Mou X (2022) New linguistic Z-number Petri nets for knowledge acquisition and representation under large group environment. Int J Fuzzy Syst 24(8):3483–3500 Wang X, Lu F, Zhou M, Zeng Q (2022) A synergy-effect-incorporated fuzzy Petri net modeling paradigm with application in risk assessment. Exp Syst Appl 199:117037 Xu XG, Xiong Y, Xu DH, Liu HC (2020) Bipolar fuzzy Petri nets for knowledge representation and acquisition considering non-cooperative behaviors. Int J Mach Learn Cybern 11:2297–2311 Yeung DS, Tsang ECC (1994) Fuzzy knowledge representation and reasoning using Petri nets. Exp Syst Appl 7(2):281–289 You XY, You JX, Liu HC, Zhen L (2015) Group multi-criteria supplier selection using an extended VIKOR method with interval 2-tuple linguistic information. Exp Syst Appl 42(4):1906–1916 Yu J, Zeng Q, Yu Y, Wu S, Ding H, Gao H, Yang J (2022) An intuitionistic fuzzy probabilistic Petri net method for risk assessment on submarine pipeline leakage failure. Ocean Eng 266:112788 Yu YX, Gong HP, Liu HC, Mou X (2023) Knowledge representation and reasoning using fuzzy Petri nets: a literature review and bibliometric analysis. Artif Intell Rev 56:6241–6265 Yue W, Wan X, Li S, Ren H, He H (2022) Simplified neutrosophic Petri nets used for identification of superheat degree. Int J Fuzzy Syst 24(8):3431–3455 Zhang H (2012) The multiattribute group decision making method based on aggregation operators with interval-valued 2-tuple linguistic information. Math Comput Modell 56(1–2):27–35

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Zhang C, Tian G, Fathollahi-Fard AM, Wang W, Wu P, Li Z (2022) Interval-valued intuitionistic uncertain linguistic cloud Petri net and its application to risk assessment for subway fire accident. IEEE Trans Autom Sci Eng 19(1):163–177

Part II

Improved FPNs for Knowledge Representation and Acquisition

Chapter 4

Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation and Acquisition

The two most important issues of expert systems are the acquisition of experts’ professional knowledge and the representation of the knowledge rules that have been identified. First, during expert knowledge acquisition processes, domain experts often demonstrate different experience and knowledge and give different types of knowledge information such as complete and incomplete, precise and imprecise, and known and unknown because of its cross-functional and multidisciplinary nature. Second, as a promising tool for knowledge representation and reasoning, the FPNs still suffer a couple of deficiencies. The parameters in current FPN models could not accurately represent the increasingly complex knowledge-based systems and the knowledge rules could not be dynamically adjustable according to propositions’ variation. In this chapter, we present a knowledge representation and acquisition approach using the fuzzy evidential reasoning (FER) approach and dynamic adaptive fuzzy Petri nets (DAFPNs) to solve the problems mentioned above. As is illustrated by a numerical example, the proposed approach can well capture experts’ diversity experience, enhance the knowledge representation power, and reason the rule-based knowledge dynamically.

4.1 Introduction An expert system is a kind of intellectual programming system which can solve problems in relevant fields with expert’s levels, use the experience and special knowledge that domain experts accumulated for many years, simulate the thought process of human experts, and solve the difficult problem that generally only experts can do (Sharma et al. 2023; Yedida et al. 2023). The building of expert systems has been characterized by capturing expert knowledge in such a way that it is possible for nonexperts to solve a particular problem using knowledge already captured and stored in the computer (Yeung and Tsang 1994; Liu et al. 2013a). Two key issues in developing an expert system are the acquisition of domain experts’ professional knowledge © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_4

63

64

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

and the representation of the transformed knowledge rules. So far, many knowledge representation methods have been developed, which include interconnected uncertain rules (Popescu and Dumitrache 2023), ontology (Lin et al. 2023), dynamic uncertain causality graph (Zhu et al. 2022), linguistic belief structures (Rajati and Mendel 2022), and FPNs (Chen et al. 1990). During the last years, FPRs have been widely used in the development of expert systems to represent, capture and store fuzzy knowledge. With the increasing complexity of knowledge-based systems, much effort has been paid to the improvement of FPRs by incorporating knowledge parameters such as threshold value, certainty factor, weight, but these parameters are difficult to acquire or extract from domain experts and it is a difficult problem to fine-tune them during the knowledge base update and maintenance phase (Liu et al. 2013b, 2016). Furthermore, the typical knowledge acquisition process is to acquire knowledge directly through interactions with experts (Sun et al. 2003). Normally, different experts may demonstrate different opinions because of their different expertise and backgrounds. They may provide different types of information for the knowledge parameters in the same fuzzy rule, some of which may be complete or incomplete, precise or imprecise, and known or unknown. Also, domain experts often use the daily life terminologies in their expressions. The given assessment grades of knowledge parameters may represent a vague concept and there may be no clear cut between the meanings of two adjacent grades. Moreover, existing knowledge acquisition tools are not effective and intuitive enough to acquire FPR parameters from domain experts. As a result, the traditional process of eliciting rules through interaction with experts may turn out to be a bottleneck, causing delays in the system’s overall development (Liu et al. 2013a). On the other hand, knowledge representation is another issue that limits the application of an expert system. FPNs are an extension of Petri net, which are a promising modeling tool to represent FPRs of a rule-based expert system. Compared to other knowledge modeling methodologies, FPNs have a couple of attractive advantages as follows (Gao et al. 2003; Ha et al. 2007; Mou et al. 2021; Liu et al. 2022). First, FPNs are formal and general graphical models of rule-based expert systems and can represent logical knowledge in an intuitive and visual way. Second, FPNs can naturally model logical and mathematical arrays and allow checking the properties of modeled systems utilizing the major characteristics of Petri nets. Third, FPNs can capture the dynamic nature of fuzzy rule-based reasoning with marking evolution or express the dynamic behavior of a system in algebraic forms. Nevertheless, the existing FPN models still suffer a couple of deficiencies. In view of the complexity of knowledge-based systems, the application of expert systems will be severely limited if it cannot acquire different kinds of domain experts’ professional knowledge and its knowledge representation models cannot express expert knowledge truly. In this chapter, we propose a new knowledge representation and acquisition model using the fuzzy evidential reasoning (FER) approach and dynamic adaptive fuzzy Petri nets (DAFPNs). The new approach can model the diversity and uncertainty of the experience and knowledge of domain experts; represent identified fuzzy knowledge rules more precisely and improve knowledge reasoning

4.2 Fuzzy Evidential Reasoning Approach

65

efficiency greatly; reason the rule-based knowledge by dynamically adjusting FPRs with the change of preconditioned propositions. The remaining part of the chapter is organized as follows. In Sects. 4.2 and 4.3, we develop the knowledge representation and acquisition model using the FER approach and DAFPNs. A numerical example is provided in Sect. 4.4 to illustrate the feasibility and validity of the proposed mode. This chapter is concluded in Sect. 4.5 with a brief summary.

4.2 Fuzzy Evidential Reasoning Approach The evidential reasoning (ER) approach was developed for dealing with decision making problems characterized by both quantitative and qualitative attributes with various types of uncertainties (Liu et al. 2011, 2023; Fang et al. 2021). Another extension to the original ER approach, Yang et al. (2006) proposed the FER approach to capture fuzziness caused by the fuzzy evaluation grades.

4.2.1 Acquisition of Rule-Based Knowledge Using Belief Structures The knowledge parameters, such as weight, threshold value, and certainty factor, can be evaluated numerically or linguistically. However, there is a high level of uncertainty involved in experts’ knowledge acquisition processes, since knowledge acquisition consists in interacting directly with domain experts and the provided information for knowledge parameters mainly based on experts’ subjective judgments. In addition, most experts are willing to express their opinions or assessments by belief degrees based on a set of expression grades. As such, in this chapter, we choose linguistic terms for the acquisition of FPR parameters and the individual evaluation grade set is defined as a fuzzy set H F as follows: HF ={H11 , H22 , H33 , H44 , H55 } ={V er y Low, Low, Moderate, H igh, V er y H igh}.

(4.1)

  To generalize the Hˆ F = Hi j , i = 1, ..., 5; j = 1, ..., 5 to fuzzy sets, we assume that a general set of fuzzy individual assessment grades {Hii }, i = 1, ..., 5, are dependent on each other and only two adjacent fuzzy individual assessment grades may intersect. Based on the experts’ opinions, we can approximate all the five individual assessment grades of knowledge parameters by trapezoidal fuzzy numbers, and their membership function values can be determined according to the historical data and the questionnaire answered by experts, as shown in Fig. 4.1 and Table 4.1. Furthermore, we define the sets H ij for i = 1, ..., 4 and j = i + 1 to 5 as trapezoidal

66

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

fuzzy sets that include fuzzy individual grade Hii , H(i+1)(i+1) , ..., H j j . If the individual assessment grades are trapezoidal fuzzy sets, every interval grade will be a trapezoidal fuzzy set as shown in Fig. 4.2. As mentioned before, the assessment grades of knowledge parameters may represent a vague concept and there may be no clear cut between the meanings of two adjacent grades. Such a problem can be solved by the FER approach, which allows domain experts to express their subjective experience in the following ways: • A certain rating such as Low, which can be written as {(H22 , 1.0)}. Such an expression is referred to as a belief structure in the FER approach. • A distribution such as Low to 0.4 and Moderate to 0.6, which means that a fuzzy rule is assessed with respect to the FPR parameter under consideration to rating Low to the degree of 0.4 and to rating Moderate to the degree of 0.6. Here the degrees of 0.4 and 0.6 represent the confidences of an expert in his/her subjective judgments and the distribution can be equivalently expressed as {(H22 , 0.4), (H33 , 0.6)}. When all the confidences are summed to one, the distribution is said to be complete; otherwise, it is said to be incomplete. • An interval such as Low-Moderate, which means that the rating of a fuzzy rule with respect to the FPR parameter under evaluation is between Low and Moderate. This can be written as {(H23 , 1.0)}.

Fig. 4.1 Fuzzy membership function for linguistic terms

Table 4.1 Ratings for linguistic terms

Linguistic terms

Fuzzy number

Very low

(0, 0, 0, 2)

Low

(1, 2, 3, 4)

Moderate

(3, 4, 6, 7)

High

(6, 7, 8, 9)

Very high

(8, 10, 10, 10)

4.2 Fuzzy Evidential Reasoning Approach

67

Fig. 4.2 Interval fuzzy grades set

• No judgment, which means the domain expert is not willing to or cannot provide an assessment for a fuzzy rule with respect to the knowledge parameter under consideration. Such judgment are referred to as total ignorance. The belief structures in the FER approach provide domain experts with an easyto-use and very flexible way to express their expertise and knowledge.

4.2.2 Group Belief Structures For each knowledge parameter, suppose there are K domain experts (TM 1 , …, TM K ) responsible for the assessment of N fuzzy rules (R1 , …, RN ) with respect to L antecedent or consequent propositions (P1 , …, PL ). For a fuzzy rule, weight is related to the antecedent proposition; certainty factor and threshold are associated with the consequent proposition. If there is only one proposition in the antecedent, the weight is set to 1 and doesn’t need to be evaluated. Each team member TM k is  given a weight vk > 0(k = 1, ..., K ) satisfying 1K vk = 1 to reflect his/her relative importance  in the expert panel.   Hi j , γikj (Rn , Pl ) , i = 1, ..., 5; j = 1, ..., 5 be the belief structure Let provided by TM k on the assessment of Rn with respect to Pl where H ii for i = 1, ..., 5 are fuzzy assessment grades defined for knowledge assessment, H ij for i = 1, ..., 4 and j = i + 1 to 5 are the interval fuzzy assessment grades between H ii and H jj , and γikj (Rn , Pl ) are the belief degrees to which Rn assessed on Pl to the intervals H ij . All the grades H ii for i = 1, ..., 5 and the intervals Hij for i = 1, ..., 4 and j =  i + 1 to 5 together form a frame of discernment, which is expressed as Hˆ F = Hi j , i = 1, ..., 5; j = 1, ..., 5 or equivalently

68

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

⎧ ⎪ ⎪ H11 H12 H13 ⎪ ⎪ ⎪ H22 H23 ⎨ ˆ HF = H33 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

H14 H24 H34 H44

⎫ H15 ⎪ ⎪ ⎪ ⎪ H25 ⎪ ⎬ H35 . ⎪ ⎪ H45 ⎪ ⎪ ⎪ H55 ⎭

(4.2)

The collective assessment of the K team members for each fuzzy rule is also a belief structure, called group belief structure, which is denoted as    X˜ n (l) = Hi j , γi j (Rn , Pl ) , i = 1, . . . , 5; j = 1, . . . , 5 , n = 1, . . . , N ; l = 1, . . . , L ,

(4.3)

where γi j (Rn , Pl ) is referred to as group belief degree and is determined by γi j (Rn , Pl ) =

K 

vk γikj (Rn , Pl ),

k=1

i =1, . . . , 5; j = 1, . . . , 5; n =1, . . . , N ; l = 1, . . . , L .

(4.4)

For each knowledge parameter, the group belief structures for the N fuzzy rules with respect to the L antecedent or consequent propositions form a fuzzy belief matrix, and the three FPR parameters’ fuzzy belief matrices can be represented as W˜ , U˜ and Th, respectively.

4.2.3 Defuzzification Based on the fuzzy belief matrix, group belief structures on the assessment of each fuzzy rule with respect to the L antecedent or consequent propositions can be aggregated into an overall belief structure using the defuzzification and weighted average methods successively. Chen and Klein (1997) proposed an easy defuzzification method for obtaining the crisp number of a fuzzy set, which is shown in Eq. (4.5): n

k=0 (bk − c)  , i = 1, . . . , 5; j = 1, . . . , 5, − c) − nk=0 (ak − d) (b k k=0

h i j = n

(4.5)

where hij is the defuzzified crisp number of H ij ; c and d are the extreme values of the range of the whole fuzzy set. The values of c and d will remain the same for the defuzzification of all linguistic terms. The values a0 and b0 are rating values at the extreme limits of each linguistic term where the membership function is 0 and

4.3 Dynamic Adaptive Fuzzy Petri Nets

69

a1 and b1 are rating values at the extreme limits of each linguistic term where the membership function is 1. Finally, the overall assessment of the fuzzy rule Rn with respect to Pl is a crisp number, called overall belief structure, which can be aggregated by X n (l) =

5 5  

h i j γi j (Rn , Pl ),

i=1 j=1

n = 1, . . . , N ; l = 1, . . . , L .

(4.6)

As to the weight parameter, the group weights of each rule’s antecedent propositions need to be defuzzified using Eq. (4.5) and then normalized using Eq. (4.7). X n (l) wl =  L , l=1 X n (l)

l = 2, . . . , L .

(4.7)

In the end, the fuzzy belief matrices can be defuzzified to crisp belief decision matrices W, U, and T h correspondingly.

4.3 Dynamic Adaptive Fuzzy Petri Nets 4.3.1 Definition of DAFPNs Considering the complexity and dynamic nature of expert systems, we define a DAFPN structure as follows. Definition 1 A DAFPN is an 11-tuple: D AF P N = (P; T ; I ; O; D; α; β; W ; U ; T h; M)

(4.8)

where • P = { p1 , p2 , ..., pm } denotes a finite nonempty set of places, P=PS ∪ PI ∪ PT . It consists of a set of starting places PS ={ p ∈ P|· p = ∅ and p · = ∅ }, a set of intermediate places PI ={ p ∈ P|· p = ∅ and p · = ∅ }, and a set of terminating places PT ={ p ∈ P|· p = ∅ and p · = ∅ }. Here, · p and p · are the set of all input transitions of p and the set of all output transitions of p, respectively. • T = {t1 , t2 , . . . , tn } denotes a finite nonempty set of transitions. • I : P × T → {0, 1} is an m × n input incidence matrix defining the directed arcs from places to transitions. Ii j = 1, if there is a directed arc from pi to t j , and Ii j = 0, if there is no directed arcs from pi to t j , for i = 1, 2, ..., m and j = 1, 2, . . . , n.

70

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

• O : T × P → {0, 1} is an m × n output incidence matrix defining the directed arcs from transitions to places. Oi j = 1, if there is a directed arc from t j to pi , and Oi j = 0, if there is no directed arcs from t j to pi , for i = 1, 2, ..., m and j = 1, 2, . . . , n. • D = {d1 , d2 , . . ., dm } denotes a finite set of propositions, P ∩ T ∩ D = ∅, |P| = |D|. • α : P → [0, 1] is an association function which maps from places to real values between 0 and 1. • β : P → D is an association function representing a bijective mapping from places to propositions. • W : P × T → [0, 1] is an input function and can be expressed as an m × ndimensional matrix. The value of an element in W, wi j ∈ [0, 1], is the weight of the input place, which indicates how much the place pi impacts its following m  wi j =1, j = 1, 2, ..., n. transition t j connected by Ii j . In this chapter, let i=1

• U : T × P → [0, 1] is an output function and can be expressed as an m × ndimensional matrix. The value of an element in U, μi j ∈ [0, 1], is the value of certainty factor, which indicates how much a transition t j impacts its output places pi if the transition fires. • T h : O → [0, 1] is an output function which assigns  a certainty value between 0 and 1 to each output place of a transition, T h = τi j m×n , i = 1, 2, ..., m; j = 1, 2, ..., n, denoting the output threshold of this place. τi j ∈ [0, 1], if there is a directed arc from t j to pi , and τi j = +∞, if there is no directed arc from t j to pi , for i = 1, 2, ..., m and j = 1, 2, . . . , n. • M is a marking of the DAFPN, M=(α( p1 ), α( p2 ), ..., α( pm ))T . The initial marking is denoted by M 0 .

4.3.2 DAFPN Representations for WFPRs In order to map WFPRs into DAFPNs, we define the WFPRs as the following new forms: Type 1 A simple weighted fuzzy production rule. R: IF a THEN c (w; τ ; μ). Type 2 A composite weighted fuzzy conjunctive rule in the antecedent. R: IF a1 AND a2 AND…AND am THEN c (w1 , w2 , ..., wm ; τ ; μ). Type 3 A composite weighted fuzzy conjunctive rule in the consequent. R: IF a THEN c1 AND c2 AND…AND cm (w; τ1 , τ2 , ..., τm ; μ1 , μ2 , ..., μm ). In the DAFPN model of a rule-based system, place represents a proposition, transition represents a fuzzy rule, and arc represents the relationship between a proposition and a rule. The rule reasoning processes can be expressed by means of firing the transitions in the DAFPN. According to the above specification, the three types of rules can be represented with DAFPN structures as shown in Figs. 4.3, 4.4 and 4.5, respectively.

4.3 Dynamic Adaptive Fuzzy Petri Nets

71

Fig. 4.3 DAFPN representation of type 1 rule

Fig. 4.4 DAFPN representation of type 2 rule

Fig. 4.5 DAFPN representation of type 3 rule

4.3.3 Execution Rules of DAFPNs Let I (t) = { p I 1 , p I 2 , . . . , p I m } with the corresponding weights w I 1 , w I 2 , . . . , w I m and let O(t) = { p O1 , p O2 , . . . , p On } with the corresponding output thresholds τ O1 , τ O2 , . . . , τ On and certainty factors μ O1 , μ O2 , . . . , μ On . The enabling and firing rules in DAFPNs are given as follows.

72

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

Enabling rule ∀ t ∈ T , t is enabled and fired if ∀ p I j ∈ I (t),     α p I j > 0 ∧ {μ(t) ≥ min(τ Ok )}, j = 1, 2, ..., m, k = 1, 2, ..., n,

(4.9)

    where α p I j , α p I j ∈ [0, 1], is the fuzzy truth value in place  pI j , pI j ∈ I (t), which indicates the truth degree of proposition d I j if β p I j =d I j ; μ(t) = m    α p I j w I j , j = 1, 2, ..., m is the equivalent fuzzy truth of input places at the j=1

enabled transition t. Firing rule After t was fired, the tokens in input places are copied and a token with fuzzy truth is put into each of the output places whose output thresholds are lesser than the equivalent fuzzy truth. The new fuzzy truth values of the output places are defined by  α( p Oi ) =

μ Oi μ(t), μ(t) ≥ τ Oi , i = 1, 2, . . . , n. 0, μ(t) < τ Oi

(4.10)

If a place has more than one input transitions and more than one of its input transitions fires, then the new fuzzy truth value of the output place is produced by the transition with the maximum fuzzy truth.

4.3.4 Concurrent Reasoning Algorithm of DAFPNs To explain the fuzzy reasoning algorithm of DAFPNs more clearly, some operators are introduced first. (1) Operator ⊕: A ⊕ B = D,

(4.11)

where A, B, and D are all m × n-dimensional  with ai j , bi j , and di j  matrices being their elements respectively, and di j = max ai j , bi j , i = 1, 2, ..., m; j = 1, 2, ..., n. (2) Operator ◦: A ◦ B = D,

(4.12)

where A, B, and D are all m × n-dimensional matrices with ai j , bi j , and di j being their elements respectively, and di j =ai j · bi j i = 1, 2, ..., m; j = 1, 2, ..., n. (3) Operator ⊗: A ⊗ B = D,

(4.13)

4.3 Dynamic Adaptive Fuzzy Petri Nets

73

where A, B, and D are all (m × p), ( p × n), (m × n)-dimensional aik ,bk j , and di j being their elements respectively, and matrices with  di j = max aik · bk j , i = 1, 2, ..., m; j = 1, 2, ..., n. 1≤k≤ p

Next, the concurrent reasoning algorithm of DAFPNs is explained in detail. Input: I, O, W, U, Th are m × n-dimensional matrices; M 0 is an m-dimensional vector. Output: M k is an m-dimensional vector, representing the final states of all statements. Step 1 Let k = 1, where k represents the time of iteration. Step 2 Compute the vector of equivalent fuzzy truth values of transitions.  (k) = W T M(k−1) ,

(4.14)

Step 3 Compute the output enabled matrix E (k) which indicates the enabled output arcs of transitions. Let yi(k) j be the comparison result between the equivalent fuzzy truth value and the output threshold of transition t i during the kth iteration, then   Y (k) = yi(k) j

m×n

= N (k) − T h,

i =1, 2, . . . , m; j = 1, 2, . . . , n, where N (k) =

(4.15)

 T  (k)   (k)   ,  , ...,  (k) m×n ◦ O.

(k) Let ei(k) j be the function of comparison result yi j , then

 (k) E (k) = ei j m×n ,

(4.16)

 1 yi j ≥ 0 , i = 1, 2, ..., m; j = 1, 2, ..., n. where ei j = 0 yi j < 0 Step 4 If E (k) is a non-zero matrix, the matrix  (k) is computed by (4.17); otherwise, go to Step 6.  (k) = E (k) ◦ U

(4.17)

Step 5 Compute new marking Mk .   Mk = Mk−1 ⊕  (k) ⊗  (k)

(4.18)

If Mk = Mk−1 then go to Step 6; otherwise let k = k + 1, go back to Step 2. Step 6 End reasoning.

74

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

4.4 An Illustrative Example In this section, we provide a numerical example to illustrate the potential applications of the proposed knowledge representation and acquisition model. An expert panel consisting of five cross-functional members identifies six fuzzy rules in a system and needs to give their opinions in terms of their knowledge parameters. Let d1 , d2 , d3 , d4 , d5 , d6 , d7 , d8 , d9 and d10 be ten propositions and the acquired FPRs are shown as follows: R1 : IF d1 and d2 and d3 THEN d5 (w11 , w21 , w31 ; τ51 ; μ51 ) R2 : IF d3 THEN d6 (w32 ; τ62 ; μ62 ) R3 : IF d3 and d4 THEN d6 and d7 (w33 , w43 ; τ63 , τ73 ; μ63 , μ73 ) R4 : IF d5 THEN d8 (w54 ; τ84 ; μ84 ) R5 : IF d6 THEN d8 (w65 ; τ85 ; μ85 ) R6 : IF d7 THEN d8 and d9 and d10 (w76 ; τ86 , τ96 , τ106 ; μ86 , μ96 , μ106 ). Based on the aforementioned translation principle, we map this knowledge base system into a DAFPN which is shown in Fig. 4.6. Due to the difficulty in precisely assessing knowledge parameters, the expert team agrees to express them using the linguistic terms defined in Table 4.1. The expert assessment information of the six fuzzy rules on each knowledge parameter is presented in Tables 4.2, 4.3 and 4.4, where incomplete assessments and ignorance information are highlighted. The five experts from different departments are assumed to be of different importance because of their different knowledge and expertise. To reflect their differences during the expertise acquisition processes, the five experts are assigned the following weights: 0.15, 0.20, 30, 0.25 and 0.10.To carry out knowledge representation and reasoning, we first use belief structures to express the experts’ individual assessments and synthesize them into group belief structures by using Eq. (4.4). The group belief structures are then defuzzified and aggregated into overall belief structures using Eqs. (4.5)–(4.6). The results are presented in the last columns of Tables 4.2, 4.3 and 4.4. For this system, suppose that the initial marking is M0 =  T 0.8 0.9 0.7 0.8 0 0 0 0 0 0 . From the DAFPN and the data in Tables 4.2, 4.3 and 4.4, we have

4.4 An Illustrative Example

75

Fig. 4.6 DAFPN of the example

⎡ ⎤ 0000 0 ⎢0 0 0 0 0⎥ ⎢ ⎥ ⎢0 0 0 0 0⎥ ⎢ ⎥ ⎢0 0 0 0 ⎥ 0⎥ ⎢ ⎢ ⎥ 0⎥ ⎢1 0 0 0 ⎥ O=⎢ ⎢0 1 1 0 0⎥ ⎢ ⎥ ⎢0 0 1 0 1⎥ ⎢ ⎥ ⎢0 0 0 1 ⎥ 0⎥ ⎢ ⎣0 0 0 0 ⎦ 0 0000 0 ⎤ 0.441 0 0 0 0 0 ⎢ 0.327 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0.232 1 0.366 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0.634 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0 0 0 1 0 0⎥ W =⎢ ⎥ ⎢ 0 0 0 0 1 0⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 1⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 0⎦ 0 0 0 000

⎡

1 ⎢1 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢ ⎢0 I =⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 0 1 0 0 0 0 0 0 0

00 00 10 10 01 00 00 00 00 00 ⎡

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 1 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 1⎦ 1

4 5 6

2 3

1 H33

(H33,0.30) (H45,0.70) 1 1 1

w32 w33

w43

w54 w65 w76

w31

(H33,0.50) (H44,0.50) H22

1 1 1

H33

1 H34

(H33,0.90)

H44

Expert panel members TM1 TM2 H44 H55

w21

Fuzzy rules 1 w11

1 1 1

1 (H12,0.30) (H33,0.60) H33

H33

H34

TM3 H45

Table 4.2 Weight assessment information of fuzzy rules

1 1 1

H33

1 H33

H22

H33

TM4 H44

1 1 1

1 H33

H23

H33

TM5 H55 {(H44, 0.40), (H45, 0.30), (H55, 0.30)} {(H33, 0.425), (H34, 0.30), (H44, 0.275)} {(H15, 0.02), (H22, 0.40), (H23, 0.10), (H33, 0.48)} 1 {(H12, 0.09), (H15, 0.03), (H33, 0.68), (H34, 0.20)} {(H15, 0.10), (H45, 0.105), (H55, 0.795)} 1 1 1

Group belief structures

1 1 1

0.634

1 0.366

0.232

0.327

Defuzzification and normalization 0.441

76 4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

μ84

μ85 μ86 μ96 μ106

4

5 6

Fuzzy rules 1 μ51 2 μ62 3 μ63 μ73

Expert panel members TM1 TM2 H55 (H45,0.80) H44 H44 H55 H55 (H34,0.20) H44 (H55,0.80) H55 (H55,0.50) (H44,0.40) H44 H55 H45 H55 H55 H55 H45 H44 H34 H55 H55 H55

H55

TM3 H55 H55 H55 H35

H44 H55 H55

H45

TM4 H55 H44 H55 H44

Table 4.3 Certainty factor assessment information of fuzzy rules

H55 H55 H55 H55

H55

TM5 H45 H44 H25 H44

{(H15, 0.04), (H45, 0.26), (H55, 0.70)} {(H44, 0.80), (H55, 0.20)} {(H25, 0.10), (H55, 0.90)} {(H34, 0.03), (H35, 0.30), (H44, 0.55), (H55, 0.12)} {(H15, 0.02), (H44, 0.08), (H45, 0.25), (H55, 0.65)} {(H34, 0.30), (H44, 0.40), (H55, 0.30)} {(H45, 0.15), (H55, 0.85)} {(H55, 1.00)} {(H15, 0.25), (H44, 0.20), (H45, 0.15), (H55, 0.40)}

Group belief structures

0.726 0.884 0.909 0.741

0.843

0.849 0.748 0.872 0.697

Defuzzification

4.4 An Illustrative Example 77

H22 H11

(H11,0.60) (H22,0.40) H22

H34

τ84 τ85

τ86

τ106

4 5

6

τ96

H22

τ73

2 3

H22

H44

H11

H22

H22 H22

H22

H23 H11

Expert panel members TM1 TM2 H11 H22

τ62 τ63

Fuzzy rules 1 τ51

H33

H22

H22 (H11,0.70) (H23,0.30) H22

H23

H22 H11

TM3 H11

Table 4.4 Threshold assessment information of fuzzy rules

(H11,0.50) (H24,0.30) H33

H11

H22 H22

H11

H22 H11

TM4 H11

H44

H22

H25

H22 H11

H22

H22 H23

TM5 (H23,0.90) {(H11, 0.70), (H15, 0.01), (H23, 0.09), (H22, 0.20)} {(H22, 0.80), (H23, 0.20)} {(H11, 0.75), (H15, 0.15), (H23, 0.10)} {(H11, 0.25), (H22, 0.45), (H23, 0.30)} {(H22, 1.00)} {(H11, 0.46), (H22, 0.45), (H23, 0.09)} {(H11, 0.34), (H22, 0.56), (H25, 0.10)} {(H11, 0.325), (H15, 0.05), (H22, 0.55), (H24, 0.075)} {(H33, 0.55), (H34, 0.15), (H44, 0.30)}

Group belief structures

0.572

0.253

0.249

0.091 0.284

0.141

0.320 0.187

0.166

Defuzzification

78 4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

4.4 An Illustrative Example

79

⎤ 0 0 0 0 0 0 ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 ⎥ ⎢ 0.849 0 U =⎢ ⎥ ⎢ 0 0.748 0.872 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0.697 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0.843 0.726 0.884 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 0.909 ⎦ 0 0 0 0 0 0.741 ⎤ ⎡ +∞ +∞ +∞ +∞ +∞ +∞ ⎢ +∞ +∞ +∞ +∞ +∞ +∞ ⎥ ⎥ ⎢ ⎢ +∞ +∞ +∞ +∞ +∞ +∞ ⎥ ⎥ ⎢ ⎢ +∞ +∞ +∞ +∞ +∞ +∞ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 0.166 +∞ +∞ +∞ +∞ +∞ ⎥ Th = ⎢ ⎥. ⎢ +∞ 0.320 0.187 +∞ +∞ +∞ ⎥ ⎥ ⎢ ⎢ +∞ +∞ 0.141 +∞ +∞ +∞ ⎥ ⎥ ⎢ ⎢ +∞ +∞ +∞ 0.091 0.284 0.249 ⎥ ⎥ ⎢ ⎣ +∞ +∞ +∞ +∞ +∞ 0.253 ⎦ +∞ +∞ +∞ +∞ +∞ 0.572 ⎡

According to the reasoning algorithm of DAFPNs, we can: (1) Compute the vector  (1) :  T  (1) = W T M0 = 0.81 0.7 0.763 0 0 0 . (2) Compute the output enabled matrix E (1) : ⎡

Y (1)

 ei j =

−∞ ⎢ −∞ ⎢ ⎢ −∞ ⎢ ⎢ −∞ ⎢ ⎢ ⎢ 0.644 (1) = N − Th = ⎢ ⎢ −∞ ⎢ ⎢ −∞ ⎢ ⎢ −∞ ⎢ ⎣ −∞ −∞

1 yi j ≥ 0 0 yi j < 0

−∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ −∞ 0.38 0.576 −∞ 0.622 −∞ −∞ −∞ −∞ −∞ −∞

−∞ −∞ −∞ −∞ −∞ −∞ −∞ −0.091 −∞ −∞

−∞ −∞ −∞ −∞ −∞ −∞ −∞ −0.284 −∞ −∞

i = 1, 2, . . . , 10; j = 1, 2, . . . , 6, thus

−∞ −∞ −∞ −∞ −∞ −∞ −∞ −0.249 −0.253 −0.572

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

80

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

⎡

0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ (1) ⎢ 1 E =⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 1 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0

Since E (1) is a non-zero matrix, the reasoning proceeds to execute the following steps. (3) Compute the matrix  (1) : ⎤ 0 0 0 000 ⎢ 0 0 0 0 0 0⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0⎥ ⎢ 0.849 0 (1) = E ◦U = ⎢ ⎥. ⎢ 0 0.748 0.872 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0.697 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 0⎦ 0 0 0 000 ⎡

 (1)

(4) Compute new marking M1 :   T  M1 = M0 ⊕  (1) ⊗  (1) = 0.8 0.9 0.7 0.8 0.688 0.666 0.532 0 0 0 . (5) For the next iteration, k = 2:  T  (2) = W T M1 = 0.81 0.7 0.763 0.688 0.666 0.532  (2) =E (2) ◦ U

4.5 Chapter Summary

81

⎤ 0 0 0 0 0 0 ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 ⎥ ⎢ 0.849 0 =⎢ ⎥ ⎢ 0 0.748 0.872 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0.697 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0.843 0.726 0.884 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 0.909 ⎦ 0 0 0 0 0 0   M2 =M1 ⊕  (2) ⊗  (2)  T = 0.8 0.9 0.7 0.8 0.688 0.666 0.532 0.580 0.484 0 . ⎡

(6) For the third iteration, k = 3:  T M3 = 0.8 0.9 0.7 0.8 0.688 0.666 0.532 0.580 0.484 0 . As the fact that M3 = M2 at the third iteration, and the reasoning process is  T finished. The final marking is 0.8 0.9 0.7 0.8 0.688 0.666 0.532 0.580 0.484 0 . Therefore, the fault diagnosis system may have the faults of d8 and d9 with their truth values being 0.580 and 0.484, respectively. As far as we know, many existing fuzzy reasoning algorithms are on the basis of reachable set of FPNs and not suitable to parallel reasoning. These conventional algorithms require the enumeration of all potential paths from the starting places to the terminating ones. The proposed algorithm adopts a matrix equation format and can implement knowledge reasoning in a parallel way. Using this algorithm, we can quickly obtain the truth degrees of all propositions from the starting places without using a reachability analysis method. From the illustrative example, we can see that the reasoning algorithm based on the DAFPNs has dynamic adaptive capacity following the changes of antecedent propositions. After the reasoning process, we can get an excellent input–output mapping of the knowledge system in different situations. Especially, for complex dynamic expert systems, the concurrent and adaptive inference advantages are more evident.

4.5 Chapter Summary In this chapter, we proposed a new knowledge representation and acquisition approach based on the FER approach and DAFPNs. Compared with traditional methods, the proposed knowledge representation and acquisition approach has the following attractions: (1) It has the capability of allowing the natural modeling of

82

4 Dynamic Adaptive Fuzzy Petri Nets for Knowledge Representation …

incomplete knowledge by using the FER method. (2) The weights of propositions are assigned to each input arc of a transition, so the same place with different transitions has different input weights. (3) The certainty factor of a rule is replaced with several output certainty factors assigned to each output arc of a transition. It avoids the shortcoming that the whole rule is assigned only one certainty factor when a rule contains two or more consequents. (4) Distinct threshold values are assigned to each proposition in the consequent parts of a composite production rule. This can prevent or reduce rule mis-firing and adjust knowledge rules dynamically with the change of antecedent propositions. (5) The improved inference model provides a new mechanism for forward reasoning. Computational complexity of the given algorithm is not affected by the number of rules and is only relative to the iterative times.

References Chen CB, Klein CM (1997) A simple approach to ranking a group of aggregated fuzzy utilities. IEEE Trans Syst Man Cybern B Cybern 27(1):26–35 Chen SM, Ke JS, Chang JF (1990) Knowledge representation using fuzzy Petri nets. IEEE Trans Knowl Data Eng 2(3):311–319 Fang R, Liao HC, Yang JB, Xu DL (2021) Generalised probabilistic linguistic evidential reasoning approach for multi-criteria decision-making under uncertainty. J Oper Res Soc 72(1):130–144 Gao MM, Zhou MC, Huang XG, Wu ZM (2003) Fuzzy reasoning Petri nets. IEEE Trans Syst Man Cybern A Syst Hum 33(3):314–324 Ha MH, Li Y, Wang XF (2007) Fuzzy knowledge representation and reasoning using a generalized fuzzy Petri net and a similarity measure. Soft Comput 11(4):323–327 Lin R, Wang H, Wang J, Wang N (2023) Knowledge representation and reuse model of civil aircraft structural maintenance cases. Exp Syst Appl 216:119460 Liu HC, Liu L, Bian QH, Lin QL, Dong N, Xu PC (2011) Failure mode and effects analysis using fuzzy evidential reasoning approach and grey theory. Exp Syst Appl 38(4):4403–4415 Liu HC, Lin QL, Mao LX, Zhang ZY (2013a) Dynamic adaptive fuzzy Petri nets for knowledge representation and reasoning. IEEE Trans Syst Man Cybern Syst 43(6):1399–1410 Liu HC, Liu L, Lin QL, Liu N (2013b) Knowledge representation and acquisition using fuzzy evidential reasoning and dynamic adaptive fuzzy Petri nets. IEEE Trans Cybern 43(3):1059– 1072 Liu HC, You JX, You XY, Su Q (2016) Fuzzy Petri nets using intuitionistic fuzzy sets and ordered weighted averaging operators. IEEE Trans Cybern 46(8):1839–1850 Liu HC, Luan X, Zhou M, Xiong Y (2022) A new linguistic Petri net for complex knowledge representation and reasoning. IEEE Trans Knowl Data Eng 34(3):1011–1020 Liu P, Li Y, Zhang X, Pedrycz W (2023) A multiattribute group decision-making method with probabilistic linguistic information based on an adaptive consensus reaching model and evidential reasoning. IEEE Trans Cybern 53(3):1905–1919 Mou X, Zhang QZ, Liu HC, Zhao J (2021) Knowledge representation and acquisition using Rnumbers Petri nets considering conflict opinions. Exp Syst 38(3):e12660 Popescu DC, Dumitrache I (2023) Knowledge representation and reasoning using interconnected uncertain rules for describing workflows in complex systems. Inform Fusion 93:412–428 Rajati MR, Mendel JM (2022) Uncertain knowledge representation and reasoning with linguistic belief structures. Inform Sci 585:471–497

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Sharma M, Patel S, Acharya UR (2023) Expert system for detection of congestive heart failure using optimal wavelet and heart rate variability signals for wireless cloud-based environment. Exp Syst 40(4):e12903 Sun B, Xu LD, Pei XM, Li HZ (2003) Scenario-based knowledge representation in case-based reasoning systems. Exp Syst 20(2):92–99 Yang JB, Wang YM, Xu DL, Chin KS (2006) The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties. Eur J Oper Res 171(1):309–343 Yedida R, Krishna R, Kalia A, Menzies T, Xiao J, Vukovic M (2023) An expert system for redesigning software for cloud applications. Exp Syst Appl 219:119673 Yeung DS, Tsang ECC (1994) Fuzzy knowledge representation and reasoning using Petri nets. Exp Syst Appl 7(2):281–289 Zhu YJ, Guo W, Liu HC (2022) Knowledge representation and reasoning with an extended dynamic uncertain causality graph under the Pythagorean uncertain linguistic environment. Appl Sci 12(9):4670

Chapter 5

Interval-Valued Intuitionistic FPNs for Knowledge Representation and Acquisition

In the highly competitive environment, capturing and disseminating of expert knowledge are significant to an organization’s success with the development of knowledgebased systems. However, in practical knowledge acquisition process, domain experts tend to express their judgments using multigranularity linguistic term sets and there usually exists uncertain and incomplete information since expert knowledge is experience-based and tacit. In addition, although the technical capabilities of expert systems based on FPNs are expanding, they still fall short of meeting the increasingly complex knowledge demands. Therefore, this chapter develops a theoretical model based on linguistic interval 2-tuples and interval-valued intuitionistic FPNs (IVIFPNs) for acquiring and representing expert knowledge to increase and sustain the competitive advantages of knowledge intensive organizations. An empirical case study in medical practice is provided to demonstrate the proposed model, and the results show that it can well capture experts’ knowledge and reuse the acquired knowledge productively.

5.1 Introduction Nowadays, knowledge management has received considerable attention from both academics and practitioners, and expert knowledge is recognized as the most important resource of enterprises, particularly in the knowledge intensive organizations. Dealing with knowledge creation, transfer, and utilization is increasingly critical for the long term sustainable competitive advantage and success of any organization. Thus, a lot of efforts have been made from companies and researchers in developing and supporting knowledge management in different organizations (Gao et al. 2023; Kaya et al. 2023). Based on these efforts, constructing expert systems has been accepted as necessary for generating the competitive capabilities of knowledge intensive organizations.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_5

85

86

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

Normally, knowledge acquisition is a group decision behavior, which cannot be completed on an individual basis (Liu et al. 2013b; Shi et al. 2022). Since expert knowledge is constituted in individuals and depends on personal experience, intuitions, and insights, domain experts may use multigranularity linguistic term sets for expressing their judgments and different experts may demonstrate different opinions considering their different expertise and backgrounds (Li et al. 2018). Furthermore, experts are often unsure of their estimations during the knowledge acquisition process because of time limit, lack of experience and data. As a result, there often exist different types of knowledge information for FPRs, some of which may be vague, uncertain, or even incomplete. The information is difficult to incorporate into the expert knowledge acquisition by using existing knowledge acquisition approaches. Whereas, the interval 2-tuple linguistic method (Zhang 2012; Liu et al. 2014b) can overcome the aforementioned limitations. Its main advantages are that domain experts can express their assessments by employing linguistic term sets with different granularity of uncertainty and various uncertainties in the judgments can be modeled by using interval 2-tuples (Liu et al. 2014c, 2015). Since its appearance, the interval 2-tuple linguistic method has been widely applied to solve many multi-criteria decision making problems (You et al. 2015; Huang et al. 2022; Yu et al. 2022; Zhang et al. 2023). Therefore, the model based on the interval 2-tuple linguistic method will be more flexible and precise to deal with expert judgments in the knowledge acquisition process. In other way, FPNs (Chen et al. 1990; Yeung and Tsang 1994) are an important modeling and computational paradigm for knowledge representation and reasoning. Compared with other knowledge representation approaches, such as knowledge graph (Wang and El-Gohary 2023), interconnected uncertain rules (Popescu and Dumitrache 2023), and ontology (Lin et al. 2023), FPNs have a couple of attractive features (refer, e.g., to Chen 2002; Gao et al. 2003; Ha et al. 2007). Even so, the traditional FPNs have been extensively criticized for having many inherent shortcomings (Hu et al. 2011; Liu et al. 2013a, 2016), thus leading to limitations in practical applications. The most important ones are stated as follows: First, the knowledge parameters, such as weight, threshold value and certainty factor, in extant FPNs are insufficient to accurately represent complex rule-based expert systems. Second, the fuzzy rules of current knowledge inference frameworks are static, which cannot be adjusted according to the variation of antecedent propositions. To improve the performance of FPNs, a number of alternative FPN models have been suggested by researchers recently (Liu et al. 2021, 2022; Mou et al. 2022; Shi et al. 2022). In this chapter, we develop an interval-valued intuitionistic FPN (IVIFPN) model to represent uncertain expert knowledge accurately and perform rule-based logic reasoning dynamically. Therefore, the IVIFPNs can be adopted to manage expert knowledge for the knowledge sharing and its conversion into organizational knowledge. Against the backdrop mentioned above, the objective of this chapter is to propose a knowledge representation and acquisition (KRA) framework by using linguistic interval 2-tuples and IVIFPNs for capturing, sharing, and leveraging expert knowledge to increase and sustain the competitive advantages of knowledge intensive organizations. Firstly, a new knowledge acquisition method based on interval 2-tuples is

5.2 Interval 2-Tuple Linguistic Variables

87

introduced to model the diversity and uncertainty of knowledge information in expert knowledge acquisition. Secondly, an IVIFPN model is developed for knowledge representation and reasoning under the interval-valued intuitionistic fuzzy environment, which gets an excellent input–output mapping of the knowledge-based system for various situations. Thirdly, a practical case study regarding lung cancer diagnosis is given to show the practicality and usefulness of the proposed KRA approach for acquiring and sharing expert knowledge. The remaining part of this chapter is structured as follows. In Sects. 5.2 and 5.3, interval 2-tuples and IVIFPNs are introduced, respectively. In Sect. 5.4, we develop the KRA model based on interval 2-tuples and IVIFPNs. A case study of developing medical diagnosis expert system is presented in Sect. 5.5 to illustrate the proposed model. Finally, conclusions of this chapter are provided in Sect. 5.6.

5.2 Interval 2-Tuple Linguistic Variables The interval 2-tuple linguistic method (Zhang 2012) is an extension of 2-tuple linguistic representation model (Herrera et al. 2000) to model various uncertainties in the assessments of decision makers. } { Definition 1 (Zhang 2012) Let S = s0 , s1 , ..., sg be a linguistic term set with granularity g + 1. An [ interval( 2-tuple )] linguistic variable is( composed ) ( of) two 2-tuples, denoted by (si , α1 ), s j , α2 , where (si , α1 ) ≤ s j , α2 , si s j and α1 (α2 ) represent the linguistic label of S and symbolic translation, respectively. The interval 2-tuple expressing equivalent information to an interval value [β1 , β2 ](β1 , β2 ∈ [0, 1], β1 ≤ β2 ) can be derived by the following function:

)] [ ( with [β1 , β2 ] = (si , α1 ), s j , α2

⎧ si , i = round(β1 · g) ⎪ ⎪ ⎪ ⎪ ⎨ s j , j = round(β2 · g)

) 1 1 , α1 = β1 − gi , α1 ∈ − 2g , 2g ⎪ ⎪ ) ⎪ ⎪ ⎩ α2 = β2 − j , α2 ∈ − 1 , 1 g 2g 2g (5.1)

where round(·) is the usual rounding operation. On the contrary, there is always an inverse function −1 such that from an interval 2-tuple it returns its equivalent interval value [β1 , β2 ](β1 , β2 ∈ [0, 1], β1 ≤ β2 ) by −1

[ )] ( (si , α1 ), s j , α2 =

[

] i j + α1 , + α2 = [β1 , β2 ]. g g

(5.2)

) ( It is worth highlighting that if (si , α1 ) = s j , α2 , the interval 2-tuple linguistic variable becomes a 2-tuple linguistic variable.

88

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

Definition 2 (Liu et al. 2014a) Let a˜ = [(r1 , α1 ), (t1 , ε1 )] and b˜ = [(r2 , α2 ), (t2 , ε2 )] be two interval 2-tuples and λ ∈ [0, 1], then the basic operation laws for interval 2-tuples are defined as follows: a˜ + b˜ =[(r1 , α1 ), (t1 , ε1 )] + [(r2 , α2 ), (t2 , ε2 )] [ −1 ] = (r1 , α1 ) + −1 (r2 , α2 ), −1 (t1 , ε1 ) + −1 (t2 , ε2 ) ; a˜ × b˜ =[(r1 , α1 ), (t1 , ε1 )] × [(r2 , α2 ), (t2 , ε2 )] (2) [ −1 ] , α ), −1 (t1 , ε1 ) · −1 (t2 , ε2 ) ; = (r1 , α1 ) · −1 (r [ 2 2−1 ] (3) λa˜ = λ[(r1 , α1 ), (t1 , ε1 )] = λ · (r1 , α1 ), λ · −1 (t1 , ε1 ) . (1)

X˜ = {[(r1 , α1 ), (t1 , ε1 )], [(r2 , α2 ), (t2 , ε2 )], be a set ..., [(rn , αn ), (tn , εn )]} of interval 2-tuples and w = (w1 , w2∑ , . . . , wn )T be their associated weighting vector, with w j ∈ [0, 1], j = 1, 2, ..., n, nj=1 w j = 1. The interval 2-tuple weighted average (ITWA) operator is defined as:

Definition 3 (Zhang 2012) Let

ITWA([(r1 , α1 ), (t1 , ε1 )], [(r2 , α2 ), (t2 , ε2 )], ..., [(rn , αn ), (tn , εn )]) ⎤ ⎡ n n ( ) ( ) w j −1 r j , α j , w j −1 t j , ε j ⎦. = ⎣ j=1

(5.3)

j=1

{ } 2012) Let S = s0 , s1 , ..., sg be a linguistic term set and Definition 4 (Zhang )] [ ( a˜ = (si , α1 ), s j , α2 be an interval 2-tuple. Then the interval 2-tuple linguistic variable can be transferred into a crisp value by aˆ = S(a) ˜ =

α1 + α2 i+j + , 2g 2

(5.4)

where S(a) ˜ ∈ [0, 1] is the score function of the interval 2-tuple a. ˜

5.3 Interval-Valued Intuitionistic Fuzzy Petri Nets 5.3.1 Definition of IVIFPNs The FPNs essentially are a graphical and mathematical modeling tool to describe production rule systems. However, fuzzy set theory only considers membership degree and is not perfect in expressing the fuzziness of a rule-based expert system (Liu et al. 2016). In contrast, the concept of interval-valued intuitionistic fuzzy sets

5.3 Interval-Valued Intuitionistic Fuzzy Petri Nets

89

(IVIFSs) (Atanassov and Atanassov 1999), characterized by membership and nonmembership functions whose values are intervals, is more useful for handling uncertain and imprecise information. Therefore, we propose an IVIFPN model in this part for knowledge representation and reasoning. Let be the set of all interval-valued intuitionistic fuzzy values (IVIFVs), the IVIFPN structure is defined as follows. Definition 5 An IVIFPN is a 9-tuple: ( ) I V I F P N = P; T ; I ; O; W ; U˜ ; R˜ I ; R˜ O ; θ˜

(5.5)

where (1) P = { p1 , p2 , ..., pm } is a finite nonempty set of propositions or called places, m is the number of propositions. (2) T = {t1 , t2 , ..., tn } is a finite nonempty set of rules or called transitions, n is the number of rules. (3) I : P × T → {0, 1} is an m × n input incidence matrix defining the directed arcs from propositions to rules. Ii j = 1, when there is a directed arc from pi to t j ; otherwise Ii j = 0, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. (4) O : P × T → {0, 1} is an m × n output incidence matrix defining the directed arcs from rules to propositions. Oi j = 1, if there is a directed arc from t j to pi ; otherwise Oi j = 0, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. (5) W : P × T → [0, 1] is an input function, which can be expressed as an m × n dimensional matrix. The value of an element in W, wi j ∈ [0, 1], is the weight of the input place, showing how much a place pi impacts its following transition m ∑ wi j =1 for j = 1, 2, . . . , n. t j connected by Ii j . Generally, i=1

(6) U˜ : P × T → is an output function and can be expressed as an m × n dimensional matrix. The value of an element in U˜ , μ˜ i j ∈ , is the certainty factor of the transition, showing how much a transition t j impacts its output place pi if the transition fires. (7) R˜ I : P × T → is an(input ) function which assigns an IVIFV to each input arc ˜ ˜ , i = 1, 2, ..., m; j = 1, 2, ..., n representing of a transition, R I = λi j m×n

the input threshold of this place. λ˜ i j ∈ , if there is a directed arc from pi to t j , and λ˜ i j =∞, if there is no directed arc from pi to t j , for i = 1, 2, . . . , m and j = 1, 2, . . . , n. (8) R˜ O : P × T → is an output(function which assigns an IVIFV to each ) output arc of a transition, R˜ O = τ˜i j m×n , i = 1, 2, ..., m; j = 1, 2, ..., n, representing the output threshold of this place. τ˜i j ∈ , if there is a directed arc from t j to pi , and τ˜i j =∞, if there is no directed arc from t j to pi , for i = 1, 2, . . . , m and j = 1, 2, . . . , n.

90

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

˜ α˜ 1 , α˜ 2 , ..., α˜ m )T , where α˜ i ∈ (9) θ˜ is a truth degree vector of the IVIFPN. θ=( denotes the interval-valued intuitionistic fuzzy truth degree of pi , for i = 1, 2, ..., m. The initial truth degree vector is represented by θ˜0 .

5.3.2 Representations of IVIFPRs To effectively represent the fuzziness and uncertainty in knowledge base systems, the interval-valued intuitionistic FPRs (IVIFPRs) are proposed in this chapter in which knowledge parameters such as confidence value, weight, and threshold are denoted by IVIFVs. The definitions of IVIFPRs are given as follows: Type 1 A simple(interval-valued ) intuitionistic fuzzy production rule. ˜ R: IF a THEN c λ; w; τ˜ ; μ˜ . Type 2 A composite interval-valued intuitionistic fuzzy conjunctive rule in the antecedent. ( ) R: IF a1 AND a2 AND…AND am THEN c λ˜ 1 , λ˜ 2 , ..., λ˜ m ; w1 , w2 , ..., wm ; τ˜ ; μ˜ . Type 3 A composite interval-valued intuitionistic fuzzy conjunctive rule in the consequent. ( ) ˜ w; τ˜1 , τ˜2 , ..., τ˜m ; μ˜ 1 , μ˜ 2 , ..., μ˜ m . R: IF a THEN c1 AND c2 AND…AND cm λ; Type 4 A composite interval-valued intuitionistic fuzzy disjunctive rule in the antecedent. R: IF a1 OR a2 OR…OR am THEN c (λ1 , λ2 , ..., λm ; w1 , w2 , ..., wm ; τ1 , τ2 , ..., τm ; μ1 , μ2 , ..., μm ) where ai and ci indicate the antecedent and consequent propositions with IVIFVs, respectively. The parameter μ˜ i is the certainty factor associated with the rule and represents the belief strength of the rule. The symbols λ˜ i and τ˜i are a set of threshold values assigned to the propositions in the antecedent and consequent parts, respectively. The set of weights given to the antecedent propositions is represented by w = {w1 , w2 , ..., wm }. In addition, the above four types of IVIFPRs can be modeled by IVIFPNs as shown in Fig. 5.1.

5.3.3 Inference Algorithm of IVIFPNs In the decision making system modeled by IVIFPNs, if all the antecedent propositions are satisfied (enabled), the IVIFPRs are fired. In an IVIFPN, place denotes a proposition, transition denotes an IVIFPR, and directed arc describes the relationship between propositions and rules. The rule reasoning process can be realized by firing the transitions in the IVIFPN. Next, the enabling and firing rules of IVIFPNs are introduced.

5.3 Interval-Valued Intuitionistic Fuzzy Petri Nets

91

Fig. 5.1 Knowledge representation based on IVIFPNs

Enabling rule For any transitions t j ∈ T , t j is enabled if and only if ( ) α( ˜ pi ) ≥ λ˜ i j ∀ pi ∈ I t j ,

(5.6)

where α( ˜ pi ) is the IVIF truth of the input place pi and λ˜ i j is the input threshold from pi to t j . Firing rule When a transition is fired, new truth degrees of its output places will be generated and can be defined as follows: [ α( ˜ pk ) =

( ) ) ( ) ( ( ) ˜ pk ) , ( μ˜ ) t j ≥ τ˜k j max μ˜ k j ⊗ μ˜ t j , α( , ∀ pk ∈ O t j , α( ˜ pk ), μ˜ t j < τ˜k j

(5.7)

˜ where μ˜ k j is the certainty ( ) factor ∑ of t j with respect to pk , λi j is the output threshold α( ˜ pi ) ⊗ wi j is the equivalent truth degree of input from t j to pk , and μ˜ t j = places at the transition when t j is enabled.

92

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

According to above definitions, a transition is enabled if all its input places have tokens and the IVIF truth associated with each token is greater than its corresponding input threshold; if the equivalent truth degree is bigger than its output thresholds, then the transition is fired by copying the tokens from its input places and depositing one token into the activated output places. Based on the reasoning principles, the concurrent inference algorithm of IVIFPNs is described as follows. Step 1 Read IVIFPN initial inputs: I, O, W, U˜ , R˜ I , R˜ O and θ˜0 . Step 2 Let k = 1. Step 3 Compute the input enabled matrix D (k) . By comparing the IVIF truth and the input threshold of transition t j in the kth iteration, the input enabled matrix D (k) is computed as ( )(k) D (k) = di j m×n = M˜ k ' [ where M˜ k ' = M˜ k−1 , M˜ k−1 , . . . , M˜ k−1

m×n

(k)

R˜ I ,

◦ I.

Step 4 If D is a non-zero matrix, the input weight matrix Eq. (5.9); otherwise, the reasoning stops. (k)

(5.8)

= D (k) ◦ W

(k)

is calculated by

(5.9)

Step 5 Compute the vector of the equivalent truth degrees of transitions by Eq. (5.10). ˜ (k) =

(

) (k) T

M˜ (k−1)

(5.10)

Step 6 Compute the output enabled matrix E (k) . The output enabled matrix E (k) is calculated based on the comparison result between the equivalent truth degree and the output threshold of transition t j in the kth iteration, i.e., ( )(k) ' E (k) = ei j m×n = ˜ (k) ' where ˜ (k) =

[(

˜ (k)

)T ( )T )T ]T ( (k) (k) ˜ ˜ , , ...,

R˜ O ,

(5.11)

◦ O.

m×n

Step 7 If E (k) is a non-zero matrix, the output certainty factor matrix ˜ (k) is obtained using Eq. (5.12); otherwise, the reasoning stops. ˜ (k) = E (k) ◦ U˜

(5.12)

Step 8 Compute the new marking θ˜k . θ˜k = θ˜k−1 ⊕

(

˜ (k) ⊗ ˜ (k)

) (5.13)

5.4 The Proposed KRA Model

93

If θ˜k = θ˜k−1 , then the reasoning is over; otherwise let k = k + 1, return to Step 3. In the above reasoning algorithm, (1) k represents the times of iteration. (2) θ˜k is an m-dimensional vector. Its components represent the final state of all propositions. (3) The “⊕”, “◦”, and “⊗” are operators from Max algebra which are defined in Chapter 4 and the “ ” operator is defined as follows: A B = D,

(5.14)

where A, B, and D are all-dimensional matrices with ai j , bi j , and di j being their elements respectively, and di j = 1 if ai j ≥ bi j , di j = 0 if ai j < bi j , i = 1, 2, ..., m, j = 1, 2, ..., n. Please refer to (Xu 2007; Zhang and Xu 2015) for the basic concepts and operations related to IFSs and IVIFSs.

5.4 The Proposed KRA Model In knowledge-intensive organizations, expert knowledge is frequently identified as a vital organizational resource in terms of the capacity of an organization to survive and the means of sustaining competitive advantage. Expert knowledge such things as the subjective insights, hunches and intuitions of domain experts is highly personal, difficult to formalize, and hard to share with others. Moreover, the acquisition of expert knowledge is a group decision behavior and cannot be accomplished on an individual basis. Different domain experts may demonstrate different opinions because of their diverse expertise and backgrounds, some of which may be vague, uncertain, and incomplete. On the other hand, the representation of expert knowledge is an issue that firms encounter and managers have to address, which limits the utilization of such knowledge. Due to the ambiguous nature of expert knowledge, an organization cannot store it beyond the mind of individuals without some process of articulation. It is difficult to communicate to others and it can be difficult to digitize. As a result, the loss of employees will translate into a loss of expert knowledge and a firm’s potential or actual performance advantage may be eroded (Li et al. 2018). Based on the foregoing, we propose a KRA model using linguistic interval 2-tuples and IVIFPNs for converting expert knowledge to explicit information so that it can be shared and used effectively to enhance the competitive advantage of knowledgeintensive organizations. In this chapter, it is suggested that the expert knowledge is described by IVIFPRs and the knowledge modeling is realized by mapping these rules into IVIFPNs. Next, the procedures of the proposed KRA modeling scheme are described. Step 1 Identity analysis objective and determine the expert system to be developed. The first step is to define the objective of expert knowledge acquisition. Giving clear and careful thought to this step is very critical to the following knowledge

94

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

representation and reasoning phases. Then, the scope of the analyzed problem should be well defined. A specific definition of the decision support system to be built will help prevent domain experts from focusing on wrong aspects during the KRA processes. Step 2 Assemble a KRA team and identify key IVIFPRs. In this step, it is necessary to establish a committee of experts, which provides group knowledge for related issues. In the expert knowledge acquisition process, several domain experts from different functional areas within an organization should be involved to form a cross-functional KRA team. Then the team should identify all possible IVIFPRs of a considered system and figure out critical ones to reduce the complexity of the rule-based system. In this step, some basic tools including brainstorming sessions and cause-effect diagrams can be adopted. Step 3 Invite experts to give knowledge parameters of IVIFPRs. After the agreement about key IVIFPRs is formed, a questionnaire survey or a depth interview to domain experts can be conducted to obtain the knowledge parameters for each WFPR, i.e., weight, certainty factor, and input (output) threshold. Normally, different types of uncertainties are inevitably involved in the subjective judgments the expert group gave due to the subjectivity and incompleteness. Such a problem can be solved by the interval 2-tuple linguistic approach, which can capture the information of different forms and accommodate uncertainties of different types under a unified framework. Considering the linguistic term set S is defined as: S = {s0 = V er y low, s1 = Low, s2 = Moderate, s3 = H igh, s4 = V er y high}, professional experts can provide their subjective judgments in the following ways: • A certain grade such as Low can be written as [(s1 , 0), (s1 , 0)]. • An interval grade such as Low-Moderate, which denotes that the grade given by the expert is between Low and Moderate. This can be expressed as [(s1 , 0), (s2 , 0)]. • No judgment, which means the expert is reluctant or unable to provide an assessment due to a lack of evidence or data. That is, the grade by this expert could be anywhere between Very Low and Very High and can be written as [(s0 , 0), (s4 , 0)]. Step 4 Aggregate the assessments of experts to obtain the values of knowledge parameters. Aggregate the linguistic evaluations of individual experts into group assessments by applying the ITWA operator to the linguistic rating matrixes of knowledge parameters. Then convert the obtained collective ratings of IVIFPR parameters into IVIFVs by using Eq. (5.2). It may be added here that the group weights of each rule’s antecedent propositions can be computed first by Eq. (5.4) and normalized further according to the derived results (Liu et al. 2013b). In this way, we can obtain wi j , μ˜ i j , λ˜ i j and τ˜i j , which indicate the weight, certainty factor, input and output thresholds of every IVIFPR. Thereby, expert knowledge of the considered system can be

5.5 Empirical Case Study

95

constructed in the form of IVIFPRs together with their parameter matrixes W, U˜ , R˜ I , and R˜ O . Step 5 Convert the decision-making rules into an IVIFPN model. The IVIFPN is composed of places, transitions, and directed arcs between places and transitions, in which place represents a proposition, transition represents a fuzzy rule, and directed arc represents the relationship between a proposition and a rule. The place may or may not contain a token associated with a truth degree belongs to IVIFVs. Here, tokens in the places represent the state of the system. According to the translation principles introduced earlier, the IVIFPRs determined can be visualized and mapped into IVIFPN structures. Step 6 Build a knowledge-based expert system based on IVIFPNs. The objective of IVIFPNs is knowledge representation and logic reasoning. Its framework denotes the knowledge structure of IVIFPRs. Hence, a knowledge-based expert system can be built on the basis of IVIFPNs. The rule reasoning process of the expert system is expressed by firing the transitions in the IVIFPNs. According to the computed inference results, decision can be made automatically. Note that if a knowledge base system is too lager, its complexity can be reduced by divided it into several sub-systems and implement reasoning algorithm of corresponding IVIFPNs respectively.

5.5 Empirical Case Study This section provides a medical application of the proposed KRA framework to demonstrate its usefulness and effectiveness for acquiring and sharing expert knowledge.

5.5.1 Background Expert knowledge, in the form of clinical experience, expertise and clinician’s clinical and social knowledge of patients, plays an important role in healthcare organizations. The reasons for this may probably have much to do with the highly subjective nature of many medical procedures, especially diagnosis of conditions where no objective measure currently exists. Medical expert knowledge encompasses not only clinical experience acquired through observations of patients alongside clinical impressions and knowledge of particular patients, but also shared understandings of ‘the way we do things around here’. Distilling experience and skill healthcare professionals in performing complex and yet successful subjective interaction with patients is both time consuming and demanding. Normally, the doctor notes a patient’s signs and symptoms, combines these with the patient’s medical history, physical examination,

96

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

and laboratory findings and then diagnoses the disease. Since patients may experience different conditions, their external and internal status can vary significantly. The patients admitted to a healthcare facility often have a broad range of conditions and are associated with imprecise, incomplete, sometimes wrong information. Hence, the diagnosis process has to be decided dynamically based on a patient’s specific status. Many kinds of factors should be considered in the process of clinical judgment and decision making. Domain knowledge required by medical experts in the clinical judgment is largely tacit and experience-based. This underlines the need for knowledge acquiring and sharing to enable healthcare organizations to bring together knowledge from distributed individuals to form a repository of organizational knowledge, retain knowledge that would otherwise be lost due to the loss of experienced staff, and improve organizational knowledge dissemination. Lung cancer is one of the most prominent causes of death in both men and women throughout the world. The diagnosis of lung cancer is made through patient history and clinical assessment of each individual’s symptoms and incapacities. Due to the subjective nature of this assessment, it is reported that more than a quarter of patients are misdiagnosed, receiving different pathological diagnoses at autopsy (Polat and Güne¸s 2008). Thus, a reliable and practical diagnosis of lung cancer is essential to reduce the current high rate of misdiagnosis and avoid the consequences of patients receiving inappropriate medication. Appling an expert system to aid lung cancer lung cancer diagnosis is particularly attractive, primarily because the medical expert knowledge is mostly tacit and the conventional clinical assessment is based on subjective observations. Therefore, lung cancer is analyzed in this example to demonstrate the proposed KRA framework.

5.5.2 Knowledge Acquisition An expert team consisting of five professional physicians, denoted as DMi (i = 1, 2, ..., 5), is established to identify all possible IVIFPRs for the lung cancer diagnosis and give their judgments for the weight, certainty factor and input/ output thresholds. For ease of exposition, seven key IVIFPRs are explained here although hundreds of rules are generated. Let di (i = 1, 2, ..., 10) be ten propositions and the meanings of these propositions are shown in Table 5.1. The acquired IVIFPRs are listed as follows: ( ) λ˜ 11 , λ˜ 41 , λ˜ 71 ; w11 , w41 , w71 ; τ˜18 ; μ˜ 18 R1 : IF d1 and d4 and d7 THEN d8 R2 : IF d2 and d4 and d7 THEN d8 R3 : IF d1 and d5 and d7 THEN d9

(

λ˜ 22 , λ˜ 42 , λ˜ 72 ; w22 , w42 , w72 ; τ˜28 ; μ˜ 28

(

λ˜ 13 , λ˜ 53 , λ˜ 73 ; w13 , w53 , w73 ; τ˜39 ; μ˜ 39

) )

5.5 Empirical Case Study Table 5.1 Propositions and their meanings

97

Propositions

Descriptions

d1

Tumor 3 cm or less in greatest dimension

d2

Tumor more than 3 cm but 7 cm or less

d3

Tumor more than 7 cm in greatest dimension

d4

No regional lymph node metastasis

d5

Metastasis in ipsilateral bronchia

d6

Metastasis in contralateral bronchia

d7

No distant metastasis

d8

Lung cancer Stage I

d9

Lung cancer Stage II

d 10

Lung cancer Stage III

R4 : IF d2 and d5 and d7 THEN d9

( ) λ˜ 24 , λ˜ 54 , λ˜ 74 ; w24 , w54 , w74 ; τ˜49 ; μ˜ 49

R5 : IF d3 and d4 and d7 THEN d9

( ) λ˜ 35 , λ˜ 45 , λ˜ 75 ; w35 , w45 , w75 ; τ˜59 ; μ˜ 59

R6 : IF d3 and d5 and d7 THEN d10

( ) λ˜ 36 , λ˜ 56 , λ˜ 76 ; w36 , w56 , w76 ; τ˜610 ; μ˜ 610

( ) λ˜ 67 , λ˜ 77 ; w67 , w77 ; τ˜710 ; μ˜ 710 . R7 : IF d6 and d7 THEN d10 In view of the difficulty in precisely assessing knowledge parameters, the KRA team members agree to evaluate them using the linguistic term sets defined in Table 5.2. Specifically, DM1 and DM5 provide their assessments in the set of five labels, A; DM2 and DM4 provide their assessments in the set of seven labels, B; DM3 provides his assessments in the set of nine labels, C. The assessment information of the seven IVIFPRs provided by the five team members is presented in Tables 5.3, 5.4, 5.5 and 5.6, where incomplete assessment information is highlighted by dashed lines. Considering their personal backgrounds and domain knowledge, the five experts are assigned the following weights: 0.15, 0.20, 0.30, 0.25 and 0.10 during the expertise acquisition process. To carry out knowledge representation and reasoning, we first use interval 2-tuples to express the KRA team members’ individual assessments and synthesize them into group interval 2-tuples by Eq. (5.3), as presented in Tables 5.7 and 5.8. The group interval 2-tuples are then transformed into IVIFVs and crisp numbers (for weight) using Eqs. (5.2) and (5.4), respectively. The results obtained are shown in Tables 5.7 and 5.8.

9

7

A ={a0 = V er y low(V L), a1 = Low(L), a2 = Moderate(M), a3 = H igh(H )

5

c7 = V er y high(V H ), c8 = E xtr eme high(E H )},

low(M L), c4 = Moderate(M), c5 = Moderately high(M H ), c6 = H igh(H ),

C ={c0 = E xtr eme low(E L), c1 = V er y low(V L), c2 = Low(L), c3 = Moderately

b6 = V er y high(V H )}.

b3 = Moderate(M), b4 = Moderately high(M H ), b5 = H igh(H ),

B ={b0 = V er y low(V L), b1 = Low(L), b2 = Moderately low (M L),

a4 = V er y high(V H )},

Linguistic terms

Cardinality

Table 5.2 Linguistic term sets adopted by the KRA team

98 5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

5.5 Empirical Case Study

99

Table 5.3 Linguistic evaluations on input thresholds KRA team members

IVIFPRs R1

R2

R3

R4

R5

R6

R7

DM1

DM2

DM3

DM4

DM5

λ11

VL

L

VL

VL-L

L

λ41

L

VL

EL-VL

VL

VL

λ71

VL

VL

VL

L

L

λ22

L-M

VL

L

VL

L

λ42

VL

L-ML

VL

VL

L

λ72

L

–

L

VL

VL

λ13

L-M

L

EL

ML

VL

λ53

L

ML

L

L

L-M

λ73

VL

L

L

L

VL

λ24

VL

ML

VL-L

VL

VL-L

λ54

L-M

VL

L

VL

L

λ74

L-M

L

L-ML

VL

L

λ35

VL

VL

VL

VL

VL

λ45

M

L

L

VL

L-M

λ75

VL

L

VL-ML

L

L-M

λ36

L

VL-L

ML

L

L

λ56

VL-M

L

L

ML

L

λ76

VL

L

L

–

VL

λ67

L

ML

VL

L-ML

VL-L

λ77

VL

L

EL-L

L-ML

VL

DM3

DM4

DM5

Table 5.4 Linguistic evaluations on output thresholds IVIFPRs

KRA team members DM1

DM2

R1

τ 18

VL

VL

VL-L

L

VL

R2

τ 28

VL-L

–

EL-VL

VL

L

R3

τ 39

L

L-ML

VL-L

L

VL

R4

τ 49

L

VL-L

VL-ML

L

L

R5

τ 59

VL

L

L

L

VL

R6

τ 610

L

ML

L

VL

L

R7

τ 710

VL

L

ML

ML

VL-L

100

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

Table 5.5 Linguistic evaluations on weights KRA team members

IVIFPRs R1

R2

R3

R4

R5

R6

R7

DM1

DM2

DM3

DM4

DM5

w11

H

H-VH

H

H

H

w41

H

MH

MH-H

H

M-H

w71

M-H

H

MH

MH

H

w22

H-VH

H

H-VH

H

H

w42

–

H

MH

MH

H

w72

H

MH

M-H

H

H

w13

H

M-H

H

H

M-H

w53

VH

H

H-VH

VH

H

w73

H

MH

MH-H

H

H

w24

VH

H-VH

H

H

H

w54

H

H

MH-VH

H-VH

VH

w74

H-VH

MH

H

MH

H

w35

H-VH

H

VH

H

VH

w45

H

MH-H

H

MH

H

w75

M-H

MH

H

H

H

w36

VH

H-VH

VH-EH

VH

H

w56

H

VH

VH

H-VH

VH

w76

H-VH

H

H

MH-VH

H

w67

VH

VH

–

H-VH

VH

w77

H

H-VH

VH

H

VH

Table 5.6 Linguistic evaluations on certainty factors IVIFPRs

KRA team members DM1

DM2

DM3

DM4

DM5

R1

μ18

VH

H

VH

MH-VH

H

R2

μ28

H

MH

H

MH-H

VH

R3

μ39

VH

VH

H-VH

H

H-VH

R4

μ49

H

H-VH

VH

VH

H-VH

R5

μ59

VH

MH

H-EH

H

H

R6

μ610

H-VH

VH

VH

–

VH

R7

μ710

VH

MH-VH

VH

VH

H-VH

5.5 Empirical Case Study

101

Table 5.7 Group evaluations on input thresholds and weights IVIFPRs

Input thresholds

Group interval 2-tuples

IVIFVs

Weights

Group interval 2-tuples

Normalized crisp values

R1

λ11

[(b1 , − 0.071), (b1 , − 0.029)]

([0.096, 0.138], [0.863, 0.904])

w11

[(b5 , − 0.046), (b5 , − 0.013)]

0.357

λ41

[(b0 , 0.038), (b0 , 0.075)]

([0.038, 0.075], [0.925, 0.963])

w41

[(b4 , − 0.013), (b5 , − 0.079)]

0.312

λ71

[(b1 , − 0.063), (b1 , − 0.063)]

([0.104, 0.104], [0.896, 0.896])

w71

[(b4 , 0.079), (b4 , 0.079)]

0.331

λ22

[(b1 , − 0.029), (b1 , 0.008)]

([0.138, 0.175], [0.825, 0.863])

w22

[(b5 , − 0.046), (b5 , 0.029)]

0.373

λ42

[(b1 , − 0.071), (b1 , − 0.038)]

([0.096, 0.129], [0.871, 0.904])

w42

[(b4 , − 0.071), (b4 , 0.079)]

0.303

λ72

[(b1 , − 0.054), (b2 , − 0.021)]

([0.113, 0.313], [0.688, 0.888])

w72

[(b4 , 0.013), (b5 , − 0.079)]

0.324

λ13

[(b1 , − 0.013), (b1 , 0.025)]

([0.154, 0.192], [0.808, 0.846])

w13

[(b4 , 0.029), (b5 , − 0.046)]

0.314

λ53

[(b1 , 0.079), (b2 , − 0.063)]

([0.246, 0.271], [0.729, 0.754])

w53

[(b5 , 0.033), (b5 , 0.071)]

0.375

λ73

[(b1 , − 0.017), (b1 , − 0.017)]

([0.150, 0.150], [0.850, 0.850])

w73

[(b4 , 0.050), (b5 , − 0.079)]

0.311

λ24

[(b1 , − 0.063), (b1 , 0.000)]

([0.104, 0.167], [0.833, 0.896])

w24

[(b5 , − 0.008), (b5 , 0.025)]

0.350

R2

R3

R4

(continued)

102

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

Table 5.7 (continued) IVIFPRs

R5

R6

R7

Input thresholds

Group interval 2-tuples

IVIFVs

Weights

Group interval 2-tuples

Normalized crisp values

λ54

[(b1, − 0.029), (b1 , 0.008)]

([0.138, 0.175], [0.825, 0.863])

w54

[(b5 , − 0.058), (b5 , 0.058)]

0.346

λ74

[(b1 , 0.004), (b1 , 0.079)]

([0.171, 0.246], [0.754, 0.829])

w74

[(b4 , 0.046), (b5 , − 0.083)]

0.304

λ35

[(b0 , 0.038), (b0 , 0.038)]

([0.038, 0.038], [0.963, 0.963])

w35

[(b5 , 0.017), (b5 , 0.054)]

0.372

λ45

[(b1 , 0.042), (b1 , 0.067)]

([0.208, 0.233], [0.767, 0.792])

w45

[(b4 , 0.046), (b4 , 0.079)]

0.313

λ75

[(b1 , − 0.029), (b1 , 0.071)]

([0.138, 0.238], [0.763, 0.863])

w75

[(b4 , 0.050), (b5 , − 0.079)]

0.315

λ36

[(b1 , 0.050), (b2 , − 0.083)]

([0.217, 0.250], [0.750, 0.783])

w36

[(b5 , 0.071), (b6 , − 0.025)]

0.355

λ56

[(b1 , 0.050), (b2 , − 0.042)]

([0.217, 0.292], [0.708, 0.783])

w56

[(b5 , 0.050), (b6 , − 0.075)]

0.341

λ76

[(b1 , − 0.058), (b2 , 0.025)]

([0.108, 0.358], [0.642, 0.892])

w76

[(b4 , 0.079), (b5 , 0.033)]

0.304

λ67

[(b1 , 0.017), (b2 , − 0.083)]

([0.183, 0.250], [0.750, 0.817])

w67

[(b4 , − 0.008), (b6 , 0.000)]

0.489

λ77

[(b0 , 0.075), (b1 , 0.025)]

([0.075, 0.192], [0.808, 0.925])

w77

[(b5 , 0.017), (b5 , 0.050)]

0.511

5.5.3 Knowledge Representation and Reasoning Based on the translation principle, the identified IVIFPRs are modeled by an IVIFPN as shown in Fig. 5.2. Then, a rule-based lung cancer diagnosis system can be built using the IVIFPN model and its reasoning algorithm.

5.5 Empirical Case Study

103

Table 5.8 Group evaluations on output thresholds and weights IVIFPRs

Output thresholds

Group interval 2-tuples

IVIFVs

Certainty factors

Group interval IVIFVs 2-tuples

R1

τ 18

[(b0 , 0.079), (b1 , − 0.050)]

([0.079, 0.117], [0.883, 0.921])

μ18

[(b5 , − 0.013), ([0.821, (b5 , 0.071)] 0.904], [0.096, 0.179])

R2

τ 28

[(b0 , 0.025), (b2 , − 0.033)]

([0.025, 0.300], [0.700, 0.975])

μ28

[(b4 , 0.071), ([0.738, (b5 , − 0.054)] 0.779], [0.221, 0.263])

R3

τ 39

[(b1 , − 0.017), (b1 , 0.054)]

([0.150, 0.221], [0.779, 0.850])

μ39

[(b5 , 0.025), ([0.858, (b6 , − 0.079)] 0.921], [0.079, 0.142])

R4

τ 49

[(b1 , − 0.025), (b2 , − 0.083)]

([0.142, 0.250], [0.750, 0.858])

μ49

[(b5 , 0.033), ([0.867, (b6 , − 0.075)] 0.925], [0.075, 0.133])

R5

τ 59

[(b1 , − 0.017), (b1 , − 0.017)]

([0.150, 0.150], [0.850, 0.850])

μ59

[(b5 , − 0.042), ([0.792, (b5 , 0.033)] 0.867], [0.133, 0.208])

R6

τ 610

[(b1 , 0.038), (b1 , 0.038)]

([0.204, 0.204], [0.796, 0.796])

μ610

[(b4 , 0.046), ([0.713, (b6 , − 0.038)] 0.963], [0.038, 0.288])

R7

τ 710

[(b1 , 0.063), (b2 , − 0.079)]

([0.229, 0.254], [0.746, 0.771])

μ710

[(b5 , 0.038), ([0.871, (b6 , − 0.038)] 0.963], [0.038, 0.129])

With the IVIFPN model and Fig. 5.2, the initial marking matrixes are generated as shown here: ⎡

1 ⎢0 ⎢ ⎢0 ⎢ ⎢1 ⎢ ⎢ ⎢0 I =⎢ ⎢0 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎣0 0

0 1 0 1 0 0 1 0 0 0

1 0 0 0 1 0 1 0 0 0

0 1 0 0 1 0 1 0 0 0

0 0 1 1 0 0 1 0 0 0

0 0 1 0 1 0 1 0 0 0

⎡ ⎤ 000 0 ⎢0 0 0 0⎥ ⎢ ⎥ ⎢0 0 0 0⎥ ⎢ ⎥ ⎢0 0 0 ⎥ 0⎥ ⎢ ⎢ ⎥ 0⎥ ⎢0 0 0 ⎥ O=⎢ ⎢0 0 0 1⎥ ⎢ ⎥ ⎢0 0 0 1⎥ ⎢ ⎥ ⎢1 1 0 0⎥ ⎢ ⎥ ⎣0 0 1 0⎦ 000 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

104

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation …

Fig. 5.2 IVIFPN of the example

⎡˜ λ11 ⎢∞ ⎢ ⎢∞ ⎢ ⎢ λ˜ ⎢ 41 ⎢ ⎢∞ R˜ I = ⎢ ⎢∞ ⎢ ⎢ λ˜ 71 ⎢ ⎢∞ ⎢ ⎣∞ ∞

∞ λ˜ 13 λ˜ 22 ∞ ∞ ∞ λ˜ 42 ∞ ∞ λ˜ 53 ∞ ∞ λ˜ 72 λ˜ 73 ∞ ∞ ∞ ∞ ∞ ∞

∞ ∞ ∞ λ˜ 24 ∞ ∞ ∞ λ˜ 35 λ˜ 36 ∞ λ˜ 45 ∞ λ˜ 54 ∞ λ˜ 56 ∞ ∞ ∞ ˜λ74 λ˜ 75 λ˜ 76 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞

⎤ ∞ ∞⎥ ⎥ ∞⎥ ⎥ ∞⎥ ⎥ ⎥ ∞⎥ ⎥ λ˜ 67 ⎥ ⎥ λ˜ 77 ⎥ ⎥ ∞⎥ ⎥ ∞⎦ ∞

5.5 Empirical Case Study

105

⎤ 0.357 0 0.314 0 0 0 0 ⎢ 0 0.373 0 0.350 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0.372 0.355 0 ⎥ ⎥ ⎢ ⎢ 0.312 0.303 0 0 0.313 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0.375 0.346 0 0.341 0 ⎥ ⎢ 0 W =⎢ ⎥ ⎢ 0 0 0 0 0 0 0.489 ⎥ ⎥ ⎢ ⎢ 0.331 0.324 0.311 0.304 0.315 0.304 0.511 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎣ 0 0 0 0 0 0 0 ⎦ 0 0 0 0 0 0 0 ⎤ ⎡ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎢∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎥ ⎥ ⎢ ⎢∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎥ ⎥ ⎢ ⎢∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎥ ⎥ ⎢ ⎥ ⎢ ⎢∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎥ ˜ RO = ⎢ ⎥ ⎢∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎥ ⎥ ⎢ ⎢∞ ∞ ∞ ∞ ∞ ∞ ∞ ⎥ ⎥ ⎢ ⎢ τ˜18 τ˜28 ∞ ∞ ∞ ∞ ∞ ⎥ ⎥ ⎢ ⎣ ∞ ∞ τ˜39 τ˜49 τ˜59 ∞ ∞ ⎦ ∞ ∞ ∞ ∞ ∞ τ˜610 τ˜710 ⎤ ⎡ 0 0 0 0 0 0 0 ⎢ 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ 0 0 0 0 0 0 0 ⎥ ⎢ U˜ = ⎢ ⎥. ⎢ 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 ⎥ ⎥ ⎢ ⎢ μ˜ 18 μ˜ 28 0 0 0 0 0 ⎥ ⎥ ⎢ ⎣ 0 0 μ˜ 39 μ˜ 49 μ˜ 59 0 0 ⎦ 0 0 0 0 0 μ˜ 610 μ˜ 710 ⎡

For a suspicious lung cancer patient, the evaluation is obtained according to the data from hospital inspection units and we suppose the initial state for the patient is θ˜0 =(([0.2, 0.2], [0.6, 0.7]) ([0.9, 0.9], [0, 0.1]) 0 0 ([0.8, 0.8], [0.1, 0.2]) ([0.3, 0.3], [0.5, 0.6]) ([0.8, 0.9], [0.1, 0.1]) According to the reasoning algorithm of IVIFPNs,

0 0 0)T .

(1) For the first iteration, k = 1: (1)

=(0 0 ([0.691, 0.751], [0.176, 0.239]) ([0.848, 0.873], [0, 0.127] 0 0 ) ([0.631, 0.741], [0.220, 0.240]))T ,

106

5 Interval-Valued Intuitionistic FPNs for Knowledge Representation … θ˜1 =(([0.2, 0.2], [0.6, 0.7]) ([0.9, 0.9], [0, 0.1]) 0 0 ([0.8, 0.8], [0.1, 0.2]) ([0.3, 0.3], [0.5, 0.6])([0.8, 0.9], [0.1, 0.1]) 0 ([0.731, 0.807], [0.075, 0.244]) ([0.549, 0.713], [0.249, 0.338]))T .

(2) For the second iteration, k = 2: θ˜2 =(([0.2, 0.2], [0.6, 0.7]) ([0.9, 0.9], [0, 0.1]) 0 0 ([0.8, 0.8], [0.1, 0.2]) ([0.3, 0.3], [0.5, 0.6])([0.8, 0.9], [0.1, 0.1]) 0 ([0.731, 0.807], [0.075, 0.244]) ([0.549, 0.713], [0.249, 0.338]))T .

As θ˜2 = θ˜1 , the reasoning process is finished. Based on the computed results, the final state of the diagnosis system for the consequent propositions are 0, ([0.731, 0.807], [0.075, 0.244]) and ([0.549, 0.713], [0.249, 0.338]), respectively. The inference results show that the proposition d 9 has the highest value compared with other terminating propositions. Therefore, the patient most likely has the disease of “Lung cancer and Stage II” with the IVIFV of ([0.731, 0.807], [0.075, 0.244]), which means that the degree that the proposition d 9 is true is between [0.731, 0.807] and the degree that the proposition of d 9 is not true is in the range of [0.075, 0.244]. Staging of cancer at the time of diagnosis is the most important predictor of survival and treatment options are made based on the stage. Thus, result of the decision can be used to choose the optimal method and perform the best therapeutic regimen on the patient behalf. From the illustrate example, it can be observed that the IVIFPN-based diagnosis system provides promising results for acquiring and representing expert knowledge in the healthcare context. Cooperating with the recommended clinical conditions, the proposed model provides a reliable and efficiency method to medical diagnosis. Moreover, for each patient, the inspection value for decision is updated so that a decision can be made in real time and fits to the practical situation. In summary, compared with other FPNs, the proposed KRA framework has the following advantages: (1) The diversity and uncertainty of expert knowledge can be well reflected and modeled using interval 2-tuples. This enables domain experts to express their judgments more realistically and makes the knowledge acquisition easier to be carried out. (2) Based on IVIFSs, the proposed model can represent imprecise and uncertain knowledge information in a flexible and effective manner. This advantage is more evident for the complex decision support systems where knowledge data are unreliable. (3) The reasoning process of an expert system can be executed by IVIFPNs in a parallel way to obtain the reasoning results automatically. Thus, the proposed modeling scheme allows the rule-based system to perform knowledge inference in an intelligent manner.

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107

5.6 Chapter Summary In this chapter, a theoretical model was proposed by combining linguistic interval 2-tuples with IVIFPNs in order to acquire and share expert knowledge for the knowledge intensive organizations. Using the proposed model, the diversity and uncertainty of domain experts’ knowledge information can be well managed using interval 2tuple linguistic variables. Then the IVIFPRs transferred from expert experience and knowledge can be represented by using IVIFPNs to perform approximate reasoning automatically. A practical application of the proposed KRA framework to the lung cancer diagnosis was presented to illustrate its feasibility and effectiveness. The results demonstrated the efficiency of the proposed method in expert knowledge representation and acquisition. Moreover, we have generated truth degrees of all the propositions, which can provide important information for medical diagnosis and treatment.

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Chapter 6

Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

FPNs have been applied in many fields as a potential modeling tool for knowledge representation and reasoning. However, there exist many deficiencies in the conventional FPNs when applied in the real-world. In this chapter, we present a new type of FPNs, called picture fuzzy Petri nets (PFPNs), to overcome the shortcomings and improve the effectiveness of the traditional FPNs. First, the proposed PFPN model adopts the picture fuzzy sets, characterized by degrees of positive membership, neutral membership, and negative membership, to depict human experts’ knowledge. As a result, the uncertainty, due to vagueness, imprecision, partial information, etc., can be well handled in knowledge representation. Second, a similarity degree-based expert weighting method is offered for consensus reaching processes in knowledge acquisition. The proposed PFPN model can manage the conflict and inconsistency among experts’ evaluations on knowledge parameters, thus making the obtained knowledge rules more accurate. Finally, a realistic example of gene regulatory network is provided to illustrate the feasibility and practicality of proposed PFPN model.

6.1 Introduction In expert systems, FPNs are often utilized as a potential modeling tool for knowledge representation and reasoning (Yeung and Tsang 1994; Chen 2002). The combination of graphical power of Petri nets and ability of fuzzy sets in expressing vague information makes FPNs suitable for modeling uncertain rule-based expert systems (Gao et al. 2003; Suraj and Fryc 2006). An FPN is a marked graphical system containing places and transitions. Due to the capacity to depict imprecise knowledge and support inference process, FPNs have caused increasing interests of both academics and practitioners and have been used in a lot of fields, such as fault diagnosis (Lin et al. 2022b), hydrogen leakage risk assessment (Kang et al. 2022), failure

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_6

109

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6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

cause analysis (Li et al. 2022), karaoke lyrics calibration (Lin et al. 2022a), and sago processing risk assessment (Selvaraj and Ramalingam 2022). However, the conventional FPNs have been pointed out as having many deficiencies in previous studies (Liu et al. 2017; Yu et al. 2022). Some scholars criticized the limitation of FPNs in knowledge representation when the knowledge parameter values are restricted to be crisp values between 0 and 1. To address this issue, some uncertainty methodologies have been incorporated into FPNs, such as the linguistic Z-numbers (Shi et al. 2022), the spherical linguistic sets (Mou et al. 2022), and the two-dimension linguistic uncertain variables (Liu et al. 2022). Recently, the concept of picture fuzzy sets (PFSs) was proposed by Cuong (2014) to describe uncertain phenomena and information. The PFSs, characterized by three membership degrees: positive, neutral and negative, are more appropriate to deal with human experts’ evaluation information. Since its introduction, the PFS has received widespread attention in academia and has been applied to many practical fields (Ding et al. 2020; Peng et al. 2022; Rong et al. 2022; Zhao et al. 2023). Knowledge acquisition plays a significant role in the development of a rule-based expert system. But there are few researches considering how to acquire knowledge parameters of FPNs in the literature. With the increasing complexity of expert systems, difficulties experienced during development particularly when acquiring knowledge from human experts. It is an even much more difficult problem to finetune the knowledge parameters of FPRs during the knowledge base update stage (Liu et al. 2013; Shi et al. 2022). Consequently, the traditional process to extract FPRs from experts can be a bottleneck, causing delay in establishing an expert system (Tsang et al. 2004; Xu et al. 2019). On the other hand, domain experts can have many differences in the evaluations of knowledge parameters as their backgrounds, attitudes and organizations, and conflicts are inevitable. Consequently, it is important to develop methods to effectively handle expert conflicts, which will lead to efficiency improvements in knowledge acquisition processes. Based on the above analyses, the objective of this chapter is to propose a new type of FPNs, called picture fuzzy Petri nets (PFPNs), to represent and acquire imprecise and uncertain expert knowledge. First, we apply PFSs to quantitatively represent the complex and uncertain knowledge in constructing an expert system. This application will practically provide an effective tool to describe uncertain information gathered in knowledge representation. Second, we use a similarity degree-based method to determine the importance weights of experts in knowledge acquisition. This adaption can take full consideration of experts’ conflicting and inconsistent opinions and facilitate consensus reaching processes. Besides, a case study about gene regulatory network is given to demonstrate the application and effectiveness of the proposed PFPN model. The remainder of this chapter is organized as follows: Sect. 6.2 introduces the basic concept of PFSs briefly. Section 6.3 develops the PFPN model and Sect. 6.4 presents a realistic example. Finally, Sect. 6.5 draws conclusions of this chapter.

6.2 Preliminaries

111

6.2 Preliminaries 6.2.1 Picture Fuzzy Sets Cuong (2014) pioneered the concept of PFSs, which is a direct extension of fuzzy sets and intuitionistic fuzzy sets (IFSs). Its definition is shown as follows. Definition 6.1 (Cuong 2014) The form of a PFS A˜ on a universe X is: A˜ =

{(

) } x, μ A˜ (x), η A˜ (x), ν A˜ (x) |x ∈ X ,

(6.1)

˜ where μ A˜ (x) ∈ [0, 1] means the degree of positive membership of x in A, ˜ η A˜ (x) ∈ [0, 1] means the degree of neutral membership of x in A, ν A˜ (x) ∈ [0, 1] ˜ and μ A˜ , η A˜ and ν A˜ satisfy means the degree of negative membership of x in A, 0 ≤ μ the following condition: + η + ν (x) (x) ˜ ˜ A A˜ (x) ≤ 1 (∀x ∈ X ). Then ( ) A π A˜ (x) = 1 − μ A˜ (x) + η A˜ (x) + ν A˜ (x) could be called the degree of refusal ˜ membership of x in A. For convenience, a˜ = (μ, η, ν) is called a picture fuzzy number (PFN), where μ(x), η(x) and ν(x) satisfy the following conditions: μ(x) ∈ [0, 1], η(x) ∈ [0, 1], ν(x) ∈ [0, 1] and μ(x) + η(x) + ν(x) ≤ 1. Definition 6.2 (Wei 2018b; Xu et al. 2019) Let a˜ = (μ, η, ν) be a PFN, then a score function S of a˜ can be represented as follows: S(a) ˜ = μ1 − ν1 ,

S(a) ˜ ∈ [−1, 1].

(6.2)

Definition 6.3 (Wei 2018b; Xu et al. 2019) Let a˜ = (μ, η, ν) be a PFN, then an accuracy function H of a˜ can be represented as follows: H (a) ˜ = μ1 +η1 +ν1 ,

H (a) ˜ ∈ [0, 1],

(6.3)

where H(a) ˜ is associated with the degree of accuracy of the PFN, which means that ˜ the more the accuracy of a. ˜ the larger the value of H(a), Definition 6.4 (Xu et al. 2019) Given two PFNs a˜ 1 = (μ1 , η1 , ν1 ) and a˜ 2 = (μ2 , η2 , ν2 ), some operational laws of PFNs can be defined as follows: (1) (2) (3) (4)

a˜ 1 ⊕ a˜ 2 = (1 − (1 − μ1 )(1 − μ2 ), η1 η2 , (ν1 + η1 )(ν2 + η2 ) − η1 η2 ); a˜ 1 ⊗ a˜ 2( = ((μ1 + η1 )(μ2 + η2 ) − η1 η2 , η1 η) 2 , 1 − (1 − ν1 )(1 − ν2 )); λa˜ 1 = 1 − (1 − μ1 )λ , η1λ , (ν1 + η1 )λ − η1λ , λ > 0; ) ( a˜ 1λ = (μ1 + η1 )λ − η1λ , η1λ , 1 − (1 − ν1 )λ , λ > 0.

Definition 6.5 (Wei 2018a) Given two PFNs a˜ 1 = (μ1 , η1 , ν1 ) and a˜ 2 = ˜ and the accuracy function H(a), ˜ (μ2 , η2 , ν2 ), and based on the score function S(a) the order relation of a1 and a2 is defined as follows:

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6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

(1) If S(a˜ 1 ) ≤ S(a˜ 2 ), the a˜ 1 is smaller than a˜ 2 , defined as a˜ 1 < a˜ 2 ; (2) If S(a˜ 1 ) = S(a˜ 2 ), then, (a) If H (a˜ 1 ) = H (a˜ 2 ), a˜ 1 and a˜ 2 indicate the same information, defined as a˜ 1 = a˜ 2 ; (b) If H (a˜ 1 ) < H (a˜ 2 ), a˜ 1 is smaller than a˜ 2 , defined as a˜ 1 < a˜ 2 . ) ( Definition 6.6 (Xu et al. 2019) Let a˜ j = μ j , η j , ν j ( j = 1, 2, ..., n) be a collection of PFNs, then the picture fuzzy weighted geometric (PFWG) operator is defined as: P F W G(a˜ 1 , a˜ 2 , . . . , a˜ n ) =

n ||

w

a˜ j j ,

(6.4)

j=1

where w = (w1 , w2 , ..., wn )T is the weight vector of {a˜ 1 , a˜ 2 , ..., a˜ n }, with w j ∈ [0, 1] n ∑ and w j = 1. The aggregated value by the PFWG operator is also a PFN, expressed as:

j=1

P F W G(a˜ 1 , a˜ 2 , ...., a˜ n ) =

\ / n n n n || ( || || w j || w j )w ( )w μj + ηj j − 1 − νj j . ηj , ηj ,1 − j=1

j=1

j=1

(6.5)

j=1

Definition 6.7 (Zhang et al. 2018) Given two PFNs a˜ 1 = (μ1 , η1 , ν1 ) and a˜ 2 = (μ2 , η2 , ν2 ), the relative projection of a˜ 1 on a˜ 2 , which describes the closeness of a˜ 1 on a˜ 2 , is defined as: R Pa˜ 2 (a˜ 1 ) =

μ1 μ2 + η1 η2 + (1 − ν1 )(1 − ν2 ) ) ( . μ22 + η22 + 1 − ν22

(6.6)

( ) = Definition 6.8 (Zhang et al. 2018) Let us suppose that R˜ 1 = r˜i1j m×n ( ) ) ( ) ( μi1j , ηi1j , νi1j and R˜ 2 = r˜i2j = μi2j , ηi2j , νi2j are two picture fuzzy m×n

m×n

m×n

evaluation matrices, then the relative projection of R˜ 1 on R˜ 2 , which describes the closeness of R˜ 1 on R˜ 2 , is defined as: | | |} {| | | | | ( ) ( ) | | | | min | R Pr˜ 2 r˜i1j − 1| | R Pr˜ 2 r˜i1j − 1| − | | 1≤i≤m,1≤ j≤n | | 1 i j i j |} |} . {| {| R PR˜ 2 R˜ 1 = | | | | ( ) ( ) mn | | | | i=1 j=1 max min | R Pr˜ 2 r˜i1j − 1| − | R Pr˜ 2 r˜i1j − 1| | | | 1≤i≤m,1≤ j≤n | 1≤i≤m,1≤ j≤n ij ij (

)

m ∑ n ∑

(6.7)

For two picture matrices Definition ) ) ( )6.9 (Zhang ( et al. 2018) ( )fuzzy evaluation ( 1 1 1 1 2 2 2 1 2 ˜ ˜ = μi j , ηi j , νi j and R = r˜i j = μi j , ηi j , νi2j , R = r˜i j m×n

m×n

the similarity degree of R˜ 1 and R˜ 2 is defined as:

m×n

) 1 (( ( )) ( ( ))) ( 1 − R PR˜ 2 R˜ 1 + 1 − R PR˜ 1 R˜ 2 . S D R˜ 1 , R˜ 2 = 2

m×n

(6.8)

6.2 Preliminaries

113

{ } Let R˜ 1 , R˜ 2 , ..., R˜ q be a set of picture fuzzy evaluation matrices. The similarity degree of an evaluation matrix R˜ k to all of the others can be defined as (Wei 2018a): ( ) S D R˜ k =

q ∑ i=1,i/=k

) ( S D R˜ k , R˜ i (k = 1, 2, . . . , q).

q −1

(6.9)

et al. 2018) Give two picture fuzzy evaluation matrices Definition ) ( )6.10 (Zhang ( ) ( ( ) k k k k k ˜ = μi j , ηi j , νi j and R˜ = r˜i j m×n = μi j , ηi j , νi j m×n , the R = r˜i j m×n

m×n

consensus degree of R˜ k to R˜ is computed by (

)

n m 1 ∑∑ C D R˜ k = mn

i=1 j=1

| |} | ( ) {| ( ) | | | | min |R Pr˜i j r˜ikj − 1| |R Pr˜i j r˜ikj − 1| − 1≤i≤m,1≤ j≤n |} |} . {| ( ) {| ( ) | | | | max min |R Pr˜i j r˜ikj − 1| − |R Pr˜i j r˜ikj − 1|

1≤i≤m,1≤ j≤n

(6.10)

1≤i≤m,1≤ j≤n

6.2.2 Defuzzification of PFNs Suppose that a˜ = (μ, η, ν) is a PFN, where π = 1 − μ − η − ν. A defuzzification method to obtain a crisp value of the PFN is introduced as follows (Garg 2017; Xu et al. 2019): Step 1 Distribute the neutral degree to the positive degree and negative degree as follows: μ' = μ +

η , 2

(6.11)

ν' = ν +

η . 2

(6.12)

Step 2 Calculate the defuzzification value y by y = μ' +

1 + μ' − ν ' π 2

(6.13)

114

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

6.3 Picture Fuzzy Petri Nets 6.3.1 Definition of PFPNs With the increasing of complexity of rule-based systems, a new FPN model called PFPNs is proposed for knowledge representation and reasoning in this section. Definition 6.11 Let ˜ be the set of all PFNs defined in the universe X. A PFPN definition can be described as follows: ( ) ˜ T h, U˜ , L W, GW , P F P N = P, T , D, I, O, M, (6.14) where (1) P, T, D, I, and O are denoted as in Eq. (6.8); ˜ (2) M:P → ˜ is a marking vector M˜ = (α˜ 1 , α˜ 2 , ..., α˜ m )T , which indicates a mapping from places to PFNs and denotes the truth degree of the place pi ( pi ∈ P). The starting vector is denoted by M˜ 0 . )T ( (3) T h:T → ˜ denotes a vector T h = λ˜ 1 , λ˜ 2 , ..., λ˜ m . The function assigns a threshold value λ˜ i expressed by a PFN to each input place of a transition; (4) U˜ :T → ˜ denotes a vector U˜ = (μ˜ 1 , μ˜ 2 , ...μ˜ n ). The element μ˜ i is expressed by a PFN, which denotes the certainty factor of the transition ti ; (5) LW:P → [0, 1] is a set of local weights of places, which can be expressed as a vector L W = {lw1 , lw2 , ..., lwm }T . Here, lwi is a real value between 0 and 1, representing the relative importance of the input place pi contributing to the transition t j ; T (6) GW:[T ] × P] → [0, 1] is an m × n input dimensional matrix GW = [ gwi j m×n . The element gwi j ∈ [0, 1] is the global weight which reflects how n ∑ much an input transition impacts an output place. Generally, gwi j = 1, j = i=1

1, 2, …, m.

6.3.2 PFPN Representations of WPFPRs In this chapter, a decision support system is supposed to be modelled by the weight picture fuzzy production rules (WPFPRs). To map WPFPRs into PFPNs, we define WPFPRs in the following new forms: Type 1 A simple(WPFPRs. ) ˜ lw; μ; ˜ gw ; R: IF a THEN c λ; Type 2 A composite weighted picture fuzzy conjunctive rule in the antecedent.

6.3 Picture Fuzzy Petri Nets

115

Fig. 6.1 PFPN representation of Type 1 rule

IF a1 AND a2 ) AND…AND am THEN c ( R: ˜λ1 , λ˜ 2 , ...λ˜ m ; lw1 , lw2 , ..., lwm ; μ; ˜ gw ; Type 3 A composite weighted picture fuzzy(conjunctive rule ) in the consequent. ˜ lw; μ; R: IF a THEN c1 AND c2 AND…AND cm λ; ˜ gw . The above three types of WPFPRs can be represented by PFPNs as shown in Figs. 6.1, 6.2 and 6.3, respectively. Fig. 6.2 PFPN representation of Type 2 rule

Fig. 6.3 PFPN representation of Type 3 rule

116

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

6.3.3 Knowledge Acquisition In this part, an approach to determine the knowledge parameters of PFPNs, i.e., global weight (GW ), local weight (LW ), threshold value (T h), and certainty factor (U˜ ), is introduced. Particularly, the conflict and inconsistency among experts’ evaluations and the vagueness in knowledge parameters are considered in the knowledge acquisition processes. Suppose that there are N WPFPRs {R1 , R2 , ..., R N } to be evaluated by K experts {T M1 , T M2 , ..., T M K } with respect to L antecedent propositions {P1 , P2 , ..., PL }. For a WPFPR, local weight and threshold value are related to the antecedent propositions; global weight and certainty factor are related to the WPFPRs. To simplify the discussion, only the process of determining the local weight (LW ) is explained here. For the local weight, supposed that an expert T (Mk provides ) the assessment of Rn with respect to Pl by a PFN denoted as γ˜ k (Pl ) = μlk , ηlk , νlk . Then a picture fuzzy evaluation vector for the N WPFPRs with respect to L antecedent propositions can ( )T be formed as X˜ k = γ˜ k (P1 ), γ˜ k (P2 ), ..., γ˜ k (PL ) . The steps to determine the LW are described as follows: Step 1 Determine the weights of experts. Let X˜ k (k = 1, 2, ...K ) be the picture fuzzy evaluation vectors of local weights provided by the experts. The similarity degree of X˜ k to all the others can be calculated by ( ) S D X˜ k =

K ∑ i=1,i/=k

) ( S D X˜ k , X˜ i K −1

.

(6.15)

Then the weight of each expert can be obtained as follows (Zhang et al. 2018): ⎧ ( ) S D ( X˜ o(1) ) ⎪ F , k = 1, ⎪ ⎪ T ⎨ ⎛ k ⎞ ⎛ k−1 ⎞ ∑ ∑ S D ( X˜ o(i) ) S D ( X˜ o(i ) ) ωk = ⎪ ⎠ − F ⎝ i=1 ⎠, k = 2, 3, ..., K , ⎪ F ⎝ i=1 T ⎪ T ⎩

(6.16)

( ) K K ∑ ∑ where ωk satisfies ωk = 1, T = S D X˜ k is the sum of all the similarity k=1 { ( ) ( ) k=1 ( )} degrees, S D X˜ o(1) , S D X˜ o(2) , ...., S D X˜ o(K ) is a decreasing ranking value ( ) ( )} { ( ) of S D X˜ 1 , S D X˜ 2 , ..., S D X˜ K . F(a) = a λ is a function to avoid the situation where the generated weights differ greatly from each other. Step 2 Aggregate all the picture fuzzy evaluation vectors. The picture fuzzy evaluation vectors X˜ k (k = 1, 2, . . . , K ) can be aggregated to obtain a collective picture fuzzy evaluation vector X˜ = (γ˜ (P1 ), γ˜ (P2 ), . . . , γ˜ (PL ))T

6.3 Picture Fuzzy Petri Nets

117

with the PFWG operator. Each element of X˜ can be computed by ( ) γ˜ (Pl ) =P F W G γ˜ 1 (Pl ), γ˜ 2 (Pl ), . . . , γ˜ K (Pl ) /K \ K ( ) K K ( )ω j || )ω j || ( j || ω j || ( )ω j j j j j μl + ηl ηl ηl 1 − νl − , ,1 − = . j=1

j=1

j=1

j=1

(6.17) Step 3 Consensus checking and improving. Once the collective picture fuzzy evaluation vector X˜ = (γ˜ (P1 ), γ˜ (P2 ), ..., γ˜ (PL ))T is gained, the consensus degree of the vector ( )T X˜ k = γ˜ k (P1 ), γ˜ k (P2 ), ..., γ˜ k (PL ) to X˜ can be computed by (

C D X˜ k

)

| | |} ( {| ( ) ) | R Pγ˜ (P ) γ˜ k (Pl ) − 1| − min | R Pγ˜ (P ) γ˜ k (Pl ) − 1| L l l 1∑ 1≤l≤L |} |} . {| ( {| ( ) ) = L l=1 max | R Pγ˜ (Pl ) γ˜ k (Pl ) − 1| − min | R Pγ˜ (Pl ) γ˜ k (Pl ) − 1| 1≤l≤L

1≤l≤L

(6.18) ( ) In general, if C D X˜ k = 0, the consensus of X˜ k to X˜ is extremely high. If ( ) C D X˜ k ≤ θ , the consensus of X˜ k to X˜ is acceptable where θ is the threshold value ( ) provided by each expert, i.e., θ = min{θ1 , θ2 , ..., θ K }. However, if C D X˜ k > θ , which means that the consensus of X˜ k to X˜ is not acceptable, the picture fuzzy evaluation vector X˜ k should be modified according to the following rules: (1) If γ˜ k (Pl ) < γ˜ (Pl ), then expert T Mk should increase his/her assessment of local weight with respect to Pl ; (2) If γ˜ k (Pl ) > γ˜ (Pl ), then expert T Mk should decrease his/her assessment of local weight with respect to Pl ; (3) If γ˜ k (Pl ) = γ˜ (Pl ), then expert T Mk will not revise his/her assessment of local weight with respect to Pl . Step 4 Determine the local weights of antecedent propositions. After consensus checking and improving, the updated collective picture fuzzy evaluation vector X˜ ' can be obtained. Then, it should be defuzzified into a crisp evaluation vector X = (x1 , x2 , ..., x L ) by using the defuzzification method given above. Next, the crisp evaluation vector X needs to be normalized by lwh =

xh , H ∑ xi

h = 1, 2, ..., H,

(6.19)

i=1

where lwh (h = 1, 2, ..., H ) are the local weights with respect to each WPFPR’s antecedent propositions.

118

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

The knowledge parameters of global weight (GW ), threshold value (T h) and certainty factor (U˜ ) can be acquired in the same way. For the GW, the collective picture fuzzy evaluation vector needs to be defuzzified and normalized as well, while the T h and the U˜ can be directly determined from their collective picture fuzzy evaluation vectors.

6.3.4 Execution Rules of PFPNs A marked PFPN is a PFPN with some places containing tokens. Let I (t) = { p I 1 , p I 2 , ..., p I m } be input places of transition t. The threshold values {λ I 1 , λ I 1 , ..., λ I m } and local weights {lw I 1 , lw I 2 , ..., lw I m } are assigned to input places, while the certainty factor and global weight of a transition t can be expressed by μ˜ t and gwt . Next, the enabling and firing rules of PFPNs are introduced. (1) Enabling Rule The transition t is enabled if ( ) α˜ p I j ≥ λ I j ,

j = 1, 2, ..., m,

(6.20)

where α( ˜ pi ) is the truth value of place pi and λ˜ I j is the threshold value of p I j . (2) Firing Rule Let p O be an output place of transition t. When transition t is fired, the truth values of input places will be copied and deposited into its output places. If p O has only one input transition, then the truth value of p O is computed by α( ˜ p O ) = P F W G(α( ˜ p I 1 ), α( ˜ p I 2 ), ..., α( ˜ p I m )) ⊗ μ˜ i ,

(6.21)

where the PFWG weight vector is W = (lw1 , lw2 , ..., lwm ). If p O has more than one fired input transitions ti (i = 1, 2..., n, i > 1), then the truth value of p O is defined by α( ˜ p O ) = P F W G(α( ˜ p O1 ), α( ˜ p O2 ), ..., α( ˜ p On )),

(6.22)

where the PFWG weight vector is W = (gw1 , gw2 , ..., gwn ) and α( ˜ p Oi ) is the truth value determined by the ith input transition.

6.3.5 Reasoning Algorithm Based on PFPNs In this section, based on the basic matrix operators introduced Chapters 4 and 5, a concurrent inference algorithm of PFPNs is proposed.

6.4 Illustrative Example

119

Input: I, O, and GW are m × n-dimensional matrices, U˜ is an n-dimensional vector, and LW, T h, and M˜ 0 are m-dimensional vectors. Output: M˜ k is an m-dimensional vector, signifying the final truth degree of all the propositions. Step 1 Let k = 1. The parameter k denotes the time of iterations. Step 2 Calculate the enabled place vector D (k) . D (k) can be obtained by comparing the truth values of places with their threshold values, which indicates the enabled input places of transitions. D (k) = M˜ k−1

T h.

(6.23)

Step 3 Calculate the token value vector of input places ˜ (k) with respect to local weight of each input place. If D (k) is a nonzero matrix, ˜ (k) can be computed by Eq. (6.24); otherwise, go to Step 7. ) ( ˜ (k) = P F W G (I ◦ W )T , M˜ k−1 ,

(6.24)

where W = [L W, L W, ..., L W ]m×n . Step 4 Calculate the enabled transition vector F (k) by Eq. (6.25), which indicates the enabled transitions of output places. F (k) = (E × I )

((

D (k)

)T

) ×I ,

(6.25)

where E = (1)1×m = [1, 1, ..., 1]. Step 5 Calculate the output truth degree vector (k) . If F (k) is a nonzero matrix, (k) can be calculated by Eq. (6.26); otherwise, go to Step 7. (k)

( = F (k) ◦

(k)

)

◦ U˜ .

(6.26)

Step 6 Compute the new marking vector Mk . ( M˜ k = M˜ k−1 ⊕ P F W G GW,

(k)

) .

(6.27)

If M˜ k = M˜ k−1 , then go to Step 7; otherwise let k = k + 1, go back to Step 2. Step 7 The reasoning is over.

6.4 Illustrative Example In this section, a realistic example regarding a gene regulatory network with activating and repressing process (Hamed 2018) is provided to show the applicability and feasibility of the PFPN model.

120

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

6.4.1 Implementation Gene regulatory network controls biological function by regulating the level of gene expression level. It is meaningful to understand the complex causal relationships within a gene regulatory network in biology system. Genes are paired into activator and repressor, and this gene pair determines the predicted target gene expression level. In what follows, the proposed PFPN model is used to model gene regulatory network to predict changes in expression level of the target gene. Let di (i = 1, 2., .., 9) be nine genes (i.e., nine propositions). The WPFPRs of the gene network are defined as follows:( ) R1 : IF d1 and d2 and d3 THEN d7 λ˜ 11 , λ˜ 21 , λ˜ 31 ; lw11 , lw21 , lw31 ; μ˜ 1 ; gw1 ; ( ) R2 : IF d4 THEN d8 λ˜ 42 ; lw42 ; μ˜ 2 ; gw2 ; ( ) R3 : IF d5 THEN d8 λ˜ 53 ; lw53 ; μ˜ 3 ; gw3 ; ( ) R4 : IF d6 THEN d8 λ˜ 64 ; lw64 ; μ˜ 4 ; gw4 ; ( ) R5 : IF d7 THEN d9 λ˜ 75 ; lw75 ; μ˜ 5 ; gw5 ; ( ) R6 : IF d8 THEN d9 λ˜ 86 ; lw86 ; μ˜ 6 ; gw6 . Based on the transition principle, the gene regulatory network with nine genes can be modeled by a PFPN as shown in Fig. 6.4. According to the gene regulatory network of biological system, the places in the PFPN with respect to their relative propositions are presented in Table 6.1. The places pi (i = 1, 2, ..., 6) are called starting places, the places p7 and p8 are called intermediate places, and the place p9 is a terminating place. To determine the knowledge parameters of local weights, global weights, threshold values, and certainty factors, five experts {T M1 , T M2 , T M3 , T M4 , T M5 } are invited to provide their judgements with respect to the six WPFPRs. As it is not easy for the experts to express unprecise knowledge information, a linguistic term set S is utilized to evaluate the knowledge parameters: S = {s0 = V er y Low, s1 = Low, s2 = Moderate, s3 = H igh, s4 = V er y H igh}. All the five linguistic terms can be approximated by PFNs as outlined in Table 6.2. For the gene regulatory network, the initial picture fuzzy evaluation vectors of local weights, global weights, threshold values, and certainty factors, provided by the five experts, are shown in Tables 6.3, 6.4, 6.5 and 6.6, respectively. Then, the above knowledge parameters of the six WPFPRs are acquired according to the proposed knowledge acquisition approach. First, the weight of each expert is obtained using Eqs. (6.15)–(6.16). In the second step, the picture fuzzy evaluation vectors of experts are synthesized into a collective picture fuzzy evaluation vector using Eq. (6.17). Next, the consensus degrees of experts can be checked according to Eq. (6.18) and improved until they are less than or equal to the consensus degree threshold value θ (θ = 0.5). Consequently, the computation results for the four knowledge parameters are displayed in Tables 6.3, 6.4, 6.5 and 6.6, respectively. Note that the collective picture fuzzy evaluation vectors

6.4 Illustrative Example

121

Fig. 6.4 PFPN model for the gene network Table 6.1 Places of the PFPN model and their propositions

Table 6.2 Linguistic term set defined by PFNs

Place (pi )

Proposition (d i )

p1

Activator gene 1 that increases gene 7 expression

p2

Activator gene 2 that increases gene 7 expression

p3

Activator gene 3 that increases gene 7 expression

p4

Repressor gene 4 that decreases gene 8 expression

p5

Repressor gene 5 that decreases gene 8 expression

p6

Repressor gene 6 that decreases gene 8 expression

p7

Repressor gene 7 that decreases gene 9 expression

p8

Repressor gene 8 that decreases gene 9 expression

p9

Target gene 9

Linguistic term

Picture fuzzy number

Very Low

(0, 0, 0.9)

Low

(0, 0.5, 0.4)

Moderate

(0, 0.9, 0)

High

(0.5, 0.4, 0)

Very High

(0.9, 0, 0)

122

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

Table 6.3 Local weights on six WPFPRs by five experts WPFPRs

1

Local weight

Initial picture fuzzy evaluation vectors

Collective picture fuzzy evaluations

Defuzzification and normalization

TM1

TM2

TM3

TM4

TM5

lw11

(0.9, 0, 0)

(0.8, 0.05, 0.05)

(0.7, 0.1, 0.1)

(0.8, 0.05, 0.05)

(0.8, (0.862, 0, 0, 0.1) 0.038)

0.468

lw21

(0.5, 0.1, 0.3)

(0.4, 0.2, 0.3)

(0.5, 0.2, 0.2)

(0.6, 0.2, 0.1)

(0.5, 0.1, 0.3)

(0.534, 0, 0.362)

0.292

lw31

(0.3, 0.4, 0.2)

(0.3, 0.4, 0.2)

(0.2, 0.4, 0.3)

(0.3, 0.3, 0.3)

(0.2, 0.2, 0.5)

(0.304, 0.240 0.27, 0.324)

2

lw42

1

3

lw53

1

4

lw64

1

5

lw75

1

6

lw86

1

Similarity degree

0.625

0.544

0.635

0.540

0.551

Expert weight

0.216

0.187

0.223

0.185

0.190

Consensus degree

0.357

0.397

0.378

0.335

0.385

of local weights and global weights should be defuzzified and normalized as presented in the last columns of Tables 6.3 and 6.4. For the gene regulatory network, the truth degrees of the starting places are set as follows: ˜ p2 ) = (0.878, 0, 0.06), α( ˜ p3 ) = (0.45, 0.305, 0.194), α( ˜ p1 ) =(0.742, 0.108, 0.06), α( α( ˜ p4 ) =(0.408, 0.204, 0.287), α( ˜ p5 ) = (0.608, 0.19, 0.110), α( ˜ p6 ) = (0.754, 0.095, 0.110).

With the PFPNs established in Fig. 6.4, we can obtain ⎡

1 ⎢1 ⎢ ⎢1 ⎢ ⎢0 ⎢ ⎢ I = ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎦ 0

⎡

0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ O = ⎢0 ⎢ ⎢0 ⎢ ⎢1 ⎢ ⎣0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

⎤ 0 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

6.4 Illustrative Example

123

Table 6.4 Global weights on six WPFPRs by five experts WPFPRs

Global weights

TM1

Initial picture fuzzy evaluation vectors TM2

TM3

TM4

TM5

Collective picture fuzzy evaluations

Defuzzification and normalization

0.297

1

gw1

2

gw2

(0.7, 0.1, 0.1)

(0.5, 0.1, (0.5, 0.3) 0.2, 0.2)

(0.6, 0.2, 0.1)

(0.2, 0.4, 0.3)

(0.518, 0.175, 0.206)

1

3

gw3

(0.7, 0.1, 0.1)

(0.6, 0.1, (0.6, 0.2) 0.1, 0.2)

(0.6, 0.1, 0.2)

(0.6, 0.1, 0.2)

(0.556, 0.1, 0.303 0.243)

4

gw4

(0.9, 0, 0)

(0.7, 0.1, (0.5, 0.1) 0.3, 0.1)

(0.3, 0.3, 0.3)

(0.6, 0.2, 0.1)

(0.829, 0, 0.07)

0.4

5

gw5

(0.6, 0.1, 0.2)

(0.6, 0.1, (0.3, 0.2) 0.5, 0.1)

(0.7, 0.1, 0.1)

(0.6, 0.2, 0.1)

(0.725, 0.103, 0.072)

0.602

6

gw6

(0.3, 0.4, 0.2)

(0.4, 0.2, (0.5, 0.3) 0.3, 0.1)

(0.3, 0.4, 0.2)

(0.3, 0.4, 0.2)

(0.352, 0.398 0.32, 0.228)

Similarity degree

0.556

0.588

0.581

0.599

0.591

Expert weight

0.189

0.201

0.198

0.209

0.203

Consensus degree

0.447

0.3598

0.459

0.345

0.36

Table 6.5 Certain factors on six WPFPRs by five experts Certainty factors

Initial picture fuzzy evaluation vectors TM1

TM2

TM3

TM4

TM5

1

μ˜ 1

(0.7, 0.3, 0)

(0.96, 0, 0)

(0.99, 0, 0)

(0.98, 0, 0)

(0.9, 0, 0) (0.966, 0, 0)

2

μ˜ 2

(0.96, 0, 0)

(0.8, 0.1, 0.1)

(0.99, 0, 0)

(0.98, 0, 0)

(0.9, 0, 0) (0.946, 0, 0.02)

3

μ˜ 3

(0.9, 0, 0) (0.9, 0, 0) (0.7, 0.3, 0)

(0.85, 0, 0)

(0.9, 0, 0) (0.907, 0, 0)

4

μ˜ 4

(0.99, 0, 0)

(0.99, 0, 0)

(0.99, 0, 0)

(0.98, 0, 0)

(0.99, 0, 0)

(0.988, 0, 0)

5

μ˜ 5

(0.99, 0, 0)

(0.99, 0, 0)

(0.7, 0.2, 0.1)

(0.7, 0.3, 0)

(0.99, 0, 0)

(0.974, 0, 0.02)

6

μ˜ 6

(0.96, 0, 0)

(0.99, 0, 0)

(0.7, 0.3, 0)

(0.9, 0.1, 0)

(0.7, 0.3, 0)

(0.988, 0, 0)

WPFPRs

Similarity degree

0.733

0.662

0.723

0.794

0.747

Expert weight

0.2

0.179

0.196

0.22

0.204

Consensus degree

0.227

0.197

0.3

0.285

0.24

Collective picture fuzzy evaluations

124

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

Table 6.6 Threshold values on six WPFPRs by five experts Threshold values

Initial picture fuzzy evaluation vectors

λ˜ 11

(0.7, (0.3, 0.1, 0.1) 0.25, 0.35)

λ˜ 21

(0.5, (0.1, (0.3, (0.4, (0.25, 0.1, 0.3) 0.1, 0.7) 0.1, 0.5) 0.1, 0.4) 0.05, 0.6)

(0.252, 0.086, 0.558)

λ˜ 31

(0.1, (0.2, (0.2, (0.1, (0.05, 0.2, 0.7) 0.1, 0.6) 0.3, 0.4) 0.3, 0.5) 0.2, 0.65)

(0.12, 0.202, 0.599)

2

λ˜ 42

(0.1, 0.05, 0.85)

(0.186, 0.025, 0.789)

3

λ˜ 53

(0.2, (0.2, (0.25, 0.1, 0.6) 0.1, 0.6) 0.05, 0.6)

(0.4, (0.2, (0.243, 0.106, 0.3, 0.2) 0.1, 0.6) 0.547)

4

λ˜ 64

(0.1, 0.15, 0.65)

(0.2, 0.15, 0.55)

(0.05, 0.15, 0.7)

(0.5, 0.25, 0.15)

(0.05, 0.25, 0.6)

(0.106, 0.184, 0.608)

5

λ˜ 75

(0.3, 0.05, 0.65)

(0.2, 0.25, 0.45)

(0.2, (0.2, (0.2, 0.1, 0.6) 0.2, 0.5) 0.01, 0.79)

(0.196, 0.019, 0.785)

6

λ˜ 86

(0.15, 0.05, 0.8)

(0.3, 0.01, 0.69)

(0.1, 0.05, 0.85)

(0.4, 0, 0.5)

(0.2, (0.188, 0.019, 0.1, 0.6) 0.793)

Similarity degree

0.582

0.647

0.633

0.56

0.653

Expert weight

0.188

0.211

0.205

0.18

0.216

Consensus degree

0.355

0.409

0.375

0.422

0.214

WPFPRs

1

TM1

⎡

0 ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢ GW = ⎢ 0 ⎢ ⎢0 ⎢ ⎢1 ⎢ ⎣0 0

TM2

TM3

(0.4, 0.05, 0.55)

0 0 0 0 0 0 0 0.297 0

TM4

TM5

Collective picture fuzzy evaluations

(0.4, (0.4, (0.1, (0.196, 0.2, 0.2, 0.3) 0.2, 0.4) 0.3, 0.6) 0.604)

(0.1, (0.3, 0.1, 0.7) 0.05, 0.65)

0 0 0 0 0 0 0 0.303 0

0 0 0 0 0 0 0 0.4 0

0 0 0 0 0 0 0 0 0.602

(0.2, 0.01, 0.79)

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎦ 0 0.398

[ ] U˜ = (0.966, 0, 0) (0.946, 0, 0.02) (0.907, 0, 0) (0.988, 0, 0) (0.974, 0, 0.02) (0.988, 0, 0)

6.4 Illustrative Example

125

[ ]T L W = 0.468 0.292 0.24 1 1 1 1 1 0 T h =[ (0.196, 0.2, 0.604) (0.252, 0.086, 0.558) (0.12, 0.202, 0.599) (0.186, 0.025, 0.789) (0.243, 0.106, 0.547) (0.106, 0.184, 0.608) (0.196, 0.019, 0.785) (0.188, 0.019, 0.793) (1, 0, 0) ], M˜ 0 =[ (0.742, 0.108, 0.06) (0.878, 0, 0.06) (0.45, 0.305, 0.194) (0.408, 0.204, 0.287) (0.608, 0.19, 0.110) (0.754, 0.095, 0.110) (0, 0, 1) (0, 0, 1) (0, 0, 1) ] . Based on the concurrent inference algorithm of PFPNs, the reasoning process for the considered system is explained as below. (1) The enabled place vector D (1) is calculated using Eq. (6.23) as follows: D (1) = M˜ 0

[ ]T Th = 1 1 1 1 1 1 0 0 0 .

(2) The token value vector of input places ˜ (1) is calculated using Eq. (6.24) as follows: ) ( ˜ (1) =P F W G (I ◦ W )T , M˜ 0 =[ (0.834, 0, 0.094) (0.408, 0.204, 0.287) (0.608, 0.19, 0.11) (0.754, 0.095, 0.11) (0, 0, 1) (0, 0, 1) ]. (3) The enabled transition vector F (1) is calculated using Eq. (6.25) as follows: F (1) = (E × I )

((

(4) The output truth degree vector (1)

( = F (1) ◦

(1)

)

D (1)

(1)

)T

) [ ] ×I = 111100 .

is calculated using Eq. (6.26) as follows:

◦ U˜

=[ (0.8006,0,0.094) (0.587, 0, 0.287) (0.7182, 0, 0.11) (0.8405, 0, 0.11) (0, 0, 1) (0, 0, 1) . (5) The new marking vector M˜ 1 is calculated using Eq. (6.27) as follows: ) ( M˜ 1 = M˜ 0 ⊕ P F W G GW, (1) =[ (0.742, 0.108, 0.06) (0.878, 0, 0.06) (0.45, 0.305, 0.194) (0.408, 0.204, 0.287)

126

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition (0.608, 0.19, 0.110) (0.754, 0.095, 0.110) (0.801, 0, 0.094) (0.72, 0, 0.168) (0, 0, 1) ].

(6) Since M˜ 1 /= M˜ 0 , we will continue to next iteration and let k = 2. [ ]T D (2) = 1 1 1 1 1 1 1 1 0 ˜ (2) =[ (0.834, 0, 0.094) (0.408, 0.204, 0.287) (0.608, 0.19, 0.11) (0.754, 0.095, 0.11) (0.801,0,0.094) (0.72,0,0.168) ] [ ] F (2) = 1 1 1 1 1 1 (2)

=[ (0.8006,0,0.094) (0.587, 0, 0.287) (0.7182, 0, 0.11) (0.8405, 0, 0.11) (0.793, 0, 0.094) (0.7056, 0, 0.168) ]

M˜ 2 =[ (0.742, 0.108, 0.06) (0.878, 0, 0.06) (0.45, 0.305, 0.194) (0.408, 0.204, 0.287) (0.608, 0.19, 0.110) (0.754, 0.095, 0.110) (0.801, 0, 0.094) (0.72, 0, 0.168) (0.757, 0, 0.125) ] (7) Since M˜ 2 /= M˜ 1 , we will continue to next iteration and k = 3. M˜ 3 =[ (0.742, 0.108, 0.06) (0.878, 0, 0.06) (0.45, 0.305, 0.194) (0.408, 0.204, 0.287) (0.608, 0.19, 0.110) (0.754, 0.095, 0.110) (0.801, 0, 0.094) (0.72, 0, 0.168) (0.757, 0, 0.125) ]. Since M˜ 3 = M˜ 2 , the reasoning process is over. The final PFNs of all the places are obtained as M˜ 3 . The expression level of the target gene d9 is (0.757, 0, 0.125), which means that the degree of positive membership is 0.757, the degree of neutral membership is 0 and the degree of negative membership is 0.125.

6.4.2 Comparisons and Discussions To show the effectiveness of the proposed PFPNs, a comparison analysis with the IFPNs (Hamed 2018) and the conventional FPNs (Chen et al. 1990) are made in this part. The expression level of the target gene d9 by using the IFPNs is (0.725, 0.149) (Hamed 2018). For the given gene regulatory network, the knowledge parameters in the FPNs are shown as follows:

6.4 Illustrative Example

127

Table 6.7 Rankings of p7 , p8 and p9 by PFPNs, IFPNs and FPNs

FPN models

Ranking results

PFPNs

p7 > p9 > p8

IFPNs

p7 > p9 > p8

FPNs

p8 > p9 > p7

[ ] U = 0.96 0.96 0.90 0.99 0.99 0.98 , [ ]T T h = 0.3 0.3 0.2 0.2 0.3 0.2 0.2 0.2 1 , [ ]T M0 = 0.8 0.9 0.6 0.5 0.7 0.8 0 0 0 . Based on the reasoning algorithm of FPNs, the result is obtained as: [ ]T M3 = 0.8 0.9 0.6 0.5 0.7 0.8 0.75 0.792 0.776 . According to above three FPN models, the ranking results of the intermediate places p7 , p8 , and the terminating place p9 are listed in Table 6.7. Firstly, we can find that the ranking results of PFPNs and IFPNs are the same. This can prove the feasibility of the proposed PFPNs. But IFSs are utilized in the IFPN model to handle uncertainty and vagueness in knowledge representation and reasoning, which are not suitable to describe uncertain information and data in some situations. For example, the expression level of the target gene d9 is (0.757, 0, 0.125) in the PFPNs and (0.725, 0.149) in the IFPNs. The neutral membership degree is ignored in the IFPNs. Thus, the proposed PFPNs have a wider range of applicability than the IFPNs. The ranking results derived by the PFPNs and the FPNs are different. The main reason is that the information concerning neutral membership degree and negative membership degree is ignored when the FPN model is used. Thus, the original information will be lost in the knowledge representation and acquisition process. Furthermore, the global weights are not considered in the traditional FPNs. This implies a lack of precision in the final reasoning results of FPNs. In summary, the PFPNs proposed in this chapter have the following advantages: (1) Using PFSs, the FPPNs are more efficient to deal with the vagueness and imprecision in knowledge representation. (2) Via a similarity degree-based expert weighting method, the conflict and inconsistency among experts’ evaluations can be handled in knowledge acquisition. As a result, the knowledge parameters of PFPNs could be determined accurately based on the opinions of different experts. (3) Compared with the reachability tree-based reasoning algorithm in FPNs, the developed inference algorithm of PFPNs adopts matrix equation format and can execute knowledge reasoning more efficiently.

128

6 Picture Fuzzy Petri Nets for Knowledge Representation and Acquisition

6.5 Chapter Summary In this chapter, a new type of FPNs, called PFPNs, is proposed to enhance ability of FPNs in knowledge representation and acquisition. In modeling a rule-based system, the proposed model can deal with imprecise knowledge information by using PFSs. Considering conflict and inconsistency among experts’ evaluations, a similarity degree-based method is adopted to derive the weights of experts objectively in the process of knowledge acquisition. Finally, a practical case of gene regulatory network is presented to illustrate the applicability and usefulness of the proposed PFPNs. The results showed that the proposed PFPN model can overcome certain disadvantages of the traditional FPNs and is efficient for knowledge representation and acquisition.

References Chen SM (2002) Weighted fuzzy reasoning using weighted fuzzy Petri nets. IEEE Trans Knowl Data Eng 14(2):386–397 Chen SM, Ke JS, Chang JF (1990) Knowledge representation using fuzzy Petri nets. IEEE Trans Knowl Data Eng 2(3):311–319 Cuong BC (2014) Picture fuzzy sets. J Comput Sci Cybern 30(4):409–420 Ding XF, Zhang L, Liu HC (2020) Emergency decision making with extended axiomatic design approach under picture fuzzy environment. Exp Syst 37(2):e12482 Gao MM, Zhou MC, Huang XG, Wu ZM (2003) Fuzzy reasoning Petri nets. IEEE Trans Syst Man Cybern A Syst Hum 33(3):314–324 Garg H (2017) Some picture fuzzy aggregation operators and their applications to multicriteria decision-making. Arab J Sci Eng 42(12):5275–5290 Hamed RI (2018) Quantitative modeling of gene networks of biological systems using fuzzy Petri nets and fuzzy sets. J King Saud Univ Sci 30(1):112–119 Kang J, Wang L, Jin H, Zhang J (2022) Proposed model of potential accident process at hydrogen refueling stations based on multi-level variable weight fuzzy Petri net. Int J Hydrogen Energy 47(67):29162–29171 Li L, Xie Y, Cen L, Zeng Z (2022) A novel cause analysis approach of grey reasoning Petri net based on matrix operations. Appl Intell 52(1):1–18 Lin YN, Yang CY, Wang SK, Chiou GJ, Shen VRL, Tung YC, Shen FHC, Cheng HC (2022a) Development and evaluation of an intelligent system for calibrating karaoke lyrics based on fuzzy Petri nets. Appl Artif Intell 36(1):3021–3048 Lin Z, Zhang J, Chen Y, Tian Q, Lin Z, Huang G (2022b) A new approach to power system fault diagnosis based on fuzzy temporal order Petri nets. Energy Rep 8:969–978 Liu HC, Liu L, Lin QL, Liu N (2013) Knowledge acquisition and representation using fuzzy evidential reasoning and dynamic adaptive fuzzy Petri nets. IEEE Trans Cybern 43(3):1059– 1072 Liu HC, You JX, Li ZW, Tian G (2017) Fuzzy Petri nets for knowledge representation and reasoning: a literature review. Eng Appl Artif Intell 60:45–56 Liu H-C, Luan X, Zhou M, Xiong Y (2022) A new linguistic Petri net for complex knowledge representation and reasoning. IEEE Trans Knowl Data Eng 34(3):1011–1020 Mou X, Mao LX, Liu HC, Zhou M (2022) Spherical linguistic Petri nets for knowledge representation and reasoning under large group environment. IEEE Trans Artif Intell 3(3):402–413

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Peng JJ, Chen XG, Tian C, Zhang ZQ, Song HY, Dong F (2022) Picture fuzzy large-scale group decision-making in a trust relationship-based social network environment. Inform Sci 608:1675– 1701 Rong Y, Liu Y, Pei Z (2022) A novel multiple attribute decision-making approach for evaluation of emergency management schemes under picture fuzzy environment. Int J Mach Learn Cybern 13(3):633–661 Selvaraj P, Ramalingam S (2022) Integrated risk assessment in sago preparation process using fuzzy petri net model. J Food Process Eng 45(8):e14046 Shi H, Liu HC, Wang JH, Mou X (2022) New linguistic Z-number Petri nets for knowledge acquisition and representation under large group environment. Int J Fuzzy Syst 24(8):3483–3500 Suraj Z, Fryc B (2006) Timed approximate Petri nets. Fund. Inform 71(1):83–99 Tsang EC, Yeung DS, Lee JW, Huang DM, Wang XZ (2004) Refinement of generated fuzzy production rules by using a fuzzy neural network. IEEE Trans Syst Man Cybern B Cybern 34(1):409–418 Wei G (2018a) Picture fuzzy Hamacher aggregation operators and their application to multiple attribute decision making. Fund Inform 157(3):271–320 Wei G (2018b) Some similarity measures for picture fuzzy sets and their applications. Iran J Fuzzy Syst 15(1):77–89 Xu XG, Shi H, Xu DH, Liu HC (2019) Picture fuzzy Petri nets for knowledge representation and acquisition in considering conflicting opinions. Appl Sci 9(5):983 Yeung DS, Tsang ECC (1994) Fuzzy knowledge representation and reasoning using Petri nets. Exp Syst Appl 7(2):281–289 Yu YX, Gong HP, Liu HC, Mou X (2022) Knowledge representation and reasoning using fuzzy Petri nets: a literature review and bibliometric analysis. Artif Intell Rev 56:6241–6265 Zhang XY, Wang XK, Yu SM, Wang JQ, Wang TL (2018) Location selection of offshore wind power station by consensus decision framework using picture fuzzy modelling. J Clean Prod 202:980–992 Zhao XK, Zhu XM, Bai KY, Zhang RT (2023) A novel failure model and effect analysis method using a flexible knowledge acquisition framework based on picture fuzzy sets. Eng Appl Artif Intell 117:105625

Chapter 7

R-Numbers Petri Nets for Knowledge Representation and Acquisition

As a vital modeling technique, FPNs have been widely used in various areas for knowledge representation and reasoning. However, the conventional FPNs have many deficiencies in representing inaccurate knowledge, acquiring knowledge parameters and conducting approximate reasoning when used in the real world. In this chapter, a new version of FPNs, called R-numbers Petri nets (RPNs), is proposed to overcome the shortcomings and enhance the effectiveness of FPNs. Based on Rnumbers, expert knowledge is depicted in the form of weighted R-numbers production rules. The interrelationships among input places (or transitions) are modelled by the R-numbers Maclaurin symmetric mean operator in the knowledge reasoning process. In addition, the conflict opinions of experts are handled with the proposed RPN model in order to obtain more precise knowledge parameters. Finally, the effectiveness and practicality of the proposed RPNs are illustrated by a realistic example concerning reliability analysis of an electric vehicle motor.

7.1 Introduction Incorporating fuzzy logic in Petri nets, the FPNs are an effective and commonly used modeling technique for knowledge representation and reasoning (Li et al. 2000; Chen 2002; Gao et al. 2003). The combination of graphical power and fuzzy inference capacity allows FPNs to cope with imprecise information in knowledge-based systems (Suraj and Fryc 2006; Ha et al. 2007). An FPN is a marked graphical system, in which transitions denote FPRs and places denote propositions. An approximate knowledge inference algorithm is executed by firing the transitions in FPNs to evaluate the truth degrees of terminating places. Therefore, it is a general graphical model of an expert system and can represent logical knowledge in an intuitive and visual way (Hu et al. 2011; Liu et al. 2013; Suraj 2013). Moreover, FPNs can capture the dynamic nature and improve the efficiency of fuzzy rule-based reasoning by marking evolution (Liu et al. 2016; Hamed 2018). In recent years, owing their strong ability © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Shi and H.-C. Liu, Fuzzy Petri Nets for Knowledge Representation, Acquisition and Reasoning, https://doi.org/10.1007/978-981-99-5154-3_7

131

132

7 R-Numbers Petri Nets for Knowledge Representation and Acquisition

to depict uncertain knowledge and support reasoning processes, FPNs have been widely used in various fields for chemical plant risk assessment (Guo and Wang 2023), subway fire risk assessment (Zhang et al. 2022), submarine pipeline leakage failure analysis (Yu et al. 2022), sago processing risk assessment (Selvaraj and Ramalingam 2022), failure mode and effect analysis (Shi et al. 2020), process equipment failure risk assessment (Wang et al. 2022), and so on. Although FPNs have been successfully applied to a variety of problems, the current models have deficiencies in knowledge representation and acquisition (Liu et al. 2022; Mou et al. 2022; Shi et al. 2022; Yu et al. 2023). First, the knowledge parameters are supposed to be fuzzy and fuzzy numbers are used in FPNs to capture experts’ opinions. However, there are situations in which fuzzy numbers are insufficient to depict expert knowledge precisely, particularly in cases where uncertainty about available information exists. The R-numbers, proposed by Seiti et al. (2019), can better cope with the risks and errors associated with fuzzy numbers when available data either come from unreliable sources or refer to events in the future. Thus, some percentage of error and risk as a confidence factor can be taken into account for this type of data. Considering the configurations such as positive risk, negative risk and negative acceptable risk, R-numbers can depict ambiguous information more comprehensively, and are more effective to model uncertainty and vagueness in realworld decision making problems (Liu et al. 2021b; Mou et al. 2021; Seiti et al. 2021). As a promising tool for describing the risks and errors of fuzzy data, the R-numbers have drawn increasing attention from academics recently (Mousavi et al. 2021, 2023; Zhao et al. 2022). On the other hand, acquiring knowledge is of significance to develop a rule-based expert system. However, there is little investigation on determining the knowledge parameters of FPNs. With the growing complexity of expert systems, it is much more difficult to acquire knowledge from domain experts (Li et al. 2018; Xu et al. 2020; Liu et al. 2021a). Generally, knowledge acquisition is to obtain knowledge through interactions with experts. In the knowledge acquisition process, experts are often diverse in their education, expertise, and understanding of the problem at hand. Hence, they tend to have different judgments on the knowledge parameters of an FPN. Such discrepancies in the opinions of experts could cause intra-group conflicts, which can potentially lead to non-cooperation and thus are not conducive to the final knowledge parameter determination (Xu et al. 2019; Liu et al. 2020). Therefore, it is necessary to eliminate the conflict among experts or reduce it under an acceptable degree in the knowledge acquisition process. According to the foregoing discussions, the aim of this chapter is to develop a new version of FPNs, called R-numbers Petri nets (RPNs), to represent and acquire expert knowledge more accurately. First, R-numbers are applied to capture the possible risks and errors of experts’ opinions on the knowledge base rules in an expert system. Second, considering the conflict opinions of experts, we introduce a preference modifying-based method for acquiring more accurate knowledge parameters of RPNs. Additionally, the interrelationships among input places (or transitions) are managed by the R-numbers Maclaurin symmetric mean (RMSM) operator for deriving more reasonable knowledge reasoning results. Finally, an application

7.2 Preliminaries

133

example concerning reliability issue of an electric vehicle motor is given to illustrate the effectiveness and capability of the proposed RPNs. The remaining sections of this chapter are structured as follows. Section 7.2 introduces the definitions of R-numbers and Sect. 7.3 develops the RPN model for knowledge representation and reasoning. A realistic example is provided in Sect. 7.4 to illustrate and validate the proposed RPNs. Section 7.5 offers the conclusions of this chapter.

7.2 Preliminaries The R-numbers were proposed by Seiti et al. (2019) to handle the risks and errors associated with fuzzy numbers in real-world decision-making problems. ˜ its R-numbers are defined in Definition 7.1 (Seiti et al. 2019) For a fuzzy number B, optimistic-pessimistic mode by considering fuzzy negative and positive risks (˜r − and − + r˜ + ), fuzzy negative and positive acceptable risks (  A R and  A R ), and fuzzy negative − + and positive risk perceptions (  R P and  R P ) for beneficial and non-beneficial values.   The R-numbers expressed by Rb B˜ for beneficial values can be expressed as:          Rb B˜ = R1b B˜ , R2b B˜ , R3b B˜ , where

 ⎧   

r˜ − − ⎪ ˜ ˜  ⎪ R B = max , l B˜ , α ⊗ 1  A R B ⊗ 1  min ⎪ 1b − ⎪ ⎪  1  R P ⎪ ⎪   ⎪ ⎪ ⎨ R2b B˜ = B˜ , (7.1)

  

 ⎪ ⎪ r˜ + + ⎪ ˜ ˜  ⎪ R3b B = min B ⊗ 1 ⊕ , u B˜ ⎪ + ⊗ 1  AR ⎪ ⎪ 1 RP ⎪ ⎪ ⎩ 0 < r − < 1, r + > 0 ˜ u B˜ is the maximum possible value of B˜ and α is a where l B˜ is the lower limit of B, number that is infinitely close to one.   = For non-beneficial values, R-numbers can be defined as: Rc B˜        R1c B˜ , R2c B˜ , R3c B˜ , where

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7 R-Numbers Petri Nets for Knowledge Representation and Acquisition



 ⎧ 

  r˜ + + ⎪ ˜ ˜  ⎪ R , l B˜ B = max , α ⊗ 1  A R B ⊗ 1  min ⎪ 1c + ⎪ ⎪  1  R P ⎪ ⎪   ⎪ ⎪ ⎨ R B˜ = B˜ 2c . (7.2)

  

 ⎪ ⎪ r˜ − − ⎪ ˜ ˜  ⎪ R3c B = min B ⊗ 1 ⊕ , u B˜ ⎪ − ⊗ 1  AR ⎪ ⎪ 1 RP ⎪ ⎪ ⎩ − r > 1, 1 < r + < 0 Definition 7.2 (Seiti et al. 2019) Based on type-2 fuzzy sets, the membership functions μ Rb B˜ (x) for beneficial values and μ Rc B˜ (x) for non-beneficial values are defined as:        ⎧ ˜  B˜  R1b B˜ , R1b B˜ ≤ x < B˜ ⎪ x  R B 1b ⎪ ⎪ ⎨          (7.3) μ Rb B˜ (x) = ˜  x  R3b B˜  B˜ , B˜ ≤ x < R3b B˜ , R B 3b ⎪ ⎪ ⎪ ⎩ 0, otherwise        ⎧ ˜  B˜  R1c B˜ , R1c B˜ ≤ x < B˜ ⎪ x  R B 1c ⎪ ⎪ ⎨          (7.4) μ Rc B˜ (x) = ˜  x  R3c B˜  B˜ , B˜ ≤ x < R3c B˜ . R B 3c ⎪ ⎪ ⎪ ⎩ 0, otherwise Definition 7.3 (Seiti et al. 2019; Mou et al. 2021) For any two   R˜ = ((a11 , a12 , a13 ), (a21 , a22 , a23 ), (a31 , a32 , a33 )) and R b˜ = numbers R(a) ((b11 , b12 , b13 ), (b21 , b22 , b23 ), (b31 , b32 , b33 )) in type-2 fuzzy forms, their operational laws are defined as:

(1)

(2) (3) (4) (5)

⎞ ⎛ (a11 + b11 , a12 + b12 , a13 + b13 ),   ⎟ ⎜ R(a) ˜ ⊕ R b˜ = ⎝ (a21 + b21 , a22 + b22 , a23 + b23 ),⎠, + b31 , a32 + b32 , a33 + b33 )   (a31   (a11 b11 , a12 b12 , a13 b13 ), (a21 b21 , a22 b22 , a23 b23 ), , R(a) ˜ ⊗ R b˜ = (a31 b31 , a32 b32 , a33 b33 )         a12 a13 a21 a22 a23 a31 a32 a33 , , , , , , , , , R(a) ˜  R b˜ = ab11 b b b b b b b b 33 32 31 23 22 21 13 12 11 λR(a) ˜ = ((λa11 , λa12 , λa13 ), (λa21 , λa22 , λa23 ), (λa31 , λa32 , λa33 )), λ > 0, R(a) ˜ λ = (a31 , a32 , a33 )λ , (a21 , a22 b, a23 )λ , (a11 , a12 , a13 )λ , λ > 0.

7.3 R-Numbers Petri Nets

135

Definition 7.4 (Seiti et al. 2019) Given two R-numbers R(a) ˜ and R( b), the distance between them is defined by ⎡ ⎛

1 ⎞⎤ 21  2 1  2 2 2 (a11 − b11 ) + (a12 − b12 ) + (a13 − b13 ) ⎢ ⎜ ⎟⎥ ⎢ ⎜ ⎟⎥ 3 ⎢ ⎜ ⎟⎥ 1 ⎟⎥



⎢ ⎜    ⎢ 1 ⎜ 2 ⎟⎥   1 d R(a), ˜ R b = ⎢ ⎜+ ⎟⎥ . (a21 − b21 )2 + (a22 − b22 )2 + (a23 − b23 )2 ⎢3⎜ ⎟⎥ 3 ⎢ ⎜ ⎟⎥ ⎢ ⎜

21 ⎟⎥ ⎣ ⎝ ⎠⎦  1  + (a31 − b31 )2 + (a32 − b32 )2 + (a33 − b33 )2 3 (7.5) Definition 7.5 (Seiti et al. 2019) For a R-number R(a) ˜ = ((a11 , a12 , a13 ), (a21 , a22 , a23 ), (a31 , a32 , a33 )), the following defuzzification operation can be performed to obtain its crisp value: 1 R(a)= ˜ (a11 + a12 + a13 + a21 + a22 + a23 + a31 + a32 + a33 ). 9

(7.6)

al. 2021). Let R(a˜ i ) Definition =  7.6  i (Mou  et  i  i i i i i i i , a13 , a22 , a23 , a32 , a33 , a21 , a31 a11 , a12 (i = 1, 2, ...., n) be n R-numbers n  wi = 1, with the weight vector w = (w1 , w2 , ..., wn )T satisfying wi ∈ [0, 1] and i=1

and g = 1,2,…,n. Then the R-numbers Maclaurin symmetric mean (RMSM) operator is expressed as:        R M S M (g) R a˜ 1 , R a˜ 2 , . . . , R a˜ n =

 1≤i 1