Space Fault Tree Theory and System Reliability Analysis 9782759825042

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Current Natural Sciences

Tiejun CUI and Shasha LI

Space Fault Tree Theory and System Reliability Analysis

The publication of this book was financially supported by the: National Fund for Academic Publication in Science and Technology (2019-G-002); National Natural Science Foundation of China (51704141); National key research and development program (2017YFC1503102).

Printed in France

EDP Sciences – ISBN(print): 978-2-7598-2499-1 – ISBN(ebook): 978-2-7598-2504-2 DOI: 10.1051/978-2-7598-2499-1 All rights relative to translation, adaptation and reproduction by any means whatsoever are reserved, worldwide. In accordance with the terms of paragraphs 2 and 3 of Article 41 of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for private use and not intended for collective use” and, on the other hand, analyses and short quotations for example or illustrative purposes, are allowed. Otherwise, “any representation or reproduction – whether in full or in part – without the consent of the author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any representation or reproduction, by any means whatsoever, will therefore be deemed an infringement of copyright punishable under Articles 425 and following of the French Penal Code. The printed edition is not for sale in mainland China. Customers in mainland China please order the print book from Science Press. ISBN of the China edition: Science Press 978-7-03-066034-3 Ó Science Press, EDP Sciences, 2020

Content Summary This book discusses the theory of the space fault tree framework, which is introduced internationally in system reliability analysis for the first time. Through the integration of the factor space theory and the theory and method of intelligence, space fault tree can be used in intelligent analysis and big data processing. This book focuses on continuous space fault tree, discrete space fault tree, inward analysis of system factor structure, function structure analysis, system reliability with respect to influencing factors, cloudization space fault tree, cloud similarity, and clustering analysis and similarity. The book is partly a summary on space fault tree theory, the existing research results of which are abundant. The definitions, methods, procedures, and simple applications of the related concepts are presented. This book is suitable for researchers who are studying and applying the theory of intelligent science and safety system engineering to address system reliability issues. It can also be used as a reference by graduate students in fields related to safety engineering.

Introduction

Space fault tree (SFT) theory is a system reliability analysis method proposed by the present authors in 2012. After five years of development, the basic framework of the SFT is complete; the framework includes continuous space fault tree, discrete space fault tree, and the corresponding data mining methods. The SFT can be satisfactorily employed for the reliability analysis of a simple system. It integrates intelligent science with big data processing technologies, including factor space theory, fuzzy structured element theory, and cloud model theory. Further, the SFT with an intelligent processing capability is formed; the SFT involves the processing of fault big data, reliability causality, reliability stability, reliability inward engineering, and reliability change process description. The SFT has good versatility, extensibility, and adaptability. Although there are some limitations, the SFT also has adequate room for further development to overcome limitations. This book describes the SFT theory system. The basic framework of the SFT is introduced, including the continuous space fault tree and the discrete space fault tree. Considering the fuzziness, randomness, and discreteness of the fault data, the factor space theory, fuzzy structured element theory, and cloud model theory are used to improve the SFT. It can apply multi-factor analysis, fault logic analysis, and fault big data processing. The main contents of this book include the following: Chapter 1 introduces the status of the fundamental research related to this subject, the focal work conducted, and the innovations achieved. Chapter 2 proposes the continuous space fault tree (CSFT), which is the most basic theory in this area. It mainly deals with fault data with a strong regularity in a laboratory environment. Its characteristic functionality and fault probability distribution are continuous. Chapter 3 proposes the discrete space fault tree (DSFT), which is developed based on CSFT. It deals mainly with irregular fault data. To this end, the fuzzy structured element theory is introduced into the discrete space fault tree to represent the fuzziness of the fault data, and the fuzzy structured element characterisation function is then constructed. Chapter 4 introduces the framework of the inward analysis of the system factor structure (IASFS); furthermore, it defines and describes item-by-item analyses (IIA) and the classification reasoning method (CRM). IIA and the CRM can obtain

DOI: 10.1051/978-2-7598-2499-1.c901 © Science Press, EDP Sciences, 2020

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Introduction

an equivalent response structure, which is the same as the response of the analysed system, with respect to changes in the working environmental factors. Chapter 5 presents the system function structure analysis theory based on the factor space theory. The relationship between the classification reasoning method and function structure analysis is discussed. The system function structures of both incomplete and complete information are analysed using these methods. Chapter 6 elucidates the logical relationship between reliability and influencing factors; in addition, it presents a corresponding analysis method to improve the SFT theory, including the inference of causal relationships, reduction of the factor dimensions, and compression of the fault data. Chapter 7 describes the use of the cloud model (CM) to transform a characteristic function into a cloudization characteristic function (CCF), allowing the SFT to express the uncertainty in data. Thus, the expressions of the fault data characteristics and rules are more accurate and reflect the discreteness, randomness, and fuzziness of the fault data. Chapter 8 proposes a new semantic simplification method based on the cloud model. Multiple cloud models are built on envelopes and the method integrates the area between them. The ratio between the overlapping area and the integral area is considered to judge the similarity among the cloud models. Chapter 9 describes the influence of multiple domain attributes on the clustering analysis of objects based on factor space. A method for representing a graphical domain attribute, called an attribute circle, is proposed for an object. An attribute circle can represent infinite domain attributes. The similarity analysis of the objects is based on the concept of attribute circle; thus, the clustering analysis method of the object set is studied and improved. Chapter 10 discusses the development and prospects of the SFT theory and summarises the existing theory of SFT and its future development. The contents discussed in this book can hopefully provide a basic theory and application method for system reliability analysis, providing novel ideas and methods for the integration of system reliability theory with intelligent science and big data technology. This book will also provide a methodology for the development of safety science and technology theory, along with system reliability, to better meet the requirements for future technology. The authors wish to thank all their friends for their valuable critiques, comments and assistance in this book. This study was partially supported by grants (Grants Nos. 51704141, 52004120 and 2017YFC1503102) from the Natural Science Foundation of China, and it also obtained the national science and technology academic book publishing fund. Special thanks to Springer press – most of the achievements in the book are systematically published through Springer – and the Chinese academic journals for their support.

Contents Content Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III V

CHAPTER 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Purpose and Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Summary of Research and Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Fault Tree Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Multi-factor Influence and Fault Big Data . . . . . . . . . . . . . . . . . 1.2.3 System Function Structure Analysis and Factor Space . . . . . . . 1.2.4 System Reliability and Influencing Factors . . . . . . . . . . . . . . . . 1.2.5 Cloud Model and Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Object Classification and Similarity . . . . . . . . . . . . . . . . . . . . . 1.3 Deficiency of System Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 3 4 7 8 11 12 16

CHAPTER 2 Continuous Space Fault Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Concepts of CSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fault Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Component Fault Probability Distribution . . . . . . . . . . . . . . . . 2.2.2 System Fault Probability Distribution . . . . . . . . . . . . . . . . . . . . 2.3 Importance Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Probability Importance Distribution . . . . . . . . . . . . . . . . . . . . . 2.3.2 Criticality Importance Distribution . . . . . . . . . . . . . . . . . . . . . . 2.4 System Fault Probability Distribution Trend . . . . . . . . . . . . . . . . . . . . 2.5 Calculation of MTLa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 26 26 28 34 34 37 50 50 53 59

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CHAPTER 3 Discrete Space Fault Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Discrete Space Fault Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Significance of DSFT Modified Using Fuzzy Structured Element . . . . . 3.3 Factor Projection Fitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Constructions and Applications of EDSFT . . . . . . . . . . . . . . . . . . . . . . 3.4.1 E-Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 E-Component Fault Probability Distribution . . . . . . . . . . . . . . . 3.4.3 E-System Fault Probability Distribution . . . . . . . . . . . . . . . . . . 3.4.4 E-Probability Importance Distribution . . . . . . . . . . . . . . . . . . . 3.4.5 E-Criticality Importance Distribution . . . . . . . . . . . . . . . . . . . . 3.4.6 E-System Fault Probability Distribution Trend . . . . . . . . . . . . . 3.4.7 E-Component Domain Importance . . . . . . . . . . . . . . . . . . . . . . 3.4.8 E-Factor Importance Distribution . . . . . . . . . . . . . . . . . . . . . . . 3.4.9 E-Factor Joint Importance Distribution . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 65 66 69 69 73 74 74 75 75 76 78 79 79 80

CHAPTER 4 Inward Analysis of System Factor Structure . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Inward Analysis of System Factor Structure . . . . . . . . . . . . . . . . . . . . . 4.2 Human–Machine Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Table Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Classification Reasoning Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Mathematical Description of Classification Reasoning Method . . . . . . . 4.6 Item-By-Item Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Mathematical Description of Item-by-Item Analyses . . . . . . . . . . . . . . . 4.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 84 86 87 88 93 94 95 98 99

CHAPTER 5 Function Structure Analysis and Factor Space . . . . . . . . . . . . . . . . . . . . . . . 5.1 Factor Analysis Method of Function Structure . . . . . . . . . . . . . . . . . . . 5.1.1 Factors and Dimension Variability . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Function Structure Analysis Space . . . . . . . . . . . . . . . . . . . . . . 5.2 Factor Logic Description of Function Structure . . . . . . . . . . . . . . . . . . 5.2.1 Axiom System of Function Structure Analysis . . . . . . . . . . . . . . 5.2.2 Minimization Method of System Function Structure . . . . . . . . . 5.3 Analysis of System Function Structure . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Analysis with Incomplete Information . . . . . . . . . . . . . . . . . . . . 5.3.2 Analysis with Complete Information . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 103 103 104 105 105 109 110 111 113 116 117

Contents

CHAPTER 6 System Reliability with Influencing Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Methodology of Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . 6.2 Analysis of Relationship between Reliability and Influencing Factors . . 6.2.1 Random Variable Decomposition Formula . . . . . . . . . . . . . . . . . 6.2.2 Causal Relationship Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Causal Concept Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Background Relationship Analysis . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Factor Dimension Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.6 Compression of Fault Probability Distribution . . . . . . . . . . . . . 6.3 Algorithm Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Random Variable Decomposition Formula . . . . . . . . . . . . . . . . . 6.3.2 Causal Relationship Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Causal Concept Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Background Relationship Analysis . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Factor Dimension Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.6 Compression of Fault Probability Distribution . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 7 Cloudization Space Fault Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Definitions of SFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Construction of Cloudization Space Fault Tree . . . . . . . . . . . . . . . . . . . 7.2.1 Basis of CLSFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Cloudization Fault Probability Distribution . . . . . . . . . . . . . . . 7.2.3 Cloudization Fault Probability Distribution Trend . . . . . . . . . . 7.2.4 Cloudization Importance Distribution Probability and Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Cloudization Factor Importance and Joint Importance Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.6 Cloudization Component Domain Importance . . . . . . . . . . . . . . 7.2.7 Cloudization Path Set Domain and Cut Set Domain . . . . . . . . . 7.2.8 Uncertainty Analysis of Reliability Data . . . . . . . . . . . . . . . . . . 7.3 Example Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Cloudization Fault Probability Distribution . . . . . . . . . . . . . . . 7.3.2 Cloudization Fault Probability Distribution Trend . . . . . . . . . . 7.3.3 Cloudization Importance Distribution Probability and Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Cloudization Importance Distribution of Factor and Factor Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Cloudization Component Domain Importance . . . . . . . . . . . . . . 7.3.6 Cloudization Path Set Domain and Cut Set Domain . . . . . . . . . 7.3.7 Uncertainty Analysis of Reliability Data . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

119 121 124 124 126 127 127 128 130 132 132 136 144 147 150 153 156 158

161 163 163 164 165 165 166 166 167 168 168 169 170 172 178 181 185 187 187 196

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Contents

CHAPTER 8 Cloud Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Similarity Algorithms of Cloud Model . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Cloud Similarity Computation Based on Envelope . . . . . . . . . . . . . . . . 8.3 Algorithm Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Analyses of Algorithm Advantage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 199 201 203 205 206 206

CHAPTER 9 Clustering Analysis and Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Preliminary Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Concepts and Properties of Attribute Circle . . . . . . . . . . . . . . . . . . . . . 9.3 Object Clustering Analysis Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Improvements of Clustering Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Example Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

207 207 209 211 213 216 220 222

CHAPTER 10 Development and Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.1 Summary of Space Fault Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.2 Future Development of Space Fault Tree . . . . . . . . . . . . . . . . . . . . . . 227

Chapter 1 Introduction 1.1

Purpose and Significance

System reliability theory is one of the basic theories of safety science. Derived from system engineering, system reliability is mainly concerned with the possibility of system faults and accidents. Owing to improvements in modern science and industrialisation, to pursue greater economic and strategic goals, some countries have intensified their research and established large or super-large systems to meet their requirements. However, it was found that a decline in reliability occurs during the operation of a system with an increase in system complexity. In this case, the original problem is that the lessons learned after an accident cannot meet the requirements of the present system safety. Because the research method of the problem is primarily applicable to a low value of system, with low system reliability and no critical consequences of a fault, such a research approach is not significant for today’s large-scale and extremely complex systems. Therefore, during the 1950s, Britain and the US first proposed the concept of safety system engineering. At this time, some concepts of system engineering were introduced into the field of safety, particularly a reliability analysis method, and applied in the military and aerospace fields. Safety system engineering is, therefore, one of the basic aspects of safety science. Safety system engineering and system reliability analyses have since been developed under the conditions of relatively simple and low complexity systems and limited data scales. However, with the development of big data technology, intelligent science, system science, and related mathematical theories, existing system reliability analysis methods have also exposed certain problems, such as big data processing; reliability causation, stability, and reverse engineering; and the description of reliability changes. At the same time, existing system reliability analysis methods are mostly targeted at specific systems. Although such an analysis is effective, there has been a lack of abstraction at the system level, making it difficult to meet the required universality, scalability and adaptability. Therefore, the system reliability analysis method that achieves the above capabilities and meets future technological requirements is needed. It is, therefore, necessary to combine the system reliability analysis with intelligent science and big data technology. DOI: 10.1051/978-2-7598-2499-1.c001 © Science Press, EDP Sciences, 2020

2

Space Fault Tree Theory and System Reliability Analysis

Space fault tree theory (SFT) [1] is a system reliability analysis method proposed by the authors in 2012. After some years of development, the preliminarily foundation of the SFT theory framework has been completed, which can satisfy the reliability analysis of a simple system, including big data processing, reliability causality, reliability stability, reliability reverse engineering, and a reliability change description, and achieves high universality, extensibility, and adaptability. Its development process integrates intelligent science and big data processing technology, including factor space theory [2], fuzzy structured element theory [3], and cloud model theory [4], among others. Although some problems remain, the SFT still has adequate room to solve these issues through further development. We hope this book will broaden the basic research field of safety science, enabling readers to better understand SFT theory, factor space theory, and their role in the system reliability analysis, seeking a theoretical development of reliability adapted to intelligent science and big data technology.

1.2 1.2.1

Summary of Research and Problems Fault Tree Research

The fault tree is an important aspect of safety system engineering and plays a crucial role in the analysis of system safety and reliability in several industries. It has been widely used and studied around the world as a systematic scientific method. The applications of the fault tree and studies conducted in different fields are reviewed in the following chapter. In medical research, the fault tree has been applied to the control of hand, foot, and mouth diseases [5]. For the safety of a laboratory-scale bioreactor, the fault tree analysis method was used to deal with hydrogen sulphide biotreatment [6]. In research into uncertainty, the problem was studied using fault tree [7]. In addition, the decision tree method has been used to represent uncertainty based on probability [8]. An uncertainty analysis of fault tree model based on basic events was also developed [9]. In addition, uncertainty in fault tree analysis was processed using a hybrid probabilistic–possibilistic framework [10]. In a study on system reliability analysis, a real-time system analysis method based on fault tree was proposed [11]. A study on a non-repairable system was also conducted using dynamic fault tree and priority AND gates [12]. In an analysis of system safety, an extended fault tree was used to analyse the safety of an autonomous system [13]. In an electrical system, a multi-valued Fisher’s fuzzy decision tree was used to diagnose the faults in analogue circuits [14]. A radiation processing system was analysed using a probabilistic fault tree [15]. The fault tree was used in the fault diagnosis of a fuel cell [16], and the reliability of the uninterrupted power supply system was analysed [17]. In the field of transportation, the risk to a railway transportation system was evaluated based on a time-sequence fault tree [18]. In addition, in a related study, methods for determining basic events through the fault tree analysis were put forward [19]. Efficient ordering heuristics have also been obtained using a binary decision diagram based on fault tree [20]. Moreover, quantitative analyses of a highly coupled dynamic fault

Introduction

3

tree were performed based on SBDD and IRVPM [21] and established based on the coupling of a structure function and a Monte Carlo simulation [22]. An effective approximate Markov chain method was proposed for a dynamic fault tree analysis [23], and a quantitative analysis of the dynamic fault tree was obtained based on a structure function [24]. Studies on fault tree, such as the safety assessment of a pile foundation [25], a formal safety assessment (FSA) model [26], an associated fault tree and an FSA system for hazardous factors [27], and a qualitative and quantitative algorithm of a complex fault tree have also been conducted in China [28]. Fuzzy mathematics and FTA have also been combined to soften the boundary problem of the fault tree [29–32]. Simultaneously, changes in system reliability influenced by multiple factors are a priority in the fields of systems engineering and safety science. Current studies include a multi-factor analysis of a vehicle clutch reliability [33]; binary state reliability calculations of complex systems [34]; performance optimisation of bounded parameters under uncertain dynamic system reliability [35]; multi-function goal-oriented repairable system reliability [36]; system reliability and failure modes at different time scales [37], substation systems [38], and power distribution systems [39]; power system reliability [40]; and a control algorithm for nonlinear hysteresis system reliability [41], among others. The authors of this book also conducted studies on the relationship between reliability and the influence of multiple factors [42–44]. The aforementioned methods have achieved satisfactory results, but are too simple to use when considering multiple influencing factors on basic events. There are two analysis modes of the fault tree, from top to bottom and from bottom to top. For either of these modes, the probability of occurrence of the top event is represented by the structure relationship among the basic events. For a system, the tree structure is constant and will not change, and thus the probability of occurrence of basic events determines the probability of occurrence of the top event. For a typical fault tree, the probability of occurrence of a basic event is constant, and thus the system fault probability, probability importance, and criticality importance are constant. The fault tree constructed in this way can only be set up under certain specified conditions. Meanwhile, a traditional fault tree cannot be used to analyse the change in influence of basic events on a system. In a word, a traditional fault tree is single and constant, making it difficult to convert into a mathematical model for analysis.

1.2.2

Multi-Factor Influence and Fault Big Data

Safety system engineering is derived from the theory of system engineering and is an important theoretical basis for safety science. It plays a central role in modern manufacturing and contemporary life, such as in manufacturing industries, mining, transportation, medicine, and the military. In particular, the complexity and related life property and strategic significance of these fields are increasing in importance [45]. However, there are certain problems in research on safety system engineering, particularly system reliability. Research has focused on the internal structure of the system and the component reliability. However, the component functionality is primarily dependent on the working environment after manufacturing completion

4

Space Fault Tree Theory and System Reliability Analysis

because the basic component properties may differ in different environments. For a single-function system with numerous components, changes in the system reliability are extremely complex with changes in the environment. The above phenomenon should not be ignored. Similarly, fault monitoring data will accumulate during daily system operation and maintenance. These data record the functional characteristics of the system. Not only do they reflect the influence of the working environment on the system reliability, they can also be used for a comprehensive analysis of system reliability. Unfortunately, fault monitoring data are a type of big data, with fuzziness, randomness, and discreteness, and are difficult to analyse. One problem in this regard is the system reliability mining used in fault big data, which should adapt to the characteristics of such data. At present, research into reliability and the processing of fault big data are being conducted, and some methods related to fuzzy mathematics have been applied, with many benefits having been derived. Studies are also being conducted on the detection of electrical system faults, including a fault classification algorithm using voltage and current phase sequence components [46], fault diagnosis systems for large power generation stations and their transmission lines [47], the diagnosis of analogue circuit faults [14], multi-machine power system stabilizer optimisation [48, 49], and fault prediction for manned spacecraft [50]. In addition, research is focusing on control system fault detection, including fault diagnosis and sliding mode fault tolerant control [51], the fuzzy control design of fault-tolerant control systems [52], fuzzy state-feedback control for photovoltaic systems [53], cooperative adaptive fuzzy output feedback control [54], fuzzy finite element methods [55], adaptive fuzzy descriptor sliding mode for uncertain nonlinear systems [56], indirect adaptive fuzzy control schemes [57], and output feedback control for large-scale nonlinear systems [58]. Fuzzy decision tree has also been studied, in addition to a T-S fuzzy neural network [59], type-2 fuzzy expected regression classifier pre-set [60], two-dimensional T-S fuzzy systems [61], and fuzzy decision tree for fault classification [62]. Some related applications in other fields include the following: a client financial risk tolerance model [63], a dynamic maintenance planning framework based on fuzzy TOPSIS and FMEA [64], fuzzy predictive temperature control [65], the implementation of fuzzy logic to simulate a process of inference on sensory stimuli [66], intelligent grading of dried figs [67], petroleum demand forecasting for Taiwan [68], and an iterative fuzzy method for group decision making [69]. However, such studies have not considered the influence of multiple factors and fault big data, simultaneously. The present authors also conducted a few studies on the relationship between reliability and multi-factor influence [42–44]. From the above studies, we can see that the processing of fault data using fuzzy mathematics is a better approach.

1.2.3

System Function Structure Analysis and Factor Space

It is universally acknowledged that systems are composed of components and subsystems. Under the premise of this known basic feature, a key issue in system design

Introduction

5

is how to guarantee that a system can accomplish a specific function. A system is an organic whole formed by components or subsystems following a certain layout; a system design simply aims at solving the layout problem. In general, a system design follows a top-down approach, starting from the whole, and through a certain function decomposition finally ends at the component or subsystem level. A top-down design can meet the requirements of the system for the function, but it is difficult to determine whether the designed system is optimal. This problem is considered based on the system function and economy. Another problem that occurs is a system is made of some specific components or subsystems, and the system functions change with the component functions; however, if the system is inaccessible, how can the system be studied or even copied? The above problems can be summarised as the problem of the system function structure. In other words, if the characteristics of the basic component functions and system functions are known, how can the structure of the system functions be learned through? Apparently, an internal structure is an equivalent structure and may not be a physical one. A system function’s structure analysis is an effective tool for recognition and understanding of a system, and research has also been conducted in this area. For example, a robust performance analysis was carried out using the structured uncertainty of a system [70]. A transfer expansion method has also been formulated based on a systems analysis of the structure modelling [71]. In addition, the vulnerability of a structure has been analysed using systems theory [72, 73], and an identification of a structure was carried out based on the multi-resolution of a fuzzy system [74]. System uncertainty has also been analysed using feedback and Bayesian networks [75]. Moreover, the behaviours of a critical infrastructure have been enabled using a system method [76]. The system of a solar inverter was analysed based on network control [77], and the reliability of a repairable system was analysed based on a goal-oriented methodology [78]. In addition, a method for analysing a system structure was put forward using a reachable effect factor, and has been applied in communication networks [79, 80]. In addition, a rock drill damping system was used in a double damping chamber [81]. A decoupling control strategy was studied by considering the current dynamic characteristics of the system function [82]. The reliability, availability, and maintainability of the system were analysed using graphical models, a safety assessment, and a risk analysis by considering the uncertainty of the system [83]. The reliability of non-repairable systems was analysed using dynamic fault tree with priority AND gates [21]. The reliability of a CNC system was analysed based on field failure data in an operating environment [84]. Although these studies have achieved satisfactory results in various fields, most of the aforementioned studies have adopted a top-down approach, by which the simplest isomer equivalent to the system cannot be obtained. By contrast, the system function structure is a type of complex reasoning process. Few related studies have been conducted in this area, however, because technical engineers have been unable to develop a rigorous logic and mathematical reasoning system. The inward analysis of the system factor structure previously put forward by the authors was also inadequate in terms of reasoning and logic. Therefore, cooperation among professional scholars and mathematicians is desperately needed.

6

Space Fault Tree Theory and System Reliability Analysis

Any system has its own specific structures, environment, and function. Such structures and environment are internal and external influences, whereas a function is the outcome. Adjusting a structure to improve its function and exploring the structure from the perspective of such function are considered a type of inward analysis, and are a complex scientific problem. Factor space theory provides a convenient platform for an analysis of the system function structure. The factors and background structure of fuzzy subsets have also been proposed [85]. In addition, the factor space was put forward to describe different concepts [86], and a monograph was published for a mathematical theory on knowledge representation [87]. The authors proposed factor spaces and factor databases, as well as a factorial analysis [88, 89]. An algorithm of knowledge mining has also been studied in the factor space [90]. In addition, an improved method for the factor analysis has also been developed [91]. Moreover, factor spaces used in data science have been proposed [92], and extension-enveloping feedback has been studied [93], in which the factors are first determined, and the factors that affect the function are then found. Such factors are categorised into structure factors (the structure and the environment), f1,…, fn, and result factors (function), g. These two sets of factors constitute the factor space that can be analysed based on the structure of the function; the factor space is denoted by (U; F = {f1,…, fn}; g), where U is the change domain of the system. If the system is specified as a mine, for example, the location is unchangeable, its content may change at any time. In this case, U is a set made up by the different conditions of this mine. Although time is a continuously changing variable, discrete observations are adequate; in addition, the property observation values of the system for various structure factors are recorded to form a set of sample points of the factor space. If mines of a specific type in different locations are considered, then U is the change in domain of this type of mine and is composed of different conditions of these mines during different periods of time. In factor space theory, only the domain and observation method of such factors as the conditions and results are needed, and a set of sample points and a factor analysis table are obtained. Using the above-mentioned elements, the function structure is analysed. A factor analysis table is comparable to an information system decision table in a rough set but differs in meaning and analysis method. Factor analysis methods [89] have also been put forward. According to a factor analysis table, a causal relationship tree from the structure factors to the result factors can be given. In [1], a factor logic method is used. According to the factor analysis table, a reasoning formula from the structure factors to the result factors can be obtained. Chapters 4 and 5 describe a combination of both methods used to develop a theory for obtaining a structure from a function, namely, the inward analysis of the system factor structure and the function structure analysis. The structure obtained is not necessarily the real structure but a logical one and is a bridge to a real structure. For a switch system, the logical structure is the real structure of the system; the function structure analysis is then equivalent to a structure analysis of a logic circuit. However, an ordinary logic circuit assignment is allowed, and n components can have 2n combinations. In [26], factor logic is assigned and is restricted by the background relationship, which is a new theory of minimization with a broader use. From the perspective of safety system engineering,

Introduction

7

safety and faults are two factors related directly to the system function, and can be viewed as two rival assignments (phases) under the same factor g. The phase space of g is X(g) = {safety, fault}; in a switch system, safety and faults can be converted into X(g) = {on, off}. The structure factors affecting an on–off switch system are the circuit components of the system. Each component Ai determines a factor fi, which is a map defined on universe U, which is still denoted as fi:U→X(fi), where fi(u) refers to the phase of system component Ai at a specific moment, and the phase space is X (fi) = {on, off}. The purposes of chapters 4 and 5 lay in an integration of the inward analysis of the system factor structure in the field of engineering and function structure analysis based on factor space. The inward analysis of the system function structure in the SFT is promoted as an intelligent reasoning method.

1.2.4

System Reliability and Influencing Factors

Reliability research is a crucial part of safety system engineering; it is based on the theory of system engineering and is an important theoretical basis of safety science. It currently plays a vital role in production and in various aspects of life, such as industry and mining, transportation, medicine, and the military. In addition, particularly in complicated fields related to life property and strategic significance, reliability research is becoming increasingly important. However, there are some errors and other problems occurring in research conducted on system reliability. Existing studies have focused on the internal structure of the system and the reliability of the components and have tried to ensure system reliability by improving component reliability and optimizing system structure. In fact, the components are made up of physical materials, and their physical, mechanical, electrical, and other properties are not static in different environments [45]. The functionality of a component that performs a function depends primarily on its work environment after the component is made. As the reason for this, the basic properties of the component material may be different in different working environments, and the parameters are basically fixed after a component design. This results in a change in the basic properties of the components when they are working in a changing environment. Even a simple, single-function system is also made up of several components. If these component reliabilities change with the working environment, the change in system reliability is quite complex. The above phenomenon is presented herein and should not be ignored. According to the French aerospace defence website, as the most fatal flaw of F-35 fighters, if the fuel exceeds a certain temperature, the fighter will not function properly [45]. In this case, it seems that the influence of environmental factors (such as temperature, humidity, air pressure, and working time) on the fighter reliability during the design phase has not been considered properly. This will cause a fighter jet to frequently fail during actual use and affect the design function. Therefore, it is necessary to study system reliability under the influence of environmental factors and study the degree of influence. Finally, we must study the functional adaptability of the entire system in terms of multi-factor changes. Some problems need to be solved in this regard.

8

Space Fault Tree Theory and System Reliability Analysis

The main issue of reliability research is how a system fails, and what causes a fault. At present, in the studied relationship between fault probability and influencing factors, most issues are expressed in a quantitative form. The problem here is that the fault occurrence is affected by multiple factors with both explicit and implicit coexistences. It is difficult to distinguish the correlations among various factors. However, substantial amounts of fault data arise in engineering, and redundant and missing data occur. Existing safety system engineering methods have difficulty solving this problem. Specifically, the causal analysis through computer reasoning for big data has yet to appear in safety system engineering. However, big data remain a challenge to system reliability research [94, 95]. Existing system reliability analysis methods are unsuitable for the above phenomena and problems are found in big data and the multi-factor influence analysis. It is difficult to find effective information regarding fault big data and the use of an intelligent scientific method for exploring the logical relationship between reliability and influencing factors. There are contradictions between the traditional reliability analysis method and fault big data with multiple factors. Although some related studies have been conducted [95–98], they remain insufficient. Such research is of great significance to the analysis of system reliability under the influence of big data and multiple factors.

1.2.5

Cloud Model and Similarity

For an actual enterprise, most safety assessments are executed using a safety check list. The advantages are a simple preparation and convenient operation. However, the effect of using a safety check list is largely dependent on the experience and knowledge of the implementer. However, in academics, some practical solutions to the problems of a safety assessment have been put forward. The solutions can be divided into the following two categories: an assessment index and the corresponding algorithm, and a reasoning method based on data and information. An assessment index and the corresponding algorithm have been studied. In addition, the safety level of a highway tunnel was studied based on the AHP-Extension synthesis method [99], and a metro tunnel construction method was selected based on the rapid graphical assessment [100]. Moreover, the SFT was put forward and applied [1]. The size-set domain and cut-set domain were also defined based on SFT [101]. The maintenance method used in system reliability was studied based on SFT [102]. The selection of a construction method was studied for a subway tunnel based on a safety consideration [103]. This type of safety assessment solution weakens the function of the index system and is mainly focused on improvements to the existing assessment algorithms. The algorithm used should adapt to qualitative and quantitative coupling, and to subjective and objective coupling. This requires a continuous development of the algorithm in complex directions, and practical engineering applications are gradually becoming difficult to achieve. The reasoning method has been researched based on data and information. The safety of a coal mine disaster has also been analysed based on the factor analysis [104]. A system reliability assessment method was given based on SFT [105]. The connectivity reliability of a directed acyclic network was also studied considering

Introduction

9

nodes and lines [106]. Numerous failures of repairable systems have been studied based on an imperfect repair model [107]. The decision criterion discovery of system reliability has also been studied [108]. The factor importance distribution is definite in the CSFT [109]. System safety classification decision rules have also been obtained when considering the scope of the attribute [110]. An accident chain model was studied based on CBM and human error [111]. The above methods are based on the given qualitative and fuzzy information to separate the relevant knowledge in terms of information, simplification, inference, and relevance. Knowledge is used to determine the safety of the system being evaluated. However, the data are limited, making it difficult to form the reasoning process of an evidence chain, which is limited by the reason method, and is rarely used. The following studies were carried out on a semantic analysis and similarity. Similarity measurements have been conducted on qualitative multi-scale locations [112]. A semantic trajectory was measured using a multidimensional similarity [113]. A new recommender framework has also been put forward, which combines semantic similarity fusion and bi-cluster collaborative filtering [114]. Noise suppression for dual-energy CT through a penalized weighted least-squares optimization with similarity-based regularization has also been considered [115]. In addition, an adaptive colour similarity function was obtained that is suitable for image segmentation and a numerical evaluation [116]. Linguistic vector similarity measures have been given and applied to linguistic information classification [117]. Although the above methods have been used, it remains difficult to solve the problem of natural disaster evaluations, which are not clear and lack proper understanding. The analysis of this type of disaster is mainly based on field investigations by experts and is determined through on-site consultations. Thus, the problem is due to the differences in the individual knowledge and experience of experts, who may have a different understanding of the same problem. The assessment results obtained by many experts may be consistent with each other, but some results may also show inconsistencies, and some experts have even reached contrasting conclusions. It is, therefore, important to evaluate the safety information based on qualitative and quantitative characteristics, fuzziness, randomness, and uncertainty, among other factors. Such factors cannot be ignored when conducting a data analysis. A cloud model can be used to express different types of information and data. A cloud model [118] is a model of uncertainty transformation from qualitative language to quantitative value. It was proposed by LI, a Chinese academic, in the 1990s; furthermore, its application has been recognised and promoted. The cloud model is as a qualitative and quantitative transformation of the uncertainty model and can fully embody the concept of language fuzziness and randomness. The cloud model is an effective tool to realise a qualitative and quantitative transformation, and the cloud generator is the key to its practical application [119]. The cloud model construction and the characteristics of the parameter settings meet the requirements of dealing with the above information problem. For a cloud model and its digital characteristics, U is set as a quantitative domain with exact values, C is a qualitative concept of U if the quantitative value x 2 U, x is a stochastic realization of the qualitative concept C, and μ(x) 2 [0,1] is a

10

Space Fault Tree Theory and System Reliability Analysis

membership degree of x to C, with random number μ having a stable tendency, denoted by l : U ! ½0; 1, 8x 2 U , x ! lðx Þ. Then, the distribution of x on domain U is called a cloud model (CM), as denoted by C(x), where x is called a cloud droplet (x, μ(x)). The digital characteristics of the CM reflect the quantitative characteristics of the qualitative concept, which are the expectation (Ex), entropy (En), and hyper entropy (He), denoted by C (Ex, En, He). Here, Ex expresses the most representative qualitative concept value in the domain space and reflects the central value of this space. In addition, En is a comprehensive measure of fuzziness and randomness. On the one hand, En reflects the range of droplets in the domain space, which can be accepted by the qualitative concept; on the other hand, En can reflect the degree of discreteness of cloud droplets. He reflects the uncertainty of En and the degree of condensation of cloud droplets in the domain space. The larger the He is, the greater the thickness of the cloud droplets [120]. Cloud Model Generator: The algorithm or hardware used to generate the cloud droplets is called the cloud model generator (CMG), which includes the forward cloud model generator (FCMG), reverse cloud model generator (RCMG), and an X and Y condition cloud model generator. The FCMG can obtain the range and distribution of quantitative data in qualitative information, with the forward and direction characteristics. Input: one-dimensional qualitative characteristic parameters (Ex, En, He) and the number of cloud droplets, N. Output: qualitative value of N cloud droplets and the certainty µq of the concept, q = 1,…, N. RCMG is used to transform numerous exact values into the appropriate qualitative linguistic values, with reverse and indirection characteristics. The reverse cloud algorithm has been improved [33]. Input: quantitative value of N droplet samples. Output: characteristic parameters (Ex, En, He) used to express the droplet samples estimated based on a qualitative concept. A more specific discussion for CM is provided in [120]. At present, the cloud model has been the subject of numerous studies and applications. The uncertainty of artificial intelligence has also been put forward [118]. The classification of the stability of the surrounding rock is given using the cloud model [119]. Prof. LI proposed a new understanding of the cloud model [121]. A cloud-model based failure mode and effect analysis were constructed for prioritization of the failures of a power transformer in a risk assessment [122]. A flatness intelligent control system was realised and studied using a hybrid of MATLAB and LabVIEW with a T-S cloud inference neural network [123]. The DG location and capacity optimization of several objectives were considered based on cloud theory adapted GA [124]. Software and applications of spatial data mining have also been described [125]. Beyond Turing machines, a method was obtained through cloud computing [126]. This research ranges from Turing machine intelligence to collective intelligence [127]. Unsupervised communities in social tagging networks have been detected using the cloud model [120]. A new subjective trust model was also given based on the cloud model [128]. The attribute importance of a classification is

Introduction

11

computed based on the cloud model [129]. User preferences were modelled in folksonomy for personalized searches [130]. A novel multi-population evolution model and its application were given to a global numerical optimization based on the cloud algorithm [131]. Image segmentation was studied based on the cloud concept analysis [132]. Knowledge discovery of the classification was given based on the cloud model and genetic algorithm [133]. An image segmentation approach was constructed based on the histogram analysis utilizing the cloud model [134]. Moreover, the trusted cloud was computed with secure resources and data colouring [135]. In addition, the evaluation process may include this situation, and many experts on the same problem have the same general opinion, but for randomness and fuzziness opinions vary. From the perspective of the cloud model, a number of expert opinions can be described as the same number of cloud models for a particular problem. These clouds can represent a point with little difference between qualitative concepts (basically the same position between clouds). However, the range and thickness of the cloud droplets are different (the cloud shapes are different). These clouds may overlap, may be contained, or may be separated. Thus, how to simplify the multiple clouds of the same concept is also a problem to be solved.

1.2.6

Object Classification and Similarity

The factor space theory was created by Prof. Wang and has been developed to a certain extent. The authors have studied issues in the field of safety system engineering, which is too difficult to handle with traditional methods [42, 43, 108, 136–139]. For example, in an investigation of the safety of an electrical system, an operator of a system on system safety indicated that when the system is below 12 °C, numerous faults will occur. After working for 70 or 80 days, there will be many faults, and the system will become seriously unstable. This example has certain characteristics. It is a multi-factor decision system. The expression of a factor is a domain value, that is, a factor is a range value. The basic data are derived from the experience of different operators, and different working times and environments make their evaluation of the system different based on their experiences. Evaluations of basic data have fuzziness. How can we determine the confidence of these evaluations and can they support each other? There are, of course, some methods that can deal with the evaluation semantics of objects. Studies have been carried out in the world on evaluating the semantic analysis and similarities [112–117]. The authors have conducted some studies on the reasoning method based on data and evaluation semantics [105, 106, 109–111]. The above methods are based on the given qualitative and fuzzy information to separate the relevant knowledge in the information, simplification, inference, and relevance. The knowledge is used to determine the safety of the system being evaluated. Because the above methods cannot analyse the influence of multiple domain attributes on the clustering analysis under the evaluation semantics, it is extremely difficult for these methods to deal with an example having the above characteristics, and it is difficult to satisfy the problem analysis using these characteristics.

12

1.3

Space Fault Tree Theory and System Reliability Analysis

Deficiency of System Reliability

(1) In this study, we pay particular attention to the internal structure of the system and the reliability of the components, and try to ensure the reliability of the system by improving the reliability of the components and optimizing the system structure. However, this does not consider the fact that all components are made up of physical materials, and their physics, mechanics, and electrical properties are not invariable in different environments. The functionality of the system component that performs a function depends largely on its working environment. The reason for this is that the basic properties of the component materials may be different under different working conditions, whereas the relevant parameters are basically fixed when designing the components. This leads to a change in the ability to perform specific functions as the basic properties change in the changing environment, and from changes in the component reliability. Furthermore, even a single-function system is composed of several components. If the reliability change in each component varies with the working environment, the reliability change in the system will be quite complicated with the change in the working environment. The above facts exist and should not be ignored. (2) The main topics of reliability research are how the system fault occurs and what causes a fault. At present, the research results mostly reflect the relationship between fault probability and influencing factors, and most of these relations are expressed in quantitative form. Others reflect the causal relationship between the cause of the fault and the fault itself. However, the main problem is that fault is affected by many factors, and both explicit and implicit factors coexist, and it is difficult to distinguish between the correlation factors. In addition, the actual field fault data are large, and there are redundancy and missing data. It is difficult to solve this with existing safety system engineering methods. In particular, the causal analysis of computer reasoning for large data has not yet appeared in safety system engineering, and the causality between the reliability and the influencing factors cannot be analysed. (3) A large amount of monitoring data will be formed during the daily use and maintenance of the system. They have large data levels, such as safety records, records of failures or accidents, and routine maintenance records. These data often reflect the functional operation characteristics of the system in an actual situation. These characteristics can generally be expressed as the number of system parameters in a certain working environment or when there are failures or accidents. It can be seen that these monitoring data can not only reflect the influence of the working environment factors on the reliability of the system but also have a large amount of data, which can be comprehensively used to analyse the reliability of the system. Therefore, the methods of adapting to big data should be studied to integrate these fault data characteristics into system reliability analysis process and results.

Introduction

13

(4) Design behaviour based on the design phase of the system does not consider the different environments that may be encountered during system working; thus, some problems will be encountered during the use of the system. In particular, the systems used in aerospace, deep-sea, and underground engineering will encounter extreme working conditions. Therefore, it is not safe to study the reliability of the whole system from the viewpoint of design. This problem can be summarised as the inverse analysis of system reliability structure. That is, the basic unit reliability characteristics of the system and the reliability characteristics of the system are known, and how to deduce the internal reliability structure of the system is difficult. Of course, the internal structure is an equivalent structure, which may not be the real physical structure. (5) The characteristics of the components in the system are different owing to the different responses of physical materials to different environments. Environmental changes lead to changes in the material properties and functional reliability of the components. The system is composed of these components, and the system reliability is also changing under a different environment. This is a common phenomenon. However, from another perspective, the change in environmental factors is a reason. The change in reliability or fault probability of the system or component is a result, that is, the fault probability changes with the changes in the environment. The environmental influence is taken as the effect of the system, and the change in the fault probability as a response of the system constitutes a movement system for reliability. Then, the stability of the fault probability and reliability cloud can be discussed. A stable reliability or fault probability is an important condition for the system to be put into practice. If the reliability or fault probability changes significantly, the system function cannot be controlled. This is a key issue to describe and analyse the reliability system using the stability theory of a movement system. (6) A structured representation is used to represent the fault process. The structure of the components and system in the SFT is represented with a classical tree topology, which is a tree topology in system engineering. Although a part of the actual system fault process can be expressed, a more general fault process is caused by the interweaving and interaction of different factors. It is obvious that the tree topology cannot be used. The interaction between factors should be represented by the generalised net topology. A tree topology can be regarded as a special case of a net topology. Therefore, it is the key to fully solve the representation relationship between the component and the system and the representation relationship between the factors and the reliability. The above phenomena and problems can be attributed to the current system reliability analysis method, which is unsuitable for the analysis of large data and multiple factors. It is difficult for existing methods to mine effective information in fault big data, and it is difficult to carry these data characteristics effectively for the system reliability analysis. These problems are a contradiction between a traditional reliability analysis method and the adaptability of the fault big data emergence and multi-factor analysis.

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1.4

Space Fault Tree Theory and System Reliability Analysis

Main Research Content

Chapter 1: Introduction. This chapter introduces the basic research status related to this book, the main goal, and the innovations provided herein. Chapter 2: Continuous Space Fault Tree. For the fault tree analysis, a basic event probability is often complicated. The probability is not constant and can even be represented by a function. To analyse the system reliability and related characteristics, we represent the probabilities of the basic events by functions. The variables of the function are n influencing factors on the basic events. We extend the top event probability from the constant value to an n + 1 dimensional space considering n influencing factors, and the probability is the n + 1st dimension. Further research on the n + 1 dimension space with related mathematical methods is described, along with a transform of the system probability analysis into the problem of mathematics. The mentioned ideas constitute the continuous space fault tree (CSFT). In a CSFT, the component fault probability distribution instead of the basic event probability and the system fault probability distribution instead of the top event probability are applied. We study an electrical system fault probability distribution and explain the related construction process. The main factors influencing the system are the working temperature c and working time t. This chapter constructs the fault probability distribution of the components and system, along with the probability importance distribution and criticality importance distribution of the components. With a partial derivation of the system fault probability distribution by c and t, we study the change trend of the fault probability. The optimal replacement schemes of components and the scheme considering the cost are obtained. Chapter 3: Discrete Space Fault Tree and Fuzzy Structured Element. Some fault data in an actual system operation have strong discretization and big data characteristics. Meanwhile, the external factors affect system reliability, and a change in factors may lead to a change in system reliability. At present, the methods used in safety system engineering lack the ability to process multi-factor influence and fault big data simultaneously. However, these are general problems to be resolved, which an actual system reliability analysis must face. In order to solve such problems, based on the discrete space fault tree (DSFT), the fuzzy structured element method is introduced to construct the fuzzy structured element discrete space fault tree (EDSFT). The method can analyse the multi-factor influence on the system reliability using the DSFT, and applying the fuzzy structured element (E) to denote the discrete characteristics of fault big data. The results of an EDSFT with E can preserve the characteristics of the original fault data distribution and lay the foundation for the analysis of fault big data. Such research is particularly suitable for the analysis of system reliability under the influence of fault big data and multiple factors. Chapter 4: Inward Analysis of System Factor Structure. This chapter introduces the framework of inward analysis of system factor structure (IASFS), which is used for understanding the changes in system reliability during the process of change in the environmental factors in which the system operates. This chapter carries on the logic inference to obtain a rule according to its corresponding relationship, and the

Introduction

15

system equivalent response structure is then formed. The combination of factors that fit the system operation is obtained. IASFS is part of the SFT framework. According to the characteristics of IASFS, the 01 space fault tree is defined, and its structure is represented by both the graph method and the table method. The IASFS method includes item-by-item analyses (IIA) and the classification reasoning method (CRM) applied in the table method. This chapter mainly defines and describes the IIA and the CRM and gives a mathematical description. It is indicated that IASFS is a human–computer cognitive body for the inward analysis of the system factor structure. The IIA and the CRM can obtain an equivalent response structure, which is the same as the response of the analysis system to the changes of the working environmental factors. Chapter 5: Function Structure Analysis. To analyse the system function structure, the system function structure analysis theory is put forward based on SFT. The factor space is more suitable for describing the cognition process of intelligence science than the qualitative Cartesian space. Based on factor logic, a justice system of the function structure analysis was built. It is proved that the system function logical structure is a minimal disjunctive normal formula from the system function analysis. The relationship is discussed between the classification reasoning method of the inward analysis of the system structure and the function structure analysis. The process of the inward analysis of the system function structure is also realised by the function structure analysis theory. The original classification reasoning method is enhanced to the level of logic mathematics. The system function structures of both incomplete information and complete information are analysed using this method. Chapter 6: System Reliability with Influencing Factors. The system reliability is related to the factors that affect system reliability in a multi-factor influencing environment. At the same time, the cumulative fault data increase rapidly as the system operates. These big data magnitude fault data also contain system reliability characteristics. Therefore, it is urgent to solve the problem of a system reliability analysis suitable for big data and multiple factors. This research adopts the method of SFT and factor space (FS). The SFT can solve system reliability analysis under the influence of many factors. The FS has the ability of large data processing and logic analysis. The logical relationship between the reliability and influencing factors is studied, and we also put forward a corresponding analysis method. Based on existing research, the SFT theory is improved; the improved theory can be used to infer a causal relationship, reduce the factor dimensions, and compress fault data. Chapter 7: Cloudization Space Fault Tree. The system fault data are different from general monitoring data, which have large discreteness, randomness, and fuzziness, that is, uncertainty. The existing characteristic function is a continuous function with finite discontinuities. The function can be considered as a kernel function of the fault data, but it is difficult to express the uncertainty of the fault data. The cloud model (CM) is used to transform the characteristic function, such that it can express the uncertainty of data, called the cloudization characteristic function (CCF). The cloudization SFT (CLSFT) is constructed using the CCF; the former enables the relevant theory and methods of the SFT to express data uncertainty. Thus, the expression of the fault data characteristics and rules are more accurate. The construction process and rationality analysis of the CLSFT are given.

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Space Fault Tree Theory and System Reliability Analysis

The concepts of the SFT are reconstructed using the cloud model. The relationship between the component fault probability and the working time and temperature is studied, and the results reflect the discreteness, randomness, and fuzziness of the fault data to a considerable extent. Chapter 8: Cloud Similarity. In the actual safety assessment, the conclusions given by experts are usually vague and uncertain. The information obtained is difficult to determine from such data, and it is usually necessary to analyse, merge, and finally obtain simplified results. To realise the above process, a new semantic simplification method is proposed, which is based on the cloud model. This method will be used to express the opinions of experts with the cloud model, and numerous expert opinions with the same number of cloud models. According to the characteristics of the cloud model, multiple cloud models are built on the envelope, and integrated in the area between them. The proportion between the overlapping area and the integral area is considered to judge the similarity of the cloud models, namely the similarity of expert opinion, to simplify such opinions. Chapter 9: Clustering Analysis and Similarity. We study the influence of multiple domain attributes on the clustering analysis of objects based on FS. The representation method of the graphical domain attribute is proposed for the object, which is called an attribute circle. An attribute circle can represent infinite domain attributes. The similarity analysis of objects is based on the concept of the attribute circle, and the definition of a graphical similarity is transformed into a numerical similarity, and the clustering analysis method of object set is then studied and improved. Considering three types of graphical overlap, the analytic solution of a similarity is obtained for a numerical calculation. The reliability evaluation semantics of the actual electrical system are listed as the study objective, and the clustering analysis method and its improvement are applied.

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Chapter 2 Continuous Space Fault Tree In fault tree analysis, the probability of basic events is often complicated. The probability is not constant and can even be represented by functions. To analyse system reliability and related characteristics, we represent the probabilities of the basic events using functions. The variables of the function are n influencing factors on the basic events. We extend the top event probability from the constant value to an n + 1 dimensional space considering n influencing factors, and the probability is in the n + 1st dimension. Further research into the n + 1 dimensional space with related mathematical methods is performed, and the system probability analysis is transformed into a mathematical problem. The above ideas constitute the space fault tree (SFT). Specifically, according to the development process of the SFT, this chapter first discusses the continuous space fault tree (CSFT). In the CSFT, a component fault probability distribution instead of a basic event probability and a system fault probability distribution instead of top event probability are applied. We research the electrical system fault probability distribution and describe the related construction process. The main factors influencing the system are the working temperature c and working time t. We construct the three-dimension fault probability distribution of the components and system, and the probability importance and criticality importance of the components. With a partial derivation of the system fault probability distribution by c and t, we study the change trend of the fault probability. The optimal replacement schemes of the components and the scheme considering the cost are obtained. The results show that the CSFT is feasible and reasonable for analysing the fault probability of the system under the influence of multiple factors, and that it is suitable for research on the characteristics of the changes in system reliability. The fault probability distribution can be solved through mathematical methods and calculations, and other relevant concepts and their construction process are given. The chapter is divided into different sections. Section 2.1 describes the concepts of the CSFT. Section 2.2 describes the fault probability distribution. Section 2.3 details and importance distribution. Section 2.4 provides the system fault probability distribution trend. Section 2.5 provides a solution to the MTLα. Finally, Section 2.6 offers some concluding remarks. DOI: 10.1051/978-2-7598-2499-1.c002 © Science Press, EDP Sciences, 2020

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24

List of symbols Symbol T X1–5 t c Pidk ðxk Þ i d dk xk k A n Pi ðx1 ; x2 ; . . .; xn Þ PT ðx1 ; x2 ; . . .; xn Þ r Kj Ig ðiÞ Igc ðiÞ PTdk TLai a TLa aT MTLα

2.1

Meaning Electric system Five components Working time Working temperature Characteristic function ith component d 2 ft; cg influencing factors Factor name Value of dk Fault rate Temperature range Number of factors Component fault probability distribution System fault probability distribution Number of cut sets Event in the jth cut set Probability importance distribution Criticality importance distribution System fault probability distribution trend Component replacement cycle Given probability System replacement cycle Given probability of system fault probability Optimal replacement cycle

Concepts of CSFT

For example, herein, we analyse a simple electric system composed of diodes. The working conditions of a diode are affected by numerous factors, including the working time t and working temperature c. These two factors are considered for an electrical system. A fault tree is shown in figure 2.1, where T represents the system and X1–5 represents five components (diodes) [1, 2]. A CSFT was developed from a classical fault tree, and the definitions below are based on the relevant concepts of a fault tree. The occurrence of a basic event is equivalent to a component fault, and the occurrence of a top event is equivalent to a system fault. Definition 2.1. Space fault tree: The component fault probability is not constant, and is determined by n influencing factors, called a component fault probability distribution, i.e. the CSFT method. In this chapter, the system structure can be simplified as T ¼ X1 X2 X3 þ X1 X4 þ X3 X5 .

Continuous Space Fault Tree

25

FIG. 2.1. – Fault tree of electrical system.

Definition 2.2. Influencing factors of a component fault: The factors cause a change in the component fault probability. In this chapter, t is the working time and c is the working temperature. Definition 2.3. Characteristic function of component fault probability: Under the influence of a single factor, the function shows a change in the component fault probability with a change in the influencing factor. This can be an elementary function or piecewise function, as denoted by Pidk ðxk Þ, where i is the ith component, d 2 ft; cg is an influencing factor, dk is the name of the factor, and xk is the value of dk. In this case, the time characteristic function of the ith component is þ1 , Pit ðtÞ ¼ 1  ekt , and the temperature characteristic function is Pic ðcÞ ¼ cosð2pc=AÞ 2 where k is the fault rate, and A is the range of temperature. Definition 2.4. Component fault probability distribution: Under the influence of n factors, the change in the component fault probability is caused by a superposition of the influences of multiple factors, where n influencing factors are mutual independent variables, and the component fault probability distribution is function  Q  value, denoted by Pi ðx1 ; x2 ; . . .; xn Þ, Pi ðx1 ; x2 ; . . .; xn Þ ¼ 1  nk¼1 1  Pidk ðxk Þ where n is the number of influencing factors. In this case, Pi ðt; cÞ ¼ 1  ð1  Pit ðtÞÞð1  Pic ðcÞÞ. Definition 2.5. System fault probability distribution: Such a distribution is obtained after a structure simplification of a fault tree and is shown in an n + 1 dimensional space. The distribution ‘ Q is changed with n factor changes, denoted by PT ðx1 ; x2 ; . . .; xn Þ ¼ rj¼1 i2Kj Pi ðx1 ; x2 ; . . .; xn Þ, where r is the number of cut sets, and Kj is the event in the jth cut set. In this case, PT ðt; cÞ ¼ P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 . Definition 2.6. Probability importance distribution: The change in degree of the system fault probability in an n + 1 dimensional space is caused by a change in the

Space Fault Tree Theory and System Reliability Analysis

26

ith component fault probability with a change in the n factors, denoted by T ðx1 ;x2 ;...;xn Þ Ig ðiÞ ¼ @P @Pi ðx1 ;x2 ;...;xn Þ . In this case, the probability importance distribution of X1 is T ðt;cÞ Ig ð1Þ ¼ @P @P1 ðt;cÞ ¼ P2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 .

Definition 2.7. Criticality importance distribution: The change rate of the system fault probability in an n + 1 dimensional space is caused by a change in the ith component nÞ fault probability with a change in n factors, denoted by Igc ðiÞ ¼ PPTi ððxx11;x;x22;...;x ;...;xn Þ  Ig ðiÞ. In P1 ðt; cÞ this case, the criticality importance distribution of X1 is Igc ð1Þ ¼  Ig ð1Þ ¼ PT ðt; cÞ 1ð1P1t ðtÞÞð1P1c ðcÞÞ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P4 P1 P3 P4 P5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5

ðP2 P3 þ P4  P2 P3 P4  P3

P 4 P 5  P2 P 3 P 5 þ P 2 P 3 P 4 P 5 Þ .

Definition 2.8. System fault probability distribution trend: A partial derivation of the system fault probability based on the influence factor dk is denoted by PTdk ¼ @PT ðx1@x;xk2 ;...;xn Þ. In this case, the system fault probability distribution trend of t is

PTt ¼ @PT@tðt;cÞ.

Definition 2.9. Component replacement cycle: To ensure that the component fault probability is continuously less than a given probability under the changing factors, a component must be replaced by a certain cycle, called a component replacement cycle, denoted by TLai , where, a is the given probability. Definition 2.10. System replacement cycle: Here, aT is the given probability of a system fault, and a system replacement cycle is a set of replacement schemes, which ensures that system fault probability is less than aT . The replacement scheme includes the replacement cycles of all components under the influencing factors, denoted by TLa ¼ fTLai g. If TLai is maximum, then TLa is the optimal replacement cycle MTLα. The CSFT proposal is beneficial to the study of the relationship between the system reliability and its influence factors. It is beneficial to deeply learn the internal relationship between the system reliability changes and the factor changes.

2.2 2.2.1

Fault Probability Distribution Component Fault Probability Distribution

The component fault probability distributions of X1–5 in a system are influenced by t and c; that is, a component fault probability distribution Pi ðt; cÞ is a function of t and c. With either t or c changing, the component fault probability distribution will change. According to the ‘or’ logic, Pi ðt; cÞ can be represented as equation (2.1) [2]:    ð2:1Þ Pi ðt; cÞ ¼ 1  1  Pit ðtÞ 1  Pic ðcÞ ;

Continuous Space Fault Tree

27

where Pi ðt; cÞ, Pit ðtÞ, and Pic ðcÞ should first be calculated. According to existing research, based on hypothesis, the fault component is non-repairable in the system, and the reparability of the system is realised by changing the fault components. Then, Pit ðtÞ can be deemed as the fault probability of a non-repairable component [3], and the component should be replaced if the fault probability reaches 0.9999, as shown in equation (2.2). Pit ðtÞ ¼ 0:9999 ¼ 1  ekt ; kt ¼ 9:2103

ð2:2Þ

For Pic ðcÞ, the components will maintain their function under the proper working temperature, and will break down above or below the proper temperature. The rule obeys the cosine curve approximately, as shown in equation (2.3). Pic ðcÞ ¼

cosð2pc=AÞ þ 1 2

ð2:3Þ

In fact, different types of components have different ranges of working time t and working temperature c. The working ranges are given, and Pit ðtÞ and Pic ðcÞ within these ranges are calculated using equations (2.2) and (2.3), as shown in table 2.1. From table 2.1, Pit ðtÞ and Pic ðcÞ are piecewise functions and are not continuous within the entire range. The piecewise functions within the entire range are shown in table 2.2. The component fault probability distribution of X1–5 can be constructed according to table 2.2 and equation (2.1). Figure 2.2 shows the component fault probability distribution of X1–5. In figure 2.2, the component fault probability distributions are different based on the influences of t and c. There are two or three ranges within the distribution whose fault probability clearly decreases owing to the replacement components. As the significance of the above analysis, if we want to ensure that the component fault probability is smaller than a given threshold value α, then it can be achieved by

TAB. 2.1 – Information of Pit ðtÞ and Pic ðcÞ of X1–5. Parameter

Range of factors

k

Component Pit ðtÞ/100%

Pic ðcÞ/100%

X1

Range of time t/day 0–50

Range of temperature c/°C 0–40

0.1842

1  e0:0:1842t

X2

0–70

10–50

0.1316

1  e0:1316t

X3

0–35

0–50

0.2632

1  e0:2632t

X4

0–60

5–45

0.1535

1  e0:1535t

X5

0–45

0–45

0.2047

1  e0:20478t

cosð2pc=40Þ þ 1 2 cosð2pðc10Þ=40Þ þ 1 2 cosð2pc=50Þ þ 1 2 cosð2pðc5Þ=40Þ þ 1 2 cosð2pc=45Þ þ 1 2

Given range

0–100

0–50

Out of this range Pit ðtÞ = 0 or 1

Out of this range Pic ðcÞ = 1

Space Fault Tree Theory and System Reliability Analysis

28

TAB. 2.2 – Functions of Pit ðtÞ and Pic ðcÞ within the entire range. Parameter X1 X2 X3

X4

Functions component 

1  e0:1842t ; t 2 ½0; 50 P1t ðtÞ ¼ 1  e0:1842ðt50Þ ; t 2 ð50; 100  1  e0:1316t ; t 2 ½0; 70 P2t ðtÞ ¼ 1  e0:1316ðt70Þ ; t 2 ð70; 100 8 < 1  e0:2632t ; t 2 ½0; 35 P3t ðtÞ ¼ 1  e0:2632ðt35Þ ; t 2 ð35; 70 : 1  e0:2632ðt70Þ ; t 2 ð70; 100  1  e0:1535t ; t 2 ½0; 60 P4t ðtÞ ¼ 1  e0:1535ðt60Þ ; t 2 ð60; 100

X5 P5t ðtÞ

Pic ðcÞ  cosð2pc=40Þ þ1

Pit ðtÞ

8 < 1  e0:2047t ; t 2 ½0; 45 ¼ 1  e0:2047ðt45Þ ; t 2 ð45; 90 : 1  e0:2047ðt90Þ ; t 2 ð90; 100

; c 2 ½0; 40 2 P1c ðcÞ ¼ 1; c 2 ð40; 50  1; c 2 ½0; 10 P2c ðcÞ ¼ cosð2pðc10Þ=40Þ þ 1 ; c 2 ð10; 50 2 þ1 P3c ðcÞ ¼ cosð2pc=50Þ ; c 2 ½0; 50 2

8
> > > > a^ ¼ y  ^bx > > > PN > < ðxq  xÞðyq  yÞ ^b ¼ q¼1 ð3:3Þ PN 2 > > q¼1 ðxq  xÞ > > > > N > 1 X > 2 > > ^ r ¼ ðyq  y^Þ2 : n  2 q¼1 An algebraic interpolation method is a polynomial fitting to find the gravity centre of discrete data in a two-dimensional plane. These kernel points are used as the basic data of the construction interpolation function, and the Kernel Function (KF) constructed is shown in equation (3.4). ^f ðxÞ ¼ a^a x a þ a^a1 x a1 þ    þ a^1 x þ a^0

ð3:4Þ

xq ; dðxq ÞÞjq ¼ 1; . . .; n  2} is the basic Here, dðxq Þ is the value of ^f ðxq Þ, and {ð^ data used to construct another interpolation function, as shown in equation (3.5). ^ xðxÞ ¼ ^bb x b þ ^bb1 x b1 þ    þ ^b1 x þ ^b0

ð3:5Þ

A triangular or normal structured element E is selected, and the fitting function is finally obtained, as shown in equation (3.6). ~ ^ FðxÞ ¼ ^f ðxÞ þ xðxÞE

ð3:6Þ

Algebraic interpolation is used to construct the ECF. In [4], the significance of FVF linearly generated by E is given, as shown in definition 3.7, taking equa^ tion (3.6) as FVF. Here, ^f ðxÞ is the KF of the constructed FVF, and xðxÞ is a positive function that reflects the degree of uncertainty of y at any point x of the interpolation function. The larger the difference is in the distribution of the discrete points, the greater their value. In addition, E is regular and symmetric, as shown in figure 3.6. ^ To obtain ECF, the representations of ^f ðxÞ, xðxÞ, and E described above are first ^ determined in DSFT. Here, f ðxÞ represents a central location of the discrete data points. Referring to the example in section 3.3, the KF of Xi considering t and c in the system is expressed as equations (3.7) and (3.8), respectively: ^f t ðtÞ ¼ P t ðtÞ ¼ 1  ekt ; kt ¼ 9:2103; i i

ð3:7Þ

^f c ðcÞ ¼ P c ðcÞ ¼ cosð2pðc  c0 Þ=AÞ þ 1 ; i i 2

ð3:8Þ

^ where xðxÞ indicates the fluctuation degree of discrete data points. According to the discretization characteristics of the data in figures 3.4 and 3.5, four curves can be

Discrete Space Fault Tree

71

FIG. 3.6 – FVF linearly generated by E. obtained with two types of KF forms, namely, the up and down envelopes between the discrete data points. Here, ^fit ðtÞ and ^fit ðtÞ for t, and ^fic ðcÞ and ^fic ðcÞ for c, are U

D

U

D

shown in figures 3.7 and 3.8, respectively. In addition, ^fit ðtÞ and ^fic ðcÞ are the up U

U

envelopes of discrete data distribution, and ^fit ðtÞ and ^fic ðcÞ are the down envelopes of D

D

discrete data distribution. ^ Figures 3.7 and 3.8 are schematic diagrams, where xðxÞ can be constructed using the corresponding KF form. The values of f ðEÞ in the two figures are modified by E, ^ and f ðEÞ is a monotonic function within [−1, 1]. Here, xðxÞ indicates the degree of fluctuation in the data, and the up and down envelopes can be used to represent these points, as shown in equations (3.9) and (3.10), respectively. 8 > ^ xðtÞ ¼ ^fit ðtÞ  ^fit ðtÞ; 1  x\0jf ðEÞ ! EðxÞ > > < U ^ ti ðtÞ ¼ ^ ð3:9Þ x xðtÞ ¼ 0; x ¼ 0jf ðEÞ ! EðxÞ; > > t t ^ ^ > ^ ðtÞ  f ðtÞ; 0\x  1jf ðEÞ ! EðxÞ xðtÞ ¼ f : i i D

8 > ^ xðcÞ ¼ ^fic ðcÞ  ^fic ðcÞ; 1  x\0jf ðEÞ ! EðxÞ > > < U ^ ci ðcÞ ¼ ^ x xðtÞ ¼ 0; x ¼ 0jf ðEÞ ! EðxÞ > > c c ^ ^ > ^ xðcÞ ¼ f ðcÞ  f ðcÞ; 0\x  1jf ðEÞ ! EðxÞ : i i

ð3:10Þ

D

Here, E can be a triangular structured element or a normal structured element. According to the actual data or experimental data collected using the DSFT, it is recommended to apply a normal structured element. For image clarity, however, figures 3.7 and 3.8 use a triangular structured element.

72

Space Fault Tree Theory and System Reliability Analysis

FIG. 3.7 – ECF with E for t.

FIG. 3.8 – ECF with E for c. The above FVFs generated by E linearly for t and c in the DSFT are shown in equations (3.11) and (3.12), respectively. The equations can be used as the ECF of Xi to construct the EDSFT. ~ t ðtÞ ¼ ^f t ðtÞ þ x ^ ti ðtÞE F i i

ð3:11Þ

Discrete Space Fault Tree

73 ~ c ðcÞ ¼ ^f c ðcÞ þ x ^ ci ðcÞE F i i

ð3:12Þ

We can see that the transformation process given above conforms to the conditions in definition 3.8; thus, the MF of equations (3.11) and (3.12) can be expressed as equations (3.13) and (3.14). y  ^fit ðtÞ lF~ t ðtÞ ðyÞ ¼ E i ^ ti ðtÞ x

!

y  ^fic ðcÞ lF~ c ðcÞ ðyÞ ¼ E i ^ ci ðcÞ x

ð3:13Þ ! ð3:14Þ

Example 3.1. Let E be a regular E. For FVF, let ~f ðxÞ ¼ sin2 x þ 1 and g~ðxÞ ¼ ~ ¼ ~f ðxÞ þ g~ðxÞE. ex þ 1 be generated by E linearly, and calculate the MF of FðxÞ 2 x ~ ~ Solution: FðxÞ ¼ f ðxÞ þ g~ðxÞE ¼ sin x þ 1 þ ðe þ 1ÞE    y  sin2 x þ 1 MF : lFðxÞ ~ ðyÞ ¼ E ex þ 1

The processes given above complete the representation of the CF in a DSFT with E, that is, an ECF. The use of an ECF is one of the important methods for obtaining the CF under discrete data. The above processes have fully explained the ECF construction. The EDSFT-related concepts and methods are specifically constructed as follows.

3.4.2

E-Component Fault Probability Distribution

According to definition 2.4, the component fault probability distribution (CFPD) is  Q  Pi ðx1 ; x2 ; . . .; xn Þ ¼ 1  nk¼1 1  Pidk ðxk Þ . In this example, n = 2 and dk 2 {c,t} using t and c instead of xk , and thus Pi ðt; cÞ ¼ 1  ð1  Pit ðtÞÞð1  Pic ðcÞÞ. The fuzzy structured element component fault probability distribution (ECFPD) is as shown in equation (3.15). ~i ðx1 ; x2 ; . . .; xn Þ ¼ 1  F

n Y k¼1

ð1  Fidk ðxk ÞÞ ¼ 1 

n Y

^ di k ðxk ÞEÞ ð3:15Þ ð1  ^fidk ðxk Þ  x

k¼1

~i ðt; cÞ ¼ 1  ð1  ^f t ðtÞ  x ^ ti ðtÞEÞ In the example, the ECFPD is F i c c ^ i ðcÞEÞ, and its expansion is as shown in equation (3.16). ð1  ^fi ðcÞ  x

Space Fault Tree Theory and System Reliability Analysis

74

~i ðt; cÞ ¼ 1  ð1  ^f t ðtÞ  x ^ ti ðtÞEÞð1  ^fic ðcÞ  x ^ ci ðcÞEÞ F i t t c c t ^ i ðtÞE þ ^fi ðcÞ þ x ^ i ðcÞE  ð^fi ðtÞ þ x ^ ti ðtÞEÞ  ð^fic ðcÞ þ x ^ ci ðcÞEÞ ¼ ^fi ðtÞ þ x ^ ti ðtÞE þ ^fic ðcÞ þ x ^ ci ðcÞE  ^fit ðtÞ^fic ðcÞ  ^fit ðtÞx ^ ci ðcÞE  ^fic ðcÞx ^ ti ðtÞE ¼ ^fit ðtÞ þ x t c ^ i ðtÞE x ^ i ðcÞE x t c ^ ^ ^ ti ðtÞ þ x ^ ci ðcÞ  ^fit ðtÞx ^ ci ðcÞ  ^fic ðcÞx ^ ti ðtÞÞE ¼ fi ðtÞ þ fi ðcÞ  ^fit ðtÞ^fic ðcÞ þ ðx t c 2 ^ i ðcÞE ^ i ðtÞx x ð3:16Þ Here, E  E in equation (3.16) is equal to E 2 . Any operation that produces the FVF linearly with E has its own MF, as shown in definition 3.8 and example 3.1. For more detailed information on the MF of the FVF, please refer to Guo’s study [4, 15].

3.4.3

E-System Fault Probability Distribution

According to definition Q the system fault probability distribution (SFPD) is ‘ 2.5, PT ðx1 ; x2 ; . . .; xn Þ ¼ rj¼1 basic event i2Kj Pi ðx1 ; x2 ; . . .; xn Þ. Because of T ¼ X1 X2 X3 þ X1 X4 þ X3 X5 , PT ðt; cÞ ¼ P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 . The fuzzy structured element system fault probability distribution (ESFPD) is as shown in equation (3.17). ~T ðx1 ; x2 ; . . .; xn Þ ¼ F

r a

Y

~i ðx1 ; x2 ; . . .; xn Þ F

ð3:17Þ

j¼1 basic event i2Kj

In the example, the ESFPD is as indicated in equation (3.18). ~T ðt; cÞ ¼ F ~2 F ~3 þ F ~4 þ F ~5  F ~2 F ~3 F ~4  F ~3 F ~4 F ~5 ~1 F ~1 F ~3 F ~1 F ~1 F F ~ ~ ~ ~ ~ ~ ~ ~ ~  F1 F2 F3 F5 þ F1 F2 F3 F4 F5

3.4.4

ð3:18Þ

E-Probability Importance Distribution

According to definition 2.6, the probability importance distribution (PID) is T ðx1 ;x2 ;...;xn Þ In the example, the PID of X1 is Ig ð1Þ ¼ Ig ðiÞ ¼ @P @Pi ðx1 ;x2 ;...;xn Þ . @PT ðt;cÞ @P1 ðt;cÞ

¼ P2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 . In addition, the fuzzy structured element probability importance distribution (EPID) is as shown in equation (3.19). ~T ðx1 ; x2 ; . . .; xn Þ @F I~g ðiÞ ¼ ~i ðx1 ; x2 ; . . .; xn Þ @F

ð3:19Þ

Discrete Space Fault Tree

75

In the example, the EPID is as shown in equation (3.20). ~T ðt; cÞ @F ~4  F ~2 F ~3 F ~2 F ~2 F ~3 þ F ~3 F ~4  F ~4 F ~5  F ~3 F ~5 þ F ~3 F ~4 F ~5 ~2 F I~g ð1Þ ¼ ¼F ~1 ðt; cÞ @F

3.4.5

ð3:20Þ

E-Criticality Importance Distribution

According to definition 2.7, the criticality importance distribution (CID) is P1 ðt;cÞ nÞ c Igc ðiÞ ¼ PPTi ðxðx11;x;x22;;x ;;xn Þ  Ig ðiÞ, and in the example, the CID of X1 is Ig ð1Þ ¼ PT ðt;cÞ  1ð1P t ðtÞÞð1P c ðcÞÞ ðP2 P3 þ P4  P2 Ig ð1Þ ¼ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P14 P1 P3 P41P5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 Þ. The criticality importance distribution of the fuzzy structured element (ECID) is as shown in equation (3.21).

~i ðx1 ; x2 ; . . .; xn Þ F  I~g ðiÞ I~gc ðiÞ ¼ ~ FT ðx1 ; x2 ; . . .; xn Þ

ð3:21Þ

In the example, the ECID of X1 is as shown in equation (3.22).    ^ t1 ðtÞE 1  ^f1c ðcÞ  x ^ c1 ðcÞE 1  1  ^f1t ðtÞ  x I~gc ð1Þ ¼  ~2 F ~3 þ F ~4 þ F ~5  F ~2 F ~3 F ~4  F ~3 F ~4 F ~5  F ~2 F ~3 F ~5 þ F ~2 F ~3 F ~4 F ~5 ~1 F ~1 F ~3 F ~1 F ~1 F ~1 F ~1 F F   ~3 þ F ~3 F ~4  F ~4 F ~5  F ~3 F ~5 þ F ~3 F ~4 F ~ ~2 F ~4  F ~2 F ~3 F ~2 F ~2 F F ð3:22Þ

3.4.6

E-System Fault Probability Distribution Trend

According to definition 2.8, the system fault probability distribution trend (SFPDT) for dk is PTdk ¼ @PT ðx1@x;xk2 ;...;xn Þ. In the example, the SFPDT for t is

PTt ¼ @PT@tðt;cÞ. This definition involves partial differentials, that is, the partial derivative of the FVF is generated linearly by E. The definition of the derivative for the FVF is shown in definition 3.9, and if gðx; yÞ is derivable by x on DX, then ~f ðxÞ is derivable on D, and its derivative function is given in equation (3.23). d ~f ðxÞ @gðx; yÞ ð3:23Þ ¼ dx @x y¼E ^ For equation (3.6), let ^f ðxÞ and xðxÞ be derivable on X, and according to ~ 0 ðxÞ ¼ ^f 0 ðxÞ þ x ^ 0 ðxÞE is then a monotonic bounded function on y in definition 3.9, F [−1, 1] and derivable. This indicates that the differential of the FVF can be transformed into the operations of the sum and multiplied between E and the partial derivative of the ordinary function. Therefore, ECFs such as those in equations (3.11) and (3.12) are also derivable.

Space Fault Tree Theory and System Reliability Analysis

76

The fuzzy structured element system fault probability distribution trend (ESFPDT) for dk is as shown in equation (3.24). ~ ~ dk ¼ @ FT ðx1 ; x2 ; . . .; xn ; yÞ j ; dk 2 fx1 ; x2 ; . . .; xn g; k ¼ 1n F y¼E T @xk

ð3:24Þ

In the example, the ESFPDT for t is as shown in equation (3.25). ~ ~ t ¼ @ FT ðt; c; yÞ j F y¼E T @t

ð3:25Þ

The fuzzy structured element component fault probability distribution trend is the same as the ESFPDT.

3.4.7

E-Component Domain Importance

The study domain is the scope of environmental change in which the system works. In the example, the domain is 0–100 days at 0 °C–50 °C. The CDI differs from the importance distributions, i.e. the component probability importance distribution (CPID) and the component criticality importance distribution (CCID). According to definitions 2.6 and 2.7, the CPID and CCID are space distributions. In the example, t and c form a space distribution surface to express the importance of the different values of t and c. The CPID and CCID represent the component importance in a certain state (t, c) in a working environment. They represent a component importance in a point of the study domain. The benefits of the CPID and CCID are accurate and especially for importance. In addition, they show the trends and changes in different states (t, c). As a disadvantage, the component importance in the study domain is insufficient. Moreover, they cannot consider the full importance of a component within the whole study domain. From another perspective, the CPID and CCID have a space distribution characteristic, and their dimensions are the working environment factors. The CDPI and CDCI obtain the full characteristic value of space distribution, which is a specific value. The CDPI and CDCI are not parallel to the CPID and CCID but are based on them. The CPID and CCID should be calculated first using a certain method (such as an integral) to obtain the CDPI and CDCI. The CDPI; ZIg ðiÞ; and CDCI, and ZIcg ðiÞ are as shown in equations (3.26) and (3.27), respectively. Z Z Z @PT ðx1 ; x2 ; . . .; xn Þ dx1 dx2 . . .dxn ZIg ðiÞ ¼  ð3:26Þ x1 x2 xn @Pi ðx1 ; x2 ; . . .; xn Þ Z Z ZIcg ðiÞ ¼

Z 

x1

x2

@PT ðx1 ; x2 ; . . .; xn Þ  Ig ðiÞdx1 dx2 . . .dxn xn @Pi ðx1 ; x2 ; . . .; xn Þ

ð3:27Þ

Discrete Space Fault Tree

77

According to P1–5 given in section 2.1, the domain importance of X1 obtained by equations (3.26) and (3.27) is calculated as follows: Z 50  C Z 100 d @PT ðt; cÞ dtdc ZIg ð1Þ ¼ @P1 ðt; cÞ 0 C 0d Z 50  C Z 100 d ¼ ðP2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 Þdtdc 0 C

0d

¼ 224:2744 Z ZIcg ð1Þ ¼ Z ¼

50  C 0 C 50  C 0 C

Z

100 d

0d 100 d

Z

@PT ðt; cÞ  Ig ð1Þdtdc @P1 ðt; cÞ

   1 1  1  P1t ðtÞ 1  P1c ðcÞ B C @ P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Adtdc 0

0d

ðP2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 Þ ¼ 228:3632

The remaining calculations of the CDPI are the same. After calculating the component importance in the study domain, for the CDPI, X1 = 224.2744, X2 = 8.3744, X3 = 174.4662, X4 = 120.7763, and X5 = 94.8089, sorted as X1 > X3 > X4 > X5 > X2; and for the CDCI, X1 = 228.3626, X2 = 8.7888, X3 = 182.2204, X4 = 121.2681, and X5 = 94.7605, sorted as X1 > X3 > X4 > X5 > X2. The fuzzy structured element component domain importance (ECDI) is the same as the above representation. However, an integral operation for E is needed. The definition of the integral operation of the FVF generated by E is given in definition 3.10. Z Z ~f ðxÞdx ¼ gðx; EÞdx ð3:28Þ D

Z

~f ðxÞdx ¼ D

D

Z D

gðx; yÞdxjy¼E

ð3:29Þ

According to equations (3.28) and (3.29), the fuzzy structured element component domain probability importance (ECDPI) and fuzzy structured element component domain criticality importance (ECDCI) are as shown in equations (3.30) and (3.31), respectively. Z Z Z ~T ðx1 ; x2 ; . . .; xn ; EÞ @F ~  ð3:30Þ dx dx . . .dxn ZIg ðiÞ ¼ ~ i ðx1 ; x2 ; . . .; xn ; EÞ 1 2 x1 x2 xn @ F c Z~Ig ðiÞ ¼

Z Z

Z 

x1

x2

~T ðx1 ; x2 ; . . .; xn ; EÞ @F  I~g ðiÞdx1 dx2 . . .dxn ~i ðx1 ; x2 ; . . .; xn ; EÞ xn @ F

ð3:31Þ

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78

The ECDPI (Z ~ Ig ð1Þ) and ECDCI (Z I~gc ð1Þ) for X1 are as follows: Z 50  C Z 100 d ~ @ FT ðt; c; EÞ Z~Ig ð1Þ ¼ dtdc ~1 ðt; c; EÞ @F 0 C 0d Z 50  C Z 100 d   ~2 F ~3 þ F ~3 F ~4  F ~4 F ~5  F ~3 F ~5 þ F ~3 F ~4 F ~5 dtdc ~4  F ~2 F ~3 F ~2 F ~2 F F ¼ 0 C

Z

Z

0d

~T ðt; c; EÞ @F  ~Ig ð1Þdtdc ~1 ðt; c; EÞ @ F 0d    1 0 ^ t1 ðtÞE 1  ^f1c ðcÞ  x ^ c1 ðcÞE 1  1  ^f1t ðtÞ  x Z 50  C Z 100 d B C BF ~1 F ~3 F ~1 F ~1 F ~1 F ~1 F ~2 F ~3 þ F ~4 þ F ~5  F ~2 F ~3 F ~4  F ~3 F ~4 F ~5  F ~2 F ~3 F ~5 þ F ~2 F ~3 F ~4 F ~5 C ¼ Adtdc @ ~1 F  0 C 0d   ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ F2 F3 þ F4  F2 F3 F4  F3 F4 F5  F2 F3 F5 þ F2 F3 F4 F

c Z~Ig ð1Þ ¼

50  C

100 d

0 C

It should be noted that for the entire study domain, the CF is likely to be discontinuous, and the solution is as shown in section 2.1. For the ECF, the solution is similar. The partial derivative of the FVF has the same form as the traditional definition of the functional derivative in [4].

3.4.8

E-Factor Importance Distribution

According to definition 3.4, the component factor importance distribution (CFID) is ;x2 ;...;xn Þ FIdi k ðx1 ; x2 ; . . .; xn Þ ¼ @Pi ðx1@x , and the system factor importance distribution k

(SFID) is FIdTk ðx1 ; x2 ; . . .; xn Þ ¼ @PT ðx1@x;xk2 ;...;xn Þ. Then, the fuzzy structured element component factor importance distribution (ECFID) and fuzzy structured element system factor importance distribution (ESFID) are as shown in equations (3.32) and (3.33), respectively. ~i ðx1 ; x2 ; . . .; xn ; yÞ @F dk jy¼E ð3:32Þ F~Ii ðxk Þ ¼ @xk ~T ðx1 ; x2 ; . . .; xn ; yÞ @F dk F~IT ðxk Þ ¼ jy¼E @xk

ð3:33Þ

Then, the ECFID and ESFID for t and c are as follows:  ~i ðt; cÞ  ~ t ðt; yÞ  @F dF t c ~ ~ ¼ 1  Fi ðc; y Þ  i FIi ðtÞ ¼ @t dt y¼E 0           1 ~1 F ~1 F ~3 F ~1 F ~1 F ~2 F ~3 ~4 ~5 ~2 F ~3 F ~4 ~3 F ~4 F ~5 @ F @ F @ F @ F @ F þ þ   C ~T ðt; cÞ B @F t @t @t @t @t @t C B F~IT ðtÞ ¼ ¼B C @ @t ~1 F ~1 F ~2 F ~3 F ~5 Þ @ðF ~2 F ~3 F ~4 F ~5 Þ A @ðF  þ @t @t y¼E   ~i ðt; cÞ ~ c ðc; yÞ @ F d F c t i ~ ðt; yÞÞ  F~Ii ðcÞ ¼ ¼ ð1  F i @c dc y¼E

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79

        0 ~ ~ ~  ~1 F ~3 F ~1 F ~1 F ~4 ~5 ~2 F ~3 F ~4 ~3 F ~4 F ~5 1 @ F1 F2 F3 @ F @ F @ F @ F þ þ   C @c @c @c @c @c :    C ~1 F ~1 F ~2 F ~3 F ~5 ~2 F ~3 F ~4 F ~5 A @ F @ F þ  @c @c y¼E

~T ðt; cÞ B @F c ¼B F~IT ðcÞ ¼ @ @c

3.4.9

E-Factor Joint Importance Distribution

According

to

definition

3.5,

@Pil ðx1 ;x2 ;...;xn Þ

the d

CFJID

is

d

FIi 1l ðx1 ; x2 ; . . .; xn Þ ¼

@PTl ðx1 ;x2 ;...;xn Þ

; l ¼ 1  n. The SFJID is FIT1l ðx1 ; x2 ; . . .; xn Þ ¼ @x1 @x2 ...@xl ; l ¼ 1  n. The fuzzy structured element component factor joint importance distribution (ECFJID) and fuzzy structured element system factor joint importance distribution (ESFJID) are as shown in equations (3.34) and (3.35), respectively. ~ l ðx1 ; x2 ; . . .; xn ; yÞ @F d1l i ð3:34Þ F~Ii ðx1 ; x2 ; . . .; xn Þ ¼ @x1 @x2 . . .@xl y¼E @x1 @x2 ...@xl

~ l ðx1 ; x2 ; . . .; xn ; yÞ @F d1l T F~IT ðx1 ; x2 ; . . .; xn Þ ¼ @x1 @x2 . . .@xl y¼E

ð3:35Þ

The ECFJID and ESFJID for t and c are thus as follows: ~ 2 ðt; cÞ  d F ~ c ðc; yÞ d F ~ t ðt; yÞ @F t;c i i ~ ~ ¼   i FIi ðt; cÞ ¼ FIi ðc; tÞ ¼ @t@c dc dt y¼E t;c F~IT ðt; cÞ

3.5

~ 2 ðt;cÞ @F

T ¼ F~IT ðc; t Þ ¼ @t@c 0 2 1 ~1 F ~2 F ~3 Þ ~1 F ~4 Þ ~3 F ~5 Þ ~1 F ~2 F ~3 F ~4 Þ @ ðF @ 2 ðF @ 2 ðF @ 2 ðF þ @t@c þ @t@c  @t@c @t@c @ A ¼ 2 ~ ~ ~ ~ 2 ~ ~ ~ ~ 2 ~ ~ ~ ~ ~ @ ðF1 F3 F4 F5 Þ @ ðF1 F2 F3 F5 Þ @ ðF1 F2 F3 F4 F5 Þ   þ @t@c @t@c @t@c y¼E

Conclusions

In this chapter, the DSFT modified by the fuzzy structured element (E) was proposed. The DSFT having the capability to express the characteristics of discrete fault data under fault big data is called an EDSFT. The method for establishing an EDSFT using the ECF was also proposed. Based on the concepts and methods of the DSFT, E was introduced, and the significance of the EDSFT was described. Here, E is used to represent the analysis results of the EDSFT. The final results obtained using EDSFT with E can preserve the confidence that the distribution of the original fault data is retained. The

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concepts and methods of the EDSFT described in this chapter include the fuzzy structured element characteristic function (EFC), fuzzy structured element component fault probability distribution (ECFPD), fuzzy structured element system fault probability distribution (ESFPD), fuzzy structured element probability importance distribution (EPID), fuzzy structured element criticality importance distribution (ECID), fuzzy structured element system fault probability distribution trend (ESFPDT), fuzzy structured element component domain importance (ECDI), fuzzy structured element factor importance distribution (EFID), and fuzzy structured element factor joint importance distribution (EFJID). The processes of these concepts are built using E, and the final EDSFT is established. Here, E represents the discrete characteristics of the fault data and remains unchanged during the EDSFT operation. In addition, E can be analysed in the final results, and the confidence interval and confidence degree of the results are obtained. The EDSFT with E can preserve the characteristics of the original fault data and lay the foundation for the analysis of fault big data. The study can provide an effective method for an analysis of the fault data of an actual system, particularly for fault big data and the system reliability analysis under the influence of multiple factors. More detailed questions can be found in document [19].

References [1] Cui T.J., Ma Y.D. (2016) Discrete space fault tree construction and application research, J. Syst. Sci. Math. Sci. 36, 1753. [2] Cui T.J., Li S.S., Ma Y.D., et al. (2017) Research on trend of failure probability in DSFT based on ANN derivation, App. Res. Comput. 34, 449. [3] Cui T.J., Ma Y.D. (2016) Inaccurate reason analysis of the factors projection fitting method in DSFT, Syst. Eng. Theory Pract. 36, 1340. [4] Guo S.C. (2004) Principle of fuzzy mathematical analysis based on structural element theory. Northeastern University Press. [5] Wu C.X., Xue X.P. (2002) Advances in the analysis of fuzzy valued functions, Fuzzy Syst. Math. 16, 1. [6] Luo C.Z. (1994) Introduction to fuzzy sets. Beijing Normal University Press. [7] Chang S.L., Zadeh L.A. (1972) On fuzzy mapping and control, IEEE Trans. SMC. 2, 30. [8] Dubois D., Prade H. (1980) Systems of linear fuzzy constraints, Fuzzy Sets Syst. 17, 37. [9] Dubois D., Prade H. (1982) Towards fuzzy differential calculus, Int. J. FSS. 8, 1. [10] Mizumoto M., Tanaka K. (1976) Algebraic properties of fuzzy numbers. International Conference on Cybernetics and Society, Washington, DC. [11] Dubois D., Prade H. (1978) Operations on fuzzy numbers, Int. J. Syst. Sci. 9, 613. [12] Nahmisa S. (1978) Fuzzy variables, Int. J. Fuzzy Sets Syst. 1, 97. [13] Wang J.K. (2010) Reliability analysis and application of coal face production system based on structural element method. Liaoning Technical University, Fuxin, p. 12. [14] Guo S.C. (2002) Structural element method in fuzzy analysis (I), (II), J. Liaoning Tech. Univ. 21, 670. [15] Yue column (2011) Development and application of fuzzy structured element theory. Liaoning Technical University, p. 6. [16] Cui T.J., Ma Y.D. (2016) The construction of fuzzy structured element characteristic function and the significance of structure elemented in DSFT, Fuzzy Syst. Math. 30, 144.

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[17] Cui T.J., Ma Y.D. (2016) SFT concept reconstruction and its significance based on Fuzzy structured element, App. Res. Comput. 33, 1957. [18] Cui T.J., Ma Y.D. (2016) The definition of components area important degree in SFT and fuzzy structured element representation, Acta Anal. Funct. Appl. 18, 413. [19] Cui T.J., Li S.S. (2019) Study on the construction and application of Discrete Space Fault Tree modified by Fuzzy Structured Element, Cluster Comput. 22, 6563.

Chapter 4 Inward Analysis of System Factor Structure Operating systems can be found in the environment. Changes in environmental factors may affect some working states of a system. Thus, what environmental factors affect a system? What ranges of environmental factors have an effect on a system, and what kind of effect do they have? Finding a certain combination of such factors to allow a system to maintain the ideal conditions is a principal issue that needs to be studied. The state of a system differs under different combinations of working environmental factors. For example, the reliability of the system may be different when considering the temperature, humidity, and pressure. For the problems mentioned above, an analysis of the system response structure based on the known environmental factors and system response characteristics, which is an inward analysis of the system factor structure, is needed. Because of the unknown internal structure of the system, this can only be studied from the response of system to outside environmental factors. At present, such research is rare. Some representative papers can be found in the literature [1–5]. However, an analysis of the system equivalent response structure has yet to be conducted. This chapter presents a framework for the inward analysis of the system factor structure under SFT. The purpose is to obtain an equivalent response structure consistent with the response of the system to the changes in the working environmental factors. This chapter mainly introduces descriptions of IIA and the CRM. The inward analysis of the system component structure is also provided. The analysis method is comparable to the inward analysis of the system factor structure. Please refer to previous relevant studies by the authors. This chapter is organised as follows: section 4.1 provides the inward analysis of the system factor structure. Section 4.2 describes human–machine cognition. Section 4.3 details the table method. Section 4.4 provides a classification of the different reasoning methods. Section 4.5 gives a mathematical description of the CRM. Section 4.6 describes an item-by-item analysis. Section 4.7 gives a mathematical description of IIA, and section 4.8 provides some concluding remarks. All proofs of the propositions are omitted owing to spatial constraints. DOI: 10.1051/978-2-7598-2499-1.c004 © Science Press, EDP Sciences, 2020

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List of symbols Symbols A1–AN Z(A1)–Z(AN) Z(T) N M • + m I im B bm B(HAb  þ ) HAb  þ S(I,B,R) R sm(I,B,R) xm S(D I,B,R)

4.1

Meaning N factors State values of the factors State of the system Number of factors Number of 0, 1 state combinations Operator representing the logic concept ‘and’ Operator representing the logic concept ‘or’ Number of items, m 2 ½1; 2N  Information table mth item in table I Reasoning table mth item in table B Logical reasoning part of B Logic form of all items of Z ðHAb Þ ¼ 1 joined by ‘and ’, and then joined with another factor by ‘or +’ Item-by-item analysis with a table system Rule mth item in table S Number of factor state 1 in item m Ascending order with xm

Inward Analysis of System Factor Structure

First, the related concepts are further explained. The CSFT is closer to a classical fault tree. It has a similar function in terms of concept and method as a classical fault tree, but was developed as a special method. The CSFT is a ‘white box’ method, which recognises the structure and component properties of the system, and allows the response behaviour of the system to be studied under an outside influence. The method allows studying from an internal system, followed by the response of the system to an external environment. Correspondingly, the DSFT does not need to recognize the internal structure or component properties of the system. Its research basis is the response of the system to changes in the outside environment, which is equivalent to a ‘black box’ method. The data sources are the actual monitoring data (such as safety checks, equipment maintenance records, or an accident investigation). Thus, the DSFT analysis of the system reliability, based on the monitoring data obtained from the response of the system to the external environment, is applied from the exterior to the interior of the system. From the internal system, the CSFT allows an understanding of the characteristics of the system reliability under different environmental factors. It was found that the working environment of the system is not conducive to the reliability of the system, and the purpose of guiding the production is reached. The DSFT determines the relationship between the environmental factors and the system reliability from

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85

the monitoring data. Although it does not analyse the specific structure of the system directly, DSFT analyses and determines the internal structure of the system through the response of the system to changes in external environmental factors. Application of the CSFT and DSFT is actually a cycle that provides both data and results. The above theory is described in [6–12]. Inward analysis of the system factor structure (IASFS) is one of the branches of the inward analysis of system structure (IASS); furthermore, the other branch is the inward analysis of system component structure (IASCS), as described in another paper. IASS, CSFT, and DSFT constitute the basic SFT framework. An IASS essentially belongs to the DSFT, but is extremely complex, and thus is given separately. The IASS attempts to understand the internal structure of a system from the response of the system to the outside environment. The IASCS is mainly used for an internal equivalent physical structure of system components. It can eventually form the simplest and most heterogeneous system with the same function as the original system. The IASFS is mainly used for the equivalent response structure of a system to analyse the response of the system to the outside environmental factors. It eventually forms the simplest system for the same response as that of the original system to changes to different factors. This chapter mainly introduces the IASFS [13,14]. First, a 01 fault tree space (01SFT) is defined. The 01SFT originates from a classical fault tree with two states. This concept is introduced to study the basic method of the IASFS, along with its features and mechanism. Using a simple morphology, the IASFS of the polymorphism is studied. The 01SFT is between the CSFT and DSFT; if there are few states, then 01SFT degenerates into the CSFT, which means the characteristic function described in section 2.1 has a limited discontinuity; if the state is infinite, then 01SFT degenerates to the DSFT, which means the state is discrete, and cannot be expressed by the characteristic function. Definition 4.1. 01SFT. The factors that affect the system response are the basic events, and the response of the system is the top event. The basic event is the working environmental factors that can change the system state; in a 01 state, the value of a factor is 0 or 1, which can express the range of factors. For example, the operating temperature range of the system is 0 °C–20 °C, and the system can have a factor of 0 at 0 °C–10 °C, and 1 at 10 °C–20 °C. This division is determined by the actual situation. However, because 01SFT can only represent two states, the range of factors can only be divided into two ranges. If all influencing factors (basic events) are represented by a 0 or 1, then these numbers can express the state of the working environment of the system. First, 01SFT is used to express the organic relationship between the changes in the environmental factors and the changes in the system state, and the equivalent response structure can then be obtained through simplification and reasoning. Two kinds of structure representation methods of the 01SFT are expressed, namely, a graph method (GM) and a table method (TM). The TM is based on a two-dimensional table structure for representing the 01SFT, and the system state is

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logically deduced by different factor conditions in the table, allowing the equivalent response structure to be obtained. The next chapter elucidates the IASFS with TM. There are currently two types of IASFS based on TM, namely, item-by-item analysis (IIA) and classification reasoning (CRM). For the GM, refer to [15].

4.2

Human–Machine Cognition

TM is essentially a type of representation of the correspondence relationship for 01SFT. The correspondence relationship is between the system state and the arrangement of the system factor state. Finally, the TM forms an information database. CRM and IIA can obtain the rules from this database, and then form the knowledge base. This process is the core of the inward analysis. The aforementioned IASFS process is a simple and concrete realization of human–machine cognition, which is for the inward analysis of the system factor structure. The following discusses the concept of human–machine cognition in factor space theory as quoted by Prof. Wang. What is human–machine cognition? Human–machine cognition is a system with a certain purpose and a certain cognitive function that receives network information. It is a software and hardware system involved in the monitoring, organization, management, and control of the system under human activities. An unmanned aerial vehicle (UAV) uses human–machine cognition, and is an aircraft controlled by software to investigate an enemy and engage in combat, while avoiding potential pilot loss. It has a cognitive function, allowing it to identify characteristics of the ground target and human beings. Its flights plans are adjusted by accepting network information. Its operational process requires human control. Although it is a type of hardware, the driving force is the software. A supermarket cash register does not have human–machine cognition. Because it only collects and records transactions, it does not have cognitive function. But if the cash register functions are expanded, adding print information, some basic algorithms of the factor space are added to it. The concept of a tight commodity and customer fashions are automatically extracted including causal reasoning rules. The knowledge is read and controlled by the sales manager or an expert. The market factors are human-made analysis combined with network information, understanding one another, improving management, and servicing customers; thus, the above-mentioned cash register system becomes human–machine cognition. There will be thousands of types of human–machine cognition. In addition, there are various industries currently applying human–machine cognition. According to the function, the target optimization (such as the development system) and the factors (such as the safety system) are balanced. There are several general rules of human–machine cognition. (1) Each specialised system structure must have a corresponding concept structure. If a human–machine cognition system masters the relevant concept structure, it will reach the level of an expert. Conversely, the same expert system established

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87

must have expert experience, grasping only the concept of the actual system structure, through which the concept of a cognitive unit can be established. (2) All human–machine cognition is built within a certain environment. The function of the cognitive body is to optimise or maintain the balance between the environmental factors and the system structure factor. The structure is generated for the adaptation function. The initiative performance of human– machine cognition is to adjust its structure (internal factor) for adaptation to the environment (external factor). (3) Each human–machine cognition must be updated with information flow from the Internet. It must have a mechanism for achieving an update, or it cannot survive. For each new sample point in the data flow, the background base should always be adjusted with a basic algorithm of the background when considering the factor space. This is a type of update mechanism. (4) In the process of building human–machine cognition, the problem of data ownership is the most difficult to avoid. Of course, this also needs to be demonstrated from a legal perspective. Human–machine cognition is a new development in factor space theory, and the relevant theory is represented in [16–23].

4.3

Table Method

The TM is a logical description, which is the logical relationship between the different combinations of working environmental factors that the system works within and the system response state, although the structure of the system is not known. With all environmental factors, the TM information table contains an enumeration of all combinations of all factor values (0, 1), which can be enumerated through all combinations. The structure of the TM information table is shown in table 4.1. As shown in table 4.1, A1–AN indicate that the system working environment has N factors that affect the system response state. Here, Z(A1)–Z(AN) and Z(T) have only two state values of ‘0, 1’. In addition, Z(A1)–Z(AN) represent the state values of the factors in the current environment, and Z(T) indicates the target state of the system, that is, normal success, i.e. 1, or fault, i.e. 0. The number of 0, 1 state combinations of N factors is M = 2N. Focus should be paid to the items in the structure information table, which should be orderly for employing IIA and the TAB. 4.1 – Structure information table of TM. Number 1 … M

Status A1 1

A2 0

… …

AN–1 0

AN 0

T 0

1

1

…

1

1

1

Logical reasons

System equivalent response structure (system factor structure).

Rule

…

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TAB. 4.2 – Factor state divide table. State value 1 0

Range of factors A1 range [a11, a12] [a12, a13]

A2 range … …

… … …

AN–1 range … …

AN range … …

CRM. The operation rules are based on the rules of a Boolean algebra operation, including mainly an ‘and operation’, ‘or operation’, and ‘absorption law’. The factor state division is shown in table 4.2, which is used with table 4.1. In table 4.2, taking factor A1 as an example, the range is [a11, a13]. Because 01SFT can only represent two states, there is only one segmentation point a12. Of course, this segmentation can be determined under concrete conditions, and the state of 1, 0 can also be determined. If the state values of factor A1 are in [a11, a12] at a certain time, then Z(A1) = 1.

4.4

Classification Reasoning Method

The system equivalent response structure is given, as shown in figure 4.1. There are five factors (A1–A5) in the environment that can affect the system reliability. Here, Z(A1)–Z(A5) denote their state values, i.e. only 0, 1. Here, T indicates the system, and Z(T) is the system state, which is only 0 or 1. A value of 1 indicates a normal system state, whereas a 0 indicates a faulty state. The factor state divide information for the example is shown in table 4.3. Table 4.3 assumes that the environmental factors affecting the system include the temperature, humidity, air pressure, electric field, and magnetic field. The change in factors is determined and divided. For Z(A1)–Z(A5), the full 0, 1 state enumeration is obtained in table 4.4, which has 32 information items. According to the 32 combinations of the working

FIG. 4.1 – System equivalent response structure.

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89

TAB. 4.3 – Factor state divide table of the example. State value 1 0

Range of factors Temperature A1 range [a11, a12] [a12, a13]

Humidity A2 range [a21, a22] [a22, a23]

Air pressure A3 range [a31, a32] [a32, a33]

Electric field A4 range [a41, a42] [a42, a43]

Magnetic field A5 range [a51, a52] [a52, a53]

environment status, the system working environment is transformed, and the system state is obtained under various combinations, which constitute the basic TM information, as shown in table 4.4. All state combinations of the factors are listed in table 4.4, and there are 25 = 32 items among the combinations of system states. Because a normal system operation is expected, then Z(T) = 1. First, the information in table 4.4 is classified. The items of Z(T) = 0 are excluded, and the items of Z(T) = 1, listed in table 4.5, are studied. Table 4.5 shows valid data table for the CRM. The items in the table are the desired system state. Because the number of factors is higher, the items are classified based on the order of the factors. Firstly, A1 in the structure of the system is studied, the items of Z(A1) = 0 in table 4.5 are removed, and the remaining items of Z(A1) = 1 are listed in table 4.6. The items where Z(T) = 1 and Z(A1) = 1 are listed in table 4.6, and then removing the items for which Z(A2) = 0, the remaining items are listed in table 4.7. Table 4.7 shows the reasoning process of the logical reasoning. The ‘•’ operator represents the logic concept ‘and’, namely two events that jointly occur in the operator simultaneously, the resulting state of which reaches 1. The ‘+’ operator represents the logic concept ‘or’, namely one of two events jointly occurring in that the operator, the resulting state of which also reaches 1. Considering item 17, Z(A1) = 1, Z(A2) = 1, and Z(A3) = 1, and under this situation, regardless of the other factor states, the system state is 1. The state of A1A2A3 can exist independently and can join the state of another factor with ‘or +’. ‘Invalid by item absorption’ indicates an operation according to the absorption law of Boolean algebra. Although items 27 and 31 can exist independently, the reasoning formulas are included by item 17, and thus they are absorbed by rule 17; finally, items 27 and 31 cannot become rules, which are indicated by ‘invalid by item absorption’. The rule obtained is A1A2A3+, as shown in table 4.7. The analysis presented above shows that the relationship among A1, A2, and A3 is ‘and’, and that the relationship between them and the other parts of the system is ‘or +’. Then, other parts of the system must have A4 or A5 or A4A5, and cannot contain A1A2A3. The items in table 4.7 are those removed from table 4.5, forming table 4.8, because the rules have been formed. Table 4.8 mainly analyses the state combination joined with ‘or’, which must contain A4 or A5 or A4A5. Comparing Z(A4) and Z(A5), the items of Z(A4) = 1 are relatively few, and thus the first study has A4. The items of Z(A4) = 0 in table 4.8 are removed to obtain table 4.9. Table 4.9 is analysed using the same method shown in table 4.7. Item 9 can form the rules through reasoning. Items 23 and 28 are absorbed by items 15 and 9,

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TAB. 4.4 – Basic information. Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

Status Z(A1) 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 1 1 1

Z(A2) 0 0 1 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1

Z(A3) 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 1 1 0 1 1

Z(A4) 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 0 1

Z(A5) 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1

Z(T) 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0

respectively. Item 29 is absorbed by the rule of table 4.10 and is invalid. The rule obtained is A1A4+ in table 4.9. Finally, the manner in which A5 exists in the system is studied. The items in table 4.9 are removed from table 4.8 to form table 4.10. The rule A3A5+ from table 4.10 is obtained via reasoning item 15. Items 24 and 26 are absorbed by item 15. As the main point, item 15 can absorb the reasoning results of item 29. Thus, regardless of where the rules are formed, they can absorb the global reasoning results.

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TAB. 4.5 – Information table under Z(T) = 1. Number

Status Z(A1) 1 0 1 0 1 0 1 1 0 1

9 15 17 23 24 26 27 28 29 31

Z(A2) 0 0 1 0 0 1 1 0 1 1

Z(A3) 0 1 1 1 1 1 1 1 1 1

Z(A4) 1 0 0 1 0 0 1 1 1 0

Z(A5) 0 1 0 1 1 1 0 1 1 1

Z(T) 1 1 1 1 1 1 1 1 1 1

TAB. 4.6 – Information table under Z(T) = 1 and Z(A1) = 1. Number

Status Z(A1) 1 1 1 1 1 1

9 17 24 27 28 31

Z(A2) 0 1 0 1 0 1

Z(A3) 0 1 1 1 1 1

Z(A4) 1 0 0 1 1 0

Z(A5) 0 0 1 0 1 1

Z(T) 1 1 1 1 1 1

TAB. 4.7 – Reasoning table under Z(T) = 1, Z(A1) = 1, and Z(A2) = 1. Number

Status

Logical reasoning

17 27

Z (A1) 1 1

Z (A2) 1 1

Z (A3) 1 1

Z (A4) 0 1

Z (A5) 0 0

Z (T) 1 1

31

1

1

1

0

1

1

A1A2A3+ exist independently A1A2A3A4+ exist independently, invalidated by item 17, absorption A1A2A3A5+ exist independently, invalidated by item 17, absorption

The results obtained after the above analysis are items 9, 15, and 17, which finally form three rules: A1A4+ , A3A5+ , A1A2A3+ . Based on these three rules, the combination of environmental factors (equivalent response structure) that the system maintains under normal working conditions is T = A1A4+ A3A5+

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TAB. 4.8 – Information table under Z(T) = 1 except Z(A1)Z(A2)Z(A3) = 1. Number

Status Z(A1) 1 0 0 1 0 1 0

9 15 23 24 26 28 29

Z(A2) 0 0 0 0 1 0 1

Z(A3) 0 1 1 1 1 1 1

Z(A4) 1 0 1 0 0 1 1

Z(A5) 0 1 1 1 1 1 1

Z(T) 1 1 1 1 1 1 1

TAB. 4.9 – Reasoning table under Z(T) = 1 and Z(A4) = 1 except Z(A1)Z(A2) Z(A3) = 1. Number

Status

Logical reasoning

9 23

Z (A1) 1 0

Z (A2) 0 0

Z (A3) 0 1

Z (A4) 1 1

Z (A5) 0 1

Z (T) 1 1

28

1

0

1

1

1

1

29

0

1

1

1

1

1

A1A4+ exist independently A3A4A5+ exist independently, invalidated by item 15, absorption A1A3A4A5+ exist independently, invalidated by item 9, absorption A2A3A4A5+ exist independently

TAB. 4.10 – Reasoning table under Z(T) = 1 except for Z(A1)Z(A2)Z(A3) = 1 and Z(A1)Z(A4) = 1. Number

Status

Logical reasoning

15 24

Z (A1) 0 1

Z (A2) 0 0

Z (A3) 1 1

Z (A4) 0 0

Z (A5) 1 1

Z (T) 1 1

26

0

1

1

0

1

1

A3A5+ exist independently A1A3A5+ exist independently, invalidated by item 15, absorption A2A3A5+ exist independently, invalidated by item 15, absorption

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FIG. 4.2 – System equivalent response structure obtained. A1A2A3. The equivalent response structure represented by 01SFT is shown in figure 4.2. According to table 4.3, the actual working conditions of the system are {[a11, a12] \ [a41, a42], [a31, a32] \ [a51, a52], and [a11, a12] \ [a21, a22] \ [a31, a32]}. When the system is working under these conditions, it is considered a normal operation.

4.5

Mathematical Description of Classification Reasoning Method

Definition 4.2. The classification reasoning method is a system that includes two types of tables: information table I and reasoning table B. Table 4.1 is a basic information table, and not one of these two table types because it is simply a data table formed by a combination of factor states, and is a grouping of information containing a number N of 0 or 1, which is composed of state Z ðA1N Þ of factor A1N . The basic information list includes the item number, combinations of factor status values, and the system state. Here, m 2 ½1; 2N  is the number of items, and N represents a number of factors affecting the environment where the analysed system operates. Definition 4.3. The basic event (A1–AN) represents the factors that affect the system state in the system environment. Definition 4.4. Information table I is the basic information table to maintain the target system status items, or to remove the items in the existing information table, which has been reasoned. Let im be an item in table I, and m be the number of items in the basic information table. Here, I = {im|Zm(T) = 1 \ im 62{the items have been

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reasoned}}. From the above definition, in the different reasoning times, the items {im} contained in I are different, and with the development of reasoning, the items are gradually reduced. Tables 4.3, 4.4, and 4.6 above are information tables. In general, the classification can be made according to the order of factors, forming table B. The formation method of a reasoning table is to remove the items in which the target factor state is Z(An) = 0, and the remaining items of Z(An) = 1 form a reasoning table for the analysis of An. Definition 4.5. The reasoning table B is a table of remaining items because of the target factor of Z(An) = 1. The structure includes the item number, combination of factor status, system state, and logical reasoning. Every item bm in table B includes im and logical reasoning. Here, B = {bm|Zm(T) = 1 \ Z (where An| An is the target factor) = 1 \ bm 62 {the items have been reasoned}}. The above tables 4.5, 4.7, and 4.8 are information tables. The logical reasoning part of B is represented by B(HAb  þ ), which represents a possible structure state. However, this item does not necessarily become the rule, because it may be absorbed by the established rules. In B(HAb  þ ), HAb  fA1 ; . . .; AN g, b is the number of items with Z(An) = 1, HAb  þ indicates that the logic form is all items of Z ðHAb Þ ¼ 1 joined by ‘and ’, and then joined with the other factor by ‘or +’. Definition 4.6. The rule obtained after the logical absorption law of B is calculated. It also expresses the final system structure. According to the laws of Boolean algebra, the operation ‘and’ is absorbed by the other ‘and’ operation that concludes less and some parameters. If bm1 < bm2 , Bm1 (HAb  þ ) (item m1 in table B) is absorbed by Bm2 (HAb  þ ), namely Bm1 (HAb  þ ) + Bm2 (HAb  þ ) = Bm1 (HAb  þ ). Thus, even though B exists independently, it cannot form the rules for judging the system structure. For the order of classification, if there are more factors, Objects can be easily classified according to the order of the factors. If the number of factors is smaller, the factor with Z ðHAb Þ ¼ 1 and the minimum value b can firstly be classified.

4.6

Item-By-Item Analyses

Referring to the example in section 4.4, according to the combination of the 32 items of the working environment, the basic information table of the TM is constituted, as shown in table 4.11. The reasoning processes are given in the logical reasoning part of the basic information in table 4.11. The logic and concept of the ‘’ and ‘+’ are the same as those in section 4.4. The specific process of the IIA is described in detail as follows. Firstly, the following two points should be noted. The order of items is in ascending order according to the number of factors of state 1. Only the items in which the system state is 1 are analysed. Because a value of 1 indicates that the system is normal,

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95

which is the desired target state, the above process can be implemented symmetrically. The item (item 1) is invalid when all factors have a state of 0. Among all factors, from item 1 to item 6, if one has a state of 1, the system state with these items is 0, and the items are invalid. By contrast, there is not one factor state that joins another factor state with ‘and’ in the system. Among all factors, from item 7 to item 16, two have states of 1. The system state is 1 when the states of A1 and A4 are 1 in item 9 and the system state is 0 when one factor state is 1; thus, the rule obtained is A1A4+ instead of A1+A4+. Similarly, the rule obtained is A3A5+ from item 15. Among all factors, from item 17 to item 26, three have states of 1. Items 17, 23, 24, and 26 can exist independently. However, items 23, 24, and 26 are invalid and absorbed by item 17, and item 17 cannot be absorbed by rules 9 and 15, and thus item 17 is a rule. Among all factors, from item 27 to item 31, four have states of 1. Item 30 can exist independently, but all items are invalid by the rule absorption. Thus, none of the items in this state are a rule. Among all factors, five have states of 1, and item 32 is invalid. The equivalent response structure of the system is the same as that discussed in section 4.4, where T = A1A4+ A3A5+ A1A2A3.

4.7

Mathematical Description of Item-By-Item Analyses

Definition 4.7. An IIA uses a table system S(I,B,R), where I is basic information, B is logical reasoning, and R is a rule. Table system S consists of several number of items s. An item is sm(I, B, R), i.e. the mth item in table S. An item includes information, logical reasoning, and a rule, where sm(I, B, R) 2 S(I, B, R). The number of items is m 2 ½1; 2N , where N indicates the number of factors in the environment of the system being analysed. For the 01 space fault tree, the state of the basic event and the system are only 0, 1, namely Z ðA1N Þ 2 ½0; 1, Z ðT Þ 2 ½0; 1, where Z ðÞ indicates the state of a basic event and the system. To show the characteristics of the fault tree, ‘+’ is used instead of ‘ [ ’, and ‘’ is used instead of ‘ \ ’. Definition 4.8. Information table I can be considered a permutation of the factor state and a collection of all possible states. It is a grouping of information containing N of 0 or 1, which is composed of state Z ðA1N Þ of factor A1N . In addition, sm(I, B, R) is the mth item in information I. The number of factors of state 1 initem m is xm . Namely, xm ¼ 1 means that the number of factors of state 1 initem m is 1. This indicates that only one factor (A1–AN) has a state of 1 initem m of I, namely, Z(An) = 1, where n 2 ½1; N  and is exactly 1. There are two prerequisites for IIA: (1) First, sm(I,B,R) is classified according to xm , and the items with the same xm are of the same class. The classes are in ascending order with respect to xm and form S(D I,B,R). (2) Only the items of

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TAB. 4.11 – Basic information table and rules in TM. Information table

Logical reasoning

Number

Z(A1)–Z(A5) A1 A2 A3 A4 All states of the factors are 0 1 0 0 0 0 One state of the factors is 1 2 1 0 0 0 3 0 1 0 0 4 0 0 1 0 5 0 0 0 1 6 0 0 0 0 Two states of the factors are 1 7 1 1 0 0 8 1 0 1 0 9 1 0 0 1

Z (T) A5

T

0

0

Invalid

0 0 0 0 1

0 0 0 0 0

Invalid

0 0 0

0 0 1

0 0 1 0 1 0

1 0 0 1 0 1

0 0 0 0 0 1

16 0 0 0 1 Three states of the factors are 1 17 1 1 1 0

1

0

Invalid Invalid A1A4+ exist independently Invalid Invalid Invalid Invalid Invalid A3A5+ exist independently Invalid

0

1

18 19 20 21 22 23

0 1 0 1 1 1

0 0 0 0 0 1

10 11 12 13 14 15

Rule

State

1 0 0 0 0 0

1 1 0 0 1 0

0 1 1 1 0 0

0 0 1 1 1 0

0 1 0 0 1 1

1 0 1 0 0 1

1 1 1 1 0 1

A1A2A3+ exist independently Invalid Invalid Invalid Invalid Invalid A3A4A5+ exist independently; however, according to the Boolean algebra absorption rule, A3A4A5+ A3A5 = A3A5, invalidated by item 15, absorption

A1A4+

A3A5+

A1A2A3+

(continued)

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TAB. 4.11 – (continued). Information table

Logical reasoning

Number

Rule

State Z(A1)–Z(A5)

Z (T)

24

A1 1

A2 0

A3 1

A4 0

A5 1

T 1

25 26

1 0

1 1

0 1

1 0

0 1

0 1

Four states of the factors are 1 27 1 1 1 1

0

1

28

1

0

1

1

1

1

29

0

1

1

1

1

1

30 31

1 1

1 1

0 1

1 0

1 1

0 1

A1A3A5+ exist independently, invalidated by item 15, absorption invalid A2A3A5+ exist independently, invalidated by item 15, absorption A1A2A3A4+ exist independently, invalidated by item 17 or 9, absorption A1A2A3A4+ exist independently, invalidated by item 15 or 9, absorption A2A3A4A5+ exist independently, invalidated by item 15, absorption A1A2A3A5+ exist independently, invalidated by item 15 or 17, absorption

Five states of the factors are 1 32 1 1 1 1 1 1 Invalid The equivalent response structure of the system: A1A4+ A3A5+ A1A2A3 = T

Z ðT Þ ¼ 1 are analysed. Usually, 1 is set to the desired system state from top to bottom in analysis S(I, B, R). Definition 4.9. Logical reasoning B is directly obtained via sm(I, B, R) of Z ðT Þ ¼ 1, and is represented by B(HAb  þ ). This indicates a possible combination of factors. However, this item will not necessarily become a rule because it may be absorbed by the existing rules. In B(HAb  þ ), HAb  fA1 ; . . .AN g, the number of b is xm . Here,

Space Fault Tree Theory and System Reliability Analysis

98

HAb  þ is a type of logical structure. Its existence is joined with ‘and ’ among all factors with a state of Z ðHAb Þ ¼ 1, and then joined with ‘or þ ’ among factors of another state. Definition 4.10. R is obtained after the logical absorption calculation of B, which is obtained by S(D I,B,R). Namely, the first B(HAb  þ ) must be determined as at least xm . According to the law of Boolean algebra, the operation ‘and’ is absorbed by the other operation ‘and’ that concludes less and some parameters. If xm1 < xm2 , then Bm1 (HAb  þ ) obtained by sm1 (I, B, R) is absorbed by Bm2 (HAb  þ ) obtained by sm2 (I, B, R), namely Bm1 (HAb  þ ) + Bm2 (HAb  þ ) = Bm1 (HAb  þ ). Thus, although A can exist independently, it cannot form the rules of the equivalent response structure of the system. Thus, parameter I considers the xm classification and ascending order. According to the above definition and process, starting from xm ¼ 1 for a one-by-one analysis, we can obtain the process shown in table 4.4. Thus, the system equivalent response structure of the original system for the environment response is obtained. During the process of mathematically describing IIA and the CRM, some of the concepts and definitions are the same. However, because of the different structure of the basic information table between the two methods, the descriptions of the processes are different. Thus, the two methods are described in sections 4.7 and 4.5, respectively, to guarantee the integrity and a clear explanation.

4.8

Conclusions

To establish the framework of IASFS, 01SFT is defined, and IIA and the CRM are focused upon. The main conclusions are as follows. (1) The framework of the IASFS is established, and the 01SFT is defined. The purpose of the IASFS is to obtain the internal structure of the system from the external response of the system. Here, the 01SFT is in between the CSFT and the DSFT; if the state is a limited number, then 01SFT degenerates into CSFT; if the state is infinite, then 01SFT degenerates into DSFT. The obtained methods and conclusions of 01SFT are conveniently applied to the CSFT and DSFT. (2) The concept of human–machine cognition is introduced. This is a system with a certain purpose, a certain cognitive function, and receives network information. It is a software and hardware system involved in the monitoring, organization, management, and control of the system under human activities. The IASFS is a concrete realization of human–machine cognition, namely, it is the software for human–machine cognition. (3) A TM-based representation method for 01SFT is proposed by combining the features of the IASFS. TM can be divided into IIA and the CRM; the chapter details the basic concepts and processes of both and describes the mathematics involved. The TM of the IASFS can obtain the simplest factor system, which

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99

has the same response as the system analysed under the same environmental changes. The simple factor system is the equivalent response structure, T = A1A4+ A3A5+ A1A2A3. Thus, the combinations of environmental factors are obtained to suit the system, and guide the actual system operation.

References [1] Bu W.S., Zu C.L., Lu C.X. (2014) Decoupling control strategy of bearingless induction motor under the conditions of considering current dynamic characteristics, Control Theory App. 31, 1561. [2] Li Y.L., Ma F., Geng X.G. (2014) Effect of annular clearance inside the double damping chamber on the dynamic characteristics of a rock drill damping system, J. Univ. Sci. Technol. Beijing 36, 1676. [3] Li M.H., Xia J.B., Chen C.Q., et al. (2012) A new system structure analysis arithmetic with reachable effect factor, Trans. Beijing Inst. Technol. 32, 135. [4] Li M.H., Xia J.B., Chen C.Q., et al. (2012) System structure analysis of communication networks, Trans. Beijing Inst. Technol. 35, 38. [5] Wang H., Xiao J. (2004) Structure identification in fuzzy system based on multi-resolution analysis, J. Syst. Simul. 16, 1630. [6] Cui T.J., Ma Y.D. (2013) Research on multi-dimensional space fault tree construction and application, Chin. Saf. Sci. J. 23, 32. [7] Cui T.J., Ma Y.D. (2014) Definition and understand on size set domain and cut set domain based on multi-dimensional space fault tree, Chin. Saf. Sci. J. 24, 27. [8] Cui T.J., Ma Y.D. (2014) Research on the maintenance method of system reliability based on multi-dimensional space fault tree, J. Syst. Sci. Math. Sci. 34, 682. [9] Cui T.J., Ma Y.D. (2014) Research on the importance element in the system under the influence of the macro factors, Math. Pract. Theory 44, 124. [10] Cui T.J., Ma Y.D. (2014) System security classification decision rules considering the scope attribute, J. Saf. Sci. Technol. 10, 6. [11] Cui T.J., Ma Y.D. (2016) Research on the number of failures of repairable systems based on imperfect repair model, Syst. Eng. Theory Pract. 36, 184. [12] Cui T.J., Ma Y.D. (2015) The method research on decision criterion discovery of system reliability, Syst. Eng. Theory Pract. 35, 3210. [13] Cui T.J. (2015) The construction of space fault tree theory and research. Liaoning Technical University, Fuxin. [14] Cui T.J., Wang P.Z., Ma Y.D. (2016) Inward analysis of system factor structure in 01 space fault tree. Syst. Eng. Theory Pract. 36, 2152. [15] Cui T.J., Wang P.Z., Ma Y.D. (2016) Structured representation methods for 01 space fault tree, J. Dalian Jiaotong Univ. 37, 82. [16] Wang P.Z. (2013) Factor space and factor data bases, J. Liaoning Eng. Technol. Univ. 32, 1. [17] Li H.X. (2000) Factor space theory and its applications in fuzzy information processing: two kinds of factor spaces canes, Comput. Math. App. 40, 835. [18] Li H.X. (2014) Factor space theory, Collected works on factor space. Liaoning Engineering and Technology University, pp. 57–161. [19] Wang P.Z. (1990) A factor spaces approach to knowledge representation, Fuzzy Sets Syst. 36, 113. [20] Wang P.Z. (1990) Factor space and knowledge representation, Approximate reasoning tools in artificial intelligence (J.L. Verdegay, M. Delgado, Eds), Verlag TUV Rheinland, pp. 97–114. [21] Liu Z.L. (2014) Factor representation of knowledge, Collected works on factor space. Liaoning Engineering and Technology University, pp. 171–175. [22] Yuan X.H., Wang P.Z. (1995) Factor space and category, Fuzzy Syst. Math. 2, 26. [23] Wang P.Z., Guo S., Bao Y.K., et al. (2015) Causality analysis in factor spaces, J. Liaoning Eng. Technol. Univ. 33, 1.

Chapter 5 Function Structure Analysis and Factor Space It is universally acknowledged that a system is composed of components and subsystems. Based upon the premise of this known basic feature, a key issue in a system design is how to guarantee that the system will accomplish a specific function. A system is an organism formed by components or subsystems following a certain layout; the system design is simply about how to solve the problem of the layout. In general, the system design follows a top-down approach, that is, it starts from the whole, and through certain function decomposition, the level of the components or subsystems is finally selected. A top-down design can meet the functional requirements of the system, but it is difficult to determine whether the designed system is optimal. This problem is considered from the aspects of the system function and economy. Another problem occurs: if some specific components or subsystems making up the systems are known, and if the system function changes with the changes in the component function, but the system is inaccessible, how can the system be studied or even copied? The above problems can be summarised as the problem of the inward analysis of the system function structure. In other words, if one knows the characteristics of the basic component functions and the system functions, how can we learn the structure of the system function through the inward analysis? The purpose of this chapter lies in an integration of the inward analysis of the system function structure and factor space function structure analysis. The inward analysis of the system function structure of the SFT is promoted to intelligent reasoning methods in chapter 4, which lack mathematical logicality. At the same time, the chapter also applies factor space theory to practical engineering analysis. This chapter further summarises the methods and uses the factor space theory to analyse the system function structure, and the derivation process that conforms to the mathematical reasoning is given [1, 2]. The structure of this chapter is as follows: section 5.1 introduces the causal analysis method for analysing the function structure. Section 5.2 introduces the factor logic of the function structure analysis. Section 5.3 analyses specific cases using the function structure analysis method. Section 5.4 provides some concluding remarks. DOI: 10.1051/978-2-7598-2499-1.c005 © Science Press, EDP Sciences, 2020

102

Space Fault Tree Theory and System Reliability Analysis List of symbols

Variable f1,…, fn U f g fi(u) Ai X(f) ∇ Δ B = F(U) [a] = fj−1(a) h m d = h/m Lf S = X = X(f1) [ … [ X(fn) xij ij F(S) (F(S), ∨, ∧, ¬, →) xi(1)j(1) ∧ … ∧ xi(k)j(k) r1 ∨ … ∨ r t p b W2 MP Σ W ΓΓ1 Γ2 Fb(S) = {p ∧ b|p 2 F(S)} F+(S) U′ = U/F Pc T F O A1–AN xj x

Symbols 1st–ith structure factors Universe set Set of structure factors Result (function) factors Express phase of system components Ai in a moment ith component Phase space Comprehensive operation Decomposition operation Background relationship between the factors Structure factor phase Number of rows of fj in function structure data table Number of rows of fj in function structure data table Decision ratio Factor logic system Symbol set of factor logic system Phase Word xij in the ith phase of factor fj Formula set Boolean algebra Conjunction, simply xi(1)j(1) … xi(k)j(k) Disjunction, general r1 +    + rt Tautology form Disjunctive normal formula Value domain Reasoning rules Axiom sets Assignment set Subsystem of the theorem Word name axiom Background axiom Formula set of factor logic Proposition set or predicate set Quotient space Subset of B Positive class Negative class System N basic events or components compose system O Component normal state Component failure state

Function Structure Analysis and Factor Space

5.1

103

Factor Analysis Method of Function Structure

A factor space is a generalization of the Cartesian coordinate space. For example, in physics, the movement of a particle in a three-dimensional space is described with four variables: x (left and right), y (up and down), z (front and back), and t (time). The activity fields of each variable form a coordinate axis, and the Cartesian coordinate system is then formed to describe the particle movement. These four variables are four observation perspectives and are used for analysis, and are the basic elements to comprehensively understand the world around them. As factors, they are the foundation of cognition. Factors are a perspective to observe and analyse objects and are an extension of variables. The observed value of each factor f does not necessarily need to be real but is a more general phase [3]. The so-called phase can be the states of the objects, their attributes, or something more general. The change domain is called the phase space, denoted by X(f). A phase space can be a Euclidean space, which can be a set composed of qualitative values of natural language. A factor is a mathematical map from U to X(f), where U is a set made up of all considered factors. A factor space is a generalised product of a phase space of a set of factors under a given universe U, which is simply denoted as (U; F = {f1,…, fn}). All Cartesian spaces belong to the factor space; this is a phase space in physics, a state space in cybernetics, or a feature space in pattern recognition. The factor space differs from ordinary Cartesian spaces in the following two aspects: the operation among the factors and the dimensional variability between factors and the phase space. Furthermore, the factor space can describe the qualitative cognition process in intelligence science.

5.1.1

Factors and Dimension Variability

When a gourmet comments on a dish, the gourmet will likely consider three factors: f1 = colour, f2 = aroma, and f3 = taste, where, X(f1) = {beautiful, general, ugly}, X(f2) = {fragrance, light, smell}, and X(f3) = {good, middle, bad}. After a comprehensive operation, some composite factors are obtained, such as f4 = colour aroma, f5 = colour taste, f6 = taste aroma, and f7 = colour aroma taste, as denoted by f4 = f1 ∇ f2, f5 = f1 ∇ f3, f6 = f2 ∇ f3, and f7 = f1 ∇ f2 ∇ f3. By contrast, there is a type of operation that can simplify the complex factors: f4 = colour aroma, f5 = colour taste, and f4 Δ f5 = colour = f1, as denoted by f1 = f4 Δ f5, and similarly, f2 = f4 ∇ f6 and f3 = f5 Δ f6. Here, ∇ and Δ are, respectively, called a comprehensive operation and a decomposition operation. The comprehensive operation is wider, and the union set of dimensions is considered, whereas a decomposition operation is narrower, and the intersection set of dimensions is considered. The factor operation should not be simply explained by logic and should not be confused with a lower phase (attributes) operation of the factors. All phase spaces of the family factors are presented as follows: X(f4) = {beautiful and fragrant, beautiful and light, beautiful and smelly, general and fragrant, general and light, general and smelly, ugly and fragrant, ugly and light, ugly and smelly};

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Space Fault Tree Theory and System Reliability Analysis

X(f5) = {fragrant good, fragrant middle, fragrant bad, light good, light middle, light bad, smelly good, smelly middle, smelly bad}; X(f6) = {beautiful good, beautiful middle, beautiful bad, general good, general middle, general bad, ugly good, ugly middle, ugly bad}; X(f7) = {beautiful fragrant good, beautiful fragrant middle, beautiful fragrant bad, beautiful light good, beautiful light middle, beautiful light bad, beautiful smelly good, beautiful smelly middle, beautiful smelly bad, general fragrant good, general fragrant middle, general fragrant bad, general light good, general light middle, general light bad, general smelly good, general smelly middle, general smelly bad, ugly fragrant good, ugly fragrant middle, ugly fragrant bad, ugly light good, ugly light middle, ugly light bad, ugly smelly good, ugly smelly middle, ugly smelly bad}. The colour, aroma, and taste are three independent factors, and these factors form the factor space, which is a Cartesian product of their phase space. In fact, these factors are not necessarily independent of one another. For the comprehensive operation and decomposition operation of related factors, a complicated mathematical theory is needed to depict the phase space. We will not discuss complicated methods further in this chapter. This chapter focuses on an effective and simple method, that is, the background relationship theory. Definition 5.1. The given factor space (U; F = {f1,…, fn}), denoted by B = F (U) = {x 2 f1(X) … fn(X)|9 u 2 U; F(u) = x}, is called the background relationship between factors, or the actual Cartesian product of the phase space of various factors. When a set of factors are independent of one another, their phase combination can be unrestricted. The background relationship shall be equal to a Cartesian product of the phase space of various factors, that is, B = f1(X) … fn(X). If B = f1(X) … fn(X). This means a special relationship exists among the factors, and this relationship determines the causal reasoning among the factors. The above is one of differences between the factor space and a Cartesian space. General Cartesian space dimensions are fixed, whereas the factor space dimensions are variable. Thus, it can provide sufficient information with minimal dimensions. This is another difference between the factor space and a Cartesian space.

5.1.2

Function Structure Analysis Space

The following describe the framework of the build factor function structure analysis theory. Definition 5.2. A factor space (U, F = (f1,…, fn), g) is called a function structure analysis space. If g is a function factor, it has a phase set with a describing function, X(g) = {y1,…, yK}; in addition, fj is a structure factor that can influence the function of the system; fj has the phase set X(fj) = {a1j,…, an(j)j}(j = 1,…,n). A sample point of a function structure analysis is a row vector (ui; f1(ui),…, fn(ui);

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g(ui)). A matrix composed of m sample points is called an m row function structure data table. Subscript i of object ui in the table is called the row footmark of the table. For a row replacement, simply replace the row footmark. For a row deletion, simply delete the row footmark from the footmark set. Definition 5.3. Consider a function structure data table for every structure factor fj and the phase of this structure factor a, and denote them by [a] = fj−1(a) = {u 2 U| fj(u) = a}. If all objects u in [a] have the same result g(u) = yk 2 X(g), then [a] is called a decision class of factor fj. The union set of all decision classes of fj is called the decision domain of function of fj. The ratio between h rows taken by the decision domain of factor fj and the m rows of the table is called the decision of fj, and is denoted by d = h/m. Definition 5.4. Suppose [a] is a decision class of factor fj, and the reasoning sentence ‘if the phase of structure factors fj is a, then its function is y’ describes a function analysis of the decision class. When the row footmarks of the decision class are deleted from the rest of the table, to exchange the function analysis sentence, a single substitution is made. The basic steps of the factor analysis algorithm are as follows: (1) A larger factor decision has more influence on the function. Take the factor with the larger decision, replace all decision classes of the factor with the function analysis sentence, and delete the corresponding row footmarks of the decision domain from U. (2) Apply step 1 to the rest of universe U repeatedly until U is empty. Take all function analysis sentences together, draw a decision tree, and then stop. It is worth noting that factor analysis and decision tree methods are similar, the difference being that the largest decision is used to choose the substitution factor instead of the minimum gain information. Regarding the requirements of the analysis table, there is no need to provide a sample frequency in the phase space. Even if a frequency distribution exists, it will not be used in the deterministic analysis; a frequency distribution is only used in uncertain function structure analyses, which will be studied in future research.

5.2 5.2.1

Factor Logic Description of Function Structure Axiom System of Function Structure Analysis

Definition 5.5 [4]. Suppose there is a factor space (U, F = (f1,…, fn)), and its background relationship is denoted as B = F(U)  X(F) = X(f1) … X(fn). The factor logic system Lf in the factor phase space is then defined as follows:

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(1) Its symbol set is S = X = X(f1) [ … [ X(fn), add symbol 1, 0 and brackets ‘(“and”)’, where X(fj) = {x1j,…, xn(j)j} is the phase space of fj (j = 1,…, n); the phase xij of each of these factors is called a word, which constitutes the basic symbol S of Lf. Footmark ij indicates that word xij is the ith phase of factor fj. (2) Its formula set F(S), generated by S, is Boolean (F(S), ∨, ∧, ¬, →); all words are called the original formula, the conjunction xi(1)j(1) ∧ … ∧ xi(k)j(k) (simply xi(1)j(1) … xi(k)j(k)) of the words is called a word set; and the disjunction r1 ∨ … ∨ rt (general r1 +    + rt) of the word set is called a disjunctive normal formula. Formula p is called a tautology if p = 1(p → 1 and 1 → p), and formula p is called a contradiction if p = 0. (3) If its axioms are Boolean logic axiom sets, the following assumed axiom is provided: Γ1. Word name axiom: The word in X(fi) = {x1j,…, xm(i)j} is called the jth family word. The same name word meets the formula x1j ∨ … ∨ xn(j)j = 1; xij ∧ xkj = 0 (i ≠ k); ¬xij = ∨{xij | i ≠ k}; Γ2. Background axiom: There is a disjunctive normal formula b 2 F(S) called the background formula, which makes p → p ∧ b and p ∧ b → p (pF(S)). If system Lf fails to point out b, then b = 1, and in this case, the background axiom becomes invalid. (4) The value domain is a binary Boolean algebra W2 = {0, 1} = {{0, 1}, ∨, ∧, ¬}. (5) Reasoning rules: MP: {p, p → q} |—q. As an explanation of definition 5.5, any logical system contains five elements: symbol set S, formula set F(S), axiom set Σ, assignment set W, and reasoning rules (sets). A set of axioms can be added to the logical system, which is called a derivative subsystem of assumption axiom set Γ. The theorem in the original system is a theorem of the subsystem, and a reasoning p that is not a theorem in the original system may become a new theorem of the subsystem, if Σ [ Γ |—p. The theorem of the subsystem is called Γ-theorem, if it meets the strong completion theorem: Γ |—p iff Γ|=p. The assumption axiom Γ of Lf is given by two sets of axioms. The word name axiom Γ1 emphasises that a word is a factor phase; words of different factors have different names. A word name axiom ensures that words with the same name comply with Boolean logic relation. Axiom xij ∧ xkj = 0 (i ≠ k) indicates a rule in which each family in a word set is not allowed to have two and more than two words, for example, x = red and green and tender and fresh, which is a contradiction. Compared to the comprehensive factor F, a word is the phase of a single factor, but its original formula is not a minimum phase; the conjunction of words x = xi(1)1 … xi(n)n is the minimum phase. For example, red, large, tender, and bright are phases of colour, height, texture, and taste, respectively. These are words in the original formula, but do not represent the atomic connotation of the factor space. The conjunction x = ‘red big tender fresh’ is just an atomic connotation. Background axiom Γ2 emphasises that the most important feature of factor logic is the background relationship B = F(U)  X(F). According to this axiom, the authenticity of a proposition only depends on its phase in B, and has nothing to do

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with the phase outside B. In other words, the formula set of the factor logic is Fb(S) = {p ∧ b|p 2 F(S)}, namely F+(S) = U′ × B = {(p, x)| p 2 F(S), x 2 B}. Definition 5.6 [5]. Here, (p, x) 2 F+(S) is considered a proposition, or p(x) = (p, x), which is called a predicate. In addition, F+(S) is called a proposition set or predicate set, and U is classified by comprehensive factor F, and the quotient space U′ = U/F = {[x]| x 2 B} is obtained. It is worth noting that there exists a special relationship between the background relationship B and universe U, denoted by [x] = {u 2 U|F(u) = x}. Because B = F (U), F is a surjection from U to B, namely, for any u 2 U, x 2 B, and F(u) = x. Thus, mapping F induces a mapping, which is still denoted as F:U′ → B: F([x]) = x. It establishes a homomorphism between (U′, [ , \ , c) and (B, ∨, ∧, ¬). This characteristic allows us to equate U′ and B. Thus, B has a dual characteristic: On the one hand, the points of B are the connotation of the atomic concept, and on the other, the points of B are also equal to the extension of the atomic concept. Here, B is the bridge for the logic to convert between algebra and geometry. Any formula can be written as a disjunctive normal formula of n word set, and as a subset of B. Each formula has a geometrical figure. Supposing x = (xi(1)1 … xi(n)n) is a point in B, word xij is in word set x if and only if the word name j of x is just xij, namely xi(j)j = xij, or more simply i(j) = i. The mapping is denoted as t:S → 2B: t(xij) = {x = xi(1)1 … xi(n)n)|i(j) = i} = Xij. This makes word xij a subset Xij of B. This mapping can be extended to FB(S), to satisfy t(p ∨ q) = t(p) [ t(q), t(p ∧ q) = t(p) \ t(q), t(¬p) = t(p)c. Definition 5.7. For any pF(S), P = t(p) is the true set of p. According to normal practice, the formula and true set are expressed by lowercase and uppercase letters, respectively. Definition 5.8. Here, B is called the interpretation domain of the logic system Lf. The mapping of v: F+(S) → W2 is the assignment, if v(p, x) = 1 and when x 2 P; v(p, x) = 0 when x 62 P. Proposition 5.1. Formula p containing q is a tautology if and only if true sets P  Q. Proof. p containing q is a tautology if and only if p → q = 1, that is, ¬p ∨ q = 1 iff t (¬p ∨ q) = B, namely Pc [ Q = B. Because p = p ∧ b and q = q ∧ b, P, Q are subsets of B, iff P  Q. Although factor logic is a derivative system of Boolean logic, it has the following features: (1) Traditional proposition logic formulas interpret the formula in F(S) as a proposition. A proposition is a sentence used to judge the authenticity, and is made up of two parts, namely the concept and object. The language structure of the proposition depends only on the concepts and has nothing to do with the objects. Thus, the formulas in F(S) are interpreted as a concept (relations are concepts as well), and the formula does not directly represent the proposition.

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(2) The proposition is defined as the match between concept and object. After the interpretation domain B of the logical system is introduced, propositions are represented as only a certainty subset of B. The proposition changes into a true domain. Assignment mapping is no longer arbitrary, but becomes a membership function of the proposition true value set on B. Some propositions needed in a function structure analysis are given below. Proposition 5.2. If there are two words with the same name in word set r = xi(1)j(1) … xi(k)j(k), then r = 0. Proof. Assume i(1) = i(2) = 1, j(1) = 1, and j(2) = 2. According to the word name axiom x11 ∧ x12 = 0, the true value of this word set is 0. The words are incompatible. Proposition 5.3. The length t of the word set x1 ∧ … ∧ xt cannot be larger than n. Proof. A non-zero word set does not allow the same word name, and there are only n kinds of different names. Thus, the length of the word is no more than n, and a word set of longer than n is a contradiction. It is not difficult to prove that the true set of word r = xij is the cylinder expansion of B from the point in the ith axis and jth point to all non-i axes: R = [{xij}X(f1) … X(fi−1)X(fi+1) … X(fn)] \ B. Similarly, the true domain of word set r = xij ∧ xkl with a length of 2 is R = [{xij} × {xkl} × {X(fj)|j ≠ i, k}] \ B. The longer the true domain is, the smaller the true set. When the length is n, the true value will become a single point set. Here, n word set x1j(1) ∧ … ∧ xnj(n) corresponds to the single point set {x} of X. Definition 5.9. If p → q = (¬p) ∨ q = 1, then p implies q. Thus, p is called the implicit formula of q, and q is a function formula of p. If in F(S) there does not exist any other implicit formula p′ ≠ p of q to make p contain p′, p is the implicit formula of q. A disjunctive normal formula is called a minimal formula if each of its word sets is an implicit formula. Proposition 5.4. If all words of word set q are in word set p, then p implies q. Proof. Suppose that all words of word set q are in word set p, then there is a word set r to make p = q ∧ r valid, and r does not include the words of q. Then, according to the homogeneity of the transformation mapping J, P = Q \ R, and then P  Q, and thus word set p implies a word set q. It is not difficult to prove that p implies q if and only if the true set of the function formula contains the true set of implicit formula P  Q. An implicit word set exists if and only if there is not a third word set whose true set is between the implicit true set and function true set. Proposition 5.5. If word xij is not in any word set of a disjunctive normal formula of ¬p, namely if any item (word set) of ¬p does not contain word xij, then word xij is an implicit formula of p.

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Proof. This proposition can be proved in two steps. First, assume B = X(F) = X(f1) … X(fn). According to background axiom Γ2, the true set of word r = xlk is R = X(f1) … X(fk−1){xlk}X(fk+1) … X(fn). Word xlk is not in any word set of the disjunctive normal formula of ¬p if and only if R \ Pc = ∅. If and only if R  P, according to proposition 5.1, r implies q. Proposition 5.2 suggests that a single word set cannot contain any word or word set except itself, and thus r implies q. Second, assume that B ≠ X(F). In this case, the question is whether the true set Xij of a word xij and Pc have no intersection; Xij does not necessarily have such an intersection if it is contained by P. When B = X(F), the proposition is true, because X(F) = X(f1) … X(fn) is the complete space containing all combinations of phases. After being segmented, every phase combination x must be in one of combinations. Now B ≠ X(F), and B is an incomplete space because some phase combinations are deleted. After its segmentation, some phase combinations cannot be found on either side. However, x is in a phase combination outside B and does not actually exist. Because phase combinations do not exist, the true set of any words and word sets can eliminate them to ensure that a proposition is tenable. Proposition 5.5′. If words are replaced by k word set (k > 1), the proposition is still true. The disjunctive formula of all implicit formulas of p is simply the logical structure expression of p. Function structure analysis indicates that for each phase yi of function g, its logical expressions yi = pi can be directly obtained from the function structure analysis table. Once the minimal disjunctive normal formula of pi is found, a simple and clear logical structure is put forward for the realization of function yi.

5.2.2

Minimization Method of System Function Structure

If the factor function structure analysis table is given, how can we obtain the minimal disjunctive normal formula of the specified function class yi? The steps for this are as follows: (1) Gather all n word sets x (remove the same values) in the table, and denote them by the background relations B. Divide these word sets in B into a positive T class and negative F class by considering whether they take result yi or not. (2) With word set length k = 1, check every word one by one. If the word does not appear in all word sets of class F, it is an implicit formula of class T. Delete all implicit formulas from n word set of class T. Repeat until all single words have been checked. (3) With word set length k: = k + 1, check every k word set one by one. If the word does not appear in all word sets of class F, it is an implicit formula of class T. Delete all of its implicit formulas from the n word set of class T, and repeat until all words have been checked. (4) Repeat the above process until all word sets of T class are deleted. Connect all implicit formulas of class T with ‘+’, and a minimal disjunctive normal formula is then obtained.

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Theorem 5.1. The system function structure obtained with the above method is a minimal disjunctive normal formula of class T. Proof. First, prove the termination of the algorithm, namely, the positive word class can be deleted. Set {x} as a point of P. It definitely does not belong to Pc. The formula of its true sets is the n word sets x = x1i(1) ∧ … ∧ xni(n). If it is not deleted in the previous comparison, then check this n-word. Because it does not appear in a negative class word set, the word set must appear in a positive class word set. Thus, word set is one implicit formula of yi, and this point can then be deleted. This shows that any positive word can be deleted during the algorithm implementation process. If every word or word set recorded is contained by P, according to propositions 5.5 and 5.5′, the word or word set is the implicit formula of p. The disjunction of all implicit formulas is a minimal disjunctive normal formula of P.

5.3

Analysis of System Function Structure

The instance used by the inward analysis of system factor structure in the SFT is also an example of the function structure analysis of the factor space, as shown in figure 5.1. In addition, a structure information table of TM is used, as shown in table 5.1

FIG. 5.1 – Analysed system model.

TAB. 5.1 – Structure information table of TM. Number 1 … M

Factor

Logical reasoning

f1 1

f2 0

… …

fN−1 0

fN 0

O 0

1

1

…

1

1

1

System equivalent response structure (system factor structure).

Rule

…

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The system O in figure 5.1 consists of five basic events (components), namely A1–A5, and five factors f = (f1, f2, f3, f4, f5); here, X(f1)–X(f5) represent their respective state phase space, and their state value is only {x1j, x0j} = {xj, xj}, where xj represents the on state of the component, and xj represents the off state. In addition, O indicates the system, and X(O) = g is the system state, namely the phase space of factor g is X(g) = {T, F}, where T indicates the system is on, whereas F indicates the system is off. For a 0, 1 state of f = (f1, f2, f3, f4, f5), the word set is composed of ten words: S = {x1, x1, x2, x2, x3, x3, x4, x4, x5, x5}. There are 32 phase sets in domain U.

5.3.1

Analysis with Incomplete Information

Example 5.1. The given switching system O is shown in figure 5.1. It consists of five components, namely A1–A5, which determine five factors f = (f1, f2, f3, f4, f5), respectively. Each factor has a phase space X(fj) = {x1j, x0j} = {xj, xj}, where j = 1, …, 5, x1j indicates that component Aj is on, and x0j shows that component Aj is off. The result factor g is the condition of system O, and the g phase space is X(g) = {T, F}, where T indicates the system is on, whereas F indicates the system is off. The word set is composed of ten words: S = {x1, x1, x2, x2, x3, x3, x4, x4, x5, x5}. The system function structure analysis is performed under an incomplete background set B. The set is composed of 20 items selected from 32 item phase sets in domain U. This phase set of 20 items constitutes the factor function structure analysis table, as shown in table 5.2. According to the function structure analysis method, the system function structure is analysed when y1 = T. Step 1: The background set B with all sample points has 20 < 32 = 25 points, which are divided into two categories: T and F: T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3 x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5}; F = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3 x4x5, x1x2x3x4x5, x1x2x3x4x5}. Step 2: When k = 1, a search is conducted for single words that do not appear in the word set of class F. The first item of F is x1x2x3x4x5, which contains the following five words: x1, x2, x3, x4, x5. The second item of F is x1x2x3x4x5, which contains the following five words: x1, x2, x3, x4, x5. The two items are merged to delete the same word: x1, x1, x2, x3, x3, x4, x5. Finally, all items are put together, and it can be seen that all ten words, x1, x2, x3, x4, x5, x1, x2, x3, x4, and x5, appear in the word set F. This type of state is not expected, and is therefore invalid. If some words do not appear in F, then we can find an implicit formula whose T length is 1. The shorter the implicit formula is, the more successful the minimization that occurs. Step 3: When k: = k + 1 = 2, and when not considering a concrete algorithm, we can see that the word set x1x4 does not appear in F; note that x1x4 is an implicit formula of T. All word sets in T that contain x1x4 are deleted. According to proposition 5.3, the five word sets that contain x1x4 from T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5,

112

Factors f1 f2 f3 f4 f5 g

U u1 x1 x2 x3 x4 x5 F

u2 x1 x2 x3 x4 x5 F

u3 x1 x2 x3 x4 x5 F

u4 x1 x2 x3 x4 x5 F

u5 x1 x2 x3 x4 x5 F

u6 x1 x2 x3 x4 x5 F

u7 x1 x2 x3 x4 x5 F

u8 x1 x2 x3 x4 x5 F

u9 x1 x2 x3 x4 x5 F

u10 x1 x2 x3 x4 x5 T

u11 x1 x2 x3 x4 x5 T

u12 x1 x2 x3 x4 x5 F

u13 x1 x2 x3 x4 x5 T

u14 x1 x2 x3 x4 x5 T

u15 x1 x2 x3 x4 x5 T

u16 x1 x2 x3 x4 x5 T

u17 x1 x2 x3 x4 x5 T

u18 x1 x2 x3 x4 x5 T

u19 x1 x2 x3 x4 x5 T

u20 x1 x2 x3 x4 x5 T

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TAB. 5.2 – Function structure analysis table of 20 phase sets.

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x1x2x3x4x5, x1x2x3x4x5} are also deleted, and the result T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5} is obtained. Word set x3x5 does not appear in F; note that x3x5 is an implicit formula of T as well. The five word sets that contain x3x5 from T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5} are deleted, and the result T = {x1x2x3x4x5} is obtained. Word set x1x2 does not appear in F; x1x2, which is also an implicit formula of T, is recorded. Five word sets that contains x1x2 are deleted from T = {x1x2x3x4x5}, and the result is then T = Ø. The reasoning is stopped at this point, the implicit formulas obtained for T added together, the minimum disjunctive normal formula of T: T = x1x4 + x3x5 + x1x2 is obtained, and system structure is O = A1A4 + A3A5 + A1A2, as shown in figure 5.2.

5.3.2

Analysis with Complete Information

Example 5.2. The same switch system O as in example 5.1 is given here. The system function structure analysis is conducted under complete background set B, and the set is composed of 32 item phases in domain U. This set of 32 item phases constitutes the factor function structure analysis table, as shown in table 5.3. According to the function structure analysis method, the system function structure is analysed when y1 = T. Step 1: The background set B made up of all sample points has 32 = 25 points, which is exactly the explanation set X = X(f). This is therefore an example of an invalid background relationship set. Classic Boolean logic can also solve the problem. The 32 points are divided into two categories: T and F. T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3 x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5}; F = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3 x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5}. In fact, a T logical expression can be obtained using disjunctive T word sets: T = x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 +

FIG. 5.2 – System component function structure.

114

Factors f1 f2 f3 f4 f5 g

U u1 x1 x2 x3 x4 x5 F

u2 x1 x2 x3 x4 x5 F

u3 x1 x2 x3 x4 x5 F

u4 x1 x2 x3 x4 x5 F

u5 x1 x2 x3 x4 x5 F

u6 x1 x2 x3 x4 x5 F

u7 x1 x2 x3 x4 x5 F

u8 x1 x2 x3 x4 x5 F

u9 x1 x2 x3 x4 x5 T

u10 x1 x2 x3 x4 x5 F

u11 x1 x2 x3 x4 x5 F

u12 x1 x2 x3 x4 x5 F

u13 x1 x2 x3 x4 x5 F

u14 x1 x2 x3 x4 x5 F

u15 x1 x2 x3 x4 x5 T

u16 x1 x2 x3 x4 x5 F

u17 x1 x2 x3 x4 x5 T

u18 x1 x2 x3 x4 x5 T

u19 x1 x2 x3 x4 x5 T

u20 x1 x2 x3 x4 x5 F

u21 x1 x2 x3 x4 x5 F

u22 x1 x2 x3 x4 x5 F

u23 x1 x2 x3 x4 x5 T

u24 x1 x2 x3 x4 x5 T

u25 x1 x2 x3 x4 x5 T

u26 x1 x2 x3 x4 x5 T

u27 x1 x2 x3 x4 x5 T

u28 x1 x2 x3 x4 x5 T

u29 x1 x2 x3 x4 x5 T

u30 x1 x2 x3 x4 x5 T

u31 x1 x2 x3 x4 x5 T

u32 x1 x2 x3 x4 x5 T

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TAB. 5.3 – Function structure analysis table of 32 phase sets.

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x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5 + x1x2x3x4x5, where + indicates a disjunction and parallel component; the word set x1x2x3x4x5 indicates x1 ∧ x2 ∧ x3 ∧ x4 ∧ x5 and the component series. Thus, the logical expression formula is the function structure formula of system O. According to the expression of T, a structure chart can be obtained. The problem here is that there are too many items and thus the structure drawn is not optimal. The structure optimization process is simply a logic minimization process of a normal disjunctive formula. Step 2: When k = 1, a single word that does not appear in class F of word sets is searched. The first item of F is x1x2x3x4x5, which contains the five words x1, x2, x3, x4, x5; The second item of F is x1x2x3x4x5, which contains the five words x1, x2, x3, x4, x5. Merging the two items and deleting the same word, the result is x1, x1, x2, x2, x3, x4, x5, x5. Finally, putting all items together, all ten words x1, x2, x3, x4, x5, x1, x2, x3, x4, x5 appear in the class F word sets. This type of state is invalid. Step 3: When k: = k + 1 = 2, the two-word set x1x4 does not appear in F but does appear in T. Record x1x4 as an implicit formula of T. Delete all word sets that contain x3x5 from T, and the result T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5} is obtained. According to proposition 5.3, delete all 5-word sets that contain x1x4 from T. Before the deletion, put together all 5-word sets that contain x1x4 and conduct a Boolean operation; because word name axiom x1j … xn(j)j = 1, namely xj + xj = 1, the result is as follows: x1 x 2 x 3 x4 x 5 þ x1 x 2 x3 x4 x 5 þ x1 x 2 x 3 x4 x5 þ x1 x2 x 3 x4 x 5 þ x1 x2 x3 x4 x 5 þ x1 x 2 x3 x4 x5 þ x1 x2 x 3 x4 x5 þ x1 x2 x3 x4 x5 ¼ ðx1 x 2 x 3 x4 x 5 þ x1 x 2 x3 x4 x 5 Þ þ ðx1 x 2 x 3 x4 x5 þ x1 x 2 x3 x4 x5 Þ þ ðx1 x2 x 3 x4 x 5 þ x1 x2 x3 x4 x 5 Þ þ ðx1 x2 x 3 x4 x5 þ x1 x2 x3 x4 x5 Þ ¼ x1 x 2 x4 x 5 ðx3 þ x 3 Þ þ x1 x 2 x4 x5 ðx3 þ x 3 Þ þ x1 x2 x4 x 5 ðx3 þ x 3 Þ þ x1 x2 x4 x5 ðx3 þ x 3 Þ ¼ x1 x 2 x4 x 5 þ x1 x 2 x4 x5 þ x1 x2 x4 x 5 þ x1 x2 x4 x5 ðx3 þ x 3 ¼ 1Þ ¼ x1 x 2 x4 ðx5 þ x 5 Þ þ x1 x2 x4 ðx5 þ x 5 Þ ¼ x1 x4 ðx5 þ x 5 ¼ 1Þ

Thus, the reason all 5-word sets containing x1x4 can be deleted is clear. After deletion, T = {x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5, x1x2x3x4x5} is obtained. Here, x3x5 does not appear in F but in T. Record x3x5 as an implicit formula of T. All 5-word sets that contain x3x5 are deleted from T, and the result T = {x1x2x3x4x5} is obtained. Except for the two 2-word sets, the remaining 2-word sets all appear in F. When k: = k + 1 = 3, the 3-word set x1x2x3 does not appear in F but in T. Record x1x2x3 as an implicit formula of T. The 5-word set x1x2x3x4x5 that contains the 3-word set x1x2x3 is deleted, and thus T = Ø. The reasoning is stopped at this point, the implicit formulas obtained for T are added together, the minimum disjunctive normal formula of T = x1x4 + x3x5 + x1x2x3 is obtained, and the system component structure O = A1A4 + A3A5 + A1A2A3, as shown in figure 5.3. Continuing the discussion, the background set of example 5.1 2 the background set of example 5.2, there must, therefore, be a relationship between the two

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FIG. 5.3 – System component function structure. minimum disjunctive normal formulas. Because there must be some restricted relations between various factors in the minimum disjunctive normal formula derived from incomplete information, which are added to the background sets, this contributes to the minimum disjunctive normal formulas. Otherwise, the results will be a type of probability distribution, which shifts from a certainty analysis to an uncertainty analysis. Analysing the minimum disjunctive normal formulas in the above two examples, they are both deterministic analysis formulas, and thus 8 > < x3 ¼ x2 T ¼ x1 x4 þ x3 x5 þ x1 x2 ¼ x1 x4 þ x3 x5 þ x1 x2 x3 ) x1 x2 ¼ x1 x2 x3 ) x3 ¼ x1 > : x3 ¼ x2 þ x1 This shows that there are linear or nonlinear relationships between component A3 and components A1 and A2 of example 5.1. The function of x3 can be represented by the combination of x1 and x2. Thus, this implicit relationship between components can solve the problem in which the number of items of the background relationship set is less than 32, which transforms the uncertainty problem into a certainty problem. The function structure analysis theory of the factor space is used to realise the inward analysis of the system function structure, and thus enhance the original classification reasoning method to the level of mathematical logic. As a result, the application of factor space theory in professional and technical fields has become a reality, and this research has laid a foundation for the enhancement of the SFT to a mathematical level. This therefore indicates the theoretical and practical significance of this study.

5.4

Conclusions

(1) The chapter proposed a system function structure analysis method based on the factor analysis, and indicated that the factor space can be used to describe the qualitative cognition process in intelligence science. This differs from an ordinary Cartesian space in two aspects: the operation among the factors and the

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117

dimensional variability of the factors and phase space; the function structure analysis space was also constructed and some definitions were given. (2) An axiom system of the system function structure analysis was built based on factor logic. Definitions, logic propositions, and proof procedures were provided, laying a foundation for an enhancement of the system function analysis method to the logic level of mathematical reasoning. Based on the axiom system, this chapter put forward a minimization method of the system function structure. It proved that the system function structure obtained by this method is a minimal disjunctive normal formula. (3) Through two specific examples, the system function structures were analysed using both incomplete and complete information. In the two examples, the minimal disjunctive normal formulas are T = x1x4 + x3x5 + x1x2 and T = x1x4 + x3x5 + x1x2x3, respectively. This indicates that some relationship exists between component A3 and components A1 and A2 of example 5.1, that is the function of x3 can be represented by the combination of x1 and x2. This implicit relationship between functions can, therefore, solve the problem in example 5.1 in which is there are fewer than 32 items in the background relationship sets. The uncertainty problem is then transformed into a certainty problem. More detailed questions can be found in document [6].

References [1] Cui T.J., Li S.S., Wang L.G. (2017) System function structure analysis in the complete and incomplete background relationship, Comput. Sci. 44, 268. [2] Cui T.J., Li S.S., Wang L.G. (2019) The simplest formula analysis method of system function structure, App. Res. Comput. 36, 27. [3] Cui T.J. Ma Y.D. (2013) Research on multi-dimensional space fault tree construction and application, Chin. Saf. Sci. J. 23, 32. [4] Wang P.Z. (2013) Factor spaces and factor data-bases, J. Liaoning Tech. Univ. (Nat. Sci.). 32, 1. [5] Wang H.D., Wang P.Z., Guo S.C. (2015) Improved factor analysis on factor spaces, J. Liaoning Tech. Univ.: Nat. Sci. 34, 539. [6] Cui T.J., Wang P.Z., Li S.S. (2017) The function structure analysis theory based on the factor space and space fault tree, Cluster. Comput. 20, 1387.

Chapter 6 System Reliability with Influencing Factors The system reliability is related to the factors that affect the system reliability in a multi-factor influencing environment. At the same time, the cumulative fault data rapidly increase as the system operates. These fault big data also contain system reliability characteristics. Therefore, it is necessary to solve the problem of the system reliability analysis suitable for big data and multiple factors. The research adopts the methods of the space fault tree (SFT) and factor space (FS), which are proposed by the authors to solve these problems. The SFT method can be used for system reliability analysis under the influence of multiple factors. The FS method applies big data processing and logic analysis. In the SFT method, the fault probability is used to represent the reliability, FS is introduced into the SFT, and the logical relationship between the reliability and influencing factors is studied. We also put forward a corresponding analysis method. Based on existing research, the SFT theory is improved here, to infer causal relationships, reduce the factor dimensions, and compress the fault data. This study is an intersection of safety system science, information science, and intelligent technology, and can provide the basic theory and application method for the reliability analysis of various industrial and mining enterprises, the military, and other fields. This chapter puts forward analysis methods of the system reliability and influencing factors and is divided into multiple sections. In section 6.1, the theory is introduced. In section 6.2, the theoretical analysis methods are described. In section 6.3, an analysis example is given. In section 6.4, some concluding remarks are provided. Sections 6.2 and 6.3 provide the main content. The analysis methods are established using an electrical system as an example, and the relationship between the reliability and the influencing factors is considered. Because of the larger number of parameters involved, if there are duplicate definition variables, they are explained with the prior definition in the section where the variables are located. To ensure that the analysis is clear, the abbreviation is not used in the discussion process. In the equations and variables, we use abbreviations without ambiguity. The concepts of the factor space were put forward by Professor DOI: 10.1051/978-2-7598-2499-1.c006 © Science Press, EDP Sciences, 2020

120

Space Fault Tree Theory and System Reliability Analysis List of symbols

Variable I L I/L F′ ψ = (U, X(F)) U ∏ ∨ ∧ X(fj) a R df f1
n−1, fn′, fn fn, g [a] γ = (A, [A]) Γ = (Γ, ∨, ∧, ¬) q Q q′ pj i B f(x) ^f (x)

Symbols Division length Study range length Resolution Boolean algebra system FS Universe Cartesian product Synthesis operation Decomposition operation fj phase space Atomic intension Background relationship Degree of distinction Influencing factors Target factor Atomic extension Concept of intension A and extension [A] Concept of Boolean algebra generated by (U, X(F)) Number of state occurrences Sum of the number of occurrences Frequency corresponding to the number of occurrences Edge distribution Set including all vertices of R Function with vector x as an independent variable Function fitted by the sample using least squares or other method

δ ξ lp rp Superscript p p k x c t c0 C A k Piq U = {u1, u2,…, um} m

Random variable with Gaussian distribution RVDF Average value of the fault probability Fault probability variance Relevant data of the fault probability Fault probability data The kth division interval Factor Using temperature Using time Temperature offset Temperature scaling Fault probability offset Component fault rate Characteristic function of the fault probability Data domain Number of objects (continued)

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(continued). Variable K D(fi) a = (a1,…, an) fMax dMax bk þ 1 q Q U* n′ k′ ui pn′ H(X(fn)) H(X(fn)|X(fk))) ΔH(fk) (f1,…, fn, g) k z1(g),…, zm(g) Γ1,…, Γm Ω1,…, Ωm A1,…, Am RVDF R c t FBRSCT IPJT

Symbols Corresponding phase of fi Value range division State Factor with dMax Highest resolution Basic concept Number of phases Sum of number of occurrences Quotient space Serial number of factors Serial number of phases Corresponding state Edge distribution Entropy Condition entropy of influencing factor fk on target factor fn Information gain of influencing factor fk on target factor fn Value pair Number of data collected Value range of target factor Division of fault data value pair Removal of value pair of redundant values from Γ1,…, Γm Fault probability distribution considering Ω1,…, Ωm Random variable decomposition formula Background relationship Use of temperature Use of time Factor background relationship state corresponding table Interior point judgment theorem

Note: Short, medium, and long are abbreviated as S, M, and L; low, medium, and high are abbreviated as Lo, M, and H; day is abbreviated as d.

Wang and his team [1–11]. Necessary modifications are made, and new definitions are added here.

6.1

Methodology of Concepts and Definitions

Definition 6.1. Set the division length I and study range length L, I/L can represent the degree of detail of the division, which is called a resolution. The smaller the resolution is, and the stronger the regularity of the data, the lower the discretization and randomness. The larger the resolution is, the greater the discretization and randomness of the data, weakening the regularity.

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Definition 6.2 [1]. Set F′ = P(F) = (F′; ∨, ∧, ¬, 0, 1) as a Boolean algebra system, denoted by X(F) = {X(f)}(f2F), where ψ = (U, X(F)) is called a FS in U. If (1) X (0) = {Ø}, (2) for any T  F and s, t 2 T, if s ∧ t = 0 (where s and t are irreducible), then X(∨{f|f 2 T}) = ∏f2TX(f) (where ∏ indicates a Cartesian product). Here, F = {f1,…, fn} = {f1 ∨ … ∨ fn} = 1 is called a full factor, 0 is called an empty factor, and ∨ and ∧ are the synthesis and decomposition operations, respectively. Definition 6.3 [2]. Given a factor set F = {f1,…, fn} in U and given that fj has the phase space X(fj)(j = 1,…, n), set X = X(f1) … X(fn). For any a = (a1,…, an) 2 X, denoted by [a] = F−1(a) = {u 2 U|F(u) = a}, [a] may be an empty set; if [a] ≠ ∅, then a is called an atomic intension; else, a is a fictitious phase. The set of all atomic intensions is denoted by R = F(U) = {a = (a1,…, an) 2 X|9u 2 U; f1(u) = a1,…, fn(u) = an}, which is called the background relationship R of F = {f1,…, fn}, also called the actual Cartesian product space. Here, F is an isomorphism from H(U, F) to R. Definition 6.4 [3]. Supposing that H(U, F) = {Ck = (uk,1,…, uk,n(k))}(k = 1,…, K), denoted by df = 1 − [n(1)(n(1) − 1) +    + n(K)(n(K) − 1)]/m(m − 1), df is called the degree of distinction of f with respect to U. Definition 6.5. Influencing factors f1
n−1 and target factor fn: According to the characteristics of the SFT analysis, the ultimate target is the fault probability of the component or system; thus, the fault probability is defined as the target factor. Factors that influence the fault probability of a component or system are defined as influencing factors. The number of influencing factors is n − 1; the target factor is the last factor in sections 6.2.2, 6.2.3, and 6.2.5. Similarly, the definitions of fn′, fn, and fn, g, are given in sections 6.2.4 and 6.2.6, respectively. Because the purpose of each chapter differs, the naming of the variables is slightly different. The influencing factors and target factor are called structure and function factors, respectively, in the last chapter. Definition 6.6. In the phase space, each matching phase is called a state. Definition 6.7. The mapping f: U → X(f) is called a factor, where U is a set of one type of object, which is called the definition domain, or domain. X(f) is the property or state (called phase) set mapped from the objects; this set is called phase space of factor. Definition 6.8. Given factor space (U, X(F)), if set R is the background relationship of factor F0 = {f1,…, fn}; then, for any a 2 R, α = (a, [a]) is called the atomic concept, and a and [a] are called the atomic intension and atomic extension of the concept α, respectively; for any A  R, denoted by [A] = [ {[a]|a 2 A}, Γ = {γ = (A, [A])|A  R}, and γ = (A, [A]) is the concept of intension A and extension [A]; Γ = (Γ, ∨, ∧, ¬) is called a Boolean algebra generated by (U, X(F)).

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123

Here, a and [a] are atoms. Because F is an isomorphic mapping from H(U, F) to R, the atoms must satisfy the Wille correspondence. Based on the concept of atoms, we can generate the entire concept algebra, which is theoretically simple. Definition 6.9. Concepts based on the object state and object relationships are called basic concepts. All basic concepts form a semi-lattice, called the semi-lattice of a basic concept. Definition 6.10. The atomic concept is the unity of the concept intension and extension obtained from the background relationship. The basic concept is the unity of the concept intension and extension obtained from the actual engineering. When the atomic concept is equivalent or included in the basic concept, the atomic concept is called a true concept. The true concept can be used to reason the causal relationship of practical problems, because of the unity of the concept intension and extension in both theory and fact. Definition 6.11. In the Cartesian product phase space formed by a factor phase space, there are some state combinations. The combinations of the phases of the factors are not common sense, and thus a becomes a fictitious state a′. Definition 6.12. The frequency state method is as follows: p¼

m X i¼1

qi0 =ui ¼

m X

q=Q=ui ;

i¼1

where q is the number of occurrences of the state, Q is the sum of the number of occurrences, and q′ is the frequency corresponding to the number of occurrences, denoted by qi0 = q=Q. The number is determined by the value classification of the corresponding fault data under factor phases. Here, ui is an object, and Σ{…Σ{q′(1)… (n)|(n) = 1,…, cn}…|(1) = 1,…, c1} = 1. Definition 6.13. Given a factor fj, denoted by pji = Σ{p(1)…(n)|(j) = i}(i = 1,…, cj), {pji|i = 1,…, cj} is called an edge distribution of fj, and thus pji = Σ{pji|i = 1,…, cj} = 1. Definition 6.14 [4]. Given factor space ψ = (U, X(F)) and F = {f1,…, fn, g}, the phase space of all factors is ordered and can be numerical by int, and such a phase space is then called the bracket space. Definition 6.15 [4]. If the background relationship R is a convex set in the bracket space, denoted as B = B(R) = {P|P is the vertex of R}, B is a set including all vertices of R, called the background base. Let R be the sample S, denoted by B (S) = {P|P is the vertex of S}, which is a set including all vertices of B, called the sample background base.

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6.2 6.2.1

Analysis of Relationship between Reliability and Influencing Factors Random Variable Decomposition Formula

The analysis of the system reliability data should proceed from two aspects. First, obtain the changing law of all data, and consider the physical attributes. The full trends are clear, such as the component reliability reduction with the working time. Second, obtain the discreteness and randomness of the local data, the human– machine-environmental influence may lead to irregular fluctuations of the reliability data. According to the random variable decomposition formula (RVDF) proposed by Professor Wang, we propose using the first item of RVDF to express the trend of the complete data; in addition, the second item expresses the data fluctuation and the third item expresses the local data uncertainty. We then construct a characteristic function to represent the reliability data (fault probability) of a component or system. The RVDF is shown in equation (6.1) [5]. n ¼ f ðxÞ þ ^f ðxÞ þ d

ð6:1Þ

where f(x) is a function with vector x as an independent variable; ^f (x) is a function fitted by the sample using the least squares or other method, and is a fine dealing with a few factors whose laws are weak; and δ is a random variable with a Gaussian distribution, which can be regarded as noise. Equation (6.1) only provides the RVDF framework. For the RVDF of the reliability data, we consider mainly the factor influencing the range of the reliability data, the division of the influencing range, and the discreteness and randomness representation of data within the division range. The factor influencing range is a reasonable range in which a change in the factor leads to a change in the reliability, which should be studied to remove the range of unreasonable data. To divide this range, consider the distribution of data points, where each division should contain at least two data points. Then, equation (6.1) is equivalent to equation (6.2). n ¼ f ðxÞ þ ^f ðxÞ þ d ¼ nðL; I ; dÞ

ð6:2Þ

In equation (6.2), the third item δ represents the discretization and randomness of the data along the fault probability axis. Here, δ obeys a Gaussian distribution, and the values of fault probability within a certain interval can be used to construct the Gaussian distribution d ¼ dðlp ; rp Þ, as shown in equation (6.3). d ¼ d ¼ p

fdpk jdpk g

1  ¼ pffiffiffiffiffiffi e 2p

p ðpi þ I k l Þ2 k p2 2r k

; 0  i  I ; 0  k  bL=I c

ð6:3Þ

where lp is the average value of the fault probability corresponding to the different factor values in the interval, as shown in equation (6.4). Here, rp is the fault probability variance corresponding to lp , as shown in equation (6.5). Here, the superscript p indicates the relevant data of the fault probability, p is the fault probability data, and k, is the kth division interval.

System Reliability with Influencing Factors ( l ¼ p

lpk jlpk

¼

I X

125 )

ðpi þ I k Þ=ðI þ 1Þ ; 0  k  bL=I c

ð6:4Þ

i¼0

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) u I u 1 X p p rk jrk ¼ t ðpi þ I k  lpk Þ2 ; 0  k  bL=I c I þ 1 i¼0

( rp ¼

ð6:5Þ

The first item in equation (6.2) indicates the regularity of the data along the factor axis. Because the SFT is currently aimed at studying the influence of working temperature c and time t on the reliability of components or systems, only c and t are discussed herein. According to [5], the function form of the first item fitted can be obtained, as shown in equation (6.6). The temperature and reliability data have a sine relationship, and the relationship between the time and reliability data is exponential.   cc0 sin C  2p þ A ð6:6Þ f ðxÞ ¼ 1  ekt where x is a factor, x 2 fc; tg; c is the working temperature; t is the working time; c0 is the temperature offset; C is the temperature scaling; A is the fault probability offset; and k is the component fault rate. The second item ^f (x) in equation (6.2) represents the residual fluctuation between f(x) and the actual data. This item mainly reflects the randomness of the data, which is expressed through the polynomial residual fitting, as shown in equation (6.7). ^f ðxÞ ¼ an x n þ an1 x n1 þ    þ a1 x 1 þ a0 ;

ð6:7Þ

where, x is the factor variable value. The characteristic function of the ith basic event (component fault) is shown in equation (6.8). Pi ¼ 1 

n Y

ð1  Piq Þ;

ð6:8Þ

q2ffactorg

Here, Piq in equation (6.8) is the characteristic function of the fault probability change affected by the qth factor of the ith component, and is equivalent to the RVDF, as in equation (6.9). p Piq ¼ nqi ¼ fiq ðxÞ þ ^fiq ðxÞ þ dqi

ð6:9Þ

Furthermore, to determine Pi for every component in the system, the fault probability distribution of the whole system can be obtained by analysing the fault tree structure, as shown in equation (6.10).

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126

PðT Þ ¼

K Y X r¼1 xi 2Er

6.2.2

qi 

X

Y

qi þ    þ ð1Þk1

1  r  s  k xi 2Er

k Y

qi ð6:10Þ

r¼1 x i 2 E 1 [    [ Ek

Causal Relationship Reasoning

In this section, the concept of the degree of distinction is used to analyse the causal relationship between the reliability and influencing factors, and the state absorption method and the state recurrence method are put forward [7, 8]. As the idea of the state absorption method, the final reasoning results contain all state information to the greatest extent possible, and the reasoning from a single influencing factor to the target factor is conducted breadth first. Step 1: Determine the degree of distinction of the factors in the original state table. Set H(U, F) = {Ck = (uk,1,…, uk,n(k))}(k = 1,…, K), df = 1 − [n(1)(n(1) − 1) +    + n(K)(n(K) − 1)]/m(m − 1). In addition, sort the degrees of distinction of the different factors. Step 2: Execute the state absorption. From the target factor, absorb only the same state to retain the states that first appear. Determine the degree of distinction of the factors in the state table after absorption. Step 3: Analyse the relationship between the influencing factors and the target factor. To retain a single influencing factor (there are two factors in the state table), the first row is the influencing factor, and the second row is the target factor. According to Step 2 for a state absorption, calculate the degree of distinction and analyse the logical relationship between the influencing factors and the target factor. Step 4: Sequentially rotate the influencing factors and the target factor to form the state table, according to Step 3, to analyse the logical relationship, until the rotation of all factors is completed. As the idea of the state recurrence method, to the greatest extent possible, the state information with a large frequency plays a leading role. The logical relationship between the influencing factors and the target factor is determined first with a large frequency in the state information, and the upper-level state table is then analysed step by step, that is, depth first. Step 1: Remove the first item occurring in all state items to retain the state items occurring twice or more, and the original states occurring twice become a state occurring once (generally remove the item of the first occurrence). Repeating the above process until the same state no longer occurs, the formation of an LL-times sub-state table is formed; the logical relationship is then analysed between the influencing factors and the target factor in this state table. Step 2: Remove the states from the LL-1 times state table that also appear in the LL times state table, obtain another LL times state table, and analyse the relationship between the influencing factors and the target factor in the state table. Step 3: If the two LL times state tables still cannot contain all states of the LL-1 times state table, go to Step 2. If all states are included, go to Step 4.

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Step 4: Remove the states from the LL-2 times state table that also appear in the LL-1 times state table, obtain another LL-1 times state table, and analyse the relationship between the influencing factors and the target factor in the state table. Step 5: If the two LL-1 times state tables still cannot contain all states of the LL2 times state table, go to Step 4. If all states are included, then continue to analyse the upper level, until the analysis reaches the (original) time state table once.

6.2.3

Causal Concept Extraction

Analyse the causal relationship between the component fault probability and the influencing factors and establish a causal concept method suitable for the fault analysis and computer processing. According to the characteristics of the SFT and FS, the following method is developed to analyse the causal concept in the SFT. Step 1: Set U = {u1, u2,…, um} is the data domain, and m is the number of objects. The qualitative phase X(fi) = {K1, K2,…,} for each factor is divided, where K represents the corresponding phase of fi, which qualitatively corresponds to the value domain division D(fi) = {(a1, a2],…, (ak−1, ak]}. Step 2: Establish the Cartesian product phase space for the factor phase space. Here, X = X(f1) ×    × X(fn) is the qualitative phase space of n factors in U, and a = (a1,…, an) 2 X is called a state. Remove any fictitious states (unreasonable states), and obtain n factors to form R in U. Step 3: According to R, form the atomic intension, and use domain U to form the atomic concept with correspondence. Step 4: Calculate the degree of distinction of each factor. In addition, sort the factors according to their degrees of distinction. Step 5: Sort the factors according to their degrees of distinction, and allow the factors to take turns to classify the objects in U. First, the object of U = {u1, u2,…, um} is divided according to the qualitative phase X(fMax) = {K1, K2,…,} of the factor fMax with the highest resolution dMax = dfQ = Max{df1 , df2 ,…, dfn }. In addition, classify {Ck = (uk,1,…, uk,n(k))}(k = 1,…, K) in U according to the appearance order of the phase (K1, K2,…,), forming a new domain U1 after classification and sorting, and obtain the basic concept bk . Set factor fMax as the factor with the second high-distinction degree dMax = Max{df1 , df2 ,…, dfQ1 , dfQ þ 1 ,…, dfn }, where the objects of U1 = {u1, u2,…, um} are divided according to the qualitative phase X(fMax) = {K1, K2,…,} of the factor fMax based on the object classification order of the last U. Then, classify {Ck = (uk,1,…, uk,n(k))}(k = 1,…, Q−1, Q+1,…, K) in U1 according to the appearance order of phase (K1, K2,…,), forming a new domain U2 after classification and sorting, and obtain the basic concept bk þ 1 . After the analysis of all factors, we can obtain all basic concepts. Step 6: Draw a figure of the semi-lattice of the basic concept to distinguish from the true concept.

6.2.4

Background Relationship Analysis

The reasoning method based on the background relationship in the FS is introduced into the SFT, and based on the characteristics of the fault data, the method is

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improved. The method can be used to analyse the relationship between the influencing factors and the fault probability. From the qualitative analysis, the relationship and degree of influence are obtained quantitatively with the fault frequency [4, 9]. The steps of this are as follows: Step 1 is the same as Step 1 in section 6.2.3. Step 2 is the same as Step 2 in section 6.2.3. Step 3: According to the background relationship, construct the FBRSCT. The table contents include state U = {u1, u2,…, um}, the influencing factor phase, the target factor phase, the number q of the phases corresponding to the states of occurrence, and the frequency q′ corresponding to the number q. The number q is determined by the value classification of the factor phases according to the fault data. The frequency is the corresponding number of occurrences divided by the sum Q of the number of occurrences. Step 4: Based on the FBRSCT, according to the background space distribution of the influencing factors and the target factor, the numbers of occurrences are in descending order and expressed by the frequency state method (definition 6.15). The number of occurrences of a state in the quotient space U* is shown in equation (6.11). ð6:11Þ p ða; ui Þ ¼ qi0 ¼ q=Q=ui Step 5: Calculate the edge distribution of each factor, as shown in equation (6.12). 0

Pkn0 ¼ p ðXk 0 ðfn0 ÞÞ ¼ Rp ðX ðf1 Þ      X ðfn ÞjXk 0 ðfn0 Þ ¼ X ðfn0 Þ; ui Þ

ð6:12Þ

where n′ is the serial number of factors n′ = [1, n]; k′ is the serial number of phase k′ = [1, k − 1]; ui is the corresponding state; and the edge distribution is pn′ = (Pn′1, Pn′2,…, Pn′k). Step 6: Analyse the influence of the influencing factors on the target factor. Here, fn′ is the influencing factor, and fn is the target factor. By comparing the relationship between p* and pn, we can understand the probability of influence of the influencing factors on the target factor in a certain state. Here, pn is as shown in equation (6.13), and p* is as shown in equation (6.14). pn ¼ ðp ðX ðf1 ÞÞ; p ðX ðf2 ÞÞ; . . .; p ðX ðfk1 ÞÞÞ

ð6:13Þ

p ðFn0 !n ðu  Þ ¼ X ðf1 Þ. . .X ðfn0 1 Þ. . .  . . .X ðfn0 þ 1 Þ. . .X ðfn Þjfn ðu  Þ ¼ Xk 0 ðfn ÞÞ ¼ p ðFn0 !n ðu  Þ ¼ X ðf1 Þ. . .X ðfn0 1 Þ. . .  . . .X ðfn0 þ 1 Þ. . .X ðfn ÞÞ=fn ðu  Þ ¼ Xk 0 ðfn ÞÞ; ð6:14Þ where Fn0 !n ðu  Þ = {Fn0 !n ðu  Þ|fn′(u*) = Xk′(fn′) and (Fn0 !n ðu  Þ = X(f1) … X(fn′−1)…*…X(fn′+1) … X(fn)).

6.2.5

fn(u*) = Xk′(fn)};

p*

Factor Dimension Reduction

If the sum of the numbers of influencing and target factors is greater than 3, the space distribution of the fault probability is then difficult to illuminate. Thus, with

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the increasing state of the basic events (component fault), the tree structure becomes complex, and the complexity growth is an order of magnitude. Finally, the SFT is difficult to actually apply. To address this problem, a method for reducing the influencing factors is presented using the information gain method of the FS. Determine whether to reduce the number of dimensions (remove the factor) by considering the degree of information gain of the influencing factors on the target factor [4, 9]. Steps 1–5 are the same as Steps 1, 2, 3, 5, and 6 in section 6.2.4, respectively. Step 6: Build a condition distribution table of the influencing factors on the target factor. The data in the table are derived from the results of Step 5. The table contains the following: p*(X(fn)|X(f1)), f1, p*(X(fn)|X(f2)), f2,…, p*(X(fn)|X(fn−1)), and fn−1; the table row is X(fi) = {K1, K2,…,}, and the condition is pn = p*(X(fn)). The condition distribution table is constructed under different conditions of the target factor, and the number is the same as the number of the X(fn) = {K1, K2,…,} phase. Step 7: Calculate the information gain of the influencing factors on the fault probability (target factor). Regarding the target factor fn, its edge phase distribution pn = {pn1,…, pnk} is a finite distribution column, the entropy of which is shown in equation (6.15). X   pnk ln p pnk ð6:15Þ H ðXðfn ÞÞ ¼  k

We can find the conditional probability p(X(fn)|X(fk)), which is also a finite distribution of k columns, and its entropy is shown in equation (6.16). X pðX ðfn ÞjX ðfk ÞÞ ln pðX ðfn ÞjX ðfk ÞÞ ð6:16Þ H ðX ðfn ÞjX ðfk ÞÞ ¼  k

This entropy is the result of a factor distribution when the factor fn is fixed to the phase X(fn). Its mathematical expectation is shown in equation (6.17), which is called the conditional entropy of the influencing factor fk on the target factor fn. X H ðX ðfn ÞjX ðfk ÞÞpðX ðfk ÞÞ ð6:17Þ H ðX ðfn ÞjX ðfk ÞÞ ¼ l

The information gain of the influencing factor fk on the target factor fn is shown in equation (6.18). DH ðfk Þ ¼ H ðfn Þ  H ðfn jfk Þ

ð6:18Þ

Step 8: Analyse the information gain of the influencing factors on the target factor. According to the dimensional reduction criteria, we choose the appropriate influencing factors to reduce the number of dimensions. The criteria of the dimensional reduction are as follows: (1) The information gain of the influencing factors on the target factor is smaller. According to the probability distribution characteristic, the factors with information gain of less than 0.05 can be deleted, that is the factor can be neglected.

130

Space Fault Tree Theory and System Reliability Analysis

(2) When the information gain difference of the two factors is less than 0.05, the influence distributions (condition probability distribution) of the two influencing factors on the target factor are basically the same. At this time, combine the two factors to reduce the influencing factors, that is they are basically under the same situation as the target factor, which means that the difference in the condition probability of the two influencing factors is less than 0.05 in the target factor, and the number of these states is greater than 0.95 of the total number of states. If the above two items are met, we can then remove the influencing factors to reduce the number of dimensions.

6.2.6

Compression of Fault Probability Distribution

The SFT analysis of the basic data can be expressed as value pairs that contain the influencing factors and the target factor, such as the working time, working temperature, and fault probability. The goal of this section is to retain the most important value pairs among all value pairs, eliminate redundant value pairs to reduce the fault data, and find the characteristic area of different fault probabilities (fault probability distributions). First, we present the background base concept and interior point judgment theorem (IPJT) in the FS. The background base can generate R and is the compression of R without information loss. Regardless of the size of the data, the number of sample background bases is always maintained at low dimensions. It is determined whether new data are interior points in the sample background base. If they are, the data are deleted; otherwise, they will be included in the sample background base. Professor Wang and his students developed the IPJT, where P is an interior point of S if and only if there is a point Q in S, and the ray PQ forms an obtuse angle with the ray Po, that is, (Q–P, o–P) < 0, where, o is a point with the average value coordinates of the all interior points [9]. Example 6.1. In figure 6.1, it is given that S contains the three points a = (1, 5), b = (2, 0), and c = (5, 4). Determine whether d = (2, 2) and e = (1, 0) are in S. Solution: o = (a + b + c)/3 = (2.7, 3), (o–d, a–d) = (0.7, 1)(−1, 3) = 2.3 > 0; (o–d, b–d) = (0.7, 1)(0, −2) = −2 < 0; (o–d, c–d) = (0.7, 1)(3, 2) = 4.1 > 0. If a negative value is obtained, then d is the interior point of S. Here, (o–e, a–e) = (1.7, 3)(0, 5) = 15 > 0; (o–e, b–e) = (1.7, 3)(1, 0) = 1.7 > 0; (o–e, c–e) = (1.7, 3)(4, 5) = 15 > 0. There is no negative value; thus, e is not an interior point of S. According to the above method, the following is the fault probability distribution calculation process based on a data compression. Step 1: Set the value pair (f1,…, fn, g) of the fault data, where f1,…, fn are the factors that affect the fault probability and are influencing factors, and g is the fault probability, which is the target factor.

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FIG. 6.1 – An example of the interior-point method. Step 2: The actual fault data are expressed as a number of pairs (f1,…, fn, g)k, where k is the number of data pairs. Step 3: Dividing the value range of the target factor, Z(g) = {z1(g),…, zm(g)| z1(g) \ … \ zm(g) = 1 and z1(g) <    < zm(g)}, where the target factor value range is [0, 1]. The range of the target factor g is divided into m subspecies z1(g),…, zm(g). Step 4: Allocate the fault data value pair (f1,…, fn, g)k into Γ1,…, Γm, Γ1 = {(f1, …, fn, g)k1|g 2 z1(g)},…, Γm = { (f1,…, fn, g)km|g 2 zm(g)}, according to the divisions of Z(g). Step 5: For each division Γ1,…, Γm, using the IPJT to remove the value pair of redundant values, form the corresponding value pair sets Ω1,…, Ωm. In addition, according to Ω1,…, Ωm, draw the fault probability distribution A1,…, Am. The initial points that can be chosen are the first three value pairs or any three value pairs, and the results are the same. Step 6: Analyse the fault probability distribution and determine the relationship of the fault probability distribution under each division. According to the characteristics of the SFT and considering that the IPJT is applicable only to convex sets, there may be an overlapping phenomenon owing to the divisions of the fault probability distribution. The two or more divisions have overlapping areas, and the overlapping areas can cause the target factor to belong to two or more different divisions, which is illogical. The distribution of the large fault probability in the SFT may cover the distribution of the small fault probability. Given the aforementioned, the processing method is presented for an overlapping fault probability distribution considering the different divisions given; the processing method is the priority to meet the small fault probability distribution. If Ai \ Aj ≠ ∅ and i < j, then Ai \ Aj = Ai \ j 2 Ai and Ai \ j 62 Aj. The fault probability distributions under different divisions after removing the overlapping areas are denoted as Z1,…, Zm, and in general, Zm = Am \ ðA1 [ ::: [ Am1 Þ.

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6.3

Algorithm Application

Example algorithm applications are given in section 2, which show the fault of five components (X1–5) in the electrical system, and the factor range within the system operation is 0–500 days at 0 °C–40 °C. In this section, our study object is X1, and its fault is related with temperature, time, or humidity.

6.3.1

Random Variable Decomposition Formula

We studied the relationship between the fault probability of X1 and the working time t and temperature c. In the fault graph, t is the abscissa, c is the ordinate, and the data points represent faults. The component fault distributions of c and t are obtained along the t and c axis projections, respectively. Figure 6.2 shows a partial distribution of the component fault for c. Setting I = 3 and L = [2, 29], where the resolution is 11.1% (resolution of greater than 1/L = 3.7%), according to equations (6.3) to (6.5), the calculated results are as shown in table 6.1. The RVDF of the reliability data with the influence of c can be expressed as equation (6.19) according to table 6.1 and equation (6.2). Equations (6.4) and (6.5) can be taken into equation (6.3) to obtain the third item δ of equation (6.19). n ¼ f ðxÞ þ ^f ðxÞ þ d ¼ nð8; 3; dðlp ; rp ÞÞ

ð6:19Þ

where lp ¼ f0:4583; 0:2847; 0:2153; 0:1875; 0:1805; 0:3334; 0:3194; 0:2986; 0:6180g and rp ¼ f0:0637; 0:0602; 0:0602; 0:0303; 0:0139; 0:0922; 0:1266; 0:0842; 0:1671g: For the first item, the fitting points are f ðx; pÞjx ¼ x pk ; p ¼ lpk ; k 2 fkj 0  k; k  8gg according to the first item of equation (6.6); executing the least    2p þ 1:15, as shown in figure 6.2. squares fitting, the result is f ðxÞ ¼ sin x34:01 81:8 The residuals between the actual data points and the fitting curve are obtained, and

FIG. 6.2 – RVDF process with influence of c.

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TAB. 6.1 – RVDF parameters with influence of c. k X range x pk lpk rpk

0 2–5 3.5000 0.4583 0.0637

1 5–8 6.5000 0.2847 0.0602

2 8–11 9.5000 0.2153 0.0602

3 11–14 12.5000 0.1875 0.0303

4 14–17 15.5000 0.1805 0.0139

5 17–20 18.5000 0.3334 0.0922

6 20–23 21.5000 0.3194 0.1266

7 23–26 24.5000 0.2986 0.0842

8 26–29 27.5000 0.6180 0.1671

the residual distribution of the component fault probability is obtained with t, as shown in figure 6.3. For the second item, according to the residual coordinates in figure 6.3 and equation (6.7), the residual variation is determined by the polynomial fitting. With an increase in the number of times, the prime coefficient gradually reaches zero, although six applications can meet the required precision. The residual fitting polynomial is ^f ðxÞ ¼ 4  108 x 6 þ 6  106 x 5  3  104 x 4 þ 0:007x 3  0:074x 2 þ 0:328x 1  0:463, as shown in figure 6.3. Through a combination of the above three items used to express the influence of c on the reliability of component X1, equation (6.19) is rewritten as equation (6.20). P1c

p ¼ nc1 ¼ f1c ðxÞ þ ^f1c ðxÞ þ dc1 x34:01 ¼ sinð 81:8  2pÞ þ 1:15  4  108 x 6 þ 6  106 x 5 3  104 x 4 þ 0.007x 3  0.074x 2 þ 0.328x 1  0.463 2 ðpc lpk Þ  p2 þ p1ffiffiffiffi e 2rk ; 0  k  8; 0  i  3; 0  p  1 2p

ð6:20Þ

In equation (6.20), the first two items show the influence of the working temperature change on the change in the component reliability data distribution, and the third item represents the changing characteristics of the reliability data locally. The above resolution is 11.1%, which means that if the data characteristic is greater than I = 3, then the resolution cannot identify the data characteristics.

FIG. 6.3 – Residual distribution of component fault probability with c.

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It was determined that the RVDF affects the reliability of the data when working time is considered. In the distribution of t, the data are changed during a 50-day period to improve data utilization; the data according to their period are merged and normalised. The 0–18 days of changes in the component fault probability are representative and are shown in figure 6.4. The data merging and normalization process are described in [6]. Figure 6.4 shows the influence of the above analysis method on the reliability when working time. Set I = 2, L = [1, 17], with a resolution of 12.5% (resolution of greater than 6.25%). The results according to equations (6.3)–(6.5) are shown in table 6.2. The RVDF of the reliability data with the influence of t can be expressed as equation (6.21) according to table 6.2 and equation (6.2). Equations (6.4) and (6.5) can be substituted into equation (6.3) to obtain the third item δ of equation (6.21). n ¼ f ðxÞ þ ^f ðxÞ þ d ¼ nð7; 2; dðlp ; rp ÞÞ

ð6:21Þ

where lp ¼ f0:7333; 0:7667; 0:7333; 0:8000; 0:9667; 0:9667; 0:9667; 1:0000g and rp ¼ f0:1247; 0:1247; 0:0943; 0:0816; 0:0471; 0:0471; 0:0471; 0g:   The fitting points are ðx; pÞjx ¼ x pk ; y ¼ lpk ; k 2 fkj0  k; k  7g according to the first item of equation (6.6); when executing the least squares fitting, the result is f ðxÞ ¼ 1  e0:32t , as shown in figure 6.4.

FIG. 6.4 – RVDF process with influence of t. TAB. 6.2 – RVDF parameters with influence of t. k X range x pk lpk rpk

0 1–3 2.0000 0.7333 0.1247

1 3–5 4.0000 0.7667 0.1247

2 5–7 6.0000 0.7333 0.0943

3 7–9 8.0000 0.8000 0.0816

4 9–11 10.0000 0.9667 0.0471

5 11–13 12.0000 0.9667 0.0471

6 13–15 14.0000 0.9667 0.0471

7 15–17 16.0000 1.0000 0

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For the second item, according to the residual given in figure 6.5 and equation (6.7), the residual fitting polynomial is ^f ðxÞ ¼ 7  107 x 6 þ 5:5  105 x 5  1:5 103 x 4 þ 0:019x 3  0:098x 2 þ 0:086x 1 þ 0:372, as shown in figure 6.5. For the combination of the above three items used to express the influence of t on the reliability of component X1, equation (6.21) is rewritten as equation (6.22). P1t

p ¼ nt1 ¼ f1t ðxÞ þ ^f1t ðxÞ þ dt1 ¼ 1  e0:32x þ  7  107 x 6 þ 5:5  105 x 5  1.5  103 x 4 þ 0.019x 3 0.098x 2 þ 0.086x 1 þ 0.372 ð6:22Þ p 2 pt l Þ ð k  p2 þ p1ffiffiffiffi e 2rk 0  k  7; 0  i  2; 0  p  1 2p

Using equations (6.8), (6.20), and (6.22), the fault probability of component X1 when influenced by c and t is given as equation (6.23), n = 2, q 2 {c, t}, then, the equation is simplified. P1

¼1

2 Q

ð1  P1q Þ ¼ 1  ð1  P1c Þð1  P1t Þ

q2fc;tg

p p ¼ 1  ð1  f1c ðcÞ  ^f1c ðcÞ  dc1 Þð1  f1t ðtÞ  ^f1t ðtÞ  dt1 Þ p p p p ¼ 1  ð1  f1c  ^f1c Þð1  f1t  ^f1t Þ þ ð1  f1t  ^f1t Þdc1 þ ð1  f1c  ^f1c Þdt1  dc1 dt1

ð6:23Þ ^f c 1

= 0.1872, = −0.0356, = 0.7220, ^f1t = Setting c = 10 °C and t = 4 d, 2 ðplpk Þ  p2 cp 1 0.0328, and d1 ¼ pffiffiffiffi e 2rk . Because c = 10, when c 2 x = [8, 11], lpk = 0.2153, 2p f1c

ðpc 0:2153Þ2

f1t

ðpt 0:7667Þ2

rpk = 0.0602, and dc1 ¼ p1ffiffiffiffi e 20:06022 . Similarly, dt1 ¼ p1ffiffiffiffi e 20:12472 . When c = 10 °C 2p 2p and t = 4 d, the fault probability of component X1 influenced by t and c is shown through equation (6.24). p

p

FIG. 6.5 – Residual distribution of component failure probability with t.

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Space Fault Tree Theory and System Reliability Analysis

ðpc 0:2153Þ2 1 P1 ¼ 0:7920 þ 0:2452  pffiffiffiffiffiffi e 20:06022 þ 0:8484 2p  2 2 ðpt 0:7667Þ2 c 0:2153Þ þ 1 ðpt 0:76672Þ 1  ðp20:0602 2 20:12472  pffiffiffiffiffiffi e 20:1247  pffiffiffiffiffiffi e 2p 2p

ð6:24Þ

Equation (6.24) shows the two factors composed of a phase space, the resolution of which is 11.1% × 12.5% = 1.38%. The theoretical resolution must be more than 3.7% × 6.25% = 0.23%. Clearly, the greater the number of factors that are applied, the stronger the distinction and the smaller the resolution. In the aforementioned study, an RVDF method was realised to express the data reliability. The RVDF can be used as a characteristic function of the SFT.

6.3.2

Causal Relationship Reasoning

Based on the discussion in section 6.3.1, 20 fault states are extracted as U = {a, b, …, t}, as shown in table 6.3. The fault state contains three factors, the first two are influencing factors, namely, the working time, f1, and working temperature, f2, and the target factor is the component fault probability, f3. They form the qualitative phase space X = X(f1) … X(f3) on U. The qualitative phase of each factor, and the corresponding value range, is as follows: X(f1) = {S, M, L} = {(0, 15] d, (15, 30] d, (30, 50] d}, X(f2) = {Lo, M, H} = {(0, 15] °C, (15, 25] °C, (25, 40] °C}, X(f3) = {Lo, M, H} = {(0, 30]%, (30, 60]%, (60, 100]%}. Note that short, medium, and long are abbreviated as S, M, and L; low, medium, and high are abbreviated as Lo, M, and H; and day is abbreviated as d. (1) Reasoning process of state absorption method. Step 1: Calculate the degree of distinction, m = 20. f1:n(1) = 6, n(2) = 7, n(3) = 7, and df1 = 1 − (6 × 5 + 7 × 6 + 7 × 6)/ (20 × 19) = 0.7; f2:n(1) = 8, n(2) = 6, n(3) = 6, and df2 = 1 − (8 × 7 + 6 × 5 + 6 × 5)/ (20 × 19) = 0.6947; f3:n(1) = 1, n(2) = 4, n(3) = 15, and df3 = 1 − (1 × 0 + 4 × 3 + 15 × 14)/ (20 × 19) = 0.4158; thus, df1 > df2 > df3. Step 2: Execute the state absorption. f1(t) = f1(s), f2(t) = f2(s), f3(t) = f3(s), and s = s [ t; f1(q) = f1(r), f2(q) = f2(r), f3(q) = f3(r), and q = q [ r; f1(n) = f1(o) = f1(p), f2(n) = f2(o) = f1(p), f3(n) = f3(o) = f1(p), and n = n [ o [ p; f1(l) = f1(m), f2(l) = f2(m), f3(l) = f3(m), and l = l [ m; f1(g) = f1(n) = f1(i), f2(g) = f2(n) = f1(i), f3(g) = f3(n) = f1(i), and g = g [ n [ i. The state after absorption is shown in table 6.4. Calculate degree of distinction, m = 13. f1:n(1) = 6, n(2) = 4, n(3) = 3, df1 = 1 − (6 × 5 + 4 × 3 + 3 × 2)/(13 × 12) = 0.6923;

U f1 f2 f3

States a S Lo H

b S Lo M

c S M Lo

d S M M

e S H H

f S H M

g M Lo H

h M Lo H

i M Lo H

j M M H

k M M M

l M H H

m M H H

n L Lo H

o L Lo H

p L Lo H

q L M H

r L M H

s L H H

t L H H

TAB. 6.4 – State after absorption. U f1 f2 f3

States a S Lo H

b S Lo M

c S M Lo

d S M M

e S H H

f S H M

g M Lo H

j M M H

k M M M

l M H H

n L Lo H

k M M

l M H

n L H

q L M H

s L H H

System Reliability with Influencing Factors

TAB. 6.3 – Basic status table.

TAB. 6.5 – State of f1 and f3. U f1 f3

States a S H

b S M

c S Lo

d S M

e S H

f S M

g M H

j M H

q L H

s L H

137

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138

f2:n(1) = 4, n(2) = 5, n(3) = 4, df2 = 1 − (4 × 3 + 5 × 4 + 4 × 3)/(13 × 12) = 0.7179; f3:n(1) = 1, n(2) = 4, n(3) = 8, df3 = 1 − (1 × 0 + 4 × 3 + 8 × 7)/(13 × 12) = 0.5641; thus, df1 > df2 > df3. Step 3: Analyse the relationship between the working time and fault probability. Here, the state table of f1 and f3 formed is as shown in table 6.5. The absorption for each state was determined from the target factor f3, a = a [ e; g = g [ j [ l; n = n [ q [ s; and b = b [ d [ f, the results of which are shown in table 6.6. Calculate degree of distinction, m = 6. f1:n(1) = 3, n(2) = 2, n(3) = 1, and df1 = 1 − (3 × 2 + 2 × 1 + 1 × 0)/ (6 × 5) = 0.7333; f3:n(1) = 1, n(2) = 2, n(3) = 3, and df3 = 1 − (3 × 2 + 2 × 1 + 1 × 0)/ (6 × 5) = 0.7333. The logical relationship between the influencing factors and the target factor is analysed. Two factors have the same degree of distinction. With a short working time, the fault probability has three states: high, medium, and low. With the medium working time, the fault probability has two states: high and medium. With a long working time, the fault probability is high. This indicates that the fault probability when the component is used for a long working time has nothing to do with the working temperature; when the component is used for a short working time, the fault probability is significantly affected by the working temperature. Step 4: Analyse the relationship between the working temperature and the fault probability. The working temperature and fault probability are used to form the state table, as shown in table 6.7. The state absorption, the results of which are shown in table 6.8, was carried out from the target factor f3, where a = a [ g [ n, d = d [ k, e = e [ l [ s, and j = j [ q. TAB. 6.6 – State after absorption. U

States a S H

f1 f3

b S M

c S Lo

g M H

k M M

n L H

TAB. 6.7 – State of f2 and f3. U f2 f3

States a Lo H

b Lo M

c M Lo

d M M

e H H

f H M

g Lo H

j M H

k M M

l H H

n Lo H

q M H

s H H

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139

TAB. 6.8 – State table after absorption. U f2 f3

States a Lo H

b Lo M

c M Lo

d M M

e H H

f H M

j M H

Calculate distinction degree, m = 7. f2:n(1) = 2, n(2) = 3, n(3) = 2, and df2 = 1 − (2 × 1 + 3 × 2 + 2 × 1)/ (7 × 6) = 0.7619; f3:n(1) = 1, n(2) = 3, n(3) = 3, and df3 = 1 − (1 × 0 + 3 × 2 + 3 × 2)/ (7 × 6) = 0.7143. With a low working temperature, the fault probability has two states: high and medium. With a medium working temperature, the fault probability has three states: high, medium and low. With a high working temperature, the fault probability has two states: high and medium. This shows that the working temperature influencing the fault probability is more extensive. The states under both high and low working temperatures will lead to a high and medium fault probability, whereas under only medium working temperature the fault probability is low. (2) Reasoning process of state recurrence method. According to the basic states in table 6.3 (once), removing the first occurrence of all states, twice the state table size is obtained, as shown in table 6.9. Remove the first occurrence of all the state items in table 6.9 to obtain the 3-times state table, as shown in table 6.10. Table 6.10 does not have the same states in forming a 3-times state table. Based on the analysis of the relationship between the influencing factors and the target factor in this 3-times state table, we can see that the low working temperature led to a high fault probability, and that this was independent of the working time. The same states in table 6.10 are removed from table 6.9, with none of the same states remaining, forming another 3-times state table, as shown in table 6.11. We can see from table 6.11 that the high or medium working temperature and the medium or long working time lead to a high fault probability, although the corresponding relationship is not obvious. The two 3-times state tables 6.10 and 6.11 have completed the analysis, that is, the twice state table of table 6.9 have completed the analysis. The same states in table 6.9 are removed from table 6.3, and another twice state table is obtained, as shown in table 6.12. TAB. 6.9 – Twice state table. U f1 f2 f3

States h M Lo H

i M Lo H

m M H H

o L Lo H

p L Lo H

r L M H

t L H H

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TAB. 6.10 – 3-Times state table. U

States i M Lo H

f1 f2 f3

p L Lo H

TAB. 6.11 – 3-Times state table. U

States m M H H

f1 f2 f3

r L M H

t L H H

Table 6.12 does not include the same state, and thus it cannot be decomposed into 3-times state tables. The logical relationship in table 6.12 is analysed. When the working time is short, the low or high working temperature corresponds with the high fault probability. The medium working time corresponds with the low or medium fault probability. A short working time has a large influence on the fault probability. The medium working temperature is most favourable to reducing the fault probability. Figure 6.6 shows the distribution of the fault probability of component X1 in section 6.3.1 for t and c. This distribution is obtained through a large number of actual data and algorithms and can reflect the actual situation. Comparing the logical relationship between t and c and the component fault probability obtained using the above two methods, figure 6.6 reflects most of the fault probability distribution characteristics. However, the data and algorithm required by the two methods described herein are simpler than those shown in figure 6.6. When the sample data are greater in number and the influencing factors are fewer, the state absorption method is relatively simple. When the sample data are not large in number, the number of influencing factors is greater; and the state recurrence method is simpler. It is not true that the more the sample data that are available, the better the analysis because the more the sample data applied, the greater the number of influencing factors that may be involved. If more reasonable TAB. 6.12 – Twice state table. U f1 f2 f3

States a S Lo H

b S Lo M

c S M Lo

d S M M

e S H H

f S H M

j M M H

k M M M

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FIG. 6.6 – Fault probability distribution of X1. (a) Probability distribution, (b) Derivatives of probability distribution to t, (c) Derivatives of probability distribution to c.

142

Space Fault Tree Theory and System Reliability Analysis

FIG. 6.6 – (continued).

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FIG. 6.6 – (continued).

143

144

Space Fault Tree Theory and System Reliability Analysis

factors cannot be found, it may lead to contradictory results. By contrast, if the sample is correct, a contradictory relationship exists between the influencing factors and the target factor; thus, there will be crucial factors that are not considered.

6.3.3

Causal Concept Extraction

Table 6.3 shows an example of applying the method. Step 1 is the same as the qualitative phase division of table 6.3. Step 2: Three factors are the working time f1, working temperature f2, fault probability f3, and constituent Cartesian product phase space of factors X = X(f1) X(f2)X(f3) = {SLoLo, SLoM, SLoH, SMLo, SMM, SMH, SHLo, SHM, SHH, MLoLo, MLoM, MLoH, MMLo, MMM, MMH, MHLo, MHM, MHH, LLoLo, LLoM, LLoH, LMLo, LMM, LMH, LHLo, LHM, LHH}, where X is divided into 27 lattices. According to common sense, considering the removal of a fictitious state as an example, f3 increases with f1, and when X(f1) = long, then X(f3) ≠ low or X(f3) ≠ medium. Thus, the states [long＊low] and [long＊medium] are fictitious, where ＊ = X(f2). The fictitious states in the phase space are six lattices {LLoLo, LLoM, LMLo, LMM, LHLo, LHM}. Step 3: Remove the six aforementioned lattices; the remaining states with 21 physical logic lattices are the intensions of the atoms. The atomic concept corresponding to table 6.3 is as follows: [SLoLo] = Ø, [SLoM] = {b}, [SLoH] = {a}, [SMLo] = {c}, [SMM] = {d}, [SMH] = Ø, [SHLo] = Ø, [SHM] = {f}, [SHH] = {e}, [MLoLo] = Ø, [MLoM] = Ø, [MLoH] = {g, h, i}, [MMLo] = Ø, [MMM] = {k}, [MMH] = {j}, [MHLo] = Ø, [MHM] = Ø, [MHH] = {l, m}, [LLoH] = {n, o, p}, [LMH] = {q, r}, [LHH] = {s, t}. Where: [SLoLo] = Ø, [SMH] = Ø, [SHLo] = Ø, [MLoLo] = Ø, [MLoM] = Ø, [MMLo] = Ø, [MHLo] = Ø, and [MHM] = Ø. The atomic concepts only have an intension and no extension and cannot constitute the atomic concept. Here, a1 = ([SLoM],{b}), a2 = ([SLoH],{a}), a3 = ([SMLo], {c}), a4 = ([SMM], {d}), a5 = ([SHM], {f}), a6 = ([SHH], {e}), a7 = ([MLoH], {g, h, i}), a8 = ([MMM], {k}), a9 = ([MMH], {j}), a10 = ([MHH], {l, m}), a11 = ([LLoH], {n, o, p}), a12 = ([LMH], {q, r}), a13 = ([LHH], {s, t}). They have both an intension and an extension and form the 13 atomic concepts with correspondence. Step 4 is the same as in Example 6.1, Step 1 in section 6.2.2. Step 5: (1) According to the sorting of the degree of factor distinction to classify the objects, as is known, dMax = df1 = Max{df1 , df2 , df3 }, fMax = f1, X(fMax) = {S, M, L}, and U = {a, b,…, t} are divided and sorted into the new classification domain U1, as shown in table 6.13. After the division of f1, U1 = C1{a, b, c, d, e, f} + C2{g, h, i, j, k, l, m} + C3{n, o, p, q, r, s, t}. According to the corresponding intension and extension of f1, we can obtain three basic concepts: β1 = (S, C1), β2 = (M, C2), and β3 = (L, C3). (2) As is known, dMax = df2 = Max{df1 , df2 , df3 } and fMax = f2, X(fMax) = {Lo, M, H}; C1{a, b, c, d, e, f} + C2{g, h, i, j, k, l, m} + C3{n, o, p, q, r, s, t} is classified and sorted to obtain a new domain U2, as shown in table 6.14.

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After C1{a, b, c, d, e, f} is divided by f2, C1{a, b, c, d, e, f} = C11{a, b} + C12{c, d} + C13{e, f}. The three basic concepts here are β11 = (SLo, C11), β12 = (SM, C12), and β13 = (SH, C13). After C2{g, h, i, j, k, l, m} is divided by f2, C2{g, h, i, j, k, l, m} = C21{g, h, i} + C22{j, k} + C23{l, m}. The three basic concepts here are β21 = (MLo, C21), β22 = (MM, C22), and β23 = (MH, C23). After C3{n, o, p, q, r, s, t} is divided by f2, C3{n, o, p, q, r, s, t} = C31{n, o, p} + C32{q, r} + C33{s, t}. The three basic concepts here are β31 = (LLo, C31), β32 = (LM, C32), and β33 = (LH, C33). (3) As is known, dMax = df3 = Max{df1 , df2 , df3 }, fMax = f3, and X(fMax) = {Lo, M, H}; C11{a, b} + C12{c, d} + C13{e, f} + C22{j, k} is classified and sorted to obtain a new domain, U3, as shown in table 6.15. After C11{a, b} is divided by f3, C11{a, b} = C111{a} + C112{b}. The two basic concepts here are β111 = (SLoH, C111), β112 = (SLoM, C112). After C12{c, d} is divided by f3, C12{c, d} = C121{c} + C122{d}. The two basic concepts here are β121 = (SMLo, C121), β122 = (SMM, C122). After C13{e, f} is divided by f3, C13{e, f} = C131{e} + C132{f}. The two basic concepts here are β131 = (SHH, C131), β132 = (SHM, C132). After C22{j, k} is divided by f3, C22{j, k} = C221{j} + C222{k}. The two basic concepts here are β221 = (MMH, C221), β222 = (MMM, C222). The basic concepts obtained above are β1 = (S, C1), β2 = (M, C1), β3 = (L, C3), β11 = (SLo, C11), β12 = (SM, C12), β13 = (SH, C13), β21 = (MLo, C21), β22 = (MM, C22), β23 = (MH, C23), β31 = (HLo, C31), β32 = (HM, C32), β33 = (HH, C33), β111 = (SLoH, C111), β112 = (SLoM, C112), β121 = (SMLo, C121), β122 = (SMM, C122), β131 = (SHH, C131), β132 = (SHM, C132), β221 = (MMH, C221), and β222 = (MMM, C222). Step 6: Draw a semi-lattice based on the basic concepts obtained from the above analysis, as shown in figure 6.7. From figure 6.7, we can obtain three different concepts: the basic concept that cannot be divided, i.e. a basic concept without deviation; the medium basic concept; and the phase concept that does not contain a fault probability, i.e. a no-fault probability basic concept. For the non-deviation basic concept, β21 = (MLo, C21), β23 = (MH, C23), β31 = (LLo, C21), β32 = (LM, C22), β33 = (LH, C23), β111 = (SLoH, C111), β112 = (SLoM, C112), β121 = (SMLo, C121), β122 = (SMM, C122), β131 = (SHH, C131), β132 = (SHM, C132), β221 = (MMH, C221), and β222 = (MMM, C222). Here, β111 , a2 , β112 , a1 , β121 , a3 , β122 , a4 , β131 , a6 , β132 , a5 , β221 , a9 , β222 , a8 . These concepts form eight groups of equivalence relations, that is, the intension and extension of the atomic concept in R is equivalent to the intension and extension of the conceptual semi-lattice analysis in the example. Here, R is derived from the combination of phase spaces, and the conceptual semi-lattice is derived from the actual example. Thus, the equivalent relationship is unity between the theoretical and practical concepts of the intension and extension, and the basic concepts are true concepts. A true concept can be used to analyse the causal relationship considering the theory and practice.

U f1 f2 f3

146

TAB. 6.13 – State of f1. States a S Lo H

b S Lo M

c S M Lo

d S M M

e S H H

f S H M

g M Lo H

h M Lo H

i M Lo H

j M M H

k M M M

l M H H

n L Lo H

o L Lo H

p L Lo H

q L M H

r L M H

s L H H

t L H H

m M H H

n L Lo H

o L Lo H

p L Lo H

q L M H

r L M H

s L H H

t L H H

m M H H

n L Lo H

o L Lo H

p L Lo H

q L M H

r L M H

s L H H

t L H H

TAB. 6.14 – State of f2. U f1 f2 f3

States a S Lo H

b S Lo M

c S M Lo

d S M M

e S H H

f S H M

g M Lo H

h M Lo H

i M Lo H

j M M H

k M M M

l M H H

TAB. 6.15 – State of f3. U f1 f2 f3

States a S Lo H

b S Lo M

c S M Lo

d S M M

e S H H

f S H M

g M Lo H

h M Lo H

i M Lo H

j M M H

k M M M

l M H H

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m M H H

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147

FIG. 6.7 – Semi-lattice of basic concept. Here, a7 β21, a10 23, a11 β31, a12 β32, a13 β33. These concepts form five groups of inclusion relationships, the former atomic concepts contain the intension with three factors, and the latter basic concepts contain the intension with two factors. Therefore, the atomic concept is a special case of the basic concept. However, their extension is the same, and thus a7 , a10 , a11 , a12 and a13 are true concepts that can be used to judge the causality. For the medium basic concept, β1 = (S, C1), β2 = (M, C1), β3 = (L, C3), β11 = (SLo, C11), β12 = (SM, C12), β13 = (SH, C13), and β22 = (MM, C22). The medium basic concept is as a connecting concept between the other two concepts and is a transient concept. The basic concepts can classify objects based on the influencing factors but not the target factor, and thus the causal concept analysis of the fault data is invalid while using them. For the no-fault probability basic concept, β1 = (S, C1), β2 = (M, C1), β3 = (L, C3), β11 = (SLo, C11), β12 = (SM, C12), β13 = (SH, C13), β21 = (MLo, C21), β22 = (MM, C22), β23 = (MH, C23), β31 = (HLo, C21), β32 = (HM, C22), and β33 = (HH, C23). These basic concepts do not contain the target factor and are not valid for a causal analysis. However, they can allow object classification to be carried out.

6.3.4

Background Relationship Analysis

The example used is similar to that in table 6.3, as shown in table 6.16. Step 1 is the same as Step 1 in section 6.3.3. Step 2: Refer to Step 2 in section 6.3.3; the fictitious states in the phase space are {LLoLo, LLoM, LMLo, LMM, LHLo, LHM}. Remove them, and the remaining states with a physical logic are as follows: {SLoLo, SLoM, SLoH, SMLo, SMM, SMH, SHLo, SHM, SHH, MLoLo, MLoM, MLoH, MMLo, MMM, MMH, MHLo, MHM, MHH, LLoH, LMH, LHH}, with 21 lattices constituting R. Step 3: According to the states of the phase space determined above, the fault data described in section 6.3.1 are counted, and the FBRSCT is formed, as shown in table 6.16. Step 4: For Q = Σq = 2067, the table displayed above is the quotient space U* of domain U; divide the number of occurrences q by Q to obtain the frequency q′ of each state. Consider the influencing factors and target factor to obtain the background space distribution in descending order:

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TAB. 6.16 – FBRSCT. U f1 f2 f3 q q′ U f1 f2 f3 q q′

States u1 S Lo Lo 4 0.0019 u12 M Lo H 240 0.1161

u2 S Lo M 20 0.0097 u13 M M Lo 0 0

u3 S Lo H 234 0.1132 u14 M M M 0 0

u4 S M Lo 16 0.0077 u15 M M H 150 0.0726

u5 S M M 33 0.0160 u16 M H Lo 0 0

u6 S M H 107 0.0518 u17 M H M 0 0

u7 S H Lo 1 0.0005 u18 M H H 225 0.1089

u8 S H M 17 0.0082 u19 L Lo H 316 0.1529

u9 S H H 222 0.1074 u20 L M H 190 0.0919

u10 M Lo Lo 0 0 u21 L H H 292 0.1413

u11 M Lo M 0 0

p*(LLoH, u19) = 0.1529; p*(LHH, u21) = 0.1413; p*(MLoH, u12) = 0.1161; p* (SLoH, u3) = 0.1132; p*(MHH, u18) = 0.1089; p*(SHH, u9) = 0.1074; p*(LMH, u20) = 0.0919; p*(MMH, u15) = 0.0726; p*(SMH, u6) = 0.0518; p*(SMM, u5) = 0.0160; p*(SLoM, u2) = 0.0097; p*(SHM, u8) = 0.0082; p*(SMLo, u4) = 0.0077; p* (SLoLo, u1) = 0.0019; p*(SHLo, u7) = 0.0005; and p*(MLoLo, u10) = p* (MLoM, u11) = p* (MMLo, u13) = p* (MMM, u14) = p* (MHLo, u16) = p* (MHM, u17) = 0. Represent the aforementioned using the frequency state method: p = 0.1529/ u19 + 0.1413/u21 + 0.1161/u12 + 0.1132/u3 + 0.1089/ u18 + 0.1074/u9 + 0.0919/ u20 + 0.0726/u15 + 0.0518/u6 + 0.0160/u5 + 0.0097/u2 + 0.0082/u8 + 0.0077/u4 + 0.0019/u1 + 0.0005/u7. Step 5: Calculate the edge distribution of each factor. (1) f1: p11 = p*(S) = p*(SLoH, u3) + p*(SHH, u9) + p*(SMH, u6) + p*(SMM, u5) + p*(SLoM, u2) + p*(SHM, u8) + p*(SMLo, u4) + p*(SLoLo, u1) + p*(SHLo, u7) = 0.1132 + 0.1074 + 0.0518 + 0.0160 + 0.0097 + 0.0082 + 0.0077 + 0.0019 + 0.0005 = 0.3164. Similarly, p12 = p*(M) = 0.2976, 1 p 3 = p*(L) = 0.3861. The edge distribution of f1 is p1 = (0.3164, 0.2976, 0.3861). (2) f2: p21 = p*(Lo) = p*(LLoH, u19) + p*(MLoH, u12) + p*(SLoH, u3) + p* (SLoM, u2) + p*(SLoLo, u1) = 0.1529 + 0.1161 + 0.1132 + 0.0097 + 0.0019 = 0.3938. Similarly, p22 = p*(M) = 0.2400, p23 = p*(H) = 0.3663. The edge distribution of f2 is p2 = (0.3938, 0.2400, 0.3663). (3) f3: p31 = p*(Lo) = p*(SMLo, u4) + p*(SLoLo, u1) + p*(SHLo, u7) = 0.0077 + 0.0019 + 0.0005 = 0.0101. Similarly, p32 = p*(M) = 0.0339 and p33 = p* (H) = 0.9561. The edge distribution of f3 is p3 = (0.0101, 0.0339, 0.9561). Step 6: Analyse the influence of influencing factors on the target factor. (1) For the relationship between f1 and f3, F1!3 ðuÞ = {F1!3 ðuÞ|f1(u*) = S and SHLo;

f3(u*) = Lo} = SLoLo + SMLo +

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p*(F1!3 ðuÞ = S*Lo) = p*(SLoLo, u1) + p*(SMLo, u4) + p*(SHLo, u7) = 0.0019 + 0.0077 + 0.0005 = 0.0101. In addition, p*(F1!3 ðuÞ = S*Lo|f 3 (u*) = Lo) = p*(F1!3 ðuÞ = S*Lo)/p*(f1(u*) = S) = 0.0101/0.3164 = 0.0319. Similarly, p*(F1!3 ðuÞ = M*Lo|f3(u*) = Lo) = 0, p*(F1!3 ðuÞ = L*Lo|f3(u*) = Lo) = 0, p*(F1!3 ðuÞ = S*M|f3(u*) = M) = 0.1071, p*(F1!3 ðuÞ = M*M|f3(u*) = M) = 0, p*(F1!3 ðuÞ = L*M|f3(u*) = M) = 0, p* (F1!3 ðuÞ = S*H|f3(u*) = H) = 0.8609, p*(F1!3 ðuÞ = M*H|f3(u*) = H) = 1, p*(F1!3 ðuÞ = L*H|f3(u*) = H) = 1. p3 = (p*(Lo), p*(M), and p* (H)) = (0.0101, 0.0339, 0.9561). The low and medium fault probabilities are changed from 0.0101 → 0.0319 and 0.0339 → 0.1071, respectively, under the influence of the short working time, and the probability is increased by approximately double. This indicated that a short working time can cause a component fault to maintain a low state. A medium or long working time has no effect on the change in low or medium fault probability, i.e. 0.0101 → 0, 0.0339 → 0. The short, medium, and long working times change the probability of a high fault probability, i.e. 0.9561 → 0.8609, 0.9561 → 1, 0.9561 → 1; although these influences are different, their magnitude is not large. The short working time affecting the fault probability is slightly reduced, and the medium or long working time leads completely to a high fault probability state. (2) For the relationship between f2 and f3, p*(F2!3 ðuÞ = *LoLo|f3(u*) = Lo) = 0.0048, p*(F2!3 ðuÞ = *MLo|f3(u*) = Lo) = 0.0321, p*(F2!3 ðuÞ = *HLo|f3(u*) = Lo) = 0.0014, p*(F2!3 ðuÞ = *LoM|f3(u*) = M) = 0.0246, p*(F2!3 ðuÞ = *MM|f3(u*) = M) = 0.0667, p* (F2!3 ðuÞ = *HM|f3(u*) = M) = 0.0224, p*(F2!3 ðuÞ = *LoH|f3(u*) = H) = 0.9705, p*(F2!3 ðuÞ = *MH|f3(u*) = H) = 0.9013, and p*(F2!3 ðuÞ = *HH| f3(u*) = H) = 0.9762. In addition, p3 = (p*(Lo), p*(M), p*(H)) = (0.0101, 0.0339, 0.9561). The working temperature affecting the fault probability is common. The probability that a low, medium, or high working temperature leads to a low fault probability is changed, i.e. 0.0101 → 0.0048, 0.0101 → 0.0321, 0.0101 → 0.0014. This indicates that a low or high working time is not conducive to a low fault probability state. The probability that a medium working temperature leads to a low fault probability is increased by about twofold. In the same way, the probability that a low or medium working temperature leads to a medium fault probability is low, although the magnitude is decreased compared with the above situation, i.e. 0.0339 → 0.0246, 0.0339 → 0.0224. The probability of a medium working temperature affecting the medium fault probability is doubled, i.e. 0.0339 → 0.0667, which is favourable. The probability of a low or high working temperature affecting the medium fault probability shows a slight increase, i.e. 0.9561 → 0.9705, 0.9561 → 0.9762, indicating that a medium or low working time leads to a high fault probability. The probability of a medium working temperature affecting the high fault probability decreases, i.e. 0.9561 → 0.9013. The probability of a medium working temperature leading to a low or medium fault probability is the largest, whereas the probability of a low or high working temperature leading to a medium

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or high fault probability is generally the largest; the former is a favourable state, whereas the latter is an adverse state.

6.3.5

Factor Dimension Reduction

This example is similar to that shown in table 6.16, as indicated in table 6.17. The fault state contains four factors: three influencing factors, i.e. working time f1, working temperature f2, and working humidity f3, and a target factor, i.e. component fault probability f4. Step 1: Divide the qualitative phase of each factor and the corresponding value range as follows: Here, X(f1) and X(f2) are the same as those in section 6.3.3. In addition, X(f3) = {K1, K2, K3} = {Lo, M, H}, D(f3) = {[a1, a2], (a2, a3], (a3, a4]} = {[0, 60]%, (60, 85] °C, (85, 100] °C}, X(f4) = {K1, K2, K3} = {Lo, M, H}, and D(f4) = {[a1, a2], (a2, a3], (a3, a4]} = {[0, 30]%, (30, 60]%, (60, 100]%}. Step 2: Set up the Cartesian product phase space as follows: X = X(f1)X(f2)X(f3) X(f4) = {SLoLoLo, SLoLoM, SLoLoH, SMLoLo, SMLoM, SMLoH, SHLo, SHLoM, SHLoH, MLoLoLo, MLoLoM, MLoLoH, MMLoLo, MMLoM, MMLoH, MHLoLo, MHLoM, MHLoH, LLoLoLo, LLoLoM, LLoLoH, LMLoLo, LMLoM, LMLoH, LHLoLo, LHLoM, LHLoH,…}. The omitted factors and their phases correspond to medium or high in the third item, X(f3), and X is divided into 81 lattices. The fictitious states can be removed through common sense. The states [long ＊ ＊ low] and [long ＊＊] are fictitious states, where ＊＊ = X(f2) X(f3). The fictitious states in the phase space are {LLoLoLo, LLoLoM, LMLoLo, LMLoM, LHLoLo, LHLoM,…}, and there are 18 lattices in total. After removing the aforementioned states, the remaining states with physical logic are as follows: {SLoLoLo, SLoLoM, SLoLoH, SMLoLo, SMLoM, SMLoH, SHLoLo, SHLoM, SHLoH, MLoLoLo, TAB. 6.17 – FBRSCT. U f1 f2 f3 f4 q q′ U f1 f2 f3 f4 q q′

States u1 S Lo Lo Lo 6 0.0029 u9 S H H H 220 0.1047

u2 S Lo Lo M 18 0.0086 u10 M Lo H H 250 0.1190

u3 S Lo M H 222 0.1057 u11 M M H H 160 0.0762

u4 S M Lo Lo 20 0.0095 u12 M H H H 230 0.1095

u5 S M M M 38 0.0181 u13 L Lo H H 339 0.1614

u6 S M H H 110 0.0524 u14 L M M H 170 0.0809

u7 S H Lo Lo 1 0.0005 u15 L H H H 303 0.1442

u8 S H Lo M 14 0.0067

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MLoLoM, MLoLoH, MMLoLo, MMLoM, MMLoH, MHLoLo, MHLoM, MHLoH, LLoLoH, LMLoH, LHLoH,…}, where R consists of 63 lattices. Step 3: The fault data are counted according to the states of the phase space given above, and the FBRSCT is formed, as shown in table 6.17. Here, Q = Σq = 2101, and table 6.17 shows the quotient space U* of domain U, dividing the number of occurrences q by Q to obtain the frequency q′ of each state. Step 4: Calculate the edge distribution of each factor. (1) For f1, p11 = p*(S) = q′(u1) +    + q′(u9) = 0.3091. Similarly, p12 = p*(M) = 0.3047, p13 = p*(L) = 0.3865. (2) For f2, p21 = p*(Lo) = 0.3976, p22 = p*(M) = 0.2371, p23 = p*(H) = 0.3656. (3) For f3, p31 = p*(Lo) = 0.0282, p32 = p*(M) = 0.2047, p33 = p*(H) = 0.7674. (4) For f4, p41 = p*(Lo) = 0.0129, p42 = p*(M) = 0.0334, p43 = p*(H) = 0.9540. Step 5: Analysis of the influence of influencing factors on the target factor. (1) The conditional probability of f1 is as follows: p*(F1!4 ðuÞ = S＊＊Lo|f4(u*) = Lo) = p*(F1!4 ðuÞ = S＊＊Lo)/p*(f1(u*) = S) = 0.0129/0.3091 = 0.0417, p* p*(F1!4 ðuÞ = L＊＊Lo|f4(u*) = (F1!4 ðuÞ = M＊＊Lo|f4(u*) = Lo) = 0, Lo) = 0, p*(F1!4 ðuÞ = S＊＊M|f4(u*) = M) = 0.1081, p*(F1!4 ðuÞ = M＊ ＊M|f4(u*) = M) = 0, p*(F1!4 ðuÞ = L＊＊M|f4(u*) = M) = 0, p*(F1!4 ðuÞ = S＊＊H|f4(u*) = H) = 0.8502, p*(F1!4 ðuÞ = M＊＊H|f4(u*) = H) = 1, and p*(F1!4 ðuÞ = L＊＊H|f4(u*) = H) = 1. (2) The conditional probability of f2 is as follows: p*(F2!4 ðuÞ = ＊Lo＊Lo|f4(u*) = Lo) = p*(F2!4 ðuÞ = ＊Lo＊Lo)/p*(f2(u*) = Lo) = 0.0029/0.3976 = 0.0073, p*(F2!4 ðuÞ = ＊H＊Lo| p*(F2!4 ðuÞ = ＊M＊Lo|f4(u*) = Lo) = 0.0401, f4(u*) = Lo) = 0.0014, p*(F2!4 ðuÞ = ＊Lo＊M|f4(u*) = M) = 0.0216, p* (F2!4 ðuÞ = ＊M＊M|f4(u*) = M) = 0.0763, p*(F2!4 ðuÞ = ＊H＊M|f4(u*) p*(F2!4 = M) = 0.0183, p*(F2!4 ðuÞ = ＊Lo＊H|f4(u*) = H) = 0.9711, ðuÞ = ＊M＊H|f4(u*) = H) = 0.8836, and p*(F2!4 ðuÞ = ＊H＊H|f4(u*) = H) = 0.9803. (3) The conditional probability of f3 is as follows: p*(F3!4 ðuÞ = ＊Lo＊Lo|f4(u*) = Lo) = p*(F3!4 ðuÞ = ＊Lo＊Lo)/p*(f2(u*) = Lo) = 0/0.0282 = 0.0073, p* p*(F3!4 ðuÞ = ＊H＊Lo|f4(u*) = (F3!4 ðuÞ = ＊M＊Lo|f4(u*) = Lo) = 0, Lo) = 0, p*(F3!4 ðuÞ = ＊Lo＊M|f4(u*) = M) = 0.5426, p*(F3!4 ðuÞ = ＊M ＊M|f4(u*) = M) = 0.0884, p*(F3!4 ðuÞ = ＊H＊M|f4(u*) = M) = 0, p* p*(F3!4 ðuÞ = ＊M＊H|f4(u*) = (F3!4 ðuÞ = ＊Lo＊H|f4(u*) = H) = 0, H) = 0.9116, and p*(F3!4 ðuÞ = ＊H＊H|f4(u*) = H) = 1. Step 6: Build the conditional distribution table of the influencing factors on the target factor, as shown in tables 6.18–6.20. Step 7: For p4 = (p*(Lo), p*(M), p*(H)) = (0.0129, 0.0334, 0.9540), calculate the entropy of the target factor, H(X(f4)) = 0.3096. A 0 value in the table indicates no participation in the calculation. (1) For the information gain calculation of f1, H(X(f4)|X(f1) = S) = −p*(X(f4) = Lo| X(f1) = S)lnp*(X(f4) = Lo|X(f1) = S) − p*(X(f4) = M|X(f1) = S)lnp*(X(f4) = M|X(f1) = S) − p*(X(f4) = H|X(f1) = S)lnp*(X(f4) = H|X(f1) = S) = −0.0417 ln0.0417 − 0.1081ln0.1081 − 0.8502ln0.8502 = 0.7372. Similarly, H(X(f4)|X

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TAB. 6.18 – Conditional distribution of low fault probability. p*(X(f4) = Lo|X(f1)) f1 X(f1) = S 0.0417 0 X(f1) = M 0 X(f1) = L p41 = p*(Lo) = 0.0129

p*(X(f4) = Lo|X(f2)) X(f2) = Lo X(f2) = M X(f2) = H

f2 0.0073 0.0401 0.0014

p*(X(f4) = Lo|X(f3)) X(f3) = Lo X(f3) = M X(f3) = H

f3 0.0073 0 0

TAB. 6.19 – Conditional distribution of medium fault probability. p*(X(f4) = M|X(f1)) f1 X(f1) = S 0.1081 0 X(f1) = M 0 X(f1) = L 4 p 2 = p*(M) = 0.0334

p*(X(f4) = M|X(f2)) X(f2) = Lo X(f2) = M X(f2) = H

f2 0.0216 0.0763 0.0183

p*(X(f4) = M|X(f3)) X(f3) = Lo X(f3) = M X(f3) = H

f3 0.5426 0.0884 0

TAB. 6.20 – Conditional distribution of high fault probability. p*(X(f4) = H|X(f1)) f1 X(f1) = S 0.8502 1 X(f1) = M 1 X(f1) = L p43 = p*(H) = 0.9540

p*(X(f4) = M|X(f2)) X(f2) = Lo X(f2) = M X(f2) = H

f2 0.9711 0.8836 0.9803

p*(X(f4) = M|X(f3)) X(f3) = Lo X(f3) = M X(f3) = H

f3 0 0.9116 1

(f1) = M) = 0, H(X(f4)|X(f1) = H) = 0, and H(X(f4)|X(f1)) = p4 × (0.7372, 0, 0) = 0.0095. For the information gain, DH ðf1 Þ = H(X(f4)) − H(X(f4)|X (f1)) = 0.3096 − 0.0095 = 0.3001. (2) For the information gain calculation of f2, H(X(f4)|X(f2) = S) = 0.2124, H(X (f4)|X(f2) = M) = 0.6271, H(X(f4)|X(f2) = H) = 0.1470, and H(X(f4)|X (f2)) = p4 × (0.2124, 0.6271, 0.1470) = 0.1639. For the information gain, DH ðf2 Þ = H(X(f4)) − H(X(f4)|X(f2)) = 0.3096 − 0.1639 = 0.1457. (3) For the information gain calculation of f3, H(X(f4)|X(f3) = S) = 0.5304, H(X (f4)|X(f3) = M) = 0.4311, H(X(f4)|X(f3) = H) = 0, and H(X(f4)|X(f3)) = p4 × (0.2124, 0.6271, 0.1470) = 0.0212. For the information gain, DH ðf3 Þ = H(X (f4)) − H(X(f4)|X(f3)) = 0.3096 − 0.0212 = 0.2884. Step 8: Analyse the information gain of the influencing factors on the target factor, and select the appropriate influencing factors for reducing the dimensions. The information gain of the fault probability considering the working time, working temperature, and working humidity are 0.3001, 0.1457, and 0.2884, respectively. That is, the influence of the working time on the fault probability is the greatest, followed by humidity and working temperature in order. The influence of the working time and humidity on the fault probability is similar, but the distributions of the influence are different. When the fault probability is low, the short working time and low humidity have the same influence, although their degrees are different. That is, there is no probability that the medium or long working time or medium or high humidity affect the low fault probability. However, there is the probability that the short working time and low humidity

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affect the low fault probability. The former has a much greater effect than the latter, i.e. 0.0417/0.0073 = 5.7 times the effect. A long working time and high humidity have no effect on a medium fault probability. The rest of the corresponding degree of influence is quite different. When the fault probability is high, the long working time and the high humidity completely determine a high fault probability, whereas the remaining factors are quite different. Therefore, although the working time and humidity have a similar effect on the fault probability, their distributions are different. The influence of two factors cannot replace each other, and thus we cannot remove the working time or humidity to reduce the influencing factors (dimensions). The working temperature has the least effect on the fault probability, and the influence of temperature is the greatest for the low, medium, and high fault probability in order. A low or medium fault probability is less affected by the working temperature, which has a substantial influence on the high fault probability. These features differ greatly from the working time and humidity effects. Thus, the working temperature cannot be removed for reducing the dimensions.

6.3.6

Compression of Fault Probability Distribution

According to the component fault statistics of section 6.3.1, including the working time, working temperature, and fault probability, the frequency of component renewal during actual production is 13 days, and the working temperature is theoretically 0 °C–40 °C. The information on 81 items is obtained. The data form is as follows: {(working time, working temperature, fault probability),…} = {(3%, 5%, 81%), (5%,7%, 88%),…}. The fault data are compressed and the fault probability distribution is plotted under different divisions. Step 1: Set the fault data value pair (working time, working temperature, fault probability) as (f1, f2, g), i.e. influencing factors f1 and f2, and target factor g. Step 2: The information obtained from section 6.3.1 is made up of 81 items, k = 81, (3%, 5%, 81%)1, (5%, 7%, 88%)2,…, (2%, 16%, 12%)81. Step 3: Divide the target factor value range as follows: Z(g) = {z1(g), z2(g), z3(g), z4(g)} = {[0, 30%), [30, 60%), [60, 80%), [80, 100%]}, the semantics of which are {does not generally occur, may occur, occurs frequently, occurs}, m = 4. Step 4: Allocate the fault data value pair (f1,…, fn, g)k according to Z(g): Γ4 = {(3, 5), (5, 7), (10, 5), (10, 10), (10, 20), (10, 30), (10, 35), (13, 10), (13, 15), (13, 25), (13, 35), (5, 35), (5, 40), (4, 35), (6, 6), (8, 5), (6, 33), (6, 38), (8, 35)}, a total of 19 value pairs. As g 2 z4(g) = [80, 100%] in the value pairs given above, remove g from the value pairs to simplify the representation, as shown below. Γ3 = {(1, 8), (3, 10), (3, 8), (5, 9), (5, 15), (7, 12), (7, 28), (9, 16), (9, 25), (10, 21), (1, 40), (3, 28), (3, 32), (1, 35), (2, 32), (2, 38), (3, 11), (3, 30), (4, 11), (4, 29), (5, 20), (5, 25), (5, 30), (6, 14), (6, 20), (7, 15), (7, 19), (8, 15), (8, 25), (9, 16), (9, 23)}, a total of 31 value pairs, where g 2 z3(g) = [60, 80%). Γ2 = {(1, 9), (1, 14), (1, 27), (1, 32), (3, 11), (3, 28), (2, 10), (2, 16), (2, 26), (2, 31), (5, 15), (5, 26), (6, 21), (1, 12), (1, 28), (3, 13), (3, 30), (3, 27), (3, 19), (4, 17), (4, 20), (4, 25), (4, 27)}, a total of 23 value pairs, where g g 2 z2(g) = [30, 60%).

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Γ1 = {(1, 14), (1, 16), (1, 27), (2, 17), (2, 24), (1, 20), (2, 20), (2, 16)}, a total of 8 value pairs, where g 2 z1(g) = [0, 30%). The value pairs of the fault data contained in Γ1, Γ2, Γ3, and Γ4 above are plotted in the same coordinate system, where the abscissa is t, the ordinate is c, and the different symbols represent different fault probability ranges, as shown in figure 6.8. From figure 6.8, the actual fault data are chaotic, and there are repeated phenomena for some of the data. For the original data samples, the information processing and compression are inconvenient, resulting in a shortcoming in that the SFT is difficult to use for practical problems. Step 5: Simplify Γ1, Γ2, Γ3, and Γ4 using the IPJT. The simplification process is similar to example 6.1 and is not given here. After simplification, Ω1, Ω2, Ω3, and Ω4 are as follows: Ω4 = {(5, 40), (4, 35), (3, 5), (10, 5), (13, 10), (13, 35)}, or 23 value pairs after simplification, i.e. g 2 z4(g) = [80%, 100%], compressed at 68%. Ω3 = {(1, 40), (1, 8), (3, 8), (5, 9), (7, 12), (9, 16), (10, 21), (9, 25)}, or 8 value pairs after simplification, i.e. g 2 z3(g) = [60%, 80%), compressed at 74%. Ω2 = {(1, 32), (1, 9), (3, 11), (5, 15), (6, 21), (5, 26), (3, 30)}, or 7 value pairs after simplification, i.e. g 2 z2(g) = [30%, 60%), compressed at 67%. Ω1 = {(1, 27), (1, 14), (2, 16), (2, 24)}, or 4 value pairs after simplification, g 2 z1(g) = [0, 30%), compressed at 50%. All data are compressed into 56 items (69%). The above method is reasonable and feasible for compressing the fault data. In addition, the data compressed by the above method can represent the information in the original dataset, which is a lossless compression. Step 6: Analyse the fault probability distribution and determine the relationships of the fault probability distributions under different divisions. Figure 6.9 shows the fault probability distributions of the components under different values of t and

FIG. 6.8 – Fault probability data graph.

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FIG. 6.9 – Fault probability distribution under different division. c. According to Step 5, for simplification, the area A4 surrounded by the blue line in figure 6.9 is g 2 z4(g) = [80%, 100%]. The area A3 surrounded by the red line is g 2 z3(g) = [60%, 80%). The area A2 surrounded by the purple line is g 2 z2(g) = [30%, 60%). In addition, the area A1 surrounded by the green line is g 2 z1(g) = [0, 30%). Because the IPJT is applied to the convex set, the distribution overlapping phenomenon is observed in figure 6.10. According to the processing overlapping method, the final fault probability distributions under the different division are obtained; the areas of the fault probability are [0, 30%) where Z1 = A1; [30%, 60%) where Z2 = A2; [60%, 80%) where Z3 = A3 \ ðA1 [ A2 Þ; and [80%, 100%] is Z4 = A4 \ ðA1 [ A2 [ A3 Þ. Figure 6.10 is the fault probability distribution after removing the overlapping area. From the figure we can see that different values of t and c will make different changes in the component fault probability. Different colours represent a different area of change in the fault probability; a non-colour area in the figure is where no valid fault data value pairs reside. There may also be data islands after a fault data compression, and the above methods can indicate such islands. The fault probability distribution obtained in this example can be expressed as follows: Ω4 = {(5, 40), (4, 35), (3, 5), (10, 5), (13, 10), (13, 35)}, g 2 z4(g) = [80%, 100%]; Ω3 = {(1, 40), (1, 8), (3, 8), (5, 9), (7, 12), (9, 16), (10, 21), (9, 25)}, g 2 z3(g) = [60%, 80%); Ω2 = {(1, 32), (1, 9), (3, 11), (5, 15), (6, 21), (5, 26), (3, 30)}, g 2z2(g) = [30%, 60%); Ω1 = {(1, 27), (1, 14), (2, 16), (2, 24)}, g 2 z1(g) = [0, 30%).

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FIG. 6.10 – No overlapping fault probability distribution. Figure 6.11 shows the fault probability distribution of the component in section 6.3.1. The fault probability distribution is generated by the neural network method according to the actual fault data. Although the method itself is comprehensive, there are inherent problems, such as the need for a large amount of data, i.e. 14 × 41 data pairs; a more complicated calculation process that must be analysed by a neural network; an inability to adapt to big data for effective data compression; and a fault probability distribution that must be represented by MATLAB. The advantages of this method are an effective improvement in the first three items: The amount of data is 14% that of the original method; in addition, the calculation is relatively simple and can be more effective for a data compression, and the representations of the fault probability distributions are simple.

6.4

Conclusions

The chapter describes the relationship between reliability and influencing factors, which can be divided into three parts. In the first part, a reliability data representation method based on RVDF is constructed. We obtain the decomposition function, which can be used for the SFT as the characteristic function. In the second part, the causal relationship reasoning method is provided among the reliability and influencing factors, causal concept extraction, and background relationship analysis methods.

System Reliability with Influencing Factors

FIG. 6.11 – Fault probability distribution with DSFT.

157

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The state absorption method and state recurrence method are proposed to analyse the causal relationship between the influencing factors and the target factor. The final reasoning result can contain the most information. It is a breadth first state absorption method. As much as possible, state information with a large frequency plays a leading role. The state recurrence method has priority in terms of depth. The causal relationships obtained basically show the known fault probability distribution. The causal concept analysis method for fault data in the SFT is constructed. The results can be divided into three concepts. The true concept can be used for a causal analysis of an example. The basic medium concept and basic no-fault probability concept are different in their processes of semi-lattice reasoning. They cannot be applied to the causal concept analysis, but only as the object classification concept according to the influencing factors. According to the characteristics of the fault data, the background relationship analysis method of the fault and influencing factors is formulated. This method can be used to analyse the fault big data and the qualitative relationship between a fault and the factors, and through the number of faults, to quantify the influencing factors on the fault probability. The third part provides a reduction in the dimensions and a data compression method. According to the information gain method in FS, the reduction dimension method of SFT was developed. As the basic idea of this method, the influencing factors can be replaced or deleted directly when the information gain of two influencing factors is small or close to the target factor. To achieve a reduction of the fault space dimension in the SFT and the conditions for dimensional reduction (1) the information gain of the influencing factors on the target factor is small and (2) the influencing factors have little difference in the information gain on the target factor. The fault data compression method is suitable for the fault probability distribution representation of the SFT, particularly for discrete fault data. This method has a strong data compression capability and is suitable for the fault data analysis of big data. The representation of the fault probability distribution is simple. The above conclusions are the main theoretical results of this chapter. Although there are more case analysis processes, quantitative conclusions, and reasoning relationships, they are not given here. This chapter presents the basic theory of the SFT. In addition to the above, some other research results are also available. Please refer to the literature [12–16].

References [1] Wang P.Z., Li H.X. (1994) Mathematical theory of knowledge representation. Tianjin Science and Technology Press, Tianjin. [2] Wang P.Z., Li H.X. (1995) A thought to fuzzy computer (II), J. Beijing Normal Univ. 303. [3] Wang P.Z. (2013) Factor spaces and factor data-bases, J. Liaoning Tech. Univ. (Nat. Sci.) 32, 1.

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[4] Qu G.H, Zeng F.H, Liu Z.L., et al. (2017) Background distribution and fuzy background relation in factor spaces, Fuzzy Syst. Math. 31, 66. [5] Wang P.Z. (2015) Factor spaces and data science, J. Liaoning Tech. Univ. (Nat. Sci.) 34, 273. [6] Yong S. (2014) Big data and new challenges for science and technology, Sci. Technol. Dev. 25. [7] Wang P.Z, Zhang X.H., Lu H.Z., et al. (1995) Mathematical theorem of truth value flow inference, Fuzzy Sets Syst. 32, 221. [8] Wang P.Z. (2002) Rules detecting and rules-data mutual enhancement based on factors space theory, Inter. J. Inf. Technol. Decis. Making. 1, 73. [9] Lv J.H., Liu H.T., Guo F.F., et al. (2017) The algorithm of extraction of background bases, Fuzzy Syst. Math. 31, 82. [10] Wang P.Z. (2014) Introduction to factor space and factor database (special report). Forum on Intelligent Science and mathematics, Huludao, June 2014. [11] Chen Y.Y., Liu Y.F., Wang P.Z. (1983) Mathematical model of comprehensive evaluation, Fuzzy Math. 3, 60. [12] Chen H., Shen Y., Cui T.-J. (2017) Research on causality reasoning method of failure probability and the influencing factors, App. Res. Comput. 34, 3656. [13] Li S.S., Cui T.J., Ma Y.D., et al. (2017) Research on concepts extraction method of causal relationship of factors in SFT, App. Res. Comput. 34, 2997. [14] Li S.S., Cui T.J., Ma Y.D., et al. (2017) Analysis of background relationship between fault and its influencing factors in SFT. App. Res. Comput. 34, 3277. [15] Cui T.J., LI S.S., Han G., Jiang F.C. (2018) On the dimensional reduction method for the fault-affecting factors based on the information gain, J. Saf. Environ. 18, 1686. [16] Li S.S., Cui T.J., Ma Y.D., et al. (2018) System reliability maintain method with controllable and uncontrollable factors, App. Res. Comput. 35, 3217. [17] Cui T.J., Li S.S. (2019) Study on the relationship between system reliability and influencing factors under big data and multi-factors, Cluster Comput. 22, 10275.

Chapter 7 Cloudization Space Fault Tree The fault data (reliability data) during the actual operation of a system are quite different from the general data and are influenced by the human–machine environment. The reliability data of the components or systems are mainly derived from two aspects. First, through a laboratory test during research phase, we can obtain a more accurate fault probability according to the statistics of the fault data of the components. Second, some fault data are obtained during the actual system operation. However, the fault data from both the laboratory and the real world have uncertainty, namely discreteness, fuzziness, and randomness. These uncertainties may be systematic errors caused by the inherent characteristics of the system or components, human errors caused by the operators and inspectors, or a random fault of the system caused by the environment. Furthermore, components under the influence of different factors have different characteristics of fault probability. The change in component fault probability is an organic superposition of the influence of multiple factors. Thus, a change in the fault probability of the system composed of these components is more complex. It is therefore necessary to achieve a method with the capabilities of applying multi-factor and uncertainty analyses for the fault data. How to determine the uncertainty of the data under the influence of multiple factors, and what factors influence the data uncertainty are questions that should be confirmed. The SFT measures the reliability with the fault probability and analyses the relationship between the system reliability and influencing factors. The characteristic function (CF) of the SFT is constructed by the relationship between the component fault probability and the influencing factors and is the basis of the SFT. The system fault data differ from the general monitoring data, which have large discreteness, randomness, and fuzziness, that is, uncertainty. The existing characteristic function is a continuous function with finite discontinuities, such as described in chapters 2 and 3. The function can be considered a kernel function of the fault data, but it is difficult to express the uncertainty of such data. In this chapter, the function obtained using the cloud model (CM) to transform the characteristic function, such that it can express the uncertainty of the data, is called the cloudization characteristic function (CCF). The cloudization SFT (CLSFT) is constructed using the CCF and enables the relevant theory and methods of SFT to express a data uncertainty. Thus, the expressions of the fault data characteristics DOI: 10.1051/978-2-7598-2499-1.c007 © Science Press, EDP Sciences, 2020

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and rules are more accurate. Firstly, the construction process and rationality analysis of CLSFT are given. Secondly, the concepts of the SFT are reconstructed using the cloud model. These definitions and methods are applied to analyse the fault data of a simple electrical component. The relationship between the component fault probability and the working time and working temperature is studied. The results reflect the discreteness, randomness, and fuzziness of the fault data to a substantial extent. In summary, this chapter provides the reference for analysing and controlling the uncertainty of the fault data and reliability in practical applications. This chapter is divided into different sections. Section 7.1 provides the related definitions. Section 7.2 describes the process, definitions, and methods of the CLSFT. Section 7.3 applies the CLSFT to analyse the given example. Section 7.4 provides some concluding remarks. List of symbols Symbols Ex En He C(Ex, En, He) lq Pid ðxÞ i dk n Pi ðx1 ; x2 ; . . .; xn Þ xk PT ðx1 ; x2 ; . . .; xn Þ Pidk PTdk Ig ðiÞ Igc ðiÞ FIi ðx1 ; x2 ; . . .; xn Þ FIT ðx1 ; x2 ; . . .; xn Þ ZIg ðiÞ ZIcg ðiÞ Pb Kj(j = 1, 2,…, r) Ei/Eii Exk , Hek , and Enk

Subscript k

Meaning Digital features of cloud, expectation Digital features of cloud, entropy Digital features of cloud, hyper entropy Cloud model Degree of certainty of concept Characteristic function of component fault probability, CF The ith component Influencing factor, dk 2 fx1 ; x2 ; . . .; xn g Number of influencing factors Component fault probability distribution, CFPD Specific values of factor dk System fault probability distribution, SFPD Component fault probability distribution trend, CFPDT System fault probability distribution trend, SFPDT Probability importance distribution, PID Criticality importance distribution, CID Component factor importance distribution/factor joint importance distribution, CFID or CFJID System factor importance distribution/factor joint importance distribution, SFID or SFJID Component domain probability importance, CDPI Component domain criticality importance, CDCI Domain boundary of component/system The jth set in r minimal cut set of a fault tree ith/iith basic event in Kj Characteristic parameters of the cloud model obtained from reverse cloud model generator considering the influence of kth influencing factor on the reliability data Corresponding parameters of the kth factors (continued)

Cloudization Space Fault Tree

163 (continued).

Symbols PTEx!dk ðx1 ; x2 ; . . .; xn Þ, PTEn!dk ðx1 ; x2 ; . . .; xn Þ, PTHe!dk ðx1 ; x2 ; . . .; xn Þ M and c Q kmax and kmin k mEn , mEx , mHe df , dd , and dr h c T X1–5 C1c and C1h

7.1

Meaning Three parameters for the uncertainty of the system reliability data considering dk Total number of cloud droplets Threshold Maximum and minimum values in the differential fault probability distribution, respectively Absolute value of deviation from 0 Number of cloud droplets not in the ½k; k Fuzziness, discreteness, and randomness of reliability data, respectively Humidity Working temperature Instance system Five components in the instance system Cloud model under the influence of the working temperature c and humidity h on the reliability of X1

Definitions of SFT

Based on the information in the second chapter, the following concepts are added to the SFT. Definition 7.1-1. Cut set domain (CSD) of one component/system: A domain where the fault probability of a component/system is greater than a predetermined or necessary probability (in the study domain). The domain is unacceptable, the fault probability is too high, and we should make the component avoid working in this domain or take measures to reduce the fault probability. Definition 7.1-2. Path set domain (PSD) of one component/system: A domain where the fault probability of a component/system is smaller than a predetermined or necessary probability (in the study domain). The domain is acceptable, the fault probability is low, and we should make the component work in this domain, taking no measures to reduce the fault probability. Definition 7.1-3. Domain boundary (DB) Pb: The contour or surface of higher dimensions with a predetermined or necessary probability is provided in the above definition.

7.2

Construction of Cloudization Space Fault Tree

The basis of the construction of the cloudization space fault tree (CLSFT) is described in section 7.2.1, and the concepts and methods are provided in sections 7.2.2–7.2.8 [1–9].

164

7.2.1

Space Fault Tree Theory and System Reliability Analysis

Basis of CLSFT

In the SFT, the basis of the system reliability analysis is the CF that expresses the relationship between a factor and component reliability. The CF in the CSFT is obtained through a laboratory test, although the occurrence of the component fault is random and fuzzy. The CF in the DSFT is obtained by processing the fault statistics data, which are displayed by the actual operation of the system. The processing method contains the data fitting or factor projection fitting method described in section 3.3. The fault data have a substantial discreteness, and the changes in the influencing factors are freer and complicated, and thus the CF of the DSFT is more uncertain than that of the CSFT. To facilitate and express the uncertainty characteristics of the fault data, an algorithm or model with the aforementioned capabilities is required instead of the CF in the SFT. The reliability of a component is generally considered to have an exponential distribution, or an exponential distribution with a stable range of peak. Theoretically, the distribution characteristics of the reliability data obtained by the experiment or actual operation should have a normal distribution around this curve (CF). The closer the data are to the curve, the greater the density of the data, and vice versa. The method for replacing the CF must have such characteristics. Li Deyi, an academic of the Chinese Academy of Sciences, proposed the CM with only the aforementioned characteristics. In [10], the cloud droplets generated by the cloud model generator (CMG) are data points obeying a normal distribution around the forward cloud model generator analysis formula (FCMGAF), which is the same as the distribution of the reliability data. The membership of the cloud droplets generated by the CM is within the range of [0, 1], which is the same range as the reliability. In addition, there are numerous types of transform of the CM that can meet the requirements of a reliability data analysis. It is therefore feasible to use the FCMGAF instead of the CF, which is called the cloud characteristic function (CCF) [6]. The CCF can be used for CLSFT analysis, considering the uncertainty, the accuracy of the methods, which is sorted the CF of CSFT > CCF > the CF of DSFT, and the operability, which is sorted as CCF > the CF of CSFT > the CF of DSFT. The main steps with regard to the CCF are as follows: the component reliability data for a factor are brought into the reverse cloud model generator (RCMG), and the characteristic parameters are obtained; these are then taken into the FCMGAF, and the FCMGAF is reduced to 1 as the CCF of this component. FCMGAF is shown in equation (7.1). lq ¼ expððxq  ExÞ2 =ð2  ðrandð1Þ  He þ EnÞ2 ÞÞ

ð7:1Þ

The normal distribution droplets generated by equation (7.1) can express the reliability change in the component for one factor, denoted by lq. The CF of the component for this factor is Pid ðxÞ = 1 − lq , and the CCF is as shown in equation (7.2). Pid ðxÞ ¼ 1  expððx  ExÞ2 =ð2  ðrandð1Þ  He þ EnÞ2 ÞÞ

ð7:2Þ

Cloudization Space Fault Tree

165

The CCF is the basis for the construction of the CLSFT; the concepts and methods of the CLSFT are based on the CCF. The CLSFT thus has the capabilities of applying multiple factors and uncertain data processing.

7.2.2

Cloudization Fault Probability Distribution

According to definition 2.4 and equation (7.2), under the influence of multiple factors, the cloudization component fault probability distribution (CCFGD) is shown in equation (7.3). According to definition 2.5 and equation (7.3), the cloudization system fault probability distribution (CSFGD) is shown in equation (7.4).  1  Pidk ðxk Þ k¼1  n  Q 1  ð1  expððxk  Exk Þ2 =ð2  ðrandð1Þ  Hek þ Enk Þ2 ÞÞÞ ¼1

Pi ðx1 ; x2 ; . . .; xn Þ ¼ 1 

n  Q

k¼1

ð7:3Þ PT ðx1 ; x2 ; . . .; xn Þ r Y a Pi ðx1 ; x2 ; . . .; xn Þ ¼ j¼1 Ei 2Kj

¼

r Y a j¼1 Ei 2Kj

" 1

n Y

# 2

2

ð1  ð1  expððxk  Exk Þ =ð2  ðrandð1Þ  Hek þ Enk Þ ÞÞÞÞ

k¼1

ð7:4Þ

7.2.3

Cloudization Fault Probability Distribution Trend

According to definition 3.3 and equation (7.3), under the influence of multiple factors, the cloudization component fault probability distribution trend (CCFPDT) is shown in equation (7.5). According to definition 2.8 and equation (7.4), the cloudization system fault probability distribution trend (CSFPDT) is shown in equation (7.6) [1]. ;x2 ;...;xn Þ Pidk ¼ @Pi ðx1@x Qn k 1 k¼1 ð1ð1expððxk Exk Þ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ ¼ @xk

PTdk ¼ @PT ðx1@x;xk2 ;...;xn Þ ‘r Q Qn ½1 k¼1 ð1ð1expððxk Exk Þ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ Ei 2Kj j¼1 ¼ @xk

ð7:5Þ

ð7:6Þ

Space Fault Tree Theory and System Reliability Analysis

166

The derivative of the fault probability distribution of the components and system can indicate a change in reliability with an influencing factor. This change can help managers and technicians predict the changing trend of the system reliability and provide the basis for the measures to ensure the reliability. The CM can be used to express data uncertainty, and equations (7.5) and (7.6) can express the fault data uncertainty with a fault probability distribution trend.

7.2.4

Cloudization Importance Distribution Probability and Criticality

According to definition 2.6 and equations (7.3) and (7.4), the cloudization probability importance distribution (CPID) is shown in equation (7.7) [3, 5]. 8 @PT ðx1 ; x2 ; . . .; xn Þ > > Ig ðiÞ¼ > > > > ‘r @P Qi ðx1 ; x2 ; . . .; xn Þ > r > Y a > j¼1 Eii 2Kj Pii ðx1 ; x2 ; . . .; xn Þ < ¼ ¼ Pii ðx1 ; x2 ; . . .; xn Þ ð7:7Þ @Pi ðx1 ; x2 ; . . .; xn Þ j¼1 Eii 2Kj > > > \ ii6 ¼ i > > Pii ðx1 ;x2 ;...;xn Þ > > n Q > > : ¼1 ð1  ð1  expððxk  Exk Þ2 =ð2  ðrandð1Þ  Hek þ Enk Þ2 ÞÞÞÞ k¼1

According to definition 2.7 and equation (7.7), the cloudization criticality importance distribution (CCID) is shown in equation (7.8). 2 3 8 > > > 6a 7 > > Y 6 r 7 > Pi ðx1 ; x2 ; . . .; xn Þ > c 6 7 > Q ‘ I P  ðiÞ ¼ ðx ; x ; . . .; x Þ ii 1 2 n > r g 6 7 > < 4 j¼1 E 2 K 5 j¼1 Eii 2Kj Pi ðx1 ; x2 ; . . .; xn Þ ii j ð7:8Þ > \ ii 6¼ i > > > > Pii ðx1 ; x2 ; . . .; xn Þ > > n > Q > > :¼ 1  ð1  ð1  expððxk  Exk Þ2 =ð2  ðrandð1Þ  Hek þ Enk Þ2 ÞÞÞÞ k¼1

7.2.5

Cloudization Factor Importance and Joint Importance Distribution

According to definition 3.4 and equations (7.3) and (7.4), the cloudization component factor importance distribution (CCFID) and cloudization system factor importance distribution (CSFID) are shown in equations (7.9) and (7.10), respectively [9]. ; . . .; xn Þ FIi ðx1 ; x2Q n @ ½1 ð1ð1expððxk Exk Þ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ k¼1 ¼ @xk

ð7:9Þ

Cloudization Space Fault Tree

167

FIT ðxh1 ; x2 ; . . .; xn Þ i ‘r Q Qn @ 1 ð1ð1expððxk Exk Þ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ ½ Ei 2Kj j¼1 k¼1 ¼ @xk

ð7:10Þ

According to definition 3.5 and equations (7.3) and (7.4), the cloudization component factor joint importance distribution (CCFJID) and cloudization system factor joint importance distribution (CSFJID) are shown in equations (7.11) and (7.12), respectively. . . .; xn Þ FIi ðx1 ; x2 ;Q n @ l ½1 ð1ð1expððxk Exk Þ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ k¼1 ¼ @x1 @x2 ...@xl FIT ðx1 ;hx2 ; . . .; xn Þ ‘r Q l ¼

7.2.6

@

j¼1

½1 Ei 2Kj

Qn k¼1

ð1ð1expððxk Exk Þ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ

ð7:11Þ

i

ð7:12Þ

@x1 @x2 ...@xl

Cloudization Component Domain Importance

The calculations of ZIg ðiÞ and ZIcg ðiÞ mainly rely on equations (7.3), (7.4), and (7.8). These differ from ZIg ðiÞ and ZIcg ðiÞ in definition 3.6. The CPID and CCID express the basic fault data with CM, and the cloud droplets obtained are discrete data; thus, the integral in definition 3.6 is unusable. The summation of all cloud droplets is used instead of the integral to express the two cloud concepts, and the cloudization component domain probability importance (CCDPI) and cloudization component domain criticality importance (CCDCI) are as shown in equations (7.13) and (7.14), respectively [4]: ZIg ðiÞ ¼

C X @PT ðx1 ; x2 ; . . .; xn Þ c¼1

ZIcg ðiÞ ¼

@Pi ðx1 ; x2 ; . . .; xn Þ

C X Pi ðx1 ; x2 ; . . .; xn Þ  Ig ðiÞ ðx ; x ; . . .; xn Þ P c¼1 T 1 2

ð7:13Þ

ð7:14Þ

The cloud droplets are randomly distributed in the domain, which is a part of the working environment consisting of change factors. In the analysis process, the number of cloud droplets needs to be the same, and thus we can simply compare the importance of the factors. Equations (7.3), (7.4), and (7.7) will be substituted into equations (7.13) and (7.14) to calculate the CCDPI and CCDCI, respectively. Pi ðx1 ;x2 ;...;xn Þ T ðx1 ;x2 ;...;xn Þ It should be noted that @P @Pi ðx1 ;x2 ;...;xn Þ and PT ðx1 ;x2 ;...;xn Þ  Ig ðiÞ are distributions and space surfaces, respectively; ZIg ðiÞ and ZIcg ðiÞ are numerical values; and the importance is the entire working environment consists of these factors.

Space Fault Tree Theory and System Reliability Analysis

168

7.2.7

Cloudization Path Set Domain and Cut Set Domain

The PSD and CSD concepts are not based on a combination of components but on the working conditions of the system environment. They are not sets of several components causing a system fault simultaneously. Because the system works under certain conditions, the components are influenced by these conditions. Their fault probabilities are constantly changing, which result in a change in the system fault probability. The PSD, CSD, and DB form the entire study domain. Curves, faces, or higher dimensions of the DB should be closed. Considering the PSD, CSD, and DB in definition 7.1, the cloudization path set domain (CPSD) and cloudization cut set domain (CCSD) can be established in the CLSFT according to the original Pb, which is the dividing line between the PSD and CSD, indicating the degree of tolerance to the system fault probability. In addition, Pb is a dividing line in the CPSD and CCSD, but not an explicit dividing line, owing to the cloud droplet distribution. If the component or system fault probability expressed by the cloud droplets is less than Pb, denoted by Pb > Pi ðx1 ; x2 ; . . .; xn Þ or PT ðx1 ; x2 ; . . .; xn Þ, then the cloud droplets are in the CPSD. If the cloud droplets are larger than Pb, they are in the CCSD [3].

7.2.8

Uncertainty Analysis of Reliability Data

The reliability data of the component or system have a certain degree of uncertainty. The study of the data uncertainty should be considered in two ways. First, what parameters can express the uncertainty? Second, how can these parameters be obtained? For the first question, we use the CM theory proposed by the academic LI, as described in section 1.2.5. The expectation (Ex) reflects the centre of the data, reducing the randomness. Based on the Ex, the entropy (En) and hyper entropy (He) can further express the uncertainty of the data. Thus, the discreteness of the data is expressed by En and He; the randomness is expressed by Ex, En, and He; and the fuzziness is expressed by En and He. For the second question, there are some groups of reliability data of the system operating within the same environment. The difference between Ex, En, and He in the CMs expressing each group is small. The degree of change is extremely small, and the data are stable. For the degree of uncertainty (i.e. the CM characteristic parameters), use the derivative method to change their degree of change, that is, the change is the derivative of CFGD based on the influencing factors. According to the system reliability data obtained by equation (7.4), the three parameters needed to calculate the uncertainty of the system are equations (7.15)– h‘ Q i (7.17) [7]: r PTEx!dk ðx1 ; x2 ; . . .; xn Þ ¼

@

j¼1

Ei 2Kj

Pi ðx1 ;x2 ;...;xn Þ

h‘ Q @ Ex Q i r n @ 1 ð1ð1expððxExÞ2 =ð2ðrandð1ÞHek þ Enk Þ2 ÞÞÞÞ ½ j¼1 k¼1 Ei 2Kj ¼ @ Ex ð7:15Þ

Cloudization Space Fault Tree

PTEn!dk ðx1 ; x2 ; . . .; xn Þ ¼

@

169

h‘ Q r

i Ei 2Kj

j¼1

Pi ðx1 ;x2 ;...;xn Þ

h‘ Q @ En Q i r n 1 ð1ð1expððxExk Þ2 =ð2ðrandð1ÞHek þ EnÞ2 ÞÞÞÞ ½ j¼1 k¼1 Ei 2Kj ¼ @ En ð7:16Þ @

PTHe!dk ðx1 ; x2 ; . . .; xn Þ ¼ ¼

@ @

h‘ Q r j¼1

h‘ Q r j¼1

i Ei 2Kj

Pi ðx1 ;x2 ;...;xn Þ

@He

½1 Ei 2Kj

Qn k¼1

i

ð1ð1expððxExk Þ2 =ð2ðrandð1ÞHe þ Enk Þ2 ÞÞÞÞ @ He

ð7:17Þ By equations (7.15)–(7.17), the degree of change in the discreteness, fuzziness, and randomness of the system reliability data is determined within the domain of the change in the environment factors. The specific implementation is realised by equations (7.18)–(7.20). If the derivative is 0, then the data are not changed; the larger the difference is with a derivative of 0, the greater the changes in the degree of the data. The total number of cloud droplets is M, the threshold is Q, k ¼ ðkmax  kmin ÞQ, and m (mEn , mEx , mHe ) is the number of cloud droplets not in ½k; k. The fuzziness df , discreteness dd , and randomness dr of the reliability data are expressed by equations (7.18)–(7.20). df ¼

ðmEn þ mHe Þ 2M

ð7:18Þ

dd ¼

ðmEn þ mHe Þ 2M

ð7:19Þ

ðmEn þ mEx þ mHe Þ 3M

ð7:20Þ

dr ¼

These three parameters are percentages, and we can determine whether they exceed the degree of tolerance with threshold Q, and then perform a specific test and evaluation.

7.3

Example Analysis

To better describe and discuss the related concepts and processes, a simple electrical system is analysed in section 2.1. The system is composed of diodes; in addition, there are five components (X1–5), and the fault of these diodes is influenced by numerous factors. In this chapter, we mainly consider the use of humidity h and temperature c. The fault tree of the system is shown again in figure 2.1.

Space Fault Tree Theory and System Reliability Analysis

170

TAB. 7.1 – CMs of component reliability. Factors

Component X1 C1c (20.11, 6.05, 1.55) C1h (44.37, 5.11, 0.55)

c/°C

h/%

X2 C2c (29.33, 5.95, 0.75) C2 h (50.75, 6.83, 0.35)

X3 C3 c (25.16, 7.90, 1.05) C3 h (39.84, 7.03, 0.11)

X4 C4 c (25.33, 5.95, 0.75) C4 h (45.90, 8.43, 1.89)

X5 C5 c (22.06, 7.37, 1.15) C5h (61.32, 5.99, 0.87)

The simplified logical structure of the system in figure 2.1 is shown in equation (7.21). T ¼ X1 X2 X3 þ X1 X 4 þ X3 X5

ð7:21Þ

The statistics of the reliability data of these components are obtained under the change in c and h. We input the statistical data into the RCMG, and obtain the characteristic parameters of the CMs expressing the reliability of these components, as shown in table 7.1. According to equation (7.21), the system reliability is shown in equation (7.22). PT ðt; cÞ ¼ P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P 2 P 3 P 5 þ P 1 P 2 P 3 P 4 P 5

ð7:22Þ

where P15 is P15 ðc; hÞ for short.

7.3.1

Cloudization Fault Probability Distribution

The reliability of X1 is influenced by c and h, and the corresponding CMs are C1c (20.11, 6.05, 1.55) and C1h (44.37, 5.11, 0.55), respectively. According to equations (7.2) and (7.3), we can obtain the CCFGD of X1 , as shown in equation (7.23). 8 2 2 c > > < P1 ðcÞ ¼ 1  expððc  20:11Þ =ð2  ðrandð1Þ  1:55 þ 6:05Þ ÞÞ ð7:23Þ P1h ðhÞ ¼ 1  expððh  44:37Þ2 =ð2  ðrandð1Þ  5:11 þ 0:55Þ2 ÞÞ > > : P ðc; hÞ ¼ 1  ð1  P c ðcÞÞ  ð1  P h ðhÞÞ 1

1

1

Set the X1 operating domain as c 2 ½0; 40 °C, h 2 ½30; 60%, and draw the CCFGD of X1 , as shown in figure 7.1. As shown in figure 7.1, the fault probability 2 ½0; 25%Þ is expressed as ‘•’; the fault probability 2 ½25%; 50%Þ is expressed as ‘*’; the fault probability 2 ½50%; 75%Þ is expressed as ‘+’; and the fault probability 2 ½75%; 100% is expressed as ‘’. If it is not specified, the symbols in the following figures are the same as in figure 7.1. The figure clearly shows the component fault probability under different conditions of c and h. The fault probability of the centre position of the figure is smaller. The factor states of the position are most suitable for the component to work. The farther away from the position it is, the higher the fault probability.

Cloudization Space Fault Tree

FIG. 7.1 – CCFGD of X1 .

171

172

Space Fault Tree Theory and System Reliability Analysis

Set the system operating domain as c 2 ½0; 50 °C, h 2 ½0; 80%, according to equations (7.2), (7.3), and (7.22), and draw the CSFGD, as shown in figure 7.2. The results of figures 7.1 and 7.2 show the cloudization fault probability of the component and system. From these figures, it is possible to determine what environment (low fault rate) is suitable for the operation of the component and system. The environment is the change domain of the factors influencing the reliability. The method provides a reference for the safety of the actual production system.

7.3.2

Cloudization Fault Probability Distribution Trend

Equation (7.23) is brought into equation (7.5) to obtain the CCFPDT, as shown in equation (7.24). " # 8 2 2 1  ð1  ½1  expððc  20:11Þ =ð2  ðrandð1Þ  1:55 þ 6:05Þ ÞÞÞ > > > @ > > > ð1  ½1  expððh  44:37Þ2 =ð2  ðrandð1Þ  5:11 þ 0:55Þ2 ÞÞÞ < Pc ¼ i @c " # 2 > 1  ð1  ½1  expððc  20:11Þ =ð2  ðrandð1Þ  1:55 þ 6:05Þ2 ÞÞÞ > > > @ > > : h ð1  ½1  expððh  44:37Þ2 =ð2  ðrandð1Þ  5:11 þ 0:55Þ2 ÞÞÞ Pi ¼ @h ð7:24Þ Setting the component operating domain as c 2 ½0; 60 °C, h 2 ½0; 80%, according to equation (7.24) the CCFPDT of X1 is drawn, as shown in figure 7.3. Note: The Z-axis in the figure is divided equally into 4 ranges, which are expressed by different symbols, from top to bottom: ‘o’, 2 [0.2, 0.1); ‘+’, 2 [0.1, 0); ‘*’, 2 [0, –0.1); ‘•’, 2 [–0.1, –0.2]. Similarly, the CCFGD of X15 is obtained, and applied to equation (7.6) to obtain and draw the CSFPDT, as shown in figure 7.4. Note: The Z-axis in the figure is divided from top to bottom as follows: ‘o’, 2 [0.1, 0.05); ‘+’, 2 [0.05, 0); ‘*’, 2 [0, − 0.05); ‘•’, 2 [−0.05, − 0.1]. In figure (b), from the top to bottom, ‘+’, 2 [1.5,1) × 10−3; ‘*’, 2 [1, 0.5) × 10−3; ‘•’, 2 [0.5, 0] × 10−3. As shown in figures 7.3 and 7.4, we can know the change in the reliability of the components and system with respect to c and h. It is possible to determine which structure factors are favourable or unfavourable to the reliability. For the derivative of the fault probability distribution, if the result is larger than 0, then the fault probability is increasing within the domain; otherwise, if it is less than 0, the fault probability is decreasing. As shown in figure 7.3, when c > 20 °C, the change in temperature reduces the reliability, and the reliability increases when c < 20 °C. When h > 45%, the change in humidity decreases the reliability; the reliability increases when h < 45%. In figure 7.4, the system reliability has no obvious regularity with the change in c, which is the result of the joint action of numerous components. The change trend of the reliability is shown in the figure according to the values of the factors. The influence of h on the system reliability is extremely small (10−3); thus, the system fault probability increases and the reliability decreases, but with only a slight difference.

Cloudization Space Fault Tree

173

FIG. 7.2 – CSFGD.

174

Space Fault Tree Theory and System Reliability Analysis

FIG. 7.3 – CCFPDT of X1 . (a) CCFPDT for c, (b) CCFPDT for h.

Cloudization Space Fault Tree

175

FIG. 7.3 – (continued).

176

Space Fault Tree Theory and System Reliability Analysis

FIG. 7.4 – CSFPDT. (a) CSFPDT for c, (b) CSFPDT for h.

Cloudization Space Fault Tree

177

FIG. 7.4 – (continued).

Space Fault Tree Theory and System Reliability Analysis

178

7.3.3

Cloudization Importance Distribution Probability and Criticality

According to equations (7.7), (7.22), (7.23), and CCFGD of X15 , the CPID of X15 is calculated as shown in equation (7.25). 8 PT ðc; hÞ ¼ P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 > > > > > þ P1 P2 P3 P4 P5 > > > > > @P > T ðc; hÞ > > Ig ð1Þ ¼ ¼ P2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 > > @P > 1 ðc; hÞ > > > > @PT ðc; hÞ > > ¼ P1 P3  P1 P3 P4  P1 P3 P5 þ P1 P3 P4 P5 < Ig ð2Þ ¼ @P2 ðc; hÞ > @PT ðc; hÞ > > > ¼ P1 P2 þ P5  P1 P2 P4  P1 P4 P5  P1 P2 P5 þ P1 P2 P4 P5 Ig ð3Þ ¼ > > @P3 ðc; hÞ > > > > > @PT ðc; hÞ > > ¼ P1  P1 P2 P3  P1 P3 P5 þ P1 P2 P3 P5 Ig ð4Þ ¼ > > @P4 ðc; hÞ > > > > > @PT ðc; hÞ > > ¼ P3  P1 P3 P4  P1 P2 P3 þ P1 P2 P3 P4 : Ig ð5Þ ¼ @P5 ðc; hÞ

ð7:25Þ

Setting the component operating domain c 2 ½0; 50 °C and h 2 ½20; 80%, the CPID of X1 is drawn, as shown in figure 7.5, where M = 300. Note: The Z-axis in the figure is divided equally into four ranges, from top to bottom: ‘o’, ‘+’, ‘*’, and ‘•’, as shown in figure 7.6. In figure 7.6, the reliability of X1 has a different influence on the system reliability under different conditions of c and h. The domain with a large probability importance is concentrated in c 2 [15, 30] °C and h 2 [35, 70]%. When the system is operating in this environment, the change reliability of X1 has the greatest influence on the system reliability. Further, according to equation (7.25) and CCFGD of X15 , the CCID of X15 is calculated as shown in equation (7.26). The CCID of X1 is shown in figure 7.6, where M = 300. 8 > Igc ð1Þ > > > > > > > > > > > > > > > Igc ð2Þ > > > > > > > > > > > > > > c > > < Ig ð3Þ > > > > > > > > > I c ð4Þ > > g > > > > > > > > > > > > > Igc ð5Þ > > > > > > > > :

¼ PPT1 ðc;hÞ ðc;hÞ  Ig ð1Þ

1ð1P c ðcÞÞð1P h ðhÞÞ

¼ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P1 4 P1 P3 P4 P1 5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 ðP2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 Þ ¼ PPT2 ðc;hÞ ðc;hÞ  Ig ð2Þ 1ð1P c ðcÞÞð1P h ðhÞÞ

¼ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P2 4 P1 P3 P4 P2 5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 ðP1 P3  P1 P3 P4  P1 P3 P5 þ P1 P3 P4 P5 Þ ¼ PPT3 ðc;hÞ ðc;hÞ  Ig ð3Þ 1ð1P c ðcÞÞð1P h ðhÞÞ

¼ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P3 4 P1 P3 P4 P3 5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 ðP1 P2 þ P5  P1 P2 P4  P1 P4 P5  P1 P2 P5 þ P1 P2 P4 P5 Þ ¼ PPT4 ðc;hÞ ðc;hÞ  Ig ð4Þ 1ð1P c ðcÞÞð1P h ðhÞÞ

¼ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P4 4 P1 P3 P4 P4 5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 ðP1  P1 P2 P3  P1 P3 P5 þ P1 P2 P3 P5 Þ ¼ PPT5 ðc;hÞ ðc;hÞ  Ig ð5Þ 1ð1P c ðcÞÞð1P h ðhÞÞ

¼ P1 P2 P3 þ P1 P4 þ P3 P5 P1 P2 P3 P5 4 P1 P3 P4 P5 5 P1 P2 P3 P5 þ P1 P2 P3 P4 P5 ðP3  P1 P3 P4  P1 P2 P3 þ P1 P2 P3 P4 Þ

ð7:26Þ

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179

FIG. 7.5 – CPID of X1 .

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Space Fault Tree Theory and System Reliability Analysis

FIG. 7.6 – CCID of X1 .

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181

Similarly, figure 7.6 shows the CCID of the reliability of X1 within the domain. We can see from figures 7.5 and 7.6 that the CPID and CCID have significant uncertainty. Although the accuracy of the results is less than that of the original SFT, the SFT has eliminated the uncertainty of data reliability in the process of constructing the CF; thus, the data characteristics cannot be passed to the final importance distribution. By contrast, the CCF based on the CM preserves the characteristics of the original data. Therefore, the importance distribution of the cloudization component using CCF can retain the characteristics of the original reliability data. Despite a sacrifice with significant uncertainty, they better reflect the uncertainty of the original data.

7.3.4

Cloudization Importance Distribution of Factor and Factor Joint

According to equations (7.22) and (7.10), the importance distributions of c and h are obtained, respectively, with M = 300. The two distributions are shown in figures 7.7 and 7.8, respectively. Note that ‘•’, 2 ½0:1; 0:01Þ; ‘*’, 2 ½0:01; 0Þ; ‘+’, 2 ½0; 0:01Þ; ‘’, 2 ½0:01; 0:15, similar to figure 7.8. In figure 7.7, the importance marked by ‘+’ and ‘’ is greater than 0, that is, the temperature change increases the system fault probability and decreases the system reliability, where ‘•’ and ‘*’ are opposite. In figure 7.7, the temperature boundary c = 25 °C, when c < 25 °C, the system reliability increases; and at c > 25 °C, the system reliability decreases, and thus the system is best controlled at below 25 °C. In particular, for c 2 [25, 40] °C and h 2 [30, 60]%, the system reliability reduces quickly; therefore, the system should not be operated in the environment. When c 2 [25, 40] °C and h 2 [30, 60]%, the system reliability increases; thus, the environment is suitable for system operation. In figure 7.8, the humidity boundary h = 40%; when h > 40%, the humidity change increases the system fault probability and reduces the system reliability; in addition, when h > 40%, the results are contrary. When c 2 [13, 33] °C and h 2 [40, 65]%, the system reliability reduces quickly; thus, the system should not be operated in the environment. When c 2 [13, 33] °C and h 2 [27, 40]%, the system reliability increases; thus, the environment is suitable for the system operation. The CSFID analysis presented above is based on the influence of one factor on the system reliability. The following analysis is regarding a multi-factor change influencing the reliability, namely, the CFJID. The results of the CFJID may differ depending on the derivative order of the factors. For this case, the CFJID is the derivative of c and h; thus, the derivative order is c first, followed by h, as donated by c → h; or vice versa, h → c. After several simulations, the two orders of CFJID are basically the same, and because the natural factors are continuous, there is no discontinuity. By contrast, the randomness of the CM also leads to results with incomplete and basically consistent characteristics. Figure 7.9 shows the SFJID with the order of c → h. Note that ‘’, 2 ½0:02; 0:01Þ; ‘*’, 2 ½0:01; 0Þ; ‘+’, 2 ½0; 0:01Þ; ‘’, 2 ½0:01; 0:02. Figure 7.9 shows a superposition with the change in c and h. The figure is clearly divided into four domains, with the humidity boundary h = 40% and the

182

Space Fault Tree Theory and System Reliability Analysis

FIG. 7.7 – CSFID for c.

Cloudization Space Fault Tree

183

FIG. 7.8 – CSFID of h.

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Space Fault Tree Theory and System Reliability Analysis

FIG. 7.9 – SFJID.

Cloudization Space Fault Tree

185

temperature boundary c = 25 °C. The change in factors in the two adjacent domains makes the change in system reliability contrary; and the opposite two domains make the system reliability consistently change. In general, considering the influence of the two-factor change on the system reliability, c 2 [0, 25] °C \ h 2 [0, 40]% and c 2 [25, 50] °C \ h 2 [40, 80]% are suitable for the system operation, whereas the remaining domains are not. However, this division is inaccurate because the CM itself is a representation of uncertainty.

7.3.5

Cloudization Component Domain Importance

CCDI is different from the CPID and CCID. The domain mentioned here is the research domain consisting of factors. The subset of the domain can also be targeted as the research domain. The CPID and CCID are not sufficiently able to express the importance of the component within the complete research domain. It is impossible to consider the full importance of the component within the whole research domain. By contrast, the importance distribution is the space distribution whose dimensions are the working environment factors. The CCDI is the entire value of the distribution and is a specific value. The CCDI and the component importance distribution are not parallel concepts. The CCDI is based on the component importance distribution, and the component importance distribution should be calculated first, applying some methods (such as integration or summation) to obtain the CCDI. According to equation (7.13), a CCFGD of X15 is used to calculate a CCDPI of X15 , as shown in equation (7.27). Here, M = 300, and the distribution figures are omitted. 8 PT ðc; hÞ ¼ P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 > > > C C @P ðc; hÞ X > P > T > > ¼ ZIg ð1Þ ¼ ðP2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 Þ > > > c¼1 @P1 ðc; hÞ c¼1 > > > C > C @P ðc; hÞ X P > T > > ðP1 P3  P1 P3 P4  P1 P3 P5 þ P1 P3 P4 P5 Þ ¼ > ZIg ð2Þ ¼ > > c¼1 @P2 ðc; hÞ > c¼1 < C C @P ðc; hÞ X P T ¼ ð3Þ ¼ ðP1 P2 þ P5  P1 P2 P4  P1 P4 P5  P1 P2 P5 þ P1 P2 P4 P5 Þ ZI > g > > c¼1 @P3 ðc; hÞ > c¼1 > > > C C @P ðc; hÞ X > P > T > > ZIg ð4Þ ¼ ðP1  P1 P2 P3  P1 P3 P5 þ P1 P2 P3 P5 Þ ¼ > > > c¼1 @P4 ðc; hÞ c¼1 > > > C > C @P ðc; hÞ X P > T > > ðP3  P1 P3 P4  P1 P2 P3 þ P1 P2 P3 P4 Þ ¼ : ZIg ð5Þ ¼ c¼1 @P5 ðc; hÞ c¼1

ð7:27Þ The results of ZIg ðiÞ of X15 are 54.5969, 3.8005, 33.1951, 24.3814, and 13.3504, respectively. These values comprehensively consider the importance of the system fault caused by each component in the working environment (c 2 [0, 50] °C and h 2 [0, 80]%). That is, within the domain, the type of CCDPI is as follows: ZIg ðX1 Þ = 54.5969 > ZIg ðX3 Þ = 33.1951 > ZIg ðX4 Þ = 24.3814 > ZIg ðX5 Þ = 13.3504 > ZIg ðX2 Þ = 13.3504. This importance is for the whole domain of the system operation, and not for a certain temperature or humidity, which is at the macro level. If the system reliability is ensured in this domain,

186

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then the component X1 is most important. This does not mean that X1 has the greatest importance under the environmental conditions (e.g. c = 35 °C and h = 40%), but it has the greatest importance with the summation within the range of c 2 [0, 50] °C and h 2 [0, 80]%. Similarly, equation (7.28) is obtained according to equations (7.7), (7.14), and (7.27). Here, M = 300, the distribution figures of which are omitted. 8

C X > P1 ðc; hÞ > >  Ig ð1Þ ZIcg ð1Þ ¼ > > > ðc; hÞ P > c¼1 T > > > 0 1 > > > 1  ð1  P1c ðcÞÞ  ð1  P1h ðhÞÞ > > C > XB C > > B P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 C > ¼ > @ A > > c¼1 > > ðP2 P3 þ P4  P2 P3 P4  P3 P4 P5  P2 P3 P5 þ P2 P3 P4 P5 Þ > > > > > > C > X > P2 ðc; hÞ > c > ZI  Ig ð2Þ ð2Þ ¼ > g > > ðc; hÞ P > c¼1 T > > > 0 1 > > > 1  ð1  P2c ðcÞÞ  ð1  P2h ðhÞÞ > > C B X > C > > ¼ B P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 C > > @ A > > c¼1 > > ðP P  P P P  P P P þ P P P P Þ > 1 3 1 3 4 1 3 5 1 3 4 5 > > > > > C > > > ZIc ð3Þ ¼ X P3 ðc; hÞ  I ð3Þ > > g g > > P ðc; hÞ > c¼1 T > < 0 1 1  ð1  P3c ðcÞÞ  ð1  P3h ðhÞÞ > C XB > C > > B P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 C > > ¼ @ A > > c¼1 > > ðP P þ P  P P P  P P P  P P P þ P P P P Þ > 1 2 5 1 2 4 1 4 5 1 2 5 1 2 4 5 > > > > > C > X > P4 ðc; hÞ > c > ZI  Ig ð4Þ ð4Þ ¼ > g > ðc; hÞ P > > c¼1 T > > > 0 1 > > > 1  ð1  P4c ðcÞÞ  ð1  P4h ðhÞÞ > X C > > B C > > B P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 C ¼ > > @ A > > c¼1 > > ðP  P P P  P P P þ P P P P Þ > 1 1 2 3 1 3 5 1 2 3 5 > > > > > C > X > P5 ðc; hÞ > > ZIcg ð5Þ ¼  Ig ð5Þ > > ðc; hÞ P > > c¼1 T > > > 0 1 > > > 1  ð1  P5c ðcÞÞ  ð1  P5h ðhÞÞ > C > X > B C > > B P1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 C ¼ > > @ A > > c¼1 : ðP3  P1 P3 P4  P1 P2 P3 þ P1 P2 P3 P4 Þ

ð7:28Þ ZIcg ðiÞ for

X15 are 45.7342, 4.4966, 27.2486, 21.6836, and 13.5333, The results of respectively. Within the domain, the CCDCI is sorted as follows: ZIcg ðX1 Þ = 45.7342 > ZIcg ðX3 Þ = 27.2486 > ZIcg ðX4 Þ = 21.6836 > ZIcg ðX5 Þ = 13.5333 > ZIcg ðX2 Þ = 4.4966.

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187

The sorting of the importance is the same as that of the corresponding non-cloudization importance [11]. However, the concepts and calculation methods of the domain importance are more suitable for the fault data obtained by the actual operation of the system.

7.3.6

Cloudization Path Set Domain and Cut Set Domain

According to definition 7.1 and equation (7.23), to calculate the CCSD and CPSD of X1, for a different Pb, the distributions of cloud droplets in the two domains are different, as shown in figure 7.10. In figure 7.10a–d, Pb is equal to 10%, 20%, 30%, and 50% (the distributions of 30% and 40% are the same), respectively. Here, ‘o’ indicates the cloud droplets with P1 ðc; hÞ < Pb, that is, the CPSD of X1; and ‘×’ indicates the cloud droplets with P1 ðc; hÞ > Pb, that is, the CCSD of X1. With an increase in Pb, the fault probability toleration of X1 increases, and the CPSD increases. By contrast, it defines the change domain corresponding to the CPSD of X1 and expresses the combination of the change domain suitable for X1 to work. For the component working in the PSD, the reliability is relatively high; otherwise, for the components working in the CSD, the reliability is relatively low. Unlike the PSD and CSD, the CPSD and CCSD do not have a clear dividing boundary. That is, they are not superficially accurate, but in essence, they reflect the uncertainty of the basic data. Thus, cloudization can better reflect the original data characteristics. According to definition 7.1 and equation (7.22), the CCSD and CPSD of the system are calculated as shown in figure 7.11. In figure 7.11a–d, Pb is equal to 10%, 30%, 40%, and 50% (the distributions of 10% and 20% are the same), respectively. The mean of ‘o’ and ‘×’ are the same as in figure 7.10. Consistent with the characteristics shown in figure 7.11, figure 7.11 illustrates the work domain suitable for the system, namely the CPSD. The fault probability of the system operating in the CPSD is lower, and the CCSD has a large fault probability in the system.

7.3.7

Uncertainty Analysis of Reliability Data

Equation (7.22) is substituted into equations (7.15)–(7.17) to obtain equations (7.29)–(7.34). PTEx!c ðc; h Þ @ ðP1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Þ ¼ @Exc

ð7:29Þ PTEn!c ðc; h Þ @ ðP1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Þ ¼ @Enc

ð7:30Þ

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Space Fault Tree Theory and System Reliability Analysis

FIG. 7.10 – CPSD and CCSD of X1 with different Pb. (a) Pb = 10%, (b) Pb = 20%, (c) Pb = 30%, (d) Pb = 50%.

Cloudization Space Fault Tree

FIG. 7.10 – (continued).

189

190

Space Fault Tree Theory and System Reliability Analysis

FIG. 7.11 – CPSD and CCSD with different Pb. (a) Pb = 10%, (b) Pb = 30%, (c) Pb = 40%, (d) Pb = 50%.

Cloudization Space Fault Tree

FIG. 7.11 – (continued).

191

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Space Fault Tree Theory and System Reliability Analysis

PTHe!c ðc; h Þ @ ðP1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Þ ¼ @Hec

ð7:31Þ PTEx!h ðc; h Þ @ ðP1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Þ ¼ @Exh

ð7:32Þ PTEn!h ðc; h Þ @ ðP1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Þ ¼ @Enh

ð7:33Þ PTHe!h ðc; h Þ @ ðP1 P2 P3 þ P1 P4 þ P3 P5  P1 P2 P3 P4  P1 P3 P4 P5  P1 P2 P3 P5 þ P1 P2 P3 P4 P5 Þ ¼ @Heh

ð7:34Þ This section only analyses the uncertainty of temperature-related system reliability data. equations (7.29)–(7.31) are used to plot the change degree of the data. M = 300, Q = 10%, and the result is shown in figure 7.12. Note that ‘+’ indicates a derivative of less than k (satisfying the requirement), and ‘o’ indicates a derivative of greater than k (not satisfying the requirement). Given the statistical data in figure 7.12, mEx ¼ 6, mEn ¼ 6, and mHe ¼ 5, according to equations (7.18)–(7.20); thus, df ¼ 1:83%, dd ¼ 1:83%, dr ¼ 1:89% are obtained. If Q = 5%, then mEx ¼ 10, mEn ¼ 5, and mHe ¼ 16, and thus df ¼ 7%, dd ¼ 7%, and dr ¼ 3:44%. We can see that the adjusted Q has an important influence on the evaluation results. The greater Q is, the greater the derivative; otherwise, the smaller the derivative. With a definite Q, the uncertainty of the reliability data is evaluated. However, the specific value of the evaluation criteria of df , dr , and dd should be determined according to the actual situation. If the three values are less than Q = 5%, the data are available; thus, when Q = 10%, the uncertainty is larger. In addition, some useful information can be obtained from the change degree of the three parameters in uncertainty. Here, ‘o’ expresses the cloud droplets with an unacceptable degree. These droplets appear to be similar in this case. This shows that the fault occurrence of the system has a large uncertainty under fixed environment conditions. This may be due to system errors, human factors, or environmental changes, and should be further analysed; however, they are not discussed here. The two main parameters Q and d for the uncertainty evaluation method of the reliability data need to be analysed concretely. The bigger M is the better, although we must pay attention to the number of calculations.

Cloudization Space Fault Tree

193

FIG. 7.12 – Uncertainty degree of temperature-related data. (a) Change degree of Ex for c, (b) Change degree of En for c, (c) Change degree of He for c.

194

Space Fault Tree Theory and System Reliability Analysis

FIG. 7.12 – (continued).

Cloudization Space Fault Tree

FIG. 7.12 – (continued).

195

Space Fault Tree Theory and System Reliability Analysis

196

7.4

Conclusions

In this section, the definitions and related methods of the CLSFT are proposed. The CLSFT inherits the ability of the SFT to analyse the reliability influenced by multiple factors and the ability of the CM to express the uncertainty of the data. Thus, the CLSFT is suitable for the analysis and processing of actual fault data. Specifically, the CCF is the basis for the cloudization of the SFT, and we analyse the significance of cloudization and the feasibility of using the CCF. The CCFGD and CSFGD are two of the most basic capabilities of the CLSFT, which can be used to analyse the fault probability of the components and system under different factors. The CCFPDT and CSFPDT are the derivatives of the CCFGD and CSFGD with respect to the influencing factors, which indicate the degree of change in the fault probability with respect to the influencing factors. The CPID and CCID can determine the importance of the probability and criticality of the components under different factors. The CFID and CFJID can obtain the degree of influence of a single factor, or multiple factors, on the fault probability. Their focus is the importance of the factors, not the importance of the components. The CCDI is different from the aforementioned importance; the former is a value used to measure the importance of the probability and criticality in the change domain for the factors. Furthermore, it is a parameter to measure the importance of the entire domain. The CPSD and CCSD can distinguish between the suitable domain and unsuitable domain of the system operating within the specified change domain of the factors. The concepts of the original path set and cut set are changed from the basic event combination to a combination of the factor change domain. The uncertainty analysis of the reliability data uses cloudization characteristic parameters as evaluation data, and specific methods are developed using df , dd , and dr . Simultaneously, a simple electrical system is given as an example in section 2.2. The system reliability and reliability data are analysed based on the above definitions and methods. Because there are more quantitative and qualitative analysis results for the example in the analysis process, the details are not given here. The analysis results show that the CLSFT can fully inherit the multi-factor analysis ability of the SFT, and the ability of the CM to analyse the data uncertainty. Finally, it can effectively analyse the reliability data as influenced by numerous factors and uncertainty. This research can provide an analysis method for the safe operation of a system, and provide a new way for the development of the SFT. More detailed questions can be found in document [12].

References [1] Li S.S., Cui T.J., Ma Y.D., et al. (2016) Research on the cloud fault probability distribution trend in SFT, J. Saf. Sci. Technol. 12, 60. [2] Li S.S., Cui T.J., Ma Y.D. (2016) Method research on fuzzy comprehensive evaluation of system reliability based on cloud model considering variable factors, Chin. Saf. Sci. J. 26, 132. [3] Cui T.J., Li S.S., Ma Y.D., et al. (2016) Reconstruction and research on size set domain and cut set domain in cloud SFT, App. Res. Comput. 33, 3582.

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[4] Cui T.J., Li S.S., Ma Y.D., et al. (2016) Construction and understanding of cloud component area importance, App. Res. Comput. 33, 3570. [5] Cui T.J., Li S.S., Ma Y.D., et al. (2017) Realization and research on cloud of probability importance distribution and key importance distribution in SFT, App. Res. Comput. 34, 1971. [6] Cui T.J., Li S.S., Ma Y.D., et al. (2016) Feasibility analysis and application of cloud model instead of characteristic function in SFT, J. Comput. App. 36, 37. [7] Li S.S., Cui T.J., Ma Y.D. (2017) Uncertainty evaluation of the reliability based on cloud model and the SFT, App. Res. Comput. 34, 3656. [8] Li S.S., Cui T.J., Ma Y.D., et al. (2017) Cloud similarity based on the envelope and its application to the safety assessment, J. Saf. Environ. 17, 1267. [9] Cui T.J., Li S.S., Ma Y.D. (2017) The realization and knowledge of the cloud factor importance and the factor joint importance in SFT, J. Saf. Environ. 17, 2109. [10] Li D.Y., Liu C.Y., Gan W.Y. (2009) A new cognitive model: cloud model, Int. J. Intell. Syst. 24, 357. [11] Cui T.J., Ma Y.D. (2015) The definition and cognition of the factor importance distribution in continuous space fault tree, Chin. Saf. Sci. J. 25, 24. [12] Li S.S., Cui T.J., Liu J. (2019) Study on the construction and application of cloudization space fault tree, Cluster Comput. 22, 5613.

Chapter 8 Cloud Similarity In an actual safety assessment, the conclusions given by experts are usually vague and uncertain. The information obtained is difficult to determine from such data, and it is usually necessary to analyse, merge, and finally simplify the results. To realise the aforementioned process, a new semantic simplification method based on the cloud model and SFT is proposed. This method will be used to express the opinions of experts using the cloud model, and numerous expert opinions when applying the same number of cloud models. According to the characteristics of the cloud model, multiple cloud models are built on the envelope, integrating the area between them. To simplify the opinions, the proportion between the overlapping area and the integral area is considered to judge the similarity of the cloud models, namely the similarity of expert opinions. In this section, a moderate risk semantic of the distance between the stable strata and the roof of a roadway is analysed using the above methods. The relationship between the two opinions given by experts is analysed, and the semantic relevance is obtained. Finally, the advantages of the algorithm are provided. The chapter is divided into sections. In section 8.1, similar algorithms of the cloud model are described. In section 8.2, a cloud similarity computation based on the envelopes is provided. In section 8.3, the application is provided. In section 8.4, the advantage of the algorithm is analysed. Finally, section 8.5 provides some concluding remarks.

8.1

Similarity Algorithms of Cloud Model

Based on the literature review [1], in China, the main cloud model similarity algorithms are as follows. Liu Changyu proposed measuring the similarity based on the distance of the cloud droplets. This method can indicate the similarity of the cloud model, although the calculations of the similarity are directly related to the number of droplets, and there is a large consumption of resources. Zhang Yong improved the above algorithm, and proposed a new algorithm based on an interval cloud. This algorithm has a large optimization in terms of the calculation accuracy and DOI: 10.1051/978-2-7598-2499-1.c008 © Science Press, EDP Sciences, 2020

200

Space Fault Tree Theory and System Reliability Analysis List of symbols

Symbol Ex En He C (Ex, En, He) L D1 , D2 l En0 lsD lxD 1  randnð1Þ  1 ki K dj Y;X j SD S\ SD1 SD2 SimðC1 ! C2 Þ D K1 L H Rc

Meaning Digital features of cloud, expectation Digital features of cloud, entropy Digital features of cloud, hyper entropy Cloud model Producing the range of cloud droplets Ordered droplet set Degree of certainty of x Mean value of En Upper envelope Lower envelope Random number in [–1, 1] Enumerating the adjustment parameters of the envelope Set of ki Cloud droplet in cloud D Vertical and horizontal coordinates of cloud droplet Number of cloud droplets Area of cloud droplets Overlapping area The first cloud model envelope area The second cloud model envelope area Similarity between cloud C1 and cloud C2 Distance from stable rock to surface roof Roadway every 10 m in length for 1 min, the infiltration volume Roadway span Stable rock thickness Compressive strength of roof rock

computational consumption. Cai Shaobin considered the digital features of a cloud model as a vector, using the cosine angle to measure the similarity between the cloud models. Although the effect of the collaborative filtering algorithm is good, the expected value of the cloud is much larger than the entropy and the hyper entropy during the process, and the roles of entropy and hyper entropy are ignored. Li Hailin proposed two types of normal cloud model similarity calculation methods. The similarity of the normal cloud model is expressed using the similarity degree of the expectation curve or the maximum boundary curve. However, this method ignores the influence of the hyper entropy, and the degree of fluctuation cannot be described. In addition, although in the literature [1–3], the cloud model similarity computing performances are improved, the influence of the number of cloud droplets on the similarity cannot be omitted. In an actual safety assessment, cloud droplets are equivalent to the opinions from experts for a specific condition. Normally, comments are the most representative of an extreme situation. The working conditions are classified by the staff members themselves under such extreme conditions. That is,

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the cloud droplets in this case are near the edge of the cloud model and are fewer in number in the internal model. This situation is obviously unsuitable for the use of the above methods.

8.2

Cloud Similarity Computation Based on Envelope

According to the semantic features of the expert evaluation during the safety assessment, a similarity calculation method based on the envelope is proposed. The method can be divided into six steps, the corresponding explanations of which are given below. The corresponding image is shown in figure 8.1 in section 8.3 Step 1: The two clouds are generated using a positive cloud generator. Each produces a certain amount of cloud droplets. Step 2: The cloud droplets generated are selected. The cloud droplets in L1 ¼ ½Ex1  3En1 ; Ex1 þ 3En1  are retained for C1 ðEx1 ; En1 ; He1 Þ. The cloud droplets in L2 ¼ ½Ex2  3En2 ; Ex2 þ 3En2  are retained for C2 ðEx2 ; En2 ; He2 Þ. Step 3: The cloud droplets in the two groups selected are sorted according to the abscissa of droplets in ascending order. Ordered droplet sets D1 and D2 are obtained. Step 4: Determine the envelope parameters. According to the characteristics of the cloud droplets generated in the cloud model, a positive generator is an exponential function l ¼ exp½ðx  ExÞ2 =ð2En02 Þ. Here, l is the degree of certainty of x, which is the degree of belonging to a certain semantic, 0  l  1. Thus, the function value of the envelope function is [0, 1], and the definition domain is L ¼ ½Ex  3En; Ex þ 3En. For the determination parameter l, set Ex as a cloud parameter in the first step; in addition, En0 is the mean value of En, namely, the expectation, where En0 is applied instead of En. To omit the influence of the cloud droplet number on the cloud similarity, the cloud similarity is computed using the envelope. According to the analysis of the characteristics of the cloud droplets, for random cloud model parameters, the parameters are generated using l ¼ exp½ðx  ExÞ2 =ð2En02 Þ in l 2 ½0; 1 and L ¼ ½Ex  3En; Ex þ 3En. There are always two isomorphic curves with the same l function but different parameters. The first isomorphism curve makes the vertical coordinate of all cloud droplets in set D lower than the corresponding horizontal coordinate of droplets in set D on the curve. The second isomorphism curve makes the vertical coordinate of all cloud droplets in set D higher than the corresponding horizontal coordinate of droplets in set D on the curve. The two envelope curves contain all droplets in D. The first curve is called the upper envelope lsD , and the second is called the lower envelope lxD . Here, lsD and lxD are the same as l, although the parameters are different. There are two aspects to consider, namely the location and shape of the cloud. If these two parameters are the same, then the two clouds are the same. The characteristic parameter Ex can indicate the location of the cloud. Either positive or inverse, Ex is easier to be obtained after the given cloud droplet distribution. Thus, Ex in lsD and lxD directly uses the cloud parameters in the first step. For En0 , the positive cloud generator is defined as En0 ¼ randnð1Þ  He þ En. En is the cloud

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Space Fault Tree Theory and System Reliability Analysis

parameter in the first step. The 1  randnð1Þ  1 makes the droplets distributed on both sides of curve l, and thus randnð1Þ  He is the key to determine the envelope. For lsD , randnð1Þ  He [ 0 ) randnð1Þ [ 0; for lxD , randn. He is the cloud characteristic parameter in the first step. Step 5: The envelopes of D1 and D2 , lsD and lxD , are the same form as l. Set lsD ¼ exp½ðx  ExÞ2 =ð2En02 Þ ¼ exp½ðx  ExÞ2 = ð2  ðK þ EnÞ2 Þ, where K ¼ ki  1=10  He, ki ¼ 1 : 10 : 1. Here, ki satisfies Y ðdj 2 DÞ\lsD ðXðdj 2 DÞÞ initially, dj is the cloud droplet in cloud D, and Y, X represent the vertical and horizontal coordinates of cloud droplet dj , separately. Similarly, set lxD ¼ exp½ðx  ExÞ2 =ð2  ðK þ EnÞ2 Þ, where K ¼ ki  1=  He, ki ¼ 1 : 10 : 1. ki satisfies Y ðdj 2 DÞ [ ls ðXðdj 2 DÞÞ. Thus, the deterD 10 minations lsD and lxD are used to determine the appropriate ki . Using MATLAB with the specific cloud set D and the above function structure, circulating ki , when the first condition is met, lsD and lxD are obtained. Step 6: Apply the cloud similarity computing method using the envelope. With the previous method of cloud similarity computing, the method of dependence on a droplet is not strong, and is mainly through the envelope integral. For a cloud C, when j ! 1, the number of dj in the cloud droplet set D will fill in the area between lsD and lxD . This means that j ! 1, and the area of the cloud droplets R is SD ¼ L ðlsD  lxD Þdx, where L ¼ ½Ex  3En; Ex þ 3En. For the two clouds, C1 ðEx1 ; En1 ; He1 Þ and C2 ðEx2 ; En2 ; He2 Þ, whether similar or not can be equivalent to the coincidence degree between SD1 and SD2 , when j1 ; j2 ! 1, namely the areas of cloud droplets D1 and D2 are compared. The following two clouds are given as an example to define and explain the similarity method. For cloud C1 ðEx1 ; En1 ; He1 Þ, Z Ex1 þ 3En1   SD1 ¼ lsD1  lxD1 dx Ex1 3En1 2 3  

  s 2  2 2  ki1 He1 =10 þ En1 Z Ex1 þ 3En1 6 exp ðx  Ex1 Þ 7 6 7  

 dx; ¼ 6  7   4 5 2 Ex1 3En1 x  exp ðx  Ex1 Þ2 He1 =10 þ En1 2  ki1 s x where ki1 and ki1 are based on the determination of ki in step 5. For cloud C2 ðEx2 ; En2 ; He2 Þ, Z Ex2 þ 3En2   SD2 ¼ lsD2  lxD2 dx Ex2 3En2 2 3  

  s 2  2 2  ki2 He2 =10 þ En2 Z Ex2 þ 3En2 6 exp ðx  Ex2 Þ 7 6 7  

 ¼ 6 7dx;  x 2  5 Ex2 3En2 4 2  exp ðx  Ex2 Þ 2  ki2 He2 =10 þ En2

where kis2 and kix2 are based on ki determined in step 5.

Cloud Similarity

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When j1 ; j2 ! 1, set the overlapping area of C1 ðEx1 ; En1 ; He1 Þ and C2 ðEx2 ; En2 ; He2 Þ as S \ ¼ SD1 \ SD2 . Namely, when j1 ; j2 ! 1, the area is filled in by the cloud droplets in set fdg ¼ fdjdj1 2 D1 ; dj2 2 D2 g. The value of S \ is obtained by the sectional integral. The breakpoints are defined as the domain endpoints and intersection points between the curves of lsD1 , lxD1 , lsD2 , and lxD2 . Obviously, if the compared order of the two clouds is different, then the similarity is different. Set similarity SimðC1 ! C2 Þ ¼ S \ =SD1 to express the similarity between cloud C1 and cloud C2 . The similarity calculation has some characteristics of the cloud model of the semantic representation. For example, for two types of semantic clouds C1 and C2 , if SD1 of C1 contains SD2 of C2 , then SD2  SD1 . It is shown that the concept of a semantic scope contained in C1 can express the concept of C2 , SimðC1 ! C2 Þ ¼ S \ =SD2 ¼ 1, and thus C1 can represent the concept of C2 . In this case, the concept of C2 is only a part of the concept of C1 , SimðC2 ! C1 Þ ¼ S \ =SD1 \1, and thus C2 cannot represent the concept of C1 . The result of the above analysis is a simplification and deleted semantic of C2 . Thus, we can summarise the two-cloud model comparison of four cases: (1) If SimðC1 ! C2 Þ ¼ SimðC2 ! C1 Þ ¼ 1, then two clouds represent the same semantic concept, and the two semantic concepts are simplified as a single concept. (2) If SimðC1 ! C2 Þ ¼ S \ =SD2 ¼ 1 and SimðC2 ! C1 Þ ¼ S \ =SD1 \1, in a semantic analysis, the representation semantic of C2 is simplified and deleted. (3) If SimðC1 ! C2 Þ\1; SimðC2 ! C1 Þ\1, then the concepts indicate that the two clouds have both the same and different parts. Through the merger method of the cloud, the semantic is merged and simplified. (4) If SimðC1 ! C2 Þ ¼ SimðC2 ! C1 Þ ¼ 0, the two semantics are completely different, which cannot be simplified.

8.3

Algorithm Application

The related research of a roadway roof fall is first described in the specific conditions and evaluations of two experts on the roadway roof risk are given. A roof fall is caused by numerous factors, and thus we should find the main factors to analyse. To determine the parameters of a cable anchor, the stable layer of the anchor should be of a certain thickness to ensure the effect. Thus, the distance from a stable rock to a surface roof and the stable rock thickness are the main factors to be considered. The span of the roadway affects the stability of the roof, and the roof falls with the long span, whereas a short span is insufficient to meet the needs of production. Thus, the span is one of the main factors. The compressive strength of the roof rock affects the roof stability. The compressive strength of the roof rock represents the bearing capacity of the roof strata. Therefore, this is also one of the main reasons to be considered. The roof water seepage reflects the poor stability of the whole roof and produces roof fall accidents. The influencing indexes of the roadway roof fall are as follows: (1) the distance from the stable rock to the surface roof, D/m; (2) groundwater seepage K1, at every 10 m length of the roadway, in

204

Space Fault Tree Theory and System Reliability Analysis TAB. 8.1 – Roadway roof fall indexes and classification.

Roof fall caused by many factors

Distance from stable rock to surface roof D/m Groundwater seepage K1/L (min10 m)−1 Roadway span L/m Stable rock thickness H/m Compressive strength of roof rock Rc/MPa

Risk level Low risk (I) >6

Moderate risk (II) 4–6

High risk (III) 2–4

Very high risk (IV) 200

4.5–5.2 20–40 100–200

5.2–6 10–20 50–100

>6

>
> > : 0  aqe  1; 0  aqf  1; aqe  aqf ; aqe ; aqf Lðaq ; a0 Þ

ð9:1Þ

Definition 9.4. Here, Lðaqe ; aqf Þ or Lðaqe ; aqf Þ represents the value domain of attribute xi

aq of object xi . The larger

xi

xi

Lðaqe ; aqf Þ xi

is, the less influence attribute aq has on object xi ;

the smaller Lðaqe ; aqf Þ is, the greater the influence of aq on object xi . xi

9.3

Object Clustering Analysis Method

For a description of the classification method, the graphical similarity between x1 and x6 is given, as shown in figure 7.3. Firstly, the concept of object similarity is given from the geometric meaning, which is the graphical similarity when considering a graphical overlap. As shown in   figure 9.2, M a2f a3f a3e a2e x1

represents a convex polygon for representing both the

attribute values of object x1 on the attribute a2 ; a3 . Figure 9.3 shows a graphical  overlap of the attribute circles of x1 and x6 , and the overlap area of M a2f a3f a3e a2e x1   f f e e and M a2 a3 a3 a2 can greatly reflect the similarity between x1 and x6 regarding x6

attribute a2, a3. Please note that the shading states of x1 and x6 are different.

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FIG. 9.3 – Similarity definition between x1 and x6 . However, it is difficult to determine the similarity between x1 and    x6 using the f f e e f f e e above methods. The overlap area of M a2 a3 a3 a2 and M a2 a3 a3 a2 in the above x1

x6

method reflects the similarity between x1 and x6 about the two attributes a2 ; a3 , and a single attribute is difficult to determine.    On the other hand, the overlap area between M a2f a3f a3e a2e x1

and M a2f a3f a3e a2e x6

needs to be calculated using complex

analytical methods, which is not satisfactory for engineering applications. Therefore, the graphical similarity is transformed into a numerical calculation method to be defined and used. From figure 9.3, there is an overlapping line segment between Lða3e ; a3f Þ and x1

f Lða3e ; a3 Þ x6

on attribute a3 . This overlapping line segment shows that attribute a3 has      that has the same effect on x1 and x6 , that a line segment L a3e ; a3f \ L a3e ; a3f x1

x6

is, x1 and x6 are similar in this line segment. Based on this idea, the similarity is defined by considering the overlapping line segment, but not the overlapping area. Definition 9.5. In system T, xi ; xj 2 U , and define Sðxi ; xj ; aq Þ as the similarity between xi and xj about attribute aq , where Sðxi ; xj ; aq Þ is determined as follows. When i = j and Sðxi ; xj ; aq Þ ¼ 1, an object is compared to itself; thus, the similarity is 1.

Clustering Analysis and Similarity

213

h i When i 6¼ j, compare the relative overlapping line segments of aq ¼ aqe ; aqf and xi h i aq ¼ aqe ; aqf . When there is no overlapping line segment between aq and aq on xj

xi

xj

Lðaq ; a0 Þ and Sðxi ; xj ; aq Þ ¼ 0, it indicates that two objects are not related to aq . When there is an overlapping line segment between aq and aq on Lðaq ; a0 Þ and xi

xj

Sðxi ; xj ; aq Þ 6¼ 0, according to the overlap, Sðxi ; xj ; aq Þ is obtained, as shown in equation (9.2).     Sðxi ; xj ; aq Þ ¼ L aqf aqe \ L aqf aqe xi xj        !          f        ¼ min aq  aqe ; aqf  aqe ; aqf  aqe ; aqf  aqe  ð9:2Þ  xj         xj xj  xi xi  xj xi  xi !                 max aqf  aqe ; aqf  aqe ; aqf  aqe ; aqf  aqe   xj        xj xj xi xi xj xi xi where 0  Sðxi ; xj ; aq Þ  1. Definition 9.6. The full similarity of xi ; xj is Sðxi ; xj Þ for C ¼ fa1 ; a2 ; . . .; an g, and Q thus Sðxi ; xj Þ ¼ nq¼1 Sðxi ; xj ; aq Þ; aq 2 C . The specific processes of the above definitions are shown in figure 9.3. Definition 9.7. The entire similarity Sðxi ; xj Þ of xi ; xj is classified by the following rules. Set kaq as the similarity threshold of xi ; xj for a single attribute aq ; in general, 0:4  kaq  0:6. Here, 1  Sðxi ; xj ; aq Þ  kaq indicates a similarity, Sðxi ; xj ; aq Þ ¼ 0 indicates a dissimilarity, and the values between themQindicate the fuzzy similarity. Q Thus, for Sðxi ; xj Þ, 1  Sðxi ; xj Þ ¼ nq¼1 Sðxi ; xj ; aq Þ  nq¼1 kaq indicates a similarity, Sðxi ; xj Þ ¼ 0 indicates a dissimilarity, and the values between them indicate a fuzzy similarity.

9.4

Improvements of Clustering Analysis

The above method simplifies the geometric similarity and obtains a simple similarity algorithm. However, the original geometric similarity is more accurate, and thus this section gives the concept of a similarity algorithm based on the geometric similarity.  In figure 9.2, M a2f a3f a3e a2e x1

represents a convex polygon representing both the

attribute values of  object x1 on attributes a2 ; a3 . The overlapping area of   f f e e f f M a2 a3 a3 a2 and M a2 a3 a3e a2e can greatly reflect the similarity between x1 and x1

x6

x6 regarding attributes a2 ; a3 . The similarity between xi and xj on attributes aq and aq þ 1 is calculated, and it is     the overlapping area of M aqf aqf þ 1 aqe þ 1 aqe and M aqf aqf þ 1 aqe þ 1 aqe . xi

xj

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214

Definition 9.8. In system T, xi ; xj 2 U , define Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ as the similarity between xi and xj about attributes aq ; aq þ 1 , as shown in equation (9.3).     f f M aqf aq þ 1 aqe þ 1 aqe \ M aqf aq þ 1 aqe þ 1 aqe xi xj    Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ ¼  f f f f M aq aq þ 1 aqe þ 1 aqe [ M aq aq þ 1 aqe þ 1 aqe xi

ð9:3Þ

xj

The different states are described as follows: (1) When there is no overlapping line segment between aq and aq on Lðaq ; a0 Þ, or xi

xj

between aq and aq on Lðaq þ 1 ; a0 Þ, Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ ¼ 0, which indicates that xi

xj

two objects are not related to aq ; aq þ 1 . (2) When there is an overlapping line segment between aq and aq on Lðaq ; a0 Þ, and xi

xj

between aq þ 1 and aq þ 1 on Lðaq þ 1 ; a0 Þ, the line segment ðaqe ; aqe þ 1 Þ does not xi

xj

intersect the line segment

xj

ðaqe ; aqe þ 1 Þ; xi xi

does not intersect the line segment

xj

in addition, the line segment ðaqf ; aqf þ 1 Þ xj

ðaqf ; aqf þ 1 Þ, xi xi

xj

and the overlap area is then

defined as shown in equations (9.4) and (9.5). This is the second overlapping state, as shown in figure 9.4.     1 f f e e f f e e M aq aq þ 1 aq þ 1 aq \ M aq aq þ 1 aq þ 1 aq ¼ ðbd  acÞ sin h; 2 xi xj h ¼ \aq a0 aq þ 1 ; d ¼ minfaqf ; aqf g; b ¼ minfaqf þ 1 ; aqf þ 1 g; xi

xj

xi

xj

ð9:4Þ

a ¼ maxfaqe ; aqe g; c ¼ maxfaqe þ 1 ; aqe þ 1 g xi

xj

xi

xj

    1 f f e e f f e e M aq aq þ 1 aq þ 1 aq [ M aq aq þ 1 aq þ 1 aq ¼ ðbd  acÞ sin h; xi xj 2 h ¼ \aq a0 aq þ 1 ; d ¼ maxfaqf ; aqf g; b ¼ maxfaqf þ 1 ; aqf þ 1 g; xi

xi

xj

xj

ð9:5Þ

a ¼ minfaqe ; aqe g; c ¼ minfaqe þ 1 ; aqe þ 1 g xi

xj

xi

xj

(3) The third state occurs when there is an overlapping line segment between aq xi

and aq on Lðaq ; a0 Þ, and aq þ 1 and aq þ 1 on Lðaq þ 1 ; a0 Þ; at the same time, when xj

xi

xj

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215

FIG. 9.4 – Overlapping states of similarity between x1 and x6 .

the line segment ðaqe ; aqe þ 1 Þ intersects the line segment ðaqe ; aqe þ 1 Þ, or the line xj

segment

ðaqf ; aqf þ 1 Þ xj xj

xj

xi

intersects the line segment

ðaqf ; aqf þ 1 Þ, xi xi

xi

the overlapping area

is defined as shown in equations (9.6) and (9.7). This is the third overlapping state, as shown in figure 9.4.     f f M aqf aq þ 1 aqe þ 1 aqe \ M aqf aq þ 1 aqe þ 1 aqe xi

ð9:6Þ

xj

cos h cos h  12 eb sin h  12 ðf  eÞ2 f =ae=b ¼ 12 hc sinh  12 ðh  gÞ2 h=cg=d

    f f M aqf aq þ 1 aqe þ 1 aqe [ M aqf aq þ 1 aqe þ 1 aqe xi xj   1 1 cos h cos h ¼ 2 dg sinh  2 ðh  gÞ2 h=cg=d  12 fa sin h  12 ðf  eÞ2 f =ae=b

ð9:7Þ

where h ¼ \aq a0 aq þ 1 , a ¼ aqe þ 1 , b ¼ aqe þ 1 , c ¼ aqf þ 1 , d ¼ aqf þ 1 , e ¼ aqe , f ¼ aqe , xj

xi

xj

xi

xi

xj

g ¼ aqf and h ¼ aqf . To ensure that equations (9.6) and (9.7) are clean and tidy, xi

xj

letters are substituted for numerical markers.

216

Space Fault Tree Theory and System Reliability Analysis

The results of the overlapping state in (2) and (3) are the analytic solution obtained by a graphical calculation; owing to a space limitation, however, the analytic procedure is abbreviated here. Definition 9.9. The entire similarity of xi ; xj is Sðxi ; xj Þ, for C ¼ fa1 ; a2 ; . . .; an g, and Qn1 thus Sðxi ; xj Þ ¼ q¼1 Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ Sðxi ; xj ; ðan ; a1 ÞÞ; aq 2 C . Definition 9.10. The complete similarity Sðxi ; xj Þ of xi ; xj is classified by the following rules. Set kaq ;aq þ 1 as the similarity threshold of xi ; xj for two attributes aq ; aq þ 1 , which is 0:4  kaq ;aq þ 1  0:6 in general. Here, 1  Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ  kaq indicates a similarity, Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ ¼ 0 indicates a dissimilarity, and the values between them indicate a fuzzy similarity. Thus, for Sðxi ; xj Þ, 1  Sðxi ; xj Þ ¼ Q Sðxi ; xj ; ðaq ; aq þ 1 ÞÞ  nq¼1 kaq ;aq þ 1 indicates a similarity, Sðxi ; xj Þ ¼ 0 indicates a dissimilarity, and the values between them indicate a fuzzy similarity.

9.5

Example Analyses

For a reliability analysis of an electrical system, seven operators who have used the system were consulted. They provided an evaluation of the system reliability. Because of their specific work, scheduling, and other factors, the environments of their operating systems are different. In fact, there are many factors affecting the probability of a component fault in the system. For example, a diode fault in an electrical system has a direct relationship with the working time, temperature, current, and voltage. If the system is analysed, the adaptive working time and temperature of each component may be different. With a change in the entire system working time and the environment temperature, the system reliability is also different [22]. Therefore, the basic environment of the system reliability evaluation is different for the above operators. The proposed methods are used to classify the reliability evaluation of these operators. If the object set classification (semantic description sets) is the same as the decision set classification (semantic result sets), then these operators are objective and can support each other regarding a system reliability evaluation. If the classification does not correspond, the evaluation of more operators should be added to further determine the accuracy of the descriptions. Based on the survey, answers given by operators regarding the system reliability suggest that the system faults occur at temperatures below 12 °C; there are more faults after the system is operational for 70 or 80 days, and critical instability occurs thereafter (owing to a limited space, the other six cases are not listed). The system is generally overhauled every 100 days, and we can set a time domain of [0, 100] days. The temperature domain is [0, 40] °C. The humidity is generally determined by the seasonal climate during the work period. Defining system T ¼ ðU ; A; C ; DÞ, the descriptions of the seven operators are given for object set U ¼ fx1 ; x2 ; . . .; x7 g, and xi is the description of the ith operator, i 2 f1; . . .; 7g. The working time, temperature, and humidity of system make up the conditional attribute set C ¼ fa1 ; a2 ; a3 g, where a1 is the temperature, a2 is the

Clustering Analysis and Similarity

217

time, and a3 is the humidity. In addition, a1 , a2 , and a3 are the continuous domain values, and are normalised according to the descriptions of the seven operators. For example, for x1 , the domains of the evaluation semantics are a working temperature of [0, 12] °C, hworking i time of [70, 95] days, and humidity. They are normalised as follows: a1 ¼ a1e ; a1f , a1e ¼ ð0  0Þ=ð40  0Þ ¼ 0, and a1f ¼ ð12  0Þ=ð40  0Þ ¼ 0:3; x1

x1

x1

similarly, a2 ¼ ½0:7; 0:95 and a3 ¼ ½0:2; 0:9 . The decision set D ¼ fd1 ; d2 ; d3 g repx1

x1

resents safety levels from 1 to 3, which are ‘unreliable,’ ‘generally reliable’, and ‘very reliable.’ Obtain the basic information decision table WðT Þ, as shown in table 9.1. Without considering the decision set D, we study the attribute circle representation of the object set and attribute set. The attribute circle x1 has also been given. The attribute circles x2–x7 are shown in figure 9.5. According to table 9.1 and definitions 9.5 and 9.6, we obtain the object similarity table, as shown in table 9.2. For object set classification, first define ka1 ¼ 0:5; ka2 ¼ 0:5; ka3 ¼ 0:5, and the similarity division of Sðxi ; xj Þ is {similarity, fuzzy similarity, dissimilarity} = {[1, 0.125], (0.125, 0), 0}. The similar objects obtained in table 9.2 are classified as follows: For the similarity, Sðx2 ; x1 Þ ¼ 0.3214, Sðx5 ; x3 Þ ¼ 0.5906, Sðx6 ; x3 Þ ¼ 0.2315, Sðx6 ; x5 Þ ¼ 0.2632, and Sðx7 ; x4 Þ ¼ 0.2592. For the fuzzy similarity, Sðx3 ; x1 Þ ¼ 0.0238, Sðx4 ; x3 Þ ¼ 0.0204, Sðx5 ; x1 Þ ¼ 0.0278, Sðx5 ; x2 Þ ¼ 0.0321, Sðx5 ; x4 Þ ¼ 0.0165, Sðx6 ; x1 Þ ¼ 0.0288, Sðx6 ; x2 Þ ¼ 0.0306, Sðx6 ; x4 Þ ¼ 0.0765, and Sðx7 ; x6 Þ ¼ 0.0245. For the dissimilarity, Sðx3 ; x2 Þ ¼ 0, Sðx4 ; x1 Þ ¼ 0, Sðx4 ; x2 Þ ¼ 0, Sðx7 ; x1 Þ ¼ 0, Sðx7 ; x2 Þ ¼ 0, Sðx7 ; x3 Þ ¼ 0, and Sðx7 ; x5 Þ ¼ 0. As the clustering analysis rules, strictly obey the similarity division and dissimilarity division, and refer to the fuzzy similarity division. For example, Sðx2 ; x1 Þ ¼ 0.3214 means that x2 ; x1 should be divided into one group, and Sðx3 ; x2 Þ ¼ 0 means x3 ; x2 should not be divided into one group. Thus, the final object set is divided into U = {{x2 ; x1 }, {x7 ; x4 }, {x5 ; x3 ; x6 }}. Considering the corresponding relation between the decision set D and the object set U in table 9.1, we

TAB. 9.1 – Basic information decision table WðT Þ. U x1 x2 x3 x4 x5 x6 x7

Factors a1 temperature [0, 0.3] [0, 0.4] [0.2, 0.7] [0.6, 0.9] [0.25, 0.7] [0.2, 0.8] [0.6, 0.9]

a2 time [0.7, 0.95] [0.8, 1] [0.5, 0.8] [0.2, 0.6] [0.5, 0.9] [0.3, 0.9] [0.2, 0.4]

D safety levels a3 humidity [0.2, 0.9] [0.3, 0.9] [0.1, 0.8] [0.2, 0.8] [0.1, 0.9] [0.3, 1] [0, 0.9]

1 low 1 2 3 2 2 3 high

218

Space Fault Tree Theory and System Reliability Analysis

FIG. 9.5 – Attribute circle of object x2–x7.

Object Attribute x1

x2

x3

x4

x5

x6

x7

x1 a1 a2 1 1 Sðx1 ; x1 Þ ¼1 0.7500 0.5000 Sðx2 ; x1 Þ ¼ 0.3214 0.1429 0.2222 Sðx3 ; x1 Þ ¼ 0.0238 0 0 Sðx4 ; x1 Þ ¼0 0.0714 0.4444 Sðx5 ; x1 Þ ¼ 0.0278 0.1250 0.3077 Sðx6 ; x1 Þ ¼ 0.0288 0 0 Sðx7 ; x1 Þ ¼0

x2

x3

a3 1

a1

a2

a3

0.8571

1

1

0.7500

0

0.8571

0.2222

0.8750

0.2143

1 Sðx2 ; x2 Þ ¼1 0 Sðx3 ; x2 Þ ¼0 0 Sðx4 ; x2 Þ ¼0 0.2 Sðx5 ; x2 Þ

0.7500

0.2500

0.8889

0

¼ 0.0321 0.1429 Sðx6 ; x2 Þ ¼ 0.0306 0 Sðx7 ; x2 Þ ¼0

x4

a1

a2

a3

0.6250

1

1

0.7143

0.1429

0.7500

0.9000

1 Sðx3 ; x3 Þ ¼1 0.1667 Sðx4 ; x3 Þ ¼ 0.0204 0.7500 Sðx5 ; x3 Þ

0.8571

0.8333

0.6667

0.1429

¼ 0.5906 0.5000 Sðx6 ; x3 Þ ¼ 0.2315 0 Sðx7 ; x3 Þ ¼0

x5

a1

a2

a3

0.8571

1

1

0.8750

0.1538

1 Sðx4 ; x4 Þ ¼1 0.1429 Sðx5 ; x4 Þ

0.5556

0.2857

0.7778

1

¼ 0.0165 0.4286 Sðx6 ; x4 Þ ¼ 0.0765 0.3333 Sðx7 ; x4 Þ ¼ 0.2592

x6

a1

a2

a3

0.7500

1

1 Sðx5 ; x5 Þ

1

0.6250

0.5921

0.7778

0.1538

¼1 0.6667 Sðx6 ; x5 Þ ¼ 0.2632 0 Sðx7 ; x5 Þ ¼0

x7

a1

a2

a3

0.6667

1

1

0.8889

0.2857

1 Sðx6 ; x6 Þ ¼1 0.1429 Sðx7 ; x6 Þ ¼ 0.0245

0.6000

a1

1

a2

a3

Clustering Analysis and Similarity

TAB. 9.2 – Object similarity.

1 1 Sðx7 ; x7 Þ ¼1

219

Space Fault Tree Theory and System Reliability Analysis

220

found that U ! D = {{x2 ; x1 } ! d1 , {x7 ; x4 } ! d3 , {x5 ; x3 ; x6 } ! d2 }. This shows that the classification of object sets is non-singular and accurate for a decision set. Similarly, in accordance with the above settings, the improved method is used to obtain the object similarity, the result of which is shown in table 9.3. Setting ka1 ¼ 0:5; ka2 ¼ 0:5; ka3 ¼ 0:5, the similarity division of Sðxi ; xj Þ is {similarity, fuzzy similarity, dissimilarity} = {[1, 0.125], (0.125, 0), 0}. The similar objects obtained in table 9.3 are classified as follows: For the similarity, Sðx2 ; x1 Þ ¼ 0.5138, Sðx4 ; x3 Þ ¼ 0.1650, Sðx5 ; x3 Þ ¼ 0.7706, Sðx6 ; x3 Þ ¼ 0.4225, Sðx7 ; x4 Þ ¼ 0.4069, and Sðx6 ; x5 Þ ¼ 0.5288. For the fuzzy similarity, Sðx3 ; x1 Þ ¼ 0.0384, Sðx5 ; x1 Þ ¼ 0.0197, Sðx6 ; x1 Þ ¼ 0.0717, Sðx5 ; x2 Þ ¼ 0.0486, Sðx6 ; x2 Þ ¼ 0.0989, Sðx5 ; x4 Þ ¼ 0.0372, Sðx6 ; x4 Þ ¼ 0.0650, and Sðx7 ; x6 Þ ¼ 0.0709. For the dissimilarity, Sðx4 ; x1 Þ ¼ 0, Sðx7 ; x1 Þ ¼ 0, Sðx3 ; x2 Þ ¼ 0, Sðx4 ; x2 Þ ¼ 0, Sðx7 ; x2 Þ ¼ 0, Sðx7 ; x3 Þ ¼ 0, and Sðx7 ; x5 Þ ¼ 0. For the final object set division, U = {{x2 ; x1 }, {x7 ; x4 }, {x5 ; x3 ; x6 }}. However, this division is inconsistent with Sðx4 ; x3 Þ ¼ 0.1650. Because the similarities among x5 ; x3 ; x6 are higher than the similarity between x4 and x3 , and x7 cannot be divided with x5 ; x3 , and the similarity between x7 and x4 is extremely high, the above division is reasonable. Considering the corresponding relation between the decision set D and the object set U in table 9.1, we can see that U ! D = {{x2 ; x1 } ! d1 , {x7 ; x4 } ! d3 , {x5 ; x3 ; x6 } ! d2 }. This shows that the object set division is non-singular and accurate for the decision set, and that the initial division of the decision set is correct and can be tested in practice. The definition of the attribute circle is put forward, and the methods for an object similarity calculation are constructed based on such a circle. The first method reduces the geometric similarity to facilitate the computations, although at the expense of accuracy. The improved method is based on the geometric similarity, and is more precise, but complex. In general, the results are the same, but the more the attributes applied, the more accurate and more complex the improved method is.

9.6

Conclusions

The attribute representation method of the factor space object was modified. In the attribute circle, the infinite multiple domain attributes can be indicated. We can then analyse the similarity of objects and transform them into numerical expressions of similarity. The similarity calculation method is improved. In the attribute circle, the overlapping area of different objects for the same attributes is defined as a similarity, which is a geometric overlapping similarity calculation method. In general, the results of the two methods are the same and the more attributes that are applied, the more accurate the improved method is, and the more complex the calculations are. We obtain the rules of object clustering, which strictly follow the similarity and dissimilarity division, and refer to the fuzzy similarity division to classify the object set. The results show that if the correspondence between object sets U and decision set D is non-singular (U ! D = {{x2 ; x1 } ! d1 , {x7 ; x4 } ! d3 , {x5 ; x3 ; x6 } ! d2 }), even if the system has different environmental influence factors, an attribute

Object Attribute x1

x2

x3

x4

x5

x6

x7

x1 a1 , a2 a2, a3 1 1 Sðx1 ; x1 Þ ¼1 0.8210 0.8549 Sðx2 ; x1 Þ ¼ 0.5138 0.1891 0.6323 Sðx3 ; x1 Þ ¼ 0.0384 0 0 Sðx4 ; x1 Þ ¼0 0.1099 0.8344 Sðx5 ; x1 Þ ¼ 0.0197 0.3450 0.8312 Sðx6 ; x1 Þ ¼ 0.0717 0 0 Sðx7 ; x1 Þ ¼0

x2 a3, a1 1

a1, a2

0.7321

1

0.3212

0

0.2150

a2, a3

x3 a3, a1

1 1 Sðx2 ; x2 Þ ¼1 0 0 0.4550 Sðx3 ; x2 Þ ¼0 0 0 0 Sðx4 ; x2 Þ ¼0 0.1756 0.6324 0.4375 Sðx5 ; x2 Þ

0.2500

0.4500

0.4859

0

¼ 0.0486 0.6155 0.3571 Sðx6 ; x2 Þ ¼ 0.0989 0 0.4226 Sðx7 ; x2 Þ ¼0

a1, a2

a2 , a3

x4 a3, a1

1 1 Sðx3 ; x3 Þ ¼1 0.4623 0.6450 0.5532 Sðx4 ; x3 Þ ¼ 0.1650 0.8750 0.9200 0.9573 Sðx5 ; x3 Þ

a1 , a2

a2, a3

x5 a3, a1

x6

a1, a2

a2, a3

a3 , a1

1

1 Sðx5 ; x5 Þ

1

a1, a2

a2 , a3

x7 a3, a1

a1, a2

a2, a3

a3, a1

1

1 Sðx7 ; x7 Þ ¼1

1

Clustering Analysis and Similarity

TAB. 9.3 – Object similarity.

1

0.6756

0

¼ 0.7706 0.7690 0.8132 Sðx6 ; x3 Þ ¼ 0.4225 0 0.8320 Sðx7 ; x3 Þ ¼0

1

1 1 Sðx4 ; x4 Þ ¼1 0.0988 0.5790 0.6500 Sðx5 ; x4 Þ ¼ 0.0372 0.4490 0.6322 Sðx6 ; x4 Þ ¼ 0.0650 0.8002 0.7060 0.7203 Sðx7 ; x4 Þ ¼ 0.4069

0.2290

0.7300

0

¼1 0.8121 0.8920 Sðx6 ; x5 Þ ¼ 0.5288 0 0.9590 Sðx7 ; x5 Þ ¼0

1

1 1 Sðx6 ; x6 Þ ¼1 0.2230 0.3454 0.9200 Sðx7 ; x6 Þ ¼ 0.0709

221

222

Space Fault Tree Theory and System Reliability Analysis

evaluation of the semantics of the system is relatively objective for every object, and can support each other, and thus these evaluation semantics are correct. If it is deemed unusual, then a semantic evaluation needs to be added to the description and further determined [23–25].

References [1] Wang P.Z. (2013) Factor spaces and factor data-bases, J. Liaoning Tech. Univ. (Nat. Sci.) 32, 1. [2] Wang P.Z., Liu Z.L., Shi Y., et al. (2014) Factor space, the theoretical base of data science, Ann. Data Sci. 1, 233. [3] Wang P.Z., Guo S.C., Bao Y.K., et al. (2015) Factorial analysis in factor space, J. Liaoning Tech. Univ. (Nat. Sci.) 34, 273. [4] Wang P.Z. (2015) Factor spaces and factor science, J. Liaoning Tech. Univ. (Nat. Sci.) 34, 273. [5] Wang P.Z. (2014) Introduction to factor space and factor Library (special report). Intelligent science and mathematics forum, Huludao, p. 6. [6] Shi Y. (2014) Big data and new challenges, Sci. Technol. Dev. 25. [7] Li H.X. (2014) The space theory of factor space theory/factors and application. Intelligent science and mathematics forum, Huludao. [8] Li D.Y. (2015) Cognitive physics (special report). Oriental thinking and fuzzy logic-international conference to commemorate the fifty anniversary of the birth of fuzzy set, Dalian, p. 8. [9] Zeng W.Y., Feng S. (2014) Approximate reasoning algorithm of interval-valued fuzzy sets based on least square method, Inf. Sci. 272, 73. [10] Zeng W.Y, Feng S. (2014) An improved comprehensive evaluation model and its application, Int. J. Comput. Intell. Syst. 7, 706. [11] Li D.Q., Zeng W.Y., Li J. (2015) Note on uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Appl. Math. Model. 39, 894. [12] Li D.Q., Zeng W.Y., Zhao Y. (2015) Note on distance measure of hesitant fuzzy sets, Inf. Sci. 32, 103. [13] Li D.Q., Zeng W.Y., Li J. (2015) New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making, Eng. Appl. of Artif. Intell. 40, 11. [14] Yu G.F., Liu W.Q., Li D.F. (2015) A method of intuitionistic linguistic decision making based on eclectic variable weight vector, Control Decis. 30, 2233. [15] Yu G.F., Liu W.Q., Shi M.T. (2015) Enterprise credit evaluation based on local variable weight model, J. Manage. Sci. 17, 85. [16] He P. (2010) Design of interactive learning system based on intuition concept space, J. Comput. 21, 478. [17] Ou Y.H. (2015) Unified theory of uncertainty theory: mathematical foundations of factor spaces (special report), Oriental thinking and fuzzy logic-international conference to commemorate the fifty anniversary of the birth of fuzzy set, Dalian, p. 8. [18] Cui T.J., Ma Y.D. (2014) System security classification decision rules considering the Scope attribute, J. Saf. Sci. Technol. 10, 6. [19] Yang J.W., He F., Cui T.J., et al. (2015) Safety analysis of coal mine disaster based on factor analysis, J. Saf. Sci. Technol. 11, 84. [20] Cui T.J., Ma Y.D. (2015) Research on the classification method about coal mine safety situation based on the factor space, Syst. Eng. Theory Pract. 35, 2891. [21] Cui T.J., Ma Y.D. (2016) Research on the number of failures of repairable systems based on imperfect repair model, Syst. Eng. Theory Pract. 36, 184. [22] Niu J.L., Cheng L.S. (2012) Classification using improved Mahalanobis-Taguchi system based on omni-optimizer, Syst. Eng. Theory Pract. 32, 1324.

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[23] Cui T.J., Ma Y.D. (2015) Definition of the attribute circle in factors space and its application in object classification, Comput. Eng. Sci. 37, 2170. [24] Cui T.J., Ma Y.D. (2015) Research on the similarity of object classification of attribute circular and application based on factors space, Fuzzy Syst. Math. 29, 56. [25] Li S.S., Cui T.J., Liu J. (2018) Research on the clustering analysis and similarity in factor space, Comput. Syst. Sci. & Eng. 33, 397.

Chapter 10 Development and Future Prospects 10.1

Summary of Space Fault Tree

This book describes the main theoretical research results of the space fault tree. The results of all existing space fault tree included in this book are given below. (1) The theory, definition, formula, and method of the continuous space fault tree (CSFT) are given within the framework of the space fault tree (SFT) theory, along with example applications of these methods. These definitions include the CSFT, influencing factors of a component fault, characteristic function of the component fault probability, component fault probability distribution, system fault probability distribution, probability importance distribution, criticality importance distribution, system fault probability distribution trend, component replacement cycle, system replacement cycle, cut set and path set domains of one component/system and the domain boundary, factor importance distribution, and factor joint importance distribution. (2) The trend of the fault probability of the components and systems was studied under different factors: the optimal replacement cycle scheme and cost scheme of the system, the acceptable factor domain of the system fault probability, the importance of factors influencing the system reliability, the system fault location method, the system maintenance rate determination and optimization, system reliability assessment methods, and the importance of the system and components. (3) The theory, definition, formula, and method of the discrete spatial fault tree (DSFT) are given in the framework of the SFT theory, along with example applications of these methods. The method of calculating the fault probability distribution under the DSFT is presented, that is, the factor projection method is presented, and an inaccurate cause of the method is analysed. Then, a more accurate method for determining the fault probability distribution is proposed using an ANN, and the trend of the fault probability is obtained using an ANN derivation. (4) Research on the inward analysis method of the system structure is detailed. The 01-type space fault tree and representation methods (table and graph DOI: 10.1051/978-2-7598-2499-1.c010 © Science Press, EDP Sciences, 2020

226

(5)

(6)

(7)

(8)

(9)

Space Fault Tree Theory and System Reliability Analysis methods) of the system are proposed to analyse the physical structure and factor structure of the system. Item-by-item analyses and the classification reasoning method are proposed for the inward analysis of the system component and factor structure, and the analysis process and mathematical definition are described. Research on a method for excavating the basic data suitable for SFT processing from actual monitoring data records is also discussed. This includes a study on the simplification, distinction, and causality of a qualitative safety assessment and monitoring records; the cost of the system adaptive transformation under a changing working environment; the importance of components in the system under the influence of environmental factors; the system reliability decision rule mining method and its improvement; and the classification and similarity of different objects and their improvement methods. The cloud model is introduced to transform the space fault tree. The cloudization space fault tree inherits the ability of the SFT to analyse multiple factors affecting the reliability, and inherits the ability of the cloud model to represent the data uncertainty. Thus, the cloudization space fault tree is suitable for the analysis and processing of the actual fault data. The concepts of cloudization include the characteristic cloud function, cloud component and system fault probability distribution, cloud component and system fault probability distribution trend, cloudization probability and criticality importance distribution, cloud factor and factor joint importance distribution, cloud domain importance, cloudization path set domain and cloudization cut set domain, and the analysis of the uncertainty of reliability data. An analysis method of the system function structure is put forward based on factor analysis. It is pointed out that the factor space can describe the qualitative cognition process in intelligent science. Based on the logic of the factors, an axiomatic system of the function structure analysis is established, and the definition, logical proposition, and proof are given. The minimization method of the system function structure is proposed. The relationship between the classification reasoning method and function structure analysis method in the factor space is discussed. The system function structure is analysed using the system function structure analysis method under complete and incomplete information. The concepts of the action path and action history are put forward. The former describes the set of states that a system or component experiences under different working conditions, which is a function of various factors. The latter describes the accumulative state in the process of an action path, and is a cumulative result. The reliability system stability is described by the motion system stability theory, and the system is divided into a function subsystem, fault tolerant subsystem, and hindrance subsystem. The functions of the three subsystems in the reliability system are discussed. According to the eight stabilities of the solutions of differential equations, five corresponding system reliability implications are explained. A similarity computation method based on the envelopes is proposed. To make the cloud model more convenient and effective for multiple attribute decision

Development and Future Prospects

227

making, the existing attribute circle is improved, and the characteristic parameters of the cloud model can be calculated. A fuzzy comprehensive evaluation method is proposed to consider the effect of various factors on the system reliability. The cloud model is embedded in an AHP, and the AHP analysis process is transformed by the cloud model. A cooperative game, the cloud AHP algorithm, and the dual cloud AHP model are constructed, and the cloud ANP model and its steps are proposed. (10) A method for determining the maintenance rate of the components in the SFT is proposed, and the influence of the working environment factors on the maintenance rate distribution of the components is analysed. Using a Markov state transition chain and SFT characteristic function, the component maintenance rate distribution of series and parallel systems is derived. According to the parallel, series, and hybrid systems composed of different types of components, the component maintenance rate distribution is calculated, and the constraints are increased. By limiting the ratio of the fault rate to repair the rate of different components, the transfer parameters are used to solve the equation, and an expression of the maintenance rate is obtained. (11) The space fault network is proposed. The space fault network can describe a more complex fault occurrence process. The causal relationship between events is described by a network topology instead of a tree topology, which is more general than a tree structure. The definition, properties, and transformation method of the space fault network are given to the SFT, and the transfer probability of a fault is also considered. The purpose is to transform the space fault network into the space fault tree and use the existing research results of the space fault tree to analyse a fault network. The general network structure, multi-directional ring network structure, and multi-directional ring network structure with the unidirectional fault network are given. The characteristics of different fault occurrence processes and their corresponding fault tree transformation methods are also analysed. Finally, the final event probability calculation method of the transformed fault network is given.

10.2

Future Development of Space Fault Tree

Safety system engineering is a combination of safety science and system engineering science. With the development of information science, intelligent science, and data science, studies on safety system engineering, particularly the reliability problem, will face more problems. At the same time, a more effective method will be accompanied by a development process. (1) As a representation of the fault data, the actual fault data are fuzzy, discrete, and stochastic, that is, they have uncertainty. An appropriate method is required to represent data with such characteristics. At present, in the space fault tree, the data are represented using the factor projection fitting, fuzzy structured element, cloud model, etc., which are integrated into the space fault tree as characteristic functions. However, there are still some shortcomings, and collaboration with experts in applied mathematics, information science, and

228

(2)

(3)

(4)

(5)

Space Fault Tree Theory and System Reliability Analysis other fields is needed to promote research into new methods and provide a more comprehensive representation of the fault data. Big data level fault data processing should be applied. Big data are among the most important features of fault data. The actual working process of the system will produce numerous working state data, which can reflect the reliability characteristics of the system. It is then necessary to use appropriate mathematical methods to represent fault big data, which not only reflects the reliability characteristics under big data, but also reduces the pressure of big data on the processing process. At present, the cloud model and fuzzy structured element method are used to deal with fault big data in the space fault tree. However, a more suitable method still needs to be studied. The logical relationship between the fault and influencing factors should be determined. Owing to the characteristics of fault data, it is difficult to form or mine quantitative functions or analytical functions with quantitative characteristics in most cases. However, some causal reasoning relations between fault and influence factors can only be obtained through data analysis. At present, the space fault tree deals with such relations based on factor space theory. Further improvement and research should be based on the development of corresponding theories such as the factor space, and integration into the space fault tree. A stability analysis of the system reliability should be conducted. The system reliability or fault probability was studied, the reliability system was abstracted as a movement system, and the reliability was analysed based on the stability theory of the motion system. The corresponding relationship between the solutions of the motion differential equations and the reliability changes of the movement system was determined. However, this was simply an attempt, and the specific problems need to be further determined; for example, the abstract system reliability should be determined in concrete forms for a movement equilibrium equation, a measurement of the movement degree, etc. The relationship between the system and component should be determined. At present, the component and system structure in the space fault tree is represented by the fault tree representation, that is, a tree diagram in system engineering. Although it can represent a part of the actual system fault, a more extensive network topology is required. The tree topology can be considered a special case of a network topology, and studying the representation of the network structure is the key to completely solving the relationship between components and the system representation. The problem remains to be further studied.

It is hoped that the publication of this book can expand the space fault tree theory internationally and contribute to further research into safety system engineering and the system reliability analysis.