# Component Importance Measure Computation Method Based Fuzzy Integral with Its Application

Authors: Shuai Lin; Yanhui Wang; Limin Jia; Yang Li
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

In view of the negative impact of component importance measures based on system reliability theory and centrality measures based on complex networks theory, there is an attempt to provide improved centrality measures (ICMs) construction method with fuzzy integral for measuring the importance of components in electromechanical systems in this paper. ICMs are the meaningful extension of centrality measures and component importance measures, which consider influences on function and topology between components to increase importance measures usefulness. Our work makes two important contributions. First, we propose a novel integration method of component importance measures to define ICMs based on Choquet integral. Second, a meaningful fuzzy integral is first brought into the construction comprehensive measure by fusion multi-ICMs and then identification of important components which could give consideration to the function of components and topological structure of the whole system. In addition, the construction method of ICMs and comprehensive measure by integration multi-CIMs based on fuzzy integral are illustrated with a holistic topological network of bogie system that consists of 35 components.


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## Body

## 1. Introduction

Recent decades have witnessed not only the rapid development on the highly integrated system of electromechanical systems, but also the significant progress on the system function [1]. Complex electromechanical systems, such as high-speed train, aircraft, and other large equipment, are composed of components with specific functions, physical and chemical connections, and behaviors, and coupled relationship through mechanic, electric, and information relationship. Due to the complexity of topological structure and functional relationship within electromechanical systems, one of the component’s failures may lead to another component’s failure, which is called fault propagation. The fault propagation of complex electromechanical systems can enlarge the negative impact due to one of components failures. In these situations, it is becoming increasingly important to take functional and topological characteristics into account when assessing the importance of components and concentrating the resources on the small subset of components that are most important to the system.


### 1.1. Previous Work

In system reliability theory, importance measures are used as effective tools to evaluate the relative importance of components and identify system weaknesses [2]. Component importance measures (CIMs) are component related indices that allow security practitioners to identify how a components fault affects the overall behavior or performance of the whole technological system and are used to evaluate the relative importance of a component. The typical CIMs include but are not limited to Birnbaum importance measure [3], Fussell-Vesely (FV) importance measure [4], and criticality importance measure [5]. Detailed descriptions and mathematical expressions for importance measures can be found in Ramirez-Marquez [6]. Using the CIMs, security practitioners can estimate or prioritize components in order of their importance value with regard to system reliability and concentrate maintenance resources on the most important components.Recent advances indicate that electromechanical systems can be virtually represented as networks, where the components of technological products are easily depicted by the nodes of complex networks and the connections between linkage components are naturally depicted by the links of complex networks [7–9]. More recently, various centrality measures (CMs) have been presented to quantify the importance of an individual in a complex network, including degree centrality (DC) [10], betweenness centrality (BC) [11], and eigenvector centrality (EC) [12]. The issue of centrality has attracted the attention of physicists, who have been extending its applications to the realm of technological networks. For example, Dan et al. [13] considered that system networked reflected the organization structure and enhanced efficiency and capability of system development and production, and, in Li et al. [14] view, these physical connections between components determine the function and structure complexity of technological products. Based on CMs of complex network, Jiang et al. [15] introduced the loads and vulnerability coefficient of nodes to study the inherent vulnerability of components and Xu et al. [16] developed a comprehensive vulnerability index to find the vulnerable structure of complex system with a network model. Meanwhile, Zong et al. [17] regarded the node betweenness and node agglomeration as the indices to evaluate the importance of the components based on the maintenance relationship network.However, all these researches focused on only one measure, such as component importance measure and one centrality measure, and every measure has its own disadvantage and limitation. In recent years, researchers study a multiattribute ranking problem to evaluate the component importance comprehensively from more than one perspective, which would be a special case of multicriteria decision-making (MCDM). MCDM refers to making decision for alternatives in the presence of multiple and conflicting criteria [18] and has many developments and applications, such as extensions of TOPSIS [19, 20], Analytic Hierarchy Process [21], K-shell decomposition [22], and entropy theory [23]. Detailed descriptions and mathematical expressions for multicriteria decision-making approaches can be found in Govindan et al. [24].


### 1.2. Problem Description

Although the above component importance measures or centrality measures have been widely applied in identifying influential components, there are some limitations and disadvantages. CIMs are built on the assumption of the independence of components and none of them has taken the impact of topological structure between components into account. CMs mentioned above focus only on the components propagation behavior of complex network and are limited to the point of the reliability analysis [25]. For these reasons, it is extremely important to research on the negative impact of these restrictions and proactively overcome them by complementation with elaborate reliability contexts on identifying influential components of electromechanical system. That is to say that CIMs or CMs cannot be applied to complex electromechanical systems that contain multiple components.If only one measure is adopted, then the rankings of identifying influential components may be different by using a different measure. In some cases, using different centrality measures may provide different results, even conflicting results [26]. MCDM has been proposed to address this problem. However, the inherent limitations and disadvantages of CIMs or CMs cannot be eliminated through integration multimeasures. Moreover, among numerous MCDM methods developed to solve real-world decision problems, fuzzy integral continues to work satisfactorily across different application areas. The weights in most developments and applications of MCDM are determined in advance, such as TOPSIS and AHP, which possess definite subjectivity. Fuzzy integral makes full use of attribute information, provides a cardinal ranking of alternatives, reduces subjective influences, and does not require attribute preferences to be independent. As a well-known classical MCDM method, fuzzy integral has received much interest from researchers and practitioners.In this paper, we try to introduce fuzzy integral theory to explore how to identify influential components. Our work makes two important contributions. First, we integrate component importance measures and centrality measures with Choquet integral to define a new kind of improved centrality measures. Second, a novel index, comprehensive measure, of a meaningful fuzzy integral-based is brought into identification of important components for which it could give consideration to function of components and topological structure of the whole system.The rest of the paper is structured as follows. Section2 provides background information about the holistic topological network and fuzzy integral theory. In Section 3, the improved centrality measures are detailed. The following is presented: how to improve centrality measures, by using Choquet integral to express functional and topological properties which are related, measuring the importance of components in electromechanical systems. In Section 4, we embed the fuzzy integral into the process of construction comprehensive measure and identification of critical components by fusing multi-improved centrality measures. Section 5 presents a case study which constructs improved centrality measures of bogie system and discusses the advantages of fusion improved centrality measures in identifying critical components based on fuzzy integral.


## 1.1. Previous Work

In system reliability theory, importance measures are used as effective tools to evaluate the relative importance of components and identify system weaknesses [2]. Component importance measures (CIMs) are component related indices that allow security practitioners to identify how a components fault affects the overall behavior or performance of the whole technological system and are used to evaluate the relative importance of a component. The typical CIMs include but are not limited to Birnbaum importance measure [3], Fussell-Vesely (FV) importance measure [4], and criticality importance measure [5]. Detailed descriptions and mathematical expressions for importance measures can be found in Ramirez-Marquez [6]. Using the CIMs, security practitioners can estimate or prioritize components in order of their importance value with regard to system reliability and concentrate maintenance resources on the most important components.Recent advances indicate that electromechanical systems can be virtually represented as networks, where the components of technological products are easily depicted by the nodes of complex networks and the connections between linkage components are naturally depicted by the links of complex networks [7–9]. More recently, various centrality measures (CMs) have been presented to quantify the importance of an individual in a complex network, including degree centrality (DC) [10], betweenness centrality (BC) [11], and eigenvector centrality (EC) [12]. The issue of centrality has attracted the attention of physicists, who have been extending its applications to the realm of technological networks. For example, Dan et al. [13] considered that system networked reflected the organization structure and enhanced efficiency and capability of system development and production, and, in Li et al. [14] view, these physical connections between components determine the function and structure complexity of technological products. Based on CMs of complex network, Jiang et al. [15] introduced the loads and vulnerability coefficient of nodes to study the inherent vulnerability of components and Xu et al. [16] developed a comprehensive vulnerability index to find the vulnerable structure of complex system with a network model. Meanwhile, Zong et al. [17] regarded the node betweenness and node agglomeration as the indices to evaluate the importance of the components based on the maintenance relationship network.However, all these researches focused on only one measure, such as component importance measure and one centrality measure, and every measure has its own disadvantage and limitation. In recent years, researchers study a multiattribute ranking problem to evaluate the component importance comprehensively from more than one perspective, which would be a special case of multicriteria decision-making (MCDM). MCDM refers to making decision for alternatives in the presence of multiple and conflicting criteria [18] and has many developments and applications, such as extensions of TOPSIS [19, 20], Analytic Hierarchy Process [21], K-shell decomposition [22], and entropy theory [23]. Detailed descriptions and mathematical expressions for multicriteria decision-making approaches can be found in Govindan et al. [24].


## 1.2. Problem Description

Although the above component importance measures or centrality measures have been widely applied in identifying influential components, there are some limitations and disadvantages. CIMs are built on the assumption of the independence of components and none of them has taken the impact of topological structure between components into account. CMs mentioned above focus only on the components propagation behavior of complex network and are limited to the point of the reliability analysis [25]. For these reasons, it is extremely important to research on the negative impact of these restrictions and proactively overcome them by complementation with elaborate reliability contexts on identifying influential components of electromechanical system. That is to say that CIMs or CMs cannot be applied to complex electromechanical systems that contain multiple components.If only one measure is adopted, then the rankings of identifying influential components may be different by using a different measure. In some cases, using different centrality measures may provide different results, even conflicting results [26]. MCDM has been proposed to address this problem. However, the inherent limitations and disadvantages of CIMs or CMs cannot be eliminated through integration multimeasures. Moreover, among numerous MCDM methods developed to solve real-world decision problems, fuzzy integral continues to work satisfactorily across different application areas. The weights in most developments and applications of MCDM are determined in advance, such as TOPSIS and AHP, which possess definite subjectivity. Fuzzy integral makes full use of attribute information, provides a cardinal ranking of alternatives, reduces subjective influences, and does not require attribute preferences to be independent. As a well-known classical MCDM method, fuzzy integral has received much interest from researchers and practitioners.In this paper, we try to introduce fuzzy integral theory to explore how to identify influential components. Our work makes two important contributions. First, we integrate component importance measures and centrality measures with Choquet integral to define a new kind of improved centrality measures. Second, a novel index, comprehensive measure, of a meaningful fuzzy integral-based is brought into identification of important components for which it could give consideration to function of components and topological structure of the whole system.The rest of the paper is structured as follows. Section2 provides background information about the holistic topological network and fuzzy integral theory. In Section 3, the improved centrality measures are detailed. The following is presented: how to improve centrality measures, by using Choquet integral to express functional and topological properties which are related, measuring the importance of components in electromechanical systems. In Section 4, we embed the fuzzy integral into the process of construction comprehensive measure and identification of critical components by fusing multi-improved centrality measures. Section 5 presents a case study which constructs improved centrality measures of bogie system and discusses the advantages of fusion improved centrality measures in identifying critical components based on fuzzy integral.


## 2. Methodological Background

### 2.1. The Holistic Topological Network

Currently, complex networks are being studied in many fields of science, such as social sciences, computer sciences, physics, biology, and economics. The majority of systems in reality can be undoubtedly described by models of complex networks. For example, Internet is a complex network composed of web sites [27, 28]. The brain is a complex network of neurons [29]. An organization is a complex network of people [30].As mentioned above, electromechanical systems are characterized by large scale, complex structure, nonlinear behavior, various working states, high coupling components, random operation environment, and so forth, which are not easy to be modeled directly to analyze global behaviors. Recently, lots of attempts have been made to model network in engineering, more specifically, electromechanical systems. It can be a complex network, in which the components are depicted by nodes and the physical connections between linkage components are depicted by links between the corresponding nodes. We refer to this representation as a topological structure and a formal model is presented by the following definitions.We defineHTN=V,E,fV,fE as aholistic topological network(1)HTN=V,E,fV,fEvi∈V,eij∈E,fvi=Afvi,Atvi⊂fV,feij=Bfeij⊂fE,i=1,…,n;j≤n,where V=v1,v2,…,vn is a set of nodes and each node represents a component. n is the number of nodes. For example, the number of nodes is 7 for a system in Figure 1. E=eij∣i=1,…,n,j∈n is a set of links and eij represents physical connection between nodes vi and vj. Depending on the nature or type of the topological property, this property may be reflexive in that eij≠eji. fV is the properties of the set of nodes, Afvi is the functional properties of vi, and Atvi is the topological properties of vi. fE is the properties of the set of edges and Bfeij is the functional properties of edge eij.Figure 1
The holistic topological network.For different systems, the properties of nodes or edges may be also different. For instant, assuming that Figure1 is the partial bogie system of CRHX, the properties of nodes and edges in theholistic topological network are as follows.(
1)
A
f
v
i
=
L
T
i
,
M
T
B
F
i
,
λ
i
.
L
T
i is the service life of node vi, which is its expected lifetime or the acceptable period of use in service. It is the time that any manufactured item can be expected to be “serviceable” or supported by its manufacturer. We can obtain this parameter from its manufacturer.λ
i is failure rate of node vi and is defined as(2)λit=limΔt→0Rit-Rit+ΔtΔt·Rit,where Rit is the probability of no failure before time t for node vi; Δt is the over-a-time-interval from failure time t to the next failure time. In this paper, failure rate λi depending on failure data is computed.M
T
B
F
i is mean time between failures of node vi and the formula is(3)MTBFi=∑fid-fiuNif,where fid is start of downtime for node vi,fiu is start of uptime for node vi, and Nif is the number of failures.(
2)
A
t
v
i
=
B
C
i
,
C
C
i
,
D
C
i
,
E
C
i. CCi is closeness centrality [31] of node vi:(4)CCi=1∑jndij,where dij denotes the distance between node vi and node vj; n is the number of nodes.B
C
i is betweenness centrality [31] of node vi:(5)BCi=∑j,k≠igjkigjk,where gjk is the number of binary shortest paths between node vj and node vk and gjki is the number of those paths that go through node vi.D
C
i is degree centrality [32] of node vi:(6)DCi=∑jNxij,where i is the focal node vi; j represents all other nodes; N is the total number of nodes; xij represents the connection between node vi and node vj. The value of xij is defined as 1 if node vi is connected to node vj and 0 otherwise.E
C
i is eigenvector centrality [12] of node vi. Let A be a n×n similarity matrix. The eigenvector centrality ECi of node vi is defined as the ith entry in the normalized eigenvector belonging to the largest eigenvalue of A. λ is the largest eigenvalue of A and n is the number of vertices: (7)A·EC=λ·ECECi=μ∑j=1naijECj,i=1,…,nwith proportionality factor μ=1/λ so that ECi is proportional to the sum of similarity scores of all nodes connected to it.(
3)
B
f
e
i
j
=
p
i
j
,
s
t
i
j
,
λ
i
j
.
p
i
j is fault propagation probability of edge eij, given by(8)pij=leij∑i≠jleij,where leij is the number of shortest paths crossing a given edge eij. Gao et al. [33] introduced the concept and computational method of fault propagation probability in detail.s
t
i
j is connection strength of edge eij and is given by [34, 35](9)stij=βsvi∣vjsvi,where svi is the number of times that operation states change in the statistical time; svi∣vj indicates the number of times that vi operation states change arising from vi in the statistical time. The operation states of vi contain the failure mode and normal operation mode of the corresponding component. β is an empirical contact duration of the type of functional dependencies between components vi and vj.λ
i
j is failure rate of edge eij. The calculation method is the same as the failure rate of node in (2).In Figure1, the edge describes the physical connection between linkage components, and the arrow of the edges expresses the failure interaction of nodes. Liu and An [22] further introduced how to determine the direction of edges which describe the coupling relation of the failure interactions.


### 2.2. Fuzzy Integral Theory

About thirty years ago the concept of fuzzy integral was proposed in Japan by Sugeno [36, 37], which in the discrete case is merely a kind of distorted mean. Although this was followed by a rather mathematically oriented research, far from application concerns, some Japanese researchers, including Sugeno himself, thought that fuzzy integrals could be applied to multicriteria evaluation: since 1985, papers have been published on supplier evaluation and improvement [38], analysis of policy decision of sustainable energy strategies [39], fusion of extreme learning machine [40], analysis of human reliability [41], and so forth. The distinguishing feature of fuzzy integral is that it is able to represent a certain kind of interaction between criteria, ranging from redundancy (negative interaction) to synergy (positive interaction). To our knowledge, there is almost no well-established method to deal with interacting criteria, and usually people tend to avoid the problem by constructing independent (or supposed to be so) criteria. This innovative feature was without any doubt the reason of its success in various fields of application [42]. A wide variety of forms have been presented for fuzzy integral, such as Choquet integral [43] and Sugeno integral.LetX=x1,x2,…,xn be a nonnull and finite set of attributes or influence factors and let PX denote the power set of X.


#### 2.2.1. Discrete Choquet Integral

Letgλ be a λ fuzzy measure in PX. The discrete Choquet integral of an element f:X→IR+ with respect to gλ is defined by(10)c∫fdgλ=∑i=1nfxi-fxi-1gλAi,where i indicates a permutation on X such that fx1≤⋯≤fxn. Also Ai=xi,…,xn is an attribute set, and fx0=0. Discrete Choquet integral is the extension of the weighted average operator, which considers the relationships among attributes or influence factors.


#### 2.2.2. Discrete Sugeno Integral

Letμ be a fuzzy measure on X and f:X→0,1 be a function. The discrete Sugeno integral of f with respect to μ is(11)s∫fdμ=⋁i=1nfxi∧μAi,where 0≤fxi≤fx2≤⋯≤fxn≤1,Ai=xi,xi+1,…,xn is an attribute set and fx0=0.


## 2.1. The Holistic Topological Network

Currently, complex networks are being studied in many fields of science, such as social sciences, computer sciences, physics, biology, and economics. The majority of systems in reality can be undoubtedly described by models of complex networks. For example, Internet is a complex network composed of web sites [27, 28]. The brain is a complex network of neurons [29]. An organization is a complex network of people [30].As mentioned above, electromechanical systems are characterized by large scale, complex structure, nonlinear behavior, various working states, high coupling components, random operation environment, and so forth, which are not easy to be modeled directly to analyze global behaviors. Recently, lots of attempts have been made to model network in engineering, more specifically, electromechanical systems. It can be a complex network, in which the components are depicted by nodes and the physical connections between linkage components are depicted by links between the corresponding nodes. We refer to this representation as a topological structure and a formal model is presented by the following definitions.We defineHTN=V,E,fV,fE as aholistic topological network(1)HTN=V,E,fV,fEvi∈V,eij∈E,fvi=Afvi,Atvi⊂fV,feij=Bfeij⊂fE,i=1,…,n;j≤n,where V=v1,v2,…,vn is a set of nodes and each node represents a component. n is the number of nodes. For example, the number of nodes is 7 for a system in Figure 1. E=eij∣i=1,…,n,j∈n is a set of links and eij represents physical connection between nodes vi and vj. Depending on the nature or type of the topological property, this property may be reflexive in that eij≠eji. fV is the properties of the set of nodes, Afvi is the functional properties of vi, and Atvi is the topological properties of vi. fE is the properties of the set of edges and Bfeij is the functional properties of edge eij.Figure 1
The holistic topological network.For different systems, the properties of nodes or edges may be also different. For instant, assuming that Figure1 is the partial bogie system of CRHX, the properties of nodes and edges in theholistic topological network are as follows.(
1)
A
f
v
i
=
L
T
i
,
M
T
B
F
i
,
λ
i
.
L
T
i is the service life of node vi, which is its expected lifetime or the acceptable period of use in service. It is the time that any manufactured item can be expected to be “serviceable” or supported by its manufacturer. We can obtain this parameter from its manufacturer.λ
i is failure rate of node vi and is defined as(2)λit=limΔt→0Rit-Rit+ΔtΔt·Rit,where Rit is the probability of no failure before time t for node vi; Δt is the over-a-time-interval from failure time t to the next failure time. In this paper, failure rate λi depending on failure data is computed.M
T
B
F
i is mean time between failures of node vi and the formula is(3)MTBFi=∑fid-fiuNif,where fid is start of downtime for node vi,fiu is start of uptime for node vi, and Nif is the number of failures.(
2)
A
t
v
i
=
B
C
i
,
C
C
i
,
D
C
i
,
E
C
i. CCi is closeness centrality [31] of node vi:(4)CCi=1∑jndij,where dij denotes the distance between node vi and node vj; n is the number of nodes.B
C
i is betweenness centrality [31] of node vi:(5)BCi=∑j,k≠igjkigjk,where gjk is the number of binary shortest paths between node vj and node vk and gjki is the number of those paths that go through node vi.D
C
i is degree centrality [32] of node vi:(6)DCi=∑jNxij,where i is the focal node vi; j represents all other nodes; N is the total number of nodes; xij represents the connection between node vi and node vj. The value of xij is defined as 1 if node vi is connected to node vj and 0 otherwise.E
C
i is eigenvector centrality [12] of node vi. Let A be a n×n similarity matrix. The eigenvector centrality ECi of node vi is defined as the ith entry in the normalized eigenvector belonging to the largest eigenvalue of A. λ is the largest eigenvalue of A and n is the number of vertices: (7)A·EC=λ·ECECi=μ∑j=1naijECj,i=1,…,nwith proportionality factor μ=1/λ so that ECi is proportional to the sum of similarity scores of all nodes connected to it.(
3)
B
f
e
i
j
=
p
i
j
,
s
t
i
j
,
λ
i
j
.
p
i
j is fault propagation probability of edge eij, given by(8)pij=leij∑i≠jleij,where leij is the number of shortest paths crossing a given edge eij. Gao et al. [33] introduced the concept and computational method of fault propagation probability in detail.s
t
i
j is connection strength of edge eij and is given by [34, 35](9)stij=βsvi∣vjsvi,where svi is the number of times that operation states change in the statistical time; svi∣vj indicates the number of times that vi operation states change arising from vi in the statistical time. The operation states of vi contain the failure mode and normal operation mode of the corresponding component. β is an empirical contact duration of the type of functional dependencies between components vi and vj.λ
i
j is failure rate of edge eij. The calculation method is the same as the failure rate of node in (2).In Figure1, the edge describes the physical connection between linkage components, and the arrow of the edges expresses the failure interaction of nodes. Liu and An [22] further introduced how to determine the direction of edges which describe the coupling relation of the failure interactions.


## 2.2. Fuzzy Integral Theory

About thirty years ago the concept of fuzzy integral was proposed in Japan by Sugeno [36, 37], which in the discrete case is merely a kind of distorted mean. Although this was followed by a rather mathematically oriented research, far from application concerns, some Japanese researchers, including Sugeno himself, thought that fuzzy integrals could be applied to multicriteria evaluation: since 1985, papers have been published on supplier evaluation and improvement [38], analysis of policy decision of sustainable energy strategies [39], fusion of extreme learning machine [40], analysis of human reliability [41], and so forth. The distinguishing feature of fuzzy integral is that it is able to represent a certain kind of interaction between criteria, ranging from redundancy (negative interaction) to synergy (positive interaction). To our knowledge, there is almost no well-established method to deal with interacting criteria, and usually people tend to avoid the problem by constructing independent (or supposed to be so) criteria. This innovative feature was without any doubt the reason of its success in various fields of application [42]. A wide variety of forms have been presented for fuzzy integral, such as Choquet integral [43] and Sugeno integral.LetX=x1,x2,…,xn be a nonnull and finite set of attributes or influence factors and let PX denote the power set of X.


### 2.2.1. Discrete Choquet Integral

Letgλ be a λ fuzzy measure in PX. The discrete Choquet integral of an element f:X→IR+ with respect to gλ is defined by(10)c∫fdgλ=∑i=1nfxi-fxi-1gλAi,where i indicates a permutation on X such that fx1≤⋯≤fxn. Also Ai=xi,…,xn is an attribute set, and fx0=0. Discrete Choquet integral is the extension of the weighted average operator, which considers the relationships among attributes or influence factors.


### 2.2.2. Discrete Sugeno Integral

Letμ be a fuzzy measure on X and f:X→0,1 be a function. The discrete Sugeno integral of f with respect to μ is(11)s∫fdμ=⋁i=1nfxi∧μAi,where 0≤fxi≤fx2≤⋯≤fxn≤1,Ai=xi,xi+1,…,xn is an attribute set and fx0=0.


## 2.2.1. Discrete Choquet Integral

Letgλ be a λ fuzzy measure in PX. The discrete Choquet integral of an element f:X→IR+ with respect to gλ is defined by(10)c∫fdgλ=∑i=1nfxi-fxi-1gλAi,where i indicates a permutation on X such that fx1≤⋯≤fxn. Also Ai=xi,…,xn is an attribute set, and fx0=0. Discrete Choquet integral is the extension of the weighted average operator, which considers the relationships among attributes or influence factors.


## 2.2.2. Discrete Sugeno Integral

Letμ be a fuzzy measure on X and f:X→0,1 be a function. The discrete Sugeno integral of f with respect to μ is(11)s∫fdμ=⋁i=1nfxi∧μAi,where 0≤fxi≤fx2≤⋯≤fxn≤1,Ai=xi,xi+1,…,xn is an attribute set and fx0=0.


## 3. Improved Centrality Measures

In this section, the novel importance measures, that is, improved centrality measures (ICMs), are first proposed. It is a series of special importance measures to find the influential components that are really crucial for the normal operation of electromechanical systems. When some components lose or weaken their functions due to a certain mode of failures, there will be a degradation of the holistic system performance. Based on centrality measures, the originality and novelty of proposed ICMs is that they evaluate the importance of a component by taking into account the functional and topological properties in the holistic system. In essence, they are comprehensive indicators in the system, which are proposed for the assessment of the most important component.


### 3.1. Influencing Factors

In order to measure the importance of components, influential factors that are capable of representing the desired function of the individual components, as well as their structure, are required. Some quantitative information is introduced into the system to improve the accuracy of the important measure of the components. For example, functional properties reflect the ability of the system to perform its intended function, and topological properties describe the stability of the inherent structure of the system.Through the project (The National High Technology Research and Development Program of China (863 Program) number 2012AA112001) of cooperation withCRRC Corporation Limited, we have discussed with engineers and maintenance personnel in detail and then obtained the influential factors of bogie system as shown in Figure 2, which include the functional properties, which are used to depict the system performance, and the topological properties, which are used to describe the influence on system structure.Figure 2
Influencing factors of the individual components.


### 3.2. Improved Centrality Measures

A large number of centrality measures have been proposed to identify influential nodes within a complex network. Examples are DC, CC, BC, and EC. However, it cannot be ignored that while most of these centrality measures have been widely used inNetwork Reliability Analysis, challenges still remain in regard to the following: none of them has taken into account the impact of functional influences of nodes and edges in the holistic topological network.Improved centrality measures are a series of importance measures which are extended from centrality measures. In a sense, ICMs can serve as better importance measures of components, as it synthesizes the components functional properties, such as usage reliability, failure rate, and connection strength, meanwhile, by taking systems structure into account, such as CC, BC, DC, and EC.The following is an efficient and universal construction method to calculate ICMs.Let us considerAfvi in (1), which consists of functional properties of nodes, Atvi, which consists of topological properties of nodes, and Bfeij, which consists of functional properties of edges.Based on the holistic topological network, functional properties of nodes and edges are integrated into the definition of centrality measures, and then ICMs are constructed, defined as follows:(12)ICMsi=c∫xsi,c∫AfidμAf,c∫BfeijdμBfdμICMxsi∈Ati,where ICMsi is the improved centrality measure of node vi. μ is the weight of influencing factors.According to (12), the first thing that we need is to calculate weight μ. Here, we use g fuzzy measure and k additive fuzzy measure to compute weight μ based on Shapley values [44] and Marichal entropy theory [45]. The maximum Marichal entropy as the objective function constructs optimization model as shown in (13) based on fuzzy theory. And then, calculating (13), we can get the parameter μ.(13)maxλ,μHMμ=∑i=1n∑S⊆A-aiγsn·hμS∪ai-μSs.t.Ii=∑k=0n-1n-k-1!k!n!∑T⊆A-ai,T=kμT∪ai-μTμA=1μM∪N=μM+μN+λμMμN∀M,N∈PA,M∩N=ϕμM∈0,1∀M∈PAλ∈-1,∞i=1,…,n.Zhang et al. [39] explained all the parameters in (13) and introduced the calculation methods in detail.Some simple examples are given to explain how ICMs perform.


#### 3.2.1. Improved Degree Centrality (IDC)

In a binary network,DCi of node vi represents the total number of the connection with node vi. However, functional properties of edges, which connect with node vi and node vi cannot be ignored. The improved degree centrality, denoted as IDCi, is given by(14)Tt=1,…,Nieij=c∫BfeijdμHeij,ifaij=1,existeij;elsenoexisteijH1=c∫T1,…,TNidμHTH2=c∫AfvidμHvIDCi=c∫DCi,H1,H2dμH,where Tteij is the aggregate value of functional properties of edge eij, which is connected to vi. The value of aij is defined as 1 if node vi is connected to vj, and 0 otherwise. H1 is the aggregate value of functional properties of all edges which are connected to vi. H2 is the aggregate value of functional properties of node vi. μ is the weight of influencing factors.


#### 3.2.2. Improved Closeness Centrality (ICC)

The improved closeness centrality of nodevi, denoted as ICCi, is defined as(15)ICCi=∑jNISPij-1,where ISPij is the improved shortest path between nodes vi and vj. Its definition and calculation are explained as follows.Given a directed graph, the length of a path is the number of edges forming it. We define the shortest path as the smallest length among all the paths connecting the source vertex to the target vertex. However, for electromechanical systems, functional properties of nodes and edges also can influence the length of the shortest path. Given thatVliest represents a set of nodes, which are in lith path from node vs to node vt, and Eliest represents a set of edges, which are in lith path from node vs to node vt, improved shortest path can be expressed as follows:(16)ISPij=minlic∫dij,Yliv,YliedμHlis.t.Wliestvj=c∫AfvjdμHvj,vj∈VliestWliestepq=c∫AfepqdμHepq,epq∈EliestYliv=c∫WliestvjdμHWliv,vj∈VliestYlie=c∫WliestepqdμHWlie,epq∈Eliest,where Wliestvj is the aggregate value of functional properties of node vj which is in path li; Wliestepq is the aggregate value of functional properties of edge epq which is in path li; Yliv is the aggregate value of Wliestvj which is in path li; Ylie is the aggregate value of Wliestepq which is in path li.


#### 3.2.3. Improved Betweenness Centrality (IBC)

The improved betweenness centrality of nodevi, denoted as IBCi, can be rewritten as(17)IBCi=∑j,k≠iISPjkviISPjk,where ISPjk is the improved shortest path between nodes vj and vk; ISPjkvi is the number of those paths that go through node vi.


## 3.1. Influencing Factors

In order to measure the importance of components, influential factors that are capable of representing the desired function of the individual components, as well as their structure, are required. Some quantitative information is introduced into the system to improve the accuracy of the important measure of the components. For example, functional properties reflect the ability of the system to perform its intended function, and topological properties describe the stability of the inherent structure of the system.Through the project (The National High Technology Research and Development Program of China (863 Program) number 2012AA112001) of cooperation withCRRC Corporation Limited, we have discussed with engineers and maintenance personnel in detail and then obtained the influential factors of bogie system as shown in Figure 2, which include the functional properties, which are used to depict the system performance, and the topological properties, which are used to describe the influence on system structure.Figure 2
Influencing factors of the individual components.


## 3.2. Improved Centrality Measures

A large number of centrality measures have been proposed to identify influential nodes within a complex network. Examples are DC, CC, BC, and EC. However, it cannot be ignored that while most of these centrality measures have been widely used inNetwork Reliability Analysis, challenges still remain in regard to the following: none of them has taken into account the impact of functional influences of nodes and edges in the holistic topological network.Improved centrality measures are a series of importance measures which are extended from centrality measures. In a sense, ICMs can serve as better importance measures of components, as it synthesizes the components functional properties, such as usage reliability, failure rate, and connection strength, meanwhile, by taking systems structure into account, such as CC, BC, DC, and EC.The following is an efficient and universal construction method to calculate ICMs.Let us considerAfvi in (1), which consists of functional properties of nodes, Atvi, which consists of topological properties of nodes, and Bfeij, which consists of functional properties of edges.Based on the holistic topological network, functional properties of nodes and edges are integrated into the definition of centrality measures, and then ICMs are constructed, defined as follows:(12)ICMsi=c∫xsi,c∫AfidμAf,c∫BfeijdμBfdμICMxsi∈Ati,where ICMsi is the improved centrality measure of node vi. μ is the weight of influencing factors.According to (12), the first thing that we need is to calculate weight μ. Here, we use g fuzzy measure and k additive fuzzy measure to compute weight μ based on Shapley values [44] and Marichal entropy theory [45]. The maximum Marichal entropy as the objective function constructs optimization model as shown in (13) based on fuzzy theory. And then, calculating (13), we can get the parameter μ.(13)maxλ,μHMμ=∑i=1n∑S⊆A-aiγsn·hμS∪ai-μSs.t.Ii=∑k=0n-1n-k-1!k!n!∑T⊆A-ai,T=kμT∪ai-μTμA=1μM∪N=μM+μN+λμMμN∀M,N∈PA,M∩N=ϕμM∈0,1∀M∈PAλ∈-1,∞i=1,…,n.Zhang et al. [39] explained all the parameters in (13) and introduced the calculation methods in detail.Some simple examples are given to explain how ICMs perform.


### 3.2.1. Improved Degree Centrality (IDC)

In a binary network,DCi of node vi represents the total number of the connection with node vi. However, functional properties of edges, which connect with node vi and node vi cannot be ignored. The improved degree centrality, denoted as IDCi, is given by(14)Tt=1,…,Nieij=c∫BfeijdμHeij,ifaij=1,existeij;elsenoexisteijH1=c∫T1,…,TNidμHTH2=c∫AfvidμHvIDCi=c∫DCi,H1,H2dμH,where Tteij is the aggregate value of functional properties of edge eij, which is connected to vi. The value of aij is defined as 1 if node vi is connected to vj, and 0 otherwise. H1 is the aggregate value of functional properties of all edges which are connected to vi. H2 is the aggregate value of functional properties of node vi. μ is the weight of influencing factors.


### 3.2.2. Improved Closeness Centrality (ICC)

The improved closeness centrality of nodevi, denoted as ICCi, is defined as(15)ICCi=∑jNISPij-1,where ISPij is the improved shortest path between nodes vi and vj. Its definition and calculation are explained as follows.Given a directed graph, the length of a path is the number of edges forming it. We define the shortest path as the smallest length among all the paths connecting the source vertex to the target vertex. However, for electromechanical systems, functional properties of nodes and edges also can influence the length of the shortest path. Given thatVliest represents a set of nodes, which are in lith path from node vs to node vt, and Eliest represents a set of edges, which are in lith path from node vs to node vt, improved shortest path can be expressed as follows:(16)ISPij=minlic∫dij,Yliv,YliedμHlis.t.Wliestvj=c∫AfvjdμHvj,vj∈VliestWliestepq=c∫AfepqdμHepq,epq∈EliestYliv=c∫WliestvjdμHWliv,vj∈VliestYlie=c∫WliestepqdμHWlie,epq∈Eliest,where Wliestvj is the aggregate value of functional properties of node vj which is in path li; Wliestepq is the aggregate value of functional properties of edge epq which is in path li; Yliv is the aggregate value of Wliestvj which is in path li; Ylie is the aggregate value of Wliestepq which is in path li.


### 3.2.3. Improved Betweenness Centrality (IBC)

The improved betweenness centrality of nodevi, denoted as IBCi, can be rewritten as(17)IBCi=∑j,k≠iISPjkviISPjk,where ISPjk is the improved shortest path between nodes vj and vk; ISPjkvi is the number of those paths that go through node vi.


## 3.2.1. Improved Degree Centrality (IDC)

In a binary network,DCi of node vi represents the total number of the connection with node vi. However, functional properties of edges, which connect with node vi and node vi cannot be ignored. The improved degree centrality, denoted as IDCi, is given by(14)Tt=1,…,Nieij=c∫BfeijdμHeij,ifaij=1,existeij;elsenoexisteijH1=c∫T1,…,TNidμHTH2=c∫AfvidμHvIDCi=c∫DCi,H1,H2dμH,where Tteij is the aggregate value of functional properties of edge eij, which is connected to vi. The value of aij is defined as 1 if node vi is connected to vj, and 0 otherwise. H1 is the aggregate value of functional properties of all edges which are connected to vi. H2 is the aggregate value of functional properties of node vi. μ is the weight of influencing factors.


## 3.2.2. Improved Closeness Centrality (ICC)

The improved closeness centrality of nodevi, denoted as ICCi, is defined as(15)ICCi=∑jNISPij-1,where ISPij is the improved shortest path between nodes vi and vj. Its definition and calculation are explained as follows.Given a directed graph, the length of a path is the number of edges forming it. We define the shortest path as the smallest length among all the paths connecting the source vertex to the target vertex. However, for electromechanical systems, functional properties of nodes and edges also can influence the length of the shortest path. Given thatVliest represents a set of nodes, which are in lith path from node vs to node vt, and Eliest represents a set of edges, which are in lith path from node vs to node vt, improved shortest path can be expressed as follows:(16)ISPij=minlic∫dij,Yliv,YliedμHlis.t.Wliestvj=c∫AfvjdμHvj,vj∈VliestWliestepq=c∫AfepqdμHepq,epq∈EliestYliv=c∫WliestvjdμHWliv,vj∈VliestYlie=c∫WliestepqdμHWlie,epq∈Eliest,where Wliestvj is the aggregate value of functional properties of node vj which is in path li; Wliestepq is the aggregate value of functional properties of edge epq which is in path li; Yliv is the aggregate value of Wliestvj which is in path li; Ylie is the aggregate value of Wliestepq which is in path li.


## 3.2.3. Improved Betweenness Centrality (IBC)

The improved betweenness centrality of nodevi, denoted as IBCi, can be rewritten as(17)IBCi=∑j,k≠iISPjkviISPjk,where ISPjk is the improved shortest path between nodes vj and vk; ISPjkvi is the number of those paths that go through node vi.


## 4. Proposed Method

The ICMs, such as IDC, ICC, and IBC, reflect the function and structure of system from one aspect and cannot comprehensively reflect the functional and topological characteristics. In some cases, the results of IDC and ICC may be different, even conflicting results. To address this issue, in this paper, comprehensive measure is introduced firstly to explore how to fuse multi-ICMs based on fuzzy integral and then identify influential components. As a well-known fuzzy integral theory, Sugeno integral and Choquet integral have received much interest from researchers and practitioners.Let us consider a decision matrixDm=ICMmn, where ICMmn is the nth improved centrality measure of node vm. If we choose Sugeno integral, the comprehensive measure of node vi, denoted as ISi, is defined as follows:(18)ISi=s∫Didμ=⋁j=1mICMij∧μAij,i=1,…,n.If we choose Choquet integral, the comprehensive measure of nodevi, denoted as ICi, is defined as follows:(19)ICi=c∫Didμ=∑j=1mICMij-ICMij-1μAij,i=1,…,n,where Aij=ICMi1,…,ICMiN and μAij is the weight for improved centrality measure.The specific steps of the method are illustrated as follows.Step 1 (construct the holistic topological network of complex electromechanical system).
We can construct a network based on Section2.1. The components are abstracted as nodes, and the connections between components are represented as edges. And then the system can be described as a network.Step 2 (calculate properties value of nodes and edges).
According to Figure2, we compute the properties of nodes and edges in combination failure data. The attributes of nodes are obtained by (2)~(7), and properties of nodes are computed by (8).Step 3 (construct and calculate the improved centrality measures by (12)).
In this step, we apply (12) to construct and calculate the different improved centrality measures, such as improved degree centrality, improved closeness centrality, and improved betweenness centrality.Step 4 (select fuzzy integral and fuse all improved centrality measures by (18) or (19)).
The alternatives with higherICi or ISi are assumed to be more important and should be given higher priority. Finally, the influence of the node is identified by the value ICi or ISi.The flow chart of the proposed methods is shown in Figure3.Figure 3
The flow chart of the proposed method.


## 5. A Case Study and Discussion

China Railway CRHX Size (CRHX) is designed for a speed of 350 km/h and each car is suspended by two bogies. The bogie system of the 350 km/h EMU train is one of the key parts of CRHX which plays an important role in sustaining the static load from the body weight of a car, carrying the suspensions, brakes, wheels, and axles and controlling wheel sets on curved and straight tracks, in accordance with Figure4. The bogie system can be highly complex due to the systematic use of new technologies and be functional relationship due to the interactions among components. With rapid increase of EMU train speed, the behavior of bogie system becomes more dynamic and uncertain, which not only affects the ride comfort of passengers but also directly relates to the reliability and safety of the train. Generally, the operation of the bogie system depends on 35 components, in accordance with Table 1.Table 1
Components in bogie system.
Number
Name
v
1
Bogie frame
v
2
Brake Caliper
v
3
Brake lining
v
4
Brake discs
v
5
Booster cylinder
v
6
Spring
v
7
Axle box body
v
8
Vertical shock absorber
v
9
Bearing
v
10
Wheel
v
11
Axle
v
12
Secondary vertical shock absorber
v
13
Railway coupling
v
14
Gearbox
v
15
Grounding device
v
16
Traction motor
v
17
Height adjusting device
v
18
Antihunting damper
v
19
Air spring
v
20
Center pin bush
v
21
Traction rod
v
22
Transverse shock absorber
v
23
Transverse backstop
v
24
Anti-side-rolling torsion bar
v
25
Control valve
v
26
Speed Sensor 1
v
27
Speed Sensor 2
v
28
LKJ2000
v
29
Device for cleaning the tread band of vehicle wheels
v
30
Acceleration sensor
v
31
Junction box
v
32
Temperature sensor bearing
v
33
Axle temperature sensor
v
34
AG37
v
35
AG43Figure 4
Structural features of bogie system.


### 5.1. The Network Model and Related Data

In this section, a case concerning the holistic topological network model of bogie system (as shown in Figure5) is established as a decision support tool for importance measures and safety assurance, to provide effective support for decision-makers to proactively understand components interactions and assess their impact to the overall system. According to the construction method of network in Section 2.1, influential factors of bogie system in Figure 2, components of bogie system in Table 1, and the holistic topological network model of bogie system are described as follows:(20)HTM=V,E,fV,fEvi∈V=v1,v2,…,v35,eij∈E=eij∣i,j∈35,Afvi=MTBFi,λi,Li⊂fV,Atvi=DCi,BCi,CCi,ECi⊂fV,Bfeij=stij,λij,pij⊂fE.Figure 5
The holistic topological network of bogie system: every nodevi has property set Afvi,Atvi, and every edge eij has property set Bfeij.The attributes of edges and nodes in (20) are calculated according to the equation in Section 2.1. Related data, which is necessary for calculation, can be collected from historical failure databases and expert elicitation. For high-speed train system, through a project of cooperation with CRRC Corporation Limited, we have obtained the historical failure databases of CRHX during 2009 to 2014 (863 Program number 2012AA112001). In these databases, each failure data record contains the failure ID numbers, the vehicle ID number, the section of failure, the failure mode, the date of failure, the environment of failure, and so forth. We deal with the data by removing some irrelevant items. However, these data cannot be directly used to compute properties and need to be preprocessed. And, a preprocessed failure data of gear box in bogie system is presented in Table 2.Table 2
The preprocessed failure data of gear box.
Number
Mileage/105 km
Failure mode
(1)
6.39853
Oil leakage
(2)
7.87662
Oil leakage
(3)
7.90238
Gear shift
(4)
10.02856
Oil leakage
(5)
11.26585
Crackle
(6)
11.29788
Oil leakage
(7)
12.39568
Oil leakage
(8)
14.02572
Crackle
(9)
15.64292
Crackle
(10)
16.39853
Oil leakage
(11)
16.64292
Crackle
(12)
16. 64292
Oil leakage
(13)
17.66824
Oil leakage
(14)
18.16762
Gear shift
(15)
18.2579
Oil leakage
(16)
19.32587
Gear shift
(17)
19.90225
Oil leakage
(18)
21.0191
Crackle
(19)
22.17948
Crackle
(20)
22.38788
Crackle
(21)
23.0191
Oil leakage
(22)
27.73312
Oil leakage
(23)
28.62366
Crackle
(24)
28.87598
Oil leakage
(25)
29.62366
Oil leakage
(26)
31.79775
Crackle
(27)
32.4842
Crackle
(28)
33.43831
Crackle
(29)
34.43831
Gear shift
(30)
34.50838
Crackle
(31)
37.66297
Oil leakage
(32)
38.66297
Gear shift
(33)
40.0016
CrackleUsing the preprocessed failure data of CRHX and (2)~(3), this paper estimates the parameters for possible faults distribution function. Based on the faults distributing functions and the necessary correction, we can obtain components’ failure rate and MTBF. The nodes failure rate and MTBF within 120 million kilometers and service life time are shown in Table 3.Table 3
The nodes functional properties.
Node
Life time/year
Failure rate
MTBF
v
1
20
0.0134
2.34
v
2
20
0.00798
1.25
v
3
15
0.0089
1.54
v
4
20
0.0045
2.21
v
5
20
0.0079
1.72
v
6
20
0.0059
1.92
v
7
20
0.0086
1.41
v
8
20
0.0081
1.69
v
9
20
0.0144
1.21
v
10
20
0.0126
1.34
v
11
20
0.0176
1.47
v
12
20
0.0079
1.65
v
13
20
0.0082
1.41
v
14
20
0.0103
1.52
v
15
20
0.0103
1.81
v
16
20
0.0078
1.45
v
17
20
0.0116
1.44
v
18
20
0.0082
1.66
v
19
15
0.0061
2.03
v
20
20
0.0052
1.55
v
21
20
0.0051
1.97
v
22
20
0.0062
1.86
v
23
20
0.0049
1.93
v
24
20
0.0051
1.77
v
25
20
0.0072
1.78
v
26
20
0.0077
1.68
v
27
20
0.0177
1.32
v
28
20
0.0187
1.43
v
29
20
0.0107
1.50
v
30
20
0.0152
1.41
v
31
20
0.0049
1.53
v
32
20
0.0165
1.32
v
33
15
0.0189
1.38
v
34
20
0.0191
1.49
v
35
20
0.0169
1.53Edges have a striking effect on critical nodes in the network. In essence, edges in holistic topological network also describe components, but these components have different properties from nodes. The fault propagation probability, connection strength, and failure rate of edges are computed according to (2), (8), and (9) based on failure date and shown as in Table 4.Table 4
Functional properties of several edges.
Edge
Connection strength
Failure rate
Fault propagation probability
Edge
Connection strength
Failure rate
Fault propagation probability
e
12
0.4
0.0051
0.0001
e
1,14
0.6
0.0068
0.0003
e
13
0.5
0.0087
0.0005
e
51
0.4
0.0053
0.0004
e
15
0.5
0.0095
0.0006
e
61
0.7
0.0071
0.0001
e
16
0.8
0.0088
0.0003
e
52
0.5
0.0067
0.0015
e
17
0.6
0.0082
0.0003
e
67
0.3
0.0104
0.0010
e
18
0.5
0.0062
0.0007
e
10,4
0.7
0.0045
0.0013
e
1,12
0.7
0.0079
0.0004
e
11,9
0.4
0.0042
0.0011
⋯
⋯
⋯
…
⋯
⋯
⋯
…


### 5.2. ICMs for Bogie System

CMs of nodes in the holistic topological network are necessary to construct improved centrality measures to assess influential components. According to (20) and complex network theory, we can obtain centrality measures of all nodes, such as BC, CC, DC, and EC in (4)~(7). Figure 6 shows the results of CMs in the network. We can see that node v1 has the highest BC, CC, DC, and EC, and it means that the most important node in the holistic topological network of a bogie system isbogie frame (the corresponding node is v1). This is consistent with the structural status of the bogie frame in the bogie system.Figure 6
CMs of nodes in the holistic topological network.In the method proposed in this paper, the assessment of influential components requires, firstly, exhaustive and systematic definition and calculation improved centrality measures for any given number of components according to Section3.2. Figure 7 is presented to demonstrate the calculation results of ICMs by combination with CMs in Figure 5 and functional properties of nodes and edges in Tables 3 and 4. From Figure 7, we can also obtain that bogie frame (the corresponding node is v1) is identified as the most important components, which is in line with practical experience.Figure 7
Improved centrality measures.In order to explain the advantages of ICMs, we compare with the identification results of ICMs and CMs. The most critical node in bogie system is allv1. However, the importance ranking of other components is also the key to ensure system safety and reliability. In fact, maintenance personnel give more attention to a series of critical components, but not only the most important component. Figure 8 indicates the ranking of all nodes. We can clearly find that v2, which is the identification result by using DC, is more important in network from a topological point of view. However, if we use improved DC to analyze the Brake Caliper, the result shows that Brake Caliper is not much more important than other nodes. This effect can be explained by the fact that ICMs of components are influenced by functional properties of nodes and edges that are linked to closer neighbors at the same time. Hence, with the functional influence of nodes and edges, the importance of nodes may change. Just as we have mentioned, ICMs is better than CMs.Figure 8
The ranking of nodes.Another interesting fact observed is that, as presented in Figure8, the ranking of all nodes is different by using IBC, ICC, IDC, and IEC. Note that the result of IDC shows that v1, traction motor v16, and v7 are the critical components. If we use ICC to identify influential components, the results reveal that v1, v7, and v6 are the most important for bogie system. However, v1, v7, and v14 are the critical components by using IBC identification, and the identification results of IEC show that v7, v25, and v31 are essential for bogie system. This research presented two aspects of reasons for this: one is different influences which are taken into consideration and another is the uncertainty and randomness when single measure is used to identify critical component. For example, DC2<IDC2; however, CC2>ICC2; DC15<IDC15, but BC15>IBC15. CMs are constructed in which topological properties are taken into consideration, and ICMs are defined in combination with functional and topological features.


### 5.3. Influential Components Based on Fuzzy Integral

In order to overcome the uncertainty and randomness which applies for single measure identifying critical component, we fuse ICMs to explore how to identify influential components based on fuzzy integral. According to Section4, Figure 9 is described to demonstrate the result of comprehensive importance calculations. The result shows that the bogie frame and axle box body (the corresponding nodes are v1,v7) are identified as the most important components, which is in line with practical experience. The more critical nodes which are identified by Sugeno integral are the same as the Choquet integral. However, the calculation of Choquet integral is smaller than Sugeno integral and there is a problem that Sugeno integral will miss the information in calculation process. Therefore, the different integrals are selected based on different operation conditions when only more critical components need to be evaluated.Figure 9
Influential components of bogie system.Meanwhile, how to calculate weightμ in fuzzy integral also is important for the accuracy of assessment results. The two methods are provided to compute weight μ, and they are g fuzzy measure and k additive fuzzy measure. Theoretically, the accuracy of computation results based on k additive fuzzy measure is higher than g fuzzy measure. g fuzzy measure analyzes weights of all influences and their influencing relationship; however, k additive fuzzy measure provides weights of all influences and their influencing relationship for each nodes. Moreover, the application of k additive fuzzy measure is limited to the problem of large quantities of computation. Fortunately, we can find that most identification results with g fuzzy measure are the same as k additive fuzzy measure in Figure 9. Therefore, the g fuzzy measure can be chosen in traditional study.


### 5.4. Discussion

To better assess the importance of a component in the holistic topological network, the method of integration ICMs is proposed in this paper, which takes functional and topological properties into account. It is much different from integration topological CMs indicators such as DC, BC, CC, and EC mentioned in Section5.2. From an experimental viewpoint, Figure 10 shows the results of integration CMs and ICMs. The ranking of all nodes by comprehensive measure, which is constructed by integration ICMs, is different from integration CMs. The weakness and shortage of CMs cannot disappear if we integrate CMs. Therefore, the accuracy of only integration CMs is still lower.Figure 10
The results of integration CMs and ICMs.Table5 presents the evaluation results of different methods. Based on863 Program (number 2012AA112001), we investigate the bogie system of CRHX in-depth, communicate with maintenance personnel, and attain the ranking of important components for using the enterprise, as shown inPractical recognition in Table 5. From the expert experience viewpoint, the most important nodes in the holistic topological network of a bogie system are v1, v7, v16, and v14, respectively. In the view of ICMs, these centrality measures can be further extended by considering suitable properties to identify the important components with respect to function and topology of the nodes and edges (in Figure 6), while, using fuzzy integral fusion ICMs, some nodes (such as v6) with low values can be also identified as the most important nodes. In addition, the evaluation result of these methods is a larger gap, especially noncritical components. These methods, such as AHP and TOPSIS, are greatly affected by human factors. The accuracy of the most critical components identification results is higher, but other components have great difference on human factors. CMs, which include DC, CC, BC, and EC, identify the critical components of structure. Therefore, there are some differences in the results of CMs and other measures. In fact, there are still differences of using Sugeno integral and Choquet integral to identify influential components. The most important components are the same, while the ranking of noncritical components is different. This is because the amount of information loss exceeds the threshold value.Table 5
The evaluation results of different methods.
Number
AHP
TOPSIS
DC
CC
BC
EC
Practical recognition
Sugeno integral
Choquet integral
g
λ
k-additive
g
λ
k-additive
(1)
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
(2)
v
16
v
7
v
31
v
7
v
7
v
25
v
7
v
7
v
7
v
7
v
7
(3)
v
14
v
14
v
7
v
25
v
14
v
31
v
14
v
6
v
14
v
14
v
14
(4)
v
31
v
9
v
14
v
14
v
31
v
7
v
16
v
14
v
16
v
16
v
16
(5)
v
7
v
16
v
10
v
6
v
25
v
30
v
31
v
29
v
31
v
9
v
9
(6)
v
9
v
8
v
6
v
5
v
16
v
6
v
9
v
16
v
25
v
31
v
31
(7)
v
19
v
12
v
16
v
8
v
2
v
26
v
25
v
2
v
2
v
32
v
25
(8)
v
25
v
23
v
26
v
16
v
10
v
5
v
6
v
8
v
10
v
6
v
6
(9)
v
2
v
22
v
2
v
29
v
6
v
8
v
32
v
5
v
6
v
25
v
22
(10)
v
10
v
21
v
8
v
2
v
11
v
14
v
22
v
20
v
29
v
22
v
32With the reduction of the importance of nodes, the difference of all methods in Table5 for importance of nodes is bigger in Figure 11. For example, the results of the three methods, such as AHP, DC, and EC, are not consistent with practical recognition for node v7. And, for nodes v14, v16, and v31, the number of methods which are not different from practical recognition, is 4, 7, and 9, respectively. We can see that most methods are effective in identifying the most critical nodes. However, the validity of the identification results of the other nodes is poor.Figure 11
The ranking of partly nodes.The methods of integration multimeasures are able to overcome the randomness and uncertainty by using a single measure. Different methods of comprehensive multimeasures are selected, and the accuracy of the identification results is different. Figure12 presents the accuracy rate of these methods in comparison with practical recognition. The accuracy rate ARi=RNi/N, where RNi indicates the number of nodes in the ith method which the ranking is consistent with the expert experience, and Ni is the number of nodes in system. We can find that Choquet integral by using k-additive has a higher accuracy rate. The method of integration ICMs with Sugeno integral misses much information in computational procedure. If the amount of information loss exceeds the threshold value, the results will be not accurate.Figure 12
The accuracy rate of these methods.Through the exploratory discussion above, it is shown that the results acquired from comprehensive measure are, to a certain extent, more reasonable and powerful than those traditional importance measures. Just as mentioned, the system function represents the interactions between components; system topology represents the structure relationship. Therefore, the comprehensive gives consideration to both topological features and physical characteristics of a holistic topological network from multiple perspectives.


## 5.1. The Network Model and Related Data

In this section, a case concerning the holistic topological network model of bogie system (as shown in Figure5) is established as a decision support tool for importance measures and safety assurance, to provide effective support for decision-makers to proactively understand components interactions and assess their impact to the overall system. According to the construction method of network in Section 2.1, influential factors of bogie system in Figure 2, components of bogie system in Table 1, and the holistic topological network model of bogie system are described as follows:(20)HTM=V,E,fV,fEvi∈V=v1,v2,…,v35,eij∈E=eij∣i,j∈35,Afvi=MTBFi,λi,Li⊂fV,Atvi=DCi,BCi,CCi,ECi⊂fV,Bfeij=stij,λij,pij⊂fE.Figure 5
The holistic topological network of bogie system: every nodevi has property set Afvi,Atvi, and every edge eij has property set Bfeij.The attributes of edges and nodes in (20) are calculated according to the equation in Section 2.1. Related data, which is necessary for calculation, can be collected from historical failure databases and expert elicitation. For high-speed train system, through a project of cooperation with CRRC Corporation Limited, we have obtained the historical failure databases of CRHX during 2009 to 2014 (863 Program number 2012AA112001). In these databases, each failure data record contains the failure ID numbers, the vehicle ID number, the section of failure, the failure mode, the date of failure, the environment of failure, and so forth. We deal with the data by removing some irrelevant items. However, these data cannot be directly used to compute properties and need to be preprocessed. And, a preprocessed failure data of gear box in bogie system is presented in Table 2.Table 2
The preprocessed failure data of gear box.
Number
Mileage/105 km
Failure mode
(1)
6.39853
Oil leakage
(2)
7.87662
Oil leakage
(3)
7.90238
Gear shift
(4)
10.02856
Oil leakage
(5)
11.26585
Crackle
(6)
11.29788
Oil leakage
(7)
12.39568
Oil leakage
(8)
14.02572
Crackle
(9)
15.64292
Crackle
(10)
16.39853
Oil leakage
(11)
16.64292
Crackle
(12)
16. 64292
Oil leakage
(13)
17.66824
Oil leakage
(14)
18.16762
Gear shift
(15)
18.2579
Oil leakage
(16)
19.32587
Gear shift
(17)
19.90225
Oil leakage
(18)
21.0191
Crackle
(19)
22.17948
Crackle
(20)
22.38788
Crackle
(21)
23.0191
Oil leakage
(22)
27.73312
Oil leakage
(23)
28.62366
Crackle
(24)
28.87598
Oil leakage
(25)
29.62366
Oil leakage
(26)
31.79775
Crackle
(27)
32.4842
Crackle
(28)
33.43831
Crackle
(29)
34.43831
Gear shift
(30)
34.50838
Crackle
(31)
37.66297
Oil leakage
(32)
38.66297
Gear shift
(33)
40.0016
CrackleUsing the preprocessed failure data of CRHX and (2)~(3), this paper estimates the parameters for possible faults distribution function. Based on the faults distributing functions and the necessary correction, we can obtain components’ failure rate and MTBF. The nodes failure rate and MTBF within 120 million kilometers and service life time are shown in Table 3.Table 3
The nodes functional properties.
Node
Life time/year
Failure rate
MTBF
v
1
20
0.0134
2.34
v
2
20
0.00798
1.25
v
3
15
0.0089
1.54
v
4
20
0.0045
2.21
v
5
20
0.0079
1.72
v
6
20
0.0059
1.92
v
7
20
0.0086
1.41
v
8
20
0.0081
1.69
v
9
20
0.0144
1.21
v
10
20
0.0126
1.34
v
11
20
0.0176
1.47
v
12
20
0.0079
1.65
v
13
20
0.0082
1.41
v
14
20
0.0103
1.52
v
15
20
0.0103
1.81
v
16
20
0.0078
1.45
v
17
20
0.0116
1.44
v
18
20
0.0082
1.66
v
19
15
0.0061
2.03
v
20
20
0.0052
1.55
v
21
20
0.0051
1.97
v
22
20
0.0062
1.86
v
23
20
0.0049
1.93
v
24
20
0.0051
1.77
v
25
20
0.0072
1.78
v
26
20
0.0077
1.68
v
27
20
0.0177
1.32
v
28
20
0.0187
1.43
v
29
20
0.0107
1.50
v
30
20
0.0152
1.41
v
31
20
0.0049
1.53
v
32
20
0.0165
1.32
v
33
15
0.0189
1.38
v
34
20
0.0191
1.49
v
35
20
0.0169
1.53Edges have a striking effect on critical nodes in the network. In essence, edges in holistic topological network also describe components, but these components have different properties from nodes. The fault propagation probability, connection strength, and failure rate of edges are computed according to (2), (8), and (9) based on failure date and shown as in Table 4.Table 4
Functional properties of several edges.
Edge
Connection strength
Failure rate
Fault propagation probability
Edge
Connection strength
Failure rate
Fault propagation probability
e
12
0.4
0.0051
0.0001
e
1,14
0.6
0.0068
0.0003
e
13
0.5
0.0087
0.0005
e
51
0.4
0.0053
0.0004
e
15
0.5
0.0095
0.0006
e
61
0.7
0.0071
0.0001
e
16
0.8
0.0088
0.0003
e
52
0.5
0.0067
0.0015
e
17
0.6
0.0082
0.0003
e
67
0.3
0.0104
0.0010
e
18
0.5
0.0062
0.0007
e
10,4
0.7
0.0045
0.0013
e
1,12
0.7
0.0079
0.0004
e
11,9
0.4
0.0042
0.0011
⋯
⋯
⋯
…
⋯
⋯
⋯
…


## 5.2. ICMs for Bogie System

CMs of nodes in the holistic topological network are necessary to construct improved centrality measures to assess influential components. According to (20) and complex network theory, we can obtain centrality measures of all nodes, such as BC, CC, DC, and EC in (4)~(7). Figure 6 shows the results of CMs in the network. We can see that node v1 has the highest BC, CC, DC, and EC, and it means that the most important node in the holistic topological network of a bogie system isbogie frame (the corresponding node is v1). This is consistent with the structural status of the bogie frame in the bogie system.Figure 6
CMs of nodes in the holistic topological network.In the method proposed in this paper, the assessment of influential components requires, firstly, exhaustive and systematic definition and calculation improved centrality measures for any given number of components according to Section3.2. Figure 7 is presented to demonstrate the calculation results of ICMs by combination with CMs in Figure 5 and functional properties of nodes and edges in Tables 3 and 4. From Figure 7, we can also obtain that bogie frame (the corresponding node is v1) is identified as the most important components, which is in line with practical experience.Figure 7
Improved centrality measures.In order to explain the advantages of ICMs, we compare with the identification results of ICMs and CMs. The most critical node in bogie system is allv1. However, the importance ranking of other components is also the key to ensure system safety and reliability. In fact, maintenance personnel give more attention to a series of critical components, but not only the most important component. Figure 8 indicates the ranking of all nodes. We can clearly find that v2, which is the identification result by using DC, is more important in network from a topological point of view. However, if we use improved DC to analyze the Brake Caliper, the result shows that Brake Caliper is not much more important than other nodes. This effect can be explained by the fact that ICMs of components are influenced by functional properties of nodes and edges that are linked to closer neighbors at the same time. Hence, with the functional influence of nodes and edges, the importance of nodes may change. Just as we have mentioned, ICMs is better than CMs.Figure 8
The ranking of nodes.Another interesting fact observed is that, as presented in Figure8, the ranking of all nodes is different by using IBC, ICC, IDC, and IEC. Note that the result of IDC shows that v1, traction motor v16, and v7 are the critical components. If we use ICC to identify influential components, the results reveal that v1, v7, and v6 are the most important for bogie system. However, v1, v7, and v14 are the critical components by using IBC identification, and the identification results of IEC show that v7, v25, and v31 are essential for bogie system. This research presented two aspects of reasons for this: one is different influences which are taken into consideration and another is the uncertainty and randomness when single measure is used to identify critical component. For example, DC2<IDC2; however, CC2>ICC2; DC15<IDC15, but BC15>IBC15. CMs are constructed in which topological properties are taken into consideration, and ICMs are defined in combination with functional and topological features.


## 5.3. Influential Components Based on Fuzzy Integral

In order to overcome the uncertainty and randomness which applies for single measure identifying critical component, we fuse ICMs to explore how to identify influential components based on fuzzy integral. According to Section4, Figure 9 is described to demonstrate the result of comprehensive importance calculations. The result shows that the bogie frame and axle box body (the corresponding nodes are v1,v7) are identified as the most important components, which is in line with practical experience. The more critical nodes which are identified by Sugeno integral are the same as the Choquet integral. However, the calculation of Choquet integral is smaller than Sugeno integral and there is a problem that Sugeno integral will miss the information in calculation process. Therefore, the different integrals are selected based on different operation conditions when only more critical components need to be evaluated.Figure 9
Influential components of bogie system.Meanwhile, how to calculate weightμ in fuzzy integral also is important for the accuracy of assessment results. The two methods are provided to compute weight μ, and they are g fuzzy measure and k additive fuzzy measure. Theoretically, the accuracy of computation results based on k additive fuzzy measure is higher than g fuzzy measure. g fuzzy measure analyzes weights of all influences and their influencing relationship; however, k additive fuzzy measure provides weights of all influences and their influencing relationship for each nodes. Moreover, the application of k additive fuzzy measure is limited to the problem of large quantities of computation. Fortunately, we can find that most identification results with g fuzzy measure are the same as k additive fuzzy measure in Figure 9. Therefore, the g fuzzy measure can be chosen in traditional study.


## 5.4. Discussion

To better assess the importance of a component in the holistic topological network, the method of integration ICMs is proposed in this paper, which takes functional and topological properties into account. It is much different from integration topological CMs indicators such as DC, BC, CC, and EC mentioned in Section5.2. From an experimental viewpoint, Figure 10 shows the results of integration CMs and ICMs. The ranking of all nodes by comprehensive measure, which is constructed by integration ICMs, is different from integration CMs. The weakness and shortage of CMs cannot disappear if we integrate CMs. Therefore, the accuracy of only integration CMs is still lower.Figure 10
The results of integration CMs and ICMs.Table5 presents the evaluation results of different methods. Based on863 Program (number 2012AA112001), we investigate the bogie system of CRHX in-depth, communicate with maintenance personnel, and attain the ranking of important components for using the enterprise, as shown inPractical recognition in Table 5. From the expert experience viewpoint, the most important nodes in the holistic topological network of a bogie system are v1, v7, v16, and v14, respectively. In the view of ICMs, these centrality measures can be further extended by considering suitable properties to identify the important components with respect to function and topology of the nodes and edges (in Figure 6), while, using fuzzy integral fusion ICMs, some nodes (such as v6) with low values can be also identified as the most important nodes. In addition, the evaluation result of these methods is a larger gap, especially noncritical components. These methods, such as AHP and TOPSIS, are greatly affected by human factors. The accuracy of the most critical components identification results is higher, but other components have great difference on human factors. CMs, which include DC, CC, BC, and EC, identify the critical components of structure. Therefore, there are some differences in the results of CMs and other measures. In fact, there are still differences of using Sugeno integral and Choquet integral to identify influential components. The most important components are the same, while the ranking of noncritical components is different. This is because the amount of information loss exceeds the threshold value.Table 5
The evaluation results of different methods.
Number
AHP
TOPSIS
DC
CC
BC
EC
Practical recognition
Sugeno integral
Choquet integral
g
λ
k-additive
g
λ
k-additive
(1)
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
v
1
(2)
v
16
v
7
v
31
v
7
v
7
v
25
v
7
v
7
v
7
v
7
v
7
(3)
v
14
v
14
v
7
v
25
v
14
v
31
v
14
v
6
v
14
v
14
v
14
(4)
v
31
v
9
v
14
v
14
v
31
v
7
v
16
v
14
v
16
v
16
v
16
(5)
v
7
v
16
v
10
v
6
v
25
v
30
v
31
v
29
v
31
v
9
v
9
(6)
v
9
v
8
v
6
v
5
v
16
v
6
v
9
v
16
v
25
v
31
v
31
(7)
v
19
v
12
v
16
v
8
v
2
v
26
v
25
v
2
v
2
v
32
v
25
(8)
v
25
v
23
v
26
v
16
v
10
v
5
v
6
v
8
v
10
v
6
v
6
(9)
v
2
v
22
v
2
v
29
v
6
v
8
v
32
v
5
v
6
v
25
v
22
(10)
v
10
v
21
v
8
v
2
v
11
v
14
v
22
v
20
v
29
v
22
v
32With the reduction of the importance of nodes, the difference of all methods in Table5 for importance of nodes is bigger in Figure 11. For example, the results of the three methods, such as AHP, DC, and EC, are not consistent with practical recognition for node v7. And, for nodes v14, v16, and v31, the number of methods which are not different from practical recognition, is 4, 7, and 9, respectively. We can see that most methods are effective in identifying the most critical nodes. However, the validity of the identification results of the other nodes is poor.Figure 11
The ranking of partly nodes.The methods of integration multimeasures are able to overcome the randomness and uncertainty by using a single measure. Different methods of comprehensive multimeasures are selected, and the accuracy of the identification results is different. Figure12 presents the accuracy rate of these methods in comparison with practical recognition. The accuracy rate ARi=RNi/N, where RNi indicates the number of nodes in the ith method which the ranking is consistent with the expert experience, and Ni is the number of nodes in system. We can find that Choquet integral by using k-additive has a higher accuracy rate. The method of integration ICMs with Sugeno integral misses much information in computational procedure. If the amount of information loss exceeds the threshold value, the results will be not accurate.Figure 12
The accuracy rate of these methods.Through the exploratory discussion above, it is shown that the results acquired from comprehensive measure are, to a certain extent, more reasonable and powerful than those traditional importance measures. Just as mentioned, the system function represents the interactions between components; system topology represents the structure relationship. Therefore, the comprehensive gives consideration to both topological features and physical characteristics of a holistic topological network from multiple perspectives.


## 6. Conclusions

This paper integrates the literature of mechatronic architecture and complex networks to define holistic topological network. And, based on the notion of complex networks, meaningful improved centrality measures (ICMs) are first brought and then comprehensive measure, with first-time, is constructed to identify important components by integrated multi-ICMs. Indeed, construction ICMs with the consideration of functional and topological properties and their relationship is the originality and novelty of proposed measures. Next, integration multi-ICMs based on fuzzy integral, that is, the combination of multiple influencing factors, is also the novelty of comprehensive measure. This paper has also shown the application of the proposed approach in reliability assessment of bogie system of CRHX EMUs. By applying the comprehensive measure, the components importance of bogie system can be evaluated at reasonable human factors. Results indicate that the ranking of critical components can be not the same in selecting different fuzzy integrals. According to the applicable environment, reasonable choices can be determined. In addition, three conclusions are drawn through an exploratory discussion:(i)
The function and topology are of the same importance in electromechanical system. If identifying critical components, these two aspects should be taken into account.(ii)
The method of integration ICMs with Choquet integral by usingk-additive has a higher accuracy rate than other methods.(iii)
The result of comprehensive evaluation is better than that of single measure identification.Of course, due to the diversification and complexity of the real electromechanical system, the model presented here is just a simplification of what happens in actual systems. Several influential factors of critical components in the model need to be further developed if some additional information can be acquired. As previously mentioned, the robustness of comprehensive measure with respect to the fault propagation damping parameter is still under discussion and valuable for further research.



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*Source: 7842596-2017-08-17.xml*