Discrete Dynamics in Nature and Society

Volume 2017 (2017), Article ID 7842596, 18 pages

https://doi.org/10.1155/2017/7842596

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

^{1}State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China^{2}School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China^{3}Beijing Research Center of Urban Traffic Information Sensing and Service Technologies, Beijing Jiaotong University, Beijing 100044, China

Correspondence should be addressed to Shuai Lin

Received 13 February 2017; Accepted 22 June 2017; Published 17 August 2017

Academic Editor: Manuel De la Sen

Copyright © 2017 Shuai Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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], -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. Section 2 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 define as a* holistic topological network*where is a set of nodes and each node represents a component. is the number of nodes. For example, the number of nodes is 7 for a system in Figure 1. is a set of links and represents physical connection between nodes and . Depending on the nature or type of the topological property, this property may be reflexive in that . is the properties of the set of nodes, is the functional properties of , and is the topological properties of . is the properties of the set of edges and is the functional properties of edge .