Mathematical Problems in Engineering

Volume 2018, Article ID 8534065, 10 pages

https://doi.org/10.1155/2018/8534065

## Dynamic Importance Analysis of Components with Known Failure Contribution of Complex Systems

School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an, China

Correspondence should be addressed to Yingjie Zhang; nc.ude.utjx@gnahzjy

Received 28 December 2017; Revised 1 March 2018; Accepted 20 March 2018; Published 24 April 2018

Academic Editor: Alessandro Lo Schiavo

Copyright © 2018 Yangfan Li 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

Importance analysis deals with the investigation of influence of individual system component on system operation. This paper mainly focuses on dynamic important analysis of components in a multistate system. Assuming that failure probabilities of system components are independent, a new time integral-based importance measure approach (TIIM) is proposed to measure the loss of system performance that is caused by each individual component. Reversely the importance of a component can be evaluated according to the magnitude of performance loss of the system caused by it. Moreover, the dynamic varying curve of importance of a component with time can be described by calculating criticality of the component at each state. On the other hand, in the proposed approach, the importance probability curve of a component is fitted by using the failure data from all components of system excluding that of the component itself so as to solve the problem of inaccurate fitting caused by small sample data. The approach has been verified by probability analysis of failure data of CNC machines.

#### 1. Introduction

Importance measures have been widely used for identifying system weaknesses and supporting system improvement activities from the design viewpoint. With the importance values of all components, proper actions can be taken on the weakest component to improve system reliability at the minimum cost or effort [1].

Every component of a system has different contribution to the system performance. Investigation of this influence is the major objective of importance analysis, which can be qualitative or quantitative. From the viewpoint of classical reliability, a system is always regarded as binary-state. That is, all components only have two states of “working perfect” or “completely failed.” This representation is suitable for systems where any deviation from perfect functioning may cause a disaster, for example, nuclear power plants [2] or aviation systems [3]. Barlow and Proschan [4] proposed an importance measure for the components in a coherent system and the basis events in a fault tree is defined. However, it is not appropriate for studying systems that work at several performance levels, for example, process investigation of gradually degrading systems. It is also not very suitable for the analysis of a complex system which has many heterogeneous factors such as hardware, software, organizational structure, and human factors [5].

Unlike binary-state systems that are either perfectly working or completely failed, newer approaches show that typical components of a system may work at many performance loss levels from zero to one, each level with a certain probability. Despite a system like CNC machine tools can be called a multistate system (MSS) that has a changing situation over time, many of the results for the binary case can be computed for multistate systems using the binary structure and reliability function concepts [6]. Researchers such as Lisnianski and Levitin [7, 8], Nourelfath et al. [9, 10], Murchland [11], and Hudson and Kapur [12] have studied it in depth.

Birnbaum [13] introduced two basic importance measures (IM), that is, structural importance (SI) and Birnbaum importance (BI), and one of the first generalizations of SI and BI has been proposed for MSS [14]. These papers introduced various definitions of several basic IM [14–17]. Each of the definitions has a specific physical meaning; for example, one type of these measures allows finding component state by its influence degree on the whole system [15], but another one allows quantifying importance of individual states of a given component on a specific system state [14, 16], while there are still some methods that permit investigating the total importance of a given component [17]. In most cases, UGF (universal generating function) approach can be used for importance analyses of high dimensional systems. However, in most cases, the system structural details are unknown, which makes it difficult to use the UGF method because this method is based on a moment generating function that is mathematically defined and depends on the working context and the system structure. It implies that these computing methods add specifics to the original definitions of IM.

In recent years, many researchers have made contributions to multistate system IM analysis. They focused on developing different approaches to permit setting limitations of the calculation methods. For example, the IM for complex systems with multiple components was proposed in [18]; Cadini et al. [19] developed Monte-Carlo method for the analysis of systems with small failure probabilities. Zhang et al. [20] proposed an approach to measure component importance of manufacturing systems based on failure losses. As an application example, a piston production line was chosen for evaluation purpose [21]. Dui et al. [22] extend the integrated importance measure (IIM) from unit time to system lifetime and to different life stages. Levitin et al. [23] proposed an approach based on performance level restrictions to evaluate the commonly used importance measures. Kvassay et al. [24] developed new approach which can be used for calculation of all types of critical states and it is based on the application of direct partial logic derivatives. In addition, many researchers focus on the importance measures of -out-of- system [25, 26]. Moreover, the most comprehensive research of current work in importance analysis was conducted in [27, 28].

This paper aims to seek an efficient method for evaluating component importance of complex system as a multistate system. With the loss of system performance of complex system, one of the open questions is how to accurately calculate the dynamic importance of a component. Traditional importance measures mainly concern the change of system performance caused by the reliability change of components, but they seldom consider the joint effect of probability distributions, transition intensities of the object component states, and the loss of system performance. As a matter of fact, the expected loss of system performance from failures is related to the expected number of component failures and the effect of system structure. It means that some new mathematical approaches for known IM need to be studied. In this paper, considering how the transition of component states affects the system’ mean time to failure (MTTF), we study the time integral importance measure (TIIM) to evaluate the importance of components of system.

The remainder of this paper is organized as follows. The traditional importance measures are analyzed in Section 2. The TIIM of component states is described in Section 3. Section 4 demonstrates the application of TIIM on CNC machine tools. The conclusions and future work are given in Section 5.

#### 2. Traditional Importance Measures

Comparison of traditional methods is shown in Table 1; although several papers [24, 27] have reviewed traditional importance measures, it is seen that these papers cover various categories with different emphases. In this paper, the dynamic characteristics of the traditional methods are compared.