Complexity

Volume 2019, Article ID 1046054, 8 pages

https://doi.org/10.1155/2019/1046054

## Fault Tree Interval Analysis of Complex Systems Based on Universal Grey Operation

^{1}School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710129, China^{2}No. 203 Research Institute of China Ordnance Industries, Xi’an 710065, China

Correspondence should be addressed to Feng Zhang; nc.ude.upwn@ydniwupwn

Received 24 July 2018; Revised 1 November 2018; Accepted 4 December 2018; Published 1 January 2019

Academic Editor: Xinggang Yan

Copyright © 2019 Feng Zhang 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

The objective of this study is to propose a new operation method based on the universal grey number to overcome the shortcomings of typical interval operation in solving system fault trees. First, the failure probability ranges of the bottom events are described according to the conversion rules between the interval number and universal grey number. A more accurate system reliability calculation is then obtained based on the logical relationship between the AND gates and OR gates of a fault tree and universal grey number arithmetic. Then, considering an aircraft landing gear retraction system as an example, the failure probability range of the top event is obtained through universal grey operation. Next, the reliability of the aircraft landing gear retraction system is evaluated despite insufficient statistical information describing failures. The example demonstrates that the proposed method provides many advantages in resolving the system reliability problem despite poor information, yielding benefits for the function of the interval operation, and overcoming the drawback of solution interval enlargement under different orders of interval operation.

#### 1. Introduction

The fault tree analysis (FTA) method is typically applied as the main method in the reliability analysis of large systems [1–3]. A fault tree is a logical block diagram composed of a top event (outcome), intermediate events, and bottom events and is used to describe the internal functional logical relationship between events. The logical relationships between the components of a system and their events are obtained based on the operating principle and fault mechanisms of the system. The top-level failure probability of the system can be obtained through the logical relationship between event layers and through data operations using the failure probability statistics of the underlying components of the system. Then, a system-level reliability evaluation can be performed [4–7]. In recent studies, researchers have made major achievements in theoretical system and engineering reliability analyses based on fault trees. The FTA method based on a probability model has seen wide application in several systems engineering fields such as aviation, aerospace, and nuclear power [8–12].

The traditional fault tree analysis method is based on a probability model; when there is a large set of failure samples and other sufficient statistical information describing the evaluated parts of a system, the uncertainty of bottom events can be quantified independently [5, 12]. However, when only small sample sets are available in an engineering analysis case, the statistics describing component failure are insufficient to accurately estimate failure distribution [13, 14]. It is thus difficult to determine the failure probability of components in many kinds of complex systems, such as landing gear retraction systems and large-scale space-borne antennae. This limits the application of the probabilistic model-based fault tree method in complex engineering applications.

Based on the fuzzy set theory, the fuzzy fault tree describes the probability of event occurrence using various fuzzy numbers, addressing the difficulties associated with precisely measuring the probability of base event occurrence due to the complexity of the environment and incomplete data [15–18]. Ding and Lisianski regarded the performance rate and corresponding probability of an event as fuzzy values and developed a reliability evaluation technique for a multistate system using the fuzzy universal generation function [19]. Li et al. introduced random fuzzy variables and proposed a hybrid universal generation function [20]. Liu and Huang proposed a fuzzy continuous-time Markov model with a finite discrete state and used it to evaluate the fuzzy state probability of multistate elements at any time [21]. However, in the fuzzy fault tree, the determination of the fuzzy value, fuzzy variable, and fuzzy state probability is highly subjective.

The interval domain is an important model in nonprobability theory: the shape of an interval domain represents the degree to which events occur in an interval model, while the size of the interval domain signifies the volatility or degree of deviation of an uncertain event. To establish an interval model, only the boundaries of an event set are required, not its internal distribution. This results in significant independence from the data compared to a conventional probability model [22–27]. However, it should be pointed out that the power exponentiation of an interval number will lead to the expansion of the interval and that different orders of operation performed on the same interval numbers can provide different expansion intervals [28, 29].

The universal grey number provides the function of the interval operation and overcomes the drawback associated with traditional interval operation, i.e., the change in solution interval with order of operation [29]. Some scholars have gradually introduced and successfully applied the universal grey operation to structural reliability research [30–33]. Luo introduced the grey range transformation into the process of model building to eliminate the incomparability of different dimensions and achieved an effective risk assessment of the ice plug phenomenon [30]. Jin et al. proposed a generalized Rayleigh quotient method based on generalized grey mathematics to represent the interval parameters in uncertain structures using generalized grey numbers [31]. Liu et al. considered the uncertainty of the interval arithmetic for the structural, nonprobabilistic reliability calculation of nonlinear systems, using the universal grey number instead of the interval parameters to overcome the impact of interval arithmetic uncertainty on reliability results [32].

Based on the advantages of the universal grey number method, a new method for solving the reliability of the top event of a fault tree is proposed in this paper to overcome the shortcomings of the existing nonprobabilistic reliability method of interval operation. The proposed method complies with the conversion rule between the interval number and universal number, and the four arithmetic operations of the universal grey number.

#### 2. Interval Analysis of Simple System Fault Trees

##### 2.1. Four Arithmetic Operations of Traditional Interval Analysis

Let ‘’ represent a real binary operation on the set of real numbers, . For and , the binary operation on interval set is defined as follows:

The four arithmetic operation rules can then be derived as follows [27]:

It can be seen in (2) that the calculation of the interval number provides an extremely wide range due to the influence of interval expansion. This is the primary drawback of the interval method.

##### 2.2. Four Arithmetic Operations of Interval Analysis Based on the Universal Grey Operator

Setting the domain as (the set of real numbers), the universal grey number set in is denoted by . Calling an element in the universal grey number, , and , where is the observed value and is the grey information portion of .

The corresponding four arithmetic operation rules are accordingly [27]

In practical applications, the universal grey number and interval number can be interchanged with each other via conversion. For the grey number , the corresponding interval number is in the form of . The interval number can be uniformly expressed as for ease of operation.

##### 2.3. Interval Analysis of Fault Trees Based on Universal Grey Operator for an OR Gate Operator

A fault tree interval analysis based on the universal grey operator is performed by using an OR gate operator with three bottom events as an example. The fault tree is shown in Figure 1.