Quantitative analysis of a fault tree with priority AND gates

A method for calculating the exact top event probability of a fault tree with priority AND gates and repeated basic events is proposed when the minimal cut sets are given. A priority AND gate is an AND gate where the input events must occur in a prescribed order for the occurrence of the output event. It is known that the top event probability of such a dynamic fault tree is obtained by converting the tree into an equivalent Markov model. However, this method is not realistic for a complex system model because the number of states which should be considered in the Markov analysis increases explosively as the number of basic events increases. To overcome the shortcomings of the Markov model, we propose an alternative method to obtain the top event probability in this paper. We assume that the basic events occur independently, exponentially distributed, and the component whose failure corresponds to the occurrence of the basic event is non-repairable. First, we obtain the probability of occurrence of the output event of a single priority AND gate by Markov analysis. Then, the top event probability is given by a cut set approach and the inclusion–exclusion formula. An efficient procedure to obtain the probabilities corresponding to logical products in the inclusion–exclusion formula is proposed. The logical product which is composed of two or more priority AND gates having at least one common basic event as their inputs is transformed into the sum of disjoint events which are equivalent to a priority AND gate in the procedure. Numerical examples show that our method works well for complex systems.     

Fault trees analysis using expert opinion based on fuzzy‐bathtub failure rates

The main objective of fault tree analysis method is to estimate the “Top Event occurrence probability”. This requires determination of failure time distribution functions also known as “Bathtub Curves” for each of the system elements/events. This paper introduces a novel method to determine the failure time distribution functions using possibility theory. For this purpose, fuzzy‐bathtub distributions using expert opinions are generated for basic events and fuzzy formulas are derived for static and dynamic gates fault tree constructions. This process completed by proposed fuzzy Monte Carlo simulation throughout the preferred operational time and uses the actual time‐to‐failure data. Accordingly, the Top Event failure curve and the reliability profile of the system are depicted based on the defuzzificated basic‐events' bathtub‐failure‐rates. The results show that the proposed method not only is feasible and powerful but can also be accurate more than the other probabilistic and possibilistic techniques because of the component failure rates follow the real failure distributions.     

A recursive, time-averaged approach for fault tree quantitative analysis

A new approach in fault tree quantitative analysis, based essentially on the recursive evaluation of time-averaged reliability parameters associated with a fault tree, is presented. The methodology is a complete one covering the following problems: the evaluation of averaged unavailabilities or unreliabilities, failure and repair rates and failure and repair intensities associated with the basic events involved in a fault tree, the evaluation of unavailability or unreliability, occurrence rates and occurrence intensities associated with the top event of a fault tree, evaluation of the importance and sensitivity associated with basic events, and implicants according to different definitions. At this stage in the development of the methodology the common cause failures are not considered. Although in practice the presented algorithms have shown, in the cases of large fault trees, difficulties related to the computing speed and memory capabilities of present personal computers, the methodology remains valuable, at least by the new theoretical results.     

Improved accuracy in quantitative fault tree analysis

The fault tree diagram defines the causes of the system failure mode or ‘top event’ in terms of the component failures and human errors, represented by basic events. By providing information which enables the basic event probability to be calculated, the fault tree can then be quantified to yield reliability parameters for the system. Fault tree quantification enables the probability of the top event to be calculated and in addition its failure rate and expected number of occurrences. Importance measures which signify the contribution each basic event makes to system failure can also be determined. Owing to the large number of failure combinations (minimal cut sets) which generally result from a fault tree study, it is not possible using conventional techniques to calculate these parameters exactly and approximations are required. The approximations usually rely on the basic events having a small likelihood of occurrence. When this condition is not met, it can result in large inaccuracies. These problems can be overcome by employing the binary decision diagram (BDD) approach. This method converts the fault tree diagram into a format which encodes Shannon's decomposition and allows the exact failure probability to be determined in a very efficient calculation procedure. This paper describes how the BDD method can be employed in fault tree quantification. © 1997 John Wiley & Sons, Ltd.     

Gastric esophageal surgery risk analysis with a fault tree and Markov integrated model

Reliability methods have been widely used in risk analysis of medical surgeries. In this study, the authors combine a fault tree with Markov models to assess time independent- and dependent factors together. Dynamics are integrated in the traditional fault tree, and meanwhile the processes of solving Markov are simplified with the modular approach. Continuous time Markov chains are adopted in evaluating the failure probability of a gastric esophageal surgery after categorizing basic events in the fault tree, and a certain time dependent variables, such as failure rate of medical equipment, surgery frequency, and rescue timeliness are involved into risk analysis. A case is studied with data collected from a general hospital, to illustrate the operational process of the proposed method. Results based on the inputs show that taking rescue actions into consideration can reduce the gap between the result of fault tree analysis and the reality. Sensitivity analysis for measuring the impacts of the above time relevant variables is conducted, as well as limitations of the Markov model are discussed.     

An enhanced component connection method for conversion of fault trees to binary decision diagrams

Fault tree analysis (FTA) is widely applied to assess the failure probability of industrial systems. Many computer packages are available, which are based on conventional kinetic tree theory methods. When dealing with large (possibly non-coherent) fault trees, the limitations of the technique in terms of accuracy of the solutions and the efficiency of the processing time become apparent. Over recent years, the binary decision diagram (BDD) method has been developed that solves fault trees and overcomes the disadvantages of the conventional FTA approach. First of all, a fault tree for a particular system failure mode is constructed and then converted to a BDD for analysis. This paper analyses alternative methods for the fault tree to BDD conversion process.For most fault tree to BDD conversion approaches, the basic events of the fault tree are placed in an ordering. This can dramatically affect the size of the final BDD and the success of qualitative and quantitative analyses of the system. A set of rules is then applied to each gate in the fault tree to generate the BDD. An alternative approach can also be used, where BDD constructs for each of the gate types are first built and then merged to represent a parent gate. A powerful and efficient property, sub-node sharing, is also incorporated in the enhanced method proposed in this paper. Finally, a combined approach is developed taking the best features of the alternative methods. The efficiency of the techniques is analysed and discussed.     

基于FMEA和FTA的老年人汽车人机界面交互设计研究

目的 提升老年人汽车界面满意度,设计适合老年人使用的汽车界面.方法 提出基于失效模式与效应分析(FMEA)、故障树分析(FTA)的研究方法,首先成立会议小组进行SHERPA分析,绘制汽车界面层次任务分析图;随后运用FMEA找出汽车界面中的潜在失效模式,计算风险优先数值;接着选取风险优先数值高的失效模式,运用FTA寻找其发生的原因,对其以图形化表示;再对故障树进行定性定量分析,计算故障树的最小切集和失效事件发生概率,找到失效发生的核心因素;最后对失效事件进行改良并对改良后的结果进行验证.结论 以老年人汽车界面为例进行研究,该方法可以有效发现老年人操作汽车界面发生错误的原因,不仅可以帮助设计师找到改良的重点,还可以使老年人获得良好的驾驶体验.     

An ordering heuristic to develop the binary decision diagram based on structural importance

Fault tree analysis is often used to assess risks within industrial systems. The technique is commonly used although there are associated limitations in terms of accuracy and efficiency when dealing with large fault tree structures. The most recent approach to aid the analysis of the fault tree diagram is the Binary Decision Diagram (BDD) methodology. To utilise the technique the fault tree structure needs to be converted into the BDD format. Converting the fault tree requires the basic events of the tree to be placed in an ordering. The ordering of the basic events is critical to the resulting size of the BDD, and ultimately affects the performance and benefits of this technique. A number of heuristic approaches have been developed to produce an optimal ordering permutation for a specific tree. These heuristic approaches do not always yield a minimal BDD structure for all trees. This paper looks at a heuristic that is based on the structural importance measure of each basic event. Comparing the resulting size of the BDD with the smallest generated from a set of six alternative ordering heuristics, this new structural heuristic produced a BDD of smaller or equal dimension on 77% of trials.     

A New Ordering Method of Basic Events in Fault Tree Analysis

The ordering of basic events is critical to fault tree analysis on the basis of binary decision diagrams (BDDs). Many attempts have been made to seek an efficient ordering result with the aim of reducing the complexity of BDD. In this article, a new ordering method, namely, priority ordering method, is proposed. The new method takes into account not only the effects of the layers of fault tree but also the repeated events, the neighboring events, and the number of events under the same gate. According to these four effects, the priorities that sort the basic events of the fault tree are defined. The new method inherits the merits of structure‐based and weight‐based methods. It is able to evaluate the basic events on the basis of the structure‐based method and the size of the subtree on the basis of the weighted‐based method. Demonstrated by the examples, the proposed priority ordering method is superior to the existing ordering methods in terms of reducing the nodes in the BDD and improving the efficiency in transforming a fault tree to a BDD. Copyright © 2011 John Wiley & Sons, Ltd.     

New methods to determine the importance measures of initiating and enabling events in fault tree analysis

Components' importance measures play a very important role in system reliability analysis. They are used to identify the weakest parts of the system for design improvement, failure diagnosis and maintenance. This paper deals with the problem of determining the importance measures of basic events in case of unreliability analysis of binary coherent and non-coherent fault trees. This type of analysis is typical of catastrophic top events, characterised by unacceptable consequences. Since the unreliability of systems with repairable components cannot be exactly calculated via fault tree, the Expected Number of Failures - which is obtained by integrating the unconditional failure frequency - is considered as it represents a good upper bound. In these cases it is important to classify events as initiators or enablers since their roles in the system are different, their sequence of occurrence is different and consequently they must be treated differently. New equations based on system failure frequency are described in this paper for determining the exact importance measures of initiating and enabling events. Simple examples are provided to clarify the application of the proposed calculation methods. Compared with the exact methods available in the literature, those proposed in this paper are easier to apply by hand and are simpler to implement in a fault tree analyser.     

荔枝包装的风险识别与分析

目的解决在荔枝包装过程中风险源难于辨识及定量分析的难题。方法提出一种基于霍尔三维因素空间和模糊故障树的风险识别与定量分析方法。构建荔枝包装的安全事件、时空结构(工位)、事故致因等3个维度的因素集,通过矩阵之间的映射关系和计算分析得到风险基本事件集合。建立荔枝包装的故障树模型,并对风险基本事件进行专家问卷评价。采用梯形模糊数及左右模糊排序法将专家的评判语言转化为风险概率值。结果经计算得到荔枝包装事故发生概率(0.0409)及风险基本事件的概率重要度。结论提出了强化标准化建设,加大技术投入等防范风险事故发生的措施。     

基于模糊故障树的军用气象物资包装可靠性分析

应用模糊故障树分析方法对军用气象物资包装可靠性进行了系统分析,简要介绍了模糊故障树分析方法的基本理论,利用专家判断和模糊集理论相结合的方法,评估了故障树底事件发生的模糊失效概率。并以"TFS-1通风干湿表包装"为例,建立了包装系统的故障树,采用下行法求解了引起顶事件发生的最小割集,定量分析计算,得出模糊失效率为0.0705,同时计算了各底事件的重要度。模糊故障树分析方法对于提高军用气象物资包装防护能力,确保物资装备质量,具有非常重要的意义。     

Development and evaluation of an uncertainty importance measure in fault tree analysis

In a fault tree analysis, an uncertainty importance measure is used to identify those basic events that significantly contribute to the uncertainty of the top-event probability. This paper defines an uncertainty importance measure of a basic event or of a group of basic events, and develops a two-stage procedure for experimentally evaluating the measure under the assumption that the probability of each basic event follows a lognormal distribution. The proposed method utilizes the Taguchi tolerance design technique with modifications. Then, the so-called contribution ratios which evaluate the main and/or interaction effects of the uncertainties of log-transformed basic-event probabilities on the uncertainty of the log-transformed top-event probability are calculated. The contribution ratios are used to estimate the defined uncertainty importance measure of a basic event or of a group of basic events. The proposed method consists of two stages for computational efficiency. In the first stage, the basic events with negligible effects on the uncertainty of the log-transformed top-event probability are screened out, and more detailed analyses are conducted in the second stage with a substantially smaller number of basic events. In addition, this paper presents an analysis method to quantify the percentage reduction in the uncertainty of the log-transformed top-event probability when the uncertainty of each basic-event probability is reduced.     

A fault tree analysis strategy using binary decision diagrams

The use of binary decision diagrams (BDDs) in fault tree analysis provides both an accurate and efficient means of analysing a system. There is a problem, however, with the conversion process of the fault tree to the BDD. The variable ordering scheme chosen for the construction of the BDD has a crucial effect on its resulting size and previous research has failed to identify any scheme that is capable of producing BDDs for all fault trees. This paper proposes an analysis strategy aimed at increasing the likelihood of obtaining a BDD for any given fault tree, by ensuring the associated calculations are as efficient as possible. The method implements simplification techniques, which are applied to the fault tree to obtain a set of ‘minimal’ subtrees, equivalent to the original fault tree structure. BDDs are constructed for each, using ordering schemes most suited to their particular characteristics. Quantitative analysis is performed simultaneously on the set of BDDs to obtain the top event probability, the system unconditional failure intensity and the criticality of the basic events.     

Criteria for evaluating protection from single points of failure for partially expanded fault trees

Fault tree analysis (FTA) is a technique that describes the combinations of events in a system which result in an undesirable outcome. FTA is used as a tool to quantitatively assess a system's probability for an undesirable outcome. Time constraints from concept to production in modern engineering often limit the opportunity for a thorough statistical analysis of a system. Furthermore, when undesirable outcomes are considered such as hazard to human(s), it becomes difficult to identify strict statistical targets for what is acceptable. Consequently, when hazard to human(s) is concerned a common design target is to protect the system from single points of failure (SPOF) which means that no failure mode caused by a single event, concern, or error has a critical consequence on the system. Such a design target is common with “by-wire” systems. FTA can be used to verify if a system is protected from SPOF. In this paper, sufficient criteria for evaluating protection from SPOF for partially expanded fault trees are proposed along with proof. The proposed criteria consider potential interactions between the lowest drawn events of a partial fault tree expansion which otherwise easily leads to an overly optimistic analysis of protection from SPOF. The analysis is limited to fault trees that are coherent and static.     

Handling Uncertainties in Fault Tree Analysis by a Hybrid Probabilistic–Possibilistic Framework

Fault tree analysis is a method largely used in probabilistic risk assessment. Uncertainties should be properly handled in fault tree analyses to support a robust decision making. While many sources of uncertainties are considered, dependence uncertainties are not much explored. Such uncertainties can be labeled as ‘epistemic’ because of the way dependence is modeled. In practice, despite probability theory, alternative mathematical structures, including possibility theory and fuzzy set theory, for the representation of epistemic uncertainty can be used. In this article, a fuzzy β factor is considered to represent the failure dependence uncertainties among basic events. The relationship between β factor and system failure probability is analyzed to support the use of a hybrid probabilistic–possibilistic approach. As a result, a complete hybrid probabilistic–possibilistic framework is constructed. A case study of a high integrity pressure protection system is discussed. The results show that the proposed method provides decision makers a more accurate understanding of the system under analysis when failure dependencies are involved. Copyright © 2015 John Wiley & Sons, Ltd.     

基于事故树的深孔爆破边坡稳定性分析

为了能够对爆破后形成的露天边坡稳定性进行合理的分析评价,结合大小鱼山岛露天爆破边坡失稳的情况,采用事故树分析方法,建立了工程爆破现场露天边坡失稳事故树模型。计算出事故树的最小割集96个,最小径集3个以及基本事件的结构重要度系数,并得到基本事件结构重要度排序,找出了导致边坡失稳的主要原因是边坡监测和边坡支护。然后根据基本事件结构重要度的排序,制定合理有序的预防措施。事故树分析方法可以全面阐述露天矿边坡失稳的各种因素和逻辑关系,并通过对结构重要性分析,提出合理预防措施,为爆破露天边坡的安全管理提供参考依据。     

基于事故树的深孔爆破边坡稳定性分析

为了能够对爆破后形成的露天边坡稳定性进行合理的分析评价,结合大小鱼山岛露天爆破边坡失稳的情况,采用事故树分析方法,建立了工程爆破现场露天边坡失稳事故树模型。计算出事故树的最小割集96个,最小径集3个以及基本事件的结构重要度系数,并得到基本事件结构重要度排序,找出了导致边坡失稳的主要原因是边坡监测和边坡支护。然后根据基本事件结构重要度的排序,制定合理有序的预防措施。事故树分析方法可以全面阐述露天矿边坡失稳的各种因素和逻辑关系,并通过对结构重要性分析,提出合理预防措施,为爆破露天边坡的安全管理提供参考依据。     

A practical method for accurate quantification of large fault trees

This paper describes a practical method to accurately quantify top event probability and importance measures from incomplete minimal cut sets (MCS) of a large fault tree. The MCS-based fault tree method is extensively used in probabilistic safety assessments. Several sources of uncertainties exist in MCS-based fault tree analysis. The paper is focused on quantification of the following two sources of uncertainties: (1) the truncation neglecting low-probability cut sets and (2) the approximation in quantifying MCSs. The method proposed in this paper is based on a Monte Carlo simulation technique to estimate probability of the discarded MCSs and the sum of disjoint products (SDP) approach complemented by the correction factor approach (CFA). The method provides capability to accurately quantify the two uncertainties and estimate the top event probability and importance measures of large coherent fault trees. The proposed fault tree quantification method has been implemented in the CUTREE code package and is tested on the two example fault trees.     

基于FTA的老年人APP交互设计

目的提升老年人对APP的满意度,帮助老年人更好地接受和使用APP。方法提出基于故障树分析(FTA)的老年人APP交互设计方法。首先通过焦点小组确定导致错误发生的原因并建立故障树,对错误发生的原因进行合理分类;随后对故障树进行定性定量分析,计算故障树的最小切集和错误发生的概率,找到导致头号事件发生的核心因素;接着进行重要度分析,计算构造重要度,根据构造重要度值对引起错误发生的基本事件进行排序,为设计改良提供理论上的优先顺序;最后以老年人医疗APP为研究案例进行设计改良,并对改良后的结果进行错误率验证。结论该方法可以有效地发现老年人操作APP时出现障碍的原因,不仅可以帮助设计师找到较为合理的改良方案,而且还可以使老年人获得良好的交互体验。