A new approach to fuzzy dynamic fault tree analysis using the weakest n-dimensional t-norm arithmetic

Ge JIANG,Hongjie YUAN,*,Peichng LI,Peng LI

aSchool of Reliability and Systems Engineering,Beihang University,Beijing 100083,China

bChina Ship Development and Design Center,China Shipbuilding Industry Corporation,Wuhan 430064,China

KEYWORDS Fuzzy dynamic fault tree analysis;Fuzzy theory;Reliability evaluation;Sequential binary decision diagrams;The weakest n-dimensional tnorm arithmetic

Abstract Dynamic fault tree analysis is widely used for the reliability analysis of the complex system with dynamic failure characteristics.In many circumstances,the exact value of system reliability is difficult to obtain due to absent or insufficient data for failure probabilities or failure rates of components.The traditional fuzzy operation arithmetic based on extension principle or interval theory may lead to fuzzy accumulations.Moreover,the existing fuzzy dynamic fault tree analysis methods are restricted to the case that all system components follow exponential time-to-failure distributions.To overcome these problems,a new fuzzy dynamic fault tree analysis approach based on the weakest n-dimensional t-norm arithmetic and developed sequential binary decision diagrams method is proposed to evaluate system fuzzy reliability.Compared with the existing approach,the proposed method can effectively reduce fuzzy cumulative and be applicable to any time-tofailure distribution type for system components.Finally,a case study is presented to illustrate the application and advantages of the proposed approach.

1.Introduction

Fault Tree Analysis(FTA)is commonly used to evaluate reliability of the complex and large scaled systems.Dynamic Fault Trees Analysis(DFTA)extend traditional FTA by defining a set of dynamic logical gates.1,2The early approach to DFTA is Markov-based method3,4,which is vulnerable to the problem of well-known state space explosion.To mitigate this predicament,some effective methods have been proposed,such as modularization methods5,6,lumping Markov chains7,8and compositional methods.9,10Such methodologies can heavily reduce the state space scale of a system to be studied under certain circumstances.However,they cannot settle the state space explosion problem fundamentally.The Sequential Binary Decision Diagrams(SBDD)approach,as one of the most ef ficient quantitative techniques,has been developed to solve the problem of state space explosion and make a great success in DFTA.11–14Meanwhile some researchers devote to the reliability analysis of complex dynamic systems based on multivalued decision diagram method.15,16In the DFTA,the failure probabilities or failure rates of components are usually considered as exact values.

In many circumstances,it is difficult to obtain precise failure probabilities or rates of the components due to insufficient data or vague characteristic of the basic events.In order to solve the problem of uncertainty in real applications,fuzzy set theory is employed to handle imprecise failure information,which was introduced by Zadeh in 1965.17Tanaka et al.18conducted pioneering research on Fuzzy Fault Tree Analysis(FFTA).On the basis of this work,many researchers utilized different approaches to analyze fuzzy system reliability.19–22In the papers mentioned above,the system reliability is evaluated via the fuzzy arithmetic operations based on sup-min convolution,which results in the fuzzy accumulation.In order to lessen the increase,Hong and Do23proposed the weakestt-norm arithmetic forfuzzy operationsbased on the sup-t-norm convolution instead of sup-min convolution,and then the weakestt-norm arithmetic has been successfully applied to the fuzzy reliability analysis in many fields.24–27

However,the weakestt-norm arithmetic is only applicable for the standard FTA,and hardly handles the DFTA.In recent years,some studies have been developed about fuzzy dynamic fault tree analysis.Li et al.28proposed an approach based on Markov model and extension principle to fuzzy reliability analysis for Computer Numerical Control(CNC)hydraulic systems.But this approach is limited to the timeto-failures of system components obeying exponential distributions and may cause the growing phenomena of fuzziness.Duan and Fan29utilized the combination of the Bayesian network and defuzzification technology to evaluate the reliability ofdatacommunication system.Yang30analyzed fuzzy dynamic fault tree through the combinational method of minimum cut sequence and triangular fuzzy number.Li et al.31proposed a new method for fuzzy dynamic fault tree analysis based on continuous-time Bayesian networks.Kabir et al.32proposed a method that combines expert elicitation and fuzzy set theory with Pandora Temporal Fault Trees(TFT)to enable dynamic analysis of systems.However,these methods above can only apply to systems with exponential distributions components and hardly deal with the case of non-exponential distributions for elements.

In this paper,a new approach to fuzzy dynamic fault tree analysis is proposed based on the developed SBDD and the weakestn-dimensionalt-norm arithmetic.The developed SBDD is used to model the dynamic fault tree,and the weakestn-dimensionalt-norm arithmetic is employed to fuzzy operation of the failure rates of basic events and calculated fuzzy probability of top event.The proposed method can be applied to any time-to-failure distribution type for components and reduce accumulating phenomena of fuzziness.

This paper is organized as follows:Section 2 presentsndimensionalt-norm and the weakestn-dimensionalt-norm arithmetic;the developed SBDD-based method is elaborated in Section 3;Section 4 proposes the method for fuzzy dynamic fault tree analysis based on the weakestn-dimensionalt-norm and developed SBDD-based method;Section 5 provides a validation example to illustrate the proposed approach; finally,Section 6 is devoted to the conclusion.

2.Fuzzy operations based on the weakest n-dimensional t-norm

2.1.Traditional fuzzy operations

The concept of fuzzy set theory was introduced by Zadeh17to deal with uncertain or vague information.A fuzzy set~Adefined on a universe of discourseXis characterized by a membership function,which takes values from the interval[0,1].A membership function provides a measure of the degree of similarity of an element inXto the fuzzy subset.

where μ~A（x）is the membership function ofX,which belongs to the fuzzy set

The early fuzzy arithmetic operations based on extension principle or interval theory may result in fuzzy accumulation.To mitigate this predicament,new fuzzy arithmetic operations based ont-norm are developed.Thet-norm is a binary operationTon the unit interval[0,1],which satisfied the axioms of commutativity,associativity,monotonicity and boundary condition.The weakestt-normTwas an important kind oft-norm is widely used due to its shape preserving characteristics while applying fuzzy arithmetic operations.

ForPositiveTrapezoidalFuzzyNumbers(PTFNs),the fuzzy arithmetic operations based onTware shown in Table 1,and the details can be studied in Refs.33–35

2.2.n-dimensional t-norm

The existing fuzzy operations such as extension principle andt-norm are only applied for the addition,multiplication,subtraction and division arithmetic of two fuzzy numbers.However,in practice,the fuzzy operation may be taken for a general function with multiple fuzzy variables.For example,the time-to-failure of an element follows Weibull distribution with three parameters,and the scale parameter,shape parameter and position parameter are all fuzzy numbers.In this case,the mission reliability evaluation of the elements will be a hugechallenge.In order to solve the problem,a new fuzzy operation arithmetic based on the weakestn-dimensionalt-norm is proposed,which can be applied for arbitrary monotonous function with multiple fuzzy variables.

Table 1 Fuzzy operations based on the weakest t-norm operations.

Before elaborating on the new fuzzy operation arithmetic,n-dimensionalt-norm is introduced,which is an extension of traditional triangular norm.To facilitate the definition forn-dimensionalt-norm,several related symbols and variables are introduced.

Let ‘≤” be a partial order forIn,whereIis a unit interval[0,1].For α = （x1，x2，···，xn）∈In,β = （y1，y2，···，yn）∈In

where （In，≤）is a completely distributive lattice.

Given three transformationpi,qi，sijfor α:In?In,where‘?”is a transformational symbol.

An-ary functionT:Incan be called as an-dimensionalt-norm(short forT-norm)if the following four axioms are satisfied.

(1)Commutativity

(2)Associativity

(3)Monotonicity

(4)Boundary condition

It is obvious that the traditional triangular norm is a special case ofT-norm whenn=2.Moreover,eachT-norm may be shown to satisfy the following inequalities:

2.3.The weakest n-dimensional t-norm arithmetic

In this part,a new fuzzy operation arithmetic based on the weakestn-dimensionalt-norm is detailed.

For a functionf（A1，A2，···，Ak，···，An）（k=1，2，···，n）

where

It is important to note that the function operations of fuzzy numbers byTWmay change the original shape of the fuzzy numbers.In order to calculate conveniently,the fuzzy operation results are assumed to be approximate trapezoidal fuzzy numbers.

When α-cuts at level α ∈ (0,1)is considered,theTWfuzzy operations turn to be

wheref1+ α（f2-f1）is the lower bound of fuzzy result,f4- α（f4-f3）is the upper bound.

Compared to traditionalt-norm,the weakestn-dimensionalt-normTWcan be applied to any fuzzy arithmetic operations betweennvariables,not limited to addition,subtraction,multiplication,and division fuzzy operations.On the other hand,an advantage of fuzzy arithmetic operations using theTWis that they give smaller fuzzy accumulation within uncertain environment,and the advantage can be used for decreasing growing phenomena of fuzziness in reliability assessment of complex systems.

3.Developed SBDD-based method

With the increasing of system,system componentsusually have dynamic failurescharacteristics.To model the dynamic characteristics,a dynamic fault tree analysis method is proposed,which uses dynamic logic gates to describe the dynamic behaviors.In practice,Priority AND(PAND)gate and Warm Spare Gate(WSP)are widely used in systems design.Some achievements have been made in the study of PAND gate and WSP.36–38Therefore,this paper will focus on the dynamic fault tree with PAND and WSP.The SBDD-based dynamic fault tree analysis method as a quantitative tool has been widely used to evaluate system reliability.The SBDD-based method includes three processes:the conversion ofthe dynamic faulttree,the establishment of SBDD model and the analysis evaluation of SBDD.

In the SBDD,the conversions for PAND and WSP are used as different signs.The sign ‘→” is used for the sequential event of PAND while the signis applied for the sequential event of WSP.The approach greatly increased the complexity of failure paths simplification and model evaluation.

In this paper,a developed SBDD-based method is proposed.The symbols for the conversions of PAND and WSP are unified.For both of PAND and WSP,we usually use the sign ‘→” to denote the failure order of the basic events.The conversion rules and establishment of SBDD model are the same to the SBDD-based Method.The details can be seen in articles.11–13The difference is reflected in the model evaluation.For the developed method,the complexity of failure paths simplification will be greatly reduced.The probability calculation of sequential events will be elaborated as follows.

(1)Sequential events ‘A → B” with two basic events

IfAandBare inputs of a PAND gate,letfA（t）andfB（t）represent the probability density function respectively.In the mission timet,the probability ofA→Bis

IfAandBare inputs of a WSP.Ais the primary component and the probability density function isfA（t）;Bis the spare component and the probability density function isfB（t）in the working state,while the probability density function isfB，α（t）in the standby state.In the mission timet,the occurrence probability ofA→Bis

And the occurrence probability ofB→Ais

(2)Sequential events ‘A1→ A2→ ···→ An” withnbasic events

For the basic eventsA1,A2，...，An，the probability density function isfi（t）(i=1，2，···，n)respectively.

If anyone is not a spare component,the probability ofA1→A2→ ···→Anin the mission timetis

If theAiis a spare component,and the corresponding primary component isAj.The probability density function ofAiisfi（t）in working state andfi，α（t）in standby state.

In the case ofi＜j,the occurrence probability of sequential events is

In the case ofi＞j,the occurrence probability of sequential events is

4.Fuzzy dynamic fault tree analysis

4.1.Fuzzy failure probability of systems

The dynamic fault tree of a system contains basic eventsA1，A2，···，An,the time-to-failures of which follow exponential distribution,Weibull distribution etc.It is assumed that eventAiobeys exponential distribution while eventAjfollows Weibull distribution.The failure probability density functions of the basic events are shown as follows:

where λiis the failure rate of eventAi,mjis the shape parameter of eventAjwhile ηjis the scale parameter.Considering the uncertainty of distribution parameters,λiand ηjare supposed to be PTFNs= （λi1，λi2，λi3，λi4）and~ηj= （ηj1，ηj2，ηj3，ηj4）.

Using the developed SBDD-based method,the failure probability of the systems can be obtained

wheregis a function.

For an element with exponential distribution,the failure probability of the component will be increased during the mission time when the failure rate is increased,i.e.dAi/dλi＞ 0;for an element with Weibull distribution,when the shape parameter is constant,the increase of the scale parameter will make failure probability of the component decrease,i.e.dAj/dηj＜ 0.Moreover,the system is often easier to fail when the components have higher failure probabilities.Therefore,

Using the weakestn-dimensionalt-norm arithmetic,the fuzzy failure probability of systems can be gotten.

For instance,whenn=4，k=2，the fuzzy failure probability of systems

where

When the time-to-failures of the events are normal distribution,lognormal distribution and others,a similar approach can be used to obtain the system fuzzy failure probability.

4.2.Importance analysis

Besides the failure probability of top event,importance analysis is also important for quantitative analysis of fault tree.The importance reflects the contribution of a basic event to top event occurrence,and a higher importance indicates that a basic event has more significant influence on the system failure.Therefore,importance analysis is helpful to look for weak links of the system,and provides improvement measures to improve system reliability.

If the failure probability of basic eventAkis a PTFN= （pk1，pk2，pk3，pk4）(k=1，2，···n),then the failure probability of top event is

It is assumed that the elementAkwill never fail,in other words,the failure rate λiturns to be zero if the life distribution ofAkis exponential distribution while the scale parameter ηjchanges to positive infinity ifAkbelongs to Weibull distribution,in the circumstances,the fuzzy failure probability of top event changes to be

In order to quantitatively evaluate the influence of the basic event on the top event,a new importance is introduced and defined as follows

5.Case study

In this paper,a missile control cabin is used to validate the effectiveness of the proposed method.The system of missile control cabin is an important part of a missile.Analyzing the structure and function of the control cabin and combining with the existing failure information,the main failure models can be obtained,as shown below:Ais sealing element failure;Bis initiating explosive device failure;Cis key electronic device failure;Dis key electronic spare device failure;Eis gyroscopic device failure;Flaser detector failure.

Analyzing the logical relationship of failure models,the dynamic fault tree of the missile control cabin can be established as shown Fig.1.

Fig.1 Fault tree of the missile control cabin.

First,the developed SBDD method is used to dynamic fault tree analysis of missile control cabin.Using the conversion method for dynamic logic gate,the new fault tree with the sequential event is obtained as Fig.2 shows.

Assume index（A→B）＜ index（C→D）＜ index（D→C）＜index（F）＜ index（B）＜ index（E）,the SBDD of the system can be established as Fig.3 shows.

From Fig.3,the disjointed failure paths are gotten as follows:

Using the extended Boolean operation to simplify the failure paths,the occurrence probability of each path is shown as follows:

Therefore,the system reliabilityPTcan be obtained

The fuzzy reliability of the missile control cabin is evaluated in two cases.One case is that all components follow exponential distributions,the other case is that some components follow Weibull distributions and some elements obey exponential distributions.

Fig.2 Fault tree of system after conversion.

Fig.3 SBDD of system.

5.1.Exponential distributions

Assume all components follow exponential distributions.Considering the fuzzy uncertainty due to the insufficient data or vague failure characteristic,the distribution parameters are assumed to be PTFNs instead of the exact values,and the values of parameters are shown in Table 2.For the warm spare structure,the dormancy factor α equals to 0.5.

Table 2 Values of distribution parameters.

Therefore,the probability of top event occurrence is

Using the proposed fuzzy dynamic fault tree analysis method,the fuzzy probability of top event is

And

Using theTWarithmetic,the fuzzy probability of the system failure at timet=10000 h andt=12000 h can be calculated to be PTFNs(0.1649,0.1773,0.193,0.205),(0.2023,0.2165,0.2356,0.2492),respectively.And the membership functions of fuzzy failure probability are presented in Fig.4 at different α-cut levels compared with the results based on Ref.28and crisp method.

Moreover,the system reliability changing over time can be gotten based on the proposed method,Ref.28method and crisp method,shown in Fig.5.

From Fig.4,it is obvious that the proposed method can effectively reduce growing phenomenon of fuzziness compared with the existing method.

The result in Fig.5 shows that the fuzzy reliability of the system based on the proposed method has a smaller upper bound and a larger lower bound,meaning that the proposed approach effectively decreases fuzzy cumulative.Compared with the crisp possibility result,the proposed method can provide detailed reliability information and have less risk for reliability evaluation.

5.2.Non-exponential distributions

According to the failure data collected and expert opinions,the time-to-failures ofA,B,E,Ffollow Weibull distributions andC,Dobey exponential distributions.Considering the fuzzy uncertainty of the parameters,the values of parameters are shown in Tables 3 and 4.For the warm spare structure,the dormancy factor α equals to 0.5.

Fig.4 Failure probability membership of system with exponential distributions at t=10000 h，12000 h.

Fig.5 Reliability curves of system with exponential distributions components change over time by different methods.

Table 3 Values of Weibull distribution parameters.

Table 4 Values of exponential distribution parameters.

Therefore,the probability of top event occurrence is

Using the proposed fuzzy dynamic fault tree analysis method,the fuzzy probability of top event is

Using theTWarithmetic,the fuzzy probability of the system failure at timet=10000 h andt=12000 h can be calculated to be PTFNs(0.2342,0.2510,0.3106,0.3370)and(0.3770,0.4004,0.4726,0.5049),respectively.And the membership functions of fuzzy failure probability are presented in Fig.6 at different α-cut levels compared with the results based on crisp method.

The results in Fig.6 show that the proposed method can be applied to the system components obey non-exponential distributions.

Fig.6 Failure probability membership of system with Weibull distributions components at t=10000 h，12000 h.

Moreover,suppose the mission time bet=10000 h,the importance degree of system components can be obtained by the proposed importance analysis method as shown in Table 5.

From Table 5,it is obvious that the initiating explosive device is the most important component in the missile control cabin system as respect to the overall reliability of the system.If the reliability of the system is to be improved,then the efforts can be concentrated on improving the reliability of the initiating explosive device and sealing element first.

Besides,the system reliability changing over time can be gotten based on the proposed method,as shown in Fig.7.

The fuzzy reliability evaluation results of the missile control cabin in two cases show that the proposed method can reduce accumulating phenomena of fuzziness effectively and be applicable to any time-to-failure distribution type for components.

Table 5 Importance degree of system components.

Fig.7 Reliability curves of system with Weibull distributions components change over time by proposed methods.

6.Conclusions

In this paper,a new fuzzy dynamic fault tree analysis method is proposed and applied to evaluate the reliability of a missile control cabin system under uncertain environment.The weakestn-dimensionalt-norm is presented to operate the function of fuzzy numbers and introduced to the fuzzy fault tree analysis.The developed SBDD method is used to model the dynamic behaviors in the system.The membership of the system failure probability at the arbitrary time can be gotten.Moreover,the system fuzzy reliability and importance rank of system components during the mission time based on proposed importance measure approach can be obtained.Compared with the existing methods,the proposed method can effectively reduce fuzzy cumulative and apply to any time-tofailure distribution type for system components.The result indicates that the proposed method is a promising approach to reliability analysis of dynamic system considering fuzzy uncertainty.The fuzzy dynamic fault tree analysis to the more complex systems,such as Multi-state systems and components obeying Weibull distributions with fuzzy shape parameter,will be investigated in our future work.

Acknowledgments

This work is supported by the National Defense Basic Scientific Research program of China(No.61325102).The authors are thankful to the editor and reviewers for their valuable comments that help us improve the quality of the article.