Cluster Computing

, Volume 22, Supplement 3, pp 6563–6577

# Study on the construction and application of discrete space fault tree modified by fuzzy structured element

Article

## Abstract

Some fault data in an actual system operation has the strong discretization and big data characteristics. Meanwhile, the external factors affect the system reliability, and the change of factors may lead to the change of system reliability. At present, the methods in safety system engineering lack the ability to process the multi-factor influence and fault big data simultaneously. But these are the general problems that the actual system reliability analysis must face to be resolved. In order to solve the problems, on the basis of Discrete Space Fault Tree (DSFT), the Fuzzy Structured Element method is introduced to construct the Fuzzy Structured Element Discrete Space Fault Tree (EDSFT). The method can analyze the multi-factor influence on system reliability with DSFT, and use Fuzzy Structured Element (E) to denote the discrete characteristics of fault big data. The results of EDSFT with E can preserve the characteristics of the original fault data distribution and lay the foundation for the analysis of fault big data. The research is particularly suitable for the analysis of system reliability under the fault big data and multi-factor influence.

## Keywords

Safety system engineering Discrete Space Fault Tree Fuzzy Structured Element Multi-factor influence Fault big data Reliability analysis

## List of symbols

$$P_{i}^{{d_{k} }} (x_{k} )$$

Characteristic function of component fault probability

i

ith component

dk

kth factor name

xk

Value of factor dk, k = 1–n

n

Number of influencing factors

$$P_{i} (x_{1} ,x_{2} , \ldots x_{n} )$$

Component fault probability distribution

$$P_{T} (x_{1} ,x_{2} , \ldots x_{n} )$$

System fault probability distribution

$$P_{i}^{{d_{k} }}$$

Component fault probability distribution trend

$$P_{T}^{{d_{k} }}$$

System fault probability distribution trend

$$I_{g} (i)$$

Probability importance distribution

$$I_{g}^{c} (i)$$

Criticality importance distribution

$$FI_{i} (x_{1} ,x_{2} , \ldots x_{n} )$$

Component factor (joint) importance distribution

$$FI_{T} (x_{1} ,x_{2} , \ldots x_{n} )$$

System factor (joint) importance distribution

$$ZI_{g} (i)$$

Component domain probability importance

$$ZI_{g}^{c} (i)$$

Component domain criticality importance

$$P_{1}^{t} (t)\;{\text{and}}\;P_{1}^{c} (c)$$

Characteristic function of X1 for t and c respectively

$$P_{i}^{t} (t)\;{\text{and}}\;P_{i}^{c} (c)$$

Characteristic function of Xi for t and c respectively

C

Sinusoid cycle

c0

Horizontal offset of sine curve

A

Vertical offset of sine curve

λ

Fault rate

$$P_{1 - 5}$$

Abbreviation for $$P_{1 - 5} (t,c)$$, respectively, the fault probability of component $$X_{1 - 5}$$

$$\prod$$

‘and’ relationship among the components

$$\coprod$$

‘or’ relationship among the components

$$\tilde{f}$$

Fuzzy valued function on X

E(x)

Membership function

$$\tilde{f}(x) = g(x,E)$$

Fuzzy valued function generated by Fuzzy Structured Element E

$$\bar{x}\;{\text{and}}\;\bar{y}$$

Average value of observation data

$$\hat{\sigma }^{2}$$

Average dispersion of variable y

N

Number of data

$$\hat{a}_{k} \sim \hat{a}_{0} \;{\text{and}}\;\hat{b}_{p} \sim \hat{b}_{0}$$

Coefficients determined by fitting

α and β

Maximum fitting times

E

Fuzzy Structured Element

$$\hat{f}(x)$$

Kernel function of fuzzy valued function

$$\hat{\omega }(x)$$

Fluctuation of data, $$\hat{\omega }_{i}^{t} (t)$$ and $$\hat{\omega }_{i}^{c} (c)$$ are the fluctuation of data for t and c respectively

$$\hat{f}_{i}^{t} (t)\;{\text{and}}\;\hat{f}_{i}^{c} (c)$$

Kernel function of Xi for t and c respectively

$$\mathop {\hat{f}_{i}^{t} }\limits_{\text{U}} (t),\;\mathop {\hat{f}_{i}^{t} }\limits_{\text{D}} (t)\;{\text{and}}\;\mathop {\hat{f}_{i}^{c} }\limits_{\text{U}} (c),\;\mathop {\hat{f}_{i}^{c} }\limits_{\text{D}} (c)$$

Up and down envelope function of Xi for t and c respectively

$$f(E) \to E(x)$$

$$f(E)$$ After Fuzzy Structured Element, the domain where the variable falls in $$E(x)$$, $$- 1 \le x < 0$$, x = 0, $$0 < x \le 1$$

$$\tilde{F}_{i}^{t} (t)\;{\text{and}}\;\tilde{F}_{i}^{c} (c)$$

Fuzzy Structured Element Characteristic Function of Xi for t and c respectively

$$\tilde{F}_{i} (x_{1} ,x_{2} , \ldots x_{n} )$$

Fuzzy Structured Element Component Fault Probability Distribution of Xi, such as $$\tilde{F}_{i} (t,c)$$

$$\tilde{F}_{T} (x_{1} ,x_{2} , \ldots x_{n} )$$

Fuzzy Structured Element System Fault Probability Distribution, such as $$\tilde{F}_{T} (t,c)$$

$$\tilde{I}_{g} (i)$$

Fuzzy Structured Element Probability Importance Distribution

$$\tilde{I}_{g}^{c} (i)$$

Fuzzy Structured Element Criticality Importance Distribution

$$\tilde{F}_{T}^{{d_{k} }}$$

Fuzzy Structured Element System Fault Probability Distribution Trend

$$Z\tilde{I}_{g} (i)$$

Fuzzy Structured Element Component Domain Probability Importance

$$Z\tilde{I}_{g}^{c} (i)$$

Fuzzy Structured Element Component Domain Criticality Importance

$$F\tilde{I}_{i} (x_{k} )$$

Fuzzy Structured Element Component Factor Importance Distribution

$$F\tilde{I}_{T} (x_{k} )$$

Fuzzy Structured Element System Factor Importance Distribution

$$F\tilde{I}_{i} (x_{1} ,x_{2} , \ldots x_{n} )$$

Fuzzy Structured Element Component Factor Joint Importance Distribution

$$F\tilde{I}_{T} (x_{1} ,x_{2} , \ldots x_{n} )$$

Fuzzy Structured Element System Factor Joint Importance Distribution

$$\tilde{F}_{1 - 5}$$

Abbreviation for $$\tilde{F}_{1\sim 5} (t,c)$$, respectively, $$P_{1\sim 5} (t,c)$$ modified by Fuzzy Structured Element

$$\mu = 1 - n$$

Number of factors in Factor Joint Importance Distribution

Kj(j = 1,2,…,r)

The jth set in r minimal cut set of fault tree

Ei/Eii

ith/iith basic event in Kj, basic event is equivalent to component

subscript k

Corresponding parameters of the kth factors

c

Using temperature

t

Using time

T

Instance system

X1–5

5 components in the instance system

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