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In this paper the combinations of maximum entropy method and Bayesian inference for reliability assessment of deteriorating system is proposed. Due to various uncertainties, less data and incomplete information, system parameters usually cannot be determined precisely. These uncertainty parameters can be modeled by fuzzy sets theory and the Bayesian inference which have been proved to be useful for deteriorating systems under small sample sizes. The maximum entropy approach can be used to calculate the maximum entropy density function of uncertainty parameters more accurately for it does not need any additional information and assumptions. Finally, two optimization models are presented which can be used to determine the lower and upper bounds of systems probability of failure under vague environment conditions. Two numerical examples are investigated to demonstrate the proposed method.

Degradation is a common cause of failure of many products. In structural reliability analysis, traditionally, the uncertain deterioration and degradation are usually modeled using lifetime distributions [

In engineering practice, time-dependent reliability analysis for modeling deterioration systems is necessary because the performance of many products is a deterioration process [

In engineering practice, it is well known that data sometimes cannot be recorded or collected precisely due to various uncertainties [

Despite many efforts, reliability assessment for deteriorating systems using traditional RV models also has some limitations. In the traditional RV models, the distributions of random variables (such as the deterioration rate) usually are assumed known and precisely determined, such as normal distribution and Weibull distribution. However, determining lifetime distributions requires sufficient data which is often impossible to acquire in engineering practice, especially for some reliable and long lifetime products such as satellites, spaceships and so on. Furthermore, these assumptions are not reasonable in the case of less data and incomplete information. For example, we may face the problem that the distribution of one random variable can be viewed as a normal or Weibull distribution under small sample sizes. In order to solve the problem and consider the advantages of both the maximum entropy method and the Bayesian inference for systems under small sample sizes and incomplete information constraints, combinations of the two methods are considered for the deteriorating systems, and the fuzzy Bayesian reliability assessment for deteriorating components is proposed in this paper.

The remainder of the paper is organized as follows: in

Bayesian inference is based on the subjective view of probability. Let _{1}, _{2},…,_{n}

In engineering practice, the value or distribution of a parameter usually cannot be determined precisely under small sample sizes or vague information conditions. For example, we often state that the critical threshold is “about 10” or “about 9–11”, and the statements “about 10” and “about 9-11” can be viewed as fuzzy numbers. Therefore, the fuzzy sets theory provides an appropriate tool for modeling the situations where some parameters are fuzzy numbers. Let

There are many kinds of membership functions that can be used in engineering applications. For illustration purposes, a special kind of fuzzy real number is introduced in the paper. We say that

Triangular fuzzy number

The α-level set of the fuzzy real number

In order to improve the accuracy in systems reliability assessment, additional assumptions should be avoided. The maximum entropy method [_{x}_{i}

From the Lagrange’s method, the _{i} is the Lagrangian multiplier for the

To solve for the Lagrange multipliers _{1}, _{2}…,_{k}

In order to assess the reliability of deteriorating components using RV models, the degradation curves of components should be determined firstly. Suppose that a linear degradation curve exists and its initial value is _{0}, the degradation at time _{i}_{f}_{f}_{0})/_{i}_{ij}_{j}_{ij}_{j}_{i}_{j}_{ij}_{i}_{j}

In reality, the degradation curves are more complicated than linear degradation curves such as power laws and exponential curves. In these situations, these complicated curves can be transformed by logarithms. For example, we can transform an exponential curve

In the situation of considering measurement errors, Equation (10) can be rewritten as:

For simplicity, we let

The different kinds of degradation curves are shown in

Different degradation curves.

From the aforementioned discussions, many commonly used degradation curves can be transformed into a linear curve by using some corresponding transformations. For simplicity and illustration purposes, the linear degradation curve

Suppose that _{f}

Degradation data.

Time t | t_{0} |
t_{1} |
t_{2} |
t_{3} |
… | t_{j} |
---|---|---|---|---|---|---|

_{1}(t_{0}) |
_{1}(t_{1}) |
_{1}(t_{2}) |
_{1}(t_{3}) |
_{1}(t_{j}) |
||

_{2}(t_{0}) |
_{2}(t_{1}) |
_{2}(t_{2}) |
_{2}(t_{3}) |
_{2}(t_{j}) |
||

… | … | … | … | … | … | |

_{n}(t_{0}) |
_{n}(t_{1}) |
_{n}(t_{2}) |
_{n}(t_{3}) |
_{n}(t_{j}) |

According to Equation (13), the pseudo lifetime _{i}_{f}_{0})/_{i}_{i}

From Equation (16), the cumulative distribution function (CDF) of

From Equation (17), the probability of failure _{0} becomes:

Generally, Equation (17) has no analytical and close form solutions. We can solve it by using numerical integration methods such as the Simpson and Romberg algorithms [

Suppose that the prior PDF of _{t}) is given by:

The integration

From Equation (20), the CDF _{T}_{t}) can be expressed as:

From Equation (21), the probability of failure _{0} becomes:

Suppose that the prior PDF of _{θ}) is given by:

For example, if the prior PDF of

Similarly, if the prior PDF of

Considering a random variable _{X}_{Y}_{1} = min{_{2} = max{

According to Equation (26), the posterior PDF _{T}_{θ}) of

From Equation (27), the CDF _{T}_{θ}) is given by:

From Equation (28), the probability of failure _{0} becomes:

For example, let _{f}_{0} = _{r}_{T}_{θ}) is given by:

From Equation (30), the CDF _{T}_{θ}) of

Similarly, when the prior PDF of _{T}_{θ}) and CDF _{T}_{θ}) are respectively given by:

In

The pseudo lifetime

From Equations (34) and (35), the lower and upper bounds of probability of failure under the

Suppose that the prior PDF of the random variable

With different

For example, suppose that the prior PDF of

In this section, two examples are analyzed to demonstrate the proposed method. The first example is associated with only one linear degradation cure, and its critical threshold is a fuzzy number. The degradation curve of the second example is a non-linear curve, and some fuzzy numbers exist in the prior distribution.

GaAs is a typical reliable and long lifetime laser which is widely used in military. Since the GaAs laser is highly reliable and very costly, there is little failure information because carrying out many experiments is impossible. Suppose the degradation data of five GaAs laser units is as shown in

Degradation data of GaAs [

0 | 500 | 1000 | 1500 | 2000 | 2500 | 3000 | 3500 | 4000 | |
---|---|---|---|---|---|---|---|---|---|

0 | 0.93 | 2.72 | 4.34 | 5.48 | 6.72 | 8.00 | 9.49 | 10.94 | |

0 | 1.22 | 2.30 | 3.75 | 4.99 | 6.07 | 7.16 | 8.42 | 9.28 | |

0 | 1.17 | 1.99 | 2.97 | 3.94 | 4.45 | 5.27 | 6.02 | 6.88 | |

0 | 0.74 | 1.85 | 2.95 | 3.92 | 5.47 | 6.50 | 7.39 | 8.09 | |

0 | 0.61 | 1.77 | 2.58 | 3.38 | 4.63 | 5.62 | 6.32 | 7.59 |

Since the degradation curve is a linear function, it can be expressed as

Degradation data of each unit and its fitted curve.

From

In reality, critical thresholds sometimes cannot be determined precisely due to various uncertainties. The more appropriate way to describe the thresholds is that _{f}_{f}

PDFs of

CDFs calculated by using the first three and four moments.

Probability of failure under different

Suppose that the degradation curve of a time-depending deteriorating system is
_{f}_{θ}

The values of

θ (unit) | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

value | 1.28 | 1.47 | 1.15 | 1.33 | 1.08 | 1.21 |

In engineering practice, the available information, such as the past data and expert opinions, is valuable information for us. In this example, we consider the situation where there is prior information for the random variable _{θ}_{θ}_{θ}

Posterior distribution for _{θ}

From Equations (27), (28), (38) and (39), the probability of failure under

Probability of failure bounds under different

α | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
---|---|---|---|---|---|---|

0.0019 | 0.0030 | 0.0049 | 0.0076 | 0.0116 | 0.0172 | |

0.0942 | 0.0698 | 0.0507 | 0.0361 | 0.0252 | 0.0172 |

Probability of failure bounds under different

α | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
---|---|---|---|---|---|---|

0.0023 | 0.0039 | 0.0063 | 0.0101 | 0.0159 | 0.0245 | |

0.1476 | 0.1088 | 0.0779 | 0.0543 | 0.0449 | 0.0245 |

Probability of failure under

Probability of failure for different

In this paper the combination of Bayesian inference and the maximum entropy method have been applied to the problem of deteriorating system reliability assessment. Due to various uncertainties and incomplete information, precise determination of all parameters is impossible in engineering applications. In this case, fuzzy sets theory and Bayesian inference as well as the maximum entropy method have shown to be useful for the case of vague environments and less data. The maximum entropy method is robust under limited data constraints because it does not need any additional assumptions. The numerical examples have shown that the probability of failure is a fuzzy number in the case that fuzzy parameters exist in systems. Furthermore, the memberships can be determined by the optimization models proposed in the paper. From the discussions and the illustrated examples, we know that the proposed method requires neither the additional assumptions nor the large sample sizes, which are needed in other commonly used methods. It should be noted that there are some limitations to the proposed method. The RV deterioration model cannot capture temporal variability associated with evolution of degradation. The extension of the method for handling multiple failure modes will be the subject of future work in our research.

Random variable

Probability density function

Cumulative distribution function

Method of moments

Maximum likelihood estimation

Maximum entropy estimation

Integral domain

Lagrange multipliers

A fuzzy number

Membership function

The

Triangular fuzzy number

Prior distribution

Maximum entropy density function

Posterior PDF

_{i}(t)

Degradation at time t of unit

_{f}

Critical threshold

Given samples

This research was partially supported by the National Natural Science Foundation of China under the contract number 51075061, and the National High Technology Research and Development Program of China (863 Program) under the contract number 2007AA04Z403.

The authors declare no conflict of interest.