# A Bayesian Optimal Design for Sequential Accelerated Degradation Testing

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## Abstract

**:**

## 1. Introduction

## 2. The Test Scheme and Bayesian Optimization

#### 2.1. The Test Scheme

- Stage 1: Both n and M are predefined according to the budget and practical situation before an ADT. The initial optimal test plan ${\eta}_{K}^{*}$$\left(\right)$ is obtained through the existing Bayesian optimal design method [19] based on the initial prior information. Then, we can make an initial decision ${D}_{K}$ to conduct the ADT at the highest stress level based on ${\eta}_{K}^{*}$.
- Stage 2: After the test under the higher stress level is completed, the corresponding degradation data can be collected. Then, the posterior information could be calculated by the Bayesian inference, which then will be treated as the prior information for the test design at the lower stress level. Hence, the optimal plan for the lower stress level can simultaneously be designed by the Bayesian method, and the decision could be correspondingly adjusted.
- Stage 3: Repeat Stage 2 for $K-1$ times to make $K-1$ decisions, and the whole sequential ADT is completed after finishing the test on the lowest level. Then, the final posterior distributions could be used to estimate the product’s reliability measures under ${S}_{0}$.

#### 2.2. ADT Model and Assumption

- (1)
- $X\left(0\right)=0$ with probability one;
- (2)
- $X\left(t\right)$ has independent increments, i.e., $X\left({t}_{2}\right)-X\left({t}_{1}\right)$ and $X\left({t}_{4}\right)-X\left({t}_{3}\right)$ are independent, for $0\le {t}_{1}<{t}_{2}\le {t}_{3}<{t}_{4}$;
- (3)
- each increment follows an IG distribution, i.e., $\Delta X\left(t\right)\sim \mathcal{IG}(\mu \Delta \mathsf{\Lambda}\left(t\right),\lambda {(\Delta \mathsf{\Lambda}\left(t\right))}^{2})$, where $\mu >0$, $\lambda >0$, $\Delta \mathsf{\Lambda}\left(t\right)=\mathsf{\Lambda}(t+\Delta t)-\mathsf{\Lambda}\left(t\right)$, $\mathsf{\Lambda}\left(t\right)$ is a given monotone increasing function of time t with $\mathsf{\Lambda}\left(0\right)=0$.

#### 2.3. Prior and Posterior Distributions

#### 2.4. Bayesian Optimal Criterion

- Step 1: A plan space $\mathit{P}$ is firstly defined, which contains R choices of test plans $\mathit{\eta}$. As for ${\mathit{\eta}}_{r}$, $r=1,2,\cdots ,R$, simulate parameter ${\mathit{\theta}}_{rp}$ from the corresponding $\pi \left(\mathit{\theta}\right)$ for ${Q}_{1}$ times ($p=1,2,\cdots ,{Q}_{1}$). Based on the simulated ${\mathit{\theta}}_{rp}$, generate degradation data ${x}_{rpq}$ from the sampling distribution (2) for ${Q}_{2}$ times ($q=1,2,\cdots ,{Q}_{2}$).
- Step 2: According to the Appendix, calculate the elements of Equation (9) based on the drawn ${\mathit{\theta}}_{rp}$ and ${x}_{rpq}$.
- Step 3: Numerically calculate the value of Equation (8) based on $\mathsf{\Phi}\left({\mathit{\eta}}_{r}\right)\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}\frac{1}{{Q}_{1}\xb7{Q}_{2}}{\displaystyle \sum _{q=1}^{{Q}_{2}}}{\displaystyle \sum _{p=1}^{{Q}_{1}}}log\left(\mathrm{det}\left({I}_{rpq}({\mathit{\eta}}_{r},{\mathit{\theta}}_{rp})\right)\right)$.

## 3. Planning of a Sequential ADT

#### 3.1. Planning ADT under the Highest Stress Level ${S}_{3}$

#### 3.2. Planning ADT under the Middle Stress Level ${S}_{2}^{{}^{\prime}}$

#### 3.3. Planning ADT under the Lowest Stress Level ${S}_{1}^{\u2033}$

## 4. Case Study

#### 4.1. Numerical Case

#### 4.1.1. The Stage of the Highest Stress Level

#### 4.1.2. The Stage of the Middle Stress Level

#### 4.1.3. The Stage of the Lowest Stress Level

#### 4.2. Static Bayesian ADT Design Method vs. Dynamic Sequential ADT Design Method

- (1)
- All of the relative deviations obtained by the proposed sequential method are less than that of the static method. That is to say our proposed sequential scheme can ensure better estimates both for the parameters and quantile lifetime.
- (2)
- There are less fluctuations in the estimates based on the proposed sequential ADT plan than that based on the static ADT plan.

## 5. Conclusions

- (1)
- The ADT plan can be dynamically adjusted with the collected ADT data.
- (2)
- The Bayesian sequential ADT design method outperform the Bayesian static design method on both the accuracy of the evaluation and the robustness of the misspecification of the initial planning parameter value.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ADT | Accelerated degradation testing |

IG | Inverse Gaussian |

ADDT | Accelerated destructive degradation tests |

MCMC | Markov chain Monte Carlo |

ALT | Accelerated life testing |

CSALT | Constant stress accelerated life testing |

Probability density function | |

CDF | Cumulative distribution function |

K | Number of accelerated stress levels |

${S}_{k}$ | k-th accelerated stress levels |

${m}_{k}$ | Number of degradation measurements on the k-th stress level |

M | Total number of the degradation measurements |

n | Sample size |

$\mathit{\eta}\left(\right)open="\{"\; close="\}">n,M,\mathit{S},\mathit{m}$ | Test plan |

${\mathit{\eta}}_{k}^{*}\left(\right)open="\{"\; close="\}">\mathit{S},\mathit{m}$ | The optimal test plan at k-th accelerated stress levels |

$X\left(t\right)$ | Degradation path |

$\mathsf{\Lambda}\left(t\right)$ | A given monotone increasing function in an IG process model |

${f}_{\mathcal{IG}}(x;\mu ,\lambda )$ | Probability density function for the degradation model |

$\mu \left(S\right)$ | An acceleration model |

$\phi \left(S\right)$ | A standardized function of S |

${X}_{D}$ | Pre-specified threshold level of degradation |

${T}_{D}$ | First-passage-time |

${x}_{ikj}$ | Degradation increment |

$\mathit{\theta}$ | Parameters of an IG process model |

$\pi \left(\mathit{\theta}\right)$ | Prior distributions |

$p\left(\mathit{\theta}\right|x)$ | Posterior distributions |

$\mathsf{\Phi}\left(\mathit{\eta}\right)$ | D-optimality |

$I(\mathit{\eta},\mathit{\theta})$ | Bayesian information matrix |

$\mathit{P}$ | Plan space |

## Appendix A

## Appendix B

**Table A1.**Stress relaxation degradation data of electrical connectors under different accelerated stress levels.

T | ID | Stress Loss | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

60 °C | 1 | 2.12 | 2.7 | 3.52 | 4.25 | 5.55 | 6.12 | 6.75 | 7.22 | 7.68 | 8.46 | 9.46 |

2 | 2.29 | 3.24 | 4.16 | 4.86 | 5.74 | 6.85 | * | 7.4 | 8.14 | 9.25 | 10.55 | |

3 | 2.4 | 3.61 | 4.35 | 5.09 | 5.5 | 7.03 | 8.24 | 8.81 | 9.629 | 10.27 | 11.11 | |

4 | 2.31 | 3.48 | 5.51 | 6.2 | 7.31 | 7.96 | 8.57 | 9.07 | 10.46 | 11.48 | 12.31 | |

5 | 3.14 | 4.33 | 5.92 | 7.22 | 8.14 | 9.07 | 9.44 | 10.09 | 11.2 | 12.77 | 13.51 | |

6 | 3.59 | 5.55 | 5.92 | 7.68 | 8.61 | 10.37 | 11.11 | 12.22 | 13.51 | 14.16 | 15 | |

85 °C | 7 | 2.77 | 4.62 | 5.83 | 6.66 | 8.05 | 10.61 | 11.2 | 11.98 | 13.33 | 15.64 | - |

8 | 3.88 | 4.37 | 6.29 | 7.77 | 9.16 | 9.9 | 10.37 | 12.77 | 14.72 | 16.8 | - | |

9 | 3.18 | 4.53 | 6.94 | 8.14 | 8.79 | 10.09 | 11.11 | 14.72 | 16.47 | 18.66 | - | |

10 | 3.61 | 4.37 | 6.29 | 7.87 | 9.35 | 11.48 | 12.4 | 13.7 | 15.37 | 18.51 | - | |

11 | 3.42 | 4.25 | 7.31 | 8.61 | 10.18 | 12.03 | 13.7 | 15.27 | 17.22 | 19.25 | - | |

12 | 5.27 | 5.92 | 8.05 | 9.81 | 12.4 | 13.24 | 15.83 | 17.59 | 20.09 | 23.51 | - | |

100 °C | 13 | 4.25 | 5.18 | 8.33 | 9.53 | 11.48 | 13.14 | 15.55 | 16.94 | 18.05 | 19.44 | - |

14 | 4.81 | 6.16 | 7.68 | 9.25 | 10.37 | 12.4 | 15 | 16.2 | 18.24 | 20.09 | - | |

15 | 5.09 | 7.03 | 8.33 | 10.37 | 12.22 | 14.35 | 16.11 | 18.7 | 19.72 | 21.66 | - | |

16 | 4.81 | 7.5 | 9.16 | 10.55 | 13.51 | 15.55 | 16.57 | 19.07 | 20.27 | 22.4 | - | |

17 | 5.64 | 6.57 | 8.61 | 12.5 | 14.44 | 16.57 | 18.7 | 21.2 | 22.59 | 24.07 | - | |

18 | 4.72 | 8.14 | 10.18 | 12.4 | 15.09 | 17.22 | 19.16 | 21.57 | 24.35 | 26.2 | - |

T | Performance Inspection Time | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

65 °C | 108 | 241 | 534 | 839 | 1074 | 1350 | 1637 | 1890 | 2178 | 2513 | 2810 |

85 °C | 46 | 108 | 212 | 408 | 632 | 764 | 1011 | 1333 | 1517 | 2586 | - |

100 °C | 46 | 108 | 212 | 344 | 446 | 626 | 729 | 972 | 1005 | 1218 | - |

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**Figure 2.**Optimal results at the stage of the highest stress level. (

**a**) Fitting surface, (

**b**) Top view.

**Figure 3.**Prior and posterior distributions under ${S}_{3}$ and prior distributions ${\pi}_{2}\left(\mathit{\theta}\right)$ under ${S}_{2}^{\prime}$. (

**a**) Parameter a, (

**b**) Parameter b, (

**c**) Parameter $\lambda $, (

**d**) Parameter $\beta $.

**Figure 4.**Optimal results at the stage of the middle stress level. (

**a**) Fitting surface, (

**b**) Top view.

**Figure 5.**Prior and posterior distributions under ${S}_{2}^{\prime}$ and prior distributions ${\pi}_{1}\left(\mathit{\theta}\right)$ under ${S}_{1}^{\u2033}$. (

**a**) Parameter a, (

**b**) Parameter b, (

**c**) Parameter $\lambda $, (

**d**) Parameter $\beta $.

**Figure 7.**Parameters evaluation results with ${\mathit{\theta}}_{T1}$. (

**a**) Parameter a, (

**b**) Parameter b, (

**c**) Parameter $\lambda $, (

**d**) Parameter $\beta $.

**Figure 9.**Parameters evaluation results with ${\mathit{\theta}}_{T2}$. (

**a**) Parameter a, (

**b**) Parameter b, (

**c**) Parameter $\lambda $, (

**d**) Parameter $\beta $.

Estimated Parameters | $\widehat{\mathit{a}}$ | $\widehat{\mathit{b}}$ | $\widehat{\mathit{\lambda}}$ | $\widehat{\mathit{\beta}}$ |
---|---|---|---|---|

Mean | $-1.8966$ | 1.7379 | 0.6337 | 0.4493 |

Variance | 0.1903 | 0.1738 | 0.1968 | 0.0178 |

Parameter | a | b | $\mathit{\lambda}$ | $\mathit{\beta}$ |
---|---|---|---|---|

${\pi}_{3}\left(\mathit{\theta}\right)$ | $Normal\phantom{\rule{0.166667em}{0ex}}(-1.90,0.19)$ | $Normal\phantom{\rule{0.166667em}{0ex}}(1.74,0.17)$ | $Gamma\phantom{\rule{0.166667em}{0ex}}(2.04,0.31)$ | $Gamma\phantom{\rule{0.166667em}{0ex}}(11.34,0.04)$ |

${\mathit{S}}_{1},{\mathit{S}}_{2},{\mathit{S}}_{3}$ (°C) | ${\mathit{m}}_{1},{\mathit{m}}_{2},{\mathit{m}}_{3}$ | $\mathbf{\Phi}\left(\mathit{\eta}\right)(\times {10}^{10})$ |
---|---|---|

50, 73, 100 | 50, 40, 30 | 3.8465 |

**Table 4.**Results of log likelihood test for samples from ${p}_{3}\left(\mathit{\theta}\right|{\mathit{x}}_{3})$.

Distribution Forms | Normal | Extreme Value | Logistic | |

Parameters | ||||

a | −1027.05 | −1268.49 | −1046.75 | |

b | −1029.14 | −1521.26 | −1032.19 | |

Distribution Forms | Gamma | Logistic | Weibull | |

Parameters | ||||

$\lambda $ | −350.384 | −546.081 | −467.518 | |

$\beta $ | 4955.42 | 4908.82 | 4721.6 |

Parameter | a | b | $\mathit{\lambda}$ | $\mathit{\beta}$ |
---|---|---|---|---|

${\pi}_{2}\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\theta}\right)$ | $Normal\phantom{\rule{0.166667em}{0ex}}(-1.98,0.12)$ | $Normal\phantom{\rule{0.166667em}{0ex}}(1.68,0.12)$ | $Gamma\phantom{\rule{0.166667em}{0ex}}(5.48,0.13)$ | $Gamma\phantom{\rule{0.166667em}{0ex}}(92.94,0.0049)$ |

${\mathit{S}}_{1}^{{}^{\prime}},{\mathit{S}}_{2}^{{}^{\prime}}$ (°C) | ${\mathit{m}}_{1}^{{}^{\prime}},{\mathit{m}}_{2}^{{}^{\prime}}$ | $\mathbf{\Phi}\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\eta}\right)(\times {10}^{10})$ |
---|---|---|

50, 73 | 55, 35 | 3.9720 |

**Table 7.**Results of log likelihood test for samples from ${p}_{2}\left(\mathit{\theta}\right|{\mathit{x}}_{2})$.

Distribution Forms | Normal | Extreme Value | Logistic | |

Parameters | ||||

a | 89.2733 | 45.8437 | −1268.49 | |

b | −872.846 | −890.798 | −1521.26 | |

Distribution Forms | Gamma | Logistic | Weibull | |

Parameters | ||||

$\lambda $ | 697.036 | 591.541 | 530.585 | |

$\beta $ | 6477.74 | 6444.57 | 6252.95 |

Parameter | a | b | $\mathit{\lambda}$ | $\mathit{\beta}$ |
---|---|---|---|---|

${\pi}_{1}\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\theta}\right)$ | $Normal\phantom{\rule{0.166667em}{0ex}}(-1.92,0.06)$ | $Normal\phantom{\rule{0.166667em}{0ex}}(1.72,0.10)$ | $Gamma\phantom{\rule{0.166667em}{0ex}}(13.19,0.0548)$ | $Gamma\phantom{\rule{0.166667em}{0ex}}(259.81,0.0017)$ |

Optimal Plan | ${\mathit{S}}_{1}^{{}^{\prime \prime}},{\mathit{S}}_{2}^{{}^{\prime}},{\mathit{S}}_{3}$ (${}^{\circ}$ C) | ${\mathit{m}}_{1}^{{}^{\prime \prime}},{\mathit{m}}_{2}^{{}^{\prime}},{\mathit{m}}_{3}$ | $\mathbf{\Phi}\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\eta}\right)(\times {10}^{10})$ |
---|---|---|---|

Sequential method | 50, 73, 100 | 55, 35, 30 | 2.5044 |

“True Values” | a | b | $\mathit{\lambda}$ | $\mathit{\beta}$ |
---|---|---|---|---|

${\mathit{\theta}}_{T1}(+2\sigma )$ | −1.02 | 2.57 | 1.52 | 0.72 |

${\mathit{\theta}}_{T2}(-2\sigma )$ | −2.77 | 0.90 | 0.10 | 0.18 |

Sequential Plan No. | ${\mathit{S}}_{1}^{{}^{\prime \prime}},{\mathit{S}}_{2}^{{}^{\prime}},{\mathit{S}}_{3}$ (°C) | ${\mathit{m}}_{1}^{{}^{\prime \prime}},{\mathit{m}}_{2}^{{}^{\prime}},{\mathit{m}}_{3}$ | $\mathbf{\Phi}\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\eta}\right)(\times {10}^{10})$ |
---|---|---|---|

1 | 54, 73, 100 | 55, 35, 30 | 4.2332 |

2 | 50, 73, 100 | 60, 30, 30 | 3.1216 |

3 | 50, 73, 100 | 60, 30, 30 | 2.8496 |

4 | 52, 73, 100 | 55, 35, 30 | 2.5196 |

5 | 50, 73, 100 | 55, 35, 30 | 2.7243 |

Sequential Plan No. | ${\mathit{S}}_{1}^{{}^{\prime \prime}},{\mathit{S}}_{2}^{{}^{\prime}},{\mathit{S}}_{3}$ (°C) | ${\mathit{m}}_{1}^{{}^{\prime \prime}},{\mathit{m}}_{2}^{{}^{\prime}},{\mathit{m}}_{3}$ | $\mathbf{\Phi}\phantom{\rule{0.166667em}{0ex}}\left(\mathit{\eta}\right)(\times {10}^{10})$ |
---|---|---|---|

1 | 60, 76, 100 | 45, 45, 30 | 4.6062 |

2 | 62, 73, 100 | 45, 45, 30 | 5.4769 |

3 | 58, 73, 100 | 50, 40, 30 | 1.4460 |

4 | 52, 73, 100 | 50, 40, 30 | 1.8926 |

5 | 54, 73, 100 | 50, 40, 30 | 2.6563 |

“True Values” | Method | ${\mathit{\epsilon}}_{\mathit{a}}$ | ${\mathit{\epsilon}}_{\mathit{b}}$ | ${\mathit{\epsilon}}_{\mathit{\lambda}}$ | ${\mathit{\epsilon}}_{\mathit{\beta}}$ | ${\mathit{\epsilon}}_{\mathbf{tp}}(\times {10}^{17})$ |
---|---|---|---|---|---|---|

${\mathit{\theta}}_{T1}$ | Static method | 0.3940 | 0.5466 | 0.5250 | 0.0582 | 215.7427 |

Sequential method | 0.1502 | 0.3450 | 0.4954 | 0.0507 | 132.9126 | |

${\mathit{\theta}}_{T2}$ | Static method | 0.7265 | 1.2556 | 0.0407 | 0.0392 | 11.572 |

Sequential method | 0.3218 | 0.8806 | 0.0388 | 0.0229 | 8.0021 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, X.; Hu, Y.; Sun, F.; Kang, R.
A Bayesian Optimal Design for Sequential Accelerated Degradation Testing. *Entropy* **2017**, *19*, 325.
https://doi.org/10.3390/e19070325

**AMA Style**

Li X, Hu Y, Sun F, Kang R.
A Bayesian Optimal Design for Sequential Accelerated Degradation Testing. *Entropy*. 2017; 19(7):325.
https://doi.org/10.3390/e19070325

**Chicago/Turabian Style**

Li, Xiaoyang, Yuqing Hu, Fuqiang Sun, and Rui Kang.
2017. "A Bayesian Optimal Design for Sequential Accelerated Degradation Testing" *Entropy* 19, no. 7: 325.
https://doi.org/10.3390/e19070325