${f}_{1,1}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{\pi}_{i}{A}_{i,j}}}$, ${f}_{2,2}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{X}_{i}^{2}{\pi}_{i}{A}_{i,j}}}$, ${f}_{3,3}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{Y}_{j}^{2}}}{\pi}_{i}{A}_{i,j}$,

${f}_{4,4}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{\pi}_{i}{C}_{i,j}}}$, ${f}_{1,2}={f}_{2,1}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{X}_{i}{\pi}_{i}{A}_{i,j}}}$, ${f}_{1,3}={f}_{3,1}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{Y}_{j}{\pi}_{i}{A}_{i,j}}}$,

${f}_{1,4}={f}_{4,1}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{\pi}_{i}{B}_{i,j}}}$, ${f}_{2,3}={f}_{3,2}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{X}_{i}{Y}_{j}{\pi}_{i}{A}_{i,j}}}$,

${f}_{2,4}={f}_{4,2}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{X}_{i}{\pi}_{i}{B}_{i,j}}}$, ${f}_{3,4}={f}_{4,3}={\displaystyle \underset{i=1}{\overset{2}{\sum}}{\displaystyle \underset{j=1}{\overset{2}{\sum}}{Y}_{j}{\pi}_{i}{B}_{i,j}}}$,

Substituting the above results into (5), we obtain

$AVar\left(\stackrel{^}{\mu}\left(1,1\right)\right)=\left(1,1,1,0\right)\cdot \Sigma \cdot {\left(1,1,1,0\right)}^{\text{T}}=\left(1,1,1,0\right)\cdot {F}_{0}^{-1}\cdot {\left(1,1,1,0\right)}^{\text{T}}=\frac{n}{{\sigma}^{2}}V$

where, $V=\frac{{E}_{1}+{E}_{2}+{E}_{3}+2\left({E}_{4}+{E}_{5}+{E}_{6}\right)}{\left|{F}_{0}\right|}$, $\left|{F}_{0}\right|$ is the determinant of matrix ${F}_{0}$.

${E}_{1}={f}_{2,2}{f}_{3,3}{f}_{4,4}+2{f}_{2,3}{f}_{3,4}{f}_{4,2}-{f}_{2,4}^{2}{f}_{3,3}-{f}_{3,4}^{2}{f}_{2,2}-{f}_{2,3}^{2}{f}_{4,4}$

${E}_{2}={f}_{1,4}{f}_{2,4}{f}_{3,3}+{f}_{1,2}{f}_{3,4}^{2}+{f}_{1,3}{f}_{2,3}^{2}{f}_{4,4}-{f}_{1,2}{f}_{3,3}{f}_{4,4}-{f}_{1,4}{f}_{2,3}{f}_{3,4}-{f}_{1,3}{f}_{2,4}{f}_{3,4}$

${E}_{3}={f}_{1,2}{f}_{2,3}{f}_{4,4}+{f}_{1,4}{f}_{2,2}{f}_{3,4}+{f}_{1,3}{f}_{2,4}^{2}-{f}_{1,4}{f}_{2,3}{f}_{2,4}-{f}_{1,2}{f}_{2,4}{f}_{3,4}-{f}_{1,3}{f}_{2,2}{f}_{4,4}$

${E}_{4}={f}_{1,1}{f}_{3,3}{f}_{4,4}+2{f}_{1,3}{f}_{1,4}{f}_{3,4}-{f}_{1,4}^{2}{f}_{3,3}-{f}_{1,1}{f}_{3,4}^{2}-{f}_{1,3}^{2}{f}_{4,4}$

${E}_{5}={f}_{1,4}^{2}{f}_{2,3}+{f}_{1,1}{f}_{2,4}{f}_{3,4}+{f}_{1,2}{f}_{1,3}{f}_{4,4}-{f}_{1,1}{f}_{2,3}{f}_{4,4}-{f}_{1,2}{f}_{1,4}{f}_{3,4}-{f}_{1,4}{f}_{1,3}{f}_{2,4}$

${E}_{6}={f}_{1,1}{f}_{2,2}{f}_{4,4}+2{f}_{1,2}{f}_{1,4}{f}_{2,4}-{f}_{1,4}^{2}{f}_{2,2}-{f}_{1,1}{f}_{2,4}^{2}-{f}_{1,2}^{2}{f}_{4,4}$

Because the total number n of samples and ${\sigma}^{2}$ are constant, the optimization model is:

$\text{Min}:V=\frac{{E}_{1}+{E}_{2}+{E}_{3}+2\left({E}_{4}+{E}_{5}+{E}_{6}\right)}{\left|{F}_{0}\right|}$ (6)

where, the decision variables are: ${X}_{1}$, ${Y}_{1}$, ${\pi}_{1}$, and $0\le {X}_{1},{Y}_{1},{\pi}_{1}\le 1$.

This nonlinear programming problem can be solved by the method of reference [15].

4. Numerical Example

In order to illustrate the above optimization process, we take 50 LEDs for accelerated degradation test. The degraded performance attribute is taken as its luminous intensity. The acceleration variable of the test is taken as voltage. Under normal working conditions, the working current of the sample is 40 mA. According to previous experience, the maximum allowable working current is 330 mA (without changing its failure mechanism). Here we consider accelerated degradation tests at two stress levels. The voltage level is ${S}_{1},{S}_{2}$. $40<{S}_{1}<330$, ${S}_{2}=330$. When the sample is working under normal conditions, if its luminous intensity is lower than 50% of its initial intensity, it will be judged as invalid. If the initial luminous intensity is set to 1, the critical value under normal conditions is ${C}_{0}=0.5$. Now, we accelerate the degradation threshold at the same time. Set the extreme critical value as ${C}_{2}=0.1$. We optimize the experimental arrangement according to the above model.

The estimation of parameters in the model can be based on previous experience or similar data. Here, we set its estimate as $\stackrel{^}{\alpha}=5.5$, $\stackrel{^}{\beta}=6$, $\stackrel{^}{\gamma}=3$, $\stackrel{^}{\sigma}=1$. We take the cut-off time of the test at two stress levels as ${\tau}_{1}={\tau}_{2}=8$. Put the above data into the optimization model (6), using the nonlinear programming method, we obtain ${X}_{1}=0.412$, ${Y}_{1}=0.646$, ${\pi}_{1}=0.557$

Thus, we have ${n}_{1}=n\cdot {\pi}_{1}=50\times 0.557\approx 28$, ${n}_{2}=n-{n}_{1}=22$.

And ${X}_{1}=\frac{{S}_{1}-{S}_{2}}{{S}_{0}-{S}_{2}}$, ${S}_{1}=210.5$ ; and ${Y}_{1}=\frac{{C}_{1}-{C}_{2}}{{C}_{0}-{C}_{2}}$, ${C}_{1}=0.24$.

Therefore, the optimal design of the test is: Take 28 samples and put them into 210.5 mA for accelerated degradation test. When the luminous intensity of the test sample—led is lower than 24% of the initial value, we determine that the sample is invalid once. When the luminous intensity of the test sample is lower than 50% of the initial value, we decide that it will fail again. The closing time of the test is $t=8$. In addition, 22 samples are placed at 330 mA for accelerated degradation test. When the luminous intensity of the test sample is lower than 24% of the initial value, we determine that the sample is invalid once. When the luminous intensity of the test sample is lower than 50% of the initial value, we decide that it will fail again. The closing time of the test is $t=8$.

5. Results and Conclusion

In this work we established an optimization design model based on degradation performance for accelerated degradation test. The model shows that the test can be accelerated by selecting the critical value level. For ease of application, we simplified the model, and presented a numerical example to illustrate the procedures of the test plan.

Acknowledgements

This research was partially supported by the PhD research startup foundation of Guizhou Normal University (Grant No. GZNUD[2017]27) & Science and Technology Foundation of Guizhou Province (LKS[2012]11), China. & Teaching Project of Guizhou Normal University in 2016: Contract No. [2016] XJ No. 09.

Cite this paper

Wu, Y. (2019) An Optimal Design of Accelerated Degradation Tests Based on Degradation Performance.*Open Journal of Statistics*, **9**, 686-694. doi: 10.4236/ojs.2019.96044.

Wu, Y. (2019) An Optimal Design of Accelerated Degradation Tests Based on Degradation Performance.

References

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https://doi.org/10.1080/00401706.1994.10485803

[2] Yu, H.F. and Tseng, S.T. (1999) Designing a Degradation Experiment. Naval Research Logistics, 46, 689-706.

https://doi.org/10.1002/(SICI)1520-6750(199909)46:6<689::AID-NAV6>3.0.CO;2-N

[3] Park, J.I. and Yum, B.J. (1997) Optimal Design of Accelerated Degradation Tests for Estimating Mean Lifetime at the Use Condition. Engineering Optimization, 28, 199-230.

https://doi.org/10.1080/03052159708941132

[4] Tseng, S.T. and Yu, H.F. (1997) A Termination Rule for Degradation Experiments. IEEE Transactions on Reliability, 46, 130-133.

https://doi.org/10.1109/24.589938

[5] Yang, G. and Yang, K. (2002) Accelerated Degradation-Tests with Tightened Critical Values. IEEE Transactions on Reliability, 51, 463-468.

https://doi.org/10.1109/TR.2002.804490

[6] Wu, S.J. and Chang, C.T. (2002) Optimal Design of Degradation Tests in Presence of Cost Constraint. Reliability Engineering & System Safety, 76, 109-115.

https://doi.org/10.1016/S0951-8320(01)00123-5

[7] Yu, H.F. and Tseng, S.T. (2004) Designing a Degradation Experiment with a Reciprocal Weibull Degradation Rate. Quality Technology & Quantitative Management, 1, 47-63.

https://doi.org/10.1080/16843703.2004.11673064

[8] Park, S.J., Yum, B. J. and Balamurali, S. (2004) Optimal Design of Step-Stress Degradation Tests in the Case of Destructive Measurement. Quality Technology & Quantitative Management, 1, 105-124.

https://doi.org/10.1080/16843703.2004.11673067

[9] Wang, X. and Xu, D. (2010) An Inverse Gaussian Process Model for Degradation Data. Technometrics, 52, 188-197.

https://doi.org/10.1198/TECH.2009.08197

[10] Ye, Z.S. and Chen, N. (2014) The Inverse Gaussian Process as a Degradation Model. Technometrics, 56, 302-311.

https://doi.org/10.1080/00401706.2013.830074

[11] Sung, S.I. and Yum, B.J. (2016) Optimal Design of Step-Stress Accelerated Degradation Tests Based on the Wiener Degradation Process. Quality Technology & Quantitative Management, 13, 367-393.

https://doi.org/10.1080/16843703.2016.1189179

[12] Yang, K. and Yang, G. (1998) Degradation Reliability Assessment Using Severe Critical Values. International Journal of Reliability, Quality and Safety Engineering, 5, 85-95.

https://doi.org/10.1142/S0218539398000091

[13] Lu, J.C. and Yang, P.Q. (1997) Statistical Inference of a Time-to-Failure Distribution Derived from Linear Degradation Data. Technometrics, 39, 391-400.

https://doi.org/10.1080/00401706.1997.10485158

[14] Lall, P., Pecht, M. and Hakim, E. (1997) Influence of Temperature on Microelectronics and System Reliability. CRC Press, London.

[15] Bazaraa, M. and Shetty, C. (1979) Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York.

[1] Boulanger, M. and Escobar, L.A. (1994) Experimental Design for a Class of Accelerated Degradation Tests . Technometrics, 36, 260-272.

https://doi.org/10.1080/00401706.1994.10485803

[2] Yu, H.F. and Tseng, S.T. (1999) Designing a Degradation Experiment. Naval Research Logistics, 46, 689-706.

https://doi.org/10.1002/(SICI)1520-6750(199909)46:6<689::AID-NAV6>3.0.CO;2-N

[3] Park, J.I. and Yum, B.J. (1997) Optimal Design of Accelerated Degradation Tests for Estimating Mean Lifetime at the Use Condition. Engineering Optimization, 28, 199-230.

https://doi.org/10.1080/03052159708941132

[4] Tseng, S.T. and Yu, H.F. (1997) A Termination Rule for Degradation Experiments. IEEE Transactions on Reliability, 46, 130-133.

https://doi.org/10.1109/24.589938

[5] Yang, G. and Yang, K. (2002) Accelerated Degradation-Tests with Tightened Critical Values. IEEE Transactions on Reliability, 51, 463-468.

https://doi.org/10.1109/TR.2002.804490

[6] Wu, S.J. and Chang, C.T. (2002) Optimal Design of Degradation Tests in Presence of Cost Constraint. Reliability Engineering & System Safety, 76, 109-115.

https://doi.org/10.1016/S0951-8320(01)00123-5

[7] Yu, H.F. and Tseng, S.T. (2004) Designing a Degradation Experiment with a Reciprocal Weibull Degradation Rate. Quality Technology & Quantitative Management, 1, 47-63.

https://doi.org/10.1080/16843703.2004.11673064

[8] Park, S.J., Yum, B. J. and Balamurali, S. (2004) Optimal Design of Step-Stress Degradation Tests in the Case of Destructive Measurement. Quality Technology & Quantitative Management, 1, 105-124.

https://doi.org/10.1080/16843703.2004.11673067

[9] Wang, X. and Xu, D. (2010) An Inverse Gaussian Process Model for Degradation Data. Technometrics, 52, 188-197.

https://doi.org/10.1198/TECH.2009.08197

[10] Ye, Z.S. and Chen, N. (2014) The Inverse Gaussian Process as a Degradation Model. Technometrics, 56, 302-311.

https://doi.org/10.1080/00401706.2013.830074

[11] Sung, S.I. and Yum, B.J. (2016) Optimal Design of Step-Stress Accelerated Degradation Tests Based on the Wiener Degradation Process. Quality Technology & Quantitative Management, 13, 367-393.

https://doi.org/10.1080/16843703.2016.1189179

[12] Yang, K. and Yang, G. (1998) Degradation Reliability Assessment Using Severe Critical Values. International Journal of Reliability, Quality and Safety Engineering, 5, 85-95.

https://doi.org/10.1142/S0218539398000091

[13] Lu, J.C. and Yang, P.Q. (1997) Statistical Inference of a Time-to-Failure Distribution Derived from Linear Degradation Data. Technometrics, 39, 391-400.

https://doi.org/10.1080/00401706.1997.10485158

[14] Lall, P., Pecht, M. and Hakim, E. (1997) Influence of Temperature on Microelectronics and System Reliability. CRC Press, London.

[15] Bazaraa, M. and Shetty, C. (1979) Nonlinear Programming: Theory and Algorithms. John Wiley & Sons, New York.