## 5 Case Study
### An Extended Case Study  
In order to ensure that the method can be applied to the SSADT scheme optimization design of other products under multi stress, the case study in Reference [39] is taken as \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & \(n\) & \(S_{1}\) & \(S_{2}\) & \(\mathbf{m}\) & \(det|M(\xi)|\) \\ \hline
1 & [5 5 5 5] & [800 850 900 950] & [6000 6500 7000 7500] & [66 62 61 61] & \(4.29*10^{24}\) \\
2 & [5 5 5 5] & [600 80 850 900 950] & [5000 5500 6000 7000] & [117 113 10 10] & \(5.26*10^{24}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 14: \(D\)-optimality optimal scheme. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Cost (Ten Thousand Yuan)** \\ \hline Expected & cost of testing (\(C_{1}\)) & 75 \\  & \(C_{d}\) & 2.5 \\  & \(C_{0}\) & 0.01/h \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Expected test cost and accelerated degradation test cost. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & \(n\) & \(S_{1}\) & \(S_{2}\) & \(\mathbf{m}\) & \(|M(\xi)|/|M_{22}(\xi)|\) \\ \hline
1 & [5 5 5 5] & [800 850 900 950] & [6000 6500 7000 7500] & [66 62 61 61] & \(1651636.372\) \\
2 & [6 6 6 6] & [800 850 900 950] & [6000 6500 7000 7500] & [40 38 36 36] & \(455879.2074\) \\
3 & [7 7 7 7 7] & [800 850 900 950] & [6000 6500 7000 7500] & [16 12 11 11] & \(31610.36\) \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Optimization results of test scheme. a case study to carry out the optimization design. The accelerated degradation model of products in Reference [39] is as follows:  
\[\left\{ \begin{array}{c}\ln\mu_{k}^{1}=A^{1}/T+B^{1}*\ln V+C^{1}\\ Y\sim N\big{(}\mu_{k}^{1}\big{(}\Delta t_{k}^{1}\big{)},\sigma_{1}^{2}\big{(} \Delta t_{k}^{1}\big{)}\big{)}\end{array} \right. \tag{29}\]  
In Equation (29), \(T\) is absolute temperature, in K; \(V\) is voltage, in V; \(A^{1}\), \(B^{1}\), \(C^{1}\) are the model parameters; the specific values of \(A^{1}\), \(B^{1}\), \(C^{1}\) and \(\sigma_{1}\) are shown in Table 15.  
Substitute the parameter values into Equation (29):  
\[\left\{ \begin{array}{c}\ln\mu_{k}^{1}=\left(-8704\right)/T+0.5*\ln V+10\\ Y\sim N\Big{(}\mu_{k}^{1}\big{(}\Delta t_{k}^{1}\big{)},\sigma_{1}^{2}\big{(} \Delta t_{k}^{1}\big{)}\Big{)}\end{array} \right. \tag{30}\]  
According to the constraint conditions in Reference [39] and the proposed optimization method, the optimization problem can be described as  
\[s.t.\quad\left\{ \begin{array}{c}\max\Big{|}\frac{M^{1}(\xi) }{|M_{2}|^{2}(\xi)|}\\ \\ S_{1}{}^{1}\leq S_{1}{}^{1}<S_{2}{}^{1}<S_{1}{}^{1}\cdots<S_{iL}{}^{1}<S_{iL}{ }^{1}<S_{imax}{}^{1}\end{array} \right.\quad variables:S_{i}{}^{1},n_{l}{}^{1},m_{l }{}^{1} \tag{31}\] \[m_{1}{}^{1}\geq m_{2}{}^{1}\geq\cdots\geq m_{l}{}^{1}\geq 0,m_{1}{}^{1} \geq\frac{m^{1}}{l^{1}}\] \[n_{l}{}^{1}\geq 3\]  
In Equation (31), \(i=1,2;l=1,2,3,4\); the specific values of \(C_{1}{}^{1},C_{d}{}^{1},C_{0}{}^{1},\Delta t\) and the values range of \(S_{1}{}^{1},S_{2}{}^{1}\) are shown in Table 16:  
The proposed method is used to optimize the test scheme of the product in reference [39], and the objective function value of the scheme under \(D\)-optimality is calculated. The results are shown in Table 17.  
The optimal scheme under \(D\)-optimality in Reference [39] is optimized under the optimization method proposed in this paper, and the results are shown in Table 18. \begin{table}
\begin{tabular}{c c c c c c} \hline \(C_{1}^{1}\) **(yuan)** & \(C_{d}^{1}\) **(yuan)** & \(C_{0}^{1}\) **(yuan)** & \(\Delta t\) **(h)** & \(T\) **(K)** & **V (V)** \\ \hline
150,000 & 3000 & 200 & 1 & 313.15-383.15 & 5–15 \\ \hline \end{tabular}
\end{table} 
Table 16: Test cost and stress range. \begin{table}
\begin{tabular}{c c c c} \hline \(n^{1}\) & \(T\) & \(V\) & \(m^{1}\) & \(|M^{1}(\xi)|/|M_{22}^{1}(\xi)|\) & \(\text{det}|M(\xi)|\) \\ \hline
[6 6 6 6] & [368.15 373.15 378.15 383.15] & [11 12 13 14] & [99 97 97 97] & 742411.69 & \(6.54*10^{13}\) \\ \hline \end{tabular}
\end{table} 
Table 17: Optimization results.