## 4 Design of Practical PADT Plan with Two Stress Variables  
### First Phase: PADT Plan with Two Stress Levels by Each Variable  
In the development stage of novel products, the general tests under the use and maximum stress levels should be performed in order to acquire basic performance and reliability information such as the acceleration model parameter. Therefore, in the first phase, a PADT plan with two stress levels (\(r_{1}=r_{2}=2\)) is presented to estimate the parameters of the acceleration model. Two stress levels are assigned as the use and maximum stress levels by each variable (see Figure 2). To estimate the unknown parameters of the model (\(\delta_{1},\delta_{2},\delta_{3},\delta_{4}\) and \(\sigma\)), the ratios (\(\pi_{11},\pi_{12},\pi_{21}\) and \(\pi_{22}\)) which were allocated to each test point, should be considered based on D-optimality criteria. Particularly, the ratios were determined to maximize the determinant of the Fisher information matrix for the model parameters. First, the maximum likelihood estimators (MLEs) \begin{table}
\begin{tabular}{c l c} \hline \hline
**Symbol** & **Description** & **Index** \\ \hline \(i\) & Index of temperature stress variable (\(i=1,2,\cdots,r_{1}\)) & - \\ \(j\) & Index of non-thermal stress variable (\(j=1,2,\cdots,r_{2}\)) & - \\ \(k\) & Index of test units & - \\ \(l\) & Index of measurement times & - \\ \(T_{i}^{\prime}\) & \(i\)-th temperature stress level (in Kelvin) & \(i\) \\ \(V_{j}^{\prime}\) & \(j\)-th non-thermal stress level (voltage, vibration, etc.) & \(j\) \\ \(N\) & Pre-determined total number of specimens & - \\ \(n_{ij}\) & Test unit at \(i\)-th and \(j\)-th stress level & \(i,j\) \\ \(t_{M,ijk}\) & Maximum test duration & \(i,j,k\) \\ \(t_{ijkl}\) & Measurement time & \(i,j,k,l\) \\ \(y_{ij,k}(t)\) & Degradation characteristic of \(k\)-th test unit at \(i\)-th and \(j\)-th stress level & \(i,j,k\) \\ \(\eta^{\prime}\left(T_{i}^{\prime},V_{j}^{\prime}\right)\) & Drift of the Wiener process model & \(i,j\) \\ \(\sigma^{2}\) & Diffusion constant of the Wiener process model & - \\ \(\omega\) & Pre-determined critical value & - \\ \(\delta_{1}^{\prime}\), \(\delta_{2}^{\prime}\), \(\delta_{3}^{\prime}\), \(\delta_{4}^{\prime}\), & Parameters of generalized Eyring model & - \\ \(T_{i}\) & \(i\)-th standardized temperature stress level & \(i\) \\ \(V_{j}\) & \(j\)-th standardized non-thermal stress level (voltage, vibration, etc.) & \(j\) \\ \(T_{M}\) & Maximum standardized temperature stress level (\(T_{M}=1\)) & - \\ \(V_{M}\) & Maximum standardized non-thermal stress level (\(V_{M}=1\)) & - \\ \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\delta_{4}\) & Modified parameters of generalized Eyring model by \(T_{i}\) and \(V_{j}\) & - \\ \(\eta\left(T_{i},\ V_{j}\right)\) & Modified drift by \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\) and \(\delta_{4}\) & \(i,j\) \\ AF & Accelerated factor & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The list of symbols and descriptions. Figure 2: The test units and stress levels of four test points in the first phase. for the unknown parameters can be obtained by calculating the first partial derivatives of \(\ln L\) in Equation (1) of each parameter and solving the simultaneous equations: \(\partial\ln L/\partial\delta_{1}=0\), \(\partial\ln L/\partial\delta_{2}=0\), \(\partial\ln L/\partial\delta_{3}=0\), \(\partial\ln L/\partial\delta_{4}=0\) and \(\partial\ln L/\partial\sigma=0\).