## 4 Design of Practical PADT Plan with Two Stress Variables
### First Phase: PADT Plan with Two Stress Levels by Each Variable
The first partial derivatives of \(\ln L\) are given as follows  
\[\frac{\partial\ln L}{\partial\delta_{1}}=\sum_{i=1}^{2}\sum_{j=1}^{2}\sum_{k= 1}^{n_{ij}}\sum_{l=1}^{m}\left[\frac{\left(\Delta y_{ijkl}-\eta_{ij}\Delta t_{ ijkl}\right)\eta_{ij}}{\sigma^{2}}\right],\]  
\[\frac{\partial\ln L}{\partial\delta_{2}}=\sum_{i=1}^{2}\sum_{j=1}^{2}\sum_{k= 1}^{n_{ij}}\sum_{l=1}^{m}\left[\frac{\left(\Delta y_{ijkl}-\eta_{ij}\Delta t_ {ijkl}\right)\eta_{ij}T_{i}}{\sigma^{2}}\right],\]  
\[\frac{\partial\ln L}{\partial\delta_{3}}=\sum_{i=1}^{2}\sum_{j=1}^{2}\sum_{k= 1}^{n_{ij}}\sum_{l=1}^{m}\left[\frac{\left(\Delta y_{ijkl}-\eta_{ij}\Delta t_ {ijkl}\right)\eta_{ij}V_{j}}{\sigma^{2}}\right],\]  
\[\frac{\partial\ln L}{\partial\delta_{4}}=\sum_{i=1}^{2}\sum_{j=1}^{2}\sum_{k= 1}^{n_{ij}}\sum_{l=1}^{m}\left[\frac{\left(\Delta y_{ijkl}-\eta_{ij}\Delta t _{ijkl}\right)\eta_{ij}T_{i}V_{j}}{\sigma^{2}}\right],\]  
\[\frac{\partial\ln L}{\partial\sigma}=\sum_{i=1}^{2}\sum_{j=1}^{2}\sum_{k=1}^{n _{ij}}\sum_{l=1}^{m}\left[-\frac{1}{\sigma}+\frac{\left(\Delta y_{ijkl}-\eta _{ij}\Delta t_{ijkl}\right)^{2}}{\sigma^{3}\Delta t_{ijkl}}\right].\]  
Then, expectations of the negative second partial derivatives of \(\ln L\) with each parameter were derived to obtain the Fisher information matrix, \(\mathbf{F}\)[42]. From above, the second partial derivatives of the \(\ln L\) are shown in Appendix A. Finally, \(\mathbf{F}\) and the determinant of \(\mathbf{F}\), \(|\mathbf{F}|\), are given by  
\[\mathbf{F}=\frac{Nt_{M}}{\sigma^{2}}\left[ \begin{array}{cccc}\sum_{i=1}^{r_ {1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}&\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{ 2}}\pi_{ij}\eta_{ij}^{2}T_{i}&\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta _{ij}^{2}V_{j}&\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}V _{j}&0\\ &\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}^{2}&\sum_{i=1} ^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}V_{j}&\sum_{i=1}^{r_{1}} \sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}^{2}V_{j}&0\\ &&\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}V_{j}^{2}&\sum_{i=1} ^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}V_{j}^{2}&0\\ &&\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}^{2}V_{j}^{2}&0 \\ &&\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}\eta_{ij}^{2}T_{i}^{2}V_{j}^{2}&0 \\ &&2m/t_{M}\end{array} \right] \tag{3}\]  
\[|\mathbf{F}|=\frac{2 mN}{\sigma^{2}}\eta_{11}^{2}\eta_{12}^{2}\eta_{21}^{2}\eta _{22}^{2}\pi_{11}\pi_{12}\pi_{21}\pi_{22}=\frac{2 mN}{\sigma^{2}}\exp(8\delta_ {1}+4\delta_{2}+4\delta_{3}+2\delta_{4})\pi_{11}\pi_{12}\pi_{21}\pi_{22}. \tag{4}\]  
The ratios \((\pi_{11},\pi_{12},\pi_{21}\) and \(\pi_{22})\) are decision variables and \(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\), \(\delta_{4}\) and \(2 mN/\sigma^{2}\) are constants in Equations (3) and (4). From Equation (4), it can be shown that \(|\mathbf{F}|\) is maximized when all decision variables are equal to 0.25. Therefore, a lemma was derived as follows. **Lemma 1**. : _The optimal specimen ratios for four test points, which are a combination of the use and the maximum stress levels with two stress variables, are equal to 0.25 based on D-optimality criteria._  
In this phase, for optimal test plans, the same specimen quantities were allocated to all test points according to Lemma 1. The parameters of the model were estimated based on information from the tests. Second Phase: The Practical PADT Plan Based on the Concept of the Compromise Plan with Three Stress Levels  
In the second phase, three stress levels with two stress variables were considered to detect the curvature effect of the acceleration model based on the concept of a compromise plan [1; 24; 37]. In addition, the lifetime information was precisely estimated through the relationship between the drift and stress variables. For these reasons, at least three stress levels are considered for most reliability tests in various manufacturing industries (Figure 3).