## 4 Design of Practical PADT Plan with Two Stress Variables
### First Phase: PADT Plan with Two Stress Levels by Each Variable
To optimize a PADT plan based on the compromise concept with three stress levels, that is, to find the decision variables of proposed PADT plans, the Fisher information matrix for the second phase should be derived in a similar fashion as the first phase. The degradation characteristic and the lifetime (or failure time) were defined according to Assumption 3. The lifetime was determined by the probability density function, \(g_{ij}(t)\), of an inverse Gaussian distribution. If \(t_{q,(i,j)}\) is the \(q\)-th quantile of the distribution at stress levels (\(T_{i},V_{j}\)), then \(t_{q,(i,j)}\) can be approximately defined as [37; 43]. \[g_{ij}(t)=\sqrt{\frac{w^{2}}{2\pi\sigma^{2}t^{3}}}\exp\biggl{[}-\frac{\left\{ \eta_{ij}t-w\right\}^{2}}{2\sigma^{2}t}\biggr{]},\quad t>0\]  
\[t_{q,(i,j)}\approx\frac{\left[z_{q}\sigma+\sqrt{z_{q}^{2}\sigma^{2}+4\eta_{ij }w}\right]^{2}}{4\eta_{ij}^{2}}\]  
where \(z_{q}\) is the \(q\)-th quantile of the standard normal distribution. In addition, given that the drift at the use conditions of the two stress variables is \(\eta_{11}=\eta(T_{1},V_{1})=\eta(0,0)=\exp(\delta_{1})\), the MLE of \(t_{q,(1,1)}\) is given by  
\[\widehat{t_{q,(1,1)}}\approx\frac{\left[z_{q}\phi+\sqrt{z_{q}^{2}\phi^{2}+4 \exp\bigl{(}\delta_{1}\bigr{)}w\bigr{]}^{2}}}{4\exp\bigl{(}2\hat{\delta}_{1} \bigr{)}}\]  
where \(\delta_{1}\) is the MLE of \(\delta_{1}\) and \(\hat{\sigma}\) is the MLE of \(\sigma\). Then, the first partial derivatives of \(t_{q,(1,1)}\) with respect to each parameter, are given by  
\[\mathbf{h}=\left[ \begin{array}{c}h_{1}\\ h_{2}\\ h_{3}\\ h_{4}\\ h_{5}\end{array} \right]=\left[ \begin{array}{c}\frac{\partial t_{q,(1,1)}}{ \partial\delta_{1}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{2}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{3}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{4}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{4}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{4}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{4}}\\ \frac{\partial t_{q,(1,1)}}{\partial\delta_{4}}\\ \end{array} \right]=\left[ \begin{array}{c}\frac{w\bigl{(}z_{q}\sigma+\sqrt{4 \exp(\delta_{1})w+z_{q}^{2}\sigma^{2}}\bigr{)}}{\exp(\delta_{1}\sqrt{4\exp( \delta_{1})w+z_{q}^{2}\sigma^{2}}\bigr{)}}-\frac{\bigl{(}z_{q}\sigma+\sqrt{4 \exp(\delta_{1})w+z_{q}^{2}\sigma^{2}}\bigr{)}^{2}}{2\exp(2\delta_{1})}\\ 0\\ 0\\ 0\\ \frac{z_{q}\sigma+\sqrt{4\exp(\delta_{1})w+z_{q}^{2}\sigma^{2}}}{2\exp(2 \delta_{1})}\biggl{(}z_{q}+\frac{z_{q}^{2}\sigma^{2}}{\sqrt{4\exp(\delta_{1}) w+z_{q}^{2}\sigma^{2}}}\biggr{)}\end{array} \right]\]  
Figure 3: The test points and the individual stress levels in the second phase. Let \(\mathbf{F}^{-1}\) be the inverse of the Fisher information matrix, then the asymptotic variance of the MLE of \(t_{q,(1,1)}\) can be obtained by the inverse of the Fisher information matrix, \(\mathbf{F}^{-1}\), and the above vector \(\mathbf{h}\) as follows. \[\mathbf{F}^{-1}=\frac{\sigma^{2}}{Nt_{M}}\left[ \begin{array}{cccc}f_{11}^{-1}& f_{12}^{-1}&f_{13}^{-1}&f_{14}^{-1}&0\\ &f_{22}^{-1}&f_{23}^{-1}&f_{24}^{-1}&0\\ &f_{33}^{-1}&f_{34}^{-1}&0\\ &&f_{44}^{-1}&0\\ &\text{\em symmetric}&f_{55}^{-1}\end{array} \right],\]  
\[\text{\em Avar}\Big{(}\widehat{t_{q,(1,1)}}\Big{)}=\mathbf{h}^{\text{T}} \mathbf{F}^{-1}\mathbf{h}\,=\frac{\sigma^{2}}{Nt_{M}}\Big{[}f_{11}^{-1}h_{1}^{ 2}+f_{55}^{-1}h_{5}^{2}\Big{]} \tag{5}\]  
where \(f_{11}^{-1}=\frac{\pi_{22}\pi_{31}\pi_{33}\eta_{22}^{2}\eta_{33}^{2}(T_{2}-1 )^{2}V_{2}^{2}+\pi_{13}\eta_{13}^{2}\Bigg{(} \begin{array}{c}\pi_{22}\pi_{33} \eta_{22}^{2}\eta_{33}^{2}T_{2}^{2}(V_{2}-1)^{2}\\ +\pi_{31}\eta_{31}^{2}(\pi_{33}\eta_{33}^{2}+\pi_{22}\eta_{22}^{2}T_{2}V_{2} ^{2})\end{array} \Bigg{)}}}{\pi_{13}\pi_{22}\pi_{31}\pi_{33}\eta_{13}^{2}\pi_{2 2}^{2}\eta_{31}^{2}\pi_{33}^{2}(T_{2}-1)^{2}(V_{2}-1)^{2}+\pi_{11}\eta_{11}^{2 }\Bigg{(}