## 3 Planning Strategy and Basic Assumptions
### Basic Assumptions  
In this paper, we made the following two-phase PADT plan assumptions:  
**Assumption 1**. : _Each stress level (\(T_{i}^{\prime}\) and \(V_{j}^{\prime}\) where \(i=1,2,\cdots,r_{1}\) and \(j=1,2,\cdots,r_{2}\)) is loaded following the CS loading scheme. \(T_{i}^{\prime}\) and \(V_{j}^{\prime}\) represent the \(i\)-th temperature stress level (in Kelvin) and \(j\)-th non-thermal stress level (voltage, vibration, etc. ), respectively. \(N\) is the pre-determined total number of specimens and each \(un\) it, and \(n_{ij}\), is allocated to each stress level such that_  
\[n_{ij}=\pi_{ij}N,\quad\sum_{i=1}^{r_{1}}\sum_{j=1}^{r_{2}}\pi_{ij}=1,\quad\pi_ {ij}\geq 0\]  
**Assumption 2**. : _The maximum test duration (\(t_{M,ijk}\)) and measurement times (\(t_{ijkl}\), \(k=1,2,\cdots,n_{ij}\), \(l=1,2,\cdots,m\)) are given, with their equivalent intervals (\(t_{l}-t_{l-1}=\Delta t\)). Thus, it is assumed that the maximum test duration is equal to \(m\Delta t\Big{(}=t_{M,ijk}\Big{)}\) for all \(i\), \(j\), and \(k\)._  
**Assumption 3**. : _The degradation (or performance) characteristic \(y_{ijk}(t)\) of the \(k\)-th unit at the stress levels (\(T_{i}^{\prime}\) and \(V_{j}^{\prime}\)) follows the WP with a drift \(\eta^{\prime}\left(T_{i}^{\prime},V_{j}^{\prime}\right)\) and diffusion constant \(\sigma^{2}\). It is assumed that a failure of the test item is reached when \(y_{ijk}(t)\) becomes greater than the predetermined critical value \(\omega\)._  
**Assumption 4**. : _The use and the maximum stress levels are given (or pre-specified) as (\(T_{1}^{\prime}\) and \(V_{1}^{\prime}\)) and (\(T_{r_{1}}^{\prime}=T_{M}^{\prime}\) and \(V_{r_{2}}^{\prime}=V_{M}^{\prime}\)), respectively. In addition, the generalized Eyring model, which is one of the life-stress relationship models, is assumed to express the relationship between the drift \(\eta^{\prime}\left(T_{i}^{\prime},V_{j}^{\prime}\right)\) and the two stress variables \(T_{i}^{\prime}\) and \(V_{j}^{\prime}\), as follows [38]._  
\[\eta^{\prime}\left(T_{i}^{\prime},V_{j}^{\prime}\right)=\delta_{1}^{\prime}\exp \left(\frac{\delta_{2}^{\prime}}{T_{i}^{\prime}}+\delta_{3}^{\prime}V_{j}^{ \prime}+\delta_{4}^{\prime}\frac{V_{j}^{\prime}}{T_{i}^{\prime}}\right)\]  
_where \(\delta_{1}^{\prime}(>0)\), \(\delta_{2}^{\prime}\), \(\delta_{3}^{\prime}\) and \(\delta_{4}^{\prime}\) are unknown constants which satisfy \(\delta_{2}^{\prime}+\delta_{4}^{\prime}V_{j}^{\prime}<0\) and \(\delta_{3}^{\prime}+\delta_{4}^{\prime}/T_{i}^{\prime}>0\) for all \(i\) and \(j\)._  
**Assumption 5**. : _For each stress variable, the use and maximum stress levels were considered to estimate the parameters of the acceleration model in the first phase, and the middle-stress level was considered to identify the validity of the model and estimate the lifetime in the second phase._  
The WP, which is one of the stochastic processes, is widely applied to degradation path modeling [25; 26; 27; 28; 39]. In addition, WP is often used with degradation test design due to its mathematical tractability [24; 29; 30; 31; 32; 40; 41]. The WP was also adopted for modeling the random behavior of the degradation characteristic in this paper. The WP, \(W(t)\) (with drift \(\eta\) and diffusion constant \(\sigma^{2}\)) has the following properties:  
(1) \(W(0)=0\),  
(2) \(\{W(t)\,|\,t\geq 0\}\) has stationary and independent increments, and  
(3) for \(0\leq t_{1}<t_{2}\), the increment, \(\Delta W\) (=\(W(t_{2})-W(t_{1})\)), is normally distributed with mean \(\eta(t_{2}-t_{1})\) and variance \(\sigma^{2}(t_{2}-t_{1})\). Let \(Y(t)\) be the degradation characteristic of the product at time \(t\), and define \(y(t)=Y(t)-Y(0)\). In this paper, the degradation increment \(y(t)\) followed a WP, and we assumed that the drift \(\eta\) depends only on the stress level, whereas the diffusion constant \(\sigma^{2}\) does not. At \(t_{ijkl}\), the degradation characteristic, \(y_{ijkl}\), follows a normal distribution with mean \(\eta_{ij}t_{ijkl}\) and variance \(\sigma^{2}t_{ijkl}\).