## 2 Problem Formulation
### Robust Optimization Model for FAJSP  
In this paper, a robust optimization model for the FAJSP is constructed. Some notations used for formulating the mathematical model are listed in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Descriptions** \\ \hline \(I\) & Index set of the operations \\ \(K\) & Index set of the machines \\ \(k_{i}\) & Set of candidate machines for operation \(i\), \(i\in I\) \\ \(CA\) & Set of operation pairs \((i,\,j)\), where operation \(i\) is the direct predecessor of \\  & operation \(j,i,j\in I\) \\ \(P_{ik}\) & Standard processing time of operation \(i\) on machine \(k\), \(i\in I\), \(k\in k_{i}\) \\ \(\widetilde{P}_{ik}\) & Actual processing time of operation \(i\) on machine \(k\), \(i\in I\), \(k\in k_{i}\) \\ \(S_{ik}\) & Start time of operation \(i\) on machine \(k\), \(i\in I\), \(k\in k_{i}\) \\ \(M\) & A large positive number \\ \(a\) & Risk level of chance constraint \\ \(\tilde{P}_{ik}^{max}\) & The upper limit of the magnitude of \(\widetilde{P}_{ik}\) fluctuations \\ \(\hat{P}_{ik}^{1-\eta}\) & The upper limit of the magnitude of \(\widetilde{P}_{ik}\) fluctuations at a confidence level of \\ \(C_{max}\) & Maximum completion time \\ \(S_{ik}\) & Start time of operation \(i\) on machine \(k\), \(i\in I\), \(k\in k_{i}\) \\ \(z_{ik}\) & \(z_{ik}=1\) if operation \(i\) is processed on machine \(k\); otherwise, \(z_{ik}=0\) \\ \(x_{ijk}\) & \(x_{ijk}=1\) if operation \(i\) starts before operation \(j\) on machine \(k\); otherwise, \(x_{ijk}=0\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notations and descriptions.  
Figure 1: Tree-like assembly structure of product A.  
The robust optimization model (Model_1) is presented as follows:  
\[minimizeC_{max} \tag{1}\]  
subject to  
\[\sum_{k\in K_{i}}z_{ik}=1,\ \ \forall i \tag{2}\]  
\[S_{ik}\leq M\cdot z_{ik},\ \ \forall i;\ \ \forall k\in K_{i} \tag{3}\]  
\[S_{jk}+M\cdot\left(3-z_{ik}-z_{jk}-x_{ijk}\right)\geq S_{ik}+\widetilde{P}_{ik},\ \ \forall i,j;k\in K_{i}\cap K_{j} \tag{4}\]  
\[S_{ik}+M\cdot\left(x_{ijk}+2-z_{ik}-z_{jk}\right)\geq S_{jk}+\widetilde{P}_{jk},\ \ \forall i,j;k\in K_{i}\cap K_{j} \tag{5}\]  
\[\sum_{k\in K_{j}}S_{jk}\geq\sum_{k^{\prime}\in K_{i}}S_{ik^{\prime}}+\sum_{k^{ \prime}\in K_{i}}z_{ik^{\prime}}\cdot\widetilde{P}_{ik^{\prime}},\forall(i,j )\in CA;k^{{}^{\prime}}\in K_{i};\forall k\in K_{j} \tag{6}\]  
\[C_{max}\geq\sum_{k\in K_{i}}S_{nk}+\sum_{k\in K_{n}}z_{nk}\cdot\widetilde{P}_ {nk},\ \ \forall k\in K_{n} \tag{7}\]  
\[S_{ik}\geq 0,\ \ \forall i;\forall k\in K_{i} \tag{8}\]  
\[x_{ijk}\in(0,1),\ \ \forall i,j;\forall k\in K_{i}\cap K_{j} \tag{9}\]  
\[z_{ik}\in(0,1),\ \ \forall i;\forall k\in K_{i} \tag{10}\]  
Equation (2) indicates that each operation can and must be assigned to only one machine from the set of candidate machines. Equation (3) states that if an operation is not assigned to a machine, its start time on that machine is set to zero. Equations (4) and (5) specify that on the same machine, a new operation can only begin after the completion of the previous operation. Equation (6) represents the assembly sequence constraint, meaning that for operations with assembly sequence requirements, the subsequent operation can only start after all its immediate predecessor operations are completed. Equation (7) provides the calculation method for the maximum completion time, where the maximum completion time equals the completion time of the final operation. Equations (8)-(10) constrain the range of values for the decision variables.  
In Model_1, the processing time \(\widetilde{P}_{ik}\) in Equations (4)-(7) is an uncertain parameter. The feasible solution of the model must satisfy all constraints under all possible uncertain scenarios within the range of \(\widetilde{P}_{ik}\). Due to the presence of these constraints, Model_1 is a semi-continuous model and cannot be solved directly. To address this issue, the chance-constrained method is employed to reformulate Model_1, converting it into a deterministic model for solving.