## 3 Formulation of Multi-Criteria Optimization Problem
### Mathematical Modeling for the DDF/SP-VPT  
Some notations used to describe the mathematical model are listed in Table 1.  
Taking the makespan, tardiness, and total factory load as the optimization objectives, the mathematical model of the DDF/SP-VPT is expressed as follows:  
\[\min:\ \mathbf{F}=(f_{1},f_{2},f_{3})^{\mathrm{T}}, \tag{1}\]  
\[f_{1}=\max_{1\leq i\leq n}C_{in_{i}}, \tag{2}\]  
\[f_{2}=\sum_{i=1}^{n}\max(C_{i}-DD_{i},0), \tag{3}\]  
\[f_{3}=\sum_{l=1}^{w}\sum_{k=1}^{m_{l}}Q^{jk}, \tag{4}\]  
Subject to:  
\[E_{r+1}^{lk}\geq E_{r}^{lk}+\sum_{i=1}^{n}\sum_{j=1}^{n_{l}}p_{ij}^{lk}*y_{ij}^ {lk},\ l=1,2,\ldots,w,\ k=1,2,\ldots,m_{l},\ r=1,2,\ldots,q_{lk}-1, \tag{5}\]  
\[\sum_{l\in\delta_{i}}z_{i}^{l}=1,\ i=1,2,\ldots,n, \tag{6}\]  
\[\sum_{l\in\delta_{i}}\sum_{k\in\Omega_{ij}^{l}}x_{ij}^{lk}=1,\ i=1,2,\ldots,n,\ j=1,2,\ldots,n_{l}, \tag{7}\] \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description** \\ \hline Parameters: & \\ \(n\) & Number of jobs \\ \(l_{i}\) & The \(i\)th job \\ \(n_{i}\) & Number of operations in job \(l_{i}\) \\ \(w\) & Number of factories \\ \(U_{l}\) & The \(l\)th factory \\ \(m_{i}\) & Number of machines in factory \(U_{l}\) \\ \(\Omega_{ij}\) & The \(j\)th operation in job \(l_{i}\) \\ \(M_{lk}\) & The \(k\)th machine in factory \(U_{l}\) \\ \(\Omega_{ij}^{l}\) & Set of machines in \(U_{l}\) that can process \(O_{ij}\) \\ \(p_{ij}^{lk}\) & Theoretical processing time of operation \(O_{ij}\) on machine \(M_{lk}\) \\ \(\tilde{p}_{ij}^{lk}\) & Actual processing time of \(O_{ij}\) on \(m_{lk}\), a random variable with \(\tilde{p}_{ij}^{lk}\sim N(p_{ij}^{lk},e_{ij}^{lk})\) \\ \(\tilde{k}_{i}\) & Set of processable factories for \(l_{i}\) \\ \(DD_{i}\) & Due date of \(l_{i}\) \\ \(L\) & A sufficiently large constant \\ Decision Variables: & \\ \(t_{l}\) & Implementation time point of the adjustment strategy, \(l=1,2,\ldots\) \\ \(x_{ij}^{lk}\) & If operation \(O_{ij}\) is processed on \(M_{lk}\), then \(x_{ij}^{lk}=1\), else \(x_{ij}^{lk}=0\) \\ \(y_{ij}^{lk}\) & If \(O_{ij}\) is the \(r\)th operation processed on \(M_{lk}\), then \(y_{ij}^{lk}=1\), else \(y_{ij}^{lk}=0\) \\ \(z_{i}^{l}\) & If \(l_{i}\) is processed in \(U_{l}\), then \(z_{i}^{l}=1\), else \(z_{i}^{l}=0\) \\ \(S_{ij}^{l}\) & Start time of operation \(O_{ij}\) \\ \(C_{ij}\) & Completion time of operation \(O_{ij}\) \\ \(S_{ij}^{lk}\) & Start processing time of operation \(O_{ij}\) on \(M_{lk}\) \\ \(C_{lk}^{lk}\) & Completion time of operation \(O_{ij}\) on \(M_{lk}\) \\ \(E_{lk}\) & Start processing time of the \(r\)th operation on \(M_{lk}\) \\ \(Q^{jk}\) & Cumulative processing load on machine \(M_{lk}\) \\ \(q_{lk}\) & Number of operations processed on machine \(M_{lk}\) \\ \(C_{i}\) & Completion time of \(l_{i}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Notations used to describe the mathematical model. \[C_{ij}=S_{ij}+\sum_{l\in\delta_{i}}\sum_{k\in\Omega_{ij}^{l}}p_{ij}^{lk}*x_{ij}^{lk} =1,\;i=1,2,\ldots,n,\;j=1,2,\ldots,n_{i}, \tag{8}\]  
\[S_{ij+1}\geq C_{ij},\;i=1,2,\ldots,n,\;j=1,2,\ldots,n_{i}-1, \tag{9}\]  
\[E_{r}^{lk}+(1-y_{ij}^{lk})*L\geq S_{ij},\;i=1,2,\ldots,n,\;j=1,2,\ldots,n_{i}, \;l\in\delta_{i}\;\;, \tag{10}\]  
\[k\in\Omega_{ij}^{l},\;r=1,2,\ldots,q_{lk}\]  
\[E_{r}^{lk}\leq S_{ij}+(1-y_{ij}^{lk})*L,\;i=1,2,\ldots,n,\;j=1,2,\ldots,n_{i}, \;l\in\delta_{i}\;\;, \tag{11}\]  
\[k\in\Omega_{ij}^{l},\;r=1,2,\ldots,q_{lk}\]  
Equations (2)-(4) represent the three optimization objectives: makespan, tardiness, and total factory load, respectively. Equation (5) specifies that each machine can process at most one operation at a time. Equation (6) ensures that all operations within the same job must be processed in the same factory. Equation (7) states that each operation can be assigned to only one machine. Equation (8) guarantees that, once a job starts processing, it cannot be interrupted. Equation (9) ensures that the start time of an operation must be no earlier than the completion time of the preceding operation within the same job.