## 3 Problem Formulation
### Problem Definition
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Descriptions** \\ \hline _Indices_ & \\ \(z\) & Index of products, \(z=1,2,\cdots,N_{P}\) \\ \(j\) & Index of jobs, \(j=1,2,\cdots,N_{Q}\) \\ \(i\) & Index of processing operations, \(i=1,2,\cdots,N_{O}\) \\ \(a\) & Index of assembly operations, \(a=1,2,\cdots,N_{A}\) \\ \(m\) & Index of processing machines, \(m=1,2,\cdots,N_{M}\) \\ \(t\) & Index of transportation vehicles, \(t=1,2,\cdots,N_{T}\) \\ \(w\) & Index of transportation vehicle load transportation, \(w=1,2,\cdots,N_{tw}\) \\ \(k\) & Index of assembly machines, \(k=1,2,\cdots,N_{K}\) \\ _Parameters_ & \\ \(\mathbf{M}\) & Set of processing machines, \(\mathbf{M}=\{M_{1},\cdots,M_{m},\cdots,M_{N_{M}}\}\) \\ \(\mathbf{V}\) & Set of transportation vehicles, \(\mathbf{V}=\{V_{1},\cdots,V_{t},\cdots,V_{N_{T}}\}\) \\ \(\mathbf{K}\) & Set of assembly machine, \(\mathbf{K}=\{K_{1},\cdots,K_{k},\cdots,K_{N_{K}}\}\) \\ \(\mathbf{P}\) & Set of products, \(\mathbf{P}=\{P_{1},\cdots,P_{2},\cdots,P_{N_{P}}\}\) \\ \(N_{p}\) & Number of products \\ \(N_{Q}\) & Number of jobs \\ \(N_{O}\) & Number of processing operations \\ \(N_{A}\) & Number of assembly operations \\ \(N_{M}\) & Number of processing machines \\ \(N_{T}\) & Number of transportation vehicles \\ \(N_{K}\) & Number of assembly machines \\ \(N_{O^{\prime}}^{m},N_{A}^{k}\) & Number of operations on processing machine \(m\) and assembly machine \(k\) \\ \(N_{tw}\) & Number of load transformations by vehicle \(t\) \\ \(G_{max}\) & Maximum load capacity of vehicle \\ \(j_{j^{\prime}},\mathbf{J_{z}}\) & Job indexed by \(j\), set of all jobs belonging to \(P_{z}\) \\ \(O_{j^{\prime}}^{\prime},\mathbf{O_{j}}\) & \(i\)-th processing operation of \(J_{j}\), set of all operations of \(J_{j}\) \\ \(A_{z}^{2},\mathbf{A_{z}}\) & \(a\)-th assembly operation and set of assembly operations of \(P_{z}\) \\ \(\mathbf{M_{j}^{i}}\) & Set of machines available for processing operation \(O_{j}^{i}\) \\ \(P_{P_{m}},P_{R_{k}}\) & Unit power consumption of processing machine \(m\) and assembly machine \(k\) in working mode \\ \(Pl_{m},P_{I_{k}}\) & Unit power consumption of processing machine \(m\) and assembly machine \(k\) in standby mode \\ \(PS_{m}\) & Unit power consumption of processing machine \(m\) in setup mode \\ \(PT_{t}\) & Unit power consumption of transportation vehicle \(t\) \\ \(Dis_{p-A}\) & Distance between processing shop and assembly shop \\ \(V_{0},V_{t}^{w}\) & Speed of transportation vehicles when unloaded and speed of transportation vehicle \(t\) during \\ \(V_{t}^{w},P(n_{t}^{k})\) & \(w\)-th load transportation \\ \(SP_{(t_{1}^{w},P_{t}^{n})}\) & Processing time of operation \(\pi_{i}^{m}\) and \(n_{a}^{k}\) \\ \(SP_{(t_{1}^{w},\pi_{i}^{m})}\) & Setup time of operation \(\pi_{i-1}^{m}\) and \(\pi_{i}^{m}\) \\ \(IP(\pi_{i-1}^{w},\pi_{i}^{m})\) & Idle time between operation \(\pi_{i-1}^{m}\) and \(\pi_{i}^{m}\) excluding setup time \\ \(IP(\pi_{a-1}^{w},\pi_{a}^{k})\) & Idle time between operation \(\pi_{a-1}^{m}\) and \(\pi_{a}^{m}\) \\ \(RT(\pi_{1}^{m})\) & Release time of job \(\pi_{1}^{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations of mathematical model. The optimization objectives of this problem are to simultaneously minimize \(C_{\text{max}}\) and \(TEC\). It is a multi-objective problem and encompasses several sub-problems, including the sequencing of processing operations, assignment of processing machines, sequencing of job transportation, assignment of vehicles, sequencing of product assembly, sequencing of assembly operations, and assignment of assembly machines. The computational complexity of the mathematical model can be calculated as \(O=((N_{\text{O}}! )\times N_{M}N_{O}\times(N_{\text{Q}}! )\times N_{\text{T}}N_{ \text{Q}}\times(N_{\text{P}}! )\times N_{\text{K}}N_{A})\). So as the problem size increases, the complexity grows exponentially.