## 3 Problem Formulation
### Mathematical Model  
The mathematical model for the three stages with two optimization objectives in the EEMsMIAJSP is constructed as follows: \begin{table}
\begin{tabular}{l l} \hline \hline
**Notations** & **Descriptions** \\ \hline \(\pi^{\text{st1}},\pi^{\text{st2}},\pi^{\text{st3}}\) & Sequence of operations on processing stage, transportation stage, and assembly stage \\ \(ST(\pi_{a}^{k}),CT(\pi_{a}^{k})\) & Start time and completion time of operation \(\pi_{a}^{k}\) \\ \(EC_{1}^{\text{st1}},EC_{1}^{\text{st1}},EC_{1}^{\text{st1}},EC_{1}^{\text{st1}}\) & Release energy consumption, setup energy consumption, processing energy consumption, and idle energy consumption in processing stage \\ \(EC_{1}^{\text{st3}},EC_{1}^{\text{st3}}\) & Assembly energy consumption and idle energy consumption in assembly stage \\ \(EC_{1}^{\text{st1}},EC_{1}^{\text{st2}},EC_{1}^{\text{st3}}\) & Energy consumption in processing stage, transportation stage, and assembly stage \\ \(C_{\text{max}}\) & Maximum completion time of schedule \\ \(TEC\) & Total energy consumption \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
Figure 1: A schematic diagram of the EEMsMIAJSP. The first objective, the maximum completion time, is the time when all products are fully assembled, calculated as follows:  
\[C_{\max}=\max\Big{\{}CT(\pi_{a}^{k})\Big{\}},\forall a,k \tag{1}\]  
The second objective, namely total energy consumption, is composed of three parts: total energy consumption in the processing stage, total energy consumption in the transportation stage, and total energy consumption in the assembly stage. The total energy consumption for each stage is calculated as follows:  
(1) Processing stage: The total energy consumption in this stage is composed of the energy consumption of the processing machines, which is determined by four modes of machines: release mode, setup mode, working mode, and idle mode. The energy consumption for each mode of the processing machines is constructed as follows:  
Release energy consumption: This refers to the energy consumption of machines before the first arrival time of the job. The release energy consumption in the processing stage is calculated as follows:  
\[EC_{R}^{s\text{f1}}=\sum_{m=1}^{N_{M}}PP_{m}\times RT(\pi_{1}^{m}) \tag{2}\]  
Setup energy consumption: This energy consumption is generated during the preparation process of the machine before entering the working mode. The setup energy consumption in the processing stage is calculated as follows:  
\[EC_{S}^{s\text{f1}}=\sum_{m=1}^{N_{M}}\sum_{i=2}^{N_{On}}PS_{m}\times SP(\pi_{ i-1}^{m},\pi_{i}^{m}) \tag{3}\]  
Working energy consumption: This energy consumption is generated when the processing machine is in working mode. The working energy consumption in the processing stage is calculated as follows:  
\[EC_{P}^{s\text{f1}}=\sum_{m=1}^{N_{M}}\sum_{i=1}^{N_{On}}PP_{m}\times P(\pi_{ i}^{m}) \tag{4}\]  
Idle energy consumption: This energy consumption is generated during the time between processing two consecutive operations, excluding setup time. The idle energy consumption in the processing stage is calculated as follows:  
\[EC_{I}^{s\text{f1}}=\sum_{m=1}^{N_{M}}\sum_{i=2}^{N_{On}}PI_{m}\times IP(\pi_{ i-1}^{m},\pi_{i}^{m}) \tag{5}\]  
The total energy consumption in the processing stage is calculated as follows:  
\[EC^{s\text{f1}}=EC_{R}^{s\text{f1}}+EC_{S}^{s\text{f1}}+EC_{P}^{s\text{f1}}+EC _{I}^{s\text{f1}} \tag{6}\]  
(2) Transportation stage: The total energy consumption in this stage consists of the energy consumption of the transportation vehicles, which is composed of two parts: loaded energy consumption and unloaded energy consumption.