## 3 Problem Description and Modeling
### Problem Modeling  
Before constructing the mathematical model, it is essential to define the parameters and decision variables used in the formulation. Table 2 lists the notations and decision variables involved in the FJSPLSP-LTCC. The objective of this study is to minimize the makespan (the maximum completion time among all jobs). The objective function is formulated as follows:  
\[\text{min}C_{max}=\text{min}\left(\max_{i\in J}C_{i}\right) \tag{2}\]  
Constraints to be considered include the following modules:  
Workpiece processing-related constraints, including  
1. Precedence constraint.  
\[C_{i,k}\geq C_{i,k-1}+p_{i,k,m},\forall i\in J,k\in O_{i},m\in M_{i,k} \tag{3}\]  
2. Uniqueness of processing.  
\[\sum\
olimits_{i\in J}\ \sum\
olimits_{k\in O_{i}}\ x_{i,k,m}\leq 1,\forall m\in M \tag{4}\]  
3. Machine assignment constraint.  
\[\sum\
olimits_{m\in M_{i,k}}\ y_{i,k,m}=1,\forall i\in J,k\in O_{i} \tag{5}\]  
AGV transportation related constraints, including:  
Figure 3: An instance of the FJSPLSP-LTCC.  
4. Transportation constraint: each operation must be transported only once.  
\[\sum\
olimits_{i\in I}\ \sum\
olimits_{k\in O_{i}}\ w_{i,k,a}\leq 1,\forall a\in A \tag{6}\]  
5. AGV assignment constraint.  
\[\sum\
olimits_{a\in A}\ z_{i,k,a}=1,\forall i\in J,k\in O_{i} \tag{7}\]  
6. Complete transportation time constraint: includes transportation, charging, and battery change time.  
\[T_{\text{transport},a}=\begin{cases}\frac{D_{(s,\text{source})}}{\overline{{ \upsilon}_{a}}}+\frac{D_{(\text{source},\text{dest})}}{\overline{{\upsilon}_{ a}}},&d_{a}=0\\ \frac{D_{(s,\text{source})}}{\overline{{\upsilon}_{a}}}+T_{\text{charged}}+ \frac{D_{(s,\text{source})}}{\overline{{\upsilon}_{a}}}+\frac{D_{(\text{source,dest})}}{\overline{{\upsilon}_{a}}},&d_{a}=1\\ \frac{D_{(s,\text{dest})}}{\overline{{\upsilon}_{a}}}+T_{\text{charged}}+ \frac{D_{(s,\text{source})}}{\overline{{\upsilon}_{a}}}+\frac{D_{(\text{source,dest})}}{\overline{{\upsilon}_{a}}},&d_{a}=2\\ \frac{D_{(s,\text{dest})}}{\overline{{\upsilon}_{a}}}+\frac{D_{(s,\text{source })}}{\overline{{\upsilon}_{a}}}+\frac{D_{(\text{source,dest})}}{\overline{{ \upsilon}_{a}}}+T_{\text{maintain}},&m_{a}=1\end{cases} \tag{8}\]  
AGV charging and energy management related constraints, including:  
7. Charging decision logic constraint.  
\[d_{a}\in\{0,1,2\},\forall a\in A \tag{9}\]  
8. Mandatory maintenance battery change constraint.  
\[m_{a}=\begin{cases}1,&s_{a}\leq s_{\text{dead}}\\ 0,&\text{else}\end{cases} \tag{10}\]  
9. Charging time linkage constraint: charging actions must be recorded with corresponding time and charger availability.  
\[T_{\text{charge},a}=\begin{cases}0,&d_{a}=0\\ T_{\text{charge}1}+\frac{D_{(s,\text{e})}}{\overline{{\upsilon}_{a}}}+T_{\text {wait}}(if\ exist),&d_{a}=1\\ T_{\text{charge}2}+\frac{D_{(s,\text{e})}}{\overline{{\upsilon}_{a}}}+T_{\text {wait}}(if\ exist),&d_{a}=2\end{cases} \tag{11}\]  
\[T_{\text{charge}2}=\frac{s_{\text{full}}-s_{a}}{P_{\text{charge}}} \tag{12}\]  
10. AGV battery update rule: AGV battery levels are updated based on consumption models.  
\[s_{a}^{\text{new}}=\begin{cases}s_{a}+T_{\text{charge}1}\cdot P_{\text{charge }},&d_{a}=1\\ s_{\text{full}},&d_{a}=2\\ s_{\text{full}},&m_{a}=1\\ s_{a}-T_{\text{transport},a}^{\text{empty}}P_{\text{empty}}-T_{\text{transport},a}^{ \text{load}}P_{\text{load}},&d_{a}=0\end{cases} \tag{13}\]  
11. Charged exclusivity constraint: each charger can serve only one AGV at a time.  
\[\left\{\begin{array}{l}B_{a_{2}}\geq B_{a_{1}}+C_{a_{1}}-M\cdot(1-y_{a_{1}, a_{2}}),\\ B_{a_{1}}\geq B_{a_{2}}+C_{a_{2}}-M\cdot y_{a_{1},a_{2}},\end{array}\right. \tag{14}\]