## 4 Continuous Casting-Hot Rolling Scheduling Model Considering Reheating Conversion Mode
### Modeling of Continuous Casting-Hot Rolling Process Scheduling Problem
Depending on the type of batches, the energy waste is calculated from Equations (6)-(21) in Section 3. However, the scheduling of continuous casting-hot rolling is limited by the delivery time and the rolling operating rate, and the scheduling of batch packages cannot occupy the rolling line indefinitely. Therefore, in addition to considering the related energy waste, the maximum completion time should also be considered during scheduling. The objective function of continuous casting and hot rolling can be expressed as the weighted coupling of energy waste and completion time. Since energy waste and completion time have different dimensions, they need to be normalized before their weighting. A min-max normalization method was used to map the energy and time data to the range [0,1]. Meanwhile, the weighting coefficients were defined as \(\alpha=\beta=0.5\). \[f^{*}=\frac{f-f_{min}}{f_{max}-f_{min}} \tag{23}\]  
where \(f\) is the raw data of gas waste and completion time, \(f^{*}\) is the normalized data, and \(f_{max}\) and \(f_{min}\) are the maximum and minimum values of the raw data. After normalization and weighting, the final objective function for continuous casting-hot rolling scheduling can be obtained:  
\[min(f_{1}^{*}\times\alpha+f_{2}^{*}\times\beta) \tag{24}\]  
Constraints:  
Once the cast starts, no interruption is allowed in the process, so there exists an end time of the cast \(c\) on the equipment \(m\) equal to its start time plus its processing time. \[t_{\text{m,c}}^{\text{f}}=t_{\text{m,c}}^{\text{s}}+T_{c,m},\forall c\in C,m\in M \tag{25}\]  
A batch cannot be interrupted during rolling, so the end time of batch \(b\) on equipment \(m\) is equal to its start time plus its processing time. \[t_{\text{m,b}}^{\text{f}}=t_{m,b}^{\text{s}}+T_{b,m},\forall b\in B,m\in M \tag{26}\]  
Equation (27) determines the vacant distance, i.e., the sum of the end time of the last CCR-type slab in the reheating furnace, and the vacant time is equal to the start time of the next DHCR or HCR type slab. \[t_{\text{m},\text{a},\text{y}}^{s}=t_{\text{m},\text{a}^{\prime},\text{y}^{\prime} }^{f}+T_{m,\text{a},\text{a}^{\prime}} \tag{27}\]  
\(\forall y\in Y_{CCR},y^{\prime}\in Y_{HCR\text{ }or\text{ }DHCR},a,a^{\prime}\in A,V_{m, \text{a},\text{a}^{\prime}}=1,m\in M_{k},k=2\)  
Equation (28) ensures that the time interval between continuous casting and hot rolling is within a certain time range, which is related to the batch type. \[T_{c,y}^{min}\leq t_{\text{m},\text{b},\text{y}}^{s}-t_{\text{m}^{\prime}, \text{c},\text{y}}^{f}\leq T_{c,y}^{max},\forall c\in b(c),y\in Y,m,m^{\prime}\in M \tag{28}\]  
Equation (29) ensures that the actual production quantity of all rolling production lines must be limited by their production capacity. \[\varepsilon_{\text{gy}}(t)\leq\beta_{\text{gy}},\forall e\in E,y\in Y,t\in T \tag{29}\]  
Equations (30) and (31) express the processing sequence between two adjacent slabs, ensuring the first-in-first-out principle of the reheating furnace. \[t_{\text{a}}^{s}+\text{SetTime}_{\text{a},\text{a}^{\prime}}+BG\Big{(}3-X_{ \text{a},\text{g}}-X_{\text{a}^{\prime},\text{g}}-X_{\text{a},\text{a}^{ \prime},\text{g}}\Big{)}<t_{\text{a}^{\prime}}^{s}\ \forall a,a^{\prime}\in A \tag{30}\]  
\[t_{\text{a}}^{f}+\text{SetTime}_{\text{a},\text{a}^{\prime}}+BG\Big{(}3-X_{ \text{a},\text{g}}-X_{\text{a}^{\prime},\text{g}}-X_{\text{a},\text{a}^{ \prime},\text{g}}\Big{)}<t_{\text{a}^{\prime}}^{f}\ \forall a,\text{a}\in A \tag{31}\]  
Equation (32) indicates that the reheating time of the slab in the reheating furnace must be within the specified time range. \[f^{*}=\frac{f-f_{min}}{f_{max}-f_{min}} \tag{23}\]0  
Equation (33) ensures that the entering-furnace time of the slab is after the continuous casting stage. A DHCR-type slab can be fed into the furnace when the slab is finished casting in the continuous casting machine, while the HCR- and CCR-type slabs can be fed into the furnace only after all the slabs in the cast have been processed. \[t_{\text{a}}^{s}\geq\left\{