## 4 Continuous Casting-Hot Rolling Scheduling Model Considering Reheating Conversion Mode
### Modeling of Continuous Casting-Hot Rolling Process Scheduling Problem  
When establishing the continuous casting-hot rolling scheduling model, not only should the capacity constraints between these two processes be considered but also the impact of the connection between cold- and hot-charging slabs on the energy consumption of the reheating furnace and the production rhythm of the continuous casting-hot rolling process. The established model is shown below. Parameters  
\(C\)--Total casts of the continuous casting process;  
\(c\)--A cast in the continuous casting process;  
\(B\)--Total batches of the hot-rolling process;  
\(b\)--A batch in the hot-rolling process;  
\(A\)--Total slabs in the reheating furnace;  
\(a\)--A slab in the reheating furnace;  
\(N\)--The total number of all batches in the hot-rolling process;  
\(Y\)--Total batch types (DHCR, HCR, CCR);  
\(y\)--One batch type;  
\(k\)--Production stage, where \(k=1\) represents the continuous casting stage and \(k=2\) represents the hot-rolling stage;  
\(M_{k}\)--Collection of all equipment in the production stage \(k\);  
\(m\)--Processing equipment;  
\(g\)--reheating furnace;  
\(T_{c,m}\), \(T_{b,m}\)--Processing time of cast \(c\) or batch \(b\) on equipment \(m\);  
\(T_{c,y}^{max}\), \(T_{c,y}^{min}\)--The maximum and minimum time intervals between continuous casting and hot rolling for cast \(c\) with batch type \(y\);  
\(i\), \(j\)--Number of batches in the hot-rolling process, \(i=1\), \(2\),..\(N\), \(j=1\), \(2\),..\(N\);  
\(Q_{b,i,j}\)--Energy waste between batch \(b_{i}\) and \(b_{j}\), \(i,j\in N\), \(b_{i}\in B\), \(b_{j}\in B\);  
\(T_{m,a,b}\)--A certain vacant time of the reheating furnace between slabs \(a^{\prime}\) and \(a\);  
SetTime\({}_{a,a^{\prime}}\)--Preparation time for two adjacent slabs \(a\) and \(a^{\prime}\) entering or exiting the reheating furnace;  
\(BG\)--A big number;  
\(T_{a}^{max}\), \(T_{a}^{min}\)--Maximum and minimum residence time of slab \(a\) in the reheating furnace;  
\(Slab_{c,a}\)--The discharge time of slab \(a\) within cast \(c\) during the continuous casting stage;  
\(b(c)\)--The cast \(c\) belongs to the hot-rolling batch \(b\);  
\(Cap\)--The maximum number of slabs that can be produced in the reheating furnace;  
\(U_{t}\)--The number of slabs in the reheating furnace at one time \(t\);  
\(\beta_{ey}\)--The maximum production capacity of hot-rolled products with specification type \(e\) and batch type \(y\) produced during the hot-rolling stage;  
Variables:\(T\)--Production time;  
\(t_{\text{m,c}}^{\text{s}}\), \(t_{\text{m,c}}^{\text{f}}\), \(t_{\text{m,b}}^{\text{s}}\), \(t_{\text{m,b}}^{\text{f}}\)--The start and finish times of cast \(c\) or batch \(b\) on equipment \(m\);  
\(t_{\text{m,c,y}}^{\text{f}}\), \(t_{\text{m,b,y}}^{\text{s}}\),\(t_{\text{m,a,y}}^{\text{f}}\)--The start and finish times of cast \(c\), batch \(b\), or slab \(a\) with batch type \(y\) on equipment \(m\);  
\(t_{\text{a}}^{\text{s}}\), \(t_{\text{a}}^{\text{f}}\)--The entry and exit time of slab \(a\) on the reheating furnace;  
\(V_{m,a,a^{\prime}}\)--True when slab \(a\) is produced after \(a^{\prime}\);  
\(X_{a,g}\)--True when slab \(a\) is produced in the reheating furnace \(g\);  
\(X_{a,a^{\prime},g}\)--True when both slabs \(a\) and \(a^{\prime}\) are produced on the heating furnace \(g\), and the slab immediately following \(a\) is \(a^{\prime}\);  
\(e_{\text{eg}}(t)\)--The actual production quantity of hot-rolled products with specification type \(e\) and batch type \(y\) produced during the hot-rolling stage in time \(t\);  
Objective function:  
From the research on the conversion mode of the slab in Section 3, the reheating furnace conversion mode can be used to obtain the minimum gas energy waste in the continuous casting-hot rolling system. \[f=min\sum_{i=1}^{N-1}Q_{b_{i},b_{i+1}},i,i+1\in N \tag{22}\]  
where \(Q_{b_{i},b_{i+1}}\) represents the energy waste between adjacent batch \(i\) and batch \(i+1\).