3.1.3. Constraints

Based on the above description, the constraints of the hot-rolled batch processing planning model are as follows:

$$\begin{array}{c} \sum\_{k=1}^{N} y\_{ik} = 1\\ \forall i \in \{1, 2, 3, \dots, N\}, \quad k \in \{1, 2, 3, \dots, M\} \end{array} \tag{2}$$

$$\begin{aligned} \sum\_{i=1}^{N} (L\_i \times y\_{ik}) &\le L\_{\text{max}}\\ \forall i \in \{1, 2, 3, \dots, N\}, \qquad k \in \{1, 2, 3, \dots, M\} \end{aligned} \tag{3}$$

$$\begin{aligned} \sum\_{i=1}^{N} \left( Z\_{ijk} \times L\_i \times y\_{ik} \right) &\le R\_{\text{max}}\\ i, j &\in \{1, 2, 3, \dots, N\}, \qquad k \in \{1, 2, 3, \dots, M\} \end{aligned} \tag{4}$$

$$\begin{aligned} 0 \le X\_{i\bar{j}k} \times \left(\mathcal{W}\_{i} - \mathcal{W}\_{\bar{j}}\right) \le \mathcal{W}\_{\text{max}}\\ i, j \in \{1, 2, 3, \dots, N\}, \quad k \in \{1, 2, 3, \dots, M\} \end{aligned} \tag{5}$$

$$\begin{array}{c} 0 \le X\_{ijk} \times \Delta G\_{i,j} \le G\_{\text{max}}\\ i, j \in \{1, 2, 3, \dots, N\}, \quad k \in \{1, 2, 3, \dots, M\} \end{array} \tag{6}$$

$$\begin{array}{c} 0 \le X\_{ijk} \ltimes \Delta H\_{i,j} \le H\_{\text{max}}\\ i, j \in \{1, 2, 3, \dots, N\}, \quad k \in \{1, 2, 3, \dots, M\} \end{array} \tag{7}$$

$$\begin{aligned} \sum\_{i=1}^{N} X\_{ijk} \cdot \sum\_{j=1}^{N} X\_{ijk} &= 1\\ \text{if, } j \in \{1, 2, 3, \dots, N\}, \quad k \in \{1, 2, 3, \dots, M\} \end{aligned} \tag{8}$$

$$\mathbf{C\_{elc}}\_{t\mathbf{c}} = \begin{cases} \mathbf{C\_{0\prime}} & t \in (0, T\_0] \\ \mathbf{C\_{1\prime}} & t \in (T\_{0\prime}, T\_1] \\ \mathbf{C\_{2\prime}} & t \in (T\_{1\prime}, T\_2] \end{cases} \tag{9}$$

$$X\_{ijk} = \begin{cases} 1, & \text{Slab } j \text{ is behind slab } i, \text{ and belong to same rolling unit } k\\ 0, & \text{else} \end{cases} \tag{10}$$

*Zijk* = ! 1, Slab *j* is behind slab *i* and *j* have same width and belong to same rolling unit *k* 0, else (11)

$$y\_{ik} = \begin{cases} 1, & \text{Slab } i \text{ belongs to rolling unit } k\\ 0, & \text{else} \end{cases} \tag{12}$$

where *F* represents the target penalty value, *N* represents the number of slabs, *T* represents the rolling period. *PW*, *PG* and *P<sup>H</sup>* are the penalty coefficients for the differences in width, thickness, and hardness among adjacent slabs of the same rolling unit, respectively. In the equations, *Yi* represents the penalty coefficient for the electricity spent on the slab *i*, *C*ele,t represent the prices of the electricity spent on the slab *i*, *Pele*,*<sup>i</sup>* is the electricity spend on the slab *i*, *L*max is the length limit for each rolling unit, *R*max is the length limit for the continuously processed slabs of the same width in the same rolling unit, *W*max, *G*max, and *H*max are the upper limits for the differences in width, thickness and hardness among adjacent slabs from the same rolling unit. Constraint (2) ensured that each slab was assigned to one rolling unit; constraints (3) and (4) limited the total length of a single rolling unit and the length of continuously processed slabs of the same width; constraint (5) ensured that slabs of the same rolling unit were arranged in the descending order of width and that the width difference between two adjacent slabs did not exceed the upper limit; constraints (6) and (7) ensured that the differences in thickness and hardness between two adjacent slabs did not exceed the upper limits, respectively; constraint (8) signified that each slab could only be processed once; constraint (9) represented the prices of electricity from different time periods; and constraints (10)–(12) were the decision variables of values of 0, 1, respectively.
