*3.1. Mathematical Model*

### 3.1.1. Model Assumptions


### 3.1.2. Objective Function

In this paper, the optimizations of quality and power consumptions from different time periods were taken into consideration. On the basis of setting a certain weight coefficient, an objective function was formulated with the goal of minimizing the differences in width, thickness, and hardness among adjacent slabs of the same rolling unit, as well as the electricity cost of the rolling process. The objective function is presented as

$$\min \quad F = \sum\_{j=1}^{N} \sum\_{i=1, i \neq j}^{N} \left\{ P^W \Delta \mathcal{W}\_{i,j} + P^G \Delta G\_{i,j} + P^H \Delta H\_{i,j} \right\} \cdot X\_{i \mid \overline{k}} + \sum\_{t=1}^{T} \sum\_{i=1}^{N} \left\{ C\_{\text{clc},t} (P\_{\text{clc},i}) \right\} \cdot Y\_i \tag{1}$$

where Δ*Wi*,*<sup>j</sup>* = *Wi* <sup>−</sup> *Wj* , <sup>Δ</sup>*Gi*,*<sup>j</sup>* <sup>=</sup> *Gi* <sup>−</sup> *Gj* , <sup>Δ</sup>*Hi*,*<sup>j</sup>* <sup>=</sup> *Hi* <sup>−</sup> *Hj* .
