## 2 Model
### Matte-Grade Optimization Control Model  
As one of the most important control parameters in copper ISA smelting, converter blowing, and anode furnace refining, matte grade determines the oxygen scheduling scheme of the ISA furnace and converter and the materials entering the ISA furnace, converter, and anode furnace. The optimization control of matte grade is to find the best matte grade when a certain amount of copper concentrate is added under the restriction of converter slag processing capacity and converter planned cold material processing capacity in the process of converter blowing to maintain the process parameters (including total oxygen, total fan blast, and flux amount) within the allowable range and minimize the smelting cost. A nonlinear mathematical model for matte-grade optimization control is established as follows:  
\[\begin{array}{l}\min{W}=C_{ISA}P_{\text{cal}}+\big{(}O_{ISA}+O_{ps}\big{)}P_{ O_{2}}+\big{(}A_{ISA}+A_{ps}+A_{an}\big{)}P_{air}+\big{(}F_{ISA}+F_{ps}\big{)}+B_{an}P_{ re}\\ \hskip 14.226378 ptG_{\min}\ \leq\ G\ \leq\ G_{\max}\\ \hskip 14.226378 ptR_{O_{2}}\ \leq\ R_{O_{2}}\ \leq\ R_{\max}^{\max}\\ \hskip 14.226378 ptS_{Fi}^{\min}\ \leq\ S_{F_{F}}\ \leq\ S_{\text{Fe}}^{\max}\\ \hskip 14.226378 ptS_{\text{CaO}}\ \leq\ S_{\text{CaO}}\ \leq\ S_{\text{CaO}}^{\max}\\ \hskip 14.226378 ptC_{i}^{\min}\ \leq\ S_{i}\ \leq\ C_{i}^{\max}(i=1,2,3,4)\\ \hskip 14.226378 ptS_{ps}^{\min}\ \leq\ S_{ps}\ \leq\ S_{ps}^{\max}\\ \hskip 14.226378 ptD^{\min}\ \leq\ D\ \leq\ D^{\max}(i=1,2,3,4)\end{array} \tag{1}\]  
where \(C_{ISA}\), \(O_{ISA}\), \(A_{ISA}\), and \(F_{ISA}\) are the coal feed amount(t), oxygen supply amount (Nm\({}^{3}\)), blast volume (Nm\({}^{3}\)), and flux amount(t) of the ISA furnace, respectively; \(O_{m}\), \(A_{ps}\), and \(F_{ps}\) are the oxygen supply volume (Nm\({}^{3}\)), blast volume (Nm\({}^{3}\)), and flux amount(t) of PS converter, respectively; \(A_{an}\) and \(B_{an}\) are the blast volume (Nm\({}^{3}\)) and the reductant mass (t) of anode furnace; \(P_{cal}\) is the unit price of coal (Yuan/t); \(P_{O_{2}}\) is the unit price of oxygen (Yuan/Nm\({}^{3}\)); \(P_{air}\) is the unit price of compressed air (Yuan/Nm\({}^{3}\)); \(P_{flux}\) is the unit price of flux (Yuan/Nm\({}^{3}\)); \(P_{re}\) is the unit price of reductant (Yuan/t); \(G\) is matte grade (%); \(G_{\max}\) and \(G_{\min}\) are the upper and lower limits values of matte grade, respectively; \(R_{O_{2}}\) is oxygen-enriched concentration (%); \(R_{O_{2}}^{max}\) and \(R_{O_{2}}^{min}\) are the upper and lower limits of \(R_{O_{2}}\) values, respectively; \(S_{Fe}\) is the slag type of SiO\({}_{2}\)/Fe (%); \(S_{Fe}^{\max}\) and \(S_{Fe}^{\min}\) are the upper and lower limits of \(S_{Fe}\) values, respectively; \(S_{CaO}\) is the slag type of SiO\({}_{2}\)/CaO (%); \(S_{CaO}^{\max}\) and \(S_{CaO}^{\min}\) are the upper and lower limits of \(S_{CaO}\) values, respectively; \(C_{i}\)\((i=1,2,3,4)\) representsthe amount of cold material, residual anode, crude copper, and anode slag processed in each converter of the converter (t); \(C^{\text{max}}\) and \(C^{\text{min}}\) are the upper and lower limits of the \(C_{i}\)\((i=1,2,3,4)\) value, respectively; \(S_{ps}\) is the amount of slag per PS furnace(t); \(S_{ps}^{\text{max}}\) and \(S_{ps}^{\text{min}}\) are the upper and lower limit values of \(S_{ps}\), respectively; \(D_{i}\)\((i=1,2,3)\) represents the amount of cold material, residual anode, and purchased crude copper processed in each anode furnace (t); \(D^{\text{max}}\) and \(D^{\text{min}}\) are the upper and lower limit values of \(D_{i}\)\((i=1,2,3)\), respectively. The relationship between these process parameters is nonlinear, and the best matte grade can be found by GA under the condition of the lowest cost in the whole process of copper smelting.