## 2 Background
DOE, on the other hand, is a structured methodology designed to elucidate the relationship between factors affecting a process and the resulting output. Together, these methodologies provide a robust framework for optimizing extraction processes. The optimization of process parameters using RSM in the extraction of copper and cobalt has proven highly effective in refining key variables. For instance, during the extraction of Cu from sulfuric solutions containing cobalt, RSM was utilized to fine-tune parameters such as the initial pH, volumetric flow rate, and extractant concentration. This approach achieved a Cu extraction efficiency of 96.73%, while simultaneously minimizing Co extraction to just 2.41% [42]. Similarly, RSM was employed to optimize the leaching conditions for cobalt dissolution from heterogenite, identifying critical factors such as the ferrous sulphate dosage, leaching time, and temperature [18]. Methodologies such as the Box-Behnken design and central composite design (CCD) have been utilized to analyze the effects of various factors on extraction processes. For example, the Box-Behnken design was used to model the leaching behavior of Cu-Co ones, considering parameters such as solid percentage, time, particle size, and Fe\({}^{2+}\) ion concentration [20]. CCD was employed to evaluate cobalt ion extraction using an aqueous two-phase system, highlighting the influence of cobalt ion concentration and volume on extraction efficiency [43]. The integration of response surface methodology (RSM) with artificial neural networks (ANN) has been investigated to enhance prediction accuracy and process optimization. In one study, ANN was combined with RSM to model the leaching behavior of Cu-Coores, achieving high correlation coefficients and low root mean square errors, highlighting the effectiveness of this integrated approach [20]. RSM has also been employed in the design of counter-current leaching systems for Cu extraction, optimizing factors such as acid consumption, iron concentration, and leaching time. This approach improved overall leaching efficiency by up to 80% [44]. The application of RSM in developing environmentally sustainable extraction techniques, such as aqueous two-phase systems, demonstrates its potential to reduce the environmental impact associated with traditional extraction methods. This approach replaces hazardous organic solvents with water, aligning with sustainable and eco-friendly practices [43]. The optimized extraction processes developed through RSM and DOE have significant industrial applications, particularly in the recovery of metals from low-grade ones and waste materials. While RSM and DOE are powerful tools for optimizing extraction processes, their integration with advanced modeling techniques, such as artificial neural networks (ANNs),can significantly enhance the robustness and applicability of these models across diverse industrial settings. Moreover, the shift towards eco-friendly extraction methods highlights the crucial importance of sustainable practices within the field of metallurgy. Although considerable progress has been made in the extraction of Cu and Co, challenges remain, including the variability in ore mineralogy and the pressing need for sustainable methodologies. Future research efforts could focus on combining these approaches with emerging technologies to further improve efficiency and promote environmental sustainability. In the context of Co leaching, RSM has been employed to model the effects of various parameters, such as acid concentration and the presence of a reducing agent, offering valuable insights into the optimal conditions for maximizing Co recovery [45]. The use of mathematical models and simulations further aids in understanding the dynamic behavior of the leaching process. For Mn, RSM has been applied to optimize the reductive leaching process, exploring various reducing agents and acid concentrations.