A multi-stage differential-multifactorial evolutionary algorithm for ingredient optimization in the copper industry

Ingredient optimization plays a pivotal role in the copper industry, for which it is closely related to the concentrate utilization rate, stability of furnace conditions, and the quality of copper production. To acquire a practical ingredient plan, which should exhibit long duration time with sufficient utilization and feeding stability for real applications, an ingredient plan optimization model is proposed in this study to effectively guarantee continuous production and stable furnace conditions. To address the complex challenges posed by this integer programming model, including multiple coupling feeding stages, intricate constraints, and significant non-linearity, a multi-stage differential-multifactorial evolution algorithm is developed. In the proposed algorithm, the differential evolutionary (DE) algorithm is improved in three aspects to efficiently tackle challenges when optimizing the proposed model. First, unlike traditional time-consuming serial approaches, the multifactorial evolutionary algorithm is utilized to optimize multiple complex models contained in the population of evolutionary algorithm caused by the feeding stability in a parallel manner. Second, a repair algorithm is employed to adjust infeasible ingredient lists in a timely manner. In addition, a local search strategy taking feedback from the current optima and considering the different positions of global optimum is developed to avoiding premature convergence of the differential evolutionary algorithm. Finally, the simulation experiments considering different planning horizons using real data from the copper industry in China are conducted, which demonstrates the superiority of the proposed method on feeding duration and stability compared with other commonly deployed approaches. It is practically helpful for reducing material cost as well as increasing production profit for the copper industry.