Gaussian Learning-Based Pareto Evolutionary Algorithm for Parallel Machine Planning in Industrial Silicon Production

This study focuses on a multi-objective heterogeneous parallel machine planning problem for industrial silicon smelting. Specifically, under the conflicting objectives of minimizing carbon emissions, rollover penalty costs, and load imbalance, the total production demand of industrial silicon is allocated monthly across multiple machines. We first establish the mathematical model of the problem accounting for real-life management requirements. To solve the model, a Gaussian learning-based Pareto evolutionary algorithm (GLPEA) is proposed. The algorithm is developed based on a nondominated sorting framework and incorporates two key innovations: (1) a generation-wise dynamic Gaussian mixture component selection strategy that adaptively fits the multimodal distribution of elite solutions, and (2) a hybrid offspring generation mechanism that integrates traditional evolutionary operators with a Gaussian sampling strategy trained on perturbed solution sets, thereby enhancing exploration capability while maintaining convergence. The effectiveness of GLPEA is validated on 40 problem instances of varying scales. Compared with NSGA-II and MOEA/D, GLPEA achieves average improvements of 5.78% and 89.23% in IGD, and 1.03% and 264.43% in HV, respectively. We make the source codes of GLPEA publicly available to facilitate future research on practical applications.