Multidimensional Numbers: A Framework for Multi-Objective Optimization in Concrete Engineering Using Machine Learning Approaches

This study presents a novel framework for multi-objective optimization in concrete engineering using multidimensional numbers and machine learning approaches. The optimization of concrete mixture proportions requires the simultaneous optimization of numerous objectives and numerous variables (such as concrete components) under highly nonlinear constraints. Multi-objective optimization is not compatible with the existing single-objective optimization models. In order to optimize the proportions of the concrete mixture, this study suggests a multi-objective optimization technique based on machine learning (ML) and different algorithmic approaches. In this study, we introduce the concept of multidimensional numbers, which incorporate both quantitative values and contextual dimensions (e.g., units, time, or error), enabling a unified representation of complex engineering parameters. By applying this framework to concrete mixture design, we simultaneously optimize conflicting criteria such as concrete strength. Seven regression models—Linear, Ridge, Lasso, Support Vector, Random Forest, Gradient Boosting, and Neural Network—were evaluated. Results demonstrate that linear models outperformed complex counterparts, with neural networks exhibiting poor performance, highlighting the dominance of linear relationships in the dataset. Visualization tools, including Taylor diagrams and 3D surface plots, further validated these findings. The study underscores the efficacy of multidimensional numbers in capturing real-world engineering complexities while emphasizing that model selection should prioritize data structure over algorithmic complexity. This approach offers practical insights for civil engineering applications, enabling cost-effective, sustainable concrete design with balanced performance metrics.