Concept-Level Interpretable SOM for Visual Analysis of High-Dimensional Non-dominated Solution Set

Multi-objective optimization problems will generate a solution set consisting of many non-dominated solutions, and the determination of the final decision needs to be based on the understanding and analysis of the non-dominated solution set. Visual analysis methods can provide the most intuitive explanations, but existing visualization methods rarely deal with high-dimensional complex real-world problems and rarely consider providing conceptual-level explanations for humans. To analyze the non-dominated solution set from the conceptual level, this paper proposes a concept-level visualization method based on an interpretable Self-Organizing Map (iSOM). Firstly, the mapping table between decision variables and concepts is constructed and divided into two types of concepts, algebraic-type and set-type. Then the conceptual data plane (C-plane) for displaying algebraic-type concepts is proposed to avoid the errors associated with the approximation of the iSOM component plane, and the conceptual set matrix (C-matrix) is proposed for visualizing and analyzing set-type data by iSOM. In addition, this paper achieves effective clustering visualization of the objective function component plane and integrates real data distributions into iSOM, providing a more informative and intuitive visualization method. Finally, the proposed concept-level visualization approach is demonstrated by constructing a real-world multi-objective multi-vehicle rescue supplies distribution problem. By taking the visualization of nondominated solution sets to the conceptual level, new insights are provided for reliable decision-making in real multi-objective optimization problems.