Knowledge Extraction in Multi-objective Optimization Problem based on Visualization of Pareto Solutions

Genetic Algorithm (GA) is one of the effective methods in the application to optimization problems. Recently, Multi-Objective Genetic Algorithm (MOGA), which is the application of Genetic Algorithm to Multi-objective Optimization Problems, is focused on in the engineering design field. In this field, the analysis of design variables in the acquired Pareto solutions, which gives the designers useful knowledge in the applied problem, is important as well as the acquisition of advanced solutions. This paper proposes a visualization method using an idea of Isomap, that visualizes manifold embedded in the high dimensional space, which was originally proposed in the field of multiple classification analysis. The proposed method visualizes the geometric distance of solutions in the design variable space considering their distance in the objective space. This method enables a user to analyze the design variables of the acquired solutions considering their relationship in the objective space. This paper applies the proposed method to the conceptual design optimization problem of hybrid rocket engine and studies the effectiveness of the proposed method. We found interesting structure in the distribution of Pareto solutions by applying the proposed method to this problem. This paper shows that the visualized result gives some knowledge on the features between design variables and fitness values in the acquired Pareto solutions.