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In the multi-objective optimization problem that appears naturally in the decision making process for the complex system, the visualization of the innumerable solutions called Pareto optimal solutions is an important issue. This paper focuses on the Pareto optimal solution visualization method using the growing hierarchical self-organizing maps (GHSOM) which is one of promising visualization methods. This method has a superior Pareto optimal solution representation capability, compared to the visualization method using the self-organizing maps. However, this method has some shortcomings. This paper proposes a new Pareto optimal solution visualization method using an improved GHSOM based on the batch learning. In the proposed method, the batch learning algorithm is introduced to the GHSOM to obtain a consistent visualization maps for a Pareto optimal solution set. Then, the symmetric transformation of maps is introduced in the growing process in the batch learning GHSOM algorithm to improve readability of the maps. Furthermore, the learning parameter optimization is introduced. The effectiveness of the proposed method is confirmed through numerical experiments with comparing the proposed method to the conventional methods on the Pareto optimal solution representation capability and the readability of the visualization maps.

A Pareto Optimal Solution Visualization Method Using an Improved Growing Hierarchical Self-Organizing Maps Based on the Batch Learning