Paper:

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

## Naoto Suzuki, Takashi Okamoto, and Seiichi Koakutsu

Chiba University

1-33 Yayoicho, Inage-ku, Chiba 263-8522, Japan

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.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.20, No.5, pp. 691-704, 2016.

- [1] A. Abraham, L. Jain, and R. Goldberg (Eds.), “Evolutionary multiobjective optimization,” Springer-Verlag, 2006.
- [2] X. Gandibleux, M. Sevaux, K. Söerensen, and V. T’kindt (Eds.), “Metaheuristics for multiobjective optimisation,” Springer-Verlag, 2004.
- [3] K. Tsuchida, H. Sato, H. E. Aguirre, and K. Tanaka, “Analysis of NSGA-II and NSGA-II with CDAS, and proposal of an enhanced CDAS mechanism,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.13, No.4, pp. 470-480, 2009.
- [4] F. Bourennani, S. Rahnamayan, and G. F. Naterer, “OGDE3: Opposition-based third generalized differential evolution,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.16, No.3, pp. 469-480, 2012.
- [5] M. Miyakawa, K. Takadama, and H. Sato, “Archive of useful solutions for directed mating in evolutionary constrained multiobjective optimization,” J. of Advanced Computational Intelligence and Intelligent Informatics (JACIII), Vol.18, No.2, pp. 221-231, 2014.
- [6] S. Obayashi and D. Sasaki, “Visualization and data mining of Pareto solutions using self-organizing map,” EMO 2003, Lecture Notes in Computer Science, Vol.2632, Springer, pp. 796-809, 2003.
- [7] A. Lotov and K. Miettinen, “Visualizing the Pareto frontier,” Multiobjective Optimization, Lecture Notes in Computer Science, Vol.5252, Springer, pp. 213-243, 2008.
- [8] M. Bagajewicz and E. Cabrera, “Pareto optimal solutions visualization techniques for multiobjective design and upgrade of instrumentation networks,” Ind. Eng. Chem. Res., Vol.42, No.21, pp. 5195-5203, 2003.
- [9] X. Blasco, J. M. Herrero, J. Sanchis, and M. Martinez, “A new graphical visualization of n-dimensional Pareto front for decision-making in multiobjective optimization,” Information Sciences, Vol.178, No.20, pp. 3908-3924, 2008.
- [10] S. Chen, D. Amid, O. M. Shir, L. Limonad, D. Boaz, A. Anaby-Tavor, and T. Schreck, “Self-organizing maps for multi-objective Pareto frontiers,” Proc. of IEEE Pacific Visualization Symp. 2013, pp. 153-160, 2013.
- [11] T. Kohonen, “Self-organized formation of topologically correct feature maps,” Biol. Cybern., Vol.43, No.1, pp. 59-69, 1982.
- [12] A. Hironaka, T. Okamoto, S. Koakutsu, and H. Hirata, “Analysis and improvements of the Pareto optimal solution visualization method using the self-organizing maps,” SICE J. of Control, Measurement, and System Integration, Vol.8, No.1, pp. 34-43, 2015.
- [13] T. Okamoto, Y. Hanaoka, E. Aiyoshi, and Y. Kobayashi, “Optimal design of buffer material in the geological disposal of radioactive wastes using the satisficing trade-off method and a self-organizing map,” Electrical Engineering in Japan, Vol.187, No.2, pp. 17-32, 2014.
- [14] N. Suzuki, T. Okamoto, and S. Koakutsu, “Visualization of Pareto optimal solution sets using the growing hierarchical self-organizing maps,” IEEJ Trans. Electronics, Information and Systems, Vol.135, No.7, pp. 908-919, 2015 (in Japanese).
- [15] A. Rauber, D. Merkl, and M. Dittenbach, “The growing hierarchical self-organizing map: Exploratory analysis of high-dimensional data,” IEEE Trans. Neural Networks, Vol.13, No.6, pp. 1331-1341, 2002.
- [16] M. Dittenbach, D. Merkl, and A. Rauber, “Organizing and exploring high-dimensional data with the growing hierarchical self-organizing map,” Proc. of the 1st Int. Conf. on Fuzzy Systems and Knowledge Discovery, Vol.2, pp. 626-630, 2002.
- [17] S. Huband, L. Barone, L. While, and P. Hingston, “A scalable multi-objective test problem toolkit,” EMO 2005, Lecture Notes in Computer Science, Vol.3410, Springer, pp. 280-295, 2005.
- [18] K. Deb, A. Pratp, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. Evol. Comput., Vol.6, No.2, pp. 182-197, 2002.
- [19] R. Storn and K. Price, “Differential evolution — a simple and efficient adaptive scheme for global optimization over continuous spaces,” Tech. Rep. of Int. Computer Science Institute, No. TR-95-012, 1995.
- [20] K. Price, R. M. Storn, and J. A. Lampinen, “Differential evolution — A practical approach to global optimization,” Springer, 2005.
- [21] H. Ishibuchi, M, Yamane, N. Akedo, and Y. Nosima, “Many-Objective and Many-Variable Test Problems for Visual Examination of Multiobjective Search,” Proc. of IEEE Congress on Evolutionary Computation 2013, pp. 1491-1498, 2013.
- [22] R. Vlennet, C. Fonteix, and I. Marc, “Multicriteria optimization using a genetic algorithm for determining a Pareto set,” Int. J. of Syst. Sci, Vol.27, No.2, pp. 255-260, 1996.
- [23] K. Deb, L. Thiele, M. Laumanns, and E. Zitzler, “Scalable Test Problems for Evolutionary Multi-Objective Optimization,” Evolutionary Multiobjective Optimization — Theoretical Advances and Applications, Springer, pp. 105-145, 2005.