A Visualization Technique for Accessing Solution Pool in Interactive Methods of Multiobjective Optimization

Ernestas Filatovas, Dmitry Podkopaev, Olga Kurasova


Interactive methods of multiobjective optimization repetitively derive Pareto optimal solutions based on decision maker's preference information and present the obtained solutions for his/her consideration. Some interactive methods save the obtained solutions into a solution pool and, at each iteration, allow the decision maker considering any of solutions obtained earlier. This feature contributes to the flexibility of exploring the Pareto optimal set and learning about the optimization problem. However, in the case of many objective functions, the accumulation of derived solutions makes accessing the solution pool cognitively difficult for the decision maker. We propose to enhance interactive methods with visualization of the set of solution outcomes using dimensionality reduction and interactive mechanisms for exploration of the solution pool. We describe a proposed visualization technique and demonstrate its usage with an example problem solved using the interactive method NIMBUS.


Multiobjective optimization, interactive methods, Pareto front visualization, dimensionality reduction, multidimensional scaling.

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Belton, V., Branke, J., Eskelinen, P., Greco, S., Molina, J., Ruiz, F., Słowiński, R. (2008); Interactive multiobjective optimization from a learning perspective, in Branke, J., Deb, K., Miettinen, K., Słowiński, R., editors, Multiobjective Optimization (Lecture Notes in Computer Science, 5252), Springer, ISBN 978-3-540-88907-6, 405–433.

Borg, I., Groenen, P. J. (2005); Modern multidimensional scaling: Theory and applications, Springer, ISBN 978-0-387-25150-9.

Branke, J., Deb, K., Miettinen, K., Słowiński, R. (2008); Multiobjective optimization: Interactive and evolutionary approaches (Lecture Notes in Computer Science 5252), Springer, ISBN 978-3-540-88908-3.

Castelletti, A., Lotov, A. V., Soncini-Sessa, R. (2010); Visualization-based multi-objective improvement of environmental decision-making using linearization of response surfaces, Environmental Modelling & Software, ISSN 1364-8152, 25(12): 1552–1564.

Dzemyda, G., Kurasova, O., Žilinskas, J. (2013); Multidimensional Data Visualization: Methods and Applications (Springer Optimization and Its Applications, 75), Springer, ISBN 978- 1-4419-0235-1.

Filatovas, E., Kurasova, O. (2011); A decision support system for solving multiple criteria optimization problems. Informatics in Education, ISSN 1648-5831, 10(2): 213–224.

Gardiner, L., Vanderpooten, D. (1997); Interactive multiple criteria procedures: Some reflections. In Climaco, J., editor, Multicriteria Analysis, Springer, ISBN 978-3-642-64500-6, 290–301.

Jaszkiewicz, A., Słowiński, R. (1999). The "light beam search" approach - an overview of methodology and applications. European Journal of Operational Research, ISSN 0377-2217, 113(2): 300–314.

Visualization in the multiple objective decision-making framework, in Branke, J., Deb, K., Miettinen, K., Słowiński, R., editors, Multiobjective Optimization (Lecture Notes in Computer Science, 5252), Springer, ISBN 978-3-540-88907-6, 195–212.

Kurasova, O., Petkus, T., Filatovas, E. (2013); Visualization of pareto front points when solving multi-objective optimization problems. Information Technology And Control, ISSN 1392-124, 42(4): 353–361.

Laukkanen, T., Tveit, T.-M., Ojalehto, V., Miettinen, K., Fogelholm, C.-J. (2010); An interactive multi-objective approach to heat exchanger network synthesis. Computers & Chemical Engineering, ISSN 0098-1354, 34(6): 943–952.

Lotov, A., Bushenkov, V., Kamenev, G. (2004); Interactive Decision Maps, Approximation and Visualization of Pareto Frontier (Applied Optimization, 89), Springer, ISBN 978-1-4419- 8851-5.

Lotov, A. V., Miettinen, K. (2008); Visualizing the Pareto frontier, in Branke, J., Deb, K., Miettinen, K., Słowiński, R., editors, Multiobjective Optimization (Lecture Notes in Computer Science, 5252), Springer, ISBN 978-3-540-88907-6, 5252: 213–243.

Luque, M., Ruiz, F., Miettinen, K. (2011); Global formulation for interactive multiobjective optimization. OR Spectrum, ISSN 0171-6468, 33(1): 27–48.

Visual PROMETHEE: Developments of the PROMETHEE & GAIA multicriteria decision aid methods, in Proceedings of IEEE International Conference on Industrial Engineering and Engineering Management, 2009, e- ISBN 978-1-4244-4870-8, 1646–1649.

Miettinen, K. (1999); Nonlinear multiobjective optimization (International Series in Operations Research & Management Science, 12), Springer, ISBN 978-1-4615-5563-6.

Miettinen, K. (2014); Survey of methods to visualize alternatives in multiple criteria decision making problems. OR Spectrum, ISSN 0171-6468, 36(1): 3–37.

Interactive nonlinear multiobjective optimization methods. In Ehrgott, M., Figueia, J., and Greco, S., editors, Multiple Criteria Decision Analysis: State of the Art Surveys, Springer, (to appear).

Miettinen, K., Mäkelä, M. (1995); Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization, ISSN 0233-1934, 34(3): 231–246.

Interactive method NIMBUS for nondifferentiable multiobjective optimization problems, in Climaco, J., editor, Multicriteria Analysis, Springer, ISBN 978-3-642-64500-6, 310–319.

Miettinen, K., Mäkelä, M. M. (2000); Interactive multiobjective optimization system WWW-NIMBUS on the Internet. Computers & Operations Research, ISSN 0305-0548, 27(7): 709–723.

Miettinen, K., Mäkelä, M. M. (2006); Synchronous approach in interactive multiobjective optimization. European Journal of Operational Research, ISSN 0377-2217, 170(3): 909–922.

Miettinen, K., Mäkelä, M. M., Kaario, K. (2006); Experiments with classification-based scalarizing functions in interactive multiobjective optimization. European Journal of Operational Research, ISSN 0377-2217, 175(2): 931–947.

Miettinen, K., Mustajoki, J., Stewart, T. J. (2014b); Interactive multiobjective optimization with NIMBUS for decision making under uncertainty. OR Spectrum, ISSN 0171-6468, 36(1): 39–56.

Introduction to multiobjective optimization: Interactive approaches, in Branke, J., Deb, K., Miettinen, K., Słowiński, R., editors, Multiobjective Optimization: Interactive and Evolutionary Approaches (Lecture Notes in Computer Science, 5252), Springer, ISBN 978-3-540-88908-3, 27–57.

Narula, S. C., Weistroffer, H. (1989); A flexible method for nonlinear multicriteria decisionmaking problems. IEEE Transactions on Systems, Man and Cybernetics, ISSN 1083-4419, 19(4): 883–887.

Naud, A., Duch, W. (2000); Interactive data exploration using MDS mapping, in Proceedings of the Fifth Conference: Neural Networks and Soft Computing, ISBN 83-908587-2-X, 255–260.

Ojalehto, V., Miettinen, K., Laukkanen, T. (2014); Implementation aspects of interactive multiobjective optimization for modeling environments: the case of GAMS-NIMBUS. Computational Optimization and Applications, ISSN 0926-6003, 58(3): 757–779.

Ojalehto, V., Miettinen, K., Mäkelä, M. (2007); Interactive software for multiobjective optimization: IND-NIMBUS, WSEAS Transactions on Computers, ISSN 2224-2872, 6(1): 87– 94.

Petkus, T., Filatovas, E., Kurasova, O. (2009); Investigation of human factors while solving multiple criteria optimization problems in computer network. Technological and Economic Development of Economy, ISSN 2029-4913, 15(3): 464–479.

Pohlheim, H. (2006); Multidimensional scaling for evolutionary algorithms – visualization of the path through search space and solution space using Sammon mapping. Artificial Life, ISSN 1064-5462, 12(2): 203–209.

Saaty, T., Ozdemir, M. (2003); Why the magic number seven plus or minus two, Mathematical and Computer Modelling, ISSN 0895-7177, 38(3–4): 33–244.

Steuer, R. E. (1989); The Tchebycheff procedure of interactive multiple objective programming, in Karpak, B., Zionts, S., editors, Multiple Criteria Decision Making and Risk Analysis Using Microcomputers (NATO ASI Series, 56), Springer, ISBN 978-3-642-74919-3, 235–249.

Valdés, J. J., Barton, A. J. (2007); Visualizing high dimensional objective spaces for multiobjective optimization: A virtual reality approach, in Proceedings of the IEEE Congress on Evolutionary Computation(CEC 2007), IEEE, ISBN 978-1-4244-1339-3, 4199–4206.

Walker, D. J., Everson, R. M., Fieldsend, J. E. (2013); Visualizing mutually nondominating solution sets in many-objective optimization. IEEE Transactions on Evolutionary Computation, ISSN 1089-778X, 17(2): 165–184.

Yoshimi, M., Kuhara, T., Nishimoto, K., Miki, M., Hiroyasu, T. (2012). Visualization of pareto solutions by spherical self-organizing map and it's acceleration on a GPU. Journal of Software Engineering & Applications, ISSN 1945-3116, 5(3): 129–137.

DOI: https://doi.org/10.15837/ijccc.2015.4.1672

Copyright (c) 2017 Ernestas Filatovas, Dmitry Podkopaev, Olga Kurasova

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