Abstract
In this paper we present an efficient and effective method of using surrogate approximations to explore the design space and capture the Pareto frontier during multiobjective optimization. The method employs design of experiments and metamodeling techniques (e.g., response surfaces and kriging models) to sample the design space, construct global approximations from the sample data, and quickly explore the design space to obtain the Pareto frontier without specifying weights for the objectives or using any optimization. To demonstrate the method, two mathematical example problems are presented. The results indicate that the proposed method is effective at capturing convex and concave Pareto frontiers even when discontinuities are present. After validating the method on the two mathematical examples, a design application involving the multiobjective optimization of a piezoelectric bimorph grasper is presented. The method facilitates multiobjective optimization by enabling us to efficiently and effectively obtain the Pareto frontier and identify candidate designs for the given design requirements.
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ABAQUS Version 5.7-1, Hibbitt, Karlsson and Sorensen, Inc., 1080 Main Street, Pawtucket, RI, (URL: http://www.hks.com/), 1997.
T. W. Athan and P. Y. Papalambros, “;A note on weighted criteria methods for compromise solutions in multiobjective optimization,”; Engineering Optimization vol. 27, no. 2, pp. 155–;176, 1996.
S. Azarm, B. J. Reynolds, and S. Narajanan, “;Comparison of two multiobjective optimization techniques with and within genetic algorithms,”; Advances in Design Automation, Las Vegas, NV, ASME, Sept. 12–;15, 1999, Paper No. DETC99/DAC-8584.
R. Balling, “;Design by shopping: A new paradigm?,”; Proc. Third World Congress of Structural and Multidisciplinary Optimization, C. L. Bloebaum and K. E. Lewis et al., eds. Buffalo, NY, University at Buffalo vol. 1, May 17–;21, 1999, pp. 295–;297.
R. J. Balling, J. T. Taber, M. R. Brown, and K. Day, “;Multiobjective urban planning using genetic algorithm,”; Journal of Urban Planning and Development vol. 125, no. 2, pp. 86–;99, 1999.
J.-F. M. Barthelemy and R. T. Haftka, “;Approximation concepts for optimum structural design—;A review,”; Structural Optimization vol. 5, pp. 129–;144, 1993.
A. J. Booker, “;Design and analysis of computer experiments,”; 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis &;;;;;;; Optimization, St. Louis, MO, AIAA vol. 1, Sept. 2–;4, 1998, pp. 118–;128. AIAA-98-4757.
D. J. Cappelleri and M. I. Frecker, “;Optimal design of smart tools for minimally invasive surgery,”; Optimization in Industry II, F. Mistree and A. D. Belegundu, eds. Banff, Alberta, Canada, ASME, July 6–;11, 1999.
D. J. Cappelleri, M. I. Frecker, and T. W. Simpson, “;Optimal design of a PZT bimorph actuator for minimally invasive surgery,”; 7th International Symposium on Smart Structures and Materials, Newport Beach, CA, SPIE, Mar. 5–;9, 1999.
D. J. Cappelleri, M. I. Frecker, T. W. Simpson, and A. Snyder, “;A metamodel-based approach for optimal design of a PZT bimorph actuator for minimally invasive surgery,”; Journal of Mechanical Design 2001, to appear.
F. Y. Cheng and D. Li, “;Multiobjective optimization design with Pareto genetic algorithm,”; Journal of Structural Engineering vol. 123, no. 9, pp. 1252–;1261, 1997.
I. Das, “;An improved technique for choosing parameters for Pareto surface generation using normal-boundary intersection,”; Proc. Third World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) C. L. Bloebaum and K. E. Lewis et al., eds., Buffalo, NY, University at Buffalo, vol. 2, May 17–;21, 1999, pp. 411–;413.
I. Das, “;Optimization large systems via optimal multicriteria component assembly,”; 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis &;;;;;;; Optimization, St. Louis, MO, AIAA, vol. 1, Sept. 2–;4, 1998, pp. 661–;669, AIAA-98-4791.
I. Das and J. E. Dennis, “;A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems,”; Structural Optimization vol. 14, no. 1, pp. 63–;69, 1997.
I. Das, and J. E. Dennis, “;Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems,”; SIAM Journal on Optimization vol. 8, no. 3, pp. 631–;657, 1998.
W. Fatikow and Rembold, U., Microsystem Technology and Microrobotics, Springer-Verlag: New York, 1997.
A. Giunta, L. T. Watson, and J. Koehler, “;A comparison of approximation modeling techniques: Polynomial versus interpolating models,”; 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis &;;;;;;; Optimization, St. Louis, MO, AIAA vol. 1, Sept. 2–;4, 1998, pp. 392–;404. AIAA-98-4758.
IEEE Group on Sonics and Ultrasonics,Transducers and Resonators Committee, IEEE Standard on Piezoelectricity (176-1978), ANSI/IEEE: New York, 1978.
E. M. Kasprazak and K. E. Lewis, “;A method to determine optimal relative weights for Pareto solution sets,”; Proc. Third World Congress of Structural and Multidisciplinary Optimization (WCSMO-3), C. L. Bloebaum and K. E. Lewis et al., eds., Buffalo, NY, University at Buffalo vol. 2, May 17–;21, 1999, pp. 408–;410.
J. R. Koehler and A. B. Owen, “;Computer experiments,”; Handbook of Statistics, S. Ghosh and C. R. Rao, eds. Elsevier Science: New York, 1996, pp. 261–;308.
J. Koski, “;Defectiveness of weighting method in multicriterion optimization of structures,”; Communications in Applied Numerical Methods vol. 1, pp. 333–;337, 1985.
Y. Li, G. M. Fadel, and M. M. Wiecek, “;Approximating Pareto curves using the hyper-ellipse,”; 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis &;;;;;;; Optimization, St. Louis, MO, AIAA vol. 3, Sept. 2–;4, 1998, pp. 1990–;2002.
E. R. Liberman, “;Soviet multi-objective mathematical programming methods: An overview,”; Management Science vol. 37, no. 9, pp. 1147–;1165, 1991.
M. D. McKay, R. J. Beckman, and W. J. Conover, “;A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,”; Technometrics vol. 21, no. 2, pp. 239–;245, 1979.
M. Meckesheimer, R. R. Barton, T.W. Simpson, and A. Booker, “;Computationally inexpensive metamodel assessment strategies,”; ASME Design Technical Conferences—;Design Automation Conference A. Diaz, ed. Pittsburgh, PA, ASME, Sept. 9–;12, 2001, DETC2001/DAC-21028.
A. Messac, J. G. Sundararaj, R. V. Tappeta, and J. E. Renaud, “;The ability of objective functions to generate non-convex Pareto frontiers,”; 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, St. Louis, MO, AIAA vol. 1, Apr. 12–;15, 1999, pp. 78–;87. AIAA-99-1211.
R. H. Myers, and D. C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley &; Sons: New York, 1995.
A. Osyczka, “;Multicriteria optimization for engineering design,”; in Design Optimization, J. S. Gero, ed. Academic Press: New York, 1985, pp. 193–;227.
A. Osyczka and S. Kundu, “;A new method to solve generalized multicriteria optimization problems using the simple genetic alrogithm,”; Structural Optimization vol. 10, no. 2, pp. 94–;99, 1995.
A. B. Owen, “;Orthogonal arrays for computer experiments, integration and visualization,”; Statistica Sinica vol. 2, pp. 439–;452, 1992.
J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn, “;Design and analysis of computer experiments,”; Statistical Science vol. 4, no. 4, pp. 409–;435, 1989.
E. J. Schaumann, R. J. Balling, and K. Day, “;Genetic algorithms with multiple objectives,”; 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis &;;;;;;; Optimization, St. Louis, MO, AIAA vol. 3, Sept. 2–;4, 1998, pp. 2114–;2123.
T. W. Simpson, T. M. Mauery, J. J. Korte, and F. Mistree. “;Comparison of response surface and kriging models for multidisciplinary design optimization,”; 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis &;;;;;;; Optimization, St. Louis, MO, AIAA vol. 1, Sept. 2–;4, 1998, pp. 381–;391, AIAA-98-4755.
T. W. Simpson, T. M. Mauery, J. J. Korte, and F. Mistree, “;Kriging metamodels for global approximation in simulation-based multidisciplinary design optimization,”; AIAA Journal 2001, to appear.
T. W. Simpson, J. Peplinski, P. N. Koch, and J. K. Allen, “;Metamodels for computer-based engineering design: Survey and recommendations,”; Engineering with Computers vol. 17, pp. 129–;150, 2001.
J. Sobieszczanski-Sobieski and R. T. Haftka, “;Multidisciplinary aerospace design optimization: Survey of recent developments,”; Structural Optimization vol. 14, pp. 1–;23, 1997.
I. M. Sobol, “;An efficient approach to multicriteria optimum design problems,”; Surveys on Mathematics for Industry vol. 1, pp. 259–;281, 1992.
W. Stadler and J. Dauer, “;Multicriteria optimization in engineering:Atutorial and survey,”; Structural Optimization: Status and Promise, M. P. Kamat, ed., AIAA: Washington, D.C., 1993, pp. 209–;249.
R. E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley: New York, 1986.
R. V. Tappeta and J. E. Renaud, “;Interactive multiobjective optimization design strategy for decision based design,”; Advances in Design Automation, LasVegas, NV, ASME, Sept. 12–;15, 1999a, Paper No. DETC99/DAC-8581.
R. V. Tappeta, and J. E. Renaud, “;Interactive multiobjective optimization procedure,”; 40th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, St. Louis, MO, AIAA vol. 1, Apr. 12–;15, 1999b, pp. 27–;41, AIAA-99-1207.
R.V. Tappeta, J. E. Renaud, A. Messac, and J. G. Sundararaj, “;Interactive physical programming: Tradeoff analysis and decision making in multicriteria optimization,”; 40th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, St. Louis, MO, AIAA vol. 1, Apr. 12–;15, 1999, pp. 53–;67, AIAA-99-1209.
J. Zhang, M. M. Wiecek, and W. Chen, “;Local approximation of the efficient frontier in robust design,”; Advances in Design Automation, Las Vegas, NV, ASME, Sept. 12–;15, 1999, Paper No. DETC99/DAC-8566.
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Wilson, B., Cappelleri, D., Simpson, T.W. et al. Efficient Pareto Frontier Exploration using Surrogate Approximations. Optimization and Engineering 2, 31–50 (2001). https://doi.org/10.1023/A:1011818803494
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DOI: https://doi.org/10.1023/A:1011818803494