多目標最佳化在許多工程領域已廣泛的被應用，最佳解所形成的Pareto集合可提供設計工程師或政策決定者了解各目標函數之間的妥協關係。近年來，可靠度最佳化利用隨機變數模擬不確定因素，並結合最佳化的概念，已成功的探討單目標設計問題面臨自然界不確定因素下所需因應的策略，然而對於Pareto集合在不確定因素下的表現卻尚無清楚的了解。

在此論文中，我們利用文獻上使用目標空間計算線性多目標問題之Pareto集合的技術，結合線性Pareto集合在考慮不確定因素下的平移特性，首先發展一套可有效的預測不同可靠度下Pareto集合(亦稱為β-Pareto集合)的方法。此方法首先利用簡算法得到任一單目標的最佳解為非凌駕角解(non-dominated extreme point)，再利用拘束條件active的判定，形成一縮減成本係數矩陣(reduce cost coefficient matrix)，用該矩陣計算出此非凌駕角解之架構向量(frame)，然後便可用最佳化的計算依序得知完整的Pareto集合。在得到一組Pareto集合後，使用一階二次可靠度方法計算非凌駕角解在不確定因素下不同β量值的移動，直到active狀態改變，此改變可經由KKT必要條件的Lagrange Multiplier得知。

除了線性問題外，本論文並延伸討論非線性多目標問題的β-Pareto集合。首先針對非線性問題的Pareto集合用區塊三明治夾擠法線性化，在針對線性化後之系統進行β-Pareto集合的演算，並利用增加線性化區間的方法控制誤差。本論文提出之方法經由數學範例驗證可有效率的預測線性或非線性系統的β-Pareto集合，使決策者能在使用最少資源的情況下得知各目標函數在不同可靠度要求下的妥協關係。

In this research we investigate design optimization problems under uncertainties with multiple objectives. The probabilistic formulation of constraints when uncertainties are considered in an optimization framework has received extensive studies in the literature. Most research use a single objective function to explore the optimum under uncertainty. In this study, we consider the same probabilistic constraint framework with multiple objective functions. The results of the multiobjective optimization under uncertainty is a set of Pareto frontiers at different reliability levels, β-Pareto. We investigate how Pareto frontiers move with different reliability values and then use a LP Pareto frontier generation method to predict β-Pareto frontiers without running multiple MOO problems. We also extend the solution method to nonlinear problems. From this work we demostrate how the proposed method can be a useful tool in decision-making under uncertainty when quick estimates of outcomes of a design decision are needed in the early stage of product design process. Compared to existing methods the proposed work show significant efficiency for general MOO under uncertainty.

[1] R. Fletcher, Practical Methods of Optimization. Wiley, 1980.
[2] P. Y. Papalambros and D. J. Wilde, Principle of Optimal Design. Cambridge, 2ed., 2000.
[3] G. B.DantzigandM.N.Thapa, LinearProgramming.Springer-Verlag,1997.
[4] I. Maros, Computational Techniques of The Simplex Method. Kluwer Academic Publishers, 2003.
[5] K. M. Miettinen, Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, 1998.
[6] V. Pareto, Manuel d'Economic Politiqu. Girard, 2ed., 1927.
[7] M. Zeleny, Compromise programming. University of South Carolina Press, 1973.
[8] S. GassandT. Saaty, The computational algorithm for the parametric objective function, Naval Research Logistics Quarterly, vol.2, p.39, 1955.
[9] L. Zadeh, Optimality and non-scalar-valued performance criteria, IEEE Transactions on Automatic Control, vol.8, pp.59 V60, 1955
[10] I. Kim, Adaptive weighted sum method for biobjective optimization: Pareto front generation, Structural Multidisciplinary Optimization, vol.29, pp.149 V158, 2005
[11] S. Marglin, Public Investment Criteria. MIT Press, 1967.
[12] H. W. Corley, A new scalar equivalence for pareto optimization, IEEE Transactions on Automatic Control, vol.25, pp.829 V830, 1980
[13] R. E. Wendell and D. N. Lee, Effciency in multiple objective optimization problems, Mathematical Programming, vol.12, pp.406 V414, 1977
[14] I. Das and J. Dennis, Normal-boundary in tersection: An alternate method for generating pareto optimal points in multicriteria optimization problems, Institute for Computer Applications in Science and Engineering, 1996.
[15] J. P. Dauer, An equivalence result for solutions of multiobjective linear programs, Computer and Operations Research, vol.7, pp.33 V39, 1980
[16] J. G. Ecker and I. A. Kouada, Finding all effcient extreme points for multiple objective linear programs, Mathematical Programming, vol.14, pp.249 V261, 1978
[17] J. G. Ecker, N. S. Hegner, and I. A. Kouada, Generating all maximal effcient faces for multiple objective linear programs, Journal of Optimization Theory and Applications, vol.30, pp.353 V381, 1980
[18] J. P. Dauer and R. J. Krueger, A multiobjective optimization model for water resource planning, Applied Mathematical Modelling, vol.4, pp.171 V175, 1980
[19] J. P. Dauer and Y.-H. Liu, Solving multiple objective linear programs in objective space, European Journal of Operational Research, vol.46, pp.350 V357, 1990
[20] D. G. Luenberger, Introduction to Linear and Nonlinear Programming. Addison-Wesley, 1973.
[21] J. P. Dauer, An alysis of the objective space in multiple objective linear programming, Journal of Mathematical Analysis and Applications, vol.126, pp.579 V593, 1987
[22] R. J. B. Wets and C. Witzgall, “Algorithm for frames and lineality spaces of cones,” Journal of Research of the National Bureau of Standards-B.Mathematics and Mathematical Physics, vol. 71B, pp. 1–7, 1967.
[23] J. P. Dauer, “On degeneracy and collapsing in the construction of the set of objective values in a multiple objective linear program,” Annals of Operations Research, vol. 47, pp. 279–292, 1993.
[24] Y. Y. Kagan, “Magnitude-frequency distribution in the european-mediterranean earthquake regions-comment,” Tectonophysics, vol. 245, pp. 101–105, 1995.
[25] G. C. Larsen and K. S. Hansen, “Database on wind characteristics-analysis of wind turbine design loads,” Riso National Laboratory Roskilde, vol. Riso-R-1473(EN), 2004.
[26] R. E. Melchers, Structural Reliability-Analysis and Prediction. Wiley, 2 ed., 2002.
[27] A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design. Wiley, 2000.
[28] A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design. John Wiley and Sons, 2000.
[29] B. Fiessler, H.-J. Neumann, and R. Rackwitz, “Quadratic limit states in structural reliability,” ASCE J Eng Mech Div, vol. 105, no. 4, pp. 661–676, 1979.
[30] J. Tu, K. K. Choi, and Y. H. Park, “A new study on reliability-based design optimization,”Journal of Mechanical Design, vol. 121, pp. 557–564, 1999.
[31] F. Levi, M. Gobbi, and G. Mastinu, “An application of multi-objective stochastic optimisation to structural design,” Structural Multidisciplinary Optimization, vol. 29, pp. 272–284, 2005.
[32] Z. Li, L. E. Izquierdo, M. Kokkolaras, S. J. Hu, and P. Y. Papalambros, “Multiobjective optimization for integrated tolerance allocation and fixture layout design in multistation assembly,” Journal of Manufacturing Science and Engineering, vol. 130, 2008.
[33] D.-J. Yu, 多目標最佳化之解集合在不確定因素下之分析及預測" analyses and predictions on the shift of pareto set of multi-objective optimization problems under uncertainty,”Master’s thesis, National Cheng Kung University, Tainan,Taiwan, June 2008.
[34] P. Y. Papalambros, “Remarks on sufficiency of constraint-bound solutions in optimal design,” ASME Journal of Mechanical Design, vol. 115, pp. 374–379, 1993.
[35] D. R. Jones, C. D. Perttunen, and B. E. Stuckman, “Lipschitzian optimization without the lipschitz constant,” Journal of Optimization Theory and Applization, vol. 79, pp. 157–181, 1993.
[36] S. Ruzika and M. M. Wiecek, “Suvey paper approximation methods in multiobjective programming,” Journal of Optimization Theory and applications, vol. 126, no. 3, pp. 473–501, 2005.
[37] X. Q. Yang and C. J. Goh, “A method for convex curve approximation,” European Journal of Operational Research, vol. 97, pp. 205–212, 1997.