產品的設計往往必須同時考量多個目標，多目標最佳化設計乃針對此一需求，以數學的角度探討各目標彼此間的權衡得失，所得之最佳解可滿足拘束條件的需求並具備最佳設計之特性，然而此最佳解並不唯一，全部最佳解的集合稱之為效率解集合(Pareto set)。在數學上求得效率解集合的方法是利用有限的解來代表整體解集合，但仍須經過多次的最佳化運算。除了多目標的考量外，工程上的問題常因各種不確定因素的存在，使得結果不如預期。可靠度最佳化設計法則將不確定因素用隨機變數的型態加入最佳化架構下，並將拘束條件以機率的型式呈現，以確保最終設計在不確定因素下仍具理想可靠度。

本論文旨在分析多目標最佳化在考量不確定因素下其效率解集合之移動，並針對線性設計空間及非線性凸集合空間，提出預測效率解移動的方法。效率解的移動在文獻上雖有觀察到，但並無針對此一移動現象進行分析或預測。為探討效率解移動的主要機制，本論文先針對線性系統加以分析，利用線性規劃中，最佳解必落在極點或邊界上的特性，藉由拘束法與一階二次可靠度計算作為本論文的研究方法。研究發現，線性系統在提升可靠度後，其效率解會跟著邊界移動，且移動的量值與可靠度大小呈比例關係。基於上述特性，本論文提出兩套預測效率解移動的方法，分別為合成空間法與區間預測法，可供處理一般線性系統多目標最佳化在提升可靠度後其效率解集合的移動預測。針對非線性系統的問題，本論文探討設計空間為凸集合的狀況，利用線性化凸集合空間的方式，將線性預測法延伸應用到非線性凸集合問題上。本論文所提出的分析及預測法，將用一個在不確定因素下之桁架多目標最佳化範例，來說明預測法於工程上的實際運用。近年來，由於環保意識的抬頭，我們亦提出將預測方法推廣到結合生命週期評估與多目標最佳化的高維度系統上的概念，期望藉由此預測方法，可有效率並準確的解決工程上多目標最佳化在不確定因素下的設計問題。

Product design is a multi-objective decision-making process that requires satisfying various engineering targets simultaneously. However different objectives might contradict with each other, resulting in engineering tradeoffs. Multi-objective optimization solves this type of engineering problems in a mathematical way. Solutions to a multi-objective optimization are not unique due to the tradeoffs and no solutions is absolutely superior to the other, i.e. non-dominant. The set of all non-dominant solutions is called a Pareto set. Obtaining a Pareto set mathematically is difficulty because there exists infinite possible solutions in a set. Practically limited number of non-dominant solutions are generated to approximate the entire set.

In addition to the multi-objective characteristic, product design is also a decision-making process under various uncertainties. In this thesis, we study the effects of uncertainty on the Pareto set and provide preliminary suggestions for prediction the change of Pareto set under uncertainties of various degrees. Uncertainties are modeled as random variables in this thesis and then incorporated into the existing multi-objective optimization framework using Reliability-Based Design Optimization methodology. The effects of uncertainty is modeled by reformulating constraints into probabilistic forms.

The modifications of constraints from deterministic to probabilistic change the Pareto set. Several reports from the literature also observed this change; however, detail analyses and predictions of why and how the Pareto set moves are yet to be discussed. In order to understand the mechanisms behind the shift of Pareto set under uncertainty, we start from linear problems, that is multi-objective optimization problems with linear objective functions and linear constraints (MOLP). Since solutions to MOLP lies on the vertex or at the convex hull of the design space, the shift of Pareto set will follow the change of the design space. Results show the shift of Pareto set directly related to the reliability requirements of the probabilistic constraints.Therefore, this research presents two efficient methods,namely global synthesis and local extraction method,to predict the shift of Pareto set. In addition we extend the methods to problems with convex feasible design space that may also include nonlinear constraints. The proposed methods have been applied to a high-dimensional multi-objective ten-bar truss design optimization example under uncertainty. Expected extension of the work is on the product life cycle impact assessment. This extension is briefly discussed at the end of this thesis and detail implementations are yet to be realized.

[1] Z. Li, M. Kokkolaras, D. Jung, P.Y. Papalambros, and D.N. Assanis. An optimization study of manufacturing variation effects on diesel injector design with emphasis on emissions. SAE Paper, 01-1560, 2004.
[2] M. Gobbi, F. Levi, and G. Mastinu. Multi-objective stochastic optimisation of the suspension system of road vehicles. Journal of Sound and Vibration,298:1055-1072, 2006.
[3] K Sinha. Reliability-based multiobjective optimization for automotive crash-worthiness and occupant safety. Structural Multidisciplinary Optimization,33:255-268, 2006.
[4] 許文政. 橋梁生命週期成本評估-構件劣化預測模式之研究. 國立中央大學, 2005.
[5] 許志義. 多目標決策. 五南圖書出版社,民國八十三年.
[6] M. Zeleny. Multiceriteria decision Marking. McGraw-Hill, 1982.
[7] J. Lin. Multiple objective problems: Pareto-optimal solutions by method of proper equality constraints. IEEE Transactions on Automatic Control, 21:641-650, 1976.
[8] S. Gass and T. Saaty. The computational algorithm for the parametric objective function. Naval Research Logistics Quarterly, 2:39, 1955.
[9] L. Zadeh. Optimality and non-scalar-valued performance criteria. IEEE Trans-actions on Automatic Control, 8:59-60, 1955.
[10] I.Y. Kim. Adaptive weighted sum method for biobjective optimization: Pareto front generation. Structural Multidisciplinary Optimization, 29:149-158, 2005.
[11] S. Marglin. Public Investment Criteria. MIT Press, 1967.
[12] I. Das and J. Dennis. Normal-boundary intersection: An alternate method for generating pareto optimal points in multicriteria optimization problems.Institute for Computer Applications in Science and Engineering, 1996.
[13] 劉稚信.機械可靠性設計.清華大學出版社, 1996.
[14] 傅鶴齡.系統工程概論.滄海書局, 2001.
[15] B.S. Blanchard. System Engineering and Management. John Wiley & Sons, 2004.
[16] J.C. Helton and F.J. Davis. Latin hypercube sampling and the propogation of uncertainty in analysis of complex systems. Reliability Engineering and System Safety, 81:23-69, 2003.
[17] A. Haldar and S. Mahadevan. Probability, Reliability, and Statistical Methods in Engineering Design. Argosy, 2000.
[18] L. Wang, R. V. Grandhi, and D. A. Hopkins. Structural reliability optimization using an efficient safety index calculation procedure. International Journal for Numerical Methods in Engineering, 38:1721-1738, 1995.
[19] Y. T. Wu, H. R. Millwater, and T. A. Cruse. Advanced probabilistic structural analysis method for implicit performance functions. AIAA Journal, 28:1663-1669, 1990.
[20] 費培之.線性和非線性規劃及其應用.四川大學出版社,1989.
[21] G. B. Dantzig. Linear Programming and Extensions. Princeton University Press, 1963.
[22] H. W. Kuhn and A. W. Tucker. Nonlinear programming. University of California Press, 1951.
[23] M. J. Best and K. Ritter. Linear Programming: Active Set Analysis Computer Programs. Prentice-Hall, 1985.
[24] R. Heijungs and S. Suh. The Computational Structure of Life Cycle Assessment.Kluwer Academic, 2002.
[25] ISO14001. Environmental Management Systems-specification with guidance for use. International Organization for Standardization, 1996.
[26] ISO900412000. Quality Management Systems- Guidelines for Performance Improvement. International Organization for Standardization, 2000.
[27] A. Azapagic and R. Clift. Life cycle assessment and multiobjective optimisation. Journal of Cleaner Production, 7:135-143, 1999.
[28] A. Azapagic and R. Clift. Allocation of environmental burdens in multiplefunction systems. Journal of Cleaner Production, 7:101-119, 1999.
[29] R.E. Millor and P.D. Blair. Input-output Analysis. Prentice-Hall, 1985.