本研究為解決離散型可靠度限制式之最佳化問題(Discrete Reliability Based Design Optimization，簡稱D-RBDO)，與離散型多目標可靠度限制式之最佳化問題(Discrete Multi-Objective Reliability Based Design Optimization，簡稱DMO-RBDO)。一般來說，實務可靠度最佳化問題通常有兩種特性：離散型的設計參數與複雜的極限狀態式(通常具備非線性及不可微分的特性)。這類問題有單迴圈(Single loop)、雙迴圈(Double loop)及去偶合(decouple)三種求解模式。在本研究提出兩個求解架構求解這兩種的問題，並與單迴圈法、雙迴圈法進行比較。
本研究提出的第一種求解架構稱為PS2，此架構結合了質群演算法(Particle Swarm Optimization，PSO)、支撐向量機(Support Vector Machine，SVM)及子集合模擬法(Subset Simulation，SS)；第二種架構為MOPS2，此架構整合了MOPSO與支撐向量回歸(Support Vector Regression，SVR)及SS。由於SS能有效率地估算小失效機率，並能被SVM或SVR作為訓練資料，進而輔助推估是否滿足可靠度限制或是估算失效機率，以節省運算資源及時間。另一方面，PSO(MOPSO)與SVM(SVR)也能互相支援，SVM能夠減少PSO尋找有效解的時間，而PSO(MOPSO)則是提供SVM(SVR)重新訓練的資料，使SVM(SVR)能有更佳的準確性。
本研究桿件之設計參數參考中國國家標準(CNS)所頒布之CNS 4435,G 3102桿件尺寸列表，對於一個十桿平面桁架與二十五桿立體桁架結構進行討論，來判斷所提出之兩種求解架構與單迴圈法、雙迴圈法在求解的品質與效率的優劣。經過比較後發現PS2、MOPS2都能較雙迴圈法找出更經濟的答案，也不會有單迴圈法會出現的誤判情形。

Reliability-based design optimization (RBDO) is concerned with designing an engineering system to minimize a cost function subject to the reliability requirement that failure probability should not exceed a threshold. Conventional RBDO methods are less than satisfactory in dealing with discrete design parameters, non-Gaussian uncertainties, and complex limit state functions (highly nonlinear, discontinuous, and non-monotonic). Methods that are flexible enough to address the concerns above, however, come at a high computational cost.
To enhance computational efficiency without sacrificing model flexibility, we propose two new RBDO frameworks to solve discrete RBDO with single and multiple objectives. The former is called PS2 as it combines Particle Swarm Optimization (PSO), Support Vector Machine (SVM), and Subset Simulation (SS), whereas the latter is called MO-PS2 which uses Support Vector Regression (SVR) to replace SVM and adopts multiobjective PSO (MOPSO). PSO is employed to solve the discrete optimization problem. SS can efficiently estimate small failure probabilities, based on which SVM is adopted to evaluate the reliability of candidate solutions using binary classification. In the multiobjective context, MOPSO searches for the non-dominated solutions while SVR is used to estimate the failure probability which is considered as another objective. Primary emphasis is placed upon the retraining mechanism between SVM (SVR) and PSO (MOPSO). The cooperation is mutually beneficial since the former helps the latter evaluate the feasibility of solutions or the objective value with high efficiency while the optimal solutions obtained by the latter help retrain the former to attain better accuracy.
Two practical cases (10-bar plane and 25-bar space trusses) are use to demonstrate the proposed frameworks: PS2 and MOPS2. The design variables are chosen from CNS 4435 and G 3102 standards. It has been shown that the proposed frameworks are superior to single and double loop approaches in terms of computation efficiency and solution quality.

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