應用特定型態質群最佳化演算法於搜尋D-optimal 最佳設計之研究

最佳化實驗設計 (optimal design) 經常被廣泛地運用在化學製造業界。現今因為工業界經常有高度複雜且伴隨限制條件的問題發生，所以在實際操作時，不對稱的實驗區域設計方法 (asymmetrical design) 便有存在之必要性。然而為了建構這樣的最佳化設計，傳統的統計方法有其不足的地方。因此當因子設計 (factorial design) 、部分因子設計 (fractional factorial design) 與反應曲面方法 (response surface methodology) 不容易被應用時，借助電腦演算求得最佳設計為一個較佳的方式。許多不同的最佳化設計準則 (optimality criteria) 以不同且有意義的英文字母來表達，而不同的最佳化設計準則分別衡量不同標準的最佳化設計。這些以電腦輔助 (computer-generated design) 求算的最佳化設計中以D最佳化設計 (D-optimal designs) 是最常被使用。在早期，Fedorov 與 Mitchell 二位學者的傳統演算法經常被應用在建構D最佳化設計上。但是如果遇到問題比較複雜或者其變數維度提高時，這些方法實際應用效率較低。在本文中，我們將利用修正後的質群演算法 (modified particle swarm optimization, MPSO) 於建構一連串D最佳化設計問題，並且在搜尋最佳解的過程中，試著避免落入區域最佳解以及找出適當方法平衡質群演算法在速度更新上的搜尋能力。此外，我們利用保留區域可行解法 (feasible solutions method, FSM) 並重新設定不可行解在實驗區域邊界上的方式來處理混合實驗 (mixture experiments) 與高複雜限制式的問題。

關鍵字

反應曲面法；最佳化設計準則；質群演算法；保留區域可行解法

並列摘要

Optimal designs are applied extensively to the process industry. Asymmetrical designs are now widely used in practice since many engineering problems usually involve complex objective functions and several constraints simultaneously. Hence, the classical designs cannot be valid for this circumstance any more. Accordingly, computer-generated designs are the legitimate choice for situations where standard factorial, fractional factorial designs or response surface designs cannot be easily employed. Design optimality criteria are characterized by letters of the alphabet and as a result, are often called alphabetic optimality criteria. An optimality criterion can be interpreted as a measure of the goodness of the design. Those designs are the most typically used type of computer-generated designs, and of course, the D-optimal class of designs is the most popular one. To construct the D-optimal designs, the Fedorov’s and Mitchell’s algorithm is broadly used. But in case of high dimensionality, this class of algorithms cannot be used adequately due to its efficiency. In this thesis, an optimization technique is introduced to generate D-optimal designs using the modified particle swarm optimization (MPSO). The modified algorithm is mostly developed from the basic particle swarm optimization (BPSO). Meanwhile, the goal of the thesis is to alleviate the risk of being trapped at a suboptimal design solution and further balance exploration and exploitation within the global search ability of MPSO. Furthermore, the constraint-handling mechanism in the mixture experiments demonstrated is tried out to resolve infeasible solutions in this study. Rejection of infeasible individuals and preservation of feasible solutions method (FSM) are used for constituting constrained optimization.

並列關鍵字

Response surface methodology (RSM) ； design optimality criteria ； particle swarm optimization (PSO) ； feasible-solution method (FSM)

參考文獻

Atkinson, A. C. and Donev, A.N. (1989). “The Construction of Exact D-optimum Experimental Designs with Application to Blocking Response Surface Designs,” JSTOR, 76, 3, 515-526.

Beck, J. V. and Arnold, K. J. (1977). Parameter Estimation in Engineering and Science, Wiley, New York.

Clerc, M. (1999). “The Swarm and The Queen: Towards a Deterministic and Adaptive Particle Swarm Optimization,” Proceedings of the Congress on Evolutionary Computation, 3, 1951-1957.

Coath, G. and Halgamuge, S. K. (2003). “A Comparison of Constraint-Handling Methods for The Application of Particle Swarm Optimization to Constrained Nonlinear Optimization Problems,” Proceedings of the Congress on Evolutionary Computation, 4, 2419-2425.

Cook, R. D. and Nachtsheim, C. J. (1980). “A Comparison of Algorithms for Constructing Exact D-Optimal Designs,” Technometrics, 22, 315-324.

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應用特定型態質群最佳化演算法於搜尋D-optimal 最佳設計之研究

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