MD最適化設計在反應曲面模式之建構與分析

本論文首先針對多反應曲面模型下，以MD最適化(multivariate D-optimal)設計建立整體變異最小的設計點組合，藉由基因演算法之搜尋，探討MD最佳化設計點之解和分析設計的效率情形。MD最適設計乃是在最小化迴歸模型參數估計聯合區間的體積，分別以二種特殊的共變異數矩陣之多變量反應變數和用估計的共變異數之雙變量反應變數為例。由每個反應變數模型之D最適化設計作為基因演算法中的初始候選集合，其結果顯示MD最適化設計會在線性模型皆相同時收斂到D最適化設計；在兩反應變數模型不同時，基因演算法亦能有效地搜尋到MD最適化設計。建議設計者可採用較高的交配率和突變率但不要使用完全交配和零突變的組合，並可採前處理方式估計共變異數矩陣。接續探討尋找具顯著的因子以達操作最佳化環境之設定，考慮反應變數間具相關性的實驗設計並重複蒐集之多變量多反應變數。我們提出將多反應變數轉換至單一統計量上，並藉由Jackknife重複抽樣方式保留原始數據的資訊，提供還原策略和直接策略進行求解，建議在因子較少時採還原策略，在因子過多時採用直接策略求解，我們以一個實際的案例說明並與已存在之方法做比較。

關鍵字

MD-最適化設計；基因演算法； Jackknife重複抽樣；反應曲面模型

並列摘要

This study first presents a genetic algorithm (GA) for identifying the exact D-optimal design for multivariate response surface models (called MD-optimal design). The MD-optimal design minimizes the volume of joint confidence region of model parameters. The covariance matrix is assumed to be some special forms in two examples of four responses and biresponse problems considered in this study. In order to obtain the initial candidate set, we first obtain a D-optimal design for each response model by using a conventional approach; then, the set of solutions obtained from the individual model is treated as the initial set in the GA. This shows that the MD-optimal designs converge toward the same D-optimal design in a single response linear model when each response has same linear model. The GA exhibits stable representation in biresponse design problems when models are not the same. It is possible for an experimenter to set high crossover rate and mutation rate except full crossover and no mutation, and estimate the variance-covariance matrix in the preprocess of the GA. Following the research, experiments with multivariate responses whose data collect repetitively were considered. We then proposed transforming multiple responses into single statistic and using Jackknife resampling to recoup the information to estimate the error variance. Direct strategy and reduce strategy were provided. We suggested to use reduce strategy when factors are small, otherwise we suggested to use direct strategy. The proposed method is illustrated with an experiment from the literature.

並列關鍵字

MD-optimal design ； Genetic algorithm ； Jackknife resampling ； Response surface model

參考文獻

[1] Montgomery, D.C., Design and Analysis of Experiments, 5th Ed. Wiley, New York, 2001.

[2] Derringer, G. and Suich R., “Simultaneous Optimization of Several Response Variables,” Journal of Quality technology, Vol. 12, pp. 214-219. 1980.

[3] Khuri, A. I. and Conlon, M., “Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions,” Technometrics, Vol. 23, No. 4, pp. 363-375. 1981.

[4] Pignatiello, JR., J. J. “Strategies for Robust Multiresponse Quality Engineering,” IIE Transactions, Vol. 25, pp. 5-25. 1993.

[5] Vining, G. A., “A Compromise Approach to Multiresponse Optimization,” Journal of Quality Technology, Vol. 30, pp. 309-313. 1998.

國際替代計量

MD最適化設計在反應曲面模式之建構與分析

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