基于响应曲面模型拟合优度的多响应优化方法
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摘要
响应曲面方法(Response Surface Methodology)是对产品质量改进的一种有效的试验设计方法。在多响应优化问题中,多个响应模型的拟合优度有很大不同,多响应优化结果的准确性和可靠性会受到影响。本文主要研究基于响应曲面模型不同拟合优度的多响应优化方法。
本文首先对响应曲面方法的一些基本理论和应用方法做了介绍,然后着重介绍解决多重响应优化问题的一些方法,包括马氏距离法、质量损失函数法、传统的满意度函数方法等,并对这些常用方法进行综合评价。
因为以上提到的常用的多响应优化方法并未考虑多响应模型的拟合优度,或者并未很好的解决不同拟合优度下的多响应优化问题,本文接下来研究基于响应曲面模型拟合优度的多响应优化方法。为了这个问题,首先应分析回归的模型拟合优度,确定一个合理的能够衡量模型的回归拟合程度的指标。由于R2会随着模型中因子个数的增加而增大,使用R2作为回归模型的拟合优度的度量不太恰当。而考虑修正的R2,即R2adj,但R2adj的优越性并没有得到理论性的证明,将其作为回归模型的拟合优度的指标也不合适。RNL被用于非线性回归模型的拟合优度指标,RNL有一定的合理性。
然后,本文以传统的满意度函数为基础,结合Kim提出的指数满意度函数的思想,提出了一种考虑拟合优度指标RBL的改进的满意度函数模型,对该模型做了具体的解释和分析,最后对两个案例应用该方法进行验证,得到了较为理想的效果。该方法考虑了响应模型的拟合优度因素,分析了不同的拟合优度对多响应优化结果的影响。该方法具有一定的合理性。
RSM(Response Surface Methodology) is an effective design of experiment method to improve products'quality. For multiple-response optimization problem, the reliability of optimization result can be impacted by different goodness of fit of the repsonse surface models. Therefore, this thesis does the research on optimization method for multiple responses based on goodness of fit for response surface.
The thesis first introduces some basic theories and application methodologies of response surface method. Then some multiple response surface optimization methods are described, including Mahalanobis distance method, quality loss functin method, traditional desirability function method, etc. Assessments are made on these common methods as well.
For the common multi-response surface optimization methods above don't consider goodness of fit of the regression models, or multi-repsonse surface optimization problem cann't be well solved in these methods, the thesis does researh on this aspect. Reliable statistic index to measure the goodness of fit for repsonse surface model should be discussed as a premise. Neither of R2 or R2adj can well measure the regression degree of models. Another rational statistic, Rnl can be used to measure the goodness of fit of a regression model.
After that, the thesis uses the traditional desirability function and the exponetial desirability function defined by Kim and Lin for reference, and proposes a modified desirability funtion method. Finally, two cases are solved by the new desirability function method, considering goodness of fit of response surface models. The new method can be applied in multiple response problems.
本文首先对响应曲面方法的一些基本理论和应用方法做了介绍,然后着重介绍解决多重响应优化问题的一些方法,包括马氏距离法、质量损失函数法、传统的满意度函数方法等,并对这些常用方法进行综合评价。
因为以上提到的常用的多响应优化方法并未考虑多响应模型的拟合优度,或者并未很好的解决不同拟合优度下的多响应优化问题,本文接下来研究基于响应曲面模型拟合优度的多响应优化方法。为了这个问题,首先应分析回归的模型拟合优度,确定一个合理的能够衡量模型的回归拟合程度的指标。由于R2会随着模型中因子个数的增加而增大,使用R2作为回归模型的拟合优度的度量不太恰当。而考虑修正的R2,即R2adj,但R2adj的优越性并没有得到理论性的证明,将其作为回归模型的拟合优度的指标也不合适。RNL被用于非线性回归模型的拟合优度指标,RNL有一定的合理性。
然后,本文以传统的满意度函数为基础,结合Kim提出的指数满意度函数的思想,提出了一种考虑拟合优度指标RBL的改进的满意度函数模型,对该模型做了具体的解释和分析,最后对两个案例应用该方法进行验证,得到了较为理想的效果。该方法考虑了响应模型的拟合优度因素,分析了不同的拟合优度对多响应优化结果的影响。该方法具有一定的合理性。
RSM(Response Surface Methodology) is an effective design of experiment method to improve products'quality. For multiple-response optimization problem, the reliability of optimization result can be impacted by different goodness of fit of the repsonse surface models. Therefore, this thesis does the research on optimization method for multiple responses based on goodness of fit for response surface.
The thesis first introduces some basic theories and application methodologies of response surface method. Then some multiple response surface optimization methods are described, including Mahalanobis distance method, quality loss functin method, traditional desirability function method, etc. Assessments are made on these common methods as well.
For the common multi-response surface optimization methods above don't consider goodness of fit of the regression models, or multi-repsonse surface optimization problem cann't be well solved in these methods, the thesis does researh on this aspect. Reliable statistic index to measure the goodness of fit for repsonse surface model should be discussed as a premise. Neither of R2 or R2adj can well measure the regression degree of models. Another rational statistic, Rnl can be used to measure the goodness of fit of a regression model.
After that, the thesis uses the traditional desirability function and the exponetial desirability function defined by Kim and Lin for reference, and proposes a modified desirability funtion method. Finally, two cases are solved by the new desirability function method, considering goodness of fit of response surface models. The new method can be applied in multiple response problems.
引文
[1]Box G E P, Wilson K B, On the experimental attainment of optimum conditions. Journal of the Royal Statistical Society,1951, Series B,13:1-45.
[2]KhuriAI, ConlonM. Simultaneous optimization of multiple response represented by polynomial regression function. Technometrics,1981,23(4):363-375.
[3]何桢,张于轩,多响应实验设计的优化方法研究,工业工程,2003,4(4)：35-38.
[4]Pignatiello J. J. Strategies for robust multiresponse quality engineering.IIE Transaction,1993, 25(3),5-15.
[5]Tsui K L. Robust design optimization for multiple characteristic problems. Int. J. Prod. Res. 1999,37(2):433-445.
[6]Ames A E, et al. Quality loss functions for optimization across multiple response surfaces. Journal Quality Technology,1997,29(3):339-346.
[7]Vining G G. A compromise approach to multiresponse optimization. Journal Quality Technology,1998,30(4):309-313.
[8]Young-Hyun Ko, Kwang-Jae Kim, Chi-Hyuck Jun, A New loss function-based method of rmultiresponse opti-mization. Journal of Qulaity Tcehnology.2005,37(1):50-59.
[9]J. L. Ribeir, E. A. Elsayed. A case study on process optimization using the gradient loss function. Int. J. Prod. Res.1995,33(12):3233-3248.
[10]V. Roshan. Joseph. Quality loss functions for nonnegative variables and their application. Hournal of Quality Technology.2004,36(2):129-138.
[11]Harrington E C. The desirability function. Industrial Quality Control,1965,21:494-498.
[12]Derringer G, Su ich R. Simultaneous optimization of several response variables. Journal Quality Technology,1980,12:214-219.
[13]Kwang-Jae Kim, Dennis K J. Lin. Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions, Applied Statistics,2000,49(3):311-325.
[14]Chiao C H, Hamada M. Analyzing experiments with correlated multiple response. Journal of Quality Tcehnology,2001,33(4):451-465.
[15]John J. Peterson. A posterior predictive approach to multiple response surface optimization. Journal of Qulaity Technology.2004,36(2):139-153.
[16]Quesada G M, Castillo E D, Peterson J J. A Bayesian approach for multiple response surface optimization in the presence of noise variables. Applied Statistics,2004,31(3):251-270.
[17]Tong C T, Lee I. Multi-response robust design by principal component analysis. Total Quality Management,1997,8(6):409-416.
[18]Antony J. Multi-response optimization in industrial experiments using Taguchi's quality function and principal component analysis. Quality and reliability engineering international, 2000,16:3-8.
[19]钟晓芳,韩之俊,利用主成分分析对多质量特性的优化设计,南京理工大学学报,2003,27(3)：301-304.
[20]Auer C, Erdbrugge M, Gobel R. Comparison of multivariate method for robust parameter design in sheet metal spinning. Applied stochastic models in business and industry,2004,20: 201-218.
[21]Quesada G M, Castillo E D. A dual-response approach to the multivariate robust parameter design problem. Technometrics,2004,46(2):176-187.
[22]Del Csatillo, E. Multiresponse process optimization via constrained confidence regions. Journal Quality Technology,1996,28(1):61-70.
[23]Tong L I, Su C T. Optimizing multi-response problem in the Taguchi mehtod by fuzzy mlutiple attribute decision making. Quality and reliability engineering intenational,1997, 13:25-34.
[24]Lu D, Antony J. Optimization of multiple responses using a fuzzy rule based inference system. Int. J. Prdo. Res.,2002,40(7):1613-1625.
[25]T. S. Arthanari. A game theory application in robust design. Quality Engineering,2005, 17:291-300.
[26]Douglas C. Montgomoery, Design and Analysis of Experiments(傅珏生等译),北京：人民邮电出版社,2009,1:177-366.
[27]Douglas C. Montgomoery,试验设计与分析(王仁官,陈荣昭译),北京：中国统计出版社,1998,617-620.
[28]C. F. Jeff Wu, Michael Hamada,试验设计与分析及参数优化(张润楚等译),北京：中国统计出版社,2003,359-362.
[29]古扎拉蒂,计量经济学(费剑平,孙春霞等译),北京：中国人民大学出版社,2005.
[30]Anscombe, F. J, et al. Graphs in Statistical Analysis. Am. Statist,1973,27.
[31]Henri Thcil, Introduction to Ecnometrics, Prentice Hall, Englowood Cliffs, N. J.,1978, 135.
[32]张世强,曲线回归的拟合优度指标的探讨,中国卫生统计,2002,19(1)：9-11.
[33]董万鹏,陈军,不锈钢流动应力模型分析,计算机仿真,2010,27(4)：80-83.
[34]高雪峰,何桢等,超高温灭菌纯牛奶杀菌工艺的多响应曲面法优化研究,吉林农业大学 学报,2007,29(1)：107-112.
[35]赵松山,对拟合优度R2的影响因素分析与评价,东北财经大学学报,2003(3)：56-58.
[36]王重,刘黎明,拟合优度检验统计量的设定方法,统计与决策,2010(5)：154-156.
[37]宗志宇,何桢,孔祥芬,多响应优化方法的比较和应用研究,数理统计与管理,2006,25 (6)：697-704.
[38]Zhen He, Jing Wang, Jinho Oh, et al, Robust optimization for multiple responses using response surface methodology, Applied stochastic models in business and industry,2010, 26:157-171.
[2]KhuriAI, ConlonM. Simultaneous optimization of multiple response represented by polynomial regression function. Technometrics,1981,23(4):363-375.
[3]何桢,张于轩,多响应实验设计的优化方法研究,工业工程,2003,4(4)：35-38.
[4]Pignatiello J. J. Strategies for robust multiresponse quality engineering.IIE Transaction,1993, 25(3),5-15.
[5]Tsui K L. Robust design optimization for multiple characteristic problems. Int. J. Prod. Res. 1999,37(2):433-445.
[6]Ames A E, et al. Quality loss functions for optimization across multiple response surfaces. Journal Quality Technology,1997,29(3):339-346.
[7]Vining G G. A compromise approach to multiresponse optimization. Journal Quality Technology,1998,30(4):309-313.
[8]Young-Hyun Ko, Kwang-Jae Kim, Chi-Hyuck Jun, A New loss function-based method of rmultiresponse opti-mization. Journal of Qulaity Tcehnology.2005,37(1):50-59.
[9]J. L. Ribeir, E. A. Elsayed. A case study on process optimization using the gradient loss function. Int. J. Prod. Res.1995,33(12):3233-3248.
[10]V. Roshan. Joseph. Quality loss functions for nonnegative variables and their application. Hournal of Quality Technology.2004,36(2):129-138.
[11]Harrington E C. The desirability function. Industrial Quality Control,1965,21:494-498.
[12]Derringer G, Su ich R. Simultaneous optimization of several response variables. Journal Quality Technology,1980,12:214-219.
[13]Kwang-Jae Kim, Dennis K J. Lin. Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions, Applied Statistics,2000,49(3):311-325.
[14]Chiao C H, Hamada M. Analyzing experiments with correlated multiple response. Journal of Quality Tcehnology,2001,33(4):451-465.
[15]John J. Peterson. A posterior predictive approach to multiple response surface optimization. Journal of Qulaity Technology.2004,36(2):139-153.
[16]Quesada G M, Castillo E D, Peterson J J. A Bayesian approach for multiple response surface optimization in the presence of noise variables. Applied Statistics,2004,31(3):251-270.
[17]Tong C T, Lee I. Multi-response robust design by principal component analysis. Total Quality Management,1997,8(6):409-416.
[18]Antony J. Multi-response optimization in industrial experiments using Taguchi's quality function and principal component analysis. Quality and reliability engineering international, 2000,16:3-8.
[19]钟晓芳,韩之俊,利用主成分分析对多质量特性的优化设计,南京理工大学学报,2003,27(3)：301-304.
[20]Auer C, Erdbrugge M, Gobel R. Comparison of multivariate method for robust parameter design in sheet metal spinning. Applied stochastic models in business and industry,2004,20: 201-218.
[21]Quesada G M, Castillo E D. A dual-response approach to the multivariate robust parameter design problem. Technometrics,2004,46(2):176-187.
[22]Del Csatillo, E. Multiresponse process optimization via constrained confidence regions. Journal Quality Technology,1996,28(1):61-70.
[23]Tong L I, Su C T. Optimizing multi-response problem in the Taguchi mehtod by fuzzy mlutiple attribute decision making. Quality and reliability engineering intenational,1997, 13:25-34.
[24]Lu D, Antony J. Optimization of multiple responses using a fuzzy rule based inference system. Int. J. Prdo. Res.,2002,40(7):1613-1625.
[25]T. S. Arthanari. A game theory application in robust design. Quality Engineering,2005, 17:291-300.
[26]Douglas C. Montgomoery, Design and Analysis of Experiments(傅珏生等译),北京：人民邮电出版社,2009,1:177-366.
[27]Douglas C. Montgomoery,试验设计与分析(王仁官,陈荣昭译),北京：中国统计出版社,1998,617-620.
[28]C. F. Jeff Wu, Michael Hamada,试验设计与分析及参数优化(张润楚等译),北京：中国统计出版社,2003,359-362.
[29]古扎拉蒂,计量经济学(费剑平,孙春霞等译),北京：中国人民大学出版社,2005.
[30]Anscombe, F. J, et al. Graphs in Statistical Analysis. Am. Statist,1973,27.
[31]Henri Thcil, Introduction to Ecnometrics, Prentice Hall, Englowood Cliffs, N. J.,1978, 135.
[32]张世强,曲线回归的拟合优度指标的探讨,中国卫生统计,2002,19(1)：9-11.
[33]董万鹏,陈军,不锈钢流动应力模型分析,计算机仿真,2010,27(4)：80-83.
[34]高雪峰,何桢等,超高温灭菌纯牛奶杀菌工艺的多响应曲面法优化研究,吉林农业大学 学报,2007,29(1)：107-112.
[35]赵松山,对拟合优度R2的影响因素分析与评价,东北财经大学学报,2003(3)：56-58.
[36]王重,刘黎明,拟合优度检验统计量的设定方法,统计与决策,2010(5)：154-156.
[37]宗志宇,何桢,孔祥芬,多响应优化方法的比较和应用研究,数理统计与管理,2006,25 (6)：697-704.
[38]Zhen He, Jing Wang, Jinho Oh, et al, Robust optimization for multiple responses using response surface methodology, Applied stochastic models in business and industry,2010, 26:157-171.