考虑模型响应不确定性的稳健参数设计
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摘要
针对模型响应不确定性的稳健参数设计问题,在高斯过程回归(Gaussian process regression, GPR)建模的框架下,结合贝叶斯超参数最大后验(Maximum a posteriori estimation, MAP)估计和多目标线性加权方法构建一个新的优化模型.首先,利用MAP方法获得最优超参数组合,构建高斯回归模型;然后,考虑响应不确定性与响应之间的交互效应,采用线性加权准则,构建多响应稳健优化模型;最后,利用聚类分析方法获得最优参数解.该方法考虑了输出响应不确定性对优化结果的影响,权衡了最优因子水平与多元质量特性之间的关系.结合实际案例和软件仿真对所提出方法进行实证研究,结果表明,该方法能够较好地兼顾输出响应的最优性和稳健性,从而实现稳健参数设计.
A new optimization model, integrating maximum a posteriori(MAP) estimation of the Bayesian approach and the linear weighting method for multi-objectives in the framework of Gaussian process regression(GPR) modeling, is proposed to solve the problem of robust parameter design for the uncertainty of response. Firstly, optimal hyperparameters are obtained by using a MAP method, and the Gaussian regression model is constructed. Then, a multi-objective optimization model is constructed by using the linear weighting criterion. Finally, the optimal parameter solution is obtained by cluster analysis. The proposed method considers the effect of output response uncertainty on the optimization results and balances the relationship between the optimal factor level and the multivariate quality characteristics.The effectiveness of the proposed method is verified through a practical industrial example combined with a simulation example. The results show that the proposed method can give good consideration to the optimal and the robust of response, so as to achieve the robust parameter design.
A new optimization model, integrating maximum a posteriori(MAP) estimation of the Bayesian approach and the linear weighting method for multi-objectives in the framework of Gaussian process regression(GPR) modeling, is proposed to solve the problem of robust parameter design for the uncertainty of response. Firstly, optimal hyperparameters are obtained by using a MAP method, and the Gaussian regression model is constructed. Then, a multi-objective optimization model is constructed by using the linear weighting criterion. Finally, the optimal parameter solution is obtained by cluster analysis. The proposed method considers the effect of output response uncertainty on the optimization results and balances the relationship between the optimal factor level and the multivariate quality characteristics.The effectiveness of the proposed method is verified through a practical industrial example combined with a simulation example. The results show that the proposed method can give good consideration to the optimal and the robust of response, so as to achieve the robust parameter design.
引文
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[13]He Z,Zhu P F,Park S H.A robust desirability function method for multi-response surface optimization considering model uncertainty[J].European J of Operational Research,2012,221(1):241-247.
[14]Yuan J,Wang K,Yu T.Reliable multi-objective optimization of high-speed WEDM process based on Gaussian process regression[J].Int J of Machine Tools and Manufacture,2008,48(1):47-60.
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[18]Helin T,Burger M.Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems[J].Inverse Problems,2015,31(8):085009(1-22).
[19]Deb K,Pratap A,Agarwal S,et al.A fast and elitist multiobjective genetic algorithm:NSGA-II[J].IEEE Trans on Evolutionary Computation,2002,6(2):182-197.
[20]何桢,朱鹏飞.基于模式搜索的渴求函数法在多响应优化中的应用[J].数学的实践与认识,2009,39(18):114-121.(He Z,Zhu P F.Application of pattern search algorithm to multiresponse optimization based on desirability functions[J].Mathematics in Practice&Theory,2009,18(18):114-121.)
[21]Ding C,He X F.Cluster structure of K-means clustering via principal component analysis[J].Pacific-Asia Conf on Advances in Knowledge Discovery and Data Mining,2004,46(4):414-418.
[22]Prasanna J,Karunamoorthy L,Venkat Raman M,et al.Optimization of process parameters of small hole dry drilling in Ti-6Al-4V using Taguchi and grey relational analysis[J].Measurement,2014,48(1):346-354.
[23]Wu C F J,Hamada M S.Experiments:Planning,analysis and optimization[M].New Jersey:John Wiley&Sons,2009.
[2]欧阳林寒,马义中,汪建均,等.基于模型不确定性的响应曲面建模[J].系统工程与电子技术,2015,37(8):1818-1824.(Ouyang L H,Ma Y Z,Wang J J,et al.Response surface modeling based on model uncertainty[J].Systems Engineering&Electronics,2015,37(8):1818-1824.)
[3]顾晓光,马义中,汪建均,等.多元质量特性的满意参数设计[J].控制与决策,2014,29(6):1064-1070.(Gu X G,Ma Y Z,Wang J J,et al.Satisfactory parameter design for multivariate quality characteristics[J].Contro and Decision,2014,29(6):1064-1070.)
[4]Murphy T E,Tsui K L,Allen J K.A review of robus design methods for multiple responses[J].Research in Engineering Design,2005,16(3):118-132.
[5]汪建均,马义中,欧阳林寒,等.多响应稳健参数设计的贝叶斯建模与优化[J].管理科学学报,2016,19(2)85-94.(Wang J J,Ma Y Z,Ouyang L H,et al.Bayesian modeling and optimization of multi-response robust parameter design[J].J of Management Sciences in China,201619(2):85-94.)
[6]何桢,宗志宇,孔祥芬.改进的满意度函数法在多响应优化中的应用[J].天津大学学报,2006,39(9):1136-1140.(He Z,Zong Z Y,Kong X F.Application of improved desirability function method to the multi-response optimization[J].J of Tianjin University,2006,39(9)1136-1140.)
[7]Ko Y H,Kim K J,Jun C H.A new loss function-based method for multiresponse optimization[J].J of Quality Technology,2005,37(1):50-59.
[8]张旭涛,何桢.多响应稳健性优化的贝耶斯分析[J].系统工程,2015,33(9):127-132.(Zhang X T,He Z.Robust optimization for multi-response using bayesian analysis[J].Systems Engineering,201533(9):127-132.)
[9]Chiao C H,Hamada M.Analyzing experiments with correlated multiple responses[J].J of Quality Technology2001,33(4):451-465.
[10]Apley D W,Kim J.A cautious approach to robust design with model parameter uncertainty[J].IIE Trans,201143(7):471-482.
[11]Ouyang L,Ma Y,Byun J H,et al.An interval approach to robust design with parameter uncertainty[J].Int J of Production Research,2016,54(11):3201-3215.
[12]Wang J,Ma Y,Ouyang L,et al.A new Bayesian approach to multi-response surface optimization integrating loss function with posterior probability[J].European J of Operational Research,2016,249(1):231-237.
[13]He Z,Zhu P F,Park S H.A robust desirability function method for multi-response surface optimization considering model uncertainty[J].European J of Operational Research,2012,221(1):241-247.
[14]Yuan J,Wang K,Yu T.Reliable multi-objective optimization of high-speed WEDM process based on Gaussian process regression[J].Int J of Machine Tools and Manufacture,2008,48(1):47-60.
[15]Rasmussen C E,Williams C K I.Gaussian process for machine learning[M].Cambridge:MIT Press,2006:1-248.
[16]Quinonero-Candela J Q,Rasmussen C E.An unifying view of sparse approximate Gaussian process regression[J].J of Machine Learning Research,2005,6:1939-1959.
[17]Shi J Q,Murray-Smith R,Titterington D M.Hierarchical Gaussian process mixtures for regression[J].Statistics and Computing,2005,15(1):31-41.
[18]Helin T,Burger M.Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems[J].Inverse Problems,2015,31(8):085009(1-22).
[19]Deb K,Pratap A,Agarwal S,et al.A fast and elitist multiobjective genetic algorithm:NSGA-II[J].IEEE Trans on Evolutionary Computation,2002,6(2):182-197.
[20]何桢,朱鹏飞.基于模式搜索的渴求函数法在多响应优化中的应用[J].数学的实践与认识,2009,39(18):114-121.(He Z,Zhu P F.Application of pattern search algorithm to multiresponse optimization based on desirability functions[J].Mathematics in Practice&Theory,2009,18(18):114-121.)
[21]Ding C,He X F.Cluster structure of K-means clustering via principal component analysis[J].Pacific-Asia Conf on Advances in Knowledge Discovery and Data Mining,2004,46(4):414-418.
[22]Prasanna J,Karunamoorthy L,Venkat Raman M,et al.Optimization of process parameters of small hole dry drilling in Ti-6Al-4V using Taguchi and grey relational analysis[J].Measurement,2014,48(1):346-354.
[23]Wu C F J,Hamada M S.Experiments:Planning,analysis and optimization[M].New Jersey:John Wiley&Sons,2009.
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