Abstract：For the robust parameter design problem of prediction deviation and variability, a new optimization model is constructed by combining the quality loss and Bayesian posterior estimation method under the framework of multivariate Gaussian process (MGP) modeling. Firstly, the hyperparameters are obtained by using the pairwise estimation method, and the MGP is constructed. Secondly, Monte Carlo simulation method is used to obtain the expected probability that the responses fall within the specified intervals. Then, the optimization model is established by using the quality loss function proposed in this paper with the expected probability as the constraint. Finally, the global optimization algorithm is used to perform global optimization, and the optimization results considering the expected probability are obtained. The example shows that proposed method comprehensively considers the impact of prediction deviation and variability on the optimization result. The optimal considering quality loss and expected probability is obtained, and realizing robust parameter design.
冯泽彪, 汪建均, 马义中. 基于多变量高斯过程模型的贝叶斯建模与稳健参数设计[J]. 系统工程理论与实践, 2020, 40(3): 703-713.
FENG Zebiao, WANG Jianjun, MA Yizhong. Bayesian modeling and robust parameter design based on multivariate Gaussian process model. Systems Engineering - Theory & Practice, 2020, 40(3): 703-713.
[1] 汪建均, 马义中. 基于GLM的双响应曲面法及其稳健设计[J]. 系统工程与电子技术, 2012, 34(11):2306-2311.Wang J J, Ma Y Z. Dual response surface methodology based on generalized linear models and its application on robust design[J]. Systems Engineering & Electronics, 2012, 34(11):2306-2311. [2] 汪建均, 马义中, 翟云焕. 相关多质量特性的优化设计[J]. 管理工程学报, 2011, 25(2):66-73.Wang J J, Ma Y Z, Zhai Y H. Optimization design of correlated multiple quality characteristics[J]. Journal of Industrial Engineering & Engineering Management, 2011, 25(2):66-73. [3] 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]. Journal of Management Sciences in China, 2016, 19(2):85-94. [4] Pignatiello J J. Strategies for robust multi-response quality engineering[J]. IIE Transactions, 1993, 25(3):5-15. [5] Vining G G. A compromise approach to multiresponse optimization[J]. Journal of Quality Technology, 1998, 30(4):309-313. [6] Ko Y H, Kim K J, Jun C H. A new loss function-based method for multiresponse optimization[J]. Journal of Quality Technology, 2005, 37(1):50-59. [7] 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 multiresponse using Bayesian analysis[J]. Systems Engineering, 2015, 33(9):127-132. [8] Chiao C H, Hamada M. Analyzing experiments with correlated multiple responses[J]. Journal of Quality Technology, 2001, 33(4):451-465. [9] Parnianifard A, Azfanizam A S, Ariffin M K A, et al. An overview on robust design hybrid metamodeling:Advanced methodology in process optimization under uncertainty[J]. International Journal of Industrial Engineering Computations, 2018, 9:1-32. [10] Beyer H G, Sendhoff B. Robust optimization——A comprehensive survey[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 196(33-34):3190-3218. [11] Ouyang L, Ma Y, Wang J, et al. A new loss function for multi-response optimization with model parameter uncertainty and implementation errors[J]. European Journal of Operational Research, 2016, 258(2):552-563. [12] He Z, Zhu P F, Park S H. A robust desirability function method for multi-response surface optimization considering model uncertainty[J]. European Journal of Operational Research, 2012, 221(1):241-247. [13] 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. [14] 张旭涛, 何桢. 基于似无关回归的多元稳健损失函数方法[J]. 数理统计与管理, 2017, 36(5):802-809.Zhang X T, He Z. A robust loss function approach for multi-response optimization based on seemingly unrelated regressions[J]. Journal of Applied Statistics & Management, 2017, 36(5):802-809. [15] 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 Journal of Operational Research, 2016, 249(1):231-237. [16] Qian P Z G, Wu H, Wu C F J. Gaussian process models for computer experiments with qualitative and quantitative factors[J]. Technometrics, 2008, 50(3):383-396. [17] Kleijnen J P C. Kriging metamodeling in simulation:A review[J]. European Journal of Operational Research, 2009, 192(3):707-716. [18] Shao W, Deng H, Ma Y, et al. Extended Gaussian Kriging for computer experiments in engineering design[J]. Engineering with Computers, 2011, 28(2):161-178. [19] Yuan J, Wang K, Yu T, et al. Reliable multi-objective optimization of high-speed WEDM process based on Gaussian process regression[J]. International Journal of Machine Tools & Manufacture, 2008, 48(1):47-60. [20] Mehdad E, Kleijnen J P C. Classic Kriging versus Kriging with bootstrapping or conditional simulation:Classic Kriging's robust confidence intervals and optimization[J]. Journal of the Operational Research Society, 2015, 66(11):1804-1814. [21] Kleijnen J P C, Mehdad E. Multivariate versus univariate Kriging metamodels for multi-response simulation models[J]. European Journal of Operational Research, 2014, 236(2):573-582. [22] Dürichen R, Wissel T, Ernst F, et al. A unified approach for respiratory motion prediction and correlation with multi-task Gaussian processes[C]//2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP), 2014:1-6. [23] Costa N R, Lourenço J. Gaussian process model——An exploratory study in the response surface methodology[J]. Quality and Reliability Engineering International, 2016, 32(7):2367-2380. [24] Alshraideh H, Del Castillo E. Gaussian process modeling and optimization of profile response experiments[J]. Quality and Reliability Engineering International, 2014, 30(4):449-462. [25] Svenson J D, Santner T J. Multiobjective optimization of expensive black-box functions via expected maximin improvement[R]. The Ohio State University, Columbus, Ohio, 2010, 32. [26] Parussini L, Venturi D, Perdikaris P, et al. Multi-fidelity Gaussian process regression for prediction of random fields[J]. Journal of Computational Physics, 2017, 336:36-50. [27] Zhou Q, Qian P Z G, Zhou S. A simple approach to emulation for computer models with qualitative and quantitative factors[J]. Technometrics, 2011, 53(3):266-273. [28] Huang H, Lin D K J, Liu M Q, et al. Computer experiments with both qualitative and quantitative variables[J]. Technometrics, 2016, 58(4):495-507. [29] Bonilla E V, Chai K M, Williams C. Multi-task Gaussian process prediction[C]//Advances in Neural Information Processing Systems, 2008:153-160. [30] Williams C K I, Rasmussen C E. Gaussian processes for machine learning[M]. Cambridge, MA:MIT Press, 2006. [31] Rebonato R, Jäckel P. The most general methodology for creating a valid correlation matrix for risk management and option pricing purposes[J]. Journal of Risk, 1999, 2(2):17-28. [32] Li Y, Zhou Q. Pairwise meta-modeling of multivariate output computer models using nonseparable covariance function[J]. Technometrics, 2016, 58(4):483-494. [33] Deb K, Pratap A, Agarwal S, et al. A fast and elitist multiobjective genetic algorithm:NSGA-II[J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2):182-197. [34] 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, 39(18):114-121. [35] Lophaven S N, Nielsen H B, Sondergaard J. DACE:A Matlab Kriging toolbox[M]. IMM, Informatics and Mathematical Modelling. The Technical University of Denmark, 2002. [36] Svenson J, Santner T. Multiobjective optimization of expensive-to-evaluate deterministic computer simulator models[J]. Computational Statistics & Data Analysis, 2016, 94:250-264. [37] Qian P Z. Sliced Latin hypercube designs[J]. Journal of the American Statistical Association, 2012, 107(497):393-399. [38] 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, 48:346-354.
2020, Vol. 40 
