基于响应曲面法的多响应稳健性参数优化方法研究
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摘要
本文主要研究了多响应稳健性参数优化方法中响应对可控因子波动的稳健性问题,目的在于通过同时考虑多响应问题的最优性和稳健性,来获得稳健最优的操作条件,从而使优化的响应对可控因子的波动具有稳健性。以此为目标,本文基于响应曲面方法(RSM)构建了多响应稳健优化模型并加以分析。本文首先介绍了响应曲面和稳健性参数设计的一些基本理论和传统方法,在此基础上提出了处理多响应稳健性参数优化的方法,具体从以下几个方面进行了研究:
首先,探讨了多响应优化中响应对可控因子波动的稳健性问题。在实际生产过程中,可控因子很难控制在某一特定值上保持恒定不变,总会在设定值附近某一小范围内做微小的波动,同时响应也会相应发生改变。基于此,建立了两种稳健性的评价指标,并提出了实现多响应稳健性参数优化的技术路线。
其次,对传统的满意度函数法进行了改进,构建了一个新的稳健满意度函数,用以衡量响应对可控因子波动的稳健性。考虑到过程的最优性和稳健性,通过将传统的满意度函数与前述稳健满意度函数进行权衡,提出了稳健优化满意度函数方法。实例分析表明,该方法可以较好的解决多响应优化的稳健性问题。
再次,基于多响应之间的相关性及多响应问题的最优性和稳健性,对传统的广义距离函数法进行了改进,并提出了修正的广义距离函数法以及稳健优化总体广义距离函数法,将多响应稳健优化问题转化为使改进的广义距离最小化问题,从而得到稳健最优的操作条件。实例分析结果验证了上述方法的可行性与有效性。
最后,通过选择合适的稳健矩阵构建了一个新的多元稳健函数,并基于过程的经济性及稳定性,将多元损失函数和多元稳健函数进行权衡,提出了一种新的稳健损失函数,以此来实现过程参数优化的低成本和高稳健性。
This dissertation mainly studies the robustness of the responses to the fluctuation of controllable factors in multi-response robust parameter optimization. Considering the optimization and robustness for multi-response problem simultaneously, this dissertation mainly focuses on obtaining a compromised robust optimum condition on which multiple responses are simultaneously optimized and insensitive to small changes of input variables. Aiming at this objective, several multi-response robust optimization models are constructed based on Response Surface Methodology (RSM). With the introduction of basic theories and traditional methods of response surface methodology and robust parameter design, robust optimization methods for multi-response are proposed in this dissertation as follows.
First, the robustness of responses to the fluctuations of input variables is discussed. Considering that input variables are hard to be fixed, and the responses will change corresponding to the fluctuation of input variables in a small region, two measures of robustness are presented, and the technical approach to the robust parameter optimization of multi-response is proposed in this dissertation.
Second, with the improvement of traditional desirability function, a new robustness desirability function is proposed to measure the robustness of responses to the fluctuations of controllable variables. Considering the optimization and robustness of a process, a new robustness optimization desirability function is presented with the tradeoff between traditional optimization desirability function and robustness desirability function mentioned above. Case study shows that the proposed method is feasible to solve the robustness problem of multi-response optimization.
Third, considering the correlations among several responses for the robust optimization of multi-response problem, corrected generalized distance function and robustness optimization overall generalized distance function are introduced. A compromised robust optimum can be obtained by finding conditions that minimize the improved generalized distance. Cases study shows the feasibility and effectiveness of these methods.
Last, with the selection of appropriate robustness matrix, a new robustness function is constructed. Based on process economy and robustness, a new robustness loss function is proposed to reduce cost and improve robustness of process parameter optimization.
首先,探讨了多响应优化中响应对可控因子波动的稳健性问题。在实际生产过程中,可控因子很难控制在某一特定值上保持恒定不变,总会在设定值附近某一小范围内做微小的波动,同时响应也会相应发生改变。基于此,建立了两种稳健性的评价指标,并提出了实现多响应稳健性参数优化的技术路线。
其次,对传统的满意度函数法进行了改进,构建了一个新的稳健满意度函数,用以衡量响应对可控因子波动的稳健性。考虑到过程的最优性和稳健性,通过将传统的满意度函数与前述稳健满意度函数进行权衡,提出了稳健优化满意度函数方法。实例分析表明,该方法可以较好的解决多响应优化的稳健性问题。
再次,基于多响应之间的相关性及多响应问题的最优性和稳健性,对传统的广义距离函数法进行了改进,并提出了修正的广义距离函数法以及稳健优化总体广义距离函数法,将多响应稳健优化问题转化为使改进的广义距离最小化问题,从而得到稳健最优的操作条件。实例分析结果验证了上述方法的可行性与有效性。
最后,通过选择合适的稳健矩阵构建了一个新的多元稳健函数,并基于过程的经济性及稳定性,将多元损失函数和多元稳健函数进行权衡,提出了一种新的稳健损失函数,以此来实现过程参数优化的低成本和高稳健性。
This dissertation mainly studies the robustness of the responses to the fluctuation of controllable factors in multi-response robust parameter optimization. Considering the optimization and robustness for multi-response problem simultaneously, this dissertation mainly focuses on obtaining a compromised robust optimum condition on which multiple responses are simultaneously optimized and insensitive to small changes of input variables. Aiming at this objective, several multi-response robust optimization models are constructed based on Response Surface Methodology (RSM). With the introduction of basic theories and traditional methods of response surface methodology and robust parameter design, robust optimization methods for multi-response are proposed in this dissertation as follows.
First, the robustness of responses to the fluctuations of input variables is discussed. Considering that input variables are hard to be fixed, and the responses will change corresponding to the fluctuation of input variables in a small region, two measures of robustness are presented, and the technical approach to the robust parameter optimization of multi-response is proposed in this dissertation.
Second, with the improvement of traditional desirability function, a new robustness desirability function is proposed to measure the robustness of responses to the fluctuations of controllable variables. Considering the optimization and robustness of a process, a new robustness optimization desirability function is presented with the tradeoff between traditional optimization desirability function and robustness desirability function mentioned above. Case study shows that the proposed method is feasible to solve the robustness problem of multi-response optimization.
Third, considering the correlations among several responses for the robust optimization of multi-response problem, corrected generalized distance function and robustness optimization overall generalized distance function are introduced. A compromised robust optimum can be obtained by finding conditions that minimize the improved generalized distance. Cases study shows the feasibility and effectiveness of these methods.
Last, with the selection of appropriate robustness matrix, a new robustness function is constructed. Based on process economy and robustness, a new robustness loss function is proposed to reduce cost and improve robustness of process parameter optimization.
引文
[1]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,天津大学学报(自然科学版),2006,39(9):1136-1140.
[2] Taguchi G, Introduction to quality engineering: designing quality into products and processes, Tokyo: Asian Productivity Organization, Tokyo, 1986.
[3] Kackar R N, Off-line quality control, parameter design, and the Taguchi method, Journal of Quality Technology, 1985, 17: 176-188.
[4] Phadke M S, Quality enginnering using robust design, New Jesery, Prentice-hall, 1989.
[5] Nair V N, Taguchi’s parameter design: A panel discussion, Technometrics, 1992, 34: 127-161.
[6] Box G E P, Signal-to-noise ratios, performance criteria, and transformations (with discussion), Technometrics, 1988, 30(1): 1-40.
[7] Tong L T, Su C T, Optimizing multi-response problems in the Taguchi method by fuzzy multiple attribute decision making, Quality and Reliability Engineering International, 1997, 13(1): 25-34.
[8] Berube J, Wu C F J, Signal-to-noise ratio and related measures in parameter design optimization: an overview, Sankhya-Series B, 2000, 62: 417-432.
[9] Joseph V R, Quality loss functions for nonnegative variables and their applications, Journal of Quality Technology, 2004, 36: 129-138.
[10] Joseph V R, Taguchi's approach to robust parameter design: A new perspective, IIE Transactions, 2007, 39(8): 805-810.
[11] Kuhnt S, Erdbrugge M, A strategy of robust parameter design for multiple responses, Statistical Modelling, 2004, 4(4): 249-264.
[12] Tay K M, Butler C, Methodologies for Experimental Design: A Survey, Comparison and Future Predictions, Quality Engineering, 1999, 11(3): 343-356.
[13] Jeyapaul R, Shahabudeen P, Krishnaiah K, Quality management research by considering multi-response problems in the Taguchi method– a review, The International Journal of Advanced Manufacturing Technology, 2004, 26(11): 1331-1337.
[14] Myers R H, Montgomery D C, Response Surface Methodology (2nd ed.), New York: John Wiley & Sons, 2002.
[15] Park K S, Kim K J, Optimizing multi-response surface problems: how to use multi-objective optimization techniques, IIE Trasactions, 2005, 37(6): 523-532.
[16] Beyer H G, Sendhoff B. Robust optimization - a comprehensive survey, Computer Methods in Applied Mechanics and Engineering, 2007, 196: 3190-3218.
[17] Kovach J, Cho B R, Solving multiresponse optimization problems using quality function-based robust design, Quality Engineering, 2008, 20(3): 346-360.
[18] Wang Jing, He Zhen, Oh J H, et al. Multi-response robust optimization using desirability function, IEEE Symposium on Advanced Management of Information for Globalized Enterprises, Tianjin, 2008. 313-315.
[19] Onur K?ksoy, A nonlinear programming solution to robust multi-response quality problem, Applied Mathematics and Computation, 2008, 196: 603-612.
[20] Shoemake A C, Tsui K L, Wu C F J, Economical experimentation methods for robust design, Technometrics, 1991, 33(4): 415-427.
[21] Myers R H, Response surface methodology in quality improvement, Communications in Statistics - Theory and Method, 1991, 20: 156~476.
[22] Myers R H, Response surface methodology - current status and future directions, Journal of Quality Technology, 1999, 31: 30-44.
[23] Myers R H, Montgomery D C, Vining G G, Borror C M, Kowalski S M, Response surface methodology: a retrospective and literature survey, Journal of Quality Technology, 2004, 36: 53-77.
[24] Michael W, Siddall J N, The optimization problem with optimal tolerance assignment and full acceptance, Journal of Mechanical Design, 1981, 103: 842-848.
[25] Bates R A, Wynn M P, Toleranceing and optimization for model - based robust engineering design, Quality and Reliability Engineering International, 1996, 12: 119-127.
[26] Kusiak A, Feng C X, Robust tolerance design for quality, Journal of Engeering for Industry, 1996, 118: 166-169.
[27]张志红,何桢,郭伟,望目特性稳健参数设计优化标准的构建,机械工程学报,2008,44(4):133-137.
[28]陈立周,于晓红,基于随机优化的工程稳健设计,北京科技大学学报,1999,21(1):57-59.
[29] Parakinson A, Mechanical design using engineering models, Journal of Mechanical Design, 1995, 117: 48~54.
[30]陈立周,基于质量-成本模型的稳健优化设计-面向21世纪的工程设计,北京:人民交通出版社,1998.
[31] Dong Z, Hu W, Xue D, New production cost - tolerance models for tolerance synthesis, Journal of Engeering for Industry, 1994, 116: 199~201.
[32] 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.
[33] Khuri A I, Cornell J A, Response Surfaces: Designs and Analyses (2nd ed.), New York: Marcel Dekker, 1996.
[34] Park S H, Robust Design and Analysis for Quality Engineering, London: Chapman & Hall, 1996.
[35] Myers R H, Carter W H, Response surface techniques for dual response systems, Technometrics, 1973, 15: 301-317.
[36] Vining G G, Myers R H, Combining Taguchi and response surface philosophies: a dual response approach, Journal of Quality Technology, 1990, 22(1): 38-45.
[37] Karen A F C, Nelson P R, Dual response optimization via direct function minimization, Journal of Quality Technology, 1996, 28(3): 26~30.
[38] Lin D K J, Tu W, Dual response surface optimization, Journal of Quality Technology, 1995, 27(1): 34-39.
[39] Del Castillo E, Montgomery D C, A nonlinear programming solution to the dual response problem, Journal of Quality Technology, 1993, 25: 199-204.
[40] Kim K, Lin D K J, Dual response surface optimization: a fuzzy modeling approach, Journal of Quality Technology, 1998, 30:1-10.
[41]何桢,张生虎,刘子先,双响应曲面方法在改进产品设计中的应用研究,系统工程理论与实践,2001,21(9):140-144.
[42]崔庆安,何桢,车建国,一种基于支持向量机的非参数双响应曲面法,天津大学学报(自然科学版),2006,39(8):1008-1014.
[43]李玉强,崔振山,陈军,阮雪榆,张冬娟,基于双响应面模型的6σ稳健设计,机械强度,2006,28(5):690-694.
[44]曹衍龙,金岳辉,基于双响应面法的公差设计研究,工程设计学报,2000,3:42-44.
[45] Harrington E C, The desirability function, Industrial Quality Control, 1965, 21: 494~498.
[46] Derringer G C, Suich R, Simultaneous optimization of several response variables, Journal of Quality Technology, 1980, 12: 214-219.
[47] Shah H K, Impact of correlated responses on the desirability function, Arizona: Arizona State University, 2001.
[48] Kim K J, Lin D K J, Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions, Applied Statistics, 2000, 49(3): 311-325.
[49] Jeong I J, Kim K J, Interactive desirability function approach to multi-response surface optimization, International Journal of Reliability, Quality and Safety Engineering, 2003, 10: 205-217.
[50] Khuri A I, Conlon M, Simultaneous optimization of multiple responses represented by polynomial regression functions, Technometrics, 1981, 23(4): 363-375.
[51] Kim K J, Lin D K J, Optimization of multiple response consideraing both location and dispersion effects, European Journal of Operational Reseach, 2006, 169(1): 133-145.
[52] Pignatiello J J, Strategies for robust multiresponse quality engineering, IIE Transaction, 1993, 25(3): 5-15.
[53] Ames A E, Mattucci N, MacDonald S, et al., Quality loss functions for optimization across multiple response surfaces, Journal of Quality Technology, 1997, 29(3): 339-346.
[54] Vining G G, A compromise approach to multiresponse optimization, Journal of Quality Technology, 1998, 30(4): 309-313.
[55] Ko Y H, Kim K J, Jun C H, A new loss function-based method for multiresponse optimization, Journal of Quality Technology, 2005, 37: 50-59.
[56] Kapur C K, Cho B R, Economic design of the specification region for multiple quality characteristics, IIE Transactions, 1996, 28(4):237~248.
[57] Artiles-leon N, A pragmatic approach to multi-response problems using loss functions, Quality Engineering, 1996~97, 9(2): 213~220.
[58]马义中,徐济超,多指标稳健设计质量特性的度量,系统工程,1998,16(6):34-37.
[59]徐济超,马义中,多指标稳健设计质量特性的度量,系统工程理论与实践,1999,19(8):45-48.
[60]马义中,赵逢禹,多元质量特性的稳健设计及其实现,系统工程与电子技术,2005,27(9):1580-1582,1596.
[61] He Zhen, Zhang Yuxuan, Multiresponse surfaces optimization based on evidential reasoning theory, The Asian Journal on Quality, 2004, 5(1): 43-51.
[62]宗志宇,何桢,孔祥芬,基于满意度函数法的多响应稳健性参数设计,系统管理学报,2007,16(5):508-512.
[63]马彦辉,何桢,赵有,基于熵权理论和渴求函数法的多响应优化设计研究,组合机床与自动化加工技术,2007,2:99-102.
[64]何桢,张于轩,多响应试验设计的优化方法研究,工业工程,2003,6(4):35~38.
[65]何桢,吕海利,用加权马氏距离法进行多响应产品设计的一种改进,组合机床与自动化加工技术,2007,2:88-91,95.
[66]宗志宇,何桢,孔祥芬,噪声因子存在下的多响应参数设计的优化,工业工程,2007,10(6):127-130,140.
[67]刘欣,岳荣先,多响应近似线性回归模型D最优稳健设计,上海师范大学学报,2007,36(4):5-10.
[68]刘欣,岳荣先,多响应近似线性回归模型的Minmax稳健设计,上海师范大学学报,2006,35(1):36-41.
[69]叶亮,潘尔顺,奚立峰,基于试验设计的健壮参数控制方法,上海交通大学学报,2006,40(6):918-921.
[70]魏世振,韩玉启,陈传明,基于信噪比的多元质量损失函数研究,管理工程学报,2004,2:4-6.
[71]张根保,何桢,刘英,质量管理与可靠性,北京:中国科学技术出版社,2005
[72] Box G E P, Hunter J S, Multifactor experimental designs for exploring response surfaces, The Annals of Mathematical Statistics, 1957, 28: 195-241.
[73] Box G E P, Draper N R, A basis for the selection of a response surface design, Journal of the American Statistical Association, 1959, 54: 622-654.
[74] Box G E P, Draper N R, The choice of a second order rotatable design, Biometrika, 1963, 50: 335-352.
[75]梁昭磊,受限条件下的试验设计研究:[博士学位论文],天津:天津大学,2008
[76]张志红,基于并行质量工程的6σ公差设计方法的研究:[博士学位论文],天津:天津大学,2005
[77]张润楚,郑海涛,兰燕等译,试验设计与分析及参数优化,北京:中国统计出版社,2003
[78] Lind E, Goldin J, Hickman J, Fitting yield and cost response surfaces, Chemical Engineering Progress, 1960, 56: 62-68.
[79] Derringer G C, A balancing act: optimization a product’s properties, Quality Progress, 1994, 27: 51-58.
[80] Del Castillo E, Montgomery D C, McCarville D R, Modified desirability function for multiple response optimization, Journal of Quality Technology, 1996, 28(3): 337-345.
[81] Park S H, Park J U, Simultaneous optimization of multiple response using weighted desirability function, Journal of the Korean Society for Quality Management, 1997, 25: 56-68.
[82] Ch’ng C K, Quah S H, Low H C, A new approach for multiple-response optimization, Quality Engineering, 2005, 17:621-626.
[83] Wu F C, Optimization of correlated multiple quality characteristics using desirability function, Quality Engineering, 2005, 17(1): 119-126.
[84] Montgomery D C, Design and Analysis of Experiments (6th ed.), New York: John Wiley & Sons, 2005.
[85]刘久富,王宁生,丁宗红等,三次设计的扩展介绍与探讨,工业工程,2002,5(1):50-54.
[86]张于轩,多响应问题的稳健性设计优化研究:[硕士学位论文],天津,天津大学,2004.
[87]李昭阳,韩之俊,田口方法和双重曲面响应(DRSM)法,数理统计与管理,2000,20(5):40-44.
[88]陈立周,稳健设计,北京:机械工业出版社,2000
[89] Myers R H, Khuri A I, Vining G G, Response surface alternative to the taguchi robust parameter design, American Statistician, 1992, 46(2): 131-139.
[90] Ding R, Lin D K J, Duan W, Dual-response surface optimization: a weighted MSE approach, Quality Engineering, 2004, 16(3): 377-385.
[91] Quesada G M, Castillo E D, A dual-response approach to the multivariate robust parameter design problem, Technometrics, 2004, 46(2): 176-187.
[92] Quesada G M, Castillo E D, Two approaches for improving the dual response method in robust parameter design, Journal of Quality Technology, 2004, 36(2): 154-168.
[93] Welch W J, Yu T K, Kang S M, Computer experiments for quality control by parameter design, Journal of Quality Technology, 1990, 22(1): 15-22.
[94] Robinson T J, Borror C M, Myers R H, Robust parameter design: a review, Quality and Reliability Engineering International, 2004, 20: 81-101.
[95] Aggarwal M L, Gupta B C, Chaudhury S R, Walker H F, Interaction graphs for a two-level combined array experiment design, Journal of Industrial Technology, 2002, 18(4): 1-10.
[96] Montgomery D C, Using fractional factorial designs for robust process development, Quality Engineering, 1991, 3: 193-205.
[97] Moore L M, McKay M D, Campbell K S, Combined array experiment design, Reliability Engineering and System Safety, 2006, 91(10-11): 1281-1289.
[98] Borkowski J J, Lucas J M, Designs of mixed resolution for process robustness studies, Technometrics, 1997, 30: 63-70.
[99] Box G E P, Jones S, Split-plot designs for robust product experimentation, Journal of Applied Statistics, 1989, 19: 3-26.
[100] Lee J H, Park S H, Combined and product array approaches in simultaneous optimization of multiple responses, Journal of the Korean Society for Quality Management, 2006, 34: 93-101.
[101] Romano D, Varetto M, Vicario G, Multiresponse robust design: a general framework based on combined array, Journal of Quality Technology, 2004, 36: 27-37.
[102] Jayaram J S R, Yaacob Ibrahim, Multiple response robust design and yield maximization, International Journal of Quality and Reliability Management, 1999, 16(9): 826-837.
[103] Box G E P, Hay W A, A statistical design for the efficient removal of trends occurring in a comparative experiment with an application in biological assay, Biometrics, 1953, 9: 304-319.
[104] Zeaiter M, Roger J M, Maurel V B, Robustness of models developed by multivariate calibration, Part I: The assessment of robustness, Trends in Analytical Chemistry, 2004, 23(2): 157-170.
[105] Carlyle W M, Montgomery D C, Runger G C, Optimization problems and methods in quality control and improvement, Journal of Quality Technology, 2000, 32: 1-17.
[106] Spendley W, Hext G R, Himsworth F R, Sequential application of simplex designs in optimization and evolutionary operation, Technometrics, 1962, 4: 441-461.
[107] Subrahmanyam M B, An extension of the simplex method to constrained nonlinear optimization, Journal of Optimization Theory and Applications, 1989, 62(2): 311-319.
[108] Fan S Shu-Kai, Zahara E, A hybrid simplex search and particle swarm optimization for unconstrained optimization, European Journal of Operational Research, 2007, 181(2): 527-548.
[109] Nelder J A, Mead R, A simplex method for function minimization, The Computer Journal, 1965, 7(4): 308-313.
[110] Barton R R, Ivey J S Jr, Nelder-Mead simplex modifications for simulation optimization, management science, 1996, 42(7): 475-498.
[111] Luersen M A, Riche R L, Globalized Nelder–Mead method for engineering optimization, Computers & Structures, 2004, 82: 2251-2260.
[112] Coope I D, Price C J, On the convergence of grid-based methods for unconstrained optimization, SIAM Journal on Optimization, 2001, 11(4):859-869.
[113] Powell M J D, Direct search algorithms for optimization calculations, Acta Numerica, 1998, 7: 287-336.
[114] Kolda T G, Lewis R M, Torczon V, Optimization by direct search: new perspectives on some classical and modern methods. SIAM Review, 2003, 45(3): 385-482.
[115] Rayward-Smith V J, Osman I H, Reeves C R, Smith G D, Modern Heuristic Search Methods, New York: John Wiley, 1996.
[116] Bonet B, Geffner H, Planning as heuristic search: new results, Lecture Notes in Computer Science, 2000, 1809: 360-372.
[117] Metropolis N, Rosenbluth A, Rosenbluth M, et.al, Equations of state calculations by fast computing machines, Journal of Chemical Physics, 1953, 21: 1087-1092.
[118] Kirkpatrick S, Gelatt C D Jr., Vecchi M P, Optimization by simulated annealing, Science, 1983, 220(4598): 671– 680.
[119] Laarhoven P J M, Aarts E H L, Simulated Annealing: Theory and Applications, London: Springer, 1987.
[120] Holland J, Adaptation in Natural and Artificial Systems, Ann Arbor: The University of Michigan Press, MI, 1974.
[121] Murata T, Ishibuchi H, Tanaka H, Multi-objective genetic algorithm and its application to flowshop scheduling, Computers in Industrial Engineering, 1996, 30: 957-968.
[122] Min H, Ko H J, Ko C S, A genetic algorithm approach to developing the multi-echelon reverse logistics network for product returns Omega, 2006, 34(1): 56-69.
[123] Angus Jeang, Economic tolerance design for quality, Quality and Reliability Engineering International, 1994, 10(1): 113-121.
[124] Myers R H, Kim Y, Griffiths K L, Response surface methods and the use of variables, Journal of Quality Technology, 1997, 29(4): 429-440.
[125] Box G E P, Draper N R, Empirical Model Building and Response Surfaces, New York: John Wiley & Son, 1987.
[126] Box G E P, Behnken D W, Some new three level designs for the study of quantitative variables, Technometrics, 1960, 2(4): 455-475.
[127]张志红,何桢,郭伟,在响应曲面方法中三类中心复合设计的比较研究,沈阳航空工业学院学报,2007,24(1):87-91.
[2] Taguchi G, Introduction to quality engineering: designing quality into products and processes, Tokyo: Asian Productivity Organization, Tokyo, 1986.
[3] Kackar R N, Off-line quality control, parameter design, and the Taguchi method, Journal of Quality Technology, 1985, 17: 176-188.
[4] Phadke M S, Quality enginnering using robust design, New Jesery, Prentice-hall, 1989.
[5] Nair V N, Taguchi’s parameter design: A panel discussion, Technometrics, 1992, 34: 127-161.
[6] Box G E P, Signal-to-noise ratios, performance criteria, and transformations (with discussion), Technometrics, 1988, 30(1): 1-40.
[7] Tong L T, Su C T, Optimizing multi-response problems in the Taguchi method by fuzzy multiple attribute decision making, Quality and Reliability Engineering International, 1997, 13(1): 25-34.
[8] Berube J, Wu C F J, Signal-to-noise ratio and related measures in parameter design optimization: an overview, Sankhya-Series B, 2000, 62: 417-432.
[9] Joseph V R, Quality loss functions for nonnegative variables and their applications, Journal of Quality Technology, 2004, 36: 129-138.
[10] Joseph V R, Taguchi's approach to robust parameter design: A new perspective, IIE Transactions, 2007, 39(8): 805-810.
[11] Kuhnt S, Erdbrugge M, A strategy of robust parameter design for multiple responses, Statistical Modelling, 2004, 4(4): 249-264.
[12] Tay K M, Butler C, Methodologies for Experimental Design: A Survey, Comparison and Future Predictions, Quality Engineering, 1999, 11(3): 343-356.
[13] Jeyapaul R, Shahabudeen P, Krishnaiah K, Quality management research by considering multi-response problems in the Taguchi method– a review, The International Journal of Advanced Manufacturing Technology, 2004, 26(11): 1331-1337.
[14] Myers R H, Montgomery D C, Response Surface Methodology (2nd ed.), New York: John Wiley & Sons, 2002.
[15] Park K S, Kim K J, Optimizing multi-response surface problems: how to use multi-objective optimization techniques, IIE Trasactions, 2005, 37(6): 523-532.
[16] Beyer H G, Sendhoff B. Robust optimization - a comprehensive survey, Computer Methods in Applied Mechanics and Engineering, 2007, 196: 3190-3218.
[17] Kovach J, Cho B R, Solving multiresponse optimization problems using quality function-based robust design, Quality Engineering, 2008, 20(3): 346-360.
[18] Wang Jing, He Zhen, Oh J H, et al. Multi-response robust optimization using desirability function, IEEE Symposium on Advanced Management of Information for Globalized Enterprises, Tianjin, 2008. 313-315.
[19] Onur K?ksoy, A nonlinear programming solution to robust multi-response quality problem, Applied Mathematics and Computation, 2008, 196: 603-612.
[20] Shoemake A C, Tsui K L, Wu C F J, Economical experimentation methods for robust design, Technometrics, 1991, 33(4): 415-427.
[21] Myers R H, Response surface methodology in quality improvement, Communications in Statistics - Theory and Method, 1991, 20: 156~476.
[22] Myers R H, Response surface methodology - current status and future directions, Journal of Quality Technology, 1999, 31: 30-44.
[23] Myers R H, Montgomery D C, Vining G G, Borror C M, Kowalski S M, Response surface methodology: a retrospective and literature survey, Journal of Quality Technology, 2004, 36: 53-77.
[24] Michael W, Siddall J N, The optimization problem with optimal tolerance assignment and full acceptance, Journal of Mechanical Design, 1981, 103: 842-848.
[25] Bates R A, Wynn M P, Toleranceing and optimization for model - based robust engineering design, Quality and Reliability Engineering International, 1996, 12: 119-127.
[26] Kusiak A, Feng C X, Robust tolerance design for quality, Journal of Engeering for Industry, 1996, 118: 166-169.
[27]张志红,何桢,郭伟,望目特性稳健参数设计优化标准的构建,机械工程学报,2008,44(4):133-137.
[28]陈立周,于晓红,基于随机优化的工程稳健设计,北京科技大学学报,1999,21(1):57-59.
[29] Parakinson A, Mechanical design using engineering models, Journal of Mechanical Design, 1995, 117: 48~54.
[30]陈立周,基于质量-成本模型的稳健优化设计-面向21世纪的工程设计,北京:人民交通出版社,1998.
[31] Dong Z, Hu W, Xue D, New production cost - tolerance models for tolerance synthesis, Journal of Engeering for Industry, 1994, 116: 199~201.
[32] 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.
[33] Khuri A I, Cornell J A, Response Surfaces: Designs and Analyses (2nd ed.), New York: Marcel Dekker, 1996.
[34] Park S H, Robust Design and Analysis for Quality Engineering, London: Chapman & Hall, 1996.
[35] Myers R H, Carter W H, Response surface techniques for dual response systems, Technometrics, 1973, 15: 301-317.
[36] Vining G G, Myers R H, Combining Taguchi and response surface philosophies: a dual response approach, Journal of Quality Technology, 1990, 22(1): 38-45.
[37] Karen A F C, Nelson P R, Dual response optimization via direct function minimization, Journal of Quality Technology, 1996, 28(3): 26~30.
[38] Lin D K J, Tu W, Dual response surface optimization, Journal of Quality Technology, 1995, 27(1): 34-39.
[39] Del Castillo E, Montgomery D C, A nonlinear programming solution to the dual response problem, Journal of Quality Technology, 1993, 25: 199-204.
[40] Kim K, Lin D K J, Dual response surface optimization: a fuzzy modeling approach, Journal of Quality Technology, 1998, 30:1-10.
[41]何桢,张生虎,刘子先,双响应曲面方法在改进产品设计中的应用研究,系统工程理论与实践,2001,21(9):140-144.
[42]崔庆安,何桢,车建国,一种基于支持向量机的非参数双响应曲面法,天津大学学报(自然科学版),2006,39(8):1008-1014.
[43]李玉强,崔振山,陈军,阮雪榆,张冬娟,基于双响应面模型的6σ稳健设计,机械强度,2006,28(5):690-694.
[44]曹衍龙,金岳辉,基于双响应面法的公差设计研究,工程设计学报,2000,3:42-44.
[45] Harrington E C, The desirability function, Industrial Quality Control, 1965, 21: 494~498.
[46] Derringer G C, Suich R, Simultaneous optimization of several response variables, Journal of Quality Technology, 1980, 12: 214-219.
[47] Shah H K, Impact of correlated responses on the desirability function, Arizona: Arizona State University, 2001.
[48] Kim K J, Lin D K J, Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions, Applied Statistics, 2000, 49(3): 311-325.
[49] Jeong I J, Kim K J, Interactive desirability function approach to multi-response surface optimization, International Journal of Reliability, Quality and Safety Engineering, 2003, 10: 205-217.
[50] Khuri A I, Conlon M, Simultaneous optimization of multiple responses represented by polynomial regression functions, Technometrics, 1981, 23(4): 363-375.
[51] Kim K J, Lin D K J, Optimization of multiple response consideraing both location and dispersion effects, European Journal of Operational Reseach, 2006, 169(1): 133-145.
[52] Pignatiello J J, Strategies for robust multiresponse quality engineering, IIE Transaction, 1993, 25(3): 5-15.
[53] Ames A E, Mattucci N, MacDonald S, et al., Quality loss functions for optimization across multiple response surfaces, Journal of Quality Technology, 1997, 29(3): 339-346.
[54] Vining G G, A compromise approach to multiresponse optimization, Journal of Quality Technology, 1998, 30(4): 309-313.
[55] Ko Y H, Kim K J, Jun C H, A new loss function-based method for multiresponse optimization, Journal of Quality Technology, 2005, 37: 50-59.
[56] Kapur C K, Cho B R, Economic design of the specification region for multiple quality characteristics, IIE Transactions, 1996, 28(4):237~248.
[57] Artiles-leon N, A pragmatic approach to multi-response problems using loss functions, Quality Engineering, 1996~97, 9(2): 213~220.
[58]马义中,徐济超,多指标稳健设计质量特性的度量,系统工程,1998,16(6):34-37.
[59]徐济超,马义中,多指标稳健设计质量特性的度量,系统工程理论与实践,1999,19(8):45-48.
[60]马义中,赵逢禹,多元质量特性的稳健设计及其实现,系统工程与电子技术,2005,27(9):1580-1582,1596.
[61] He Zhen, Zhang Yuxuan, Multiresponse surfaces optimization based on evidential reasoning theory, The Asian Journal on Quality, 2004, 5(1): 43-51.
[62]宗志宇,何桢,孔祥芬,基于满意度函数法的多响应稳健性参数设计,系统管理学报,2007,16(5):508-512.
[63]马彦辉,何桢,赵有,基于熵权理论和渴求函数法的多响应优化设计研究,组合机床与自动化加工技术,2007,2:99-102.
[64]何桢,张于轩,多响应试验设计的优化方法研究,工业工程,2003,6(4):35~38.
[65]何桢,吕海利,用加权马氏距离法进行多响应产品设计的一种改进,组合机床与自动化加工技术,2007,2:88-91,95.
[66]宗志宇,何桢,孔祥芬,噪声因子存在下的多响应参数设计的优化,工业工程,2007,10(6):127-130,140.
[67]刘欣,岳荣先,多响应近似线性回归模型D最优稳健设计,上海师范大学学报,2007,36(4):5-10.
[68]刘欣,岳荣先,多响应近似线性回归模型的Minmax稳健设计,上海师范大学学报,2006,35(1):36-41.
[69]叶亮,潘尔顺,奚立峰,基于试验设计的健壮参数控制方法,上海交通大学学报,2006,40(6):918-921.
[70]魏世振,韩玉启,陈传明,基于信噪比的多元质量损失函数研究,管理工程学报,2004,2:4-6.
[71]张根保,何桢,刘英,质量管理与可靠性,北京:中国科学技术出版社,2005
[72] Box G E P, Hunter J S, Multifactor experimental designs for exploring response surfaces, The Annals of Mathematical Statistics, 1957, 28: 195-241.
[73] Box G E P, Draper N R, A basis for the selection of a response surface design, Journal of the American Statistical Association, 1959, 54: 622-654.
[74] Box G E P, Draper N R, The choice of a second order rotatable design, Biometrika, 1963, 50: 335-352.
[75]梁昭磊,受限条件下的试验设计研究:[博士学位论文],天津:天津大学,2008
[76]张志红,基于并行质量工程的6σ公差设计方法的研究:[博士学位论文],天津:天津大学,2005
[77]张润楚,郑海涛,兰燕等译,试验设计与分析及参数优化,北京:中国统计出版社,2003
[78] Lind E, Goldin J, Hickman J, Fitting yield and cost response surfaces, Chemical Engineering Progress, 1960, 56: 62-68.
[79] Derringer G C, A balancing act: optimization a product’s properties, Quality Progress, 1994, 27: 51-58.
[80] Del Castillo E, Montgomery D C, McCarville D R, Modified desirability function for multiple response optimization, Journal of Quality Technology, 1996, 28(3): 337-345.
[81] Park S H, Park J U, Simultaneous optimization of multiple response using weighted desirability function, Journal of the Korean Society for Quality Management, 1997, 25: 56-68.
[82] Ch’ng C K, Quah S H, Low H C, A new approach for multiple-response optimization, Quality Engineering, 2005, 17:621-626.
[83] Wu F C, Optimization of correlated multiple quality characteristics using desirability function, Quality Engineering, 2005, 17(1): 119-126.
[84] Montgomery D C, Design and Analysis of Experiments (6th ed.), New York: John Wiley & Sons, 2005.
[85]刘久富,王宁生,丁宗红等,三次设计的扩展介绍与探讨,工业工程,2002,5(1):50-54.
[86]张于轩,多响应问题的稳健性设计优化研究:[硕士学位论文],天津,天津大学,2004.
[87]李昭阳,韩之俊,田口方法和双重曲面响应(DRSM)法,数理统计与管理,2000,20(5):40-44.
[88]陈立周,稳健设计,北京:机械工业出版社,2000
[89] Myers R H, Khuri A I, Vining G G, Response surface alternative to the taguchi robust parameter design, American Statistician, 1992, 46(2): 131-139.
[90] Ding R, Lin D K J, Duan W, Dual-response surface optimization: a weighted MSE approach, Quality Engineering, 2004, 16(3): 377-385.
[91] Quesada G M, Castillo E D, A dual-response approach to the multivariate robust parameter design problem, Technometrics, 2004, 46(2): 176-187.
[92] Quesada G M, Castillo E D, Two approaches for improving the dual response method in robust parameter design, Journal of Quality Technology, 2004, 36(2): 154-168.
[93] Welch W J, Yu T K, Kang S M, Computer experiments for quality control by parameter design, Journal of Quality Technology, 1990, 22(1): 15-22.
[94] Robinson T J, Borror C M, Myers R H, Robust parameter design: a review, Quality and Reliability Engineering International, 2004, 20: 81-101.
[95] Aggarwal M L, Gupta B C, Chaudhury S R, Walker H F, Interaction graphs for a two-level combined array experiment design, Journal of Industrial Technology, 2002, 18(4): 1-10.
[96] Montgomery D C, Using fractional factorial designs for robust process development, Quality Engineering, 1991, 3: 193-205.
[97] Moore L M, McKay M D, Campbell K S, Combined array experiment design, Reliability Engineering and System Safety, 2006, 91(10-11): 1281-1289.
[98] Borkowski J J, Lucas J M, Designs of mixed resolution for process robustness studies, Technometrics, 1997, 30: 63-70.
[99] Box G E P, Jones S, Split-plot designs for robust product experimentation, Journal of Applied Statistics, 1989, 19: 3-26.
[100] Lee J H, Park S H, Combined and product array approaches in simultaneous optimization of multiple responses, Journal of the Korean Society for Quality Management, 2006, 34: 93-101.
[101] Romano D, Varetto M, Vicario G, Multiresponse robust design: a general framework based on combined array, Journal of Quality Technology, 2004, 36: 27-37.
[102] Jayaram J S R, Yaacob Ibrahim, Multiple response robust design and yield maximization, International Journal of Quality and Reliability Management, 1999, 16(9): 826-837.
[103] Box G E P, Hay W A, A statistical design for the efficient removal of trends occurring in a comparative experiment with an application in biological assay, Biometrics, 1953, 9: 304-319.
[104] Zeaiter M, Roger J M, Maurel V B, Robustness of models developed by multivariate calibration, Part I: The assessment of robustness, Trends in Analytical Chemistry, 2004, 23(2): 157-170.
[105] Carlyle W M, Montgomery D C, Runger G C, Optimization problems and methods in quality control and improvement, Journal of Quality Technology, 2000, 32: 1-17.
[106] Spendley W, Hext G R, Himsworth F R, Sequential application of simplex designs in optimization and evolutionary operation, Technometrics, 1962, 4: 441-461.
[107] Subrahmanyam M B, An extension of the simplex method to constrained nonlinear optimization, Journal of Optimization Theory and Applications, 1989, 62(2): 311-319.
[108] Fan S Shu-Kai, Zahara E, A hybrid simplex search and particle swarm optimization for unconstrained optimization, European Journal of Operational Research, 2007, 181(2): 527-548.
[109] Nelder J A, Mead R, A simplex method for function minimization, The Computer Journal, 1965, 7(4): 308-313.
[110] Barton R R, Ivey J S Jr, Nelder-Mead simplex modifications for simulation optimization, management science, 1996, 42(7): 475-498.
[111] Luersen M A, Riche R L, Globalized Nelder–Mead method for engineering optimization, Computers & Structures, 2004, 82: 2251-2260.
[112] Coope I D, Price C J, On the convergence of grid-based methods for unconstrained optimization, SIAM Journal on Optimization, 2001, 11(4):859-869.
[113] Powell M J D, Direct search algorithms for optimization calculations, Acta Numerica, 1998, 7: 287-336.
[114] Kolda T G, Lewis R M, Torczon V, Optimization by direct search: new perspectives on some classical and modern methods. SIAM Review, 2003, 45(3): 385-482.
[115] Rayward-Smith V J, Osman I H, Reeves C R, Smith G D, Modern Heuristic Search Methods, New York: John Wiley, 1996.
[116] Bonet B, Geffner H, Planning as heuristic search: new results, Lecture Notes in Computer Science, 2000, 1809: 360-372.
[117] Metropolis N, Rosenbluth A, Rosenbluth M, et.al, Equations of state calculations by fast computing machines, Journal of Chemical Physics, 1953, 21: 1087-1092.
[118] Kirkpatrick S, Gelatt C D Jr., Vecchi M P, Optimization by simulated annealing, Science, 1983, 220(4598): 671– 680.
[119] Laarhoven P J M, Aarts E H L, Simulated Annealing: Theory and Applications, London: Springer, 1987.
[120] Holland J, Adaptation in Natural and Artificial Systems, Ann Arbor: The University of Michigan Press, MI, 1974.
[121] Murata T, Ishibuchi H, Tanaka H, Multi-objective genetic algorithm and its application to flowshop scheduling, Computers in Industrial Engineering, 1996, 30: 957-968.
[122] Min H, Ko H J, Ko C S, A genetic algorithm approach to developing the multi-echelon reverse logistics network for product returns Omega, 2006, 34(1): 56-69.
[123] Angus Jeang, Economic tolerance design for quality, Quality and Reliability Engineering International, 1994, 10(1): 113-121.
[124] Myers R H, Kim Y, Griffiths K L, Response surface methods and the use of variables, Journal of Quality Technology, 1997, 29(4): 429-440.
[125] Box G E P, Draper N R, Empirical Model Building and Response Surfaces, New York: John Wiley & Son, 1987.
[126] Box G E P, Behnken D W, Some new three level designs for the study of quantitative variables, Technometrics, 1960, 2(4): 455-475.
[127]张志红,何桢,郭伟,在响应曲面方法中三类中心复合设计的比较研究,沈阳航空工业学院学报,2007,24(1):87-91.