One of the key issues in robust parameter design is to configure the controllable factors to minimize the variance due to noise variables. However, it can sometimes happen that the number of control variables is greater than the number of noise variables. When this occurs, two important situations arise. One is that the variance due to noise variables can be brought down to zero The second is that multiple optimal control variable settings become available to the experimenter. A simultaneous confidence region for such a locus of points not only provides a region of uncertainty about such a solution, but also provides a statistical test of whether or not such points lie within the region of experimentation or a feasible region of operation. However, this situation requires a confidence region for the multiple-solution factor levels that provides proper simultaneous coverage. This requirement has not been previously recognized in the literature. In the case where the number of control variables is greater than the number of noise variables, we show how to construct critical values needed to maintain the simultaneous coverage rate. Two examples are provided as a demonstration of the practical need to adjust the critical values for simultaneous coverage. 1. Introduction Robust Parameter Design (RPD) is also called Robust Design or Parameter Design in the literature [1, 2]. The concept of RPD was introduced in the United States by Genichi Taguchi in the early 1980s. It is a methodology that takes both the mean and variance into consideration for product or process optimization. Taguchi [3] divided the predictor variables into two categories: control variables and noise variables. Control variables are easy to control while noise variables are either difficult to control or uncontrollable at a large scale. In practice, we would like to find a range of control variables such that (1) the variance caused by the change of noise variables is minimized and (2) the mean response is close to target. Multiple optimization design and analysis methods have been developed to achieve these two goals simultaneously, ranging from the traditional Taguchi methods to the more sophisticated response surface alternatives. (See [2, 4, 5] for detailed reviews on these methods.) Under certain conditions, Myers et al. [6] proposed a way to construct a confidence region of the control variables where the variability transmission by the noise variables is minimized to zero. Although they only focused on the variance part, there are many such situations in which focus is placed
References
[1]
V. N. Nair, “Taguchi's parameter design: a panel discussion,” Technometrics, vol. 34, no. 2, pp. 127–161, 1992.
[2]
T. Robinson, C. Borror, and R. H. Myers, “Robust parameter design: a review,” Quality and Reliability Engineering International, vol. 20, no. 1, pp. 81–101, 2004.
[3]
G. Taguchi, Introduction to Quality Engineering, UNIPUB/Kraus International, White Plains, NY, USA, 1986.
[4]
R. H. Myers, A. I. Khuri, and G. G. Vining, “Response surface alternatives to the taguchi robust parameter design approach,” The American Statistician, vol. 46, no. 2, pp. 131–139, 1992.
[5]
R. H. Myers, D. C. Montgomery, and C. M. Anderson-Cook, Response Surface Metholodology—Process and Product Optimization Using Designed Experiments, John Wiley & Sons, New York, NY, USA, 3rd edition, 2009.
[6]
R. H. Myers, Y. Kim, and K. L. Griffiths, “Response surface methods and the use of noise variables,” Journal of Quality Technology, vol. 29, no. 4, pp. 429–440, 1997.
[7]
J. Kunert, C. Auer, M. Erdbrügge, and R. Ewers, “An experiment to compare Taguchi's product array and the combined array,” Journal of Quality Technology, vol. 39, no. 1, pp. 17–34, 2007.
[8]
T. J. Robinson, S. S. Wulff, D. C. Montgomery, and A. I. Khuri, “Robust parameter design using generalized linear mixed models,” Journal of Quality Technology, vol. 38, no. 1, pp. 65–75, 2006.
[9]
H. B. Goldfarb, C. M. Borror, D. C. Montgomery, and C. M. Anderson-Cook, “Using genetic algorithms to generate mixture-process experimental designs involving control and noise variables,” Journal of Quality Technology, vol. 37, no. 1, pp. 60–74, 2005.
[10]
E. Del Castillo and S. Cahya, “A tool for computing confidence regions on the stationary point of a response surface,” The American Statistician, vol. 55, no. 4, pp. 358–365, 2001.
[11]
G. E. P. Box and S. Jones, “Design products that are robust to environment,” Tech. Rep. 56, Center for Quality and Productivity Improvement, University of Wisconsin, Madison, WI, USA, 1990.
[12]
J. M. Lucas, “How to achieve a robust process using response surface methodology,” Journal of Quality Technology, vol. 26, no. 4, pp. 248–260, 1994.
[13]
R. G. Miller, Simultaneous Statistical Inference, Springer, New York, NY, USA, 2nd edition, 1981.
[14]
N. H. Timm, Applied Multivariate Analysis, Springer, New York, NY, USA, 2002.
[15]
A. Cheng, “Confidence Regions for Optimal Control Variables for Robust Parameter Design Experiments,” , Ph.D.dissertation, Temple University, Department of Statistics, 2011.
[16]
G. P. Y. Clarke, “Approximate confidence limits for a parameter function in nonlinear regression,” Journal of the American Statistical Association, vol. 82, no. 397, pp. 221–230, 1987.
[17]
S. Kotz and N. L. Johnson, “Multivariate t-distribution,” in Encyclopedia of Statistical Sciences, vol. 6, pp. 129–130, John Wiley & Sons, Hoboken, NJ, USA, 1982.
[18]
D. C. Montgomery, Design and Analysis of Experiments, John Wiley & Sons, New York, NY, USA, 7th edition, 2009.
[19]
J. B. Kristensen, X. Xu, and H. Mu, “Process optimization using response surface design and pilot plant production of dietary diacylglycerols by lipase-catalyzed glycerolysis,” Journal of Agricultural and Food Chemistry, vol. 53, no. 18, pp. 7059–7066, 2005.
[20]
J. Wu and M. Hamada, Experiments-Planning, Analysis, and Parameter Design Optimization, John Wiley & Sons, New York, NY, USA, 2000.
[21]
J. Singh, D. D. Frey, N. Soderborg, and R. Jugulum, “Compound noise: evaluation as a robust design method,” Quality and Reliability Engineering International, vol. 23, no. 3, pp. 387–398, 2007.
[22]
X. S. Hou, “On the use of compound noise factor in parameter design experiments,” Applied Stochastic Models in Business and Industry, vol. 18, no. 3, pp. 225–243, 2002.
[23]
A. C. Rencher, Multivariate Statistical Inference and Applications, John Wiley & Sons, New York, NY, USA, 1998.
[24]
C. R. Rao, Linear Statistical Inference and its Application, John Wiley & Sons, New York, NY, USA, 2nd edition, 1973.
[25]
J. J. Peterson, S. Cahya, and E. Del Castillo, “A general approach to confidence regions for optimal factor levels of response surfaces,” Biometrics, vol. 58, no. 2, pp. 422–431, 2002.
