A semi-parametric method for optimizing multi-response problems: A case study on improving the quality of plastic injection machines

Document Type : Original Article

Authors

1 Researcher of Social and Economic Statistics, Allameh Tabatabai University, Tehran, Iran

2 Allameh Tabatabaei University

Abstract
Multi-response optimization performed by the response procedure method is very common. Before optimization, we need to select and fit the appropriate model for each response. A major problem that may occur due to incorrect fitting of models and failure to reach optimal solutions is model misidentification. The solid regression model method, which is a semi-parametric method for estimating d, can have a better performance than both parametric and non-parametric estimation methods against model misclassification. In this research, the use of a robust regression model method is proposed to improve the model estimation and the appropriate fit of each of the answers will be investigated by one of the multivariate optimization methods, namely the utility function. In the following, an applied study is presented to compare parametric, nonparametric and semi-parametric methods. The results of this study show that the performance of the stable regression model is more appropriate in many situations as well as in the modeling stage than the other two methods. Therefore, the optimization results with a stable regression model are much more reliable.

Keywords


[1] Mead, R , & Pike, D .J. (1975), A Review of Response Surface Methodology from aBiometric Viewpoint , Biometrics, 31(4), 803-851.
[2] Box, G. E. P. & Wilson, K. B. (1951), On the Experimental Attainment of Optimum conditions, Journal of the Royal Statistical Society, Series B (Methodological), 13(1),1–15.
[3] Vining, G .G., & Bohn, L. L. (1998), Response Surfaces for the Mean and Variance Using a Nonparametric Approach, Journal of Quality Technology, 30, 282-291.
 [4] Nadaraya, E. (1964),On Estimating Regression, Theory of Probability and Its Applications,9,141-142.
[5] Watson, G. (1964), Smoothing Regression Analysis, Sankhya, Series A26 ,359-372.
[6] Fan, J., & Gijbels, I. (1996), Local Polynomial Modeling and Its Applications, Chapman and Hall, London.
 
[7] Fan, J. & Gijbels, I. (2000), Local polynomial fitting, In: Schimek, M.G. (Ed.), Smoothing and Regression: Approaches, Computation, and Application, Wiley, NewYork,229-276.
[8] Anderson-Cook, C .M., & Prewitt, K. (2005), Parametric Methods for Modeling Data from Response Surface Designs”, Journal of Modern Applied Statistical Methods,4 ,106-119.
[9] Pickle, S. M. (2006), Semiparametric Techniques for Response Surface Methodology Ph.D. Dissertation. Department of Statistics, Virginia Polytechnic Institute State University Blacksburg, VA.
[10] Pickle, S. M., & Robinson, T. J., & Birch, J. B. & Anderson-Cook, C. M. (2006), A SemiParametric Approach to Robust Parameter Design, Journal of Statistical Planning and Inference.
[11] Hardle, W. (1990), Applied Nonparametric Regression, Cambridge Univ, Press, London.
[12] Mays, J. E. & Birch, J. B. (1998), Smoothing Considerations in Nonparametric and Semiparametric Regression”, Technical Report Number 98-2, Department of Statistics, Virginia Polytechnic Institute State University, Blacksburg, VA.
 
[13] Einsporn, R. & Birch, J. B. (1993), Model Robust Regression: Using Nonparametric Regression to Improve Parametric Regression Analysis, Technical Report 93-5. Department of Statistics, Virginia Polytechnic Institute State University, Blacksburg,VA.
 
[14] Mays, J .E., & Birch, J. B. & Starnes, B.A.(2001), Model Robust Regression: Combining Parametric, Nonparametric, and Semiparametric Methods, Journal of Nonparametric Statistics, 13, 245-277.
[15] Robinson, P. M. (1988), Root-N-Consistent Semiparametric Regression, Econometrica,56,931-954.
[16] Hardle, W., & Muller, M., & Sperlich, S. & Werwatz, A. (2004), Nonparametric and Semiparametric Models, Springer, Berlin.
 
[17] Rencher, A. C. (2002), Method of multivariate analysis, John Wiley and Sons, Inc.,New York.
 
[18] Khuri, A. I. (1996). Simultaneous Optimization of Multiple Responses Represented by Polynomial Regression Functions , Technometrics , 23, 363-375.
 
[19] Wan, W. & Birch, j. B. (2011), A Semiparametric Technique for the Multi-response Optimization Problem , Journal of Quality and Reliabillity Engineering International, 27, 47-59.
 
[20] Shah, K. H. & Montgomery, D. C. & Carlyle, W. M. (2004), Response Surface Modeling and Optimization in Multiresponse Experiments Using Seemingly Unrelated Regressions, Quality Engineering, 16(3), 387-397.