Sains Malaysiana 51(5)(2022): 1577-1586

http://doi.org/10.17576/jsm-2022-5105-25

A New Optimization Scheme for Robust Design Modeling with Unbalanced Data

(Skema Pengoptimuman Baru bagi Pemodelan Reka Bentuk Teguh dengan Data Tak

Seimbang)

ISHAQ BABA^{1, 2}, HABSHAH MIDI^1,*, GAFURJAN IBRAGIMOV¹ & SOHEL RANA³

¹Department of Mathematics and Statistics, Faculty of Science and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia

²Department of Mathematical Sciences, Faculty of Science, Taraba State University Jalingo

P.M.B. 1164 Jalingo, Taraba State, Nigeria

³Department of mathematical and Physical Sciences, East West University, Dhaka-1212, Bangladesh

Diserahkan: 5 Februari 2021/Diterima: 20 September 2021

Abstract

The Lin and Tu (LT) optimization scheme which is based on mean squared error (MSE) objective function is the commonly used optimization scheme for estimating the optimal mean response in robust dual response surface optimization. The ordinary least squares (OLS) method is often used to estimate the parameters of the process location and process scale models of the responses. However, the OLS is not efficient for the unbalanced design data since this kind of data make the errors of a model become heteroscedastic, which produces large standard errors of the estimates. To remedy this problem, a weighted least squares (WLS) method is put forward. Since the LT optimization scheme produces a large difference between the estimates of the mean response and the experimenter actual target value, we propose a new optimization scheme. The OLS and the WLS are integrated in the proposed scheme to determine the optimal solution of the estimated responses. The results of the simulation study and real example indicate that the WLS is superior when compared with the OLS method irrespective of the optimization scheme used. However, the combination of WLS and the proposed optimization scheme (PFO) signify more efficient results when compared to the WLS combined with the LT optimization scheme.

Keywords: Optimization; robust design; unbalanced data; weighted least squares

Abstrak

Skema pengoptimuman Lin dan Tu (LT) yang berdasarkan fungsi objektif min kuasadua ralat (MSE) sering digunakan dalam skema pengoptimuman bagi menganggarkan min gerak balas optimum dalam pengoptimuman permukaan berganda teguh. Kaedah kuasadua terkecil biasa (OLS) sering digunakan untuk menganggarkan parameter model proses lokasi dan model proses skala bagi gerak balas. Walau bagaimanapun, kaedah OLS tidak cekap bagi data reka bentuk yang tak seimbang kerana data yang begini membuatkan ralat model menjadi heteroskedastik dan menghasilkan penganggar ralat piawai besar. Untuk mengatasi masalah ini, kaedah kuasadua terkecil berpemberat (WLS) dicadangkan. Kami mencadangkan skema pengoptimuman baru disebabkan skema pengoptimuman LT menghasilkan perbezaan yang besar antara penganggar min gerak balas dan nilai sebenar sasaran penyelidik. Kaedah OLS dan WLS digabungkan dalam skema yang dicadangkan bagi menentukan penyelesaian optimum bagi gerak balas yang dianggarkan. Keputusan kajian simulasi dan contoh sebenar menunjukkan bahawa kaedah WLS mengatasi kaedah OLS tanpa mengira skema pengoptimuman yang digunakan. Walau bagaimanapun, gabungan WLS dan skema pengoptimuman yang dicadang (PFO) menunjukkan keputusan yang lebih cekap apabila dibandingkan dengan WLS yang digabungkan dengan skema pengoptimuman LT. 

Kata kunci: Data tak seimbang; kuasadua terkecil berpemberat; pengoptimuman; reka bentuk teguh

RUJUKAN

Box, G.E. & Draper, N.R. 1987. Empirical Model-Building and Response Surfaces. New York: John Wiley & Sons.

Boylan, G.L. & Cho, B.R. 2013. Comparative studies on the high-variability embedded robust parameter design from the perspective of estimators. Computers & Industrial Engineering 64(1): 442-452.

Chelladurai, S.J.S., Murugan, K., Ray, A.P., Upadhyaya, M., Narasimharaj, V. & Gnanasekaran, S. 2021. Optimization of process parameters using response surface methodology: A review. Materials Today: Proceedings 37: 1301-1304.

Cho, B.R. & Park, C. 2005. Robust design modeling and optimization with unbalanced data. Computers & Industrial Engineering 48(2): 173-180.

Copeland, K.A. & Nelson, P.R. 1996. Dual response optimization via direct function minimization. Journal of Quality Technology 28(3): 331-336.

Ding, R., Lin, D.K. & Wei, D. 2004. Dual-response surface optimization: A weighted MSE approach. Quality Engineering 16(3): 377-385.

Dong, S. 2006. Methods for Constrained Optimization. Massachusetts: Massachusetts Institute of Technology.

Ghalanos, A. & Theussl, S. 2012. Rsolnp: General non-linear optimization using augmented Lagrange multiplier method. R package version, 1.

Goethals, P.L. & Cho, B.R. 2011. Solving the optimal process target problem using response surface designs in heteroscedastic conditions. International Journal of Production Research 49(12): 3455-3478.

Jeong, I.J., Kim, K.J. & Chang, S.Y. 2005. Optimal weighting of bias and variance in dual response surface optimization. Journal of Quality Technology 37(3): 236-247.

Kackar, R.N. 1985. Off-line quality control, parameter design, and the Taguchi method. Journal of Quality Technology 17(4): 176-188.

Kim, K.J. & Lin, D.K.J. 1998. Dual response surface optimization: A fuzzy modeling approach. Journal of Quality Technology 30(1): 1-10.

Lee, S.B. & Park, C. 2006. Development of robust design optimization using incomplete data. Computers & Industrial Engineering 50(3): 345-356.

Lee, D.H., Yang, J.K. & Kim, K.J. 2018. Dual-response optimization using a patient rule induction method. Quality Engineering 30(4): 610-620.

Lin, D.K. & Tu, W. 1995. Dual response surface optimization. Journal of Quality Technology 27(1): 34-39.

Myers, R.H. & Montgomery, D.C. 1995. Response Surface Methodology: Process and Product Optimization using Designed Experiments. https://doi.org/10.2307/1270613

Midi, H. & Aziz, N.A. 2019. High breakdown estimator for dual response optimization in the presence of outliers. Sains Malaysiana 48(8): 1771-1776.

Midi, H., Ismaeel, S.S., Arasan, J. & Mohammed, M.A. 2021. Simple and fast generalized-M (GM) estimator and its application to real data set. Sains Malaysiana 50(3): 859-867.

Nair, V.N., Abraham, B., MacKay, J., Box, G., Kacker, R.N., Lorenzen, T.J., Lucas, J.M., Myers, R.H., Vining, G.G., Nelder, J.A. & Phadke, M.S. 1992. Taguchi’s parameter design: A panel discussion. Technometrics 34(2): 127-161.

Park, C. & Cho, B.R. 2003. Development of robust design under contaminated and non-normal data. Quality Engineering 15(3): 463-469.

Park, C. & Leeds, M. 2016. A highly efficient robust design under data contamination. Computers & Industrial Engineering 93: 131-142.

Park, C., Ouyang, L., Byun, J.H. & Leeds, M. 2017. Robust design under normal model departure. Computers & Industrial Engineering 113: 206-220.

Rana, S., Midi, H. & Imon, A.R. 2012. Robust wild bootstrap for stabilizing the variance of parameter estimates in heteroscedastic regression models in the presence of outliers. Mathematical Problems in Engineering 2012: 730328.

Shin, D.K., Gürdal, Z. & Griffin, O.H.J. 1990. A penalty approach for nonlinear optimization with discrete design variables. Engineering Optimization+ A35 16(1): 29-42.

Taguchi, G. & Wu, Y.I. 1980. Introduction to Off-Line Quality Control Systems. Nagoyo: Central Japan Quality Control Association.

Vining, G.G. & Myers, R.H. 1990. Combining taguchi and response surface philosophies: A dual response approach. Journal of Quality Technology 22(1): 38-45.

Wan, Z., Zhang, S.J. & Wang, Y.L. 2009. Penalty algorithm based on conjugate gradient method for solving portfolio management problem. Journal of Inequalities and Applications 2009: Article ID. 970723.

Ye, Y. 1989: Solnp users' guide -a nonlinear optimization program in matlab. Department of Management Sciences, College of Business Administration, University of Iowa.

^*Pengarang untuk surat-menyurat; email: habshahmidi@gmail.com