Mbegbu , Ihenyen, Edionwe

Keywords: Semi-parametric regression, response surface methodology, multiple response optimization, kernel weights, surface roughness.

Abstract: For a machined mild steel material of certain physical and chemical properties, the surface roughness, prescribed in four different parameters each of which is desired to be as minimal as possible, is known to be a function of the machining (cutting) parameters including cutting speed (m/min), feed rate (mm/rev) and depth of cut (mm).This functional relationship between each of the roughness parameters and machining parameters must be represented by a model and subsequently optimized to determine the optimal values of the machining parameters that minimize the parameters of surface roughness. In Response Surface Methodology (RSM), Model Robust Regression 2 (MRR2) is a good choice of a statistical model. MRR2 is a hybrid model obtained from the combination of both the classical parametric Ordinary Least Squares (OLS) and a nonparametric Local Linear Regression (LLR) via a mixing parameter. LLR portion of MRR2 utilizes kernel weights derived from the simplified product Gaussian function. A motivation for this paper is derived from the fact that, since the OLS residuals are the equivalence of the response that the LLR portion is designed to estimate, then the kernel weight at each data point should reflect the relative magnitude of the OLS residual at each data point. In order to improve on the performance of MRR2, we therefore propose a robustification of the kernel weights using two different linearly transformed residuals vectors from the OLS component. Data from real experiments, statistical literature as well as simulation study were analyzed. Comparison of results shows that the MRR2 that utilizes the proposed robustified kernel weights outperforms OLS, LLR and the MRR2 that utilizes existing kernel weights by considerably wide margins. For the minimization of surface roughness of a machined mild steel material (EN10) in particular, the optimal cutting speed, feed rate and depth of cut of 254.3979m/min, 0.1774mm/rev and 0.4388mm, respectively, obtained by MRR2 utilizing one of the proposed techniques for robustifying kernel weights gave a desirability of 99.4%. This implies that the optimal value of each of the four roughness parameters collectively meets 99.4% of the process requirements.

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