Effect of Size of the Step in the Response Surface Methodology using Nonlinear Test Functions

Authors: Jesús Everardo Olguín Tiznado, Rafael García Martínez, Claudia Camargo Wilson, Juan Andrés López Barreras, Everardo Inzunza González, Javier Ordorica Villalvazo

Abstract:

The response surface methodology (RSM) is a collection of mathematical and statistical techniques useful in the modeling and analysis of problems in which the dependent variable receives the influence of several independent variables, in order to determine which are the conditions under which should operate these variables to optimize a production process. The RSM estimated a regression model of first order, and sets the search direction using the method of maximum / minimum slope up / down MMS U/D. However, this method selects the step size intuitively, which can affect the efficiency of the RSM. This paper assesses how the step size affects the efficiency of this methodology. The numerical examples are carried out through Monte Carlo experiments, evaluating three response variables: efficiency gain function, the optimum distance and the number of iterations. The results in the simulation experiments showed that in response variables efficiency and gain function at the optimum distance were not affected by the step size, while the number of iterations is found that the efficiency if it is affected by the size of the step and function type of test used.

Keywords: RSM, dependent variable, independent variables, efficiency, simulation

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1330549

References:

[1] Montgomery, D.C., Design and Analysis of Experiments, 6th edition, John Wiley & Sons, 2005.
[2] Jack P. C. Kleijnen, Desing and Analisys of Simulation Experiments. Springer. New York, 2008, p.p. 101-110.
[3] Box, G.E.P., Statistics as a catalyst to learning by scientific method, part II-a discussion. Journal of Quality Technology 31 (1), p.p. 1-38, 1999.
[4] Khuri, A.I., Multiresponse Surface methodology. In: Ghosh, S., Rao, C.R. (Eds), Handbook of Statistics, Vol. 13, Elsevier, Amsterdam, 1996.
[5] Khuri, A.I., Cornell J.A., Response Surfaces: Desings and Analyses, second ed. Marcel Dekker, Inc., New York, 1996.
[6] Myers, R.H., Response surface methodology -current status and future directions. Journal of Quality Technology 31 (1), 30-74. 1999.
[7] Myers, R.H., Montgomery, D.C., Anderson-Cook, C.M., Response surface methodology: process and product optimization using designed experiments, third edition, John Wiley & Sons, 2009.
[8] Sztendur, E., Precision of the path of steepest ascent in response surface methodology. Doctoral dissertation, Victoria University of Technology, School of Computer Science and Mathematics, Melbourne, Australia. 2005
[9] García, S., Levine, J., González F., Multi Niche Parallel GP with Junkcode Migration Model, 1999. (www.aiai.ed.ac.uk/~johnl/papers/garcía_eurogp03.ps)
[10] Dent, D. Paprzycki, M. y Kucaba-Pietal. Testing Convergence of Nonlinear Systems Solvers. Proceedings of the First Southern Symposium on Computing. The University of Southern Mississippi, 1998.
[11] Parkinson and Hutchinson., An investigation into the eficiency of variants on the simplex method. F. A. Lootsma, editor, Numerical Methods for Non-linear Optimization, pages 115-135, 1972.
[12] Sanchez, L.J., A Method of experimental optimization. New México State University, Las Cruces, Nuevo México, 1991.
[13] García Martínez, R., Master in Science dissertation: Response surface methodology evaluation.. Technological Institute of Ciudad Juárez, Mexico, 1993.