Effect of Size of the Step in the Response Surface Methodology using Nonlinear Test Functions
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.
[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.
[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.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:52677", author = "Jesús Everardo Olguín Tiznado and Rafael García Martínez and Claudia Camargo Wilson and Juan Andrés López Barreras and Everardo Inzunza González and Javier Ordorica Villalvazo", title = "Effect of Size of the Step in the Response Surface Methodology using Nonlinear Test Functions", 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", volume = "6", number = "6", pages = "1064-6", }