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Nirmal Tatavalli Mittadar^{1}, **Michael Nikolaou**^{2}, and Demetre J. Economou^{1}. (1) University of Houston, 8181 Fannin Apt#2628, Houston, TX 77054, (2) Chemical Engineering, University of Houston, 4800 Calhoun Road, Houston, TX 77204-4004

Abstract

The response surface methodology (RSM) is a collection of tools for statistical design of experiments and numerical optimization techniques, used to optimize processes and products ADDIN EN.CITE <EndNote><Cite><Author>Myers</Author><Year>2004</Year><RecNum>13</RecNum><record><rec-number>13</rec-number><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Myers, Raymond H. </author><author>Montgomery, Douglas C. </author><author>Vinning, G. Geoffrey</author><author>Borror, Connie M. </author><author>Kowalski, Scott M. </author></authors></contributors><titles><title> Response Surface Methodology: A retrospective and Literature Survey</title><secondary-title>Journal of Quality Technology</secondary-title></titles><periodical><full-title>Journal of Quality Technology</full-title></periodical><pages>53-70</pages><volume>36</volume><number>1</number><dates><year>2004</year></dates><urls></urls></record></Cite></EndNote>(Myers, Montgomery et al. 2004). RSM entails a succession of experimentation/optimization cycles, repeated until an optimal solution is determined or resources have been exhausted. In each cycle, experiments are conducted at selected values of the decision variables, and the experimental results are used to construct an input-output model (response surface) that will suggest the experimental conditions for the next cycle. Parametric or nonparametric models can be used to approximate the relationship between decision variables (inputs) and the response (outputs) of interest. A nonparametric class of models that has been successful for complicated response surfaces with multiple optima employs kriging approximators ADDIN EN.CITE <EndNote><Cite><Author>Isaaks</Author><Year>1989</Year><RecNum>14</RecNum><record><rec-number>14</rec-number><ref-type name="Book">6</ref-type><contributors><authors><author>Isaaks, E. H.</author><author>Srivastava, R. M.</author></authors></contributors><titles><title>An Introduction to Applied Geostatistics</title></titles><dates><year>1989</year></dates><pub-location>New York</pub-location><publisher>Oxford University Press</publisher><urls></urls></record></Cite></EndNote>(Isaaks and Srivastava 1989) that have been used for design and analysis of computer experiments (DACE) ADDIN EN.CITE <EndNote><Cite><Author>Santner</Author><Year>2003</Year><RecNum>15</RecNum><record><rec-number>15</rec-number><ref-type name="Book">6</ref-type><contributors><authors><author>Santner, Thomas J. </author><author>Williams, Brian J.</author><author>Notz, William I.</author></authors></contributors><titles><title>The Design and Analysis of Computer Experiments</title></titles><dates><year>2003</year></dates><publisher>Springer Series in Statistics</publisher><urls></urls></record></Cite></EndNote>(Santner, Williams et al. 2003). DACE models have been used in conjunction with global optimization methodologies inspired by Kushner's criterion ADDIN EN.CITE <EndNote><Cite><Author>Kushner</Author><Year>1964</Year><RecNum>16</RecNum><record><rec-number>16</rec-number><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Kushner, H. J.</author></authors></contributors><titles><title>A New Method of Locating the Maximum of an Arbitrary Multipeak Curve in the Presence of Noise</title><secondary-title>Journal of Basic Engineering</secondary-title></titles><periodical><full-title>Journal of Basic Engineering</full-title></periodical><pages>97-106</pages><volume>86</volume><dates><year>1964</year></dates><urls></urls></record></Cite></EndNote>(Kushner 1964). According to standard methodology, the structure of a DACE model is determined by a preselected trend function and by the experimental data collected, whereas the values of the DACE model parameters are determined using a maximum-likelihood estimation (MLE) method detailed in ADDIN EN.CITE <EndNote><Cite><Author>Pardo-Igu'zquiza</Author><Year>1998</Year><RecNum>10</RecNum><record><rec-number>10</rec-number><ref-type name="Journal Article">17</ref-type><contributors><authors><author>Eulogio Pardo-Igu'zquiza</author></authors></contributors><titles><title>Maximum Likelihood Estimation of Spatial Covariance Parameters</title><secondary-title>Mathematical Geology</secondary-title></titles><periodical><full-title>Mathematical Geology</full-title></periodical><pages>95-108</pages><volume>30</volume><number>1</number><dates><year>1998</year></dates><urls></urls></record></Cite></EndNote>(Pardo-Igu'zquiza 1998). However, given that the DACE model structure relies on interpolation, on many occasions MLE produces a DACE response surface that is void of any useful information, in that the response surface is composed of the preselected trend function along with narrow peaks and valleys in small neighborhoods around points where experiments have been conducted. Such a model is far from the actual response surface and cannot be used for optimization. This phenomenon has been observed before ADDIN EN.CITE <EndNote><Cite><Author>Sasena</Author><Year>2002</Year><RecNum>17</RecNum><record><rec-number>17</rec-number><ref-type name="Thesis">32</ref-type><contributors><authors><author>Sasena, M. J.</author></authors></contributors><titles><title>Flexibility and Efficiency Enhancements for Constrained Global Design Optimization with Kriging Approximations</title><secondary-title>Department of Mechanical Engineering</secondary-title></titles><volume>PhD Thesis</volume><dates><year>2002</year></dates><publisher>University of Michigan</publisher><urls></urls></record></Cite></EndNote>(Sasena 2002) and ADDIN EN.CITE <EndNote><Cite><Author>Lin</Author><Year>2004</Year><RecNum>12</RecNum><record><rec-number>12</rec-number><ref-type name="Thesis">32</ref-type><contributors><authors><author>Yao Lin</author></authors><tertiary-authors><author>Prof. Farrokh Mistree</author></tertiary-authors></contributors><titles><title>An Efficient Robust Concept Exploration Method and Sequential Exploratory Experimental Design</title><secondary-title>Mechanical Engineering</secondary-title></titles><pages>780</pages><volume>Ph.D.</volume><dates><year>2004</year><pub-dates><date>July</date></pub-dates></dates><pub-location>Atlanta</pub-location><publisher>Georgia Institute of Technology</publisher><urls></urls></record></Cite></EndNote>(Lin 2004). The suggested remedy in ADDIN EN.CITE <EndNote><Cite><Author>Sasena</Author><Year>2002</Year><RecNum>17</RecNum><record><rec-number>17</rec-number><ref-type name="Thesis">32</ref-type><contributors><authors><author>Sasena, M. J.</author></authors></contributors><titles><title>Flexibility and Efficiency Enhancements for Constrained Global Design Optimization with Kriging Approximations</title><secondary-title>Department of Mechanical Engineering</secondary-title></titles><volume>PhD Thesis</volume><dates><year>2002</year></dates><publisher>University of Michigan</publisher><urls></urls></record></Cite></EndNote>(Sasena 2002) of conducting two experiments at points close to each other works well for one decision variable, but does not easily extend to the more interesting case of multiple decision variables. In fact, the general suggestion in literature when dealing with multiple decision variables is to start RSM with a large number of experiments, in order to capture the shape of the response surface. For example it is suggested in ADDIN EN.CITE <EndNote><Cite><Author>Schonlau</Author><Year>1997</Year><RecNum>8</RecNum><record><rec-number>8</rec-number><ref-type name="Conference Proceedings">10</ref-type><contributors><authors><author>M. Schonlau</author><author>Welch, W J.</author><author>Jones, D R.</author></authors></contributors><titles><title>A Data Analaytic Approach to Bayesian Global Optimization</title><secondary-title>American Statistical Association Proceedings, Section of Physical Engineering Sciences</secondary-title></titles><pages>186-191</pages><dates><year>1997</year></dates><urls></urls></record></Cite></EndNote>(Schonlau, Welch et al. 1997) to start with number of experiments equal to 10 times the number of active variables. This practice may lead to inefficiencies or may limit applications to the cases where real time experiments are not costly.

In this work, we present a refinement of the MLE strategy that produces DACE response surfaces that can be better used to design experiments for optimization purposes. The proposed refinement relies on use of informative priors about the model parameters.

As a motivating example, we revisit the example in ADDIN EN.CITE <EndNote><Cite><Author>Sasena</Author><Year>2002</Year><RecNum>17</RecNum><record><rec-number>17</rec-number><ref-type name="Thesis">32</ref-type><contributors><authors><author>Sasena, M. J.</author></authors></contributors><titles><title>Flexibility and Efficiency Enhancements for Constrained Global Design Optimization with Kriging Approximations</title><secondary-title>Department of Mechanical Engineering</secondary-title></titles><volume>PhD Thesis</volume><dates><year>2002</year></dates><publisher>University of Michigan</publisher><urls></urls></record></Cite></EndNote>(Sasena 2002). The DACE model based on traditional MLE strategy is shown in Figure 1

We demonstrate the effectiveness of the proposed approach through a simulation study on a plasma etching reactor where we want to identify process conditions like pressure and reactor configuration, for example, location of induction coils that would result in an uniform etch rate in the presence of non-trivial constraints like etch rate must be greater than a specified minimum. Specifically, we consider here a case of Argon plasma with a single induction coil located on the sides. The configuration is illustrated in Figure 5_{ }

ADDIN EN.REFLIST Isaaks, E. H. and R. M. Srivastava (1989). __An Introduction to Applied Geostatistics__. New York, Oxford University Press.

Kushner, H. J. (1964). "A New Method of Locating the Maximum of an Arbitrary Multipeak Curve in the Presence of Noise." __Journal of Basic Engineering__ **86**: 97-106.

Lin, Y. (2004). An Efficient Robust Concept Exploration Method and Sequential Exploratory Experimental Design. __Mechanical Engineering__. Atlanta, Georgia Institute of Technology. **Ph.D.: **780.

Myers, R. H., D. C. Montgomery, et al. (2004). " Response Surface Methodology: A retrospective and Literature Survey." __Journal of Quality Technology__ **36**(1): 53-70.

Pardo-Igu'zquiza, E. (1998). "Maximum Likelihood Estimation of Spatial Covariance Parameters." __Mathematical Geology__ **30**(1): 95-108.

Santner, T. J., B. J. Williams, et al. (2003). __The Design and Analysis of Computer Experiments__, Springer Series in Statistics.

Sasena, M. J. (2002). Flexibility and Efficiency Enhancements for Constrained Global Design Optimization with Kriging Approximations. __Department of Mechanical Engineering__, University of Michigan. **PhD Thesis**.

Schonlau, M., W. J. Welch, et al. (1997). __A Data Analaytic Approach to Bayesian Global Optimization__. American Statistical Association Proceedings, Section of Physical Engineering Sciences.

Figure SEQ Figure \* ARABIC 1: The traditional MLE methodology results in a peak and valley behavior in the neighborhood of the design sites.

Figure SEQ Figure \* ARABIC 2: The new strategy based on informative prior results in a superior approximation.

Figure SEQ Figure \* ARABIC 3:. The graph above shows the probability that the response is greater than or equal to 9.0 using the model constructed by the traditional method. Clearly the probabilities estimated using the model do not represent the reality.

Figure SEQ Figure \* ARABIC 4: The graph above shows the probability that the response is greater than or equal to 9.0 using the model constructed by the new MLE strategy.

Figure SEQ Figure \* ARABIC 5: The reactor configuration used for the simulation. We assume axis symmetry so only one of the radius of the reactor is shown.

Figure SEQ Figure \* ARABIC 6: The predicted non-uniformity from the dace model based on the traditional MLE.

Figure SEQ Figure \* ARABIC 7 The estimate of the probability that the mean flux of Argon ions at the wafer surface is greater than 2.5x10^{20}m^{-2} as a function pressure and coil location using the dace model for mean flux constructed by the traditional MLE strategy.

Figure SEQ Figure \* ARABIC 8:- The predicted non-uniformity as a function of pressure and coil location using the dace model for non-uniformity constructed using the modified MLE strategy

Figure SEQ Figure \* ARABIC 9:-The estimate of the probability that the mean flux of Argon ions at the wafer surface is greater than 2.5x10^{20}m^{-2} as a function pressure and coil location using the dace model for mean flux constructed using the modified MLE strategy.