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Proposal:
THE
RESPONSE
SURFACE
METHODOLOGY
Nuran
Bradley
Department
of
Mathematical
Sciences
Indiana
University
of
South
Bend
E -‐mail
Address:
nbradley@iusb.edu
Submitted
to
the
faculty
of
the
Indiana
University
South
Bend
in
partial
fulfillm ent
of
requirements
for
the
degree
of
MASTER
OF
SCIENCE
in
APPLIED
MATHEMATICS
&
COMPUTER
SCIENCE
Advisor
Dr.
Yi
Cheng
Department
of
Mathematical
Science
Committee:
Dr.
Zhong
Guan
Dr.
Dana
Vrajitoru
I
Abstract
The experimentation plays an important role in Science, Engineering, and Industry. The experimentation is an application of treatments to experimental units, and then measurement of one or more responses. It is a part of scientific method. It requires observing and gathering information about how process and system works. In an experiment, some input x’s transform into an output that has one or more observable response variables y. Therefore, useful results and conclusions can be drawn by experiment. In order to obtain an objective conclusion an experimenter needs to plan and design the experiment, and analyze the results.
There are many types of experiments used in real-world situations and problems. When treatments are from a continuous range of values then the true relationship between y and x’s might not be known. The approximation of the response function y = f (x1, x2,…,xq) + ε is called Response Surface Methodology. This thesis puts emphasis on designing, modeling, and analyzing the Response Surface Methodology. The three types of Response Surface Methodology, the first-order, the second-order, and the mixture models, will be explained and analyzed in depth. The thesis will also provide examples of application of each model by numerically and graphically using computer software.
II
TABLE OF CONTENTS
1. Introduction 1
2. Literature Reviews 2
3. Response Surface Methods and Designs 3
4. First-Order Model 4 4.1 Analysis of a first-order response surface 4
4.2 Designs for fitting the first-order model 5
4.3 My Objective of first-order model 5
5. Second-Order Model 5
5.1 Analysis of a second-order response surface 5
5.2 Designs for fitting the second-order model 6
5.3 My Objective of second-order model 7
6. Mixture-Model 7
6.1 Analysis of a mixture experiment 7
6.2 Designs for fitting the mixture model 8
6.3 My Objective of mixture model 8
7. Conclusion 9
8. Bibliography 10
III
List of Figures
Figure 3-1 Response Surface plot 3
Figure 3-2 Contour plot 4
Figure 6-1 Constrained factor space for mixtures with q = 2 and q = 3 7
1
1. Introduction
As an important subject in the statistical design of experiments, the Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response (Montgomery 2005). For example, the growth of a plant is affected by a certain amount of water x1 and sunshine x2. The plant can grow under any combination of treatment x1 and x2. Therefore, water and sunshine can vary continuously. When treatments are from a continuous range of values, then a Response Surface Methodology is useful for developing, improving, and optimizing the response variable. In this case, the plant growth y is the response variable, and it is a function of water and sunshine. It can be expressed as
y = f (x1, x2) + ε
The variables x1 and x2 are independent variables where the response y depends on them. The dependent variable y is a function of x1, x2, and the experimental error term, denoted as ε. The error term ε represents any measurement error on the response, as well as other type of variations not counted in f. It is a statistical error that is assumed to distribute normally with zero mean and variance σ2. In most RMS problems, the true response function f is unknown. In order to develop a proper approximation for f, the experimenter usually starts with a low-order polynomial in some small region. If the response can be defined by a linear function of independent variables, then the approximating function is a first-order model. A first-order model with 2 independent variables can be expressed as
εβ
ββ +++= 22110 xxy
If there is a curvature in the response surface, then a higher degree polynomial should be used. The approximating function with 2 variables is called a second-order model: εβ ββ ββ β ++++++= 2112 2 2222 2 111122110 x xxxxxy
In general all RSM problems use either one or the mixture of the both of these models. In each model, the levels of each factor are independent of the levels of other factors. When the levels of each factor are not independent then a mixture model is appropriate for designing an RMS model.
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