note

1. Design of Experiments:
Response Surface Methods
Zeynep Gökçe İşlier
Yeditepe University
May 9, 2021
Zeynep Gökçe İşlier Yeditepe University
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2. Outline
1. 22 Factorial Design
2. Addition of Center Points
3. Central Composite Design
4. Method of Steepest Ascent
5. Second Order Response Surface Model
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3. Response Surface Methods
I Primary focus of previous topics is factor screening
I Two-level factorials, fractional factorials are widely used
I Objective of RSM is optimization
I RSM dates from the 1950s; early applications in chemical
industry
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4. The Simplest Case: 22
Design
I “-” and “+” denote the low
and high levels of a factor,
respectively
I Low and high are arbitrary
terms
I Geometrically, the four runs
form the corners of a square
I Factors can be quantitative
or qualitative, although their
treatment in the final model
will be different
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5. Chemical Process Example
Factor Replicate
A B
Treatment
Combination I II III Total
− − A low, B low 28 25 27 80
+ − A high, B low 36 32 32 100
− + A low, B high 18 19 23 60
+ + A high, B high 31 30 29 90
I A=reactant concentration
I A=catalyst amount
I y=recovery
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6. Analysis Procedure for a Factorial Design
I Estimate factor effects
I Formulate model
I With replication, use full model
I With an unreplicated design, use normal probability plots
I Statistical testing (ANOVA)
I Refine the model
I Analyze residuals (graphical)
I Interpret results
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7. Estimation of Factor Effects
I A = ȳA+ − ȳA− = ab+a
2n − b+1
2n
I B = ȳB+ − ȳB− = ab+b
2n − a+1
2n
I AB = ab+1
2n − a+b
2n
I The effect estimates are: A = 8.33, B = −5.00, AB = 1.67
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8. Estimation of Factor Effects
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9. Statistical Testing - ANOVA
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10. Statistical Testing - ANOVA
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11. Refine Model
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12. Residuals and Diagnostic Checking
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13. The Response Surface
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14. Addition of Center Points to a 2k
Designs
I Based on the idea of replicating some of the runs in a factorial
design
I Runs at the center provide an estimate of error and allow the
experimenter to distinguish between two possible models:
I First-order model (interaction)
y = β0 +
Pk
i=1 βi xi +
Pk
i=1
Pk
j>i βij xi xj +
I Second-order model
y = β0 +
Pk
i=1 βi xi +
Pk
i=1
Pk
ji βij xi xj +
Pk
i=1 βii x2
i +
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15. 22
Design with Center Points
I ȳF = ȳC → no curvature
I The hypotheses are:
I H0 :
Pk
i=1 βii = 0
I H1 :
Pk
i=1 βii 6= 0
I SSPureQuad = nF nC (ȳF −ȳC )2
nF +nC
I This sum of squares has a
single degree of freedom
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16. Example
I nC = 5
I Usually between 3 and 6
center points will work well
I Design-Expert provides the
analysis, including the F test
for pure quadratic curvature
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17. ANOVA for Example
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18. Central Composite Design
I If curvature is significant, augment the design with runs to
create a central composite design. The CCD is a very effective
design for fitting a second-order response surface model
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19. Practical Use of a Center Points
I Use current operating conditions as the center point
I Check for “abnormal” conditions during the time the
experiment was conducted
I Check for time trends
I Use center points as the first few runs when there is little or
no information available about the magnitude of error
I Center points and qualitative factors?
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20. Center Points and Qualitative Factors
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21. Steps in RSM
I Find a suitable approximation for y = f (x) using LS {maybe a
low - order polynomial}
I Move toward the region of the optimum
I When curvature is found, find a new approximation for
y = f (x) {generally a higher order polynomial} and perform
the “Response Surface Analysis”}
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22. RSM is s Sequential Procedure
I Factor screening
I Finding the region of the
optimum
I Modeling Optimization of
the response
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23. Steps in RSM
I Screening
I y = β0 + β1x1 + β2x2 + β12x1x2 +
I Steepest ascent
I y = β0 + β1x1 + β2x2 +
I Optimization
I y = β0 + β1x1 + β2x2 + β12x1x2 + β11x2
1 + β22x2
2 +
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24. The Method of Steepest Ascent
I A procedure for moving
sequentially from an initial
“guess” towards to region of
the optimum
I Based on the first order
model
I ŷ = β̂0 + β̂1x1 + β̂2x2
I Steepest ascent is a gradient
procedure
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25. An Example of Steepest Ascent
I ŷ = 40.44 + 0.775x1 + 0.325x2
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26. An Example of Steepest Ascent
I An approximate step size
and path can be determined
graphically
I Formal methods can also be
used
I Types of experiments along
path
I Single runs
I Replicated runs
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27. Results from the Example
I The step size is 5 minutes of reaction time and 2 degrees F
I What happens at the conclusion of steepest ascent
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28. Example
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29. The Second-Order Response Surface Model
I y = β0 + β1x1 + β2x2 + β12x1x2 + β11x2
1 + β22x2
2 +
I These models are used widely in practice
I The Taylor series analogy
I Fitting the model is easy, some nice designs are available
I Optimization is easy
I There is a lot of experimental evidence that they work well
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30. Example
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31. Example
I ŷ = 79.94 + 0.99x1 + 0.52x2 + 0.25x1x2 − 1.38x2
1 − 1.00x2
2
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32. An Example of Steepest Ascent
I The contour plot is given in
the natural variables
I The optimum is at about 87
minutes and 176.5 degrees
I Formal optimization
methods can also be used
(particularly when k 2)
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33. Multiple Responses
I The previous example illustrated three response variables
(yield, viscocity, and molecular weight)
I Multiple responses are common in practice
I Typically, we want to simultaneously optimize all responses, to
find a set of conditions where certain product properties are
achieved
I A simple approach is to model all responses and overlay
contour plots
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34. Design for Fitting RSM
I For the first-order model, two-level factorials (and fractional
factorials) augmented with center points are appropriate
choices
I The central composite design is the most widely used design
for fitting the second-order model
I Selection of a second-order design is an interesting problem
I There are numerous excellent second-order designs available
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35. Other Aspects RSM
I Robust parameter design and process robustness studies
I Find levels of controllable variables that optimize mean
response and minimize variability in the response transmitted
“noise” variables
I Original approaches due to Taguchi
I Modern approach based on RSM
I Experiments with mixtures
I Special type of RSM problem
I Design factors are components (ingredients) of a mixture
I Response depends only on the proportions
I Many applications in product formulation
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