Transcript Slide 1

Industrial Applications of
Response Surface Methodolgy
John Borkowski
Montana State University
Pattaya Conference on Statistics
Pattaya, Thailand
Outline of the Presentation
1.
2.
3.
4.
5.
6.
The Experimentation Process
Screening Experiments
2k Factorial Experiments
Optimization Experiments
Mixture Experiments
Final Comments
The Experimentation Process
Defining Experimental Objectives
 Researchers often discover after running an
experiment that the data are insufficient to meet
objectives

The first and most important step in an
experimental strategy is to clearly state the
objectives of the experiment.

The objective is a precise answer to the question
“What do you want to know when the experiment
is complete?”
2. Screening Experiments

The experimenter wants to determine which
process variables are important from a list of
potentially important variables.
Screening experiments are economical because
a large number of factors can be studied in a
small number of experimental runs.
The factors that are found to be important will be
used in future experiments. That is, we have
screened out the important factors from the list.
2. Screening Experiments
 Common screening experiments are
1. Plackett-Burman designs
2. Two-level full-factorial (2k) designs
3. Two-level fractional-factorial (2k-p) designs
 Plackett-Burman designs allow you to study as
many as k-1 factors in k points where
k = 12, 20, 24…
(k is a multiple of 4 but not a power of 2)
Example 1: Screening 6 Factors
Response: Plastic Hardness
Factor Levels
Factors
-1
+1
(X1) Tension Control
Manual Automatic
(X2) Machine
#1
#2
(X3) Throughput (liters/min)
10
20
(X4) Mixing
Single Double
(X5) Temperature
200o
250o
(X6) Moisture
20 %
30 %
Randomly assign 6 columns to the 6 factors
and then randomize the run order
Analysis of the Screening Design Data Using SAS
3.
2k Factorial Experiments
 A 2k factorial design is a k-factor design
such that
 The 2k experimental runs are the 2k
combinations of + and – factor levels
 Each factor has two levels coded + and -
 The 2k experimental runs may also be
replicated
Example 2: 23 Design with 3 Replicates
(Montgomery 2005)
 An engineer is interested in the effects of
– cutting speed (A),
– tool geometry (B),
– cutting angle (C)
on the life (in hours) of a machine tool
 Two levels of each factor were chosen
 Three replicates of a 23 design were run
 Run order was randomized
Analysis using SAS: Main Effects Model (r2=.50)
Analysis using SAS: Interaction Model (r2=.76)
Maximize Hours at A=-1 B=+1 C=+1
Unreplicated 2k Designs
 When the design is unreplicated (n=1 for each
of the 2k factor treatments), it is necessary to
“pool” interaction terms to form an error term
for hypothesis testing.
 Three steps are recommended:
1. Estimate all effects for the full-factorial model.
2. Make a normal probability plot of these estimated
effects. “Outlier” effects can be pooled together.
3. Run the ANOVA using the pooled error term.
Example 3: An Unreplicated 24 Design
(Montgomery and Myers 2002)
 An engineer studied four factors believed to affect
the filtration rate (Y) of a chemical product:
– temperature (A),
– pressure
(B),
– concentration of formaldehyde (C)
– stirring rate (D)
 Two levels of each factor were chosen
 An unreplicated 23 design were run
 Run order was randomized
The Unreplicated 24 Design
The Design and Data
Step 1: Estimates
Step 2: Normal probability plot of effects (Minitab)
Probability Plot of Effects
Normal
99
A
95
A
90
AD
Percent
80
C
70
60
50
40
D
30
20
10
5
AC
1
-20
-10
0
Effects
10
20
Step 3: The ANOVA with a Pooled Error (r2=.97)
Main Effects Plots (using Minitab)
Main Effects Plot (data means) for Rate
Temperature
Concentration
80
75
Mean of Rate
70
65
60
-1
1
Stirring Rate
80
75
70
65
60
-1
1
-1
1
Interaction Plots (using Minitab)
Temperature*Concentration Interaction Plot
Temperature*Stirring Rage Interaction Plot
100
90
90
70
Mean
Mean
80
60
Temperature
-1
1
50
80
Temperature
-1
1
70
60
40
-1
1
Concentration
-1
1
Stirring Rate
4. Optimization Experiments
 The experimenter wants to model (fit a response surface)
involving a response y which depends on process input
variables 1, 2, … k.
 Because the exact functional relationship between y and
1, 2, … k is unknown, a low order polynomial is used
as an approximating function.
 Before fitting a model, 1, 2, … k are coded as x1, x2,
…, xk. For example:
i = 100
xi = -1
150
0
200
+1
4. Optimization Experiments
The experimenter is interested in determining:
1. Values of the input variables 1, 2, … k. that
optimize the response y (known as the optimum
operating conditions).
2. An operating region that satisfy operating specifications
for y.
 A common approximating function is the quadratic or
second-order model:
f (x)  b0 
k
b x
i 1
i i

k-1
k
 b x
i 1
j i 1
ij
j

k
b x
i 1
2
ii i
Example 3: Approximating Functions
 The experimental goal is to maximize process yield (y).
 A two-factor 32 experiment with 2 replicates was run with:
Temperature
1: Uncoded Levels 100o 150o 200o
x1 Coded Levels
-1
0 +1
Process time (min) 2: Uncoded Levels 6
x2 Coded Levels -1
8 10
0 +1
True Function: y = e(.5x1 – 1.5x2)
Fitted function (from SAS) 
Predicted Maximum Yield (y) at x1= +1 , x2= -1
(or, Temperature = 200o , Process Time = 6 minutes)
CONTOUR PLOTS
TRUE FUNCTION
QUADRATIC APPROX.
Central Composite Design
Box-Behnken Design
(CCD)
(BBD)
Factorial, axial, and
center points
Centers of edges and
center points
Example 4: Central Composite Design (Myers 1976)
 The experimenter wants to study the effects of
• sealing temperature
(x1)
• cooling bar temperature (x2)
• polethylene additive
(x3)
on the seal strength in grams per inch of breadwrapper stock (y).
 The uncoded and coded variable levels are
-
-1
0
1
255o
55o
1.1%
285o
64o
1.7%

.
x1
x2
x3
204.5o
39.9o
.09%
225o
46o
.5%
305.5o
70.1o
2.11%
Example 4: Central Composite Design
Canonical Analysis of Quadratic Model (using SAS)
Ridge Analysis of Quadratic Model (using SAS)
Predicted Maximum at
x1=-1.01
x2=0.26
x3=0.68
5. Mixture Experiments
 A mixture contains q components where xi is the
proportion of the ith component (i=1,2,…, q)
 Two constraints exist:
0 ≤ xi ≤ 1 and
Σ xi = 1
Simplex Coordinate System
Mixture Experiment Models
 Because the level of the final component can written as
xq = 1 – (x1 + x2 + + xq-1)
any response surface model used for independent factors
can be reduced to a Scheffé model. Examples include:
Example of a 3-Component Mixture Design
Analysis of a 3-component Mixture Experiment
4-Component Mixture Experiment with Component Level
Constraints (McLean & Anderson 1966)
Response: Flare Brightness
6. Final Comments





Screening experiments
2k and 2k-p experiments
Optimization experiments
Mixture experiments
Other applications:
• Fractional factorial designs
• Path of steepest ascent (descent)
• Experiments with blocking
• Experiments with restrictions on randomization