Transcript Balanced Incomplete Block Designs

Lecture 17
• Today: Start Chapter 9
• Next day: More of Chapter 9
Example
• Chemical Engineer is interested in maximizing yield of a process
• 2 variables influence process yield:
– reaction time (x1) and reaction temperature (x2)
• Current operating conditions have reaction time at 35 minutes and
temperature at 155 oF, which give a yield of about 40%
• Best operating conditions may be far from current conditions
Response Surface Methods
• Often the goal of experimentation is to optimize (say maximize) a
system response
• When there are only a few quantitative factors, response surface
methodology can be use for understanding the relationship between the
response and input factors
• Experimentation strategy is sequential
Models
• Relationship between input variables and response:
– y  f ( x1 , x2 ,...,xk )  
• If the expected response is denoted E(y)=f(x1, x2,…, xk), then
f(x1, x2,…, xk) is called the response surface
Models
• First order model to approximate f:
– y   0    i xi  
k
i 1
• Second order model to approximate f:
k
k
k
i 1
i j
i 1
– y   0    i xi    ij xi x j    ii xi2  
• Typically a lower order polynomial is used to approximate the local
response surface
General Idea
•
If there are many factors to consider, perform a screening design first (e.g., 2k-p
fractional factorial design) to screen out unimportant factors
•
Response surface methodology involves experimentation, modeling, data
analysis and optimization
•
First run a sequence of small experiment designs to fit a first order model to
identify the experiment region that is near or contains the optimum
•
Next an experiment design is performed to estimate the second order model
close to the optimum
•
Designs to estimate the response surface are called response surface designs
General Idea
• Design that estimates a first order model is called a first order design
• Design that estimates a second order model is called a second order
design
• Analysis is performed using regression
Coding the Variables
• It is convenient to code the variables (as we have done so far)
• For example consider a factor with 3 equally spaced levels
• Let zi be the mid-point of the levels
• The 3 levels are: Z: zi-c, zi, zi+c
• Transformation: x  (Z  zi )
c
• 3 coded levels: X: -1, 0, 1
First Order Designs
• Will run a series of first order designs until near the optimum
• When there is substantial curvature, first order model becomes
ineffective for approximating the surface
• How do we know if there is curvature?
First Order Designs
• Must have more than 2 level factors to check for curvature
• Solution is to add experiment trials at the center of the experimental
region (e.g., x=(0,0,0,0…0) )
• Designs combine factorials and center points
Example
• Engineer decide that reaction time should be investigated in the area of
the operating conditions
– time = 35 minutes (z1=30 or 40)
– temperature = 155 oF (z2=150 or 160)
• Will include center points at the current operating conditions
• Coded variables:
Example
• Design and responses:
Natural Variables
z1
z2
30
150
30
160
40
150
40
160
35
155
35
155
35
155
35
155
35
155
Coded Variables
x1
x2
-1
-1
-1
+1
+1
-1
+1
+1
0
0
0
0
0
0
0
0
0
0
Response
y
39.3
40.0
40.9
41.5
40.3
40.5
40.7
40.2
40.6
Curvature Check
• Have nf trials at the factorial design points (e.g., -1 and +1
combinations)
• Have nc trials at the center point
• Motivation for test:
• Test:
Example
• Fit regression line to data, include a quadratic term for curvature check
Source
X1
X2
X12
X1 X2
Residual
DF
1
1
1
1
4
Sum of Sq.
2.4025
0.4225
0.0027
0.0025
0.1720
Mean Sq.
2.4025
0.4225
0.0027
0.0025
0.0430
F
55.8721
9.8255
0.06331
0.05814
P-Value
0.0017
0.0350
0.8137
0.8213
Method of Steepest Ascent (climbing the hill)
• Linear effects for first order model estimated by least squares:
• Take partial derivative with respect to each variable:
• Direction of steepest ascent:
Method of Steepest Ascent
• Several experiment trials are taken along the line from the center point
of the design, in the direction of the steepest ascent until no further
increase is observed
• The location where the maximum has occurred is the center point of
the next first order design
• Design should have nc trials at the center point
• If curvature is detected, augment the design with additional trials so
that the second order model can be estimated
Method of Steepest Ascent