Transcript Ghada - American University in Cairo

The American University in Cairo
Interdisciplinary Engineering Program
ENGR 592: Probability & Statistics
Factorial &
Central Composite Designs
k
2
Presented to:
Dr. Lotfi K. Gaafar
592 Class
Presented by:
Ghada Moustafa Gad
Factorial Designs
Factorial Design
Full Factorial Design
2k Full Factorial Design
2k Fractional Factorial
Design
Mirror Image Fold over
Design
Allow the effect of
each and every
factor to be tested
and estimated
independently with
the interactions
also assessed.
Factorial Designs
Factorial Design
Full Factorial Design
2k Full Factorial Design
2k Fractional Factorial
Design
Mirror Image Fold over
Design
A factorial design in
which every setting
of every factor
appears with every
setting of every
other factor
Factorial Designs
Factorial Design
Full Factorial Design
2k Full Factorial Design
2k Fractional Factorial
Design
Mirror Image Fold over
Design
Designs having all
input factors set at
two levels each.
These levels are
called high/+1 and
low/-1
Factorial Designs
Factorial Design
Full Factorial Design
2k Full Factorial Design
2k Fractional Factorial
Design
Mirror Image Fold over
Design
Only an adequately
chosen fraction of the
treatment
combinations required
for the complete
factorial experiment
are selected to be run
Factorial Designs
Factorial Design
Full Factorial Design
2k Full Factorial Design
2k Fractional Factorial
Design
Mirror Image Fold over
Design
Factorial with the
number of runs in the
follow up experiment
equal to the original.
Fractional factorial
designs are augmented
by reversing the signs
of all the columns of
the original design
matrix
2k Full Factorial Design
# of runs required = 2 # of factors
# of Factors
# of Runs
2
4
3
8
4
16
5
32
6
64
7
128
2k Full Factorial Design
Standard Order Matrix 22
Trial
1
X1
-1
X2
-1
2
+1
-1
3
-1
+1
4
+1
+1
2k Full Factorial Design
Analysis Matrix 22
Dot product for any pair of columns is 0
Trial
1
I
+1
X1
-1
X2
-1
X1*X2
+1
2
+1
+1
-1
-1
3
+1
-1
+1
-1
4
+1
+1
+1
+1
Balanced Property
Fractional Factorial Design
23 = 8 runs
23-1 = 4 runs
½
space
Trial
X1
X2
X1X*X
3 2
1
-1
-1
+1
2
+1
-1
-1
3
-1
+1
-1
4
+1
+1
+1
Fractional Factorial Design
23 = 8 runs
23-1 = 4 runs
½
space
Trial
X1
X2
X1X*X
3 2
1
-1
-1
+1
2
+1
-1
-1
3
-1
+1
-1
4
+1
+1
+1
Fractional Factorial Design
Blocking Effect
Resolution
A schedule for conducting
runs of an experimental
study such that any effects
on the experimental results
due to a known change in
raw materials, operators,
etc. become concentrated
in the levels of the blocking
variable
Fractional Factorial Design
Blocking Effect
Resolution
It is the length of the smallest
interaction among the set
of defining relations. It
describes the degree to
which the estimated main
effects are confounded with
the estimated interactions.
Factorial Design Features
Ideal for screening design objective
Simple and economical for small number of
factors.
2k fractional factorial designs if properly
chosen to can be balanced and orthogonal.
Fractional Factorial designs has low number of
runs compared to high information obtained.
Most popular designs
Factorial Design Features
A two-level experiment can not fit quadratic effects
Case Example:
Fold-over Fractional Factorial Design
Set Objectives
Select Variables & Levels
The aim of the study is to find
the factors affecting the time
to peddle a bicycle up a hill.
Screening experiment.
Select Design
Evaluate Results
Case Example:
Fold-over Fractional Factorial Design
Set Objectives
Select Variables & Levels
Select Design
Evaluate Results
Case Example:
Fold-over Fractional Factorial Design
Set Objectives
Select Variables & Levels
7 factors  27= 128
Limitation  8 runs
Select Design
Evaluate Results
Case Example:
Fold-over Fractional Factorial Design
4
5
6
7
Resolution III
73
4
22
Case Example:
Fold-over Fractional Factorial Design
Set Objectives
Select Variables & Levels
Select Design
2 and 4 are significant.
4 confounded by 12 ?
1 & 14 could be significant?

Fold over design
Evaluate Results
Case Example:
Fold-over Fractional Factorial Design
4
5
6
Resolution III
Resolution IV
7
Central Composite Designs
CCD fall under the classical quadratic designs
category where fractional plan is used to fit a
second order equation
They start with a factorial or a fractional factorial
design (with center points) and then star points or
axial points are added to estimate curvature
Central Composite Designs
Rotatability
Most important criterion
Means that the standard error value of the points
located at same distance from the center of the region
is the same.
It is a measure of uncertainty of a predicted response
CCD Designs
Circumscribed
Central
Composite
Face Centered
Central
Composite
Inscribed
Central
Composite
CCD General Features
Most types are rotatable
Minimizes the error of prediction.
Good lack of fit detection.
Suitable for blocking.
Good graphical analysis through simple data
patterns.
Provides information on variable effects and
experimental error with minimum number of runs.
Sequential construction of higher order designs
from simpler designs to estimate curvature effects.
Case Example: CCD
Set Objectives
Select Variables & Levels
Select Design
Evaluate Results
The aim is to find the best
ratio of the two admixtures to
be used as a super plasticizer
for cement to obtain optimal
workability.
Response surface methodology
Case Example: CCD
Set Objectives
Select Variables & Levels
Select Design
Evaluate Results
W/C
0.33
0.35
% BL
0.12
0.18
% SNF
0.08
0.12
Case Example: CCD
Set Objectives
Select Variables & Levels
Select Design
Since RSM

High quality prediction
Larger process space

Circumscribed Central
Composite Design

Evaluate Results
Extremes generated are
reasonable =>O.K.
Case Example: CCC
Thank you…
Questions?