Partition Experimental Designs for
Sequential Process Steps:
Application to Product Development
Leonard Perry, Ph.D., MBB, CSSBB, CQE
Associate Professor & ISyE Program Chair
Industrial & Systems Engineering (ISyE)
University of San Diego
1
Example: Lens Finishing Processes

A company desires to improve their lens finishing process.
Experimental runs must be limited due to cost concerns.
Process One:
Four Factors
Process Two:
Six
Factors
Controllable factors
Controllable factors
x1
x2
x1
xk
x2
...
...
Inputs
Manufacturing
Process #1
Outputs, y
Inputs
Manufacturing
Process #2
z2
Outputs, y
...
...
z1
xk
zr
Uncontrollable factors
z1
z2
zr
Uncontrollable factors
What type of design do you recommend?
2
Objective of Partition Designs


To create a experimental design capable of
handling a serial process consisting of
multiple sequential processes that possess
several factors and multiple responses.
Advantages:



Output from first process may be difficult to
measure.
Potential interaction between sequential
processes
Reduction of experimental runs
3
Partition Design
Controllable factors
Controllable factors
x1
x1
x2
xk
x2
...
...
Inputs
Manufacturing
Process #1
Outputs, y
Inputs
Manufacturing
Process #2
z2
z1
zr
Design Matrix #1
x2
1
1
-1
-1
1
z2
zr
Uncontrollable factors
Uncontrollable factors
x1
1
-1
1
-1
1
Outputs, y
...
...
z1
xk
Design Matrix #2
+
x3
1
-1
1
1
-1
x4
1
1
1
-1
-1
Responses
=
R1
34.43
19.94
4.695
-31.34
13.37
R2
12.4
2.751
32.35
-18.5
-8.625
4
Partition Design: Assumptions




Process/Product Knowledge required
Screening Experiment required
Resources limited, minimize runs
Sparsity-of-Effect Principle
5
Partition Design: Methodology
1.
2.
3.
4.
Perform Screening
Experiment for Each
Individual Process
Construct Partition Design
Perform Partition Design
Experiment
Perform Partition Design
Analysis
a)
b)
c)
d)
5.
Select Significant Effects for
Each Response
Build Empirical Model for
Each Response
Calculate Partition Intercept
Select Significant Effects for
Intercept
Build Final Empirical Model
6
Review: Experimental Objectives

Product/Process Characterization




Product/Process Improvement




Find the setting for factors that create a desired output or response
Determine model equation to relate factors and observed response
Designs: 2k Factorial, 2k Factorial with Center Points
Product/Process Optimization



Determine which factors are most influential on the observed response.
“Screening” Experiments
Designs: 2k-p Fractional Factorial, Plackett-Burman Designs
Determine an operating or design region in which the important factors
lead to the best possible response. (Response Surface)
Designs: Central Composite Designs, Box-Behnken Designs, D-optimal
Product/Process Robustness


Explore settings that minimize the effects of uncontrollable factors
Designs: Taguchi Experiments
7
Example: First-order Partition Design

Two factors significant in each process



Total of k = 4 factors
Potential Interaction between processes
Partition Design

N = 5 runs (N = k - 1) (Saturated Design)
8
Step 1:
Perform Screening Experiment


Process 1:
Significant Factors:


Factor A
Factor B


Process 2:
Significant Factors:


Factor C
Factor D
9
Step 2:
Construct Partition Design

Partition Design: Design Criteria

First-order models




Orthogonal
D-optimal
Minimize Alias Confounding
Second-order models



D-efficiency
G-efficiency
Minimize Alias Confounding
10
Step 2:
Construct Partition Design

First-order Design (Res III or Saturated)



Orthogonal
D-optimality
Minimize Alias Confounding
Step 1
x1
1
-1
1
-1
x2 x3 x4
1
1
-1
-1
Step 2
x1
1
-1
1
-1
x2
1
1
-1
-1
x3 x4
1
-1
-1
1
Step 3
x1
1
-1
1
-1
x2
1
1
-1
-1
x3
1
-1
-1
1
x4
1
1
-1
-1
Step 4
x1
1
-1
1
-1
-1
x2
1
1
-1
-1
-1
x3
1
-1
-1
1
1
X4
1
1
-1
-1
1
11
Step 2:
Construct Partition Design
Model
Model
Error
Error
X'X=
(5x5)
Det(X'X)=
1024
5
1
1
1
1
inv(X'X)= 0.25
(5x5) 0.00
-0.13
Det(X'X)= -0.13
0.000977 0.00
Term
A-A
B-B
C-C
D-D
Aliases
BD CD ABC
AB BC BD ABC ABD BCD
AB AD BC BD ABC ABD BCD
AB AC BCD
1
5
1
1
1
1
1
5
-3
1
1
1
-3
5
1
1
1
1
1
5
0.00
0.25
-0.13
-0.13
0.00
-0.13
-0.13
0.50
0.38
-0.13
-0.13
-0.13
0.38
0.50
-0.13
0.00
0.00
-0.13
-0.13
0.25
Alias Matrix=inv(X'X)X'Z
Est. AB AC AD BC BD CD
Int
0.0 0.0 0.0 -1.0 0.0 0.0
A
0.0 0.0 0.0 0.0 -1.0 1.0
B
1.0 0.0 0.0 1.0 1.0 0.0
C
1.0 0.0 1.0 1.0 1.0 0.0
D
-1.0 1.0 0.0 0.0 0.0 0.0
12
Step 3:
Perform Partition Design Experiment



Planning is key
Requires increased coordination between
process steps
Identification of Outputs and Inputs
Run Order Std Order
1
2
2
1
3
4
4
5
5
3
Partition
A
1
-1
1
-1
1
1
B
1
1
-1
-1
1
Partition
C
1
-1
1
1
-1
2
D
1
1
1
-1
-1
Responses
R1
R2
34.4
31.93
19.9
21.75
4.7
11.72
-31.3
1.838
13.4
-27.64
13
Step 4:
Perform Partition Design Analysis
For Each Response:
A. Select Significant Effects
B. Build Empirical Model
C. Calculate Partition Intercept Response
D. Select Significant Effects for Intercept
Response
14
Step 4a:
Select Significant Effects
Source
Model
A-A
B-B
D-D
Residual
Cor Total
R-Squared
Adj R-Squared
Sum of
Squares
2428.235
246.0159
1022.879
515.6769
0.029403
2428.264
df
3
1
1
1
1
4
Mean
Square
809.4116
246.0159
1022.879
515.6769
0.029403
F
p-value
Value
Prob > F
27528.08 0.0044
8366.999 0.0070
34788.12 0.0034
17538.17 0.0048
0.999988
0.999952
15
Step 4a:
Select Significant Effects
STEP 4a - First Partition
Sum of
Source
Squares
Model
1912.56
A-A
365.77
B-B
1266.76
Residual
515.71
Cor Total
2428.26
R-Squared
Adj R-Squared
df
2
1
1
2
4
Mean
Square
956.28
365.77
1266.76
257.85
F
Value
3.71
1.42
4.91
p-value
Prob > F
0.2124
0.3558
0.157
0.7876
0.5752
16
Step 4b:
Build Empirical Model
Final Equation in Terms of Coded Fact
R1 =
3.15
8.85 * A
16.48 * B
17
Step 4c:
Calculate Partition Intercept Response
Calculations
Run 1
Int1i = - 8.85A - 16.47B + y1i
Int1i = - 8.85A - 16.47B + y1i
for i= 1 to N
Int11 = - 8.85(1) - 16.47(1) + 34.4
Int11 = 9.101
A
1
-1
1
-1
1
B
1
1
-1
-1
1
C
1
-1
1
1
-1
D
1
1
1
-1
-1
R1
34.4
19.9
4.7
-31.3
13.4
Int1
9.101
12.318
12.317
-6.011
-11.959
18
Step 4: Partition Analysis

Repeat for Second Partition
A.
B.
C.
A
1
-1
1
-1
1
Select Significant Effects
Build Empirical Model
Calculate Partition Intercept Response
B
1
1
-1
-1
1
C
1
-1
1
1
-1
D
1
1
1
-1
-1
R1
34.4
19.9
4.7
-31.3
13.4
R2
31.93
21.75
11.72
1.838
-27.64
Int1
9.101
12.318
12.317
-6.011
-11.959
Int2
9.298
11.794
-10.912
11.794
-5.008
19
Step 4d:
Select Significant Effects for Intercept
Source
Model
AC
Residual
Cor Total
R-Squared
Adj R-Squared
Sum of
Squares
491.1211
491.1211
24.58514
515.7063
df
1
1
3
4
Mean
F
p-value
Square
Value
Prob > F
491.1211 59.92901 0.0045
491.1211 59.92901 0.0045
8.195048
0.9523
0.9364
20
Step 5:
Build Final Empirical Model
Final Equation in Terms of Coded Factors:
R1
1.635938
7.335938
14.95844
10.62094
=
A
B
AC
R2
1.974
4.919
14.875
9.934
=
C
D
BD
21
Case Study: Biogen IDEC
• Q8 Design Space
– Link input parameters with quality attributes over broad range
• Traditional Design of Experiments (DOE)
– Systematic approach to study effects of multiple factors on process
performance
– Limitation: not applied to multiple sequential process steps; does not
account for the effects of upstream process factors on downstream process
outputs
Controllable factors
Controllable factors
x1
x1
xk
x2
...
...
Manufacturing
Process #1
Manufacturing
Process #2
Outputs, y
...
...
z2
Outputs, y
Inputs
Inputs
z1
xk
x2
zr
Uncontrollable factors
z1
z2
zr
Uncontrollable factors
22
Case Study: Biogen IDEC
Partition Design: Experimental
Controllable factors
x1
x2
Controllable factors
xk
...
Harvest
pH 4.5 pool
pH 5.75 pool
pH 7 pool
x1
x2
z2
xk
...
x2 x1
20 Protein-A
eluate pools
Protein-A
...
z1
Controllable factors
Uncontrollable factors
z1
z2
20 CEX
eluate pools
...
CIEX
...
zr
xk
...
zr
Uncontrollable factors
z2 z1
zr
Uncontrollable factors

Resolution IV: 1/16 fractional factorial for whole design

Each partition: full factorial

Harvest pH included in Protein A partition

Each column: 16 expts + 4 center points = 20 expts
23
Partition Design: Designs
Protein-A Chromatography Step
Experiment
Harvest
pH
Load
Capacity
(%)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
5.75
4.5
7
7
4.5
4.5
7
4.5
7
5.75
5.75
4.5
7
4.5
7
4.5
7
4.5
7
5.75
75
120
30
30
30
30
30
30
120
75
75
30
120
120
120
120
30
120
120
75
Wash I
Conc.
(mM)
2100
0
0
0
0
4200
4200
4200
4200
2100
2100
0
0
0
0
4200
4200
4200
4200
2100
Elution
velocity
(cm/hr)
Mab Eluate
from
Experiment #
262.5
75
75
450
75
75
75
450
75
262.5
262.5
450
75
450
450
75
450
450
450
262.5
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Cation Chromatography Step
Elution
Load
Wash
NaCl
Capacity
volume
Conc.
(%)
(CV)
(mM)
70
3
155
110
2
185
30
4
185
110
2
185
30
2
125
110
4
185
110
2
125
30
2
185
30
2
185
70
3
155
70
3
155
110
4
125
110
4
125
30
4
185
30
2
125
30
4
125
30
4
125
110
2
125
110
4
185
70
3
155
Elution
pH
5.5
6
6
5
5
5
6
6
5
5.5
5.5
6
5
5
6
6
5
5
6
5.5
24
CIEX Step HCP ANOVA Comparison: Main Effects
Partition Model Results
Traditional Model Results
Input Parameter
% of Total
Sum of
Squares
Load HCP
[f(Harvest pH, ProA Wash I)]
83.3
CIEX Elution pH
6.6
Load
HCP2
5.8
Load HCP * Elution pH
1.6
CIEX Elution [NaCl]
1.1
CIEX Elution pH2
0.7
CIEX Elution [NaCl] * CIEX Elution pH
0.5
Load HCP * CIEX Elution [NaCl]
0.2
CIEX Load Capacity
0.1
R2
0.96
Input Parameter
% of Total
Sum of
Squares
Harvest pH
32.6
Pro A Wash I Conc.
17.7
Harvest pH * Pro A Wash I
15.6
CIEX Elution pH
10.1
Harvest pH * CIEX Elution pH
8.8
Pro A Wash I. * CIEX Elution pH
4.6
CIEX Load Capacity
3.9
Pro A Wash I. Conc. * CIEX Elution NaCl
1.8
CIEX Elution [NaCl]
1.5
Harvest pH * CIEX Elution [NaCl]
1.2
CIEX Elution [NaCl] * CIEX Elution pH
0.2
Adjusted
R2
0.95
R2
0.99
Predicted
R2
0.92
Adjusted R2
0.99
Predicted R2
0.96
• Partition model identified same significant main factors and their relative rank
in
25 significance
CIEX Step HCP ANOVA Comparison: Interactions
Partition Model Results
Traditional Model Results
Input Parameter
% Sum
of
Squares
Input Parameter
% of Total
Sum of
Squares
Load HCP [f(A,C)]
83.3
Harvest pH
32.6
CIEX Elution pH
6.6
Pro A Wash I Conc.
17.8
Load HCP2
5.8
Harvest pH * Pro A Wash I conc
15.6
Load HCP * CIEX Elution pH
1.6
CIEX Elution pH
10.1
Elution [NaCl]
1.1
Harvest pH * CIEX Elution pH
8.8
Elution pH2
0.7
Pro A Wash I. Conc.* CIEX Elution pH
4.6
Elution [NaCl] * CIEX Elution pH
0.5
CIEX Load Capacity
3.9
Load HCP * CIEX Elution [NaCl]
0.2
ProA Wash 1. * CIEX Elution NaCl
1.8
SPXL Load Capacity (mg/ml)
0.1
Elution [NaCl]
1.5
Harvest pH * CIEX Elution [NaCl]
1.2
CIEX Elution [NaCl] * CIEX Elution pH
0.2
• Partition model able to identify interactions between process steps
26
Summary of Partition Designs
Controllable factors
Controllable factors
Controllable factors
x1
x1
x1
x2
xk
x2
Manufacturing
Process #1
Outputs, y
Inputs
z2
Outputs, y
Inputs
Manufacturing
Process #3
...
zr
Uncontrollable factors
z1
z2
xk
...
Manufacturing
Process #2
...
z1
x2
...
...
Inputs
xk
Outputs, y
...
zr
Uncontrollable factors
z1
z2
zr
Uncontrollable factors
Experimental design capable of handling a serial process


Sequential process steps that possess several factors and multiple
responses
Potential Advantages





27
Links process steps together: identify upstream operation effects
and interactions to downstream processes.
Better understanding of the overall process
Potentially less experiments
No manipulation of uncontrollable parameters necessary
References








D. E. Coleman and D. C. Montgomery (1993), ‘Systematic Approach to Planning
for a Designed Industrial Experiment’, Technometrics, 35, 1-27.
Lin, D.J.K. (1993). "Another Look at First-Order Saturated Designs: The pefficient Designs," Technometrics, 35: (3), p284-292.
Montgomery, D.C., Borror, C.M. and Stanley, J.D., (1997). “Some Cautions in
the Use of Plackett-Burman Designs,” Quality Engineering, 10, 371-381.
Box, G. E. P. and Draper, N. R. (1987) Empirical Model Building and Response
Surfaces, John Wiley, New York, NY
Box, G. E. P. and Wilson, K. B. (1951), “On the Experimental Attainment of
Optimal Conditions,” Journal of the Royal Statistical Society, 13, 1-45.
Hartley, H. O. (1959), “Smallest composite design for quadratic response
surfaces,” Biometrics 15, 611-624.
Khuri, A. I. (1988), “A Measure of Rotatability for Response Surface Designs,”
Technometrics, 30, 95-104.
Perry, L. A., Montgomery, and D. C, Fowler, J. W., " Partition Experimental
Designs for Sequential Processes: Part I - First Order Models ", Quality and
Reliability Engineering International, 18,1.
28