partition experimental designs for sequential process steps: application to product development...

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

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Example: Lens Finishing Processes

A company desires to improve their lens finishing process. Experimental runs must be limited due to cost concerns.

What type of design do you recommend?

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ManufacturingProcess #1

Controllable factors

Uncontrollable factors

Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .

Process One:Four Factors

Process Two:Six Factors

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

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Partition Design

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Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .

x1 x21 1-1 11 -1-1 -11 1

x3 x41 1-1 11 11 -1-1 -1

R1 R234.43 12.419.94 2.7514.695 32.35

-31.34 -18.513.37 -8.625

Design Matrix #1 Design Matrix #2 Responses

+ =

Partition Design: Assumptions

Process/Product Knowledge required Screening Experiment required Resources limited, minimize runs Sparsity-of-Effect Principle

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Partition Design: Methodology

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1. Perform Screening Experiment for Each Individual Process

2. Construct Partition Design3. Perform Partition Design

Experiment4. Perform Partition Design

Analysisa) Select Significant Effects for

Each Responseb) Build Empirical Model for

Each Responsec) Calculate Partition Interceptd) Select Significant Effects for

Intercept5. Build Final Empirical Model

Screening Experiments for each Process

Process 1

Process 2

Process n

Partition Design Analysis

Select Significant

Effects

Perform Partition Design

Experiment

Construction of

Partition Design

Build Empirical

Model

Calculate Partition Intercept

Select Significant

Effects

Build Final Empirical Model

Review: Experimental Objectives

Product/Process Characterization Determine which factors are most influential on the observed response. “Screening” Experiments Designs: 2k-p Fractional Factorial, Plackett-Burman Designs

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 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

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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)

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Step 1: Perform Screening Experiment Process 1: Significant Factors:

Factor A Factor B

Process 2: Significant Factors:

Factor C Factor D

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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

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Step 2: Construct Partition Design First-order Design (Res III or Saturated)

Orthogonal D-optimality Minimize Alias Confounding

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Step 1 Step 2 Step 3 Step 4

x1 x2 x3 x4 1 1 -1 1 1 -1

-1 -1

x1 x2 x3 x4 1 1 1 -1 1 -1 1 -1 -1

-1 -1 1

x1 x2 x3 x4 1 1 1 1 -1 1 -1 1 1 -1 -1 -1

-1 -1 1 -1

x1 x2 x3 X4 1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1

Step 2: Construct Partition Design

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Term AliasesModel A-A BD CD ABCModel B-B AB BC BD ABC ABD BCDError C-C AB AD BC BD ABC ABD BCDError D-D AB AC BCD

X'X= 5 1 1 1 1(5x5) 1 5 1 1 1

Det(X'X)= 1 1 5 -3 11024 1 1 -3 5 1

1 1 1 1 5 Est. AB AC AD BC BD CDInt 0.0 0.0 0.0 -1.0 0.0 0.0

inv(X'X)= 0.25 0.00 -0.13 -0.13 0.00 A 0.0 0.0 0.0 0.0 -1.0 1.0(5x5) 0.00 0.25 -0.13 -0.13 0.00 B 1.0 0.0 0.0 1.0 1.0 0.0

-0.13 -0.13 0.50 0.38 -0.13 C 1.0 0.0 1.0 1.0 1.0 0.0Det(X'X)= -0.13 -0.13 0.38 0.50 -0.13 D -1.0 1.0 0.0 0.0 0.0 0.00.000977 0.00 0.00 -0.13 -0.13 0.25

Alias Matrix=inv(X'X)X'Z

Step 3: Perform Partition Design Experiment Planning is key Requires increased coordination between

process steps Identification of Outputs and Inputs

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Run Order Std Order A B C D R1 R21 2 1 1 1 1 34.4 31.932 1 -1 1 -1 1 19.9 21.753 4 1 -1 1 1 4.7 11.724 5 -1 -1 1 -1 -31.3 1.8385 3 1 1 -1 -1 13.4 -27.64

Partition 2 ResponsesPartition 1

Step 4: Perform Partition Design AnalysisFor Each Response:

A. Select Significant Effects

B. Build Empirical Model

C. Calculate Partition Intercept Response

D. Select Significant Effects for Intercept Response

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Step 4a: Select Significant Effects

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Sum of Mean F p-valueSource Squares df Square Value Prob > FModel 2428.235 3 809.4116 27528.08 0.0044 A-A 246.0159 1 246.0159 8366.999 0.0070 B-B 1022.879 1 1022.879 34788.12 0.0034 D-D 515.6769 1 515.6769 17538.17 0.0048

Residual 0.029403 1 0.029403Cor Total 2428.264 4

R-Squared 0.999988Adj R-Squared 0.999952

Step 4a: Select Significant Effects

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STEP 4a - First PartitionSum of Mean F p-value

Source Squares df Square Value Prob > FModel 1912.56 2 956.28 3.71 0.2124 A-A 365.77 1 365.77 1.42 0.3558 B-B 1266.76 1 1266.76 4.91 0.157



Step 4b: Build Empirical Model

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Final Equation in Terms of Coded Factors:

R1 =3.158.85 * A

16.48 * B

Step 4c:Calculate Partition Intercept Response

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A B C D R1 Int11 1 1 1 34.4 9.101-1 1 -1 1 19.9 12.3181 -1 1 1 4.7 12.317-1 -1 1 -1 -31.3 -6.0111 1 -1 -1 13.4 -11.959

Calculations

Int1i = - 8.85A - 16.47B + y1i

for i= 1 to N

Run 1

Int1i = - 8.85A - 16.47B + y1i

Int11 = - 8.85(1) - 16.47(1) + 34.4

Int11 = 9.101

Step 4: Partition Analysis

Repeat for Second PartitionA. Select Significant Effects

B. Build Empirical Model

C. Calculate Partition Intercept Response

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A B C D R1 R2 Int1 Int21 1 1 1 34.4 31.93 9.101 9.298-1 1 -1 1 19.9 21.75 12.318 11.7941 -1 1 1 4.7 11.72 12.317 -10.912-1 -1 1 -1 -31.3 1.838 -6.011 11.7941 1 -1 -1 13.4 -27.64 -11.959 -5.008

Step 4d:Select Significant Effects for Intercept

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Sum of Mean F p-valueSource Squares df Square Value Prob > FModel 491.1211 1 491.1211 59.92901 0.0045 AC 491.1211 1 491.1211 59.92901 0.0045



Step 5:Build Final Empirical Model

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Final Equation in Terms of Coded Factors:

R1 = R2 =1.635938 1.9747.335938 A 4.919 C14.95844 B 14.875 D10.62094 AC 9.934 BD




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .

• 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

Case Study: Biogen IDEC




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .

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Protein-A



x 1 x 2 x k

z 1 z 2 z r

. . .

. . .

CIEX



x 1 x 2 x k

z 1 z 2 z r

. . .

. . .

pH 4.5 poolpH 5.75 poolpH 7 pool

20 Protein-A eluate pools

20 CEXeluate pools

Harvest



x 1 x 2 x k

z 1 z 2 z r

. . .

. . .

Case Study: Biogen IDECPartition Design: Experimental

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

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Partition Design: Designs

ExperimentHarvest

pH

Load Capacity

(%)

Wash I Conc. (mM)

Elution velocity (cm/hr)

1 5.75 75 2100 262.52 4.5 120 0 753 7 30 0 754 7 30 0 4505 4.5 30 0 756 4.5 30 4200 757 7 30 4200 758 4.5 30 4200 4509 7 120 4200 75

10 5.75 75 2100 262.511 5.75 75 2100 262.512 4.5 30 0 45013 7 120 0 7514 4.5 120 0 45015 7 120 0 45016 4.5 120 4200 7517 7 30 4200 45018 4.5 120 4200 45019 7 120 4200 45020 5.75 75 2100 262.5

Protein-A Chromatography Step

Mab Eluate from

Experiment #

Load Capacity

(%)

Wash volume

(CV)

Elution NaCl Conc. (mM)

Elution pH

1 70 3 155 5.52 110 2 185 63 30 4 185 64 110 2 185 55 30 2 125 56 110 4 185 57 110 2 125 68 30 2 185 69 30 2 185 5

10 70 3 155 5.511 70 3 155 5.512 110 4 125 613 110 4 125 514 30 4 185 515 30 2 125 616 30 4 125 617 30 4 125 518 110 2 125 519 110 4 185 620 70 3 155 5.5

Cation Chromatography Step

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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

Adjusted R2 0.95

Predicted R2 0.92

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


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

R2 0.99

Adjusted R2 0.99

Predicted R2 0.96

Traditional Model Results Partition Model Results

CIEX Step HCP ANOVA Comparison: Main Effects

• Partition model identified same significant main factors and their relative rank in significance

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Input Parameter% Sum

of Squares

Load HCP [f(A,C)] 83.3

CIEX Elution pH 6.6

Load HCP2 5.8

Load HCP * CIEX Elution pH 1.6

Elution [NaCl] 1.1

Elution pH2 0.7

Elution [NaCl] * CIEX Elution pH 0.5

Load HCP * CIEX Elution [NaCl] 0.2

SPXL Load Capacity (mg/ml) 0.1

Input Parameter% of

Total Sum of Squares

Harvest pH 32.6

Pro A Wash I Conc. 17.8

Harvest pH * Pro A Wash I conc 15.6

CIEX Elution pH 10.1

Harvest pH * CIEX Elution pH 8.8

Pro A Wash I. Conc.* CIEX Elution pH 4.6


ProA Wash 1. * CIEX Elution NaCl 1.8

Elution [NaCl] 1.5

Harvest pH * CIEX Elution [NaCl] 1.2

CIEX Elution [NaCl] * CIEX Elution pH

0.2

Traditional Model Results Partition Model Results

• Partition model able to identify interactions between process steps

CIEX Step HCP ANOVA Comparison: Interactions

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Summary of Partition Designs

Experimental design capable of handling a serial process Sequential process steps that possess several factors and multiple

responses

Potential Advantages 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




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .




Inputs

Outputs, y

x1 x2 xk

z1 z2 zr

. . .

. . .

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 p-efficient 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.

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