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Post-Fractionated Strip-Block Designs: A Tool for Robustness Applications

and Multistage Processes

Carla A. Vivacquavivacqua@cae.wisc.edu

University of Wisconsin-Madison

Federal University of Rio Grande do Norte-Brazil

Søren BisgaardUniversity of Massachusetts-Amherst

Harold J. SteudelUniversity of Wisconsin-Madison

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Outline

• Motivation

• Research Question

• Battery Cells Case Study

• New Arrangement: Post-Fractionated Strip-Block Designs

• Conclusions

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Motivation

• Competitive environment requires:– Design of high-quality products and processes at

low cost

• Design of experiments (DOE) plays a critical role

4

Research Question

• How to reduce costs of experimentation? – Robust Design

• Products insensitive to different sources of variation

– Multistage Processes

5

Battery Cells Case Study

Task 2

Task 1

Task n

Storage Process

End

Begin

AssemblyProcess

• Defective rate: 5%

• Cause of cells rejection: high open circuit voltage (OCV)

• Consequences of high OCV: self-discharging, leading to low performance or dead cells.

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Process Characteristics

• Two shifts for production• One storage room• Storage cycle: at least five days• Six factors for investigation

– Assembly process: A, B, C, D– Storage process: E, F

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Approach 1

• Completely randomized design

• 26 = 64 independent trials

• 64 changes in assembly configuration– Could not be run in one shift

• 64 changes in storage conditions– Data collection: 64 * 5 = 320 days

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Approach 2

Assembly Variables Storage Variables

A B C DE

F} 22 full factorial design

24 full factorial design

16 trials

• Advantages: – only 16 changes in the assembly configuration– only 4 changes in the storage configuration

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RunAssembly

Variables (24)

Storage Variables (22)

Storage Conditions

(1) (2) (3) (4)

(1)

(2)

(16)

Strip-Block Design

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Scenario

• Space restrictions in storage room

• Only 8 sub-lots can be placed in the storage room simultaneously

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State-of-the-Art Approach – Use of Fractional Factorials

- + - + E

A B C D ABCD - - + + F

- - - - + X X X X

+ + - - + X X X X

+ - + - + X X X X

- + + - + X X X X

+ - - + + X X X X

- + - + + X X X X

- - + + + X X X X

+ + + + + X X X X

Row Design

Column Design

Generator: D = ABCResolution IV design

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New Approach: Post-Fractionated Strip-Block Design

- + - + E- - + + F

A B C D ABCD + - - + EF- - - - + X X+ - - - - X X- + - - - X X+ + - - + X X- - + - - X X+ - + - + X X- + + - + X X+ + + - - X X- - - + - X X+ - - + + X X- + - + + X X+ + - + - X X- - + + + X X+ - + + - X X- + + + - X X+ + + + + X X

Row Design

Column Design

Generator: EF = ABCDResolution VI design

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Post-Fractionated Strip-Block Design (2)

- + - + E

A B C D ABC BCD - - + + F

- - - - - - X

+ - - - + - X

- + - - + + X

+ + - - - + X

- - + - + + X

+ - + - - + X

- + + - - - X

+ + + - + - X

- - - + - + X

+ - - + + + X

- + - + + - X

+ + - + - - X

- - + + + - X

+ - + + - - X

- + + + - + X

+ + + + + + X

Row Design

Column Design

Generators: E = ABC, F = BCDReduces to a split-plot design

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Maximum Post-Fractionation Order

• Base strip-block design: 2k-p x 2q-r

• Maximum value for post-fractionation order to preserve the strip-block structure:

f = min(k-p, q-r) - 1.

Ex.: 24 x 22 base design

f = min(4, 2) – 1 = 2 – 1 = 1

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Analysis of Post-Fractionated Strip-Block Designs

• Compute main effects and interactions

• Not all effects with same precision

• Group effects with same variance

• Separate analyses for each stratum

• Four different strata

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Contrast Estimates

Contrast Estimate Contrast Estimate Contrast Estimate Contrast EstimateA -0.00331 E 0.00456 AE -0.00331 EF= ABCD -0.00181

B -0.00169 F -0.03056 AF 0.00131

C 0.00456 BE 0.00331

D 0.00656 BF 0.00244

AB -0.00381 CE -0.00219

AC 0.00369 CF -0.00231

AD -0.00156 DE 0.00256

BC 0.00006 DF 0.00369

BD -0.00294 ABE -0.00031

CD 0.00081 ABF -0.00244

ABC -0.00206 ACE 0.00244

ABD 0.00069 ACF 0.00006

ACD -0.00031 ADE -0.00156

BCD= AEF 0.00006 ADF -0.00169

Row Stratum Column Stratum Interaction Stratum Post-Fraction Stratum

f = 1 basic generator of post-fraction

k-p = 4 basic generators of

row design

Remaining Contrasts

q-r = 2 basic generators of column design

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Variances

2224

)ˆ( RCRfrq

NRVar

2224

)ˆ( RCCfpk

NCVar

2^ 4

)( RCNRCVar

222 224

)ˆ( RCCfpk

Rfrq

NFVar

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Conclusions

• Post-fractionated strip-block designs

– Cost-effective method to gather knowledge about products and processes

– Attention to conduct appropriate analysis

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Before vs. After Implementation

0%

1%

2%

3%

4%

5%

6%

7%

Battery Lot

Perc

en

t H

igh

OC

V R

eje

cts

New Stomper

New percentage of rejects 0.92%Improvement of 82%