DEMA2008

Post-Fractionated Strip-Block Designs with Applications to Robust Design

and Multistage Processes

Carla A. VivacquaUniversidade Federal do Rio Grande do Norte (UFRN) - Brazil

vivacqua@ccet.ufrn.br

Søren BisgaardUniversity of Massachusetts Amherst (UMASS) – USA

University of Amsterdam – The Netherlands

bisgaard@som.umass.edu

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Outline

• Introduction:– Strip-block designs– Battery cells case study

• New Arrangement: Post-Fractionated Strip-Block Design

• Analysis• Conclusions

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Introduction

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

at low cost

• Six Sigma initiatives:– Design of experiments (DOE) plays a critical

role

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Research Question

• How to reduce costs of experimentation? – Robust Design

• Products insensitive to different sources of variation

– Multistage Processes

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Project Home

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The Problem

• High percentage of rejected batteries

• Annual losses of over $154,000

• 2 millions batteries scraped annually

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Customer Requirements

• High performance batteries

• Specification limits for the critical to quality issues:– Open Circuit Voltage (OCV) [1.00V, 1.38V]– Impedance [2Ω, 8Ω]

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Various Types of Batteries

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

2000/07 2000/08 2000/09 2000/10 2000/11 2000/12 2001/01

Hig

h O

CV

Rej

ects

($)

5A

10U

10SU

312U

312SU

13U

13SU

Total

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Battery Cells Case Study

Task 2

Task 1

Task n

Curing Process

End

Begin

AssemblyProcess

• Defective rate: 5%

• Cause of cells rejection: high OCV

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

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Objective

• Identify settings of process variables leading to high quality battery cells– Close to target– Least amount of variation

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Process Characteristics

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

– Assembly process: A, B, C, D– Curing 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 curing conditions– Data collection: 64 * 5 = 320 days

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(16)

(2)

(1)

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

Curing Variables (22)

Curing ConditionsAssembly Variables

(24)

Fully Randomized Arrangement

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Approach 2

EF

A B C D

Storage VariablesSub-plots

Assembly VariablesWhole-plots

} 22 full factorial design with 16 replicates

24 full factorial design

Requires 16 changes in assembly

configuration

Still requires 64 changes in the storage configuration

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(16)

(2)

(1)

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

Storage Variables (22 with 16 replicates)Storage ConditionsAssembly

Variables (24)

Run

Split-Plot Design

Whole Plot Sub-Plot

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Approach 3

EF

A B C D

Curing VariablesAssembly Variables

} 22 full factorial design

24 full factorial design16 trials

• Advantages:

– only 16 changes in the assembly configuration– only 4 changes in the curing configuration

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(16)

(2)

(1)

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

Curing Variables (22)Curing ConditionsAssembly

Variables (24)

Run

Strip-Block Design

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Strip-Block Experiment

Order E- F- E+ F- E- F+ E+ F+

11 - - - - X X X X15 + - - - X X X X5 - + - - X X X X3 + + - - X X X X9 - - + - X X X X13 + - + - X X X X2 - + + - X X X X12 + + + - X X X X8 - - - + X X X X4 + - - + X X X X1 - + - + X X X X6 + + - + X X X X14 - - + + X X X X7 + - + + X X X X10 - + + + X X X X16 + + + + X X X X

Run

Curing Conditions

Based on Factors E & F

Assembly Factors

A B C D

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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

Effect Estimate Effect Estimate Effect Estimate Effect 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 generator of post-fraction

k-p = 4 basic factors of row design

Remaining effects

q-r = 2 basic factors 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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Results

• Based on the analysis of the OCV mean only and taking into account that the problem is cells with high OCV the recommended levels would be:

A high level (+) B high level (+)

C low level (-) D low level (-)

F high level (+)

28DEMA2008

Results – cont.• Considering the OCV sub-lot variability

and other variables of interest, the recommended settings are:

A low level (-) B low level (-)

C low level (-) D low level (-)

E low level (-) F high level (+)

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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

• Catalogs of maximum resolution post-fractionated strip-block designs– 16-run and 32-run designs– Up to 11 factors

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Summary

• Strip-block experiments: – Reduction of experimentation costs– Easy to execute– Logically suitable to available resources

and restrictions

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Before vs. After Implementation

0%

1%

2%

3%

4%

5%

6%

7%

Battery Lot

Pe

rce

nt

Hig

h O

CV

Re

jects

New Stomper

80% reduction on defective rate and 75% reduction on process variability!!!!

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Questions?

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