Carla A. Vivacqua vivacqua@cae.wisc University of Wisconsin-Madison

1. Post-Fractionated Strip-Block Designs: A Tool for Robustness Applications and Multistage Processes Carla A. Vivacqua vivacqua@cae.wisc.edu University of Wisconsin-Madison Federal University of Rio Grande do Norte-Brazil Søren Bisgaard University of Massachusetts-Amherst Harold J. Steudel University of Wisconsin-Madison

2. Outline • Motivation • Research Question • Battery Cells Case Study • New Arrangement: Post-Fractionated Strip-Block Designs • Conclusions

3. 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 Begin Task 1 • 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. Task 2 Assembly Process Task n Storage Process End

6. 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

7. 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

8. Approach 2 } 22 full factorial design • Advantages: • only 16 changes in the assembly configuration • only 4 changes in the storage configuration 24 full factorial design 16 trials

9. Strip-Block Design

10. Scenario • Space restrictions in storage room • Only 8 sub-lots can be placed in the storage room simultaneously

11. State-of-the-Art Approach – Use of Fractional Factorials Generator: D = ABC Resolution IV design

12. New Approach: Post-Fractionated Strip-Block Design Generator: EF = ABCD Resolution VI design

13. Post-Fractionated Strip-Block Design (2) Generators: E = ABC, F = BCD Reduces to a split-plot design

14. 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

15. 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

16. q-r = 2 basic generators of column design k-p = 4 basic generators of row design Remaining Contrasts Contrast Estimates f = 1 basic generator of post-fraction

17. Variances

18. Conclusions • Post-fractionated strip-block designs • Cost-effective method to gather knowledge about products and processes • Attention to conduct appropriate analysis

19. Before vs. After Implementation New percentage of rejects  0.92% Improvement of 82%