1 Post-Fractionated Strip-Block Designs: A Tool for Robustness Applications and Multistage Processes Carla A. Vivacqua 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 Task 2 Task 1 Task n Storage Process End Begin Assembly Process 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.

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 2 6 = 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 Assembly VariablesStorage Variables A B C D EFEF } 2 2 full factorial design 2 4 full factorial design 16 trials Advantages: – only 16 changes in the assembly configuration – only 4 changes in the storage configuration

9 Run Assembly Variables (2 4 ) Storage Variables (2 2 ) Storage Conditions (1)(2)(3)(4) (1) (2) (16) 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: 2 k-p x 2 q-r Maximum value for post-fractionation order to preserve the strip-block structure: f = min(k-p, q-r) - 1. Ex.: 2 4 x 2 2 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 Contrast Estimates 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

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%