1. Chalmers University of Technology Split-plot designs Martin Arvidsson

2. Chalmers University of Technology Improvement work performed at Cochlear BAS

3. Chalmers University of Technology Two important inventions contribute to the successfulness of the hearing aid device The titanium implant The vibrator

4. Chalmers University of Technology A simple test performed at Cochlear BAS to evaluate a new supplier of components The objective of the test was to evaluate whether washers from a new supplier could be used Altogether 120 vibrators where produced, 60 with washers ordinary used and 60 with washers from a new potential supplier The order in which the 120 vibrators was produced was randomised

5. Chalmers University of Technology Details of the improvement work The objective of the project is to improve the production yield of the vibrators The vibrator is made up by a rather large number of components The assembly process of vibrators include a rather large number of operations The assembly process requires that measurement equipment work satisfactory

6. Chalmers University of Technology Individual value plot

7. Chalmers University of Technology Individual value plot – two outliers removed

8. Chalmers University of Technology Time series plot to investigate whether the process was stable during the test

9. Chalmers University of Technology Histogram of the”populations”

10. Chalmers University of Technology Complete randomisation Randomisation of run order Resetting of all factor levels between each experiment

11. Chalmers University of Technology Randomizing Problem: Systematic dependence between the experiments. Solution: Make the experiments in random order. orderExp. nr ABCY 8 5 1 2 4 7 6 3 1 2 3 4 5 6 7 8 - + - + - + - + - - + + - - + + - - - - + + + + 53.8 51.8 47.4 47.8 50.6 51.8 48.2 48.6

12. Chalmers University of Technology Resetting of factor levels

13. Chalmers University of Technology If factors are not reset between each experiment, contrasts will have unequal variance! Responses are not independent!

14. Chalmers University of Technology Four different process conditions Eight batches of raw material Split-plot designs: A Composite Material Example Manufacturing process of composite material y – bending strengthresponse variable A – curing temperature B – pressure C – holding time control factors (process variables) D – proportion of hardener E – thermo-plastic content F – proportion of epoxy G – material ageing H – process type noise factors y = f (A,B,C,D,E,F,G,H) ?

15. Chalmers University of Technology Experimental design Process variables (control factors) A Curing temperature B Pressure C Holding time Incoming material (noise factors) D Proportion of hardener E Thermo-plastic content F Proportion of epoxy G Material aging H Type of process Process Product

16. Chalmers University of Technology Confounding pattern

17. Chalmers University of Technology Contrasts!

18. Chalmers University of Technology Analysis of the experiment G contrasts B BG

19. Chalmers University of Technology Confounding pattern

20. Chalmers University of Technology Error structure of a Strip-Block Experiment εsεs ε s1 ε s2 ε w1 ε w2 ε w3 ε w4 εwεw ε 32 ε1ε1 ε

21. Chalmers University of Technology

22. Variances of the contrasts

23. Chalmers University of Technology Identification of location effects B, G and BG was determined to be active based on engineering knowledge and the normal plots Process factors Factors and interactions associated with incoming material Interactions between ”process factors” and ”incoming material factors”

24. Chalmers University of Technology Model B ≈ 1.4

25. Chalmers University of Technology Conclusions The storage time of the incoming material (G) is causing variation in the bending strength of the composite material. If the pressure (B) is set at high level the bending strength is made insensitive to the storage time.

26. Chalmers University of Technology Randomisation and split-plot View randomisation as an insurance against unknown factors - buy as much as you can afford It is not always advisable to reset all factor levels between each experiment! –Can be very time consuming and expensive –Split-plot designs allow some contrasts of interest to be estimated with great precision. This characteristic can, for example, be useful in robust design experiments