Stat 470-18

1. Stat 470-18 • Today: Start Chapter 10

2. Additional Homework Question

3. Chapter 10:Robust Parameter Design • Robust parameter design is an experimentation technique which aims to reduce system variation and also optimize the mean system response • Idea is to use control factors to make the system robust to the influences of noise factors

4. Example: Leaf Spring Experiment (p. 438) • Experiment was conducted to investigate the impact of a heat treatment process on truck leaf springs where the target height of the springs is 8 inches • Experiment considered 5 factors, each at 2 levels: • B: High heat treatment • C: Heating time • D: Transfer time • E: Hold down time • Q: Quench oil temperature • In regular production Q is not controllable, but can be in the experiment

5. Example: Leaf Spring Experiment (p. 438) • 25-1 fractional factorial design was performed: I=BCDE • Experiment has 3 replicates

6. Noise Factors • Noise factors are factors that impact the system response, but in practice are not controllable • Examples include environmental factors, differing user conditions, variation in process parameter settings, … • Example: refrigerators are manufactured so that the interior temperature remains close to some target • Section 10.3 discusses different types of noise factors…please read

7. Variance Reduction Via Parameter Design • Let x denote the control factor settings and z denote the noise factor settings • Relationship between the system response and the factors: y = f(x,z) • If noise factors impact the response, then variation in the levels of z will transmit this variance to the response, y • If some noise and control factors interact, can potentially adjust levels of control factors to dampen impact of noise factor variation

8. Variance Reduction Via Parameter Design • Suppose there is one noise factor and two control factors • What is variance of y in practice? • What does this imply?

9. Cross Array Strategy • We will consider two types of design/analysis techniques for robust parameter design • The first one uses location-dispersion modeling (e.g., have a model for the mean response and another for the variance) similar to the epitaxial layer growth experiment in Chapter 3 • The design strategy for this technique is based on a cross array

10. Cross Array Strategy • Consider the leaf spring example • We can view this experiment as the combination of two separate experimental designs • Control array: design for the control factors • Noise array: design for the noise factors • Cross array: design consisting of all level combinations between the control array and the noise array • If there are N1 runs in the control array and N2 trials in the noise array, then the cross array has N1 N2 trials

11. Cross Array Strategy • Design for control factors: • Design for noise factors:

12. Cross Array Strategy • The responses are modeled using the location-dispersion approach • The models include ONLY the control factors • At each control factor setting, and are used as measures of location and dispersion • Factors that impact the mean are called location factors and those that impact the variance are dispersion factors • Location factors that are not dispersion factors are called adjustment factors

13. Example: Leaf Spring

14. Example: Leaf Spring

15. Example: Leaf Spring

16. Example: Leaf Spring • Location Model: • Dispersion Model: • Level settings:

17. Two-Step Optimization Procedures • Nominal the best problem: • Select the levels of the dispersion factors to minimize the dispersion • The select the levels of the adjustment factors to move the process on target • Larger (Smaller) the better problem: • Select levels of location factors to optimize process mean • Select levels of dispersion factors that are not location factors to minimize dispersion • Leaf Spring Example was a nominal the best problem

18. Response Modeling • There may be several noise factors and control factors in the experiment • The cross array approach identifies control factors to help adjust the dispersion and location models, but does not identify which noise factors interact with which control factors • Cannot deduce the relationships between control and noise factors • The response model approach explicitly model both control and noise factors in a single model (called the response model)

19. Response Modeling • Steps: • Model response, y, as a function of both noise and control factors (I.e., compute regression model with main effects and interactions of both types of factors) • To adjust variance: • make control by noise interaction plots for the significant control by noise interactions. The control factor setting that results in the flattest relationship gives the most robust setting. • construct the variance model, and choose control factor settings that minimize the variance

20. Example: Leaf Spring Experiment (p. 438) • 25-1 fractional factorial design was performed: I=BCDE • Experiment has 3 replicates

21. Example: Leaf Spring Experiment (p. 438) • 25-1 fractional factorial design was performed: I=BCDE

22. Example: Leaf Spring Experiment (p. 438)

23. Example: Leaf Spring Experiment (p. 438)

24. Example: Leaf Spring Experiment (p. 438) • Response Model:

25. Example: Leaf Spring Experiment (p. 438)

26. Example: Leaf Spring Experiment (p. 438) • Variance Model: