Lecture 15

• Today: Finish FFSP designs, 10.1-10.5

• Next day: 10.6-10.9

• Read 10.2-10.3!!!

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

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

Example: Leaf Spring Experiment (p. 438)

• 2 5-1 fractional factorial design was performed:

I=BCDE

• Experiment has 3 replicates

B Control Factors C D E

-1 +1 -1 -1 -1 +1 -1 -1 -1 -1 +1 +1 +1 -1 +1 -1 +1 +1 -1 -1 +1 +1 -1 +1 +1 +1 +1 -1 +1 -1 -1 +1 7.78

8.15

7.50

7.59

7.94

7.69

7.56

7.56

-1 7.78

8.18

7.56

7.56

8.00

8.09

7.62

7.81

Q Level

7.81

7.88

7.50

7.75

7.88

8.06

7.44

7.69

7.50

7.88

7.50

7.63

7.32

7.56

7.18

7.81

+1 7.25

7.88

7.56

7.75

7.44

7.69

7.18

7.50

7.12

7.44

7.50

7.56

7.44

7.62

7.25

7.59

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

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

Variance Reduction Via Parameter Design

• Suppose there is one noise factor and two control factors

y

   

x

1  

x

2

z

 

z

  • What is variance of y in practice?

• What does this imply?

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

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 N 1 runs in the control array and N 2 then the cross array has N 1 N 2 trials trials in the noise array,

Cross Array Strategy

• The responses are modeled using the location-dispersion approach • The models include ONLY the control factors location and dispersion

y i

ln(

s i

2 ) • Factors that impact the mean are called

location factors

impact the variance are

dispersion factors

and those that • Location factors that are not dispersion factors are called adjustment factors

Example: Leaf Spring

B Control Factors C D E

-1 +1 -1 +1 -1 +1 -1 +1 +1 +1 -1 -1 +1 +1 -1 -1 +1 +1 +1 +1 -1 -1 -1 -1 -1 +1 +1 -1 +1 -1 -1 +1

y i

7.540

7.902

7.520

7.640

7.670

7.785

7.372

7.660

ln  

i

-2.4075

-2.6488

-6.9486

-4.8384

-2.3987

-2.9392

-3.2697

-4.0582

Example: Leaf Spring

Effect

B C D E BC=DE BD=CE BE=CD

Estimate (Location)

0.22125

0.17625

0.02875

0.10375

0.01725

0.01975

-0.03525

Estimate (Dispersion)

0.1350

2.1802

-1.0444

-0.6499

-0.5259

0.7995

1.1852

Example: Leaf Spring

Qu a n t i l e s o f S ta n d a rd No rm a l

Example: Leaf Spring

• Location Model:  7 .

6360  0 .

1106

x B

 0 .

0881

x C

 0 .

0519

x E

• Dispersion Model:

z

ˆ   3 .

6886  1 .

0901

x C

• Level settings:

Example: Leaf Spring

• Location Model:  7 .

6360  0 .

1106

x B

 0 .

0881

x C

 0 .

0519

x E

• Dispersion Model:

z

ˆ   3 .

6886  1 .

0901

x C

• Level settings: