Lecture 16

• Today: 10.6-10.9

• Next day:

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

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)

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

Example: Leaf Spring Experiment (p. 438)

• 25-1 fractional factorial design was performed: I=BCDE

• Experiment has 3 replicates

Control Factors Q LevelB C D E -1 +1-1 -1 -1 -1 7.78 7.78 7.81 7.50 7.25 7.12+1 -1 -1 +1 8.15 8.18 7.88 7.88 7.88 7.44-1 +1 -1 +1 7.50 7.56 7.50 7.50 7.56 7.50+1 +1 -1 -1 7.59 7.56 7.75 7.63 7.75 7.56-1 -1 +1 +1 7.94 8.00 7.88 7.32 7.44 7.44+1 -1 +1 -1 7.69 8.09 8.06 7.56 7.69 7.62-1 +1 +1 -1 7.56 7.62 7.44 7.18 7.18 7.25+1 +1 +1 +1 7.56 7.81 7.69 7.81 7.50 7.59

Example: Leaf Spring Experiment (p. 438)

• 25-1 fractional factorial design was performed: I=BCDEB C D E Q Y

-1 -1 -1 -1 -1 7 .7 91 -1 -1 1 -1 8 .0 7

-1 1 -1 1 -1 7 .5 21 1 -1 -1 -1 7 .6 3

-1 -1 1 1 -1 7 .9 41 -1 1 -1 -1 7 .9 5

-1 1 1 -1 -1 7 .5 41 1 1 1 -1 7 .6 9

-1 -1 -1 -1 1 7 .2 91 -1 -1 1 1 7 .7 3

-1 1 -1 1 1 7 .5 21 1 -1 -1 1 7 .6 5

-1 -1 1 1 1 7 .4 01 -1 1 -1 1 7 .6 2

-1 1 1 -1 1 7 .2 01 1 1 1 1 7 .6 3

Example: Leaf Spring Experiment (p. 438)

Effect EstimateB 0.221C -0.176D -0.029E 0.104Q -0.260BC=DE -0.017BD=CE -0.020BE=CD -0.035BQ 0.085CQ 0.165DQ -0.054EQ 0.027BCQ=DEQ -0.010BDQ=CEQ 0.040BEQ=CDQ -0.047

Example: Leaf Spring Experiment (p. 438)

Quantiles of Standard Normal

Effe

ct E

stim

ate

s

-1 0 1

-0.2

-0.1

0.0

0.1

0.2

Example: Leaf Spring Experiment (p. 438)

• Response Model:

Example: Leaf Spring Experiment (p. 438)

Q

me

an

of Y

7.5

7.6

7.7

7.8

7.9

-1 1

C

-11

Example: Leaf Spring Experiment (p. 438)

• Variance Model:

Design Strategy for the Response Model