Modeling and Multivariate Methods - SAS

606 Visualizing, Optimizing, and Simulating Response Surfaces Chapter 24
The Simulator
Now, the defect rate is down to about 0.004, much improved. Further reduction in the defect rate can occur
by continued investigation of the parametric profiles, making changes to the distributions, and rerunning
the simulations.
As the defect rate is decreased further, the mean defect rates across the factors become relatively less reliable.
The accuracy could be improved by reducing the ranges of the factors in the Profiler a little so that it
integrates the distributions better.
This level of fine-tuning is probably not practical, because the experiment that estimated the response
surface is probably not at this high level of accuracy. Once the ranges have been refined, you may need to
conduct another experiment focusing on the area that you know is closer to the optimum.
Stochastic Optimization Example
This example is adapted from Box and Draper (1987) and uses Stochastic Optimization.jmp. A chemical
reaction converts chemical “A” into chemical “B”. The resulting amount of chemical “B” is a function of
reaction time and reaction temperature. A longer time and hotter temperature result in a greater amount of
“B”. But, a longer time and hotter temperature also result in some of chemical “B” getting converted to a
third chemical “C”. What reaction time and reaction temperature will maximize the resulting amount of
“B” and minimize the amount of “A” and “C”? Should the reaction be fast and hot, or slow and cool?
Figure 24.57 Chemical Reaction
The goal is to maximize the resulting amount of chemical “B”. One approach is to conduct an experiment
and fit a response surface model for reaction yield (amount of chemical “B”) as a function of time and
temperature. But, due to well known chemical reaction models, based on the Arrhenius laws, the reaction
yield can be directly computed. The column Yield contains the formula for yield. The formula is a function
of Time (hours) and reaction rates k1 and k2. The reaction rates are a function of reaction Temperature
(degrees Kelvin) and known physical constants θ 1 , θ 2 , θ 3 , θ 4 . Therefore, Yield is a function of Time and
Temperature.
Figure 24.58 Formula for Yield
You can use the Profiler to find the maximum Yield. Open Stochastic Optimization.jmp and run the
attached script called Profiler. Profiles of the response are generated as follows.