This document discusses selecting variables and designing experiments for response surface modeling. It recommends beginning with identifying all possible factors, then controlling some and selecting others to experiment with using a design with three levels per variable, like a Box-Behnken design. Response surface modeling fits a quadratic surface to the experimental results rather than optimizing one variable at a time, which can miss the true optimal conditions. The goal is to approximate the maximum or minimum response across variable levels through designed experiments and modeling rather than sequential optimization of individual factors.

1. Selecting experimental variables
for response surface modeling
Special topics in Industrial Chemistry
Seppo Karrila
November 2014

2. Executive summary
• When you are given a problem
– Try to list all possible effects
– Which effects can you eliminate?
– Which can you control?
• For “experimental optimization”
– Typically use three levels for each factor
– Use an experimental design, like Box-Behnken, and fit
a “response surface model”
– Do not try to optimize one factor at a time, it is wrong

3. Our “model problem”
• What is the gas consumption of a car?
– How can you compare between different cars?
• Once you have a car, how should you drive it
for best gas economy?

4. Things to note
• You do NOT HAVE a mathematical model!
– You can not use numerical optimization
• Instead you need to design experiments
– How can we do that?

5. First step, what effects do we know?
• Weight
• Route
• Speed
• Condition of car
• Weather
• Type of gasoline

6. More variables or influencing factors
• Weight
– Car itself
– Passengers
• Number, weight
– Cargo loading
• Weight
– Gas tank full/empty
• Weight of gas
• Route
– Uphill, downhill, level?
– Stops at lights, intersections?
– Distance to drive
– Type or road
• Asphalt
• Unpaved dirt road
• Muddy
• Off-road
– You might have many alternatives…
• Speed
• Condition of car
– Old or new car?
– Engine runs well?
– Maintenance, oil, air
filter, lubrication?
– Type and condition of
tires, tire pressure?
• Weather
– Rain?
– Sunny
• Air conditioning on?
– Windy
• Type of gasoline
– You might have many
alternatives…

7. Anything missing
• Add driving style
– Aggressive, hard acceleration, hard braking
– Other extreme: Smooth and nice

8. How can anyone determine
consumption numbers?
• They “standardize” a test
– Achieve reproducible results
• A test that every time gives different numbers for the
same test subject is not very useful
• Instead of driving on the road, the tests are
performed in a garage
– Control of resistance at speed
• No effects from road type, wind, temperature, sun,
rain, uphill, downhill, traffic lights, …

9. A car on a test bench

10. Your problem
• Maybe the numbers from standardized testing
are “wrong” for your purposes
– The many effects we listed are specific to location,
driver, typical use of the car, etc.
• How would you make your own test?

11. Which factors to choose?
• You want to run some experiments…
• What can you select?
– Example: do test only if day is sunny, without wind
or heavy rain
– This way you can remove many variables!
• What can you adjust?
– Load, speed, driving style, condition, route,…
– Still too many variables, simplify with “given
assumptions”

12. For example
• Fix load: 2 people + luggage for a total of 200kg, 20 L of gas
in tank
• Fix distance: start with cold engine, drive to Big C and back,
around 9:30 am on Sundays
• Fix weather: but only if day is sunny without heavy winds
• Car well maintained, standard tires that came with the car
• Use cheapest gas
• ….
• Finally: two variables, tire pressure, driving style (speed)

13. Now what?
• You have “standardized” test conditions
• You have two variables left
– You must design experiments
– Tire pressures: min < pressure < max
– Driving styles: slow, normal, fast, crazy fast
• Expectations?
– High pressure, hard tires, least resistance, best mileage
(hypothesis, don’t strictly believe your expectations)
– Going too slow is bad, if you stand still you don’t get
anywhere but you burn gas
– Going crazy fast is bad, you accelerate hard and then you
have to brake hard, but you don’t get there much faster

14. So why not always
highest pressure in
the tires?
• Too hard tires:
discomfort,
punctured tires
– Make notes of
punctures or
other complaints

15. Full experimental design
• Select some levels of tire pressure: T1,T2, T3,
T4
• This is called “full factorial design”
Slow Normal Fast Crazy
T1    
T2    
T3    
T4    

16. The trouble
• We “assumed” something for most factors,
but kept speed and tire pressure
– Still we would need 16 experiments!
– If the experiments are done only on sunny Sunday
mornings, starting with a cold engine, this might
take a whole year!
• This is the reason for incomplete, or reduced,
or fractional experimental designs

17. Reduce number of factor levels
• It is typical to use three levels per factor
– Two levels are too little to find best value in range
– The “middle point” can be dropped out, and we
now have only 8 experiments
Slow Normal Fast
T1   
T2  
T3   

18. Why three levels?
• Fit a parabola, it will estimate the maximum
or minimum:
Minimum
Maximum

19. With many factors…
• A typical experimental design is Box-Behnken
• Here, three factors
• Three levels each
• 13 experiments
• A “full factorial”
would have 27
experiments !

20. A good online source for help
• http://www.itl.nist.gov/div898/handbook/index.htm

21. Strategy
• Explore the factors with three levels
• When you have an estimate of what is close to
optimal
– You can use a three-level design again, but with
smaller steps between levels of a factor
• For example:
– Tire pressure in range 20 to 35  experiments 
31 is near optimal  next experiments for range
29 to 33

22. Response surface models
• Two inputs
• The output level
is drawn like a map
– Level curves or
3D plots of the
fitted surface
– Fit is with “parabola”,
i.e. a second degree
polynomial

23. Common mistake
• Example: biogas production depends on pH
and temperature
– First hold pH fixed, find “best” temperature
– Then find best pH at the “best” temperature
• The correct way is designed experiments and
response surface.
– Why, what was wrong?

24. Path to highest point? This is the
problem…

25. Summary
• Begin by listing all factors that might be
important
– Can you prevent some from affecting
– Which ones can you control and experiment with
• Design experiments
– Usually three levels per factor
• Response surfaces for approximate optimization
– Often Box-Behnken design and quadratic fit
• Don’t try “optimizing one factor at a time.” It is a
fundamentally flawed, wrong, incorrect,
uneducated, silly approach.