Selecting experimental variables for response surface modeling

Special topics in Industrial ChemistrySeppo Karrila

November 2014

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

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?

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?

First step, what effects do we know?

• Weight

• Route

• Speed

• Condition of car

• Weather

• Type of gasoline

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…

Anything missing

• Add driving style

– Aggressive, hard acceleration, hard braking

– Other extreme: Smooth and nice

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, …

A car on a test bench

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?

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”

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)

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

So why not always highest pressure in

the tires?

• Too hard tires: discomfort, punctured tires

– Make notes of punctures or other complaints

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

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

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

Why three levels?

• Fit a parabola, it will estimate the maximum or minimum:

Minimum

Max

imu

m

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 !

A good online source for help

• http://www.itl.nist.gov/div898/handbook/index.htm

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

Response surface models

• Two inputs

• The output level is drawn like a map

– Level curves or3D plots of the fitted surface

– Fit is with “parabola”,i.e. a second degreepolynomial

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?

Path to highest point? This is the problem…

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.