Design of Experiments:Taguchi Methods
By Peter Woolf ([email protected])University of Michigan
Michigan Chemical ProcessDynamics and ControlsOpen Textbook
version 1.0
Creative commons
Existing plantmeasurements
Physics, chemistry, and chemicalengineering knowledge & intuition Bayesian network models to
establish connections
Patterns of likelycauses & influences
Efficient experimental design totest combinations of causes
ANOVA & probabilistic models to eliminateirrelevant or uninteresting relationships
Process optimization (e.g. controllers,architecture, unit optimization,
sequencing, and utilization)
Dynamicalprocess modeling
You have been called in as a consultant to find out how to optimize a client’s CSTRreactor system to both minimize product variation and also to maximize profit. Afterexamining the whole dataset of 50 variables, you conclude that the most likely fourvariables for controlling profitability are the impeller type, motor speed for the mixer,control algorithm, and cooling water valve type. Your goal now is to design anexperiment to systematically test the effect of each of these variables in the currentreactor system.
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Each time you have to change the system setup, youhave to stop much of the plant operation, so it means asignificant profit loss.
Scenario
How should we design our experiment?
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
Option 1: Factorial design to test all possible combinationsA, 300,PID, BB, 300,PID, BC, 300,PID, B
A, 300,PI, BB, 300,PI, BC, 300,PI, BA, 350,PI, BB, 350,PI, BC, 350,PI, BA, 400,PI, BB, 400,PI, BC, 400,PI, B
A, 300,P, BB, 300,P, BC, 300,P, BA, 350,P, BB, 350,P, BC, 350,P, BA, 400,P, BB, 400,P, BC, 400,P, B
A, 300,PID, GB, 300,PID, GC, 300,PID, GA, 350,PID, GB, 350,PID, GC, 350,PID, GA, 400,PID, GB, 400,PID, GC, 400,PID, G
A, 300,PI, GB, 300,PI, GC, 300,PI, GA, 350,PI, GB, 350,PI, GC, 350,PI, GA, 400,PI, GB, 400,PI, GC, 400,PI, G
A, 300,P, GB, 300,P, GC, 300,P, GA, 350,P, GB, 350,P, GC, 350,P, GA, 400,P, GB, 400,P, GC, 400,P, G
Total experiments=(3 impellers)(3 speeds)(3 controllers)(2 valves)=54Can we get similar information with fewer tests?How do we analyze these results?
A, 350,PID, BB, 350,PID, BC, 350,PID, BA, 400,PID, BB, 400,PID, BC, 400,PID, B
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
Option 2: Taguchi Method of orthogonal arraysMotivation: Instead of testing all possible combinations ofvariables, we can test all pairs of combinations in somemore efficient way. Example: L9 orthogonal arrayKey Feature:Compare any pair ofvariables (P1, P2, P3,and P4) across allexperiments and you willsee that eachcombination isrepresented.
Option 2: Taguchi Method of orthogonal arraysArrays can be quite complicated. Example: L36 array
Each pair of combinations is tested at least onceFactorial design: 323=94,143,178,827 experimentsTaguchi Method with L36 array: 36 experiments (~109 x smaller)
Option 2: Taguchi Method of orthogonal arraysWhere do we these arrays come from?1) Derive them
• Small arrays you can figure out by hand using trial and error(the process is similar to solving a Sudoku)
• Large arrays can be derived using deterministic algorithms (seehttp://home.att.net/~gsherwood/cover.htm for details)
2) Look them up• Controls wiki has a listing of some of the more common designs• Hundreds more designs can be looked up online on sites such
as: http://www.research.att.com/~njas/oadir/index.html
How do we choose a design?The key factors are the # of parameters and the number of levels (states)that each variable takes on.
Option 2: Taguchi Method of orthogonal arrays# parameters:
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
# levels: 3 3 3 2 =Impeller, speed, algorithm, valve = 4
~3
L9
BFPI400C9
BFPID350C8
GP300C7
GPID400B6
BFP350B5
BFPI300B4
GP400A3
GPI350A2
BFPID300A1
ValveControlMotorSpeed
ImpellerExperiment
No valvetype 3, so thisentry is filled atrandom in abalanced way
Option 3: Random Design: Surprisingly, randomly assigningexperimental conditions will with high probability create a nearoptimal design.• Choose the number of experiments to run (this can be trickyto do as it depends on how much signal recovery you want)• Assign to each variable a state based on a uniform sample(e.g if there are 3 states, then each is chosen with 0.33probability)Random designs tend to work poorly for small experiments(fewer than 50 variables), but work well for large systems.
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
http://groups.csail.mit.edu/drl/journal_club/papers/CS2-Candes-Romberg-05.pdfhttp://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1614066
For more information on these methods see the following resources
When do we use which method?Option 1: Factorial DesignSmall numbers of variables with few states (1 to 3)Interactions between variables are strong and importantEvery variable contributes significantlyOption 2: Taguchi MethodIntermediate numbers of variables (3 to 50)Few interactions between variablesOnly a few variables contributes significantlyOption 3: Random DesignMany variables (50+)Few interactions between variablesVery few variables contributes significantly
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
Once we have a design, how do we analyze the data?
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
19.117.918.116.217.915.816.217.016.1
Yield
BFPI400C9BFPID350C8GP300C7GPID400B6BFP350B5BFPI300B4GP400A3GPI350A2BFPID300A1
ValveControlMotorSpeed
ImpellerExpt. 1) Plot the dataand look at it
2) ANOVA1-way: effect of
impeller2-way: effect of
impeller andmotor speed
Test multiplecombinations
Once we have a design, how do we analyze the data?
These variables can take the following values:Impellers: model A, B, or CMotor speed for mixer: 300, 350, or 400 RPMControl algorithm: PID, PI, or P onlyCooling water valve type: butterfly or globe
Scenario
19.117.918.116.217.915.816.217.016.1
Yield
BFPI400C9BFPID350C8GP300C7GPID400B6BFP350B5BFPI300B4GP400A3GPI350A2BFPID300A1
ValveControlMotorSpeed
ImpellerExpt. 3) Bin yield andperformFisher’sexact test orChi squaredtest to see ifany effect issignificant
Field case study:Polyurethane quality control
Polyurethane manufacturing involves many steps, someof which involve poorly understood physics orchemistry.
Three dominant factors of product quality are:1) Water content2) Chloroflourocarbon-11 (CFC-11) concentration3) Catalyst type
Case modified from Lunnery, Sohelia R., and Joseph M. Sutej. "Optimizing a PU formulation by theTaguchi Method. (polyurethane quality control)." Plastics Engineering 46.n2 (Feb 1990): 23(5).
11,122Isocyanate type25,352CFC-11, wt%0.5, 1,52Water, wt%S1, S2, S33Surfactant type1,2,3,4,55Catalyst packageA,B,C, D4Polyol type
Description# factorsFactors andLevels
Field case study:Polyurethane quality control
Case modified from Lunnery, Sohelia R., and Joseph M. Sutej. "Optimizing a PU formulation by theTaguchi Method. (polyurethane quality control)." Plastics Engineering 46.n2 (Feb 1990): 23(5).
A 3 S2 25 11B 1 S2 35 12C 3 S1 35 12D 1 S1 25 11B 3 S1 25 12A 2 S1 35 11D 3 S2 35 11C 2 S2 25 12
B 3 S3 25 11A 4 S3 35 12D 3 S2 35 12C 4 S2 25 11A 3 S2 25 12B 5 S2 35 11C 3 S3 35 11D 5 S3 25 12
Experiment design using a modified L16 array
Case modified from Lunnery, Sohelia R., and Joseph M. Sutej. "Optimizing a PU formulation by theTaguchi Method. (polyurethane quality control)." Plastics Engineering 46.n2 (Feb 1990): 23(5).
Field case study:Polyurethane quality control
Design modified from an L25array to better account for thenumber of states of eachvariable.
Note not all pairs involvingcatalyst are tested--this iseven sparser
Experimental Procedure:Reactivity profile and friability (subjective rating) weredetermined from hand-mix foams prepared in 1-gal paper cans.Free rise densities were measured on core samples of openblow foams. Height of rise at gel, final rise height, and flow ratiowere determined in a flow tube.
Case modified from Lunnery, Sohelia R., and Joseph M. Sutej. "Optimizing a PU formulation by theTaguchi Method. (polyurethane quality control)." Plastics Engineering 46.n2 (Feb 1990): 23(5).
Field case study:Polyurethane quality control
Data AnalysisANOVA to identify significant factors, followed by linearregression to identify optimal conditions
Extreme Example:Sesame Seed Suffering
You have just produced 1000x 55 gallon drums ofsesame oil for sale to your distributors.
One barrel of sesame oil sells for $1000, while each assay forinsecticide in food oil costs $1200 and takes 3 days. Tests forinsecticide are extremely sensitive. What do you do?
Just before you are to ship the oil, one of youremployees remembers that one of the oil barrels wastemporarily used to store insecticide and is almostsurely contaminated. Unfortunately, all of the barrelslook the same.
Extreme Example:Sesame Seed SufferingSolution: Extreme multiplexing. LikeTaguchi methods but optimized for verysparse systems
Example solution w/ 8 barrels
1 2 53 4 6 7 8Mix samples from eachbarrel and test mixturesMix 1,2,3,4 --> Sample AMix 1,2,5,6 -> Sample BMix 1,3,5,7 -> Sample C
A,B,C poison barrel-,-,- 8+,-,- 4-,+,- 6-,-,+ 7+,+,- 2+,-,+ 3-,+,+ 5+,+,+ 1
Result: Using only 3 testsyou can uniquely identifythe poison barrel!
Is this enough tests?
Extreme Example:Sesame Seed SufferingSolution: Extreme multiplexing. LikeTaguchi methods but optimized for verysparse systems
Solution w/ 1000 barrels
1 2 53 4 6 7 8Mix samples from eachbarrel and test mixturesExperiments required=Log2(1000)=~10
Solution w/ 1,000,000 barrelsExperiments required=Log2(1,000,000)=~20
Optimal experiments can be extremely helpful!
Take Home Messages• Efficient experimental design helps to
optimize your process and determinefactors that influence variability
• Factorial designs are easy to construct,but can be impractically large.
• Taguchi and random designs oftenperform better depending on size andassumptions.