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Design of Experiments: Taguchi Methods By Peter Woolf ([email protected]) University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons
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Page 1: Lecture 25

Design of Experiments:Taguchi Methods

By Peter Woolf ([email protected])University of Michigan

Michigan Chemical ProcessDynamics and ControlsOpen Textbook

version 1.0

Creative commons

Page 2: Lecture 25

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

Page 3: Lecture 25

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?

Page 4: Lecture 25

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

Page 5: Lecture 25

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.

Page 6: Lecture 25

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)

Page 7: Lecture 25

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.

Page 8: Lecture 25

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

Page 9: Lecture 25

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

Page 10: Lecture 25

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

Page 11: Lecture 25

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

Page 12: Lecture 25

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

Page 13: Lecture 25

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).

Page 14: Lecture 25

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).

Page 15: Lecture 25

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

Page 16: Lecture 25

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

Page 17: Lecture 25

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.

Page 18: Lecture 25

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?

Page 19: Lecture 25

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!

Page 20: Lecture 25

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.