Unit 1 (part 2): The Taguchi Method: Minimizing the Number of
Experiments
Concepts:
Students should explore establishing experiments to measure specified
dependent variables, and be able to calculate the necessary experiments.
Objectives:
Students will learn how to determine the number of experiments necessary
for a given number of variables with given number of levels.
Useful websites:
The Michigan Chemical Process Dynamics and Controls Open Text Book:
http://controls.engin.umich.edu/wiki/index.php/Design_of_experiments_via
_taguchi_methods:_orthogonal_arrays
Wikipedia: Genichi Taguchi: http://en.wikipedia.org/wiki/Genichi_Taguchi
A more complex site with many array choices:
http://www2.research.att.com/~njas/oadir/index.html
Activity Sheets:
Taguchi Exercise
Engage:
Review the concepts of experiments, dependent and independent variables
and defining the number of levels of each variable. Review the answers
from the previous class to the following questions:
1. Why would we want to define the number of variable we will control
before doing an experiment?
2. Why would we want to minimize the number of experiments?
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron
Explore:
1. Review the ‘ice cream experiments’ from the teacher background with
students in the classroom. Lead the students through the development
of the final answer of the number of experiments.
2. During the ice cream example, show the Taguchi selection table and
discuss how to look up the number of experiments using the table (by
specifying the number of input variables and number of levels for
each).
3. Have students work through the questions on the “Taguchi Exercise”
worksheet.
4. Working in pairs, ask each student team to come up with an
experiment, listing each independent variable and the specific levels
they would test. Try to come up with 8-12 input variables with 3-4
levels: then determine the number of experiments required.
Explain:
1. Discuss the results of the ice cream experiment.
2. Review the solution of the Taguchi Exercise worksheet.
3. Determine the number of possible input variables we could control in
growing algae in small tanks, and the number of experiments we need
in full factorial and Taguchi methods.
Extend:
The Taguchi method can also be used to determine which input
variable, when changed, has the most dramatic effect on the dependent
variable we’re testing. This is often called the standard deviation analysis.
This will be covered soon.
Evaluate:
1. Use student solutions to the Taguchi Experiment Exercise
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron
Teacher Notes:
Design of experiments (DoE) or experimental design is very commonly used to figure
out how a number of variables work together toward some result. As we have seen, the
disadvantage is that adding a variable or a level result in a big increase in the number of
experiments.
The Taguchi Method is a way to minimize the number of experiments needed to
determine the effect of independent variables with specific numbers of levels. The basic
process is
-
-
Define the number of variables and levels
Look up the proper Taguchi Array
o The arrays can be derived or developed, but they are readily available to
look up
Perform the experiments as indicated
Analyze the results to determine the optimal combination of independent
variables.
Types of experimentation:
Full factorial experiments involve testing each possible combination of variables. In
cases with a small number of variables, like our ice cream example, there is usually no
need to reduce the number of experiments. When we have a medium amount of variables
and levels, the Taguchi method allows us to test a specific, smaller number of
combinations, then analyze the results to find the optimal solution. With a large number
of input variables, we might choose to do a random test, testing some random
combinations of variables.
As a guideline:
 Factorial design: suitable for 1-3 input variables
 Taguchi method: suitable for 3 to 50 inputs variables
 Random experiments: suitable for over 50 input variables.
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron
Let’s revisit our experiment with making the best ice cream cone given these inputs:
Cone: sugar cone or regular cone
Ice cream: vanilla or chocolate
Topping: dip, sprinkles, none
Texture: soft, regular
Number of tests = 2 levels * 2 levels * 3 levels * 2 levels = 24 experiments
Vanilla, sugar cone, dip, soft
Vanilla, reg cone, dip, soft
Chocolate, sugar cone, dip, soft
Vanilla, reg cone, dip, soft
Vanilla, sugar cone, dip, reg tex
Vanilla, reg cone, dip, reg tex
Chocolate, sugar cone, dip, reg tex
Vanilla, reg cone, dip, reg tex
Vanilla, sugar cone, sprinkles, soft
Vanilla, reg cone, sprinkles, soft
Chocolate, sugar cone, sprinkles, soft
Vanilla, reg cone, sprinkles, soft
Vanilla, sugar cone, sprinkles, reg tex
Vanilla, reg cone, sprinkles, reg tex
Chocolate, sugar cone, sprinkles, reg tex
Vanilla, reg cone, sprinkles, reg tex
Vanilla, sugar cone, none, soft
Vanilla, reg cone, none, soft
Chocolate, sugar cone, none, soft
Vanilla, reg cone, none, soft
Vanilla, sugar cone, none, reg tex
Vanilla, reg cone, none, reg tex
Chocolate, sugar cone, none, reg tex
Vanilla, reg cone, none, reg tex
Using the Taguchi selection chart:
We’ll use the Taguchi method to reduce the number of experiments we need to do to
determine the best ice cream cone. To use the Taguchi method, we refer to the Taguchi
selection chart:
(see
http://controls.engin.umich.edu/wiki/index.php/Design_of_experiments_via_taguchi_met
hods:_orthogonal_arrays for more detail)
Number of parameters: 4
Number of levels (the maximum number): 3
Taguchi chart tells us to use: L9 array.
We look up the L9 array and it shows us which
combinations to test:
L9:
What about blocks with no values? Use
a random value… for example, under
Cone in row 7, we can put either Sugar
or Regular. There really is no
statistical advantage to one or the
other; often one is cheaper or easier to
test. As we will see, we may be able to
duplicate a test…
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron
L9 (filled in)
Exp
1
2
3
4
5
6
7
8
9
Cone
Sugar
Sugar
Sugar
Regular
Regular
Regular
Ice cream
Vanilla
Chocolate
Vanilla
Chocolate
Vanilla
Chocolate
L9 (filled in, with random values in blank boxes)
Exp
Cone
Ice cream
1 Sugar
Vanilla
2 Sugar
Chocolate
3 Sugar
Vanilla
4 Regular
Vanilla
5 Regular
Chocolate
6 Regular
Vanilla
7 Sugar
Vanilla
8 Regular
Chocolate
9 Regular
Vanilla
Topping
Dip
Sprinkles
None
Sprinkles
None
Dip
None
Dip
Sprinkles
Texture
Soft
Regular
Topping
Texture
Dip
Sprinkles
None
Sprinkles
None
Dip
None
Dip
Sprinkles
Soft
Regular
Regular
Soft
Soft
Regular
Regular
Soft
Soft
Regular
Regular
Soft
Soft
Note that we were able to duplicate a few tests: rows 3 and 7 are the same, and rows 4
and 9 are the same.
Officially, the number of experiments we’ll need to do = 9
Realistically, the number is 7 (we’ll record our results of row 3 and row 4 in their
identical rows 7 and 9)


Full factorial test: 24 experiments
Taguchi: 9 experiments (or in this case, 7 will work)
Another example: if we have 23 inputs to control with 2 – 3 levels,


Full factorial test: 3*3*3*3*3*… (23 times) = 323 experiments =
94,143,178,827 experiments
Taguchi: the selection chart shows we would use an L36, or 36 experiments
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron
Running the Experiments:
First, determine how you will measure the dependent variable. In most cases, you want a
maximum or minimum number: which experiment grew the most algae, which
experiment resulted in the lowest cost. For our ice cream example, you may have to
assign the deliciousness of the ice cream cone a scale of 1 – 100.
Use the input variable values indicated in the Taguchi chart to run each experiment. For
example, on the above ice cream experiment, you would eat:
 Vanilla soft-serve on a sugar cone, with dip,
 Chocolate, regular ice cream on a sugar cone with sprinkles,
 Vanilla, regular ice cream on a sugar cone (plain),
 etc.
… and record the deliciousness of the ice cream cone in each of the 9 cases.
It is often useful to do multiple tests of each case (whether using factorial or Taguchi).
Analyzing the results:
The question: if we only test 9 combinations out of the possible 24, how can we know
which is best?
Let’s assume we made 2 ice cream cones testing each of our test conditions, and collected
the following data (and averaged all of the experiments for each setup):
L9 (filled in, with random values in blank boxes) Each level is named 1, 2 or 3 (for the next table)
Exp
1
2
3
4
5
6
7
8
9
Cone
Sugar 1
Sugar 1
Sugar 1
Regular 2
Regular 2
Regular 2
Sugar 1
Regular 2
Regular 2
Ice cream
Vanilla 1
Chocolate 2
Vanilla 1
Vanilla 1
Chocolate 2
Vanilla 1
Vanilla 1
Chocolate 2
Vanilla 1
Topping
Dip 1
Sprinkles 2
None 3
Sprinkles 2
None 3
Dip 1
None 3
Dip 1
Sprinkles 2
Texture
Soft 1
Regular 2
Regular 2
Soft 1
Soft 1
Regular 2
Regular 2
Soft 1
Soft 1
1
88
78
82
77
57
80
72
68
77
2
86
77
84
81
54
80
84
62
81
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron
avg
87
77.5
83
79
55.5
80
78
65
79
Next, average each factor value. For example, notice that experiments 1, 6 and 8 all had
dip, 2, 4 and 9 have sprinkles and 3, 5 and 7 have nothing.
Level 1
Level 2
(87+77.5+83+78) / 4=
(79+55.5+80+65+79) / 5=
81.375
71.7
Ice cream
81
66
Topping
77.333
78.5
Texture
73.1
79.625
Cone
Level 3
72.166
The simple averages show that, in this case,
 Cone: level 1 is better than level 2 (or sugar is better than plain)
 Ice cream: level 1 is better than level 2 (vanilla is better than chocolate)
 Topping: level 2 is slightly better than level 1 or level 3 (sprinkles are slightly
better than dip, which is better than nothing) and
 Regular ice cream is preferred over soft serve.
Our ideal cone should be: vanilla, regular ice cream on a sugar cone with sprinkles!
Notice that this wasn’t one of the test cases, but our data shows this should be the best.
(You might want to make a vanilla, regular ice cream on a cone with sprinkles to test
your experimental results!)
© 2010, Ohio Northern University: Dr. Debra Gallagher, Dr. Robert Verb, Dr. Ken Reid and Ben McPheron