Taguchi Design of Experiments
• Many factors/inputs/variables must be taken into
consideration when making a product especially a brand new
one
• The Taguchi method is a structured approach for determining
the ”best” combination of inputs to produce a product or
service
• Based on a Design of Experiments (DOE) methodology for
determining parameter levels
• DOE is an important tool for designing processes and
products
• A method for quantitatively identifying the right inputs and
parameter levels for making a high quality product or service
• Taguchi approaches design from a robust design perspective
Taguchi method
• Traditional Design of Experiments focused on how
different design factors affect the average result level
• In Taguchi’s DOE (robust design), variation is more
interesting to study than the average
• Robust design: An experimental method to achieve
product and process quality through designing in an
insensitivity to noise based on statistical principles.
Robust Design
• A statistical / engineering methodology that aim at
reducing the performance “variation” of a system.
• The input variables are divided into two board
categories.
• Control factor: the design parameters in product or
process design.
• Noise factor: factors whoes values are hard-to-control
during normal process or use conditions
The Taguchi Quality Loss Function
• The traditional model for quality losses
• No losses within the specification limits!
Cost
Scrap Cost
LSL
Target
USL
• The Taguchi loss function
• the quality loss is zero only if we are on target
4
Example (heat treatment process for
steel)
• Heat treatment process used to harden steel
components
Parameter
number
1
Parameters
Level 1 Level 2 unit
Temperature
760
900
OC
2
Quenching rate
35
140
OC/s
3
Cooling time
1
300
s
4
Carbon contents
1
6
Wt% c
5
Co 2 concentration
5
20
%
• Determine which process parameters have the greatest
impact on the hardness of the steel components
Taguchi method
• To investigate how different parameters affect
the mean and variance of a process
performance characteristic.
• The Taguchi method is best used when there
are an intermediate number of variables (3 to
50), few interactions between variables, and
when only a few variables contribute
significantly.
Two Level Fractional Factorial Designs
• As the number of factors in a two level factorial design
increases, the number of runs for even a single replicate of
the 2k design becomes very large.
• For example, a single replicate of an 8 factor two level
experiment would require 256 runs.
• Fractional factorial designs can be used in these cases to
draw out valuable conclusions from fewer runs.
• The principle states that, most of the time, responses are
affected by a small number of main effects and lower order
interactions, while higher order interactions are relatively
unimportant.
Half-Fraction Designs
• A half-fraction of the 2k design involves running only half of
the treatments of the full factorial design. For example,
consider a 23 design that requires 8 runs in all.
• A half-fraction is the design in which only four of the eight
treatments are run. The fraction is denoted as 2 3-1with the
“-1 " in the index denoting a half-fraction.
• In the next figure: Assume that the treatments chosen for
the half-fraction design are the ones where the interaction
ABC is at the high level (1). The resulting 23-1 design has a
design matrix as shown in Figure (b).
Half-Fraction Designs
No. of runs = 8
No. of runs = 4
No. of runs = 4
23
2 3-1
I= ABC
2 3-1
I= -ABC
Half-Fraction Designs
• The effect, ABC , is called the generator or word
for this design
• The column corresponding to the identity, I , and
column corresponding to the interaction , ABC
are identical.
• The identical columns are written as I= ABC and
this equation is called the defining relation for the
design.
Quarter and Smaller Fraction Designs
• A quarter-fraction design, denoted as 2k-2 , consists of a
fourth of the runs of the full factorial design.
• Quarter-fraction designs require two defining relations.
• The first defining relation returns the half-fraction or
the 2 k-1design. The second defining relation selects
half of the runs of the 2k-1 design to give the quarterfraction.
• Figure a, I= ABCD  2k-1. Figure b, I=AD 2k-2
Quarter and Smaller Fraction Designs
I= ABCD
24-1
I=AD
24-2
Taguchi's Orthogonal Arrays
• Taguchi's orthogonal arrays are highly fractional orthogonal
designs. These designs can be used to estimate main
effects using only a few experimental runs.
• Consider the L4 array shown in the next Figure. The L4
array is denoted as L4(2^3).
• L4 means the array requires 4 runs. 2^3 indicates that the
design estimates up to three main effects at 2 levels each.
The L4 array can be used to estimate three main effects
using four runs provided that the two factor and three
factor interactions can be ignored.
Taguchi's Orthogonal Arrays
L4(2^3)
2III3-1
I = -ABC
Taguchi's Orthogonal Arrays
• Figure (b) shows the 2III3-1 design (I = -ABC,
defining relation ) which also requires four runs
and can be used to estimate three main effects,
assuming that all two factor and three factor
interactions are unimportant.
• A comparison between the two designs shows
that the columns in the two designs are the same
except for the arrangement of the columns.
Taguchi’s Two Level Designs-Examples
L4 (2^3)
L8 (2^7)
Taguchi’s Three Level Designs- Example
L9 (3^4)
Analyzing Experimental Data
• To determine the effect each variable has on
the output, the signal-to-noise ratio, or the SN
number, needs to be calculated for each
experiment conducted.
• yi is the mean value and si is the variance. yi is
the value of the performance characteristic for
a given experiment.
signal-to-noise ratio
Worked out Example
• A microprocessor company is having difficulty with its
current yields. Silicon processors are made on a large
die, cut into pieces, and each one is tested to match
specifications.
• The company has requested that you run experiments
to increase processor yield. The factors that affect
processor yields are temperature, pressure, doping
amount, and deposition rate.
• a) Question: Determine the Taguchi experimental
design orthogonal array.
Worked out Example…
• The operating conditions for each parameter and
level are listed below:
•A: Temperature
•A1 = 100ºC
•A2 = 150ºC (current)
•A3 = 200ºC
•B: Pressure
•B1 = 2 psi
•B2 = 5 psi (current)
•B3 = 8 psi
•C: Doping Amount
•C1 = 4%
•C2 = 6% (current)
•C3 = 8%
•D: Deposition Rate
•D1 = 0.1 mg/s
•D2 = 0.2 mg/s (current)
•D3 = 0.3 mg/s
Selecting the proper orthogonal array by
Minitab Software
Example: select the appropriate design
Example: select the appropriate design
Example: enter factors’ names and levels
Worked out Example…
a) Solution: The L9 orthogonal array should be used.
The filled in orthogonal array should look like this:
This setup allows the testing of all four variables without having to run 81 (=34)
Selecting the proper orthogonal array by
Minitab Software
Worked out Example…
• b) Question: Conducting three trials for each
experiment, the data below was collected.
Compute the SN ratio for each experiment for
the target value case, create a response chart,
and determine the parameters that have the
highest and lowest effect on the processor
yield.
Worked out Example…
Experi
ment
Numbe
r
Temper
ature
1
2
3
4
5
6
7
8
9
100
100
100
150
150
150
200
200
200
Doping
Pressur Amoun
e
t
2
5
8
2
5
8
2
5
8
4
6
8
6
8
4
8
4
6
Deposit
ion
Rate
Trial 1
0.1
0.2
0.3
0.3
0.1
0.2
0.2
0.3
0.1
87.3
74.8
56.5
79.8
77.3
89
64.8
99
75.7
Trial 2
Trial 3
Mean
82.3
70.7
54.9
78.2
76.5
87.3
62.3
93.2
74
70.7
63.2
45.7
62.3
54.9
83.2
55.7
87.3
63.2
80.1
69.6
52.4
73.4
69.6
86.5
60.9
93.2
71
Standar
d
deviatio
n
8.5
5.9
5.8
9.7
12.7
3
4.7
5.9
6.8
Enter data to Minitab
Worked out Example…
• b) Solution:
For the first treatment,
Experiment
Number
1
2
3
4
5
6
7
8
9
A
(temp)
1
1
1
2
2
2
3
3
3
80.12
SN i  10 log 8.5 2  19.5
B
(pres)
1
2
3
1
2
3
1
2
3
C
(dop)
1
2
3
2
3
1
1
2
1
D
(dep)
1
2
3
3
1
2
2
3
1
T1
87.3
74.8
56.5
79.8
77.3
89
64.8
99
75.7
T2
82.3
70.7
54.9
78.2
76.5
87.3
62.3
93.2
74
T3
70.7
63.2
45.7
62.3
54.9
83.2
55.7
87.3
63.2
SNi
19.5
21.5
19.1
17.6
14.8
29.3
22.3
24.0
20.4
Worked out Example
• Shown below is the response table. calculating an average SN
value for each factor. A sample calculation is shown for Factor
B (pressure):
Experiment
Number
1
2
3
4
5
6
7
8
9
A
B
(temp) (pres)
1
1
1
2
1
3
2
1
2
2
2
3
3
1
3
2
3
3
C
(dop)
1
2
3
2
3
1
1
2
1
D
(dep)
1
2
3
3
1
2
2
3
1
SNi
19.5
21.5
19.1
17.6
14.8
29.3
22.3
24.0
20.4
Worked out Example
SNB1  19.5  17.6  22.3  19.8
3
SNB3  19.1  29.3  20.4  22.9
3
Level
1
2
3

Rank
SNB2  21.5  14.8  24.0  20.1
3
A (temp) B (pres) C (dop)
20
19.8
24.3
20.6
20.1
19.8
22.2
22.9
18.7
2.2
3.1
5.5
4
3
2
D (dep)
18.2
24.4
20.2
6.1
1
The effect of this factor is then calculated by determining the range:
  Max  Min  22.9 19.8  3.1
Deposition rate has the largest effect on the processor yield
and the temperature has the smallest effect on the processor yield.
Example solution by Minitab
Example: determine response columns
Example Solution
Example: Main Effect Plot for SN
ratios
Differences between SN and Means
response table
Main effect plot for means
Mixed level designs
• Example: A reactor's behavior is dependent upon impeller
model, mixer speed, the control algorithm employed, and the
cooling water valve type. The possible values for each are as
follows:
• Impeller model: A, B, or C
• Mixer speed: 300, 350, or 400 RPM
• Control algorithm: PID, PI, or P
• Valve type: butterfly or globe
• There are 4 parameters, and each one has 3 levels with the
exception of valve type.
Mixed level designs
Available designs
Select the appropriate design
Factors and levels
Enter factors and levels names
Design matrix