A Primer On The Taguchi Method
A Primer On The Taguchi Method
A Primer On The Taguchi Method
TAGUCHI
METHOD
SECOND EDITION
Ranjit K. Roy
Copyright © 2010 Society of Manufacturing Engineers
987654321
ACKNOWLEDGMENTS
This book would not be in circulation today without the Society
of Manufacturing Engineers undertaking continued printing of the
first edition after the original publisher folded technical publish-
ing activities. For this second edition, I am indebted to Rosemary
Csizmadia of SME, who was relentless in her conviction of market
demand for the book even in difficult economic times. I would also
like to thank Ellen Kehoe of SME for her grasp of language and
insight into the technology in editing the manuscript.
I am grateful to my professional and business associates, Larry
Smith, Mike Comerford, Greg Adams, Larry Tracey, Dave White,
and Jay Chandra, for their trust in my ability to support their clients
seeking training and application of the Taguchi technique. I would
like to express my sincere thanks to Larry Smith, in particular, who
spent an extraordinary amount of time on a meticulous review, and
to Andrea Stamps, Kush Shah, Side Zhao, Fred Schenkelberg, and
Pradeep Kumar for their detailed and constructive suggestions.
Finally, I thank my wife, Krishna, who has put up with my
dedication to the life of an independent consultant, trainer, and
author for more than two decades.
Ranjit K. Roy
January 2010
Contents
Preface .......................................................................................................................... ix
Abbreviations and Symbols ........................................................................... xii
3 Measurement of Quality
THE QUALITY CHARACTERISTIC ........................................................................................ 25
VARIATION AS A QUALITY YARDSTICK .............................................................................. 26
COST OF VARIATION .......................................................................................................... 27
QUALITY AND VARIATION ................................................................................................. 27
THE QUALITY WE ARE AFTER ............................................................................................ 29
vi A Primer on the Taguchi Method
7 Loss Function
DERIVATION OF LOSS FUNCTION .................................................................................... 185
AVERAGE LOSS FUNCTION FOR PRODUCT POPULATION ............................................... 189
APPLICATION OF LOSS FUNCTION CONCEPTS ............................................................... 189
LOOKS OF PERFORMANCE IMPROVEMENT ..................................................................... 196
EXERCISES ................................................................................................................. 202
BACKGROUND
Mankind has always had a fascination with quality. Today’s
technology is testimony to man’s incessant desire to provide a
higher level of quality in products and services to increase market
share and profits. Sometimes quality is essential. A pacemaker that
controls heart action must operate continuously and precisely. An
erratic pacemaker is valueless, useless, and dangerous.
Driven by the need to compete on price and performance and
to maintain profitability, quality-conscious manufacturers are
increasingly aware of the need to optimize products and processes.
Quality achieved by means of design optimization is found by many
manufacturers to be cost effective in gaining and maintaining a
competitive position in the world market.
Response
68
Similarly, the main effect for 64 Sugar level 65
chocolate chips is 22.5%. It 60
is important to note that, in 56 55
this example, only the main 52 S1
effects are analyzed; no at- 48
44 45
tempt is made to analyze the
40
interactions between the fac- C1 C2
tors. Interactions may or may Chocolate chips level
not be present. The relative
influence of the factors and Figure 1-1. Factor effects
interactions between various
factors included in the study
can be quantitatively determined by using the analysis of variance
(ANOVA). This procedure is described in Chapter 6.
For the present, the degree of interactions for a 2 × 2 experi-
ment can be determined from Figure 1-1, which graphs the response
against one factor (C) for two levels of the second factor (S). Because
the lines for the two levels, S1 and S2, are almost parallel, the factors
(S and C) are said to be independent, and little or no interaction is
assumed to exist. Nonparallel lines would indicate the presence of
some interaction. Highly skewed lines or a higher angle between
lines (need not be intersecting) would indicate strong interaction
between the two factors. Figure 1-1 indicates only a slight interac-
tion between the two factors (sugar and chocolate chips).
In the above example there were only two factors, each at two
different levels. It would be rather easy to manufacture four types
of cookies reflecting all possible combinations of the factors under
study and to subject them to a market survey.
For a full factorial design, the number of possible designs, N, is
m
N=L (1.1)
where L = number of levels for each factor and m = number of factors.
4 A Primer on the Taguchi Method
EXERCISES
1-1. What are the three main disadvantages of the conventional
design of experiments approach as compared with Taguchi’s
method?
1-2. Which one of the two factor effect graphs in Figure 1-2 indi-
cates the existence of an interaction between the two factors
of an experiment?
1-3. A product involves three primary parameters at three differ-
ent levels of each. To optimize the product, a full factorial
design is planned for experimental evaluations. How many
possible design configurations need to be tested to achieve
the objective?
1-4. Draw a factor graph for the experiment shown in Table 1-3
and discuss the results.
Quality Through Product and Process Optimization 7
B2 B1
Response
Response
B1 B2
A1 A2 A1 A2
Factor A Factor A
B2 65 45 B2 =
Average A1 = A2 =
Response to A
(A2 – A1)
2 Taguchi Approach to
Quality and Cost
Improvement
BACKGROUND
After the Second World War, Allied forces found the quality of
the Japanese telephone system to be extremely poor and totally
unsuitable for long-term communication purposes. To improve
the system to a state-of-the-art level, the Allied command rec-
ommended that Japan establish research facilities similar to the
Bell Laboratories in the United States. The Japanese founded the
Electrical Communication Laboratories (ECL), with Dr. Genichi
Taguchi in charge of improving R&D productivity and enhancing
product quality. Taguchi observed that a great deal of time and
money was expended in engineering experimentation and testing,
with little emphasis on the process of creative brainstorming to
minimize the expenditure of resources.
Taguchi started to develop new methods to optimize the
process of engineering experimentation. He developed the
techniques that are now known as the Taguchi Methods. His
greatest contribution lies not in the mathematical formulation
of the design of experiments (DOE) but rather in the accompa-
nying philosophy. His approach is more than a method to lay
out experiments. It is a concept that has produced a unique
and powerful quality improvement discipline that differs from
traditional practices.
Two completely opposing points of view are commonly held
about Taguchi’s contribution to the statistical design of experi-
ments. One view holds that his contribution to the field of quality
control is one of the most significant developments of the last
few decades. The other view maintains that many of the ideas
10 A Primer on the Taguchi Method
TAGUCHI PHILOSOPHY
Taguchi espoused an excellent philosophy for quality control
in the manufacturing industries. Indeed, his doctrine is creating
an entirely different breed of engineers who think, breathe, and
live quality. He has, in fact, given birth to a new quality culture
in this country. Ford Motor Company, for example, decreed in the
early 1990s that all Ford Motor and suppliers’ engineers be trained
in the Taguchi methodology and that these principles be used
to resolve quality issues. Taguchi’s philosophy has far-reaching
consequences, yet it is founded on three very simple and funda-
mental concepts. The whole of the technology and techniques arise
entirely out of these three ideas. These concepts are:
1. Quality should be designed into the product and not
inspected into it.
2. Quality is best achieved by minimizing the deviation from a
target. The product should be so designed that it is immune
to uncontrollable environmental factors.
3. The cost of quality should be measured as a function of
deviation from the standard, and the losses should be
measured system-wide.
Taguchi built on W.E. Deming’s observation that 85% of poor
quality is attributable to the manufacturing process and only 15%
to the worker. Hence, Taguchi developed manufacturing systems
that were “robust” or insensitive to daily and seasonal variations
of environment, machine wear, and other external factors. The
three principles were his guides in developing these systems,
Taguchi Approach to Quality and Cost Improvement 11
Old school
(No loss range)
5
Sony Japan
3
Frequency
Sony U.S.
0
LAL T UAL
LAL = Lower acceptable limit Color density
UAL = Upper acceptable limit
T = Target value
Nominal diameter
4 "
2
" "
" "
0 " "
LAL Bearing diameter UAL
ANALYSIS OF RESULTS
In the Taguchi method, the results of the experiments are ana-
lyzed to achieve one or more of the following three objectives:
1. To determine the trend of influence of factors and interac-
tions under study.
2. To identify the significant factors and their relative influ-
ences on the variability of results.
20 A Primer on the Taguchi Method
AREAS OF APPLICATION
Analysis
In the design of engineering products and processes, analytical
simulation plays an important role, transforming a concept into
the final product design. The Taguchi approach can be utilized to
arrive at the best parameters for the optimum design configura-
tion with the least number of analytical investigations. Although
there are several methods available for optimization, using such
simulations when the factors are continuous, the Taguchi method
is the method that treats factors at discrete levels. Frequently this
approach significantly reduces computer time.
Process Development
Manufacturing processes typically have a large number of
factors that influence the final outcome. Identification of their
individual contributions and their intricate interrelationships is
essential in the development of such processes. The Taguchi con-
cepts used in such projects have helped many U.S. and Japanese
companies realize significant cost savings in recent times.
Validation Testing
For many products, proper validation testing requires as-
surance of performance under numerous application factors
22 A Primer on the Taguchi Method
Problem Solving
Production and manufacturing problems related to variations,
rework, and rejects are common in industry. While many such
issues may be resolved by common problem-solving disciplines,
some require special techniques. Fortunately, the solution often
is obtainable by properly adjusting many influencing factors
rather than searching for innovative means. The Taguchi DOE
is a powerful technique to investigate such technical issues and
determine data-driven permanent solutions.
Limitations
The most severe limitation of the Taguchi method is the need
for proactive thinking and working as a group to address the qual-
ity improvement issues early in the product/process development.
The technique is most effective when applied before the design of
the product/process system is released. After the design variables
are determined and their nominal values are specified, experimen-
tal design may not be cost effective. Also, though the method has
wide-ranging applications, there are situations in which classical
techniques are better suited; for example, in simulation studies
involving factors that vary in a continuous manner, such as the
torsional strength of a shaft as a function of its diameter, the
Taguchi method may not be the best choice.
EXERCISES
2-1. There are two types of losses that society incurs because of the
poor quality of a product. What are these losses?
2-2. Explain why the old definition of cost of quality is inad-
equate.
2-3. What is the most important idea of Taguchi’s concept of achiev-
ing higher product quality?
24 A Primer on the Taguchi Method
COST OF VARIATION
Early in his research, Dr. Taguchi observed that unexpected
variation was common to all manufacturing processes and that it
was the primary cause for rejection of parts. Parts were rejected
upon inspection when they did not conform to a predefined speci-
fication. Rejection increases the cost of production. Often, 100%
inspection is excessively costly or impractical; thus, a defective
part may reach a customer and lead to warranty costs and cus-
tomer dissatisfaction. Taguchi held that variation is costly even
beyond the immediate factory production cost and that excessive
variation causes loss of quality. He contended that the cure for
quality loss is reduction of variation. Thus, he recommended that
effort should be directed toward minimizing variation, with less
emphasis placed on production within fixed tolerance limits.
Target value
(b)
Average value on target.
Little variation around target value.
Frequency
Target value
(c)
from the target and (b) variation with respect to others in the
group. In Figure 3-2, a typical quality measure of a product (similar
to Bearing Dimension discussed in Chapter 2) is compared with
the desired state. Note that the product mean value is off target
and that the variation around the mean is large, though within
upper and lower acceptance limits. A much narrower distribution
Measurement of Quality 29
4
Desired condition
Mean = Target
Std. dev. — Low
• •
3
Frequency
2
• •
Current status
Mean < > Target
Std. dev. — High
1
• •
0 • •
LAL Target Mean UAL
Quality measure
4 "
"
" "
"
Color quality
"
0 "
VC B B B VD
Voltage
COMMON TERMINOLOGY
The technique for laying out the conditions of experiments
when multiple factors are involved has been known to statisti-
cians for a long time. The technique was first introduced by
Measurement of Quality 33
A. Eggs A1 A2
B. Butter B1 B2
C. Milk C1 C2
D. Flour D1 D2
E. Sugar E1 E2
EXERCISES
3-1. How does Taguchi’s view of quality differ from the conventional
practice?
3-2. How does variation affect cost and quality?
3-3. What are the main causes of variation?
3-4. How is a product design optimized?
3-5. How does Taguchi make the design less sensitive to the noise
factors?
3-6. What are orthogonal arrays?
3-7. What is implied by the term parameter design and what is its
significance in achieving higher product quality?
4 Attractions and Benefits
of the Taguchi Method
Some
Let’s try this Brainstorming!
thinking
1. What is the Quality Characteristic?
2. What are the Design Parameters?
Test
More
Let’s try that
thinking
• • •
Test
Analysis of results
Confirmation
• • •
test
UP-FRONT THINKING
The value of brainstorming in product development or for
solving complex problems is well known, yet it was rarely used for
engineering problems. Brainstorming prior to an experiment is a
necessary requirement in the Taguchi approach; however, Taguchi
does not give any guidelines for conducting brainstorming for an
experiment. The content and outcome of a brainstorming session
is largely dependent on the nature of a project and, as such, is a
technique learned primarily by experience. Most application spe-
cialists consider brainstorming to be the most important element
in deriving benefits from the Taguchi method.
Taguchi brings a new breadth to planning experimental stud-
ies. Experimenters think through the whole process before starting
40 A Primer on the Taguchi Method
the tests. This helps to decide which factors are likely to be most
important, how many experiments are needed, and how the results
would be measured and analyzed—before actually conducting any
experiment. Figure 4-1 shows the typical steps followed by experi-
menters—some initial thinking, followed by some testing, which, in
turn, is followed by some more thinking, and so on. In the Taguchi
approach, the complete plan of how to test, what to test, and when
to analyze the results will all be decided beforehand. Ideally, an ex-
periment planning (brainstorming) session will rely on the collective
experience of the group to determine the factors to be selected for
testing in an appropriate design. Practice of the Taguchi method
fosters a team approach to design optimization because participa-
tion of people from engineering, manufacturing, testing, and other
activities may be necessary for complete variable identification.
EXPERIMENTAL EFFICIENCY
In most cases, the Taguchi experiment design using an orthogo-
nal array requires the least number of test runs. A full factorial
experiment with 15 factors at two levels each is performed with
a test matrix with 32,768 (215) test runs. A fractional factorial
experiment with an orthogonal array suitable for 15 two-level
factors consists of only 16 test runs.
The experimental efficiency Taguchi offers can be described
using the following analogy. Assume that you are asked to catch a
big fish from a lake with a circular net. You are also told that the
fish usually stays around its hideout, but you have no knowledge
of where this place is. How do you go about catching this fish?
Thinking analytically, you may first calculate the area of the net
and the lake and then lay out an elaborate scheme to cover the
entire lake. You may find, after all this planning, that you need
the whole day to locate the spot where the fish is. Wouldn’t it
be nice to have a fish finder that could tell you the approximate
locations of where to throw your net? The Taguchi approach in
experimental studies, to a great extent, works like a fish finder.
It tells you which areas to try first, and then from the results of
the trials you determine, with a high degree of certainty, the most
probable location of the fish.
Attractions and Benefits of the Taguchi Method 41
LONG-TERM BENEFITS
Most of the benefits of quality improvement effort in the
design stage come after the product is put in use. The reduced
variation, a characteristic that is designed in through the optimum
combination of the factors, will yield consistent performance of
the product. This means that more of the products will perform
as designed. There will be happier customers and, therefore, less
warranty costs and increased sales.
With the target value of 9.00 volts, the above measured values pro-
duce characteristics as shown below (also see Tables 4-1 and 4-2).
Average value = 8.67
Standard deviation = 0.37
Mean square deviation (MSD) = [(8.1 − 9.0)2 + (9.25 − 9.0)2
+ … + (8.90 − 9.0)2]/10
= 0.23
S/N ratio = −10 log10 (MSD)
= 6.36
After the experiment, a batch of 10 batteries showed the fol-
lowing characteristics:
AFTER EXPERIMENT
Voltages
9.10 8.93 8.69 8.92 9.08
8.08 9.02 8.91 9.15 9.25
Average value = 8.99
Standard deviation = 0.1598
Mean square deviation = 0.023
S/N ratio = 16.37
2.0
1.5
• •
Frequency
1.0
• •
0.5
• • • •
0.0
9.0
Volts
±.40
±.63
±1.0
REQUIRED TOLERANCES
Manufacturer tolerance = 9 ± 0.63
Supplier tolerance = 9 ± 0.45
NOTE: If these tolerances are held, there will be no nonfunctional part in the
customer's hands. For the same cost, the manufacturer will maintain satisfied
customers and quality products in the field.
volt, that is, between 8.00 and 10.00 volts. If the voltage is beyond
this range, customers request a refund ($1.25).
Taguchi’s approach to the computation of cost savings is based
on determining the refund cost associated with the variation of
the batteries, as measured by the mean square deviation (MSD)
46 A Primer on the Taguchi Method
CALCULATION OF LOSS
PROBLEM DEFINITION
Target value of quality characteristic (m) = 9.00
Tolerance of quality characteristic = 1.00
Cost of rejection at production (per unit) = $1.25
Units produced per month (total) = 100,000
S/N ratio of current design/part = 6.37
S/N ratio of new design/part = 16.37
BEFORE EXPERIMENT:
Loss/unit due to deviation from target in current design = $0.288
AFTER EXPERIMENT:
Loss/unit due to deviation from target in new design = $0.028
MONTHLY SAVINGS:
If production is maintained at the improved condition,
then based on 100,000 units/month = $25,950.90
from the target voltage. Obviously, the greater the variation the
more likely that some batteries will exceed the limits of customer
acceptance. With the above information, the loss is computed as
$.288 per battery for the sample before the experiment and $.028
per unit for the sample after the experiment. Because 100,000
units are manufactured per month, the total savings per month
is estimated to be $25,950.90 (Figure 4-3).
Attractions and Benefits of the Taguchi Method 47
EXERCISES
4-1. The Taguchi method is considered a technique that helps build
quality into a product or process. Explain what aspect of quality
it influences and how.
4-2. Compare the roles of the Taguchi method with that of statisti-
cal process control (SPC) in a manufacturing process. Explain
how the Taguchi method can influence decisions in the SPC
activities.
5 Working Mechanics of the
Taguchi Design of Experiments
BASIC METHODOLOGY
The technique of laying out the conditions (designs) of experi-
ments involving multiple factors was first proposed by Sir Ronald
A. Fisher of England in the 1920s. The method is popularly known
as the factorial design of experiments. A full factorial design will
identify all possible combinations for a given set of factors. Because
most industrial experiments usually involve a significant number
of factors, a full factorial design results in a large number of experi-
ments. For example, in an experiment involving seven factors, each
at two levels, the total number of combinations will be 128 (27). To
reduce the number of experiments to a practical level, only a small
set from all of the possibilities is selected. The method of selecting
a limited number of experiments that produces the most informa-
tion is known as a fractional factorial experiment. Although this
shortcut method is well known, there are no general guidelines for
its application or the analysis of the results obtained by performing
the experiments. Dr. Taguchi’s approach complements these two
important areas. First, he clearly defined a set of OAs, each of which
can be used for many experimental situations. Second, he devised
a standard method for analysis of the results. The combination of
standard experimental design techniques and analysis methods in
the Taguchi approach produces a higher degree of consistency and
reproducibility of the predicted performance.
Before discussing how the Taguchi approach reduces the num-
ber of experiments, it is helpful to understand how all possible
combinations result from a set of factors.
Suppose we are concerned about one factor, A (say, tempera-
ture). If we were to study the effect of A on a product at two levels,
say, 400°F and 500°F, then two tests become necessary:
Level 1 = A1 (400°F) and Level 2 = A2 (500°F)
Consider now two factors, A and B, each at two levels (A1, A2
and B1, B2). This produces four combinations because at A1, B can
assume values B1 and B2, and at A2, B can again assume values
B1 and B2.
Symbolically, these combinations are expressed as follows:
A1(B1, B2), A2(B1, B2), or as A1B1, A1B2, A2B1, A2B2
Working Mechanics of the Taguchi Design of Experiments 51
With three factors, each at two levels, there are 23 (8) possible
experiments, as described in the previous section. If A, B, and C
represent these factors, the eight experiments can be expressed
as follows:
Example 5-1
An experimenter has identified three controllable factors for
a plastic molding process. Each factor can be applied at two levels
(Table 5-3). The experimenter wants to determine the optimum
combination of the levels of these factors as well as the contribu-
tion of each to product quality.
Experiment Design
There are three factors, each at two levels, thus an L4 will be
suitable, per Table 5-1. An L4 OA with spaces for the factors and
their levels is shown in Table 5-4. This configuration is a conve-
nient way to lay out a design. Because an L4 has three columns,
the three factors can be assigned to these columns in any order.
Having assigned the factors, their levels can also be indicated in
the corresponding column.
There are four independent experimental conditions in an L4.
These conditions are described by the numbers in the rows. For an
experienced user of the technique, an array with factors assigned
as shown in Table 5-2 contains all of the necessary information;
200°F
9 sec.
2
LEVEL
150°F
6 sec.
1
DESCRIPTION
temperature
VARIABLE
pressure
Injection
Set time
Mold
COLUMN REPETITION
EXPERIMENT 1 2 3 1 2 3 •••
1 1 1 1 30
2 1 2 2 25
3 2 1 2 34
4 2 2 1 27
56 A Primer on the Taguchi Method
Replication
In this approach, all of the trial conditions will be run in a
random order. One way to decide the order is to randomly pull
one trial number at a time from a set of trial numbers, including
repetitions. Often a new setup will be required for each run. This
increases the cost of the experiment.
Repetition
Each trial is repeated as planned before proceeding to the next
trial run. The trial run sequence is selected in a random order.
For example, given the trial sequence 2, 4, 3, and 1, three suc-
cessive runs of trial 2 are made, followed by three runs of trial 4,
and so on. This procedure reduces setup costs for the experiment.
However, a setup error is unlikely to be detected. Furthermore,
the effect of external factors such as humidity, tool wear, and so
on, may not be captured during the successive runs if the runs
are short in duration.
Analysis of Results
Although, a detailed analysis of the results will be discussed in
Chapter 6, a brief description and objectives of such an analysis
are introduced here.
Following the specifications as prescribed above, the experi-
menter conducted the four trials. The molded products were then
evaluated, and the results, in terms of a quality characteristic, Y,
were measured as shown below:
Y1 = 30, Y2 = 25, Y3 = 34, Y4 = 27
Working Mechanics of the Taguchi Design of Experiments 57
Quality Characteristics
In a previous chapter, the quality characteristics were de-
scribed as:
• bigger is better
• smaller is better
• nominal is best
For the molding process example, higher strength of the mold-
ed plastic part is desired and thus “bigger is better.” From Figure
5-1, the A2 B1 C2 will likely produce the best result and therefore
represents the optimum condition except for the possible effect
of interactions between the factors.
In terms of the actual design factors, the probable optimum
condition becomes:
A2 that is, injection pressure at 350 psi
B1 that is, mold temperature at 150°F
C2 that is, set time at 9 sec.
Working Mechanics of the Taguchi Design of Experiments 59
33
32.0
32
31
30.5
Response
30
29.5
29
28.5
28
27.5
27
26.0
26
25
A1 A2 B1 B2 C1 C2
Factors
Variance
The variance of each factor is determined by the sum of the
square of each trial sum result involving the factor, divided by the
degrees of freedom of the factor. Thus:
VA S A fA (for factor A )
VB SB fB (for factor B )
VC SC fC (for factoor C )
Ve Se fe (for error terms)
Variance Ratio
The F-ratio is the variance of the factor divided by the error
variance.
FA VA Ve
FB VB Ve
FC VC Ve
Fe Ve Ve 1
Percent Influence
The percent influence of each factor is the ratio of the factor
sum to the total, expressed in percent.
PA S A s 100 ST
PB SB s 100 ST
PC SC s 100 ST
Pe Se s 100 ST
Working Mechanics of the Taguchi Design of Experiments 61
or ST Y12 Y42
Y1 Y2 Y3 Y4 4
2
2
302 252 342 272
30 25 34 27 4
3410
3364
46
For the molding process experiment, the totals of the factors are:
A1 = 30 + 25 = 55 A2 = 34 + 27 = 61
B1 = 30 + 34 = 64 B2 = 25 + 27 = 52
C1 = 30 + 27 = 57 C2 = 25 + 34 = 59
therefore, the total variance of each factor is:
S A A12 N A1 A22 N A 2
C.F .
552 2 612 2
3364
1512.5 1860.5
3364 9.0
0
SB B12 N B1 B22 N B 2
C.F . 36.0
and
SC C12 N C1 C22 N C 2
C.F . 1 0
The error variance
Se ST
S A SB SC
46
9
36
1 0 (in this case)
Variance
V A S A fA 9 1 9
VB SB fB 36 1 36
VC SC fC 1 1 1
Ve Se fe 0 0 indetermin
nate
Note that if the experiment included repetitions, say 2, then:
fT = 4 × 2 – 1 = 7
fe = 7 – 1 – 1 – 1 = 4
where Se need not equal zero, depending on test results, and Ve
need not be zero.
Variance Ratio
FA VA Ve is indeterminate because Ve = 0. Similarly, FB and
FC are indeterminate (Table 5-6). However, Ve can be combined
(pooled) with another small variance, VC, to calculate a new error
Ve that can then be used to produce meaningful results. The pro-
cess of disregarding an individual factor’s contribution and then
subsequently adjusting the contributions of the other factors is
Working Mechanics of the Taguchi Design of Experiments 63
therefore,
Yopt 29 30.5
29 32
29 29.5
29
34.0
which is the result obtained in trial 3.
When the optimum is not one of the trial runs already com-
pleted, this projection should be verified by running a confirmation
test(s). Confirmation testing is a necessary and important step
in the Taguchi method as it validates assumptions used in the
analysis. Generally speaking, the average result from the confir-
mation tests should agree with the optimum performance, Yopt,
estimated by the analysis. The correlation can also be established
in statistical terms, reflecting the level of confidence, influence
of number of confirmation tests, and so on. The procedure for
calculating the confidence interval of the optimum performance
is discussed in Chapter 6.
Example 5-3
Number of factors = 8 through 11
Number of levels for each = 2
Use array L12
Experiment Design
Use L12 in Table A-3 for this example. Assign factors 1 through
11 in the 11 columns available, in any order. Express the 12 ex-
perimental conditions by using the 12 rows of the OA. Note that
L12 is a special array prepared for study of the main effects only
(not suitable for study of interaction between factors). In this
array, the interaction effects of factors assigned to any two col-
umns are mixed with all other columns, which renders the array
unsuitable for interaction studies. (Use L16, L32, and L64 shown in
Appendix A to design experiments with higher numbers of two-
level factors.)
80 80 80
T2 T2
76 78
76 75
75
Response
Response
72 73 72
68 68 70
T1 T1
67
64 64
62
60 60
H1 H2 H1 H2
With interaction Without interaction
(a) (b)
Linear Graphs
Linear graphs are made up
1 2
of numbers, dots, and lines, as
shown in Figure 5-3 for an 3
L4 array, where a dot and its
assigned number identifies Figure 5-3. Linear graph for L4 array
a factor, a connecting line
between two dots indicates
interaction, and the number assigned to the line indicates the col-
umn number in which interaction effects will be compounded.
In designing experiments with interactions, the triangu-
lar tables are essential; the linear graphs are complementary
to the tables. For most industrial experiments, interactions
between factors are minor and the triangular tables suffice.
The following example shows how these two tools are used for
experimental design.
Example 5-6
In a baking experiment designed to determine the best recipe
for a pound cake, five factors and their respective levels were
identified, as presented in Table 5-12.
Among these factors, milk (factor C) was suspected to interact
with eggs (A) and butter (B). An experiment was designed to study
the interactions A × C and B × C in addition to the main effects
of factors A, B, C, D, and E.
72 A Primer on the Taguchi Method
Column Assignment
In designing experiments with interactions, the columns to
reserve to study interactions must be identified first. We have
Working Mechanics of the Taguchi Design of Experiments 73
Description of Combinations
The eight trial conditions contained in Table 5-13 can be de-
scribed individually. Tables 5-14 and 5-15 show trial runs 1 and 2,
respectively. The other trial runs can be similarly described. Note
that the numbers in the columns where interactions are assigned
(columns 3 and 6 in Table 5-13) are not used in the description
of trial run 2 (Table 5-15). Normally the interaction column does
not need to appear in the description and thus is deleted from the
74 A Primer on the Taguchi Method
Analysis of Results
The analysis of data including interactions follows the same
steps as are taken when there is no interaction. The objectives are
the same: (1) determine the optimum condition, (2) identify the in-
dividual influence of each factor, and (3) estimate the performance
at the optimum condition. The methods for objectives 2 and 3 are
the same as before. For the optimum condition, interactions intro-
duce a minor change in the manner in which the optimum levels
of factors are identified. To develop a clear understanding of how
the optimum condition is selected, the main effects are discussed
here in detail. (The details of ANOVA will be covered in Chapter
6, but only the results of a computer analysis will be presented.)
76 A Primer on the Taguchi Method
A s C 1 67.75
A s C 2 62.50
The calculations for each factor and level are in Table 5-16(a).
The difference between the average value of each factor at levels
2 and 1 indicates the relative influence of the effect. The larger
the difference (magnitude), the stronger the influence. The sign of
the difference obviously indicates whether the change from level 1
to 2 increases or decreases the result. The main effects are shown
visually in Figure 5-4. Figure 5-5 shows the interaction effects of
A × C and B × C.
Ignoring interaction effects for the moment, notice that Table
5-16(a) and Figure 5-4 show an improvement at level 2 only for
factors A and B, while level 2 effects for C, D, and E cause a de-
crease in quality. Hence, the optimum levels for the factors based
on the data are A2, B2, C1, D1, and E1. Coincidentally, trial 6 tested
78 A Primer on the Taguchi Method
78
74.25
74
70.00
Response
70 68.75
65.50
66
66.00
62 64.25 64.75
61.50
58 60.25
56.00
54
A1 A2 C1 C2 B1 B2 D1 D2 E1 E2
68 67.75
66
Response
65.75
64 64.50
62 62.50
60
(A x C)1 (A x C)2 (B x C)1 (B x C)2
Interactions
Interaction Effects
To determine whether the interaction is present, a proper
interpretation of the results is necessary. The general approach
is to separate the influence of an interacting member from the
influences of the others. In this example, A × C and B × C are
the interactions with C common to both. The information about C
Working Mechanics of the Taguchi Design of Experiments 79
82
80 78.50
78
76
B2
74
72 70.50
70
Response
70.00
68
68.50 A2
66 65.00
64
62 A1
60 59.00
58
58.00 B1
56
54 53.00
52
C1 C2 C1 C2
Interactions
Yopt T A2
T C1
T B2
T
D
T E
T
1 1
Yopt T A1
T C1
T ¨ª A s C ·¹
T
1
B
T D
T E
T
2 1 1
T 64
4.25
T 68.75
T 67.75
T
65.125
.875 3.625 2.625 9.125 4.875 .375
84.875
Yopt T A1C1
T B 2
T D1
T E1
T
T 70.5
T 74.25
T 70
T 65.5
T
65.125 5.375 9.125 4.875 0.375
84.875
82 A Primer on the Taguchi Method
Key Observations
• In designing experiments with interactions, triangular
tables or linear graphs should be used for column as-
signments. To select the appropriate OA, the types of
interactions and their degrees of freedom will have to be
considered. The following steps are recommended for the
experiment design process:
1. Select the array based on factors and interactions and
their levels. The degrees of freedom of the OA must
equal or exceed the DOF of factors and interactions.
2. Assign factors to the column arbitrarily when no in-
teraction is included. In case interaction is part of the
study, treat interacting factors first and reserve columns
based on the triangular table to study interaction.
3. Describe trial conditions by reading across the OA with
factors and interactions assigned to the columns.
Working Mechanics of the Taguchi Design of Experiments 83
× ×
FACTOR/
INTERACTION A C E B B D D
COLUMN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
EXPERIMENT
1 1 1 1 1 1 0 0 1 0 1 0 0 0 0 0
2 1 1 1 1 1 0 0 2 0 2 0 0 0 0 0
3 1 1 1 2 2 0 0 1 0 1 0 0 0 0 0
4 1 1 1 2 2 0 0 2 0 2 0 0 0 0 0
5 1 2 2 1 1 0 0 1 0 2 0 0 0 0 0
6 1 2 2 1 1 0 0 2 0 1 0 0 0 0 0
7 1 2 2 2 2 0 0 1 0 2 0 0 0 0 0
8 1 2 2 2 2 0 0 2 0 1 0 0 0 0 0
9 2 1 2 1 2 0 0 1 0 1 0 0 0 0 0
1 0 2 1 2 1 2 0 0 2 0 2 0 0 0 0 0
1 1 2 1 2 2 1 0 0 1 0 1 0 0 0 0 0
1 2 2 1 2 2 1 0 0 2 0 2 0 0 0 0 0
1 3 2 2 1 1 2 0 0 1 0 2 0 0 0 0 0
1 4 2 2 1 1 2 0 0 2 0 1 0 0 0 0 0
1 5 2 2 1 2 1 0 0 1 0 2 0 0 0 0 0
1 6 2 2 1 2 1 0 0 2 0 1 0 0 0 0 0
Table 5-18. L16 design with nine two-level factors and five interactions—
Example 5-8
A A B A A
× × × × ×
FACTOR/
INTERACTION A E E F F G D D H C C I B B
COLUMN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
EXPERIMENT
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
3 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
4 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1
5 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
6 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1
7 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1
8 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2
9 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 0 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1
1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1
1 2 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2
1 3 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
1 4 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2
1 5 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2
1 6 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1
A1 A2 A1 A3 A2
Factors
Figure 5-8. Main effects of a factor with two and three levels
Steps 3
1 2
1. From the linear graph
for L8, select a set of
6
three interacting col- 5
•
umns (Figure 5-9). 7
Example: columns 1, 4
2, and 3.
2. Select any two col- Figure 5-9. Groups of interacting columns
for level upgrading
umns. Suppose 1 and
2 are selected.
3. Combine the two columns row by row, by following the rules
of Table 5-19, to get a combined column such as shown in
Table 5-17. Replace the original columns 1, 2, and 3 by the
new column that has just been prepared.
Example 5-9
Design an experiment to accommodate one factor at four levels
and four others at two levels each.
Variables: A, B, C, D
Interactions: None
Levels: A = 4; B, C, D = 2
Experiment Design
Factor A has four levels and 3 DOF. The other four two-level
factors each have 1 DOF. The total DOF is 7. An L8 OA, shown in
Table 5-20, that has 7 DOF, appears suitable.
Working Mechanics of the Taguchi Design of Experiments 89
Building Columns
The first three columns of an L8 can be combined to produce a
four-level column following the procedure previously described.
Step 1. Start with an original L8 and select a set of three
interacting columns, say 1, 2, and 3.
Step 2. Ignore column 3 (Table 5-21).
Step 3. Combine column 1 and 2 into a new column. Follow
the procedure as shown by Tables 5-22 and 5-23.
Step 4. Assign the four-level factor to this new column and the
others to the remaining original two-level columns,
as shown in Tables 5-24 and 5-25.
The experimental conditions and the subsequent analysis are
handled in a manner similarly to the techniques described before.
90 A Primer on the Taguchi Method
Table 5-26. Rules for preparation of an eight-level column for an L16 array
COLUMN
FIRST SECOND THIRD NEW COLUMN
1 1 1 1
1 1 2 2
1 2 1 3
1 2 2 4
2 1 1 5
2 1 2 6
2 2 1 7
2 2 2 8
NEW COLUMN
COLUMN 1 2 4 8 9 10 11 12 13 14 15
EXPERIMENT
1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 2 2 2 2 2 2 2 2
3 1 1 2 2 1 1 1 1 2 2 2 2
4 1 1 2 2 2 2 2 2 1 1 1 1
5 1 2 3 1 1 1 2 2 1 1 2 2
6 1 2 3 1 2 2 1 1 2 2 1 1
7 1 2 4 2 1 1 2 2 2 2 1 1
8 1 2 4 2 2 2 1 1 1 1 2 2
9 2 1 5 1 1 2 1 2 1 2 1 2
1 0 2 1 5 1 2 1 2 1 2 1 2 1
1 1 2 1 6 2 1 2 1 2 2 1 2 1
1 2 2 1 6 2 2 1 2 1 1 2 1 2
1 3 2 2 7 1 1 2 2 1 1 2 2 1
1 4 2 2 7 1 2 1 1 2 2 1 1 2
1 5 2 2 8 2 1 2 2 1 2 1 1 2
1 6 2 2 8 2 2 1 1 2 1 2 2 1
L16 has 15 DOF and therefore is suitable for the design. The
three sets of interacting columns used for column upgrading are
1 2 3, 4 8 12, and 7 9 14. The column preparation and assignment
follows these steps.
experience, the run conditions are easily read from the array.
But for the inexperienced, and for large arrays, translating the
array notations into actual descriptions of the factor levels may
be subject to error. Computer software [7] is available to reduce/
eliminate chances of such errors. A printout of the trial conditions
for sample trial runs is shown in Table 5-30(d).
Main Effect Plots for Three-Level and Four-Level Factors
The analysis of experimental data follows the same steps as
before. The results of a single test run at each of the 16 conditions
are shown in Table 5-30(e). The main effects of the factors are
presented in Table 5-30(f); the effects for the three- and four-level
98 A Primer on the Taguchi Method
COMBINATION DESIGN
Consider an experiment involving three three-level factors and
two two-level factors. An experiment design could consider an L16
Working Mechanics of the Taguchi Design of Experiments 99
OA with three columns for each of the three-level factors and two
additional columns for the two-level factors. Such a design will
utilize 11 of the available 15 columns and require 16 trial runs for
the experiment. Alternatively consider the L9 OA. Three columns
satisfy the three-level factors. If the fourth column can be used to
accommodate two two-level factors, then L9 with only nine trial
runs could be used.
Indeed, it is possible to combine two two-level factors into a single
three-level factor, with some loss of confidence in the results and loss
of opportunity to study interactions. The procedure is given below.
100 A Primer on the Taguchi Method
82 80.75
80 78.50
78 77.50
76 78.00
Response
75.00
74 72.75 74.75
72
70 70.50
68 69.25
67.50
66
64
L1 L2 L3 L4 L1 L2 L3 L1 L2 L3
Coating Sand compaction Gating
The total data are analyzed with the two factors X and Y
treated as one, (XY). The analysis yields the main effect of (XY).
The individual effect of the constituents X and Y is then obtained
as follows:
Main effect of X XY 1
XY 2 and
Main effect of Y XY 1
XY 3
Working Mechanics of the Taguchi Design of Experiments 103
Example 5-11
In a simple study to determine the influence of alcohol con-
sumption during different types of meals on blood alcohol content
(BAC), the test parameters were defined as:
Controllable factor—
A: Type of meals
(two levels: A1 = light snack, A2 = steak dinner)
Noise factor—
N: Type of alcohol
(two levels: N1 = hard liquor, N2 = light beer)
Assume that the individuals under observation have no control
over the drinks he/she will be served, but do have control over the
meals consumed before going to a party. The effects of alcohol con-
sumption when meals are of the type described are found to be:
A1N1 = 50, A2N1 = 30, A1N2 = 25, and A2N2 = 20
The numbers shown are the likelihood of exceeding the BAC limit.
From the above data, the control factor and noise effects can
be plotted by taking two data points at a time from the set of four
data [Fig. 5-12(a) and 5-12(b)]. These plots showing the effects of
one factor at various levels of the other are called interaction plots.
The angle between the lines, if present, indicates the strength of
the presence of interaction. The first of the two plots [Fig. 5-12(a)]
indicates that there is interaction between meals and alcohol
consumption because the lines are not parallel. The second plot
[Fig. 5-12(b)], however, is of most interest. This graph presents
the same interaction showing the effects of noise (alcohol) at two
levels of the control factor (meals). From the slope of the two lines,
it is obvious that the influence of alcohol is much less (shallower
line) when a steak dinner is consumed (A2). So, given an option,
106 A Primer on the Taguchi Method
A1N1 = 50
Level of intoxication
N1 = Hard liquor
A2N1 = 30
A1N2 = 25
N2 = Light beer
A2N2 = 20
Figure 5-12(a). Type of meal and alcohol interaction plot (factor along x-
axis)—Example 5-11
A1N1 = 50
Level of intoxication
A1 = Light snack
A2N1 = 30
A2 = Steak dinner A1N2 = 25
A2N2 = 20
Figure 5-12(b). Alcohol and type of meal interaction plot (noise along x-
axis)—Example 5-11
Working Mechanics of the Taguchi Design of Experiments 107
or
MSD = .029395, a small value.
The S/N ratio is calculated as:
S/N
10 log10 MSD
10 log10 .029395
15.31
S/N values for all rows are shown in Table 5-33.
In analysis, the S/N ratio is treated as a single data point at
each of the test run conditions. Normal procedure for studies of the
main effects will follow. The only difference will be in the selection
of the optimum levels. In S/N analysis, the value of MSD or the
greatest value of S/N represents a more desirable condition.
TWO-STEP OPTIMIZATIONS
In this approach, product and process designs are achieved by
adjusting factor levels to reduce variability. The process follows two
distinct steps, with the assumption that reduction of variability
is more important than being on the target:
1. Reduce variability by adjusting the levels of factors deter-
mined to be influential
2. Adjust performance mean to target by adjusting those fac-
tors with less influence on variability
Dr. Taguchi recommends the two-step optimization strategy
when multiple factors influence the outcome. The following ex-
ample demonstrates how robust factor levels are determined when
there is only one major noise factor.
Working Mechanics of the Taguchi Design of Experiments 111
FACTOR RESULTS
TRIAL A B – C D E F Noise N1 Noise N2
1 1 1 1 1 1 1 1
For each trial condition:
2 1 1 1 2 2 2 2
3 1 2 2 1 1 2 2 3 sample results were
4 1 2 2 2 2 1 1 exposed to noise
condition N1.
5 2 1 2 1 2 1 2
6 2 1 2 2 1 2 1 3 sample results were
7 2 2 1 1 2 2 1 exposed to noise
condition N2.
8 2 2 1 2 1 1 2
Figure 5-13(a). Experimental design for six two-level factors and noise
exposure—Example 5-12
Noise N2 13.49 14.82 13.31 14.97 14.15 14.13 14.88 13.39 14.30 13.98 15.35 12.93
15.50
15.00
14.50
A2
14.00 B2
13.50 C1
13.00
12.50
C2
B1
12.00 A1
11.50
11.0
N1 Noise N2 N1 Noise N2 N1 Noise N2
15.50
15.00
14.50
F1
14.00
D1 E1
13.50
13.00
12.50 D2 E2
12.00 F2
11.50
11.0
N1 Noise N2 N1 Noise N2 N1 Noise N2
Figure 5-13(c). Calculated noise and control factor interactions and plots
(N×A, N×B, ..., N×F)—Example 5-12
used to obtain plot of A1 line. Plots for all other factor level effects
are obtained in the same manner.
Review of the noise and control factor interaction plots [Fig.
5-13(c)] shows that the plots for factors B, C, D, and F have more
angle between the lines, indicating that there is significant in-
teraction. Because, for robust design, the line with a shallower
angle to horizontal is likely to produce less variation, levels B1, C1,
Working Mechanics of the Taguchi Design of Experiments 115
14.00
13.80
13.60
13.40
13.20
13.00
12.80
12.60
12.40
12.20
A1 Factor A2 B1 Factor B2 C1 Factor C2
14.00
13.80
13.60
13.40
13.20
13.00
12.80
12.60
12.40
12.20
D1 Factor D2 E1 Factor E 2 F1 Factor F2
Figure 5-13(d). Calculated factor average effects and plots (Main effects of
A, B, C, D, E, and F)
D2, and F2 are the choices for these factors. Factors A and E are
considered to have interaction of lesser degree and are treated by
analysis using the main effects of factors.
Main effects of factor are plotted from the calculated average
effects using the trial result averages [last column in Fig. 5-13(b)],
as shown in Figure 5-13(d). The levels of the remaining two fac-
116 A Primer on the Taguchi Method
tors, A and E, now can be identified from the lower values (QC =
smaller is better) of the factor average effects as A1 and E2.
1.50
1.45
1.40
1.35
1.30
1.25
1.20
1.15
1.10
1.05
A1 Factor A2 B1 Factor B2 C1 Factor C2
1.50
1.45
1.40
1.35
1.30
1.25
1.20
1.15
1.10
1.05
D1 Factor D2 E1 Factor E 2 F1 Factor F2
RESULTS (y)
Tr A B – C D E F N1 N2 S/N RATIO
1 1 1 1 1 1 1 1 11.5 11.8 11.3 14.1 14.5 13.8 –22.21
2 1 1 1 2 2 2 2 9.2 8.7 8.2 9.3 10.7 9.6 –19.38
3 1 2 2 1 1 2 2 11.7 11.8 11.5 14.3 14.4 14.1 –22.30
4 1 2 2 2 2 1 1 12.7 12.7 12.6 15.6 15.6 15.4 –23.03
5 2 1 2 1 2 1 2 13.8 13.5 13.8 13.3 12.8 12.4 –22.46
6 2 1 2 2 1 2 1 13.2 13.5 13.4 16.2 16.6 16.4 –23.50
7 2 2 1 1 2 2 1 12.6 12.9 12.1 15.4 15.8 14.8 –22.93
8 2 2 1 2 1 1 2 12.3 11.7 12 15.1 14.3 14.2 –22.50
Grand averages => –22.29
–21.20
–21.40
–21.60
–21.80
–22.00
–22.20
–22.40
–22.60
–22.80
–23.00
A1 Factor A2 B1 Factor B2 C1 Factor C2
–21.20
–21.40
–21.60
–21.80
–22.00
–22.20
–22.40
–22.60
–22.80
–23.00
D1 Factor D2 E1 Factor E 2 F1 Factor F2
FACTOR AND
# INTERACTION DOF SS V F S P (%)
1 A: Injection pressure 1 2.486 2.486 6.031 2.073 18.96
2 B: Mold closing sp. 1 1.277 1.277 3.099 .865 7.91
3 Interaction A × B 1 2.277 2.277 5.525 1.865 17.05
4 C: Mold pressure (1) (.278) Pooled
5 D: Back pressure 1 .915 .915 2.222 .503 4.60
6 E: Screw speed (1) (.545) Pooled
7 F: Spear temperature 1 3.156 3.156 7.657 2.743 25.09
Other/Error 2 .824 .412 26.38
Total 7 10.937 100%
Control Noise
Number of factors
factors factors
R&D Adv. Eng. Design & Devel. Test & Valid. Mfg. & Prod.
1
Outer array
2
NOISE FACTORS
2
COLUMN
EXPERIMENT
1
4
Inner array
with four combinations of the three noise factors, tests each of the
eight trial conditions four times. The experiment design with inner
and outer array is shown by Table 5-35. Note that for the outer array,
column 3 represents both the third noise factor and the interaction
of the first and second noise factor. Note also the arrangement of
each array, with the noise (outer) array perpendicular to the inner
array. The complete design is shown by Table 5-36.
For most simple applications, the outer array describes the
noise conditions for the repetitions. This formal arrangement
of the noise factors and the subsequent analysis influences the
combination of the controllable factors for the optimum condition.
The use of S/N ratio in analysis is strongly recommended.
124 A Primer on the Taguchi Method
EXPERIMENT
8
COLUMN
FACTOR
LEVEL 1 LEVEL 2
DESCRIPTION
Eggs 2 eggs 3 eggs
2
2
1
5
1
2
TYPE OF BAKING
OVEN: TIME: HUMIDITY
1. Gas 1. +5 min. 1. 80%
2. Electric 2. –5 min. 2. 60%
COLUMN 1 2 3
EXPERIMENT
1 1 1 1
R1
2 1 2 2
R2
3 2 1 2
R3
4 2 2 1
R4
EXPERIMENT
DESIGN
TYPES OF ANALYSIS
• Nominal is best
• Smaller is better
• Bigger is better
Analysis of Results
Analysis of results follows either paths (Fig. 5-16) of repetitions
or no repetition. Generally, for a single observation for each trial
condition, the standard analysis approach is followed. When there
are repetitions of the trial runs, whether by outer array designed
noise condition, or under random noise condition, S/N analysis
should be performed. The final analysis for the optimum condition
is based on one of the three characteristics of quality—greatest,
smallest, or nominal.
EXERCISES
5-1. Identify each element (8, 2, 7, and so on) of the notation for
an orthogonal array L8 (27).
5-2. Design an experiment to study four factors, A, B, C, and D,
and three interactions, A × C, C × D, and A × D. Select the
orthogonal array and identify the columns for the three inter-
actions.
5-3. An experiment with three two-level factors yielded the follow-
ing results. Determine the average effect of factor C at levels
C1 and C2.
Working Mechanics of the Taguchi Design of Experiments 127
fe fT
fA
fB
fC
Analysis of Variance (ANOVA) 131
Sum of Squares
The sum of squares is a measure of the deviation of the ex-
perimental data from the mean value of the data. Summing each
squared deviation emphasizes the total deviation. Thus,
n
2
ST ¤ Yi
Y
i 1
where Y is the average value of Yi.
Similarly, the sum of squares of deviations, ST, from a target
value, Y0, is given by
n
2 2
ST ¤ Yi
Y
i 1
n Y
Y 0
(6-1-1)*
* n
2
ST ¤ Yi
Y0
i 1
n
2
¤ Yi
Y Y
Y0
i 1
n
2 2
¤ ¨ Yi
Y
i 1
ª©
2 Yi
Y Y
Y0 Y
Y0 ·
¹̧
n n n
2 2
¤ Yi
Y
i 1
¤ 2 Y
Y Y
Y ¤ Y
Y
i 1
i 0
i 1
0
n n n
because ¤ Y Y
¤Y ¤Y nY nY 0
i 1
i
i 1
i
i 1
n
2 2
and ¤ Y Y
i 1
0 n Y Y0
n
2 2
The above equation becomes ST ¤ Yi
Y
i 1
n Y
Y0 .
132 A Primer on the Taguchi Method
Sum of squares
Variance
Degrees of freedom
or V ST f
When the average sum of squares is calculated about the
mean, it is called the general variance. The general variance, S2,
is defined as:
1 n 2
S 2 ¤ Yi
Y
n i 1
(6-1-2)
* 1 2
Sm
n ª̈ Y1
Y0 Yn Y0 ·¹
which can also be expressed as:
1 2
Sm
n ª̈ Y1 Y2 Yn
nY0 ·¹
1¨ 2
or Sm
nª
nY
nY0
·
¹
n2 ¨ 2
or Sm
n ª
Y
Y0 ·¹
Sm nm2
134 A Primer on the Taguchi Method
Y1 – Y0 = 3 Y4 – Y0 = 4
Y2 – Y0 = 5 Y5 – Y0 = 6
Y3 – Y0 = 7 Y6 – Y0 = 8
where Y0 is a target value, then
ST 32 52 72 42 62 82
199
2
Sm 3 5 7 4 6 8 6
332 6
181.5
Y 3 5 7 4 6 8 6
5 .5
2 2
and Se ¨© 3
Y 5
Y ·
ª ¹̧
2 2 2
¨3
5.5 5
5.5 8
5.5 ·
ª ¹
1 7 .5
ONE-WAY ANOVA
One-Factor One-Level Experiment
When one-dimensional experimental data (one response
variable) are analyzed using ANOVA, the procedure is termed
a one-way analysis of variance. The following problem is an ex-
ample of a one-way ANOVA. Later, ANOVA will be extended to
multidimensional problems.
Example 6-1
To obtain the most desirable iron castings for an engine block,
a design engineer wants to maintain the material hardness at 200
BHN. To measure the quality of the castings being supplied by
the foundry, the hardness of 10 castings chosen at random from
a lot is measured, as displayed in Table 6-1.
136 A Primer on the Taguchi Method
The analysis:
fT = total number of results – 1
= 10 – 1 = 9
Y0 = desired value = 200
the mean value is:
¥ 240 190 210 230 220 ´
Y ¦ 10
§180 195 205 215 215 µ¶
210
2 2 2
then ST 240
200 190
200 210
200
2 2 2
230
200 220
200 180
200
2 2 2
195
200 205
200 215
200
2
215
200
4000
2
and
Sm n Y
Y0 2
10 210
200 1000
Se ST
Sm 4000
1000 3000
And the variance is calculated as follows:
VT ST fT 4000 9 444.44
Vm 1000 1 1000
Ve ST
Sm fe 4000
1000 9 333.33
These results are summarized in Table 6-2. Table 6-3 represents
a generalized format of the ANOVA table.
Analysis of Variance (ANOVA) 137
Variance Ratio
The variance ratio, commonly called the F statistic, is the ratio
of variance due to the effect of a factor and variance due to the
error term. (The F statistic is named after Sir Ronald A. Fisher.)
This ratio is used to measure the significance of the factor under
investigation with respect to the variance of all of the factors in-
cluded in the error term. The F value obtained in the analysis is
compared with a value from standard F-tables for a given statisti-
cal level of significance. The tables for various significance levels
and different degrees of freedom are available in most handbooks
of statistics. Tables B-1 through B-5 in Appendix B provide a brief
list of F factors for several levels of significance.
To use the tables, enter the DOF of the numerator to deter-
mine the column and the DOF of the denominator to determine
the row. The intersection is the F value. For example, the value of
F.10 (5, 30) from the table is 2.0492, where 5 and 30 are the DOF of
the numerator and denominator, respectively. When the computed
F value is less than the value determined from the F-tables at
the selected level of significance, the factor does not contribute
to the sum of squares within the confidence level. Computer
software, such as [7], simplifies and speeds the determination of
the level of significance of the computed F values.
The F values are calculated by:
Fm Vm Ve
Fe Ve Ve 1 (6-7)
and for a factor A it is given by:
FA VA Ve (6-8)
Sea Se Ve (6-9)
If factors A, B, and C, having DOF fA, fB, and fC, are included in an
experiment, their pure sum of squares are determined by:
S Aa S A
fA s Ve (6-10)
SBa SB
fB s Ve
SCa SC
fC s Ve
Sea Se fA fB fC s Ve (6-11)
Percent Contribution
The percent contribution for any factor is obtained by dividing
the pure sum of squares for that factor by ST and multiplying the
result by 100. The percent contribution is denoted by P and can
be calculated using the following equations:
Pm Sma s 100 ST
PA S Aa s 100 ST
PB SBa s 100 ST
PC SCa s 100 ST
Pe Sea s 100 ST (6-12)
The pure sum of squares obtained using Eqs. (6-9) and (6-10)
is shown below:
140 A Primer on the Taguchi Method
* Taguchi considers deviation from the target more significant than that about the mean. The
cost of quality is measured as a function of the deviations from the target. Therefore, Taguchi
eliminates the variation about the mean from Eq. (6-14) by redefining ST as follows:
n
2
ST ¤ Yi
Y0
C.F. Se S A
i 1
or Se ST
S A
Analysis of Variance (ANOVA) 143
Confidence Intervals
The calculations shown in the ANOVA table are only estimates
of the population parameters. These statistics are dependent
on the size of the sample being investigated. As more castings
Table 6-7. ANOVA table for cylinder block castings from two sources—
Example 6-2
PURE
VARIANCE SUM
SOURCE (MEAN VARIANCE OF PERCENT
OF SUM OF SQUARE), RATIO, SQUARES, CONTRIBUTION,
VARIATION f SQUARES V F S’ P
Factor (A) 1 500.00 500.00 2.81 321.89 8.68
Error (e) 18 3206.00 178.00 1.00 3384.11 91.30
Total 19 3706.00 100.00
Analysis of Variance (ANOVA) 145
ne
Number of trials
¨DOF of mean always 1 + ·
©ªDOF of all facctors used in the estimate ¹̧
To determine the C.I. for the estimated value of the mean for
the above data, we proceed as follows:
¨240 190 215 215 ·
E m © ¸ 20
ª 197 202 195 201¹
205
The number of experiments is 20, and there are two factors, m
and A, involved in the estimates. Therefore,
20 20
ne 10
fA fm 1 1
146 A Primer on the Taguchi Method
Therefore, it can be stated that there is a 90% probability that the true
value of the estimated mean will lie between 197.68 and 212.32.
The confidence interval can similarly be calculated for other
statistics.
TWO-WAY ANOVA
The one-way ANOVA discussed above included one factor
with two levels. This section extends ANOVA to experimental
data of two or more factors with two or more levels. The following
examples illustrate the procedure.
Example 6-3
The wear characteristics of two brands of tires (factor B),
“Wearwell” and “Superwear,” are to be compared. Several factors
such as load, speed, and air temperature have significant effect
on the useful life of tires. The problem will be limited to only one
among these factors, that is, temperature (factor A). Let Tw and Ts
represent winter (low) and summer (high) temperatures, respec-
tively. Tire life (response characteristic) is measured in hours of
operation at constant speed and load. The experiment design for
this example is given in Table 6-8. This is called a two-factor two-
level experiment. It has four possible trial runs, and the results
of each run can be interpreted as follows:
With A at A1 and B at B1, the life is = 70 hr
With A at A1 and B at B2, the life is = 75 hr
With A at A2 and B at B1, the life is = 65 hr
With A at A2 and B at B2, the life is = 60 hr
The analysis of the data follows exactly the same procedures
presented in the previous example. In this case, the total degrees
Analysis of Variance (ANOVA) 147
SA × B = SAB – SA – SB
ST = Se + SA + SB + SA × B
Assuming
Y0 = target value = 0
ST = sum of squares of all eight data points – C.F.
Y12 Y82
C.F.
702 722 752 772 652 622 602 612
36720.5
37028.0
36720.5
307.5
The contribution of each factor is shown below:
A1 = 142 + 152 = 294 A2 = 127 + 121 = 248
B1 = 142 + 127 = 269 B2 = 152 + 121 = 273
Analysis of Variance (ANOVA) 151
Table 6-12. ANOVA table for tire wear with repetitions—Example 6-4
PURE
VARIANCE SUM
SOURCE (MEAN VARIANCE OF PERCENT
OF SUM OF SQUARE), RATIO, SQUARES, CONTRIBUTION,
VARIATION f SQUARES V F S’ P
A 1 264.50 264.50 117.50 262.25 85.28
B 1 2.00 2.00 0.89 –0.25 –0.08
A×B 1 32.00 32.00 14.22 29.75 9.68
Error (e) 4 9.00 2.25 1.00 15.75 5.12
Total 7 307.50 100.00
Table 6-13. ANOVA table for tire wear with repetitions and pooling—
Example 6-4
PURE
VARIANCE SUM
SOURCE (MEAN VARIANCE OF PERCENT
OF SUM OF SQUARE), RATIO, SQUARES, CONTRIBUTION,
VARIATION f SQUARES V F S’ P
A 1 264.50 264.50 120.20 262.30 85.30
B Pooled
A×B 1 32.00 32.00 14.50 29.80 9.69
Error (e) 5 11.00 2.20 1.00 15.40 5.01
Total 7 307.50 100.00
154 A Primer on the Taguchi Method
Similarly
B1 143 B1 35.75
B2 204 B2 51.00
D1 187 D1 46.75
D2 160 D2 40.00
E1 172 E1 43.00
E2 175 E2 43.75
156 A Primer on the Taguchi Method
Computation of Interaction
Interaction effects are always mixed with the main effects
of the factors assigned to the column designated for interaction.
The relative significance of the interaction effects is obtained by
ANOVA, just as are the relative significance of factor effects. To
determine whether two factors, A and C, interact, the following
calculations are performed.
Level totals and their averages for A and C:
A1C1 y1 y2 2 42 50 2 92 2 46.0
A1C2 y3 y4 2 36 45 2 81 2 40.5
A2C1 y5 y6 2 35 55 2 90 2 45.0
A2C2 y7 y8 2 30 54 2 84 2 42.0
51
46.75
45.0
Response
35.75
A1 A2 C1 C2 B1 B2 D1 D2 E1 E2
Factors
52.50
49.50
B2
46.0
A1
Response
45.0 A2 42.0
38.50 40.5
B1 33.0
C1 C2 C1 C2
Factors
422 502 362 542
15051.125
599.88
Analysis of Variance (ANOVA) 159
SA
A1
A2 2
173
174
2
0.125
N A1 NA 2
4 4
SB
B1
B2 2
143
204
2
465.125
N B1 NB 2
4 4
SC
C1
C2 2
182
165
2
36.125
N C1 N C2 4 4
160 A Primer on the Taguchi Method
SD
D1
D2 2
187
160 2 91.125
N D1 N D2 4 4
SE
E1
E2 2
172
175 2 1.125
N E1 N E2 4 4
2
S A sC
A s C
A s C 176
171
1 2
2
3.125
N A sC N A sC
1 2
4 4
2
S B sC
B s C
B s C 176
171
1 2
2
3.125
N B sC N B sC
1 2
4 4
Se ST
S A SB SC SD SE S A sC SB sC
599.88
0.125 465.125 36.125 91.125
91.125 3.125 3.125
599.88
599.88 0
where
NA = total number of experiments where factor A1 is present
1
NB = total number of experiments where factor B1 is present
1
A1 = sum of results (Yi) where factor A1 is present
B1 = sum of results (Yi) where factor B1 is present
Step 5. Total and factor degrees of freedom (DOF):
DOF total = number of test runs minus 1
or fT = n – 1 = 8 – 1 = 7
DOF of each factor is 1 less than the number of levels:
fA = (number of levels of factor A) – 1
=2–1=1
fB = (number of levels of factor B) – 1
=2–1=1
fC = (number of levels of factor C) – 1
=2–1=1
Analysis of Variance (ANOVA) 161
Pooling
Note that in Step 7 the effects of factors A and E and interac-
tions B×C and A×C are small, totaling slightly more than 1%
(1.2%). These factors are pooled to obtain new, non-zero estimates
of Se and fe.
Sum of squares of error term:
Let: Se = SA + SE + SA × C + SB × C
then Se ST
SB SC SD 599.9
592.4 7.5
larger effects until the total pooled DOF equals approximately half
of the total DOF. The larger DOF for the error term, as a result of
pooling, increases the confidence level of the significant factors.
Note that as small factor effects are pooled, the percentage
contributions and the confidence level of the remaining factors
decrease (PC = 5.71 versus PC = 6.02). By pooling, the error term
is increased and, in comparison, the other factors appear less in-
fluential. The greater the number of factors pooled, the worse the
unpooled factor effects look. Then we must consider why column
effects are pooled.
Error variance represents the degree of inter-experiment error
when the DOF of the error term is sufficiently large. When the
error DOF is small or zero, which is the case when all columns
of the OA are occupied and trials are not repeated, small column
effects are successively pooled to form a larger error term (this
is known as a pooling-up strategy). The factors and interactions
that are now significant, in comparison with the larger magnitude
of the error term, are now influential. Taguchi prefers this strat-
egy as it tends to avoid the mistake (alpha mistake) of ignoring
helpful factors.
A large error DOF naturally results when trial conditions are
repeated and standard analysis is performed. When the error DOF
is large, pooling may not be necessary. Therefore, one could repeat
Analysis of Variance (ANOVA) 165
the experiment and avoid pooling, but to repeat all trial conditions
just for information on the error term may not be practical.
A sure way to determine if a factor or interaction effect should
be pooled is to perform a test of significance (1 – confidence level).
But what level of confidence do you work with? No clear guidelines
are established. Generally, factors are pooled if they do not pass
the test of significance at the confidence level assumed for the
experiment. A factor is considered significant if its experimental
F-ratio exceeds the standard table value at a confidence level. A
common practice is to subjectively assume a confidence level be-
tween 85% and 99%, with 90% or 95% being a popular selection.
Consider factor C in Example 6-5, which has 5.7% influence (19.267
F-ratio). When tested for significance, this factor shows more than
99% confidence level and thus should not be pooled.
From the ANOVA table
FC = 19.267
From the F-table, find the F value at
n1 = DOF of factor C = 1
n2 = DOF of error term = 4
at a confidence level (say, the 99% confidence level).
F = 21.198 (from Table C-4)
As FC from the experiment (19.267) is smaller than the F-table
value (21.198), factor C should be pooled.
SUMMARY RESULTS
Description of the factor = Factor C
Column the factor is assigned to =2
Variance ratio for this factor = 19.267
DOF of the factor =1
DOF of error term, fe =4
Confidence level % = 99
Based on the level of confidence desired (99%), the following
is recommended:
“Pool this factor”
166 A Primer on the Taguchi Method
Thus for factor C at level C1, the C.I. is calculated by first deter-
mining the F factor:
n2 = 4
Ne = 8/(1+1) = 4
F (1,4) = 7.7086 at 95% confidence level
Analysis of Variance (ANOVA) 167
T B1
T D 2
T C 2
T
43.375 35.75
43.375 40
43.375 41.25
43.375
30.25
Note that the optimum condition for the “smaller is better” quality
characteristic is B1 C2 D2. The average values at these conditions
were previously calculated as summarized in Table 6-19.
168 A Primer on the Taguchi Method
C.I. p F 1, n s V
2 e Ne
where
F (1,n2) = F value from the F-table at a required confidence
level and at DOF 1 and error DOF n2
Ve = variance of error term (from ANOVA)
Ne = effective number of replications
Total number of results or number of S/N ratios
DOF of mea an =1, always + DOF of all factors
included in the estimate of the mean
Three factors, B1, C2, and D2, are included in calculating the
estimate of the performance at the optimum condition. There-
fore, the effective number of replications, the F value, and the
confidence intervals are calculated as shown below. A confidence
level of 85% to 99% is the normal range of selection for common
industrial experiments. A 90% confidence level is arbitrarily se-
lected for the following calculations.
Analysis of Variance (ANOVA) 169
n2 =4
Ne = 8/(1+3) = 2
F (1,4) = 4.5448 at the 90% confidence level
Ve = 1.88
C.I. = ±2.067 at the 90% confidence level
Therefore, the result at the optimum condition is 22.0 ± 2.067 at
the 90% confidence level.
SUMMARY RESULTS
SB
B1
B2 2
429
612
2
1395.375
N B1 N B2 24
VB SB fB 1395.375 1 1395.375
FB VB Ve 1395.375 39.72 35.126
SBa SB
Ve s fB 1395.375
39.72 1355.65
PB 100 s SBa ST 100 s 1355.655 2571.63 52.70%
factors not designed into the experiment that influence the out-
come. These uncontrollable factors are called the noise factors, and
their effect on the outcome of the quality characteristic under test
is termed “noise.” The signal-to-noise ratio (S/N ratio) measures
the sensitivity of the quality characteristic being investigated in
a controlled manner to those influencing factors (noise factors)
not under control. The concept of S/N originated in the electri-
cal engineering field. Taguchi effectively applied this concept to
establish the optimum condition from the experiments.
The aim of any experiment is always to determine the highest
possible S/N ratio for the result. A high value of S/N implies that
the signal is much higher than the random effects of the noise fac-
tors. Product design or process operation consistent with highest
S/N always yields the optimum quality with minimum variance.
From the quality point of view, there are three typical catego-
ries of quality characteristics:
1. Smaller is better; for example, minimum shrinkage in a
cast iron cylinder block casting.
2. Nominal is best; for example, dimension of a part consis-
tently achieved with modest variance.
3. Bigger is better; for example, maximum expected life of a
component.
The S/N analysis is designed to measure quality characteristics.
MSD Y12 Y22 YN2 N (6-17)
174 A Primer on the Taguchi Method
2 2
MSD Y1
Y0 Y2
Y0 YN
Y0
2
N (6-18)
The bigger is better quality characteristic:
MSD 1 Y12 1 Y22 1 YN2 N (6-19)
The MSD is a statistical quantity that reflects the deviation from
the target value. The expressions for MSD are different for differ-
ent quality characteristics. For the nominal is best characteristic,
the standard definition of MSD is used. For the other two charac-
teristics, the definition is slightly modified. For smaller is better,
the unstated target value is zero. For larger is better, the inverse
of each large value becomes a small value and, again, the unstated
target is zero. Thus, for all three MSD expressions, the smallest
magnitude of MSD is being sought. In turn, this yields the greatest
discrimination between controlled and uncontrolled factors. This
is Taguchi’s measure for robust product or process design.
Alternate forms of definitions of the S/N ratios exist ([6], pp.
172-173), particularly for the nominal is best characteristic. The
definition in terms of MSD is preferred as it is consistent with
Taguchi’s objective of reducing variation around the target. Con-
version to S/N ratio can be viewed as a scale transformation for
convenience of better data manipulation.
Target value
X = 60.2 X = 75
S/N = –23.05 S/N = –25.82
A B
0 20 40 60 80 100 120
value that equals the target value, but it has a wide spread around
it. On the other hand, for set A, the spread around its average is
smaller, but the average itself is quite far from the target. Which
one of the two is better? Based on average value, the product shown
by observation B appears to be better. Based on consistency, prod-
uct A is better. How can one credit A for less variation? How does
one compare the distances of the averages from the target? Surely,
comparing the averages is one method. Use of the S/N ratio offers
an objective way to look at the two characteristics together.
has the same units as the original recorded data. The degrees of
freedom for the experiment (DOF column in ANOVA table) is 11
(4 × 3 − 1).
Comparing the standard analysis with the analysis using the
S/N ratio [Table 6-25(b)], note that the average value of the re-
sults is replaced by the S/N ratio. The S/N ratios are then used to
compute the main effects as well as the estimated performance at
the optimum condition. Notice also that the degrees of freedom
for the experiment is 3. This difference in DOF produces a big
difference in the way the two analyses compute ANOVA, that is,
the percentage contribution of the factors involved (for spring
rate, the value is 23.6% from standard analysis as compared with
48.79% from S/N analysis). Likewise, the other factors will have
different magnitudes of contribution in the two methods.
In estimating the result at the optimum condition, only the
factors that will have significant contributions are included. In this
case, both methods selected level 1 of factors in columns 1 (spring
rate) and 2 (cam profile). This may not always be true.
When the S/N ratio is used, the estimated result can be con-
verted back to the scale of units of the original observations. For
example, the expected result in terms of S/N ratio is −29.9425
[Table 6-25(b), bottom line]. This is equivalent to an average
performance, Y, which is calculated as follows:
Because
S N
10 log MSD
and
MSD Y12 YN2 N for smaller is better
Yexpected
2
Therefore,
MSD 10(
S/N ) /10 10
(
29.9425 ) /10 986.8474
or
Yexpected MSD 1 / 2 986.8474 1 / 2 31.41
which is comparable to 30.5 shown at the bottom of Table 6-25(a).
182 A Primer on the Taguchi Method
EXERCISES
6-1. In an experiment involving four factors (A, B, C, and D) and
one interaction (A × B), each trial condition is repeated three
times and the observations recorded as shown in Table 6-26.
Determine the total sum of squares and the sum of squares
for factor A.
6-2. Assuming the “bigger is better” quality characteristic, trans-
form the results of trial 1 (Table 6-26) into the corresponding
S/N ratio.
6-3. Table 6-27 shows the product of ANOVA performed on the
observed results of an experiment. Determine the following
from the ANOVA table.
a. Percent influence of the clearance factor.
b. Degrees of freedom of the speed factor.
c. Error degrees of freedom.
d. Influence of noise factors and all other factors not included
in the experiment.
e. Confidence interval (90%) of the performance at the opti-
mum condition (use F-table for 90% confidence level).
Analysis of Variance (ANOVA) 183
Target value
•
•
15
Loss in dollars
10
• •
5
• •
• •
0 •
–6 –4 –2 0 2 4 6
Y (quality characteristics)
Target value
20 • •
N N
° K=2 °
• •
15
Loss in dollars
K=3
K=1
• ° N N ° •
10
• •
° N N
°
• •
5
• ° ° •
N N
•
° °• •
•
°• •N •°
N N
0
–6 –4 –2 0 2 4 6
Y (quality characteristics)
L0
or k
$2
Therefore, Eq. (7-1) for this case becomes:
L0
L Y
$2
Y
Y0 2
The above equation now completely defines the loss function
in terms of the deviation from the target value.
TEST DATA
Before Experiment
11.80 12.30 12.20 12.4 12.1 12.2 11.9 11.8 11.85 12.15
After Experiment
11.9 12.2 12.1 12.2 12.1 12.1 11.9 11.95 11.95 12.1
Other Data:
Target value = 12.00 in.
Tolerance = ±0.35 in.
Cost of rejection = $20.00
Production rate = 1500 per month
Solution
For this application, the expression of loss in terms of the
MSD will be used.
and
L(Y) = k(MSD) for multiple samples
Using Eq. (7-1), the constant k is determined as follows:
L = k(Y − Y0)2
When all parts are made just outside of the specifications,
that is, when
Y =Y0 ± Tolerance,
then L = k(Y0 ± Tolerance − Y0)2
But the loss L in this case equals the cost of rejecting a part
($20.00), and the tolerance is 0.35.
or 20 = k(.35)2
or k = 20/(.35)2 = 163.265
Therefore, from Eq. (7-4),
L = 163.265 (MSD) (7-5)
Using the data from samples before the experiment,
2 2
MSD ¨11.8
12 12.3
12 · 10
ª ¹
0.0475
The MSD and other statistical parameters for this example, as
shown in Tables 7-1, 7-2, and 7-3, are obtained by using the soft-
ware in [7]. The format of the design descriptions and the results
are presented in the manner displayed by the software.
From Eq. (7-5) the average loss per unit is calculated as:
L = 163.265 × .0475 = 7.754 (in dollars)
Using the data from samples after the experiment,
2 2
MSD ¨11.9
12 12.2
12 · 10
ª ¹
0.0145
From Eq. (7-6), the average loss per unit is calculated to be:
L = 163.265 × .0145 = 2.367 (in dollars)
192 A Primer on the Taguchi Method
Example 7-2
Dryer Motor Belt (Manufacturer/Supplier Tolerance)
Alarmed by a high rate of warranty repairs of drive belts for
one of its products, the distributor sought to reduce such defects.
The field reports suggested that the problem was mainly caused
by the lack of adjustment of tension in the drive belt. To correct
194 A Primer on the Taguchi Method
customer is the one who uses the product and experiences its per-
formance. In this example, the distributor and the customers are
considered to be the end users. The customer and the manufacturer
may have a supplier (not identified) for the motor and belt assem-
bly. The relationships among the three can be represented in the
following way.
Supplier m Manufacturer m Customer
(Belt + motor) (Washer) (Washer in use)
Tolerance required:
(unknown) (unknown) (±15 lbs.)
From Eq. (7-1) we have
L(Y) = k(Y − Y0)2 = k (Tolerance)2
where
Tolerance = Y(max. or min.) − Y0
2.0
1.5
Frequency
1.0
0.5
0.0
100
S
M
C
REQUIRED TOLERANCES
Manufacturer tolerance = 100 ± 9.18
Supplier tolerance = 100 ± 5.3
NOTE: If these tolerances are held, there will be no nonfunctional part in the
customer’s hands. For the same cost, the manufacturer will maintain satisfied
customers and quality products in the field.
COLUMN 1 2 3
TRIAL
Trial 1 1 1 1
Trial 2 1 2 2
Trial 3 2 1 2
Trial 4 2 2 1
Table 7-4(b). Fuel pump noise study (Result: Main effect and ANOVA)
Original Observations and Their S/N Ratios
Quality Characteristic: Nominal is Best
REPETITION R1 R2 R3 R4 R5 R6
TRIAL S/N
1 67.00 85.00 87.00 65.00 59.00 76.00 –20.71
2 65.00 65.00 66.00 54.00 73.00 58.00 –18.99
3 54.00 45.00 56.00 45.00 63.00 46.00 –25.89
4 56.00 67.00 45.00 54.00 56.00 74.00 –23.36
Main Effects
COLUMN FACTOR LEVEL 1 LEVEL 2 L 2 – L1 LEVEL 3 LEVEL 4
1 Seal thickness –19.85 –24.63 –4.78 00.00 00.00
2 Rotor chuck type –23.30 –21.18 2.12 00.00 00.00
3 Finger to drive –22.04 –22.44 –0.41 00.00 00.00
ANOVA Table
SUM OF
COLUMN FACTOR DOF SQUARES VARIANCE F PERCENT
1 Seal thickness 1 22.801 22.801 82.97
2 Rotor chuck type 1 4.512 4.512 16.43
3 Finger to drive 1 0.164 0.164 00.60
All other/error 0
Total: 3 27.480 100.00
Current Condition
Average performance aligned with the target value (assumed
for simplistic calculation) is:
200 A Primer on the Taguchi Method
Table 7-4(c). Fuel pump noise study (Optimum and cost savings)
Estimate of Optimum Condition of Design/Process
Quality Characteristic: Nominal is Best
FACTOR DESCRIPTION LEVEL DESCRIPTION LEVEL CONTRIBUTION
Seal thickness Present design 1 2.3875
Rotor chuck type New design 2 1.0625
Finger to drive clearance Present design 1 0.2025
Contribution from all factors (total) 3.6524
Current grand average of performance –22.2375
Expected result at optimum condition –18.5850
This estimate includes only those variables that have a significant contribution; that is, pooled variables
are excluded from the estimate. Estimates may also be made with variables of choice.
CALCULATION OF LOSS
PROBLEM DEFINITION
Target value of quality characteristic (m) = 70.00
Tolerance of quality characteristic = 20.00
Cost of rejection at production (per unit) = $45.00
Units produced per month (total) = 20000
S/N ratio of current design/part = –20.71
S/N ratio of new design/part = –18.585
COMPUTATION OF LOSS USING
TAGUCHI LOSS FUNCTION
Loss function: L(y) = 0.11 × (MSD) Also L(y) = K × (y – m)2
BEFORE EXPERIMENT:
Loss/unit due to deviation from target
in current design = $12.953
AFTER EXPERIMENT:
Loss/unit due to deviation from target
will be reduced from $12.953 to = $7.941
MONTHLY SAVINGS:
If production is maintained at the improved
condition, then based on 20000 units/month = $100,246.90
Improved Condition
After completing the experimental study for the smaller is
better quality characteristic, the performance at optimum condi-
tion (improved condition) expressed in S/N ratio was estimated
to be:
S/N = 32.081 (or MSD = 1614.73)
1614.73
3348.88
s $1.00 assumed loss in
current condition
$0.483
Thus, Savings (1.00 – 0.483) = 51.7 cents for every dollar spent
at current condition.
202 A Primer on the Taguchi Method
S current
Cp s C p current
S improved
and
S current
C pk s C pk current
S improved
that is
13.402
C p and C pk s1
9.306
1.44
The plot of variation reduced by adopting optimum design
along with statistics calculated above is shown in Figure 7-4 (graph
from [7]). The reduction of variation is expected to lower the rejec-
tion and warranty items, which results in cost savings expressed in
terms of percentage of the loss at the current condition. A single
figure like this can capture the essence of improvement expected
and represent it graphically for all to understand.
EXERCISES
7-1. The manufacturer of a 10.5-volt smoke alarm battery employed
the Taguchi method to determine the better design param-
eters. The experimenters estimated the signal-to-noise (S/N)
ratio for the proposed design to be 6.3. Based on a sample
inspection of the current production process, the S/N ratio was
calculated to be 5.2. The analysis of warranty showed that
when the battery voltage was beyond (10.50 ± 0.75) volts,
the smoke alarm malfunctioned and customers returned the
batteries for $6.50 each. Determine the monthly savings that
the proposed new design is expected to generate if 20,000
units are manufactured each month.
Loss Function 203
Team on site 5
Run Tests to
Facilitator off site Confirm
Solutions
Project team 4
Analyze Test
3 Results and
Carry Out Prescribe
Planned Tests Solutions
and Collect
Results
2
Design and
1 Describe Test
Hold Recipes
Experiment
Planning
Discussions
Team Leader
For the successful completion of a Taguchi case study, the ap-
pointment of a team leader, from among the project team members,
is necessary. The team leader must recognize the need for a brain-
storming session and call for such a session. The leader should
try to hold the session on neutral ground on a pre-announced day.
The leader should ensure the participation of all team members
with responsibilities for the product/process.
Session Facilitator
The session should be facilitated by someone with a good
working knowledge of the Taguchi methodologies. Engineers or
statisticians dedicated to helping others apply this tool often make
better facilitators. A facilitator need not be a participator unless
the project leader facilitates the session. The facilitator initiates
and leads the discussion but never dominates it.
Brainstorming
for Design of Experiments
Determine:
– Control factors
– Noise factors
– Factor levels
Scope of project:
– How many experiments
– How many repetitions
Assign tasks:
– Who does what
Input Output
System
Note: OUTPUT of previous subprocess is INPUT to the next. For example, Batter is output
of the Mixing process but input to the Baking process.
Example Description
“We have been experiencing high rejects and warranty
from our plastic molding process. This study is undertaken
to determine process parameters that will reduce our scrap
rate. The improved process design is also expected to keep our
customers satisfied and affect our bottom line.”
The project description may be composed during or after the
planning session. Below are a few questions to help describe your
projects and define the objectives.
— What are the reasons for performing this project?
— What is it that you want to accomplish with this project?
— What specific objectives/goals you wish to achieve from
this project?
If the study involves baking pound cakes, the objectives may
be considered be to: (a) improve taste, (b) increase moistness, (c)
prolong shelf life.
212 A Primer on the Taguchi Method
Long List
Brainstorm, solicit, and list ideas and suggestions about how
to make improvements and what are the possible sources of influ-
ence. Realize that, by now, all involved on the team already know
what you are after and what are the objectives. The goal here is
to capture a quantity of ideas and list them. All ideas gathered do
not necessarily make valid factors. However, all suggestions and
ideas solicited must be collected without concern for validity. The
time for scrutiny and consideration for study will come later.
Below are sample questions that may initiate thoughts about
factors:
• What are some of the actions you can take to improve and
satisfy performance objectives?
• What are variables (materials/environmental factors/con-
stituents/settings/parameters, etc.) that may influence the
outcome of the project?
• If you have done some process studies and have prepared
cause-and-effect diagrams (fishbone or Ishikawa diagrams),
what are some of the factors that were identified?
If you have a number of people on your project team, this is
a good time to ask ideas from each and every person. You do not
want to leave “any stone unturned.”
If you are alone on the project, or working with a few members
on your team, it is a good idea to pause and attempt to collect
as many ideas as possible. For the preliminary list of ideas, the
longer the list the better chance you have to capture all possible
influencing factors.
Qualified List
After you have captured ideas and have a quantity of them
listed, you will need to qualify them and identify the valid factors
and noise factors by scrutinizing each from the long list. Use the
216 A Primer on the Taguchi Method
Study List
Here you would select factors that you wish to include in the
study. Often this list will be shorter than the qualified list.
When you conduct an effective brainstorming with your team,
it is very likely that you will identify and qualify a larger number
Brainstorming—An Integral Part of the Taguchi Philosophy 217
Example:
Note: Optionally, list interactions and noise factors you wish to include in your study.
Also, indicate the inner and outer arrays used for the experiment design and how
the control factors and noise factors will be assigned to the columns of the arrays.
Based on the final design, indicate the test sample size requirements.
EXERCISES
8-1. In an experiment involving the study of an automobile door
design, two criteria were used for evaluation purposes. Deflec-
tion at a fixed point in the door was measured to indicate the
stiffness, and the door closing effort was subjectively recorded
on a scale of 0 to 10.
a. Develop a scheme to define an overall evaluation criterion.
b. Explain why the overall evaluation may be useful.
8-2. During the brainstorming session for a Taguchi experiment,
a large number of factors were initially identified. Discuss the
type of information that needs to be considered to determine
the number of factors for the experiment, and state how you
will proceed to select these factors.
8-3. A group of manufacturing engineers identified the following
process parameters for an experimental investigation:
• Fourteen two-level factors (not all considered important)
• One interaction between two factors (considered important)
• Three noise factors at two levels each (considered important)
If the total number of trial runs (samples) is not to exceed
32, design the experiment and indicate the sizes of the
inner and outer arrays.
8-4. Brainstorm and carry out the experiment planning session follow-
ing the steps discussed in this chapter. The experimental design
222 A Primer on the Taguchi Method
Problem Description
Engineers and production specialists in a supplier plant wish to
optimize the production of foam seats for automobile manufacturers.
The improvement project has been undertaken because there have
been complaints from the customer about the quality of the delivered
parts. The main defects found in the foam parts are: (1) excessive
shrinkage, (2) too many voids, (3) inconsistent compression set, and
(4) varying tensile strength. There appears to be general agreement
that these are the primary objectives; however, there is no consensus
as to their relative importance (weighting). Most of the individuals
involved are aware that just satisfying one of the criteria may not
always satisfy the others. It is believed that a process design that
produces parts within the acceptable ranges of all of the objective
criteria would be preferable.
Conventional wisdom will dictate that a designed experiment be
analyzed separately using the readings for each of the objectives
(criteria of evaluations). This way, four separate analyses will have to
be performed and optimum design conditions determined. Because
each of these optimums is based only on one objective, there is no
guarantee that they all will prescribe the same factor levels for the
optimum condition. To release the design, however, only one com-
bination of factor levels is desired. Such design must also satisfy all
objectives in a manner consistent with the consensus priority estab-
lished by the project team members.
Combining all of the evaluation criteria into a single index (OEC),
which includes the subjective as well as the objective evaluations, and
also incorporates the relative weightings of the criteria, may produce
the design being sought. Of course, even if the experiment is analyzed
using the overall evaluation criteria (OEC), separate analysis may
still be performed for individual objectives.
Discussions and investigations into possible causes of the sub-
quality parts revealed many variables (not all are necessarily factors),
Brainstorming—An Integral Part of the Taguchi Philosophy 223
such as: (a) chemical ratio, (b) mold temperature, (c) lid close time,
(d) pour weight, (e) discoloration of surface, (f) humidity, (g) index-
ing, (h) flow rate, (i) flow pressure, (j) nozzle cleaning time, (k) type
of cleaning agent, and so on. Most project team members suspect
that there are interactions between the chemical ratio and the pour
weight, and between the chemical ratio and the flow rate. Past studies
also indicated possible nonlinearity in the influence of the chemical
ratio, and thus, four levels of this factor are also desirable for the
experiment. But because there have been no scientific studies done
in the recent past, any objective evidence of interaction or nonlin-
earity is not available. Because of the variability from part to part, it
is a common practice to study a minimum of three samples for any
measurements. The funding and time available for the project is such
that only 30 to 35 samples can be molded. (Your plan and answer
may vary from others’.)
9 Examples of Taguchi
Case Studies
APPLICATION BENCHMARKS
Experiments designed and carried out according to the Taguchi
methodology are generally referred to as case studies. Perhaps
they are called case studies to indicate that they are well planned
experiments and not simply a few tests to investigate the effects
of varying one or more factors at a time. The term case study
may also be used to signify that such planned experiments have
been fruitfully carried out, that the results have been analyzed to
determine the optimum combination of the factors under study,
and that tests to confirm the optimum conditions have been con-
ducted. But what does a case study look like? What are the steps
to be followed in completing a case study?
In Chapters 5 and 6, the mechanics of the Taguchi design of
experiments and the procedure for the analysis of the experi-
mental data were discussed in detail. Those chapters included
several application examples (case studies). The examples in
this chapter are representative of practical problems the author
has encountered during his associations with various industries
and clients.
A typical application of the method will include the following
five major steps (see also Figure B-1):
1. A brainstorming session
2. Designing the experiment
3. Conducting the experiment
4. Analyzing the results
5. Running the confirmation test
226 A Primer on the Taguchi Method
The L8 array is shown in Table 9-1(b). Note that only six col-
umns define the test condition, with the zeros in the unused column
(column 7) showing that no condition is implied. The two-level ar-
ray, L8, describes eight trial conditions. The design may be created
manually, but a computer program will perform such computations
in a matter of seconds and without mathematical errors.
The results of the eight trial conditions, with one run per trial
condition, are shown in Table 9-2(a). Examples in this chapter
utilized computer software [7], which displays up to six repetitions
and their averages. These observed results are used to compute
the main effects of the individual factors [Table 9-2(b)]. Because
228 A Primer on the Taguchi Method
Table 9-2. Original data and their averages (Results and analysis)—
Example 9-1
(a) Original Observations and Their Averages
Quality Characteristic: Smaller is Better
Results: Up to Six Repetitions Shown
REPETITION R1 R2 R3 R4 R5 R6
TRIAL AVERAGE
1 45.00 45.00
2 34.00 34.00
3 56.00 56.00
4 45.00 45.00
5 46.00 46.00
6 34.00 34.00
7 39.00 39.00
8 43.00 43.00
(b) Main Effects
COLUMN FACTOR LEVEL 1 LEVEL 2 L2 – L1 LEVEL 3 LEVEL 4
1 Valve guide clearance 45.00 40.50 –4.50 00.00 00.00
2 Upper guide length 39.75 45.75 6.00 00.00 00.00
3 Valve geometry 40.25 45.25 5.00 00.00 00.00
4 Seat concentricity 46.50 39.00 –7.50 00.00 00.00
5 Lower valve length 44.50 41.00 –3.50 00.00 00.00
6 Valve face runout 44.75 40.75 –4.00 00.00 00.00
(c) ANOVA Table
SUM OF
COLUMN FACTOR DOF SQUARES VARIANCE F PERCENT
1 Valve guide clear. (1) (40.50) Pooled
2 Upper guide length 1 72.00 72.00 2.011 9.96
3 Valve geometry (1) (50.00) Pooled
4 Seat concentricity 1 112.50 112.50 3.142 21.10
5 Lower guide length (1) (24.50) Pooled
6 Valve face runout (1) (32.00) Pooled
the factors have only two levels, the main effects are shown under
the two columns marked Level 1 and Level 2. The third column
labeled (L2 – L1) contains the difference between the main effects
at Level 1 and Level 2. A minus sign (in the difference column)
indicates a decrease in noise as the factor changes from Level 1 to
Level 2. A positive value, on the other hand, indicates an increase
in noise. A quick inspection of the difference column permits
selection of the optimum combination, for example, the “smaller
is better” characteristic. A negative sign in the column (L2 – L1)
indicates Level 2 of the factor is desirable, while a positive value
indicates Level 1 is the choice. This quick inspection is a sufficient
test only when two levels are involved and when all factors are
considered significant.
If the desired characteristic is “bigger is better,” then the level
selection criteria will be the reverse of the scheme given above;
positive values indicate Level 2, and all negative values will in-
dicate the choice of Level 1 for the optimum condition. In this
example with all factors, the optimum condition for “smaller is
better” is levels 2, 1, 1, 2, 2, and 2 for factors in columns 1 through
6, respectively. The sign (±) directs the selection of levels, while
the magnitude suggests the strength of the influence of the factor.
The quantitive measure of the influence of individual factors is
obtained from ANOVA [Table 9-2(c)].
ANOVA follows procedures outlined in Chapter 6. No new data
or decisions on the part of the experimenter are required. This
is an ideal situation for standard computer routines. The results
of ANOVA are shown in Table 9-2(c). A review of the percent
column shows that Upper Guide (9.96%) and Seat Concentricity
(21.10%) are significant. The other insignificant factors are pooled
(combined) with the error term. Based on information from the
ANOVA Table 9-2(c), the mean performance at optimum condi-
tion and the confidence interval are calculated as shown in Tables
9-3(a) and 9-3(b), respectively.
The last step in the analysis is to estimate the performance at the
optimum condition. Normally only the significant factors are used
for this estimate. An examination of main effects indicates which
levels will be included in the optimum condition. In addition, ANOVA
230 A Primer on the Taguchi Method
F 1, n2 Ve
C.I.
Ne
where F(n1, n2) = computed value of F with n1 = 1, n2 = error DOF
at a desired confidence level
Ve = error variance
Ne = effective number of replications
Based on: F = 3.2999999, n1 = 1, n2 = 5, Ve = 35.8, Ne = 2.6667 [from 8/(1+2)]
The confidence interval C.I. = ±6.656011, which is the variation of the estimated result
at the optimum condition; that is, the mean of the result, m, lies between (m + C.I.) and
(m – C.I.) at 89.93% confidence level.
Solution—Example 9-2
With six factors and one interaction involved in this study, an
L8 orthogonal array (OA) is suitable for the experimental design.
The first step is to decide where to assign the interacting factors
and which column to reserve for their interaction. The table of
interaction (Table A-6) for two-level orthogonal arrays shows that
columns 1, 2, and 3 form an interacting group. The two interact-
ing factors are therefore assigned to columns 1 and 2. Column 3
is kept aside for their interaction. The remaining four factors are
then assigned to any of the four remaining columns. The completed
design, with descriptions of factors, their levels, and the orthogonal
array, are shown in Tables 9-4(a) and 9-4(b). Eight crankshafts
were fabricated to the specifications described by the eight trial
conditions. Each sample was tested for durability (life). Because
longer life was desirable, the quality characteristic applicable in
this case was “bigger is better.”
The observed durability, the main effects, and the unpooled
ANOVA are shown in Table 9-5. The study of the main effects in-
dicates some interaction between the factors. This is shown by the
magnitude 1.25 in the column labeled (L2 – L1) in Table 9-5(b). This
value is of the same order of magnitude as the values 3.25, –6.25,
4.25, and so on. But is the interaction significant? The answer to
this question can be obtained from the percentage column of the
ANOVA table [Table 9-5(c)]. The interaction under column 3 is
only 0.74%. Contributions below 5% are generally not considered
significant. The interaction and the factor in column 5, which has
1.46 in the percentage column, are pooled. The pooled ANOVA is
232 A Primer on the Taguchi Method
3 Interaction N/A
Example 9-3
Automobile Generator Noise Study
Engineers identified one four-level factor and four two-level
factors for experimental investigation to reduce the operating
noise of a newly released generator. Taguchi methodologies were
followed to lay out the experiments and analyze the test results.
Examples of Taguchi Case Studies 235
Solution—Example 9-3
The factors in this example present a mixed-level situation.
Although experiment design is simplified if all factors have the
same level, it is not always possible to compromise the factor level.
For instance, if a factor influence is believed to be nonlinear, it
should be assigned three or more levels. The factor and its influ-
ence are assumed to be continuous functions. If, however, the
factor assumes discrete levels such as design type 1, design type
2, and so on, then the influence is a discrete function, and each
discrete step (level) must be incorporated in the design. The four-
level factor in the example is discrete. Because the four-level factor
has 3 DOF, and four two-level factors each have 1 DOF, the total
DOF for the experiment is 7. An L8 with seven two-level columns
and 7 DOF was selected for the design. The first step provides for
the four-level factor. Columns 1, 2, and 3 of L8 are used to prepare
a four-level column. This new four-level column now replaces col-
umn 1 and is assigned to the four-level factor. As columns 2 and 3
were used to prepare column 1 as a four-level column, they cannot
be used for any other factor. Thus, the four two-level factors are
assigned to the remaining columns 4, 5, 6, and 7. The design and
the modified OA are shown in Tables 9-7(a) and 9-7(b).
One run at each trial condition was tested in the laboratory,
and the performance was measured in terms of a noise index.
The index ranged between 0 (low noise) and 100 (loud noise). The
lower value of this index was desirable. The data and calculated
main effects are shown in Tables 9-8(a) and 9-8(b), respectively.
Note that the four-level factor in column 1 (Casement Structure)
has its main effects at the four levels. This factor has 3 DOF
as noted in the ANOVA table [Table 9-9(a)] under the column
marked DOF.
The ANOVA table clearly shows that the factor in column 6
(Contact Brushes) has the smallest sum of squares and hence the
least influence. This factor is pooled and the new ANOVA is in Table
9-9(a). Using the significant contributors, the estimated performance
at the optimum condition was calculated as 49.375. In this case, the
optimum condition is trial 1 (Levels 1 1 1 1 1). The result for trial 1
was 50 [Table 9-8(a)]. The difference between the trial result and the
236 A Primer on the Taguchi Method
2 (Unused)
3 (Unused)
Solution—Example 9-4
The smallest three-level OA, L9, has four three-level columns.
With three three-level factors in this study, the L9 is appropriate
for the design. The factors are placed in the first three columns,
leaving the fourth column unused. The factors, their levels, and
the modified OA are shown in Tables 9-10(a) and 9-10(b).
The performance of the engine tested under various trial condi-
tions was measured in terms of the deviation of the speed from a
nominal idle speed. A smaller deviation represented a more stable
condition. Three separate observations were recorded for each trial
condition, as shown in Table 9-11(a). The signal-to-noise (S/N)
ratio was used for the analysis of the results. The main effects,
optimum condition, and ANOVA table are shown in Tables 9-11
and 9-12. Based on the error DOF and variance, the confidence
238 A Primer on the Taguchi Method
4 (Unused)
Note: Three three-level factors studied.
Objective: Determine best engine setting.
Characteristic: Smaller is better (speed deviation).
COLUMN 1 2 3 4
TRIAL
Trial 1 1 1 1 0
Trial 2 1 2 2 0
Trial 3 1 3 3 0
Trial 4 2 1 3 0
Trial 5 2 2 1 0
Trial 6 2 3 2 0
Trial 7 3 1 2 0
Trial 8 3 2 3 0
Trial 9 3 3 1 0
Table 9-11. Engine idle stability study (Main effects and ANOVA)—
Example 9-4
(a) Original Observations and Their S/N Ratios
Quality Characteristic: Smaller is Better
Results: Up to Six Repetitions Shown
REPETITION R1 R2 R3 R4 R5 R6
TRIAL S/N
1 20.00 25.00 26.00 –27.54
2 34.00 36.00 26.00 –30.19
3 45.00 34.00 26.00 –31.10
4 13.00 23.00 22.00 –25.96
5 36.00 45.00 35.00 –31.81
6 23.00 25.00 34.00 –28.87
7 35.00 45.00 53.00 –33.06
8 56.00 46.00 75.00 –35.60
9 35.00 46.00 53.00 –33.12
(b) Main Effects
COLUMN FACTOR LEVEL 1 LEVEL 2 L2 – L1 LEVEL 3 LEVEL 4
1 Indexing –29.61 –28.88 0.73 –33.93 00.00
2 Overlap –28.85 –32.53 –3.69 –31.03 00.00
3 Spark advance –30.67 –29.76 0.91 –31.99 00.00
(c) ANOVA Table
SUM OF
COLUMN FACTOR DOF SQUARES VARIANCE F PERCENT
1 Indexing 2 44.636 22.318 932.80 61.26
2 Overlap 2 20.541 10.271 429.27 28.15
3 Spark advance 2 7.565 3.783 158.10 10.33
All other/error 2 0.05 0.26
Total: 8 72.79 100.00
Overlap 0% 1 1.9522
F 1, n2 Ve
C.I.
Ne
where F(n1, n2) = computed value of F with n1 = 1, n2 = error DOF
at a desired confidence level
Ve = error variance
Ne = effective number of replications
Based on: F = 5.999996, n1 = 1, n2 = 2, Ve = 2.392578E,
Ne = 1.285714 [from 9/(1+6)]
The confidence interval C.I. = ±0.3341461, which is the variation of the estimated
result at the optimum condition; that is, the mean of the result, m, lies between (m + C.I.)
and (m – C.I.) at 89.77% confidence level.
Example 9-6
Study Leading to Selection of Worst-Case Barrier Test Vehicle
To assure that the design of a new vehicle complies with all
of the applicable Federal Motor Vehicle Safety Standards (FM-
VSS) requirements, engineers involved in the crashworthiness
development of a new vehicle design want to determine the worst
combination of vehicle body style and options. This vehicle is to be
used as the test specimen for laboratory validation tests instead of
242 A Primer on the Taguchi Method
Example 7-3. The factors, their levels, and the modified OA are
shown in Tables 9-15(a) and 9-15(b). The description of the trial
conditions derived from the designed experiment served as the
specifications for the test vehicle. For barrier tests, the specimens
are prototype vehicles built either on the production line or are
handmade, one-of-a-kind test vehicles. In either case, the cost for
preparing the test vehicles could easily run in the hundreds of
thousands of dollars. Proper specification, in a timely manner, is
crucial to the cost effectiveness of the total vehicle development
program. For the purpose of the tests, eight vehicles were built on
the production line following the specifications that correspond
to the eight trial conditions. These vehicles were barrier tested
and the results recorded in terms of a predefined occupant injury
244 A Primer on the Taguchi Method
COLUMN 1 2 3 4 5 6 7
TRIAL
Trial 1 1 0 0 1 1 1 1
Trial 2 1 0 0 2 2 2 2
Trial 3 2 0 0 1 1 2 2
Trial 4 2 0 0 2 2 1 1
Trial 5 3 0 0 1 2 1 2
Trial 6 3 0 0 2 1 2 1
Trial 7 4 0 0 1 2 2 1
Trial 8 4 0 0 2 1 1 2
index. The results and the analyses are shown in Tables 9-16 and
9-17. By using eight test vehicles, the engineers were able to learn
the worst vehicle configuration. This information was then used
to adhere to several of the compliance regulations.
Example 9-7
Airbag Design Study
Engineers involved in the development of an impact-sensi-
tive inflatable airbag for automobiles identified nine four-level
Examples of Taguchi Case Studies 245
F 1, n2 Ve
C.I.
Ne
where F(n1, n2) = computed value of F with n1 = 1, n2 = error DOF
at a desired confidence level
Ve = error variance
Ne = effective number of replications
Based on: F = 5.999996, n1 = 1, n2 = 2, Ve = 24.125,
Ne = 1.333333 [from 8/(1+5)]
The confidence interval C.I. = ±10.4
Solution—Example 9-7
Because the experiment involves nine four-level factors, an
L32 with nine four-level columns and one two-level column was
selected for the design. Because there is no two-level factor in this
design, the two-level column (column 1) of the OA shown in Table
9-18(b) is set to zero. The factors, their levels, and the analyses
are shown in Tables 9-18(a) through 9-20(b). The study was done
using a theoretical simulation of the system. The trial conditions
were used to set up the input conditions for the computer runs.
The results of the computer runs at each of the trial conditions
Examples of Taguchi Case Studies 247
Table 9-19. Airbag design study (Main effects and ANOVA)—Example 9-7
(a) Main Effects
COLUMN FACTOR LEVEL 1 LEVEL 2 L2 – L1 LEVEL 3 LEVEL 4
2 Steering column rotation 6.19 6.13 –0.07 5.00 5.94
3 Steering column crush 6.25 4.94 –1.32 6.00 6.06
stiffness
4 Knee bolster stiffness 6.38 5.88 –0.50 5.25 5.75
5 Knee bolster location 4.94 5.94 1.00 6.38 6.00
6 Inflation rate 6.50 6.06 –0.44 4.94 5.75
7 Development time 6.19 5.56 –0.63 6.38 5.13
8 Vent size 5.63 6.56 0.93 5.06 6.00
9 Bag size (E-7 mm) 5.88 5.19 –0.69 6.13 6.06
10 Maximum bag pressure 6.63 5.50 –0.88 5.00 6.38
F 1, n2 Ve
C.I.
Ne
where F(n1, n2) = computed value of F with n1 = 1, n2 = error DOF
at a desired confidence level
Ve = error variance
Ne = effective number of replications
Based on: F = 2.600, n1 = 1, n2 = 19, Ve = 1.851974,
Ne = 2.461539 [from 32/(1+12)]
The confidence interval C.I. = ±1.3
COLUMN 1 2 3
TRIAL
Trial 1 1 1 1
Trial 2 1 2 2
Trial 3 2 1 2
Trial 4 2 2 1
(c) Original Observations and Their S/N Ratios
Quality Characteristic: Nominal is Best
REPETITION R1 R2 R3 R4 R5 R6
TRIAL S/N
1 12.00 14.00 11.00 –6.37
2 18.00 16.00 15.00 –8.46
3 14.00 15.00 15.00 –1.76
4 19.00 18.00 15.00 –11.47
Example 9-10
Electronic Connector Spring Disengagement Force Study
A manufacturer of precision electronic switch assemblies was
experiencing high rejects with one of its connectors. This con-
nector consists of insertion of a solid, screw-machined pin into a
flexible sleeve. The design created a compliant sleeve to generate
sufficient spring force between the sleeve and the gold-plated,
stamped metal pin similar to that shown in Figure 9-1. The plant
has been producing the pin for several years. But recently, for
some causes unknown, there has been higher than acceptable vari-
Examples of Taguchi Case Studies 253
All other/error 0
Total: 3 96.250 100.00
(c) Estimate of Optimum Condition of Design/Process:
For Nominal is Best Characteristic
FACTOR LEVEL
DESCRIPTION DESCRIPTION LEVEL CONTRIBUTION
Lower rail section Proposed design 2 1.2800
Upper rail section Open section 1 3.8300
Cross member Reinforced design 2 2.7850
Contribution from all factors (total) 7.8950
Current grand average of performance –6.1350
Expected result at optimum condition 1.7600
This estimate includes only those variables that have a significant contribution; that is, pooled variables
are excluded from the estimate. Estimates may also be made with variables of choice.
Solution—Example 9-10
The experiment was designed using a standard L-8 array by
assigning seven two-level factors in the order shown (Table 9-24).
Five sets of samples, with multiple fabricated parts in each set,
were tested in each of the eight trial conditions. Description of
an example trial condition (#3) is shown in Table 9-25. Upon
completion of the tests, results (Table 9-26) were analyzed using
S/N of results for the “bigger is better” quality characteristic.
Computer software [7] was used to perform the analysis and
draw conclusions.
Table 9-26. Experimental results and S/N for trials (Bigger is better)—
Example 9-10
CONDITIONS SAMPLE SAMPLE SAMPLE SAMPLE SAMPLE SAMPLE
#1 #2 #3 #4 #5 #6
TRIAL S/N RATIO
1 1.57 1.69 1.685 1.74 1.821 4.584
2 3.335 3.425 3.62 2.815 2.773 9.933
3 1.991 2.036 2.428 2.521 3.037 7.31
4 1.27 1.295 1.303 1.29 1.192 2.062
5 3.275 3.735 4.167 4.132 2.915 10.982
6 1.288 1.256 1.342 1.286 1.277 2.204
7 2.091 1.986 1.927 1.925 1.97 5.92
8 1.348 1.5 1.425 1.345 1.418 2.945
Avg. = 5.742
All results: Avg. = 2.111, Std. dev. = .89
40.3
2.1
G: E: D: A: F: Error
Spring Spring Sleeve Spring Metal
OD c. rad. ID gap hardness
(Date: 2/2/2009-File:DITECH0A.Q4W)
Recreated from Qualitek-4 screenshot
For more examples with S/N analysis, noise factors, and OEC,
readers are referred to [2–5, 8].
xii A Primer on the Taguchi Method
BIBLIOGRAPHY
American Supplier Institute, Inc. 1985. 3rd Supplier Symposium
on the Taguchi Method, Oct. 8, 1985. Dearborn, Mich.
Baker, Thomas B., and Clausing, Donald P. 1984. Quality engi-
neering—by design. 40th Annual Rochester Section Quality
Control Conference, March 6, 1984.
Burgam, Patrick M. 1985. Design of experiments—the Taguchi
way. Manufacturing Engineering May 1985:44–46.
Byrne, Diane M., and Taguchi, Shin. 1987. The Taguchi approach
to parameter design. Quality Progress December 1987.
Cochran, W.G., and Cox, G.M. 1992. Experimental designs, 2nd
ed. New York: John Wiley & Sons.
Gunter, Berton. 1987. A perspective on the Taguchi methods.
Quality Progress June 1987:44–52.
Iman, Ronald L., and Conover, W.J. 1983. A modern approach to
statistics. New York: John Wiley & Sons.
Peace, Glen Stuart. 1992. Taguchi methods. New York: Addison-
Wesley.
Phadke, Madhav S. 1989. Quality engineering using robust design.
Englewood Cliffs, N.J.: PTR Prentice Hall.
Quinlan, Jim. 1985. Product improvement by application of Tagu-
chi methods. Midvale, Ohio: Flex Products, Inc.
Sullivan, Lawrence P. 1987. The power of Taguchi methods. Qual-
ity Progress 12(6):76–79.
Taguchi, G., and Konishi, S. 1987. Orthogonal arrays and linear
graphs—tools for quality engineering. Dearborn, Mich.: Ameri-
can Supplier Institute, Inc.
Wu, Yuin, and Moore, Willie Hobbs. 1986. Quality engineering—
product and process optimization. Dearborn, Mich.: American
Supplier Institute, Inc.
TO OUR GRANDSON, CIARAN, AND OUR GRANDDAUGHTER, KAMALA
Glossary 295
GLOSSARY
ANOVA (Analysis of Variance)
An analysis of variance is a table of information that displays
the contributions of each factor.
Controllable Factor
A design variable that is considered to influence the response
and is included in the experiment. Its level can be controlled
by the experimenter.
Design of Experiment
A systematic procedure to lay out the factors and conditions
of an experiment. Taguchi employs specific partial factorial
arrangements (orthogonal arrays) to determine the optimum
experiment design.
Factorial Experiment
A systematic procedure in which all controllable factors except
one are held constant as the variable factor is altered discretely
or continuously.
Error
Amount of variation in the response caused by factors other
than controllable factors included in the experiment.
Inner Array
Describes the combination of control factors and layout of the
design of experiment.
Interaction
Two factors are said to have interaction with each other if
the influence of one on the response function is dependent
on the value of the other.
Linear Graph
A graphical representation of relative column locations of fac-
tors and their interactions. Linear graphs were developed by
Dr. Taguchi to assist in assigning different factors to columns
of the orthogonal array.
Loss Function
A mathematical expression proposed by Dr. Taguchi to quan-
titatively determine the additional cost to society caused by
the lack of quality in a product. This additional cost is viewed
296 A Primer on the Taguchi Method
Signal Factor
A factor that influences the average value but not the vari-
ability in response.
S/N (Signal-to-Noise) Ratio
Ratio of the power of a signal to the power of the noise (error).
A high S/N ratio means that there is high sensitivity with the
least error of measurement. In Taguchi analysis using S/N
ratios, a higher value is always desirable regardless of the
quality characteristic.
System Design
The design of a product or process using special Taguchi
techniques.
Taguchi Design
A methodology to increase quality by optimizing system design,
parameter design, and tolerance design. This text deals with
system design.
Target Value
A value that a product is expected to possess. Most often this
value is different from what a single unit actually exhibits.
For a 9-volt transistor battery, the target value is 9 volts.
Tolerance Design
A sophisticated version of parametric design that is used
to optimize tolerance, reduce costs, and increase customer
satisfaction.
Variables (or Factors or Parameters)
These words are used synonymously to indicate the control-
lable factors in an experiment. In the case of a plastic molding
experiment, molding temperature, injection pressure, set time,
and so on, are the factors.
Variation Reduction
Variation in the output of a process produces nonuniformity
in the product and is perceived as an important criteria for
quality. Reduced variation increases customer satisfaction
and reduces warranty cost arising from variation. To achieve
better quality, a product must perform optimally and should
have less variation around the desired critical characteristic
for quality.
WHAT READERS ARE SAYING...
“…a clear, step-by-step guide to the Taguchi design of experi-
ments method. The careful descriptions, calculations and examples
demonstrate the versatility of these practical and powerful tools.”
— Fred Schenkelberg, consultant, FMS Reliability, Los Gatos,
Calif.
“Dr. Roy’s book lists many application examples that can help
engineers use the Taguchi method effectively.”
— Dr. Side Zhao, control engineer, NACCO Materials Handling
Group, Portland, Ore.
REFERENCES
[1] Fisher, R[onald] A. 1951. The design of experiments. Ed-
inburgh: Oliver & Boyd.
[2] Nutek, Inc. DOE application resources. http://nutek-us.com/
wp-free.html
[3] Nutek, Inc. Experiment design tips. http://nutek-us.com/wp-
tip.html
[4] Nutek, Inc. Experiment planning steps. http://nutek-
us.com/wp-exptplanning.html
[5] Nutek, Inc. OEC description. http://nutek-us.com/wp-oec.
html
[6] Ross, Philip J. 1988. Taguchi techniques for quality engi-
neering. New York: McGraw-Hill.
[7] Roy, Ranjit K. 1996. Qualitek-4, software for automatic
design and analysis of Taguchi experiments. Bloomfield
Hills, Mich.: Nutek, Inc. Limited-capability working copy
downloadable from www.Nutek-us.com/wp-q4w.html
[8] Roy, Ranjit K. 2001. Design of experiments using the Tagu-
chi approach: 16 steps to product and process improvement.
New York: John Wiley & Sons.
[9] Taguchi, Genichi. 1987. System of experimental design.
New York: UNIPUB, Kraus International Publications.
[10] Wu, Yuin. 1986. Orthogonal arrays and linear graphs.
Dearborn, Mich.: American Supplier Institute, Inc.
Appendix A
3
1 2
(a) 1 (b)
2
3
3 5 5 4
1
7
6
2 4 7
6
1 3
L4 1 2
L8
1
3 5
7 3 14
9
2 4 13
6 2 15
2 L16 11
3 6 5 7
L8
5 10
1 4
4 8
12
6
7 9
3 14
7 1
15 8
13
14 10 7
L16 9
13 L16 11
12 10
1 11 6 5 15
3
5 2
2 4 12 8
6 4
CONDITION
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
4 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
5 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
6 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1
7 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1
8 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2
9 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
10 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1
11 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1
12 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2
13 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1
14 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2
15 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2
16 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1
17 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
18 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
19 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1
20 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2
21 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1
22 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2
23 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2
24 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1
25 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
26 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
27 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 1
28 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
29 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2
30 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1
31 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1
32 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2
Appendix A
265
266
Table A-5(a). Orthogonal arrays L64 (two-level)*
L64 (263)
COLUMN
CONDITION
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
5 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
7 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
8 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
9 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
10 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
11 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1
12 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1
13 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1
14 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1
15 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2
16 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2
17 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
18 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
19 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1
20 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1
21 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1
22 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1
23 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2
24 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2
25 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1
26 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1
27 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2
28 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2
29 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2
30 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2
31 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1
32 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1
33 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
34 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2
35 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
36 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
37 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1
38 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1
39 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2
40 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2
41 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1
42 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1
Appendix A
43 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2
44 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2
45 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2
46 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2
267
(continued)
268
Table A-5(a). Orthogonal arrays L64 (two-level)* (continued)
L64 (263)
CONDITION
47 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1
48 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1
49 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
50 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
51 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
52 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
53 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2
54 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2
55 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1
56 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1
57 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2
58 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2
59 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1
60 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1
61 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1
62 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1
63 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2
64 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1
1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1
2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1
2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2
2 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2 1 1 1 1
1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
Appendix A
2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1
1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1
1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 2 2 2 1 1
269
2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2
(continued)
270
Table A-5(b). Orthogonal array L64* (continued)
L64 (263)
1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2
2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1
1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1
2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2
1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2
2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1
1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2
2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1
1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1
2 2 1 1 1 1 2 2 1 1 2 2 2 2 1 1 1 1 2 2 2 2 1 1 2 2 1 1 1 1 2 2
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1
2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 1
1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2
2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1
1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1
2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2
1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2
2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1
1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2
2 1 2 1 2 1 2 1 2 1 2 2 1 2 1 2 1 2 1 1 1 2 1 2 1 2 1 2 2 1 2 1
1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1
2 1 2 1 1 2 1 2 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 1 2 1 2
1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2
2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1
1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2
2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1
1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1
2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2
1 2 2 1 2 1 1 2 1 2 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2
2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1
1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1
2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1
2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2
1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1 2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2
2 1 1 2 1 2 2 1 1 2 2 1 2 1 1 2 1 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1
Appendix A
271
272
Table A-6. Triangular table for two-level orthogonal arrays*
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
CONDITION 1 2 3 4
1 1 1 1 1
2 1 2 2 2
3 1 3 3 3
4 2 1 2 3
5 2 2 3 1
6 2 3 1 2
7 3 1 3 2
8 3 2 1 3
9 3 3 2 1
(b)
COLUMN L18 (21 × 37)
CONDITION 1 2 3 4 5 6 7 8
1 1 1 1 1 1 1 1 1
2 1 1 2 2 2 2 2 2
3 1 1 3 3 3 3 3 3
4 1 2 1 1 2 2 3 3
5 1 2 2 2 3 3 1 1
6 1 2 3 3 1 1 2 2
7 1 3 1 2 1 3 2 3
8 1 3 2 3 2 1 3 1
9 1 3 3 1 3 2 1 2
10 2 1 1 3 3 2 2 1
11 2 1 2 1 1 3 3 2
12 2 1 3 2 2 1 1 3
13 2 2 1 2 3 1 3 2
14 2 2 2 3 1 2 1 3
15 2 2 3 1 2 3 2 1
16 2 3 1 3 2 3 1 2
17 2 3 2 1 3 1 2 3
18 2 3 3 2 1 2 3 1
Note: Like the L12 (211), this is a specially designed array. An interaction is built in between the first two
columns. This interaction information can be obtained without sacrificing any other column. Interactions
between three-level columns are distributed more or less uniformly to all the other three-level columns,
which permits investigation of main effects. Thus, it is a highly recommended array for experiments.
* Reprinted with permission of the American Supplier Institute, Inc.
274 A Primer on the Taguchi Method
Appendix A 275
276 A Primer on the Taguchi Method
5 2 1 2 3 4
6 2 2 1 4 3
7 2 3 4 1 2
8 2 4 3 2 1
9 3 1 3 4 2
10 3 2 4 3 1
11 3 3 1 2 4
12 3 4 2 1 3
13 4 1 4 2 3
14 4 2 3 1 4
15 4 3 2 4 1
16 4 4 1 3 2
278
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
3 2 2 2 7 6 6 6 11 10 10 10 15 14 14 14 19 18 18 18
Appendix A
1 1
(19) 18 18
21 20
1
(20) 18
279
19
* Reprinted with permission of the American Supplier Institute, Inc.
Appendix B
Brainstorming
1. What are we after?
2.
A
3.
4.
5.
120 2.7478 2.3473 2.1300 1.9923 1.8959 1.8238 1.7675 1.7220 1.6843
(continued)
10 12 15 20 24 30 40 60 120 d
60.195 60.705 61.220 61.740 62.002 62.265 62.529 62.794 63.061 63.328
9.3916 9.4081 9.4247 9.4413 9.4496 9.4579 9.4663 9.4746 9.4829 9.4913
5.2304 5.2156 5.2003 5.1845 5.1764 5.1681 5.1597 5.1512 5.1425 5.1337
3.9199 3.8955 3.8689 3.8443 3.8310 3.8174 3.8036 3.7986 3.7753 3.7607
3.2974 3.2682 3.2380 3.2067 3.1905 3.1741 3.1573 3.1402 3.1228 3.1050
2.9369 2.9047 2.8712 2.8363 2.8183 2.8000 2.7812 2.7620 2.7423 2.7222
2.7025 2.6681 2.6322 2.5947 2.5723 2.5555 2.5351 2.5142 2.4928 2.4708
2.5380 2.5020 2.4642 2.4246 2.4041 2.3830 2.3614 2.3391 2.3162 2.2926
2.4163 2.3789 2.3396 2.2983 2.2768 2.2547 2.2320 2.2085 2.1843 2.1592
2.3226 2.2841 2.2435 2.2007 2.1784 2.1554 2.1317 2.1072 2.0818 2.0554
2.2482 2.2087 2.1671 2.1230 2.1000 2.0762 2.0516 2.0261 1.9997 1.9721
2.1878 2.1474 2.1049 2.0597 2.0360 2.0115 1.9861 1.9597 1.9323 1.9036
2.1376 2.0966 2.0532 2.0070 1.9827 1.9576 1.9315 1.9043 1.8759 1.8462
2.0954 2.0537 2.0095 1.9625 1.9377 1.9119 1.8852 1.8572 1.8280 1.7973
2.0593 2.0171 1.9722 1.9243 1.8890 1.8728 1.8454 1.8168 1.7867 1.7551
2.0281 1.9854 1.9399 1.8913 1.8656 1.8388 1.8108 1.7816 1.7507 1.7182
2.0009 1.9577 1.9117 1.8624 1.8362 1.8090 1.7805 1.7506 1.7191 1.6856
1.9770 1.9333 1.8868 1.8368 1.8103 1.7827 1.7537 1.7232 1.6910 1.6567
1.9557 1.9117 1.8647 1.8142 1.7873 1.7592 1.7298 1.6988 1.6659 1.6308
1.9367 1.8924 1.8449 1.7938 1.7667 1.7382 1.7083 1.6769 1.6433 1.6074
1.9197 1.8750 1.8272 1.7756 1.7481 1.7193 1.6890 1.6569 1.6228 1.5862
1.9043 1.8503 1.8111 1.7590 1.7312 1.7021 1.6714 1.6389 1.6042 1.5668
1.8903 1.8450 1.7964 1.7439 1.7159 1.6864 1.6554 1.6224 1.5871 1.5490
1.8775 1.8319 1.7831 1.7302 1.7019 1.6721 1.6407 1.6073 1.5715 1.5327
1.8548 1.8200 1.7708 1.7175 1.6890 1.6589 1.6272 1.5934 1.5570 1.5176
1.8550 1.8090 1.7596 1.7059 1.6771 1.6468 1.6147 1.5805 1.5437 1.5036
1.8451 1.7989 1.7492 1.6951 1.6662 1.6356 1.6032 1.5687 1.5313 1.4906
1.8359 1.7895 1.7395 1.6852 1.6560 1.6252 1.5925 1.5575 1.5198 1.4784
1.8274 1.7808 1.7306 1.6759 1.6465 1.6155 1.5825 1.5472 1.5090 1.4670
1.8195 1.7727 1.7223 1.6673 1.6377 1.6065 1.5732 1.5376 1.4989 1.4564
1.7627 1.7146 1.6624 1.6052 1.5741 1.5411 1.5056 1.4572 1.4248 1.3769
1.7070 1.6574 1.6034 1.5435 1.5107 1.4755 1.4373 1.3952 1.3476 1.2915
1.6524 1.6012 1.5450 1.4821 1.4472 1.4094 1.3676 1.3203 1.2646 1.1926
1.5987 1.5458 1.4871 1.4206 1.3832 1.3410 1.2951 1.2400 1.1686 1.0000
284 A Primer on the Taguchi Method
120 3.9201 3.0718 2.6802 2.4472 2.2900 2.1750 2.0867 2.0164 1.9588
(continued)
10 12 15 20 24 30 40 60 120 d
241.88 243.91 245.95 248.01 249.05 250.09 251.14 252.20 253.25 254.32
19.396 19.413 19.429 19.446 19.454 19.462 19.471 19.479 19.487 19.496
8.7855 8.7446 8.7029 8.6602 8.6385 8.6166 8.5944 8.5720 8.5494 8.5265
5.9644 5.9117 5.8578 5.8025 5.7744 5.7459 5.7170 5.6878 5.6581 5.6281
4.7351 4.6777 4.6188 4.5581 4.5272 4.4957 4.4638 4.4314 4.3984 4.3650
4.0600 3.9999 3.9381 3.8742 3.8415 3.8082 3.7743 3.7398 3.7047 3.6688
3.6365 3.5747 3.5108 3.4445 3.4105 3.3758 3.3404 3.3043 3.2674 3.2298
3.3472 3.2840 3.2184 3.1503 3.1152 3.0794 3.0428 3.0053 2.9669 2.9276
3.1373 3.0729 3.0061 2.9365 2.9005 2.8637 2.8259 2.7872 2.7475 2.7067
2.9782 2.9130 2.8450 2.7740 2.7372 2.6996 2.6609 2.6211 2.5801 2.5379
2.8536 2.7876 2.7186 2.6464 2.6090 2.5705 2.5309 2.4901 2.4480 2.4045
2.7534 2.6866 2.6169 2.5436 2.5055 2.4663 2.4259 2.3842 2.3410 2.2962
2.6710 2.6037 2.5331 2.4589 2.4202 2.3803 2.3392 2.2966 2.2524 2.2064
2.6021 2.5342 2.4630 2.3879 2.3487 2.3082 2.2664 2.2230 2.1778 2.1307
2.5437 2.4753 2.4035 2.3275 2.2878 2.2468 2.2043 2.1601 2.1141 2.0658
2.4935 2.4247 2.3522 2.2756 2.2354 2.1938 2.1507 2.1058 2.0589 2.0096
2.4499 2.3807 2.3077 2.2304 2.1898 2.1477 2.1040 2.0584 2.0107 1.9604
2.4117 2.3421 2.2686 2.1906 2.1497 2.1071 2.0629 2.0166 1.9681 1.9168
2.3779 2.3080 2.2341 2.1555 2.1141 2.0712 2.0264 1.9796 1.9302 1.8780
2.3479 2.2776 2.2033 2.1242 2.0825 2.0391 1.9938 1.9464 1.8963 1.8432
2.3210 2.2504 2.1757 2.0960 2.0540 2.0102 1.9645 1.9165 1.8657 1.8117
2.2967 2.2258 2.1508 2.0707 2.0283 1.9842 1.9380 1.8895 1.8380 1.7831
2.2747 2.2036 2.1282 2.0476 2.0050 1.9605 1.9139 1.8649 1.8128 1.7570
2.2547 2.1834 2.1077 2.0267 1.9838 1.9390 1.8920 1.8424 1.7897 1.7331
2.2365 2.1649 2.0889 2.0075 1.9643 1.9192 1.8718 1.8217 1.7684 1.7110
2.2197 2.1479 2.0716 1.9898 1.9464 1.9010 1.8533 1.8027 1.7488 1.6906
2.2043 2.1323 2.0558 1.9736 1.9299 1.8842 1.8361 1.7851 1.7307 1.6717
2.1900 2.1179 2.0411 1.9586 1.9147 1.8687 1.8203 1.7689 1.7138 1.6541
2.1768 2.1045 2.0275 1.9446 1.9005 1.8543 1.8055 1.7537 1.6981 1.6377
2.1646 2.0921 2.0148 1.9317 1.8874 1.8409 1.7918 1.7396 1.6835 1.6223
2.0772 2.0035 1.9245 1.8389 1.7929 1.7444 1.6928 1.6373 1.5766 1.5089
1.9926 1.9174 1.8364 1.7480 1.7001 1.6491 1.5943 1.5343 1.4673 1.3893
1.9105 1.8337 1.7505 1.6587 1.6084 1.5543 1.4952 1.4290 1.3519 1.2539
1.8307 1.7522 1.6664 1.5705 1.5173 1.4591 1.3940 1.3180 1.2214 1.0000
286 A Primer on the Taguchi Method
120 5.1524 3.8046 3.2270 2.8943 2.6740 2.5154 2.3948 2.2994 2.2217
(continued)
10 12 15 20 24 30 40 60 120 d
968.63 976.71 984.87 993.10 997.25 1001.4 1005.6 1009.8 1014.0 1018.3
39.398 39.415 39.431 39.448 39.456 39.465 39.473 39.481 39.490 39.498
14.419 14.337 14.253 14.167 14.124 14.081 14.037 13.992 13.947 13.902
8.8439 8.7512 8.6565 8.5599 8.5109 8.4613 8.4111 8.3604 8.3092 8.2573
6.6192 6.5246 6.4277 6.3285 6.2780 6.2269 6.1751 6.1225 6.0693 6.0153
5.4613 5.3662 5.2687 5.1684 5.1172 5.0652 5.0125 5.9589 4.9045 4.8491
4.7611 4.6658 4.5678 4.4667 4.4150 4.3624 4.3089 4.2544 4.1989 4.1423
4.2951 4.1997 4.1012 3.9995 3.9472 3.8940 3.8398 3.7844 3.7279 3.6702
3.9639 3.8682 3.7694 3.6669 3.6142 3.5604 3.5055 3.4493 3.3918 3.3329
3.7168 3.6209 3.5217 3.4186 3.3654 3.3110 3.2554 3.1984 3.1399 3.0798
3.5257 3.4296 3.3299 3.2261 3.1725 3.1176 3.0613 3.0035 2.9441 2.8828
3.3736 3.2773 3.1772 3.0728 3.0187 2.9633 2.9063 2.8478 2.7874 2.7249
3.2497 3.1532 3.0527 2.9477 2.8932 2.8373 2.7797 2.7204 2.6590 2.5955
3.1469 3.0501 2.9493 2.8437 2.7888 2.7324 2.6742 2.6142 2.5519 2.4872
3.0602 2.9633 2.8621 2.7559 2.7006 2.6437 2.5850 2.5242 2.4611 2.3953
2.9862 2.8890 2.7875 2.6808 2.6252 2.5678 2.5085 2.4471 2.3831 2.3163
2.9222 2.8249 2.7230 2.6158 2.5598 2.5021 2.4422 2.3801 2.3153 2.2474
2.8664 2.7689 2.6667 2.5590 2.5027 2.4445 2.3842 2.3214 2.2558 2.1869
2.8173 2.7196 2.6171 2.5089 2.4523 2.3937 2.3329 2.2695 2.2032 2.1333
2.7737 2.6758 2.5731 2.4645 2.4076 2.3486 2.2873 2.2234 2.1562 2.0853
2.7348 2.6368 2.5338 2.4247 2.3675 2.3082 2.2465 2.1819 2.1141 2.0422
2.6998 2.6017 2.4984 2.3890 2.3315 2.2718 2.2097 2.1446 2.0760 2.0032
2.6682 2.5699 2.4665 2.3567 2.2989 2.2389 2.1763 2.1107 2.0415 1.9677
2.6396 2.5412 2.4374 2.3273 2.2693 2.2090 2.1460 2.0799 2.0099 1.9353
2.6135 2.5149 2.4110 2.3005 2.2422 2.1816 2.1183 2.0517 1.9811 1.9055
2.5895 2.4909 2.3867 2.2759 2.2174 2.1565 2.0928 2.0257 1.9545 1.8781
2.5675 2.4688 2.3644 2.2533 2.1946 2.1334 2.0693 2.0018 1.9299 1.8527
2.5473 2.4484 2.3438 2.2324 2.1735 2.1121 2.0477 1.9796 1.9072 1.8291
2.5286 2.4295 2.3248 2.2131 2.1540 2.0923 2.0276 1.9591 1.8861 1.8072
2.5112 2.4120 2.3072 2.1952 2.1359 2.0739 2.0089 1.9400 1.8664 1.7867
2.3882 2.2882 2.1819 2.0677 2.0069 1.9429 1.8752 1.8028 1.7242 1.6371
2.2702 2.1692 2.0613 1.9445 1.8817 1.8152 1.7440 1.6668 1.5810 1.4822
2.1570 2.0548 1.9450 1.8249 1.7597 1.6899 1.6141 1.5299 1.4327 1.3104
2.0493 1.9447 1.8326 1.7085 1.6402 1.5660 1.4835 1.3883 1.2684 1.0000
288 A Primer on the Taguchi Method
120 6.8510 4.7865 3.9493 3.4706 3.1735 2.9559 2.7918 2.6629 2.5586
(continued)
10 12 15 20 24 30 40 60 120 d
6055.8 6106.3 6157.3 6208.7 6234.6 6260.7 6286.8 6313.0 6339.4 6366.0
99.399 99.415 99.432 99.449 99.458 99.466 99.474 99.483 99.491 99.501
27.229 27.052 26.872 26.690 26.598 26.505 26.411 26.316 26.221 26.125
14.546 14.374 14.198 14.020 13.929 13.838 13.745 13.652 13.558 13.463
10.051 9.8883 9.7222 9.5527 9.4665 9.3793 9.2912 9.2020 9.1118 9.0204
7.8741 7.7183 7.5590 7.3958 7.3127 7.2285 7.1432 7.0568 6.9690 6.8801
6.6201 6.4691 6.3143 6.1554 6.0743 5.9921 5.9084 5.8236 5.7372 5.6495
5.8143 5.6668 5.5151 5.3591 5.2793 5.1981 5.1156 5.0316 4.9460 4.8588
5.2565 5.1114 4.9621 4.8080 4.7290 4.6486 4.5667 4.4831 4.3978 4.3105
4.8402 4.7059 4.5582 4.4054 4.3269 4.2469 4.1653 4.0619 3.9965 3.9090
4.5393 4.3974 4.2509 4.0990 4.0209 3.9411 3.8596 3.7761 3.6904 3.6025
4.2961 4.1553 4.0096 3.8584 3.7805 3.7008 3.6192 3.5355 3.4494 3.3608
4.1003 3.9603 3.8154 3.6646 3.5868 3.5070 3.4253 3.3413 3.2548 3.1654
3.9394 3.8001 3.6557 3.5052 3.4274 3.3476 3.2656 3.1813 3.0942 3.0040
3.8049 3.6662 3.5222 3.3719 3.2940 3.2141 3.1319 3.0471 2.9595 2.8684
3.6909 3.5527 3.4089 3.2588 3.1808 3.1007 3.0182 2.9330 2.8447 2.7528
3.5931 3.4552 3.3117 3.1615 3.0835 3.0032 2.9205 2.8348 2.7459 2.6530
3.5082 3.3706 3.2273 3.0771 2.9990 2.9185 2.8354 2.7493 2.6597 2.5660
3.4338 3.2965 3.1533 3.0031 2.9249 2.8442 2.7608 2.6742 2.5839 2.4893
3.3682 3.2311 3.0880 2.9377 2.8594 2.7785 2.6947 2.6077 2.5168 2.4212
3.3098 3.1729 3.0299 2.8976 2.8011 2.7200 2.6359 2.5484 2.4568 2.3603
3.2576 3.1209 2.9780 2.8274 2.7488 2.6675 2.5831 2.4951 2.4029 2.3055
3.2106 3.0740 2.9311 2.7805 2.7017 2.6202 2.5355 2.4471 2.3542 2.2559
3.1681 3.0316 2.8887 2.7380 2.6591 2.5773 2.4923 2.4035 2.3099 2.2107
3.1294 2.9931 2.8502 2.6993 2.6203 2.5383 2.4530 2.3637 2.2695 2.1694
3.0941 2.9579 2.8150 2.6640 2.5848 2.5026 2.4170 2.3273 2.2325 2.1315
3.0618 2.9256 2.7827 2.6316 2.5522 2.4699 2.3840 2.2938 2.1984 2.0965
3.0320 2.8959 2.7530 2.6017 2.5223 2.4397 2.3535 2.2529 2.1670 2.0642
3.0045 2.8685 2.7256 2.5742 2.4946 2.4118 2.3253 2.2344 2.1378 2.0342
2.9791 2.8431 2.7002 2.5487 2.4589 2.3860 2.2992 2.2079 2.1107 2.0062
2.8005 2.6649 2.5216 2.3689 2.2880 2.2034 2.1142 2.0194 1.9172 1.8047
2.6318 2.4961 2.3523 2.1978 2.1154 2.0285 1.9360 1.8363 1.7263 1.6006
2.4721 2.3363 2.1915 2.0346 1.9500 1.8600 1.7629 1.6557 1.5330 1.3805
2.3209 2.1848 2.0385 1.8783 1.7908 1.6964 1.5923 1.4730 1.3246 1.0000
290 A Primer on the Taguchi Method
120 8.1790 5.5393 4.4973 3.9207 3.5482 3.2849 3.0874 2.9330 2.8083
(continued)
10 12 15 20 24 30 40 80 120 d
24224 24426 24630 24836 24940 25044 25148 25253 25359 25465
199.40 199.42 199.43 199.45 199.46 199.47 199.47 199.48 199.49 199.51
43.686 43.387 43.085 42.778 42.622 42.466 42.308 42.149 41.989 41.829
20.967 20.705 20.438 20.167 20.030 19.892 19.752 19.611 19.468 19.325
13.618 13.384 13.146 12.903 12.780 12.656 12.530 12.402 12.274 12.144
10.250 10.034 9.8140 9.5888 9.4741 9.3583 9.2408 9.1219 9.0015 8.8793
8.3803 8.1764 7.9578 7.7540 7.6450 7.5345 7.4225 7.3088 7.1933 7.0760
7.2107 7.0149 6.8143 6.6082 6.5029 6.3961 6.2875 6.1772 6.0649 5.9505
6.4171 6.2274 6.0325 5.8318 5.7292 5.6248 5.5186 5.4104 5.3001 5.1875
5.8467 5.6613 5.4707 5.2740 5.1732 5.0705 5.9659 4.8592 4.7501 4.6385
5.4182 5.2363 5.0489 4.8552 4.7557 4.6543 4.5508 4.4450 4.3367 4.2256
5.0855 4.9063 4.7214 4.5299 4.4315 4.3309 4.2282 4.1229 4.0149 3.9039
4.8199 4.6429 4.4600 4.2703 4.1726 4.0727 3.9704 3.8655 3.7577 3.6465
4.6034 4.4281 4.2468 4.0585 3.9614 3.8619 3.7600 3.6553 3.5473 3.4359
4.4236 4.2498 4.0698 3.8826 3.7859 3.6867 3.5850 3.4803 3.3722 3.2602
4.2719 4.0994 3.9205 3.7342 3.6378 3.5388 3.4372 3.3324 3.2240 3.1115
4.1423 3.9709 3.7929 3.6073 3.5112 3.4124 3.3107 3.2058 3.0971 2.9839
4.0305 3.8599 3.6827 3.4977 3.4017 3.3030 3.2014 3.0962 2.9871 2.8732
3.9329 3.7631 3.5866 3.4020 3.3062 3.2075 3.1058 3.0004 2.8906 2.7762
3.8470 3.6779 3.5020 3.3178 3.2220 3.1234 3.0215 2.9159 2.8058 2.6904
3.7709 3.6024 3.4270 3.2431 3.1474 3.0488 2.9467 2.8408 2.7302 2.6140
3.7030 3.5350 3.3600 3.1764 3.0807 2.9821 2.8799 2.7736 2.6625 2.5455
3.6420 3.4745 3.2999 3.1165 3.0208 2.9221 2.8198 2.7132 2.6016 2.4837
3.5870 3.4199 3.2456 3.0624 2.9967 2.8679 2.7654 2.6585 2.5463 2.4276
3.5370 3.3704 3.1953 3.0133 2.9176 2.8187 2.7160 2.6099 2.4960 2.3765
3.4916 3.3252 3.1515 2.9685 2.8728 2.7738 2.6709 2.5633 2.4501 2.3297
3.4499 3.2839 3.1104 2.9275 2.8318 2.7327 2.6296 2.5217 2.4078 2.2867
3.4117 3.2460 3.0727 2.8899 2.7941 2.6949 2.5916 2.4834 2.3689 2.2469
3.3765 3.2111 3.0379 2.8551 2.7594 2.6601 2.5565 2.4479 2.3330 2.2102
3.3440 3.1787 3.0057 2.8230 2.7272 2.6278 2.5241 2.4151 2.2997 2.1760
3.1167 2.9531 2.7811 2.5984 2.5020 2.4015 2.2958 2.1838 2.0635 1.9318
2.9042 2.7419 2.5705 2.3872 2.2989 2.1874 2.0789 1.9622 1.8341 1.6885
2.7052 2.5439 2.3727 2.1881 2.0890 1.9839 1.8709 1.7459 1.6055 1.4311
2.5188 2.3583 2.1868 1.9998 1.8983 1.7891 1.6691 1.5325 1.3637 1.0000
INDEX
Advertising 22 210
Analysis of results v vii 4
6 19 50
56 57 75
124 126 154
196 237
Analysis of variance (ANOVA) 3 4 20
39 41 59
63 75 119
129 154 158
172 182 295
Appendixes 261 281
Application techniques/areas 1 5 10
21 23 37
49 50 107
123 124 135
172 189 190
205 208 225
226 293 294
296
Battery examples 26 27 41
42 44–47 199
202 203 297
Bearings 16 17 127
230
Bell Laboratories 9
Brainstorming 9 22 37
39 40 54
205–209 212 214–216
221 225 226
230 231
Confirmation experiment/test 20 37 39
57 65 82
169 225
Conformance (to specifications) 11 12 17
27 47
Consistency 5 12 15
19 22 27
29 30 41
50 103 104
173 176 259
Control/controllable factor(s) 20 31 35
41 54 103–105
110 112 113
121–124 129 173
174 216 219
295 298
Cookie baking example 2–4
Costbenefit/Cost saving 1 4 6
9 17 21–23
29 31 37
41 42 44–47
56 72 189
190 192 194–196
201 202 243
255 259 297
298
Cost of quality 10 13 23
142 185
Cost(s) 4 11 13
17 21–23 27
31 47 56
72 74 104
107 126 186
194–196 203 207
243 296
Crash/crush testing 241 243 247
Customer(s) 2 13–15 17
27 29–31 41
45–47 87 185
188 190 194
197 207 211
222 297 298
Data-driven solution 5 22 75
107 108 110
122 124 129–132
135 137 146
174 178 198
213
Degree of freedom (DOF) viii 59 61
72 73 82
83 86 88
96 129 130
134 138 145
Dummy treatment 86 93 95
96
Engine components 16 17 26
93 127 135
226 230 236
Error viii 56 59–63
97 129 135
137–139 148 149
152 153 164
165 295 297
Evaluation criteria vii 25 74
208 212–214 221
222
Examples/studies 2–4 15 16
25 26 30–34
40 41 44–47
54 55 71
72 74–76 86
93 105 106
111 135–137 141
144 146 147
149 150 152
153 178–180 190
192 193 196
198–200 202 210
Examples/studies (Cont.)
211 212 213
221–223 226 230
234 236 238
241 244 247
252 259
Exercises(end of chapters) 6 23 35
48 126 182
202 221
Experiment conditions/layout vii 1 6
9 18 19
21 22 32–35
38 40 49–51
53–55 57 59
65 66 71
82 89 96
122 126 129
149 205–211 218
220–223 225 226
234 253 294–296
Factor effect(s) 2 6 20
39 50 56–58
77–79 82 105
112 113 115
116 118 120
Glossary 295
Influence/contribution viii 3 4
6 11–13 19–21
24 26 30
39 41 53
54 57 59
60 63 68
75 77–79 82
83 86 103–105
107 110 120–124
130 137 139
140 143 144
Influence/contribution (Cont.)
153 156 158
165 172 173
182 215 216
219 223 226
229–231 235 238
242 245 257
259 295–297
Injection molding 111
Inner array 35 122 123
221 295 296
Interaction 3 6 19
49 53 67–75
77–85 87–89 91
92 96 99
103–105 110–113 116
118 124 126
149 156–158 164
165 209 218
219 223 226
231 232 239
247 295 296
Intersecting lines 3 79 157
Japan v 4 9
15 16 21
Latin squares 18
Life cycle/expectancy 11 22 25
146 152 173
194 211 212
231
Linear graph 71 73 82
84 88 92
261 262 264
293 294 296
Logarithmictransformation vii 6 35
39 42 108
109 173–177 181
182
Lossfunction vii viii 6
13–15 17 18
22 23 27
41 42 46
47 185–190 198
296
Lower acceptable limit (LAL) 14–16 28 201
Main effect(s) 2 3 20
39 53 58
63 67–69 71
75 77 80
92 97 100
102 103 109
110 114–118 121
149 156 158
163 166 170
178 181 182
226 227 229
231 235 237
240 247 255
Marketing and advertising 22 38 205
207 208 210
Market research 2–4
Mean squar edeviation (MSD) viii 42 45
108 173 189
296
Mean square variance viii 59 129
130 135 161
Mixed levels of factors 38 49 53
67 85 86
235
Molding process 54 57 58
61 64 86
111 210 211
297 298
Multiple factor/level 1–6 12 18
19 22 25
31 32 34
38 40 49–51
53 54 58
61 65 67
70–73 77 79
80 83–89 91–99
105 110–112 126
127 130 141
146 154 159
170 178 196
209 212 214
217–219 221 223
226 227 229
231 234–237 239
242 244 246
247 254 259
261
Multiple sample(s) 6 35 39
103 106 107
191 220 254
Noise effect/factor 6 12 22
35 38 39
103–105 107 110–113
121–124 126 129
173 182 209
215 216 218–221
259 296
Noise (sound) 26 178 196
226 234 235
Nonparallel lines 3 68 105
O
Objective evaluation 25 41 174
176 190 207
213 222 223
Off-line quality control 11 296
One-way ANOVA 135 141
Optimization 1 5 6
9 20–22 30
35 37 40
53 100 112
116 118 121
185 189 190
210 222 231
238 247 253
293 297
Optimum condition 1 12 14
15 20 21
30–33 35 37
39 41 53
54 57 58
63–65 75 77
78 80–83 93
97 103 107
110 120 121
123 126 127
156–158 167–169 172–175
178 181 182
187 196 201
202 218 222
225 229 230
232 235–238 240
245 247 255
295 296
Orthogonal array (OA) viii 5 12
18 19 21
23 34 35
38 40 49
54 65 72
108 122 126
129 154 196
214 219 231
261–265 267 270
274–278 293–296
Outer array 6 12 35
122 123 126
219–221 296
Overall Evaluation Criteria (OEC) vii 25 213
214 221 222
258
Parallel lines 3 68 79
82 157
Parameter design 12 31 35
294 296 297
Pareto 111 215 253
Partial factorial experiment 4 49 50
51 129 295
Perception of quality 30 190
Performance v vii 1
5 6 12–17
20 21 25
26 30 31
35 39 41
49 53 57
63–65 75 80–82
103 104 108
110 111 120
127 167 168
172 175 178
Performance (Cont.)
181 182 185
189 195 196
198 199 201
208 210 212
215 229 230
232 235–238 240
242 245 250
255 297 298
Pooling/pooled effects 62 63 120
152 153 161–165
167 229 231
232 235
Poorquality 10 11 13
17 23 24
27 30 185
188
Process parameter 12 93 211
221
Product/process development 21–23 294
Random order 39 54 56
66 73 74
84 96 140
226
References 293
Reject/repair/rework 11 13 14
17 22 23
27 47 111
187 190 191
193–195 202 211
252
Repetitions/replications/runs 12 19 20
39 40 51
56 57 61–63
65 72–74 77
79 81 93
96–100 107–110 121–124
126 129 130
145 146 148
154 160 166–170
Repetitions/replications/runs (Cont.)
172 174 178
214 221 227
235 239 240
246 296
Reproducibility 50 172
Responses/results 2 3 12
20 26 130
135 141 146
148 210 213
230 295–297
Robust design vii viii 6
11 12 22
23 30 103
104 114 117
121 172 174
185 219 220
294 297
Service cost(s) 11 23 30
193–195
Signal-to-noise (S/N) ratio viii 6 12
20 21 23
35 39 42
107–110 118 119
121 123 125
Taguchi (Cont.)
methods/techniques v–vii 9 10
19 21–23 31
32 37 39–41
48–50 54 59
65 68 89
104 124 202
205 207 225
226 234 293
294 297
quality strategy 11 30
Target value 5 10–15 17
22 26 30
31 42 103
107 110 116
121 174 182
185–187 198 296
297
Taylor’s theorem 187
Team approach 22 23 40
205 207 208
210 215–217 220
222 223 253
Television examples 15 16 31
32
Warranty 11 23 27
41 47 187
188 193 202
203 211 298
World War II 9 18